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(centroid_(k_1,1),centroid_(k_1,2),centroid_(k_1,~ 3)):(k_1=1...RowsOf(centroid)):(10)~p0 0 ~gc-1 1 ? 0 0 -530 1365 ~S1 ? 16711680 ? ? (centroid_(k_1,1),~ centroid_(k_1,2),centroid_(k_1,3)):(k_1=1...RowsOf(centroid)):(~ 10)~p0 0 ~gc-1 1 ? 0 0 -530 1365 ~t~p0 3 ~Q ]|Expr|[#b @`bb#_b#_b#_}`fb"#})$# b'4" *~: ;bP8&c54JSolution | |Procedure,Z}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0 0 ~Q ]|Expr|[#b @`bb#_b#_b#_})b$.# b'4" *~: ;bP8&c0!*The reference| | Cartesian coordiante system ,Hxyz,I is located at the back end| | of the tank with origin at the center of the cross section,N| | The x and y define the horizontal and vertical cross,M sectional| | axes,L respectively while the z axis is along the length of | |the tank,N Irrespective of the amout of fuel in the tank,L the| | fuel volume will be symmetric with respect to the vertical axis| | ,Hy,I,N Also because the tank is uniform along its length,L| | the centroid will be located at the midway point along the z| | axis,N Therefore,L the only centroidal position that needs | |to be calculated is that of y bar,N The ,Bfree,B surface or | |fuel level measured from the bottom of the tank is described | |by the expression}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0 1 ~A(ysurf=2*b*'a/100-b)~p0 2 ~d~A(ysurf=-0.60000000000000009)~p0 3 ~sb/_!! } b00(! c#T"_c/__c/__! ""} ^ _~Q ]|Expr|[#b @`bb#_b#_b#_})-# b'4" *~: ;bP8&c0!*The equation for| | ybar is given as}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0 2 ~A(ybar=(Integral(y*d*A):(A))/(Integral(d*A):(A)))~p0 3 ~Q ]|Expr|[#b @`bb#_b#_b#_})b _# b'4" *~: ;bP8&c0!*The differential| | area dA is a horizontal strip defined as x dy,N To write x | |as a function of y we use the equation of ellipse in Cartesian| | coordinates given as}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 4 ~A(x^2/a^2+y^2/b^2=1)~p0 5 ~A(x=sqrt(a^2*(-y^2/b^2+1)))~p0 6 ~d~sb/_!! } )/! c#T" c"L" c"\" c%X" c'8 _c/__c/__! !} ^ _~A(~ x=a*sqrt(-y^2/b^2+1))~p0 7 ~sb/_!! } "&! c#T"!c"4!_c/__c/__} ^ _~A(x=(a*sqrt(-y^2+b^2))/~ b)~p0 8 ~sb/_!! } #'! c#T"!c"H"_c/__c/__!} ^ _~Q ]|Expr|[#b @`bb#_b#_b#_})b!)# b'4" *~: ;bP8&c0!*Notice that the| | positive root for x defines the right half of the ellipse while| | the negative root defines the left half,N Thus,L we introduce| | new variables named ,Bxleft,B and ,Bxright,B as follows,N| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 6 ~A(xleft=-x)~p0 7 ~d~A(xleft=-(a*sqrt(-y^2+b^2))/b)~p0 8 ~sb/_!! } b00&! c#T"_c/__c/__!"} ^ _~A(xright=x)~p0 7 ~d~A(xright=(a*sqrt(-y^2+b^2))/b)~p0 8 ~sb/_!! } b00&! c#T"_c/__c/__!"} ^ _~A(ybar=((Integral(y*xleft*~ d*y):(ysurf):(-b))+(Integral(y*xright*d*y):(-b):(ysurf)))/((Integral(~ xleft*d*y):(ysurf):(-b))+(Integral(xright*d*y):(-b):(ysurf))))~p0 7 ~d~A(ybar=((Integral(y*(-(a*sqrt(-y^2+b^2))/b)*d*y):(-0.60000000000000009):(~ -b))+(Integral(y*(a*sqrt(-y^2+b^2))/b*d*y):(-b):(-0.60000000000000009)))/~ ((Integral((-(a*sqrt(-y^2+b^2))/b)*d*y):(-0.60000000000000009):(~ -b))+(Integral((a*sqrt(-y^2+b^2))/b*d*y):(-b):(-0.60000000000000009))))~p0 8 ~sb/_! } b00* c#T"_c/__c/__! " #"} ^ _~A(ybar=((EvaluateAt(~ Integral((a*sqrt(-y^2+b^2)*y)/b*d*y)):(y=-b):(y=-0.60000000000000009))+~ (EvaluateAt(Integral((-(a*sqrt(-y^2+b^2)*y)/b)*d*y)):(y=-0.60000000000000009):(~ y=-b)))/((EvaluateAt((a*(Integral(sqrt(-y^2+b^2)*d*y)))/b):(y=~ -b):(y=-0.60000000000000009))+(EvaluateAt(-(a*(Integral(sqrt(~ -y^2+b^2)*d*y)))/b):(y=-0.60000000000000009):(y=-b))))~p0 9 ~sb/_! } !& c#T"!c"\"_c/__c/__} ^ _~A(ybar=-2/3*((b^2-0.3600000000000001)^~ (3/2)*a)/((2*((-0.30000000000000004*sqrt(b^2-0.3600000000000001)-~ 1/2*b^2*arcsin(0.60000000000000009*1/b))*a)/b+1/2*'p*a*b)*b))~p0 10 ~sb/_!! } !&! c#T"!c"\"_c/__c/__} ^ _~A(ybar=-4.0418317629510501*~ 1/(4.6666666666666661*(-1.125*arcsin(0.40000000000000008)-0.41243181254602568)+~ 2.625*'p))~p0 11 ~sb/_!! } b00*! c#T"!c"T!_c/__c/__! ""} ^ _~Q ]|Expr|[#b @`bb#_b#_b#_})2# b'4" *~: ;bP8&c0!*Hence,L the centroid| | in Cartesian coordinates is located at| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0 12 ~A(xbar=0)~p0 13 ~d~A(ybar=-0.97123568463949195)~p0 13 ~sb/_!! } $&! c#T"!c"T!_c/__c/__} ^ _~A(zbar=L/2)~p0 13 ~d~A(zbar=15)~p0 14 ~sb/_!! } $&! c#T"!c"\"_c/__c/__} ^ _~Q ]|Expr|[#b @`bb#_b#_b#_})b!\# b'4" *~: ;bP8&c0!*We need to know| | where the free surface intersects the tank boundary to show | |the fuel volume inside the tank as depicted in the figure above| |,N This is done using a cylindrical coordinate system ,Hr "!Symbol^:!&c0 q| |: &c55* z,I&c0!* with origin at the center of the cross section| | as before,N The equation describing the location of free surface| | is given as}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 1 ~A(rsurf=ysurf/(sin('q)))~p0 2 ~d~A(rsurf=-0.60000000000000009*1/(sin('q)))~p0 255 ~sb/_!! } b00&! c#T"_c/__c/__!"} ^ _~Q ]|Expr|[#b @`bb#_b#_b#_})b E# b'4" *~: ;bP8&c0!*The point of | |intersect at a given section is defined by the coordinates ,H| |rint,L "!Symbol^:!q: int,I defined as}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0 2 ~A(ysurf=2*b*'a/100-b)~p0 2 ~A(ysurf=-0.60000000000000009)~p0 3 ~sb/_!! } b00(! c#T"_c/__c/__! ""} ^ _~A(rint=ysurf/(sin('qint)))~p0 3 ~d~A(rint=r)~p0 4 ~A('q='qint)~p0 255 ~d~A(rint=(a*b)/sqrt(b^2*(cos('q))^2+a^2*(sin('q))^2))~p0 5 ~A(ysurf/(sin('qint))=(a*b)/sqrt(b^2*(cos('qint))^2+a^2*(sin(~ 'qint))^2))~p0 6 ~sb/_!! } b00(! c#T"_c/__c/__! ""} ^ _~A((ysurf/(sin('qint)))^~ 2=((a*b)/sqrt(b^2*(cos('qint))^2+a^2*(sin('qint))^2))^2)~p0 6 ~A(ysurf^2/(sin('qint))^2=(a^2*b^2)/(b^2*(cos('qint))^2+a^2*(~ sin('qint))^2))~p0 7 ~sb/_!! } "&! c#T"!c%X"_c/__c/__} "&! c#T" c%X"_c/__c/__} ^ _~ ~A(ysurf^2/(a^2*b^2*(sin('qint))^2)=1/(b^2*(cos('qint))^2+a^2*~ (sin('qint))^2))~p0 8 ~sb/_!! } *+! c#T"!c"\" c"H"_c/__c/__! !} ^ _~A(ysurf^2/(a^~ 2*b^2)=(sin('qint))^2/(b^2*(cos('qint))^2+a^2*(sin('qint))^2))~p0 9 ~sb/_!! } *-! c#T" c"\"!c"H#"c%X"_c/__c/__! !} ^ _~A((ysurf^~ 2/(a^2*b^2))^(-1)=((sin('qint))^2/(b^2*(cos('qint))^2+a^2*(sin(~ 'qint))^2))^(-1))~p0 9 ~A((a^2*b^2)/ysurf^2=(b^2*(cos('qint))^2)/(sin('qint))^2+a^2)~p0 10 ~sb/_!! } "&! c#T"!c%X"_c/__c/__} "&! c#T" c%X"_c/__c/__} ^ _~ ~A((a^2*b^2)/ysurf^2-a^2=(b^2*(cos('qint))^2)/(sin('qint))^2)~p0 11 ~sb/_!! } *+! c#T"!c"L"!c%X"_c/__c/__! !} ^ _~A(((a^2*b^2)/~ ysurf^2-a^2)/b^2=(cos('qint))^2/(sin('qint))^2)~p0 12 ~sb/_!! } *+! c#T"!c"\" c"H"_ !! !} ^ _~A(((a^2*b^2)/ysurf^~ 2-a^2)/b^2=(cot('qint))^2)~p0 13 ~sb/_!! } #'! c#T"!c"\"_c/__c/__!} ^ _~A('qint=arccot(sqrt(~ ((a^2*b^2)/ysurf^2-a^2)/b^2)))~p0 14 ~d~sb/_!! } )-! c#T"!c%X" c#L"!c'8 _c/__c/__! !} ^ _~A('qint=~ arccot(1/6*sqrt(1029)))~p0 15 ~sb/_!! } b00*! c#T"_c/__c/__! " #"} ^ _~Q ]|Expr|[#b @`bb#_b#_b#_})8# b'4" *~: ;bP8&c0!*Which we can now| | substitute into the expression derived above for rint,N| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0 15 ~A(rint=ysurf/(sin('qint)))~p0 15 ~A(rint=1/6*sqrt(1065)*ysurf)~p0 16 ~sb/_!! } b00&! c#T"_c/__c/__!"} ^ _~A(rint=-0.10000000000000001*~ sqrt(1065))~p0 17 ~sb/_!! } b00&! c#T"_c/__c/__!"} ^ _~Q ]|Expr|[#b @`bb#_b#_b#_})3# b'4" *~: ;bP8&c0!*The figure shown| | above is obtained using the following equations| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0 2 ~A(r)~p0 3 ~A(r=(a*b)/sqrt(b^2*(cos('q))^2+a^2*(sin('q))^2))~p0 4 ~A(r=5.25*1/sqrt(2.25*(cos('q))^2+12.25*(sin('q))^2))~p0 5 ~d~sb/_!! } b00*! c#T"!c"\"_c/__c/__! ""} ^ _~A(r=3.2634337744161441)~p0 6 ~sb/_!! } $&! c#T"!c"H"_c/__c/__} ^ _~A(rsurf)~p0 3 ~A(rsurf=-0.60000000000000009*1/(sin('q)))~p0 4 ~sb/^!! } b00(! c#T"!c'8 _c/__c/__!"} ^ _~A(rsurf=-3.2634337744161441)~p0 5 ~sb/_!! } $&! c#T"!c"T!_c/__c/__} ^ _~A(rint)~p0 3 ~A(rint=-0.10000000000000001*sqrt(1065))~p0 4 ~sb/^!! } b00(! c#T"!c'8 _c/__c/__!"} ^ _~A(rint=-3.2634337744161446)~p0 5 ~sb/_!! } $&! c#T"!c"T!_c/__c/__} ^ _~A('qint)~p1 3 ~A('qint=arccot(1/6*sqrt(1029)))~p0 4 ~sb/^!! } b00(! c#T"!c'8 _c/__c/__!"} ^ _~A(centroid=(xbar,~ ybar,zbar))~p0 3 ~d~A(centroid=(0,-0.97123567982541281,15))~p0 4 ~sb/_!! } $&! c#T"!c&T#_c/__c/_^} ^ _~Q ]|Expr|[#b @`bb#_b#_b#_}`fb"V})'# b'4" *~: ;bP8&c0!*Volume Centroid| | Notebook Declarations}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 0 ~Q ]|Expr|[#b @`bb#_b#_b#_})## b'4" *~: ;bP8&c0!*Additional Variables| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 1 ~V?v0 (rsurf)~p0 2 ~V?f144 (RowsOf)~p0 2 ~V?m0 (centroid)~p0 2 ~V?c0 ('qint)~p0 2 ~V?f130 (FromCylindrical)~p0 2 ~V?v0 (rint)~p0 2 ~V?v0 ('q)~p0 2 ~V?v0 (r)~p0 2 ~V?c0 ('a)~p0 2 ~V?v0 (ysurf)~p0 2 ~V?f0 (lower)~p0 2 ~V?v0 (xleft)~p0 2 ~V?v0 (xright)~p0 2 ~V?v0 (zbar)~p0 2 ~V?v0 (xbar)~p0 2 ~V?v0 (ybar)~p0 2 ~V?v0 (A)~p0 2 ~V?c0 (L)~p0 2 ~Q ]|Expr|[#b @`bb#_b#_b#_})!# b!(" *~: ;,&c0 Trigonometry| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 1 ~V?c1 ('p)~p0 2 ~Q ]|Expr|[#b @`bb#_b#_b#_})## b!U" *~: ;,&c0 Function Declarations| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 2 ~V?f256 (sin)~p0 3 ~V?f257 (cos)~p0 3 ~V?f258 (tan)~p0 3 ~V?f261 (sec)~p0 3 ~V?f260 (csc)~p0 3 ~V?f262 (cot)~p0 3 ~V?f272 (arcsin)~p0 3 ~V?f273 (arccos)~p0 3 ~V?f274 (arctan)~p0 3 ~V?f276 (arccsc)~p0 3 ~V?f277 (arcsec)~p0 3 ~V?f278 (arccot)~p0 3 ~Q ]|Expr|[#b @`bb#_b#_b#_})## b V" *~: ;,&c0 Trig rules| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 2 ~Q ]|Expr|[#b @`bb#_b#_b#_})## b!1" *~: ;,&c0 basic identities| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Hs(sin(-?x)):(-sin(?x))~p0 4 ~Hs(arcsin(-?x)):(-arcsin(?x))~p0 4 ~Hs(cos(-?x)):(cos(?x))~p0 4 ~Hs(tan(-?x)):(-tan(?x))~p0 4 ~Hs(arctan(-?x)):(-arctan(?x))~p0 4 ~Hs(sin('p)):(0)~p0 4 ~Hs(sin(?n*'p)):(0)~p0 4 ~Hs(cos('p/2)):(0)~p0 4 ~Hs(cos(?n/2*'p)):(0)~p0 4 ~Hs(tan(?n*'p)):(0)~p0 4 ~Hs(cot(?n/2*'p)):(0)~p0 4 ~Hs(arccos(1)):(0)~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_}))# b"0" *~: ;,&c0 transform to,Ofrom| | Basic trigs}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Ht(tan(?x)):((sin(?x))/(cos(?x)))~p0 4 ~Ht((sin(?x))/(cos(?x))):(tan(?x))~p0 4 ~Ht(csc(?x)):(1/(sin(?x)))~p0 4 ~Ht(1/(sin(?x))):(csc(?x))~p0 4 ~Ht(sin(?x)):(1/(csc(?x)))~p0 4 ~Ht(sec(?x)):(1/(cos(?x)))~p0 4 ~Ht(cos(?x)):(1/(sec(?x)))~p0 4 ~Ht(cot(?x)):((cos(?x))/(sin(?x)))~p0 4 ~Ht((cos(?x))/(sin(?x))):(cot(?x))~p0 4 ~Ht((cos(?x))/(sin(?x))^?n):((cos(?x))/(sin(?x))^(?n-1)*cot(?x))~p0 4 ~Ht((cos(?x))^?n/(sin(?x))^?n):((cot(?x))^?n)~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})## b!(" *~: ;,&c0 trig Addition| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Ht(cos(?x+?y)):(cos(?x)*cos(?y)-sin(?x)*sin(?y))~p0 4 ~Ht(sin(?x+?y)):(cos(?x)*sin(?y)+sin(?x)*cos(?y))~p0 4 ~Ht(sin(?x)+sin(?y)):(2*cos(1/2*(?x-?y))*sin(1/2*(?x+?y)))~p0 4 ~Ht(sin(?x)-sin(?y)):(2*cos(1/2*(?x+?y))*sin(1/2*(?x-?y)))~p0 4 ~Ht(cos(?x)+cos(?y)):(2*cos(1/2*(?x-?y))*cos(1/2*(?x+?y)))~p0 4 ~Ht(cos(?x)-cos(?y)):(2*sin(1/2*(-?x+?y))*sin(1/2*(?x+?y)))~p0 4 ~Ht(tan(?x)+tan(?y)):((sin(?x+?y))/(cos(?x)*cos(?y)))~p0 4 ~Ht(tan(?x)-tan(?y)):((sin(?x-?y))/(cos(?x)*cos(?y)))~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})## b!:" *~: ;,&c0 trig Multiplication| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Ht(cos(?y)*sin(?x)):(1/2*(sin(?x-?y)+sin(?x+?y)))~p0 4 ~Ht(cos(?x)*cos(?y)):(1/2*(cos(?x-?y)+cos(?x+?y)))~p0 4 ~Ht(sin(?x)*sin(?y)):(1/2*(cos(?x-?y)-cos(?x+?y)))~p0 4 ~Ht((cos(?x))^2):(1/2*(cos(2*?x)+1))~p0 4 ~Ht((sin(?x))^2):(1/2*(-cos(2*?x)+1))~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})'# b"9" *~: ;,&c0 multiple,Mof,Margument| | identities}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Ht(sin(2*?x)):(2*sin(?x)*cos(?x))~p0 4 ~Ht(cos(2*?x)):(2*(cos(?x))^2-1)~p0 4 ~Ht(tan(2*?x)):((2*tan(?x))/(-(tan(?x))^2+1))~p0 4 ~Ht(cos(?n*?x)):(cos(?x)*cos((?n-1)*?x)-sin(?x)*sin((?n-1)*?x))~p0 4 ~Ht(sin(?n*?x)):(cos((?n-1)*?x)*sin(?x)+cos(?x)*sin((?n-1)*?x))~p0 4 ~Ht(sin(1/2*?x)):(sqrt((1-cos(?x))/2))~p0 4 ~Ht(cos(1/2*?x)):(sqrt((1+cos(?x))/2))~p0 4 ~Ht(tan(1/2*?x)):((1-cos(?x))/(sin(?x)))~p0 4 ~Ht(tan(1/2*?x)):((sin(?x))/(cos(?x)+1))~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})'# b!^" *~: ;,&c0 transform squares| | of trigs}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Ht((sin(?x))^2):(1-(cos(?x))^2)~p0 4 ~Ht((cos(?x))^2):(1-(sin(?x))^2)~p0 4 ~Ht((tan(?x))^2):((sec(?x))^2-1)~p0 4 ~Ht((sec(?x))^2):((tan(?x))^2+1)~p0 4 ~Ht((csc(?x))^2):((cot(?x))^2+1)~p0 4 ~Ht((cot(?x))^2):((csc(?x))^2-1)~p0 4 ~Hs((cos(?x))^2+(sin(?x))^2):(1)~p0 4 ~Hs(-(cos(?x))^2-(sin(?x))^2):(-1)~p0 4 ~Hs(-(sec(?x))^2+(tan(?x))^2):(-1)~p0 4 ~Hs((sec(?x))^2+(tan(?x))^2):(1)~p0 4 ~Hs(-(cot(?x))^2+(csc(?x))^2):(1)~p0 4 ~Hs((cot(?x))^2-(csc(?x))^2):(-1)~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})## b!C" *~: ;,&c0 Complex Identities| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Ht(sin(i*?x)):(i*sinh(?x))~p0 4 ~Ht(cos(i*?x)):(cosh(?x))~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})!# b$L" *~: ;bP8&c0!*Hyperbolic| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 1 ~V?f293 (sech)~p0 2 ~V?f288 (sinh)~p0 2 ~V?f289 (cosh)~p0 2 ~V?f290 (tanh)~p0 2 ~Q ]|Expr|[#b @`bb#_b#_b#_})%# b$L" *~: ;bP8&c0!*Logarithms ,F Powers| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 1 ~V?f32 (log)~p0 2 ~V?f307 (ln)~p0 2 ~V?f291 (exp)~p0 2 ~V?c2 (e)~p0 2 ~Q ]|Expr|[#b @`bb#_b#_b#_})## b$L" *~: ;bP8&c0!*Standard Rules| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 1 ~Q ]|Expr|[#b @`bb#_b#_b#_})!# b'4" *~: ;bP8&c0!*Algebra| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 2 ~Ht(?a*sqrt(-?y^2/?b^2+1)):(?a/?b*sqrt(-?y^2+?b^2))~p0 3 ~Q ]|Expr|[#b @`bb#_b#_b#_})%# b$L" *~: ;bP8&c0!*Logarithms ,F Powers| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 2 ~Ht(?x^(-?y)):(1/?x^?y)~p0 3 ~Hs(exp(?z)):(e^?z)~p0 3 ~Hs(e^(ln(?x))):(?x)~p0 3 ~Hs(10^(log(?x))):(?x)~p0 3 ~Hs(?y^(log_?y(?x))):(?x)~p0 3 ~Hs(ln(e^?x)):(?x)~p0 3 ~Hs(log(10^?x)):(?x)~p0 3 ~Hs(log_?y(?y^?x)):(?x)~p0 3 ~He(ln(?u*?v)):(ln(?u)+ln(?v))~p0 3 ~He(log(?u*?v)):(log(?u)+log(?v))~p0 3 ~He(log_?y(?u*?v)):(log_?y(?u)+log_?y(?v))~p0 3 ~He(ln(?u/?v)):(ln(?u)-ln(?v))~p0 3 ~He(log(?u/?v)):(log(?u)-log(?v))~p0 3 ~He(log_?y(?u/?v)):(log_?y(?u)-log_?y(?v))~p0 3 ~He(ln(?u^?v)):(?v*ln(?u))~p0 3 ~He(log(?u^?v)):(?v*log(?u))~p0 3 ~He(log_?y(?u^?v)):(?v*log_?y(?u))~p0 3 ~He(ln(sqrt(?u))):(1/2*ln(?u))~p0 3 ~He(log(sqrt(?u))):(1/2*log(?u))~p0 3 ~He(log_?y(sqrt(?u))):(1/2*log_?y(?u))~p0 3 ~Q ]|Expr|[#b @`bb#_b#_b#_})!# b$L" *~: ;bP8&c0!*Trigonometry| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 2 ~Q ]|Expr|[#b @`bb#_b#_b#_})+# b$L" *~: ;bP8&c0!*Simplify ,M negation| | and common zeros}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Hs(sin(-?x)):(-sin(?x))~p0 4 ~Hs(cos(-?x)):(cos(?x))~p0 4 ~Hs(tan(-?x)):(-tan(?x))~p0 4 ~Hs(sin('p)):(0)~p0 4 ~Hs(sin(?n*'p)):(0)~p0 4 ~Hs(cos(1/2*'p)):(0)~p0 4 ~Hs(cos(?n/2*'p)):(0)~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})'# b$L" *~: ;bP8&c0!*Transform to basic| | types}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Ht(tan(?x)):((sin(?x))/(cos(?x)))~p0 4 ~Ht(csc(?x)):(1/(sin(?x)))~p0 4 ~Ht(sin(?x)):(1/(csc(?x)))~p0 4 ~Ht(sec(?x)):(1/(cos(?x)))~p0 4 ~Ht(cos(?x)):(1/(sec(?x)))~p0 4 ~Ht(cot(?x)):((cos(?x))/(sin(?x)))~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})## b$L" *~: ;bP8&c0!*Trig Addition| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Ht(cos(?x+?y)):(cos(?x)*cos(?y)-sin(?x)*sin(?y))~p0 4 ~Ht(sin(?x+?y)):(cos(?x)*sin(?y)+sin(?x)*cos(?y))~p0 4 ~Ht(cos(2*?x)):(2*(cos(?x))^2-1)~p0 4 ~Ht(sin(2*?x)):(2*cos(?x)*sin(?x))~p0 4 ~Ht(sin(?n*?x)):(cos((?n-1)*?x)*sin(?x)+cos(?x)*sin((?n-1)*?x))~p0 4 ~Ht(cos(?n*?x)):(cos(?x)*cos((?n-1)*?x)-sin(?x)*sin((?n-1)*?x))~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})/# b$L" *~: ;bP8&c0!*Transform ,M into| | another flavor of trig function}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Ht((sin(?x))^2):(1-(cos(?x))^2)~p0 4 ~Ht((cos(?x))^2):(1-(sin(?x))^2)~p0 4 ~Ht((tan(?x))^2):((sec(?x))^2-1)~p0 4 ~Ht((sec(?x))^2):((tan(?x))^2+1)~p0 4 ~Ht((csc(?x))^2):((cot(?x))^2+1)~p0 4 ~Ht((cot(?x))^2):((csc(?x))^2-1)~p0 4 ~Hs((sin(?x))^2+(cos(?x))^2):(1)~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})b @# b%4" *~: ;bP8&c0!*substituting | |z,]tan,Hx,O2,I into a rational function in sin,Hx,I and cos,H| |x,I}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Hs(cos(2*arctan(?z))):((1-?z^2)/(1+?z^2))~p0 4 ~Hs(sin(2*arctan(?z))):(2*?z/(1+?z^2))~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})## b$L" *~: ;bP8&c0!*Other rules| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Ht((cos(?x))^2):(1/2*(cos(2*?x)+1))~p0 4 ~Ht((sin(?x))^2):(1/2*(-cos(2*?x)+1))~p0 4 ~Ht(cos(?x)*sin(?x)):(1/2*sin(2*?x))~p0 4 ~Hs(sin(arccos(?x))):(sqrt(-?x^2+1))~p0 4 ~Hs(cos(arcsin(?x))):(sqrt(-?x^2+1))~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})!# b!T" *~: ;bP8&c0!*Hyperbolic| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 2 ~Q ]|Expr|[#b @`bb#_b#_b#_})+# b$L" *~: ;bP8&c0!*Simplify ,M negation| | and common zeros}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Hs(sinh(-?x)):(-sinh(?x))~p0 4 ~Hs(cosh(-?x)):(cosh(?x))~p0 4 ~Hs(tanh(-?x)):(-tanh(?x))~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})'# b$L" *~: ;bP8&c0!*Transform into | |other types}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Ht((sinh(?x))^2):((cosh(?x))^2-1)~p0 4 ~Ht((cosh(?x))^2):(1+(sinh(?x))^2)~p0 4 ~Ht((tanh(?x))^2):(1-(sech(?x))^2)~p0 4 ~Ht(sinh(?x)):((e^?x-e^(-?x))/2)~p0 4 ~Ht(cosh(?x)):((e^?x+e^(-?x))/2)~p0 4 ~Ht(tanh(?x)):((e^?x-e^(-?x))/(e^?x+e^(-?x)))~p0 4 ~Ht(tanh(?x)):((sinh(?x))/(cosh(?x)))~p0 4 ~Hs((cosh(?x))^2-(sinh(?x))^2):(1)~p0 4 ~Hs(-(cosh(?x))^2+(sinh(?x))^2):(-1)~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})%# b$L" *~: ;bP8&c0!*Other hyperbolic| | rules}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Ht((cosh(?x))^2):(1/2*(cosh(2*?x)+1))~p0 4 ~Ht((sinh(?x))^2):(1/2*(cosh(2*?x)-1))~p0 4 ~Ht(sinh(2*?x)):(2*cosh(?x)*sinh(?x))~p0 4 ~Ht(cosh(2*?x)):(2*(cosh(?x))^2-1)~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})## b'4" *~: ;bP8&c0!*Calculus Rules| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 1 ~Q ]|Expr|[#b @`bb#_b#_b#_})## b$L" *~: ;bP8&c0!*Integration Rules| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0 2 ~Q ]|Expr|[#b @`bb#_b#_b#_}))# b$8" *~: ;bP8&c0!*after Partial Fraction| | Decomposition integration}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Hs(ln(?x+?b*i)):((-i*arctan(?x/?b)+1/2*ln(?x^2+?b^2))+1/2*'p*~ i)~p0 4 ~Hs(ln(?x+i)):((-i*arctan(?x)+1/2*ln(?x^2+1))+1/2*'p*i)~p0 4 ~Hs(ln(?x-?b*i)):((i*arctan(?x/?b)+1/2*ln(?x^2+?b^2))+1/2*'p*~ i)~p0 4 ~Hs(ln(?x-i)):((i*arctan(?x)+1/2*ln(?x^2+1))+1/2*'p*i)~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})## b!:" *~: ;,&c0 Common Integrals| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0 3 ~Q ]|Expr|[#b @`bb#_b#_b#_})'# b"0" *~: ;,&c0 irrational containing| | ;bP9&c0!2ax;,&c0 ,K;bP9&c0!2b}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 4 ~Hs(Integral(?x*sqrt(?x+?b)*d*?x)):(2/15*(3*?x-2*?b)*(?x+?b)^~ (3/2)+(Integral(0)))~p0 5 ~Hs(Integral(?x*sqrt(?a*?x+?b)*d*?x)):((2*(3*?a*?x-2*?b)*sqrt(~ (?a*?x+?b)^3))/(15*?a^2)+(Integral(0)))~p0 5 ~Hs(Integral(?x*sqrt(-?x+?b)*d*?x)):(2/15*(-3*?x-2*?b)*(-?x+?b)^~ (3/2)+(Integral(0)))~p0 5 ~Hs(Integral(?x*sqrt(-?a*?x+?b)*d*?x)):((2*(-2*?b-3*?a*?x)*sqrt(~ (?b-?a*?x)^3))/(15*?a^2)+(Integral(0)))~p0 5 ~Hs(Integral(sqrt(?x+?b)/?x*d*?x)):(2*sqrt(?x+?b)+?b*(Integral(~ 1/(sqrt(?x+?b)*?x)*d*?x)))~p0 5 ~Hs(Integral(sqrt(-?x+?b)/?x*d*?x)):(2*sqrt(?b-?x)+?b*(Integral(~ 1/(sqrt(-?x+?b)*?x)*d*?x)))~p0 5 ~Hs(Integral(sqrt(?a*?x+?b)/?x*d*?x)):(2*sqrt(?a*?x+?b)+?b*(Integral(~ 1/(sqrt(?a*?x+?b)*?x)*d*?x)))~p0 5 ~Hs(Integral(sqrt(-?a*?x+?b)/?x*d*?x)):(2*sqrt(?b-?a*?x)+?b*(Integral(~ 1/(sqrt(-?a*?x+?b)*?x)*d*?x)))~p0 5 ~Hs(Integral(1/sqrt(?x+?b)*d*?x)):(2*sqrt(?x+?b)+(Integral(0)))~p0 5 ~Hs(Integral(1/sqrt(-?x+?b)*d*?x)):(-2*sqrt(-?x+?b)+(Integral(~ 0)))~p0 5 ~Hs(Integral(1/sqrt(?a*?x+?b)*d*?x)):(2*sqrt(?a*?x+?b)/?a+(Integral(~ 0)))~p0 5 ~Hs(Integral(1/sqrt(-?a*?x+?b)*d*?x)):(2*(-sqrt(?b-?a*?x)/?a)+~ (Integral(0)))~p0 5 ~Hs(Integral(1/(?x*sqrt(?x+?b))*d*?x)):(Conditional((ln(abs((~ sqrt(?x+?b)-sqrt(?b))/(sqrt(?x+?b)+sqrt(?b)))))/sqrt(?b)+(Integral(~ 0)),(?b>0);2*(arctan(sqrt((?x+?b)/(-?b))))/sqrt(abs(?b))+(Integral(~ 0)),(?b<0)))~p0 5 ~Hs(Integral(1/(?x*sqrt(-?x+?b))*d*?x)):(Conditional((ln(abs(~ (sqrt(-?x+?b)-sqrt(?b))/(sqrt(-?x+?b)+sqrt(?b)))))/sqrt(?b)+(Integral(~ 0)),(?b>0);2*(arctan(sqrt((-?x+?b)/(-?b))))/sqrt(abs(?b))+(Integral(~ 0)),(?b<0)))~p0 5 ~Hs(Integral(1/(?x*sqrt(?a*?x+?b))*d*?x)):(Conditional(1/sqrt(~ ?b)*ln(abs((sqrt(?a*?x+?b)-sqrt(?b))/(sqrt(?a*?x+?b)+sqrt(?b))))+~ (Integral(0)),(?b>0);2/sqrt(abs(?b))*arctan(sqrt((?a*?x+?b)/(-~ ?b)))+(Integral(0)),(?b<0)))~p0 5 ~Hs(Integral(1/(?x*sqrt(-?a*?x+?b))*d*?x)):(Conditional(1/sqrt(~ ?b)*ln(abs((sqrt(-?a*?x+?b)-sqrt(?b))/(sqrt(-?a*?x+?b)+sqrt(?b))))+~ (Integral(0)),(?b>0);2/sqrt(abs(?b))*arctan(sqrt((-?a*?x+?b)/~ (-?b)))+(Integral(0)),(?b<0)))~p0 5 ~Hs(Integral(1/(?x^2*sqrt(?x+?b))*d*?x)):(-sqrt(?x+?b)/(?b*?x)+~ (-1/2*1/?b)*(Integral(1/(sqrt(?x+?b)*?x)*d*?x)))~p0 5 ~Hs(Integral(1/(?x^2*sqrt(-?x+?b))*d*?x)):(-sqrt(?b-?x)/(?b*?x)+~ (1/2*1/?b)*(Integral(1/(sqrt(-?x+?b)*?x)*d*?x)))~p0 5 ~Hs(Integral(1/(?x^2*sqrt(?a*?x+?b))*d*?x)):(-sqrt(?b+?a*?x)/~ (?b*?x)+1/2*(-?a)/?b*(Integral(1/(sqrt(?b+?a*?x)*?x)*d*?x)))~p0 5 ~Hs(Integral(1/(?x^2*sqrt(-?a*?x+?b))*d*?x)):(-sqrt(?b-?a*?x)/~ (?b*?x)+1/2*?a/?b*(Integral(1/(sqrt(?b-?a*?x)*?x)*d*?x)))~p0 5 ~Q ]|Expr|[#b @`bb#_b#_b#_})%# b!^" *~: ;,&c0 irrational containing| | $^;bP9&c0!2x_^;bP:&c0!*2}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 4 ~Hs(Integral(1/sqrt(1-?x^2)*d*?x)):(arcsin(?x)+(Integral(0)))~p0 5 ~Hs(Integral(1/(?x*sqrt(?x^2-1))*d*?x)):(arcsec(?x)+(Integral(~ 0)))~p0 5 ~Hs(Integral(sqrt(?x^2+?b)*d*?x)):(1/2*(?x*sqrt(?x^2+?b)+?b*ln(~ abs(?x+sqrt(?x^2+?b))))+(Integral(0)))~p0 5 ~Hs(Integral(sqrt(-?x^2+?b^2)*d*?x)):((?x*sqrt(?b^2-?x^2))/2+~ ?b^2/2*arcsin(?x/?b)+(Integral(0)))~p0 5 ~Hs(Integral(sqrt(-?x^2+?b^2)*?x*d*?x)):(-(?b^2-?x^2)^(3/2)/3+~ (Integral(0)))~p0 5 ~Hs(Integral(sqrt(-?x^2+?b)*d*?x)):(1/2*(?x*sqrt(?b-?x^2)+?b*~ arcsin(?x/sqrt(?b)))+(Integral(0)))~p0 5 ~Hs(Integral(1/sqrt(?x^2+?b)*d*?x)):(ln(abs(?x+sqrt(?x^2+?b)))+~ (Integral(0)))~p0 5 ~Hs(Integral((?b-?x^2)^(3/2)*d*?x)):(1/4*(?x*sqrt((?b-?x^2)^3)+~ (3*?b*?x)/2*sqrt((?b-?x^2)^3)+(3*?b^2)/2*arcsin(?x/sqrt(?b)))+~ (Integral(0)))~p0 5 ~Q ]|Expr|[#b @`bb#_b#_b#_})$# b!^" *~: ;,&c0 transcendental functions| |"!Caslon 3 Roman^:!;),N}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 4 ~Hs(Integral(e^?x*sin(?x)*d*?x)):(e^?x/2*(-cos(?x)+sin(?x))+(Integral(~ 0)))~p0 5 ~Hs(Integral(e^?x*sin(?b*?x)*d*?x)):(e^?x/(1+?b^2)*(sin(?b*?x)-~ ?b*cos(?b*?x))+(Integral(0)))~p0 5 ~Hs(Integral(e^?x*cos(?x)*d*?x)):(e^?x/2*(cos(?x)+sin(?x))+(Integral(~ 0)))~p0 5 ~Hs(Integral(e^?x*cos(?b*?x)*d*?x)):(e^?x/(1+?b^2)*(cos(?b*?x)+~ ?b*sin(?b*?x))+(Integral(0)))~p0 5 ~Hs(Integral(e^(-?x)*sin(?x)*d*?x)):(e^(-?x)/2*(-sin(?x)-cos(~ ?x))+(Integral(0)))~p0 5 ~Hs(Integral(e^(-?x)*sin(?b*?x)*d*?x)):(e^(-?x)/(1+?b^2)*(-sin(~ ?b*?x)-?b*cos(?b*?x))+(Integral(0)))~p0 5 ~Hs(Integral(e^(-?x)*cos(?x)*d*?x)):(e^(-?x)/2*(-cos(?x)+sin(~ ?x))+(Integral(0)))~p0 5 ~Hs(Integral(e^(-?x)*cos(?b*?x)*d*?x)):(e^(-?x)/(1+?b^2)*(-cos(~ ?b*?x)+?b*sin(?b*?x))+(Integral(0)))~p0 5 ~Hs(Integral(e^(?a*?x)*sin(?x)*d*?x)):(e^(?a*?x)/(?a^2+1)*(?a*~ sin(?x)-cos(?x))+(Integral(0)))~p0 5 ~Hs(Integral(e^(?a*?x)*sin(?b*?x)*d*?x)):(e^(?a*?x)/(?a^2+?b^~ 2)*(?a*sin(?b*?x)-?b*cos(?b*?x))+(Integral(0)))~p0 5 ~Hs(Integral(e^(?a*?x)*cos(?x)*d*?x)):(e^(?a*?x)/(?a^2+1)*(?a*~ cos(?x)+sin(?x))+(Integral(0)))~p0 5 ~Hs(Integral(e^(?a*?x)*cos(?b*?x)*d*?x)):(e^(?a*?x)/(?a^2+?b^~ 2)*(?a*cos(?b*?x)+?b*sin(?b*?x))+(Integral(0)))~p0 5 ~Hs(Integral(e^(-?a*?x)*sin(?x)*d*?x)):(e^(-?a*?x)/(?a^2+1)*(~ -?a*sin(?x)-cos(?x))+(Integral(0)))~p0 5 ~Hs(Integral(e^(-?a*?x)*sin(?b*?x)*d*?x)):(e^(-?a*?x)/(?a^2+?b^~ 2)*(-?a*sin(?b*?x)-?b*cos(?b*?x))+(Integral(0)))~p0 5 ~Hs(Integral(e^(-?a*?x)*cos(?x)*d*?x)):(e^(-?a*?x)/(?a^2+1)*(~ -?a*cos(?x)+sin(?x))+(Integral(0)))~p0 5 ~Hs(Integral(e^(-?a*?x)*cos(?b*?x)*d*?x)):(e^(-?a*?x)/(?a^2+?b^~ 2)*(-?a*cos(?b*?x)+?b*sin(?b*?x))+(Integral(0)))~p0 5 ~Hs(Integral(?x*ln(?x)*d*?x)):((?x^2/2*ln(?x)-?x^2/4)+(Integral(~ 0)))~p0 5 ~Hs(Integral(?x^?n*ln(?x)*d*?x)):(?x^(?n+1)*((ln(?x))/(?n+1)-~ 1/(?n+1)^2)+(Integral(0)))~p0 5 ~Hs(Integral(?x*ln(?a*?x)*d*?x)):(?x^2*((ln(?a*?x))/2-1/4)+(Integral(~ 0)))~p0 5 ~Hs(Integral(?x^?n*ln(?a*?x)*d*?x)):(?x^(?n+1)*((ln(?a*?x))/(~ ?n+1)-1/(?n+1)^2)+(Integral(0)))~p0 5 ~Q ]|Expr|[#b @`bb#_b#_b#_})## b!^" *~: ;,&c0 trigonometric functions| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 4 ~Hs(Integral(?x*sin(?x)*d*?x)):(sin(?x)-?x*cos(?x)+(Integral(~ 0)))~p0 5 ~Hs(Integral(?x*sin(?a*?x)*d*?x)):(1/?a^2*sin(?a*?x)-?x/?a*cos(~ ?a*?x)+(Integral(0)))~p0 5 ~Hs(Integral(?x*cos(?x)*d*?x)):(cos(?x)+?x*sin(?x)+(Integral(~ 0)))~p0 5 ~Hs(Integral(?x*cos(?a*?x)*d*?x)):(1/?a^2*cos(?a*?x)+1/?a*?x*~ sin(?a*?x)+(Integral(0)))~p0 5 ~Hs(Integral(?x^?m*sin(?x)*d*?x)):(-?x^?m*cos(?x)+?m*(Integral(~ ?x^(?m-1)*cos(?x)*d*?x)))~p0 5 ~Hs(Integral(?x^?m*sin(?a*?x)*d*?x)):(-1/?a*?x^?m*cos(?a*?x)+~ ?m/?a*(Integral(?x^(?m-1)*cos(?a*?x)*d*?x)))~p0 5 ~Hs(Integral(?x^?m*cos(?x)*d*?x)):(?x^?m*sin(?x)-?m*(Integral(~ ?x^(?m-1)*sin(?x)*d*?x)))~p0 5 ~Hs(Integral(?x^?m*cos(?a*?x)*d*?x)):(1/?a*?x^?m*sin(?a*?x)-?m/~ ?a*(Integral(?x^(?m-1)*sin(?a*?x)*d*?x)))~p0 5 ~Hs(Integral((sin(?x))^?n*d*?x)):(-((sin(?x))^(?n-1)*cos(?x))/~ ?n+(?n-1)/?n*(Integral((sin(?x))^(?n-2)*d*?x)))~p0 5 ~Hs(Integral((cos(?x))^?n*d*?x)):((sin(?x)*(cos(?x))^(?n-1))/~ ?n+(?n-1)/?n*(Integral((cos(?x))^(?n-2)*d*?x)))~p0 5 ~Q ]|Expr|[#b @`bb#_b#_b#_})%# b$4" *~: ;bP8&c0!*Derivatives of | |Integrals}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Ht(Diff(?x)*(Integral(?y*d*?x))):(?y)~p0 4 ~V?c4 (i)~p0 1 ~V?d16 (d)~p0 1 ~V?v0 (k)~p0 1 ~V?c0 (n)~p0 1 ~V?c0 (c)~p0 1 ~V?c0 (b)~p0 1 ~V?c0 (a)~p0 1 ~V?v0 (z)~p0 1 ~V?v0 (y)~p0 1 ~V?v0 (x)~p0 1 ~c3 20 -1 19 -1 48 -1 109 -1 ~c17 22 -1 21 -1 14 -1 366 -1 15 -1 103 -1 28 -1 104 -1 13 -1 367 -1 34 -1 370 -1 38 -1 106 -1 40 -1 107 -1 42 -1 110 -1 ~c5 24 -1 23 -1 12 -1 112 -1 50 -1 108 -1 ~c3 29 -1 28 -1 15 -1 14 -1 ~c1 34 -1 33 -1 ~c1 35 -1 34 -1 ~c2 36 -1 35 -1 207 -1 ~c2 39 -1 38 -1 36 -1 ~c2 41 -1 40 -1 36 -1 ~c4 43 -1 42 -1 29 -1 39 -1 41 -1 ~c1 44 -1 43 -1 ~c4 45 -1 44 -1 322 -1 323 -1 131 -1 ~c3 46 -1 45 -1 14 -1 13 -1 ~c1 49 -1 46 -1 ~c3 51 -1 50 -1 12 -1 112 -1 ~c2 54 -1 53 -1 57 -1 ~c3 57 -1 56 -1 14 -1 15 -1 ~c3 62 -1 61 -1 58 -1 60 -1 ~c1 64 -1 63 -1 ~c1 65 -1 64 -1 ~c1 66 -1 65 -1 ~c1 68 -1 67 -1 ~c1 69 -1 68 -1 ~c1 70 -1 69 -1 ~c2 71 -1 70 -1 153 -1 ~c1 72 -1 71 -1 ~c4 73 -1 72 -1 57 -1 13 -1 14 -1 ~c2 76 -1 75 -1 73 -1 ~c2 77 -1 76 -1 57 -1 ~c3 81 -1 80 -1 13 -1 14 -1 ~c13 82 -1 81 -1 14 -1 366 -1 13 -1 367 -1 15 -1 103 -1 28 -1 104 -1 72 -1 98 -1 60 -1 101 -1 ~c2 84 -1 83 -1 54 -1 ~c13 85 -1 84 -1 14 -1 366 -1 13 -1 367 -1 15 -1 103 -1 28 -1 104 -1 72 -1 98 -1 60 -1 101 -1 ~c2 87 -1 86 -1 77 -1 ~c1 88 -1 87 -1 ~c2 90 -1 89 -1 73 -1 ~c23 92 -1 91 -1 48 -1 109 -1 14 -1 366 -1 15 -1 103 -1 28 -1 104 -1 13 -1 367 -1 34 -1 370 -1 38 -1 106 -1 40 -1 107 -1 42 -1 110 -1 12 -1 112 -1 50 -1 108 -1 ~e