Convert`TeX`ExpressionToTeX;
System`Convert`TeXFormDump`StripIdentityBoxes[System`Convert`CommonDump`boxes_]:=System`Convert`CommonDump`boxes//.{RowBox[{System`Convert`CommonDump`x_}]:>System`Convert`CommonDump`x,InterpretationBox[TemplateBox[_,"NumericalApproximation",___],System`Convert`CommonDump`x_,___]:>Block[{BoxForm`UseApproximations=False},MakeBoxes[System`Convert`CommonDump`x,TraditionalForm]],(AdjustmentBox|InterpretationBox|TagBox|DynamicBox|TooltipBox)[System`Convert`CommonDump`x_,___]:>System`Convert`CommonDump`x}



Initial/:Verbatim[TagSetDelayed][Initial[sym_],lhs_,rhs_]:=With[{new=Block[{sym},TagSetDelayed[sym,lhs,rhs];
First@Language`ExtendedDefinition[sym]],protect=Unprotect[sym]},sym;
Replace[new,Rule[values_,n:Except[{}]]:>(values[sym]=Prepend[values[sym],n]),{2}];
Protect@protect;];
Initial[Convert`TeX`ExpressionToTeX]/:Convert`TeX`ExpressionToTeX[e__]/;!TrueQ@$TeX:=Block[{$TeX=True},Convert`TeX`ExpressionToTeX[e]];

(*System`Convert`TeXFormDump`maketex[s_String]/;!StringMatchQ[s,"\""~~___~~"\""]&&SyntaxQ[s,TeXForm]:=Replace[s,{n_/;StringMatchQ[n,NumberString]:>n,w_?wordQ:>"\\operatorname{"<>w<>"}"}]

wordQ[s_String]:=Length@StringSplit[s,WordBoundary]==1
*)


Initial[FresnelC]/:MakeBoxes[FresnelC[a_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[FresnelC][a],TraditionalForm];
Initial[CosIntegral]/:MakeBoxes[CosIntegral[a_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[CosIntegral][a],TraditionalForm];
Initial[AiryAiPrime]/:MakeBoxes[AiryAiPrime[a_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[AiryAiPrime][a],TraditionalForm];
Initial[AiryAi]/:MakeBoxes[AiryAi[a_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[AiryAi][a],TraditionalForm];
Initial[BesselI]/:MakeBoxes[BesselI[a_,n_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[BesselI][a,n],TraditionalForm];

Initial[BesselJZero]/:MakeBoxes[BesselJZero[a_,n_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[BesselJZero][a,n],TraditionalForm];
Initial[BesselJZero]/:MakeBoxes[BesselJZero[a_,n_,x0_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[BesselJZero][a,n,x0],TraditionalForm];

Initial[BesselJ]/:MakeBoxes[BesselJ[a_,n_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[BesselJ][a,n],TraditionalForm];

Initial[Erfi]/:MakeBoxes[Erfi[a_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[Erfi][a],TraditionalForm];
Initial[ExpIntegralEi]/:MakeBoxes[ExpIntegralEi[a_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[ExpIntegralEi][a],TraditionalForm];
Initial[HypergeometricU]/:MakeBoxes[HypergeometricU[a_,b_,c_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[HypergeometricU][a,b,c],TraditionalForm];
Initial[LaguerreL]/:MakeBoxes[LaguerreL[a_,b_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[LaguerreL][a,b],TraditionalForm];
Initial[LaguerreL]/:MakeBoxes[LaguerreL[a_,b_,x_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[LaguerreL][a,b,x],TraditionalForm];

Initial[HermiteH]/:MakeBoxes[HermiteH[a_,b_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[HermiteH][a,b],TraditionalForm];

Initial[Hypergeometric1F1]/:MakeBoxes[Hypergeometric1F1[a_,b_,c_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[Hypergeometric1F1][a,b,c],TraditionalForm];

Initial[Hypergeometric2F1]/:MakeBoxes[Hypergeometric2F1[a_,b_,c_,d_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[Hypergeometric2F1][a,b,c,d],TraditionalForm];
Initial[HypergeometricPFQ]/:MakeBoxes[HypergeometricPFQ[a_,b_,c_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[HypergeometricPFQ][a,b,c],TraditionalForm];
Initial[Erfc]/:MakeBoxes[Erfc[a_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[Erfc][a],TraditionalForm];
Initial[Erf]/:MakeBoxes[Erf[a_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[Erf][a],TraditionalForm];
Initial[Erf]/:MakeBoxes[Erf[a_,b_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[Erf][a,b],TraditionalForm];

Initial[ExpIntegralE]/:MakeBoxes[ExpIntegralE[a_,b_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[ExpIntegralE][a,b],TraditionalForm];

ParabolicCylinderD;
Initial[ParabolicCylinderD]/:MakeBoxes[ParabolicCylinderD[a_,b_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[ParabolicCylinderD][a,b],TraditionalForm];
Initial[ProductLog]/:MakeBoxes[ProductLog[a_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[ProductLog][a],TraditionalForm];
Initial[ProductLog]/:MakeBoxes[ProductLog[a_,b_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[ProductLog][a,b],TraditionalForm];

DawsonF;
Initial[DawsonF]/:MakeBoxes[DawsonF[a_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[DawsonF][a],TraditionalForm];
Initial[EllipticF]/:MakeBoxes[EllipticF[a_,b_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[EllipticF][a,b],TraditionalForm];
Initial[EllipticPi]/:MakeBoxes[EllipticPi[a_,b_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[EllipticPi][a,b],TraditionalForm];
Initial[EllipticPi]/:MakeBoxes[EllipticPi[a_,b_,c_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[EllipticPi][a,b,c],TraditionalForm];

Initial[Gamma]/:MakeBoxes[Gamma[a_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[Gamma][a],TraditionalForm];
Initial[Gamma]/:MakeBoxes[Gamma[a_,b_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[Gamma][a,b],TraditionalForm];
Initial[Gamma]/:MakeBoxes[Gamma[a_,b_,c_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[Gamma][a,b,c],TraditionalForm];

Initial[LogIntegral]/:MakeBoxes[LogIntegral[a_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[LogIntegral][a],TraditionalForm];

Initial[PolyLog]/:MakeBoxes[PolyLog[a_,b_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[PolyLog][a,b],TraditionalForm];
Initial[PolyLog]/:MakeBoxes[PolyLog[a_,b_,c_],TraditionalForm]/;$TeX:=MakeBoxes[Defer[PolyLog][a,b,c],TraditionalForm];

?EllipticPi

Mathematica 10 results for Charlwood’s integrals

<<Rubi`

1

Clear[x]
mma=Integrate[ArcSin[x] Log[x],x]

TeXForm[mma]

rubi=Int[ArcSin[x] Log[x],x]

TeXForm[rubi]

2

Integrate[((x ArcSin[x])/(Sqrt[1-(x^2)])),x]

TeXForm[%]

Int[((x ArcSin[x])/(Sqrt[1-(x^2)])),x]

TeXForm[%]

3

integrand=ArcSin[(Sqrt[x+1])-(Sqrt[x])]
TeXForm[integrand]

res=Integrate[integrand,x]

TeXForm[res]

ClearAll[x]
integrand=ArcSin[Sqrt[x+1]-Sqrt[x]];
res=Int[integrand,x]
TeXForm[res]

4

integrand=Log[1+x(Sqrt[1+(x^2)])]
TeXForm[integrand]

res=Integrate[integrand,x]

TeXForm[res]

integrand=Log[1+x(Sqrt[1+(x^2)])]
res=Int[integrand,x];
TeXForm[res]

5

integrand=(((Cos[x])^2)/(Sqrt[((Cos[x])^4)+((Cos[x])^2)+1]));
TeXForm[integrand]

integrand=(((Cos[x])^2)/(Sqrt[((Cos[x])^4)+((Cos[x])^2)+1]));
res=Integrate[integrand,x]
TeXForm[res]

<<Rubi`

integrand=(((Cos[x])^2)/(Sqrt[((Cos[x])^4)+((Cos[x])^2)+1]));
res=Int[integrand,x]
TeXForm[res]

6

integrand=Tan[x](Sqrt[1+((Tan[x])^4)]);
TeXForm[integrand]

integrand=Tan[x](Sqrt[1+((Tan[x])^4)]);
res=Integrate[integrand,x];
TeXForm[res]

ClearAll[x]
integrand=Tan[x] Sqrt[1+Tan[x]^4];
res=Int[integrand,x]

TeXForm[res]

7

integrand=((Tan[x])/(Sqrt[((Sec[x])^3)+1]));
TeXForm[integrand]

integrand=((Tan[x])/(Sqrt[((Sec[x])^3)+1]));
res=Integrate[integrand,x]
TeXForm[res]

ClearAll[x]
integrand=Tan[x]/Sqrt[Sec[x]^3+1];
res=Int[integrand,x]
TeXForm[res]

8

integrand=(Sqrt[((Tan[x])^2)+2Tan[x]+2]);
TeXForm[integrand]

integrand=(Sqrt[((Tan[x])^2)+2Tan[x]+2]);
res=Integrate[integrand,x];
TeXForm[res]

integrand=Sqrt[Tan[x]^2+2 Tan[x]+2];
res=Int[integrand,x]
TeXForm[res]

9

integrand=Sin[x]ArcTan[(Sqrt[Sec[x]-1])];
TeXForm[integrand]

integrand=Sin[x]ArcTan[(Sqrt[Sec[x]-1])];
res=Integrate[integrand,x]
TeXForm[res]

integrand=Sin[x] ArcTan[Sqrt[Sec[x]-1]];
res=Int[integrand,x];
TeXForm[res]

10

integrand=(((x^3)Exp[ArcSin[x]])/(Sqrt[1-(x^2)]));
TeXForm[integrand]

integrand=(((x^3)Exp[ArcSin[x]])/(Sqrt[1-(x^2)]));
res=Integrate[integrand,x]
TeXForm[res]

integrand=(x^3 Exp[ArcSin[x]])/Sqrt[1-x^2];
res=Int[integrand,x];
TeXForm[res]

11

Integrate[(x*Log[1+x^2]*Log[x+Sqrt[1+x^2]])/Sqrt[1+x^2],x]

12

Integrate[ArcTan[x+Sqrt[1-x^2]],x]

13

Integrate[x*ArcTan[x+Sqrt[1-x^2]]/Sqrt[1-x^2],x]

14

Integrate[ArcSin[x]/(1+Sqrt[1-x^2]),x]

15

Integrate[Log[x+Sqrt[1+x^2]]/(1-x^2)^(3/2),x]

16

Integrate[ArcSin[x]/(1+x^2)^(3/2),x]

17

Integrate[Log[x+Sqrt[x^2-1]]/(1+x^2)^(3/2),x]

18

Integrate[Log[x]/(x^2*Sqrt[x^2-1]),x]

19

Integrate[Sqrt[1+x^3]/x,x]

20

Integrate[x*Log[x+Sqrt[x^2-1]]/Sqrt[x^2-1],x]