1,1,51,75,0.033964,"\int x^4 \cos ^{-1}(a x) \, dx","Integrate[x^4*ArcCos[a*x],x]","\frac{1}{5} x^5 \cos ^{-1}(a x)-\frac{\sqrt{1-a^2 x^2} \left(3 a^4 x^4+4 a^2 x^2+8\right)}{75 a^5}","-\frac{\left(1-a^2 x^2\right)^{5/2}}{25 a^5}+\frac{2 \left(1-a^2 x^2\right)^{3/2}}{15 a^5}-\frac{\sqrt{1-a^2 x^2}}{5 a^5}+\frac{1}{5} x^5 \cos ^{-1}(a x)",1,"-1/75*(Sqrt[1 - a^2*x^2]*(8 + 4*a^2*x^2 + 3*a^4*x^4))/a^5 + (x^5*ArcCos[a*x])/5","A",1
2,1,54,69,0.0379106,"\int x^3 \cos ^{-1}(a x) \, dx","Integrate[x^3*ArcCos[a*x],x]","\frac{8 a^4 x^4 \cos ^{-1}(a x)-a x \sqrt{1-a^2 x^2} \left(2 a^2 x^2+3\right)+3 \sin ^{-1}(a x)}{32 a^4}","\frac{3 \sin ^{-1}(a x)}{32 a^4}-\frac{x^3 \sqrt{1-a^2 x^2}}{16 a}-\frac{3 x \sqrt{1-a^2 x^2}}{32 a^3}+\frac{1}{4} x^4 \cos ^{-1}(a x)",1,"(-(a*x*Sqrt[1 - a^2*x^2]*(3 + 2*a^2*x^2)) + 8*a^4*x^4*ArcCos[a*x] + 3*ArcSin[a*x])/(32*a^4)","A",1
3,1,42,54,0.0303652,"\int x^2 \cos ^{-1}(a x) \, dx","Integrate[x^2*ArcCos[a*x],x]","\frac{1}{3} x^3 \cos ^{-1}(a x)-\frac{\sqrt{1-a^2 x^2} \left(a^2 x^2+2\right)}{9 a^3}","\frac{\left(1-a^2 x^2\right)^{3/2}}{9 a^3}-\frac{\sqrt{1-a^2 x^2}}{3 a^3}+\frac{1}{3} x^3 \cos ^{-1}(a x)",1,"-1/9*(Sqrt[1 - a^2*x^2]*(2 + a^2*x^2))/a^3 + (x^3*ArcCos[a*x])/3","A",1
4,1,42,45,0.0175487,"\int x \cos ^{-1}(a x) \, dx","Integrate[x*ArcCos[a*x],x]","\frac{-a x \sqrt{1-a^2 x^2}+2 a^2 x^2 \cos ^{-1}(a x)+\sin ^{-1}(a x)}{4 a^2}","-\frac{x \sqrt{1-a^2 x^2}}{4 a}+\frac{\sin ^{-1}(a x)}{4 a^2}+\frac{1}{2} x^2 \cos ^{-1}(a x)",1,"(-(a*x*Sqrt[1 - a^2*x^2]) + 2*a^2*x^2*ArcCos[a*x] + ArcSin[a*x])/(4*a^2)","A",1
5,1,26,26,0.0075359,"\int \cos ^{-1}(a x) \, dx","Integrate[ArcCos[a*x],x]","x \cos ^{-1}(a x)-\frac{\sqrt{1-a^2 x^2}}{a}","x \cos ^{-1}(a x)-\frac{\sqrt{1-a^2 x^2}}{a}",1,"-(Sqrt[1 - a^2*x^2]/a) + x*ArcCos[a*x]","A",1
6,1,51,51,0.0171726,"\int \frac{\cos ^{-1}(a x)}{x} \, dx","Integrate[ArcCos[a*x]/x,x]","-\frac{1}{2} i \text{Li}_2\left(-e^{2 i \cos ^{-1}(a x)}\right)-\frac{1}{2} i \cos ^{-1}(a x)^2+\cos ^{-1}(a x) \log \left(1+e^{2 i \cos ^{-1}(a x)}\right)","-\frac{1}{2} i \text{Li}_2\left(-e^{2 i \cos ^{-1}(a x)}\right)-\frac{1}{2} i \cos ^{-1}(a x)^2+\cos ^{-1}(a x) \log \left(1+e^{2 i \cos ^{-1}(a x)}\right)",1,"(-1/2*I)*ArcCos[a*x]^2 + ArcCos[a*x]*Log[1 + E^((2*I)*ArcCos[a*x])] - (I/2)*PolyLog[2, -E^((2*I)*ArcCos[a*x])]","A",1
7,1,34,27,0.0139018,"\int \frac{\cos ^{-1}(a x)}{x^2} \, dx","Integrate[ArcCos[a*x]/x^2,x]","a \log \left(\sqrt{1-a^2 x^2}+1\right)-a \log (x)-\frac{\cos ^{-1}(a x)}{x}","a \tanh ^{-1}\left(\sqrt{1-a^2 x^2}\right)-\frac{\cos ^{-1}(a x)}{x}",1,"-(ArcCos[a*x]/x) - a*Log[x] + a*Log[1 + Sqrt[1 - a^2*x^2]]","A",1
8,1,31,34,0.0177613,"\int \frac{\cos ^{-1}(a x)}{x^3} \, dx","Integrate[ArcCos[a*x]/x^3,x]","\frac{a x \sqrt{1-a^2 x^2}-\cos ^{-1}(a x)}{2 x^2}","\frac{a \sqrt{1-a^2 x^2}}{2 x}-\frac{\cos ^{-1}(a x)}{2 x^2}",1,"(a*x*Sqrt[1 - a^2*x^2] - ArcCos[a*x])/(2*x^2)","A",1
9,1,67,56,0.0278446,"\int \frac{\cos ^{-1}(a x)}{x^4} \, dx","Integrate[ArcCos[a*x]/x^4,x]","-\frac{1}{6} a^3 \log (x)+\frac{a \sqrt{1-a^2 x^2}}{6 x^2}+\frac{1}{6} a^3 \log \left(\sqrt{1-a^2 x^2}+1\right)-\frac{\cos ^{-1}(a x)}{3 x^3}","\frac{a \sqrt{1-a^2 x^2}}{6 x^2}+\frac{1}{6} a^3 \tanh ^{-1}\left(\sqrt{1-a^2 x^2}\right)-\frac{\cos ^{-1}(a x)}{3 x^3}",1,"(a*Sqrt[1 - a^2*x^2])/(6*x^2) - ArcCos[a*x]/(3*x^3) - (a^3*Log[x])/6 + (a^3*Log[1 + Sqrt[1 - a^2*x^2]])/6","A",1
10,1,41,58,0.029743,"\int \frac{\cos ^{-1}(a x)}{x^5} \, dx","Integrate[ArcCos[a*x]/x^5,x]","\frac{a x \sqrt{1-a^2 x^2} \left(2 a^2 x^2+1\right)-3 \cos ^{-1}(a x)}{12 x^4}","\frac{a \sqrt{1-a^2 x^2}}{12 x^3}+\frac{a^3 \sqrt{1-a^2 x^2}}{6 x}-\frac{\cos ^{-1}(a x)}{4 x^4}",1,"(a*x*Sqrt[1 - a^2*x^2]*(1 + 2*a^2*x^2) - 3*ArcCos[a*x])/(12*x^4)","A",1
11,1,72,80,0.0807304,"\int \frac{\cos ^{-1}(a x)}{x^6} \, dx","Integrate[ArcCos[a*x]/x^6,x]","\frac{1}{40} \left(-3 a^5 \log (x)+\frac{a \sqrt{1-a^2 x^2} \left(3 a^2 x^2+2\right)}{x^4}+3 a^5 \log \left(\sqrt{1-a^2 x^2}+1\right)-\frac{8 \cos ^{-1}(a x)}{x^5}\right)","\frac{a \sqrt{1-a^2 x^2}}{20 x^4}+\frac{3}{40} a^5 \tanh ^{-1}\left(\sqrt{1-a^2 x^2}\right)+\frac{3 a^3 \sqrt{1-a^2 x^2}}{40 x^2}-\frac{\cos ^{-1}(a x)}{5 x^5}",1,"((a*Sqrt[1 - a^2*x^2]*(2 + 3*a^2*x^2))/x^4 - (8*ArcCos[a*x])/x^5 - 3*a^5*Log[x] + 3*a^5*Log[1 + Sqrt[1 - a^2*x^2]])/40","A",1
12,1,82,120,0.0674205,"\int x^4 \cos ^{-1}(a x)^2 \, dx","Integrate[x^4*ArcCos[a*x]^2,x]","-\frac{16 x}{75 a^4}-\frac{8 x^3}{225 a^2}-\frac{2 \sqrt{1-a^2 x^2} \left(3 a^4 x^4+4 a^2 x^2+8\right) \cos ^{-1}(a x)}{75 a^5}+\frac{1}{5} x^5 \cos ^{-1}(a x)^2-\frac{2 x^5}{125}","-\frac{16 x}{75 a^4}-\frac{8 x^3}{225 a^2}-\frac{2 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{25 a}-\frac{16 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{75 a^5}-\frac{8 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{75 a^3}+\frac{1}{5} x^5 \cos ^{-1}(a x)^2-\frac{2 x^5}{125}",1,"(-16*x)/(75*a^4) - (8*x^3)/(225*a^2) - (2*x^5)/125 - (2*Sqrt[1 - a^2*x^2]*(8 + 4*a^2*x^2 + 3*a^4*x^4)*ArcCos[a*x])/(75*a^5) + (x^5*ArcCos[a*x]^2)/5","A",1
13,1,74,98,0.0460325,"\int x^3 \cos ^{-1}(a x)^2 \, dx","Integrate[x^3*ArcCos[a*x]^2,x]","\frac{\left(8 a^4 x^4-3\right) \cos ^{-1}(a x)^2-a^2 x^2 \left(a^2 x^2+3\right)-2 a x \sqrt{1-a^2 x^2} \left(2 a^2 x^2+3\right) \cos ^{-1}(a x)}{32 a^4}","-\frac{3 \cos ^{-1}(a x)^2}{32 a^4}-\frac{3 x^2}{32 a^2}-\frac{x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{8 a}-\frac{3 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{16 a^3}+\frac{1}{4} x^4 \cos ^{-1}(a x)^2-\frac{x^4}{32}",1,"(-(a^2*x^2*(3 + a^2*x^2)) - 2*a*x*Sqrt[1 - a^2*x^2]*(3 + 2*a^2*x^2)*ArcCos[a*x] + (-3 + 8*a^4*x^4)*ArcCos[a*x]^2)/(32*a^4)","A",1
14,1,63,82,0.0558906,"\int x^2 \cos ^{-1}(a x)^2 \, dx","Integrate[x^2*ArcCos[a*x]^2,x]","-\frac{4 x}{9 a^2}-\frac{2 \sqrt{1-a^2 x^2} \left(a^2 x^2+2\right) \cos ^{-1}(a x)}{9 a^3}+\frac{1}{3} x^3 \cos ^{-1}(a x)^2-\frac{2 x^3}{27}","-\frac{2 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{9 a}-\frac{4 x}{9 a^2}-\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{9 a^3}+\frac{1}{3} x^3 \cos ^{-1}(a x)^2-\frac{2 x^3}{27}",1,"(-4*x)/(9*a^2) - (2*x^3)/27 - (2*Sqrt[1 - a^2*x^2]*(2 + a^2*x^2)*ArcCos[a*x])/(9*a^3) + (x^3*ArcCos[a*x]^2)/3","A",1
15,1,57,60,0.02777,"\int x \cos ^{-1}(a x)^2 \, dx","Integrate[x*ArcCos[a*x]^2,x]","-\frac{x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{2 a}+\frac{\left(2 a^2 x^2-1\right) \cos ^{-1}(a x)^2}{4 a^2}-\frac{x^2}{4}","-\frac{x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{2 a}-\frac{\cos ^{-1}(a x)^2}{4 a^2}+\frac{1}{2} x^2 \cos ^{-1}(a x)^2-\frac{x^2}{4}",1,"-1/4*x^2 - (x*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(2*a) + ((-1 + 2*a^2*x^2)*ArcCos[a*x]^2)/(4*a^2)","A",1
16,1,35,35,0.0170346,"\int \cos ^{-1}(a x)^2 \, dx","Integrate[ArcCos[a*x]^2,x]","-\frac{2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{a}+x \cos ^{-1}(a x)^2-2 x","-\frac{2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{a}+x \cos ^{-1}(a x)^2-2 x",1,"-2*x - (2*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/a + x*ArcCos[a*x]^2","A",1
17,1,73,73,0.0203883,"\int \frac{\cos ^{-1}(a x)^2}{x} \, dx","Integrate[ArcCos[a*x]^2/x,x]","-i \cos ^{-1}(a x) \text{Li}_2\left(-e^{2 i \cos ^{-1}(a x)}\right)+\frac{1}{2} \text{Li}_3\left(-e^{2 i \cos ^{-1}(a x)}\right)-\frac{1}{3} i \cos ^{-1}(a x)^3+\cos ^{-1}(a x)^2 \log \left(1+e^{2 i \cos ^{-1}(a x)}\right)","-i \cos ^{-1}(a x) \text{Li}_2\left(-e^{2 i \cos ^{-1}(a x)}\right)+\frac{1}{2} \text{Li}_3\left(-e^{2 i \cos ^{-1}(a x)}\right)-\frac{1}{3} i \cos ^{-1}(a x)^3+\cos ^{-1}(a x)^2 \log \left(1+e^{2 i \cos ^{-1}(a x)}\right)",1,"(-1/3*I)*ArcCos[a*x]^3 + ArcCos[a*x]^2*Log[1 + E^((2*I)*ArcCos[a*x])] - I*ArcCos[a*x]*PolyLog[2, -E^((2*I)*ArcCos[a*x])] + PolyLog[3, -E^((2*I)*ArcCos[a*x])]/2","A",1
18,1,98,74,0.1608572,"\int \frac{\cos ^{-1}(a x)^2}{x^2} \, dx","Integrate[ArcCos[a*x]^2/x^2,x]","2 i a \text{Li}_2\left(-i e^{i \cos ^{-1}(a x)}\right)-2 i a \text{Li}_2\left(i e^{i \cos ^{-1}(a x)}\right)-\frac{\cos ^{-1}(a x) \left(\cos ^{-1}(a x)+2 a x \left(\log \left(1+i e^{i \cos ^{-1}(a x)}\right)-\log \left(1-i e^{i \cos ^{-1}(a x)}\right)\right)\right)}{x}","2 i a \text{Li}_2\left(-i e^{i \cos ^{-1}(a x)}\right)-2 i a \text{Li}_2\left(i e^{i \cos ^{-1}(a x)}\right)-\frac{\cos ^{-1}(a x)^2}{x}-4 i a \cos ^{-1}(a x) \tan ^{-1}\left(e^{i \cos ^{-1}(a x)}\right)",1,"-((ArcCos[a*x]*(ArcCos[a*x] + 2*a*x*(-Log[1 - I*E^(I*ArcCos[a*x])] + Log[1 + I*E^(I*ArcCos[a*x])])))/x) + (2*I)*a*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] - (2*I)*a*PolyLog[2, I*E^(I*ArcCos[a*x])]","A",0
19,1,43,43,0.0344833,"\int \frac{\cos ^{-1}(a x)^2}{x^3} \, dx","Integrate[ArcCos[a*x]^2/x^3,x]","\frac{a \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{x}+a^2 \log (x)-\frac{\cos ^{-1}(a x)^2}{2 x^2}","\frac{a \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{x}+a^2 \log (x)-\frac{\cos ^{-1}(a x)^2}{2 x^2}",1,"(a*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/x - ArcCos[a*x]^2/(2*x^2) + a^2*Log[x]","A",1
20,1,152,124,0.6672043,"\int \frac{\cos ^{-1}(a x)^2}{x^4} \, dx","Integrate[ArcCos[a*x]^2/x^4,x]","-\frac{-i a^3 x^3 \text{Li}_2\left(-i e^{i \cos ^{-1}(a x)}\right)+i a^3 x^3 \text{Li}_2\left(i e^{i \cos ^{-1}(a x)}\right)-a^3 x^3 \cos ^{-1}(a x) \log \left(1-i e^{i \cos ^{-1}(a x)}\right)+a^3 x^3 \cos ^{-1}(a x) \log \left(1+i e^{i \cos ^{-1}(a x)}\right)+a^2 x^2-a x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)+\cos ^{-1}(a x)^2}{3 x^3}","\frac{1}{3} i a^3 \text{Li}_2\left(-i e^{i \cos ^{-1}(a x)}\right)-\frac{1}{3} i a^3 \text{Li}_2\left(i e^{i \cos ^{-1}(a x)}\right)-\frac{2}{3} i a^3 \cos ^{-1}(a x) \tan ^{-1}\left(e^{i \cos ^{-1}(a x)}\right)+\frac{a \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{3 x^2}-\frac{a^2}{3 x}-\frac{\cos ^{-1}(a x)^2}{3 x^3}",1,"-1/3*(a^2*x^2 - a*x*Sqrt[1 - a^2*x^2]*ArcCos[a*x] + ArcCos[a*x]^2 - a^3*x^3*ArcCos[a*x]*Log[1 - I*E^(I*ArcCos[a*x])] + a^3*x^3*ArcCos[a*x]*Log[1 + I*E^(I*ArcCos[a*x])] - I*a^3*x^3*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] + I*a^3*x^3*PolyLog[2, I*E^(I*ArcCos[a*x])])/x^3","A",0
21,1,69,87,0.0463258,"\int \frac{\cos ^{-1}(a x)^2}{x^5} \, dx","Integrate[ArcCos[a*x]^2/x^5,x]","\frac{1}{3} a^4 \log (x)-\frac{a^2}{12 x^2}+\frac{a \sqrt{1-a^2 x^2} \left(2 a^2 x^2+1\right) \cos ^{-1}(a x)}{6 x^3}-\frac{\cos ^{-1}(a x)^2}{4 x^4}","\frac{1}{3} a^4 \log (x)-\frac{a^2}{12 x^2}+\frac{a \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{6 x^3}+\frac{a^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{3 x}-\frac{\cos ^{-1}(a x)^2}{4 x^4}",1,"-1/12*a^2/x^2 + (a*Sqrt[1 - a^2*x^2]*(1 + 2*a^2*x^2)*ArcCos[a*x])/(6*x^3) - ArcCos[a*x]^2/(4*x^4) + (a^4*Log[x])/3","A",1
22,1,122,201,0.0739825,"\int x^4 \cos ^{-1}(a x)^3 \, dx","Integrate[x^4*ArcCos[a*x]^3,x]","\frac{1125 a^5 x^5 \cos ^{-1}(a x)^3+2 \sqrt{1-a^2 x^2} \left(27 a^4 x^4+136 a^2 x^2+2072\right)-30 a x \left(9 a^4 x^4+20 a^2 x^2+120\right) \cos ^{-1}(a x)-225 \sqrt{1-a^2 x^2} \left(3 a^4 x^4+4 a^2 x^2+8\right) \cos ^{-1}(a x)^2}{5625 a^5}","-\frac{16 x \cos ^{-1}(a x)}{25 a^4}-\frac{8 x^3 \cos ^{-1}(a x)}{75 a^2}-\frac{3 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a}+\frac{6 \left(1-a^2 x^2\right)^{5/2}}{625 a^5}-\frac{76 \left(1-a^2 x^2\right)^{3/2}}{1125 a^5}+\frac{298 \sqrt{1-a^2 x^2}}{375 a^5}-\frac{8 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a^5}-\frac{4 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a^3}+\frac{1}{5} x^5 \cos ^{-1}(a x)^3-\frac{6}{125} x^5 \cos ^{-1}(a x)",1,"(2*Sqrt[1 - a^2*x^2]*(2072 + 136*a^2*x^2 + 27*a^4*x^4) - 30*a*x*(120 + 20*a^2*x^2 + 9*a^4*x^4)*ArcCos[a*x] - 225*Sqrt[1 - a^2*x^2]*(8 + 4*a^2*x^2 + 3*a^4*x^4)*ArcCos[a*x]^2 + 1125*a^5*x^5*ArcCos[a*x]^3)/(5625*a^5)","A",1
23,1,115,167,0.0730247,"\int x^3 \cos ^{-1}(a x)^3 \, dx","Integrate[x^3*ArcCos[a*x]^3,x]","\frac{8 \left(8 a^4 x^4-3\right) \cos ^{-1}(a x)^3+3 a x \sqrt{1-a^2 x^2} \left(2 a^2 x^2+15\right)-24 a x \sqrt{1-a^2 x^2} \left(2 a^2 x^2+3\right) \cos ^{-1}(a x)^2-24 a^2 x^2 \left(a^2 x^2+3\right) \cos ^{-1}(a x)-45 \sin ^{-1}(a x)}{256 a^4}","-\frac{45 \sin ^{-1}(a x)}{256 a^4}-\frac{3 \cos ^{-1}(a x)^3}{32 a^4}-\frac{9 x^2 \cos ^{-1}(a x)}{32 a^2}+\frac{3 x^3 \sqrt{1-a^2 x^2}}{128 a}-\frac{3 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{16 a}+\frac{45 x \sqrt{1-a^2 x^2}}{256 a^3}-\frac{9 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{32 a^3}+\frac{1}{4} x^4 \cos ^{-1}(a x)^3-\frac{3}{32} x^4 \cos ^{-1}(a x)",1,"(3*a*x*Sqrt[1 - a^2*x^2]*(15 + 2*a^2*x^2) - 24*a^2*x^2*(3 + a^2*x^2)*ArcCos[a*x] - 24*a*x*Sqrt[1 - a^2*x^2]*(3 + 2*a^2*x^2)*ArcCos[a*x]^2 + 8*(-3 + 8*a^4*x^4)*ArcCos[a*x]^3 - 45*ArcSin[a*x])/(256*a^4)","A",1
24,1,95,136,0.0539857,"\int x^2 \cos ^{-1}(a x)^3 \, dx","Integrate[x^2*ArcCos[a*x]^3,x]","\frac{9 a^3 x^3 \cos ^{-1}(a x)^3+2 \sqrt{1-a^2 x^2} \left(a^2 x^2+20\right)-9 \sqrt{1-a^2 x^2} \left(a^2 x^2+2\right) \cos ^{-1}(a x)^2-6 a x \left(a^2 x^2+6\right) \cos ^{-1}(a x)}{27 a^3}","-\frac{x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{3 a}-\frac{4 x \cos ^{-1}(a x)}{3 a^2}-\frac{2 \left(1-a^2 x^2\right)^{3/2}}{27 a^3}+\frac{14 \sqrt{1-a^2 x^2}}{9 a^3}-\frac{2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{3 a^3}+\frac{1}{3} x^3 \cos ^{-1}(a x)^3-\frac{2}{9} x^3 \cos ^{-1}(a x)",1,"(2*Sqrt[1 - a^2*x^2]*(20 + a^2*x^2) - 6*a*x*(6 + a^2*x^2)*ArcCos[a*x] - 9*Sqrt[1 - a^2*x^2]*(2 + a^2*x^2)*ArcCos[a*x]^2 + 9*a^3*x^3*ArcCos[a*x]^3)/(27*a^3)","A",1
25,1,85,99,0.0439378,"\int x \cos ^{-1}(a x)^3 \, dx","Integrate[x*ArcCos[a*x]^3,x]","\frac{3 a x \sqrt{1-a^2 x^2}+\left(4 a^2 x^2-2\right) \cos ^{-1}(a x)^3-6 a x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2-6 a^2 x^2 \cos ^{-1}(a x)-3 \sin ^{-1}(a x)}{8 a^2}","\frac{3 x \sqrt{1-a^2 x^2}}{8 a}-\frac{3 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{4 a}-\frac{3 \sin ^{-1}(a x)}{8 a^2}-\frac{\cos ^{-1}(a x)^3}{4 a^2}+\frac{1}{2} x^2 \cos ^{-1}(a x)^3-\frac{3}{4} x^2 \cos ^{-1}(a x)",1,"(3*a*x*Sqrt[1 - a^2*x^2] - 6*a^2*x^2*ArcCos[a*x] - 6*a*x*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2 + (-2 + 4*a^2*x^2)*ArcCos[a*x]^3 - 3*ArcSin[a*x])/(8*a^2)","A",1
26,1,60,60,0.0188016,"\int \cos ^{-1}(a x)^3 \, dx","Integrate[ArcCos[a*x]^3,x]","\frac{6 \sqrt{1-a^2 x^2}}{a}-\frac{3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{a}+x \cos ^{-1}(a x)^3-6 x \cos ^{-1}(a x)","\frac{6 \sqrt{1-a^2 x^2}}{a}-\frac{3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{a}+x \cos ^{-1}(a x)^3-6 x \cos ^{-1}(a x)",1,"(6*Sqrt[1 - a^2*x^2])/a - 6*x*ArcCos[a*x] - (3*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/a + x*ArcCos[a*x]^3","A",1
27,1,101,101,0.022775,"\int \frac{\cos ^{-1}(a x)^3}{x} \, dx","Integrate[ArcCos[a*x]^3/x,x]","-\frac{3}{2} i \cos ^{-1}(a x)^2 \text{Li}_2\left(-e^{2 i \cos ^{-1}(a x)}\right)+\frac{3}{2} \cos ^{-1}(a x) \text{Li}_3\left(-e^{2 i \cos ^{-1}(a x)}\right)+\frac{3}{4} i \text{Li}_4\left(-e^{2 i \cos ^{-1}(a x)}\right)-\frac{1}{4} i \cos ^{-1}(a x)^4+\cos ^{-1}(a x)^3 \log \left(1+e^{2 i \cos ^{-1}(a x)}\right)","-\frac{3}{2} i \cos ^{-1}(a x)^2 \text{Li}_2\left(-e^{2 i \cos ^{-1}(a x)}\right)+\frac{3}{2} \cos ^{-1}(a x) \text{Li}_3\left(-e^{2 i \cos ^{-1}(a x)}\right)+\frac{3}{4} i \text{Li}_4\left(-e^{2 i \cos ^{-1}(a x)}\right)-\frac{1}{4} i \cos ^{-1}(a x)^4+\cos ^{-1}(a x)^3 \log \left(1+e^{2 i \cos ^{-1}(a x)}\right)",1,"(-1/4*I)*ArcCos[a*x]^4 + ArcCos[a*x]^3*Log[1 + E^((2*I)*ArcCos[a*x])] - ((3*I)/2)*ArcCos[a*x]^2*PolyLog[2, -E^((2*I)*ArcCos[a*x])] + (3*ArcCos[a*x]*PolyLog[3, -E^((2*I)*ArcCos[a*x])])/2 + ((3*I)/4)*PolyLog[4, -E^((2*I)*ArcCos[a*x])]","A",1
28,1,139,122,0.1183461,"\int \frac{\cos ^{-1}(a x)^3}{x^2} \, dx","Integrate[ArcCos[a*x]^3/x^2,x]","-\frac{\cos ^{-1}(a x)^3}{x}+3 a \left(2 i \cos ^{-1}(a x) \left(\text{Li}_2\left(-i e^{i \cos ^{-1}(a x)}\right)-\text{Li}_2\left(i e^{i \cos ^{-1}(a x)}\right)\right)-2 \text{Li}_3\left(-i e^{i \cos ^{-1}(a x)}\right)+2 \text{Li}_3\left(i e^{i \cos ^{-1}(a x)}\right)+\cos ^{-1}(a x)^2 \left(\log \left(1-i e^{i \cos ^{-1}(a x)}\right)-\log \left(1+i e^{i \cos ^{-1}(a x)}\right)\right)\right)","6 i a \cos ^{-1}(a x) \text{Li}_2\left(-i e^{i \cos ^{-1}(a x)}\right)-6 i a \cos ^{-1}(a x) \text{Li}_2\left(i e^{i \cos ^{-1}(a x)}\right)-6 a \text{Li}_3\left(-i e^{i \cos ^{-1}(a x)}\right)+6 a \text{Li}_3\left(i e^{i \cos ^{-1}(a x)}\right)-\frac{\cos ^{-1}(a x)^3}{x}-6 i a \cos ^{-1}(a x)^2 \tan ^{-1}\left(e^{i \cos ^{-1}(a x)}\right)",1,"-(ArcCos[a*x]^3/x) + 3*a*(ArcCos[a*x]^2*(Log[1 - I*E^(I*ArcCos[a*x])] - Log[1 + I*E^(I*ArcCos[a*x])]) + (2*I)*ArcCos[a*x]*(PolyLog[2, (-I)*E^(I*ArcCos[a*x])] - PolyLog[2, I*E^(I*ArcCos[a*x])]) - 2*PolyLog[3, (-I)*E^(I*ArcCos[a*x])] + 2*PolyLog[3, I*E^(I*ArcCos[a*x])])","A",0
29,1,92,102,0.2124841,"\int \frac{\cos ^{-1}(a x)^3}{x^3} \, dx","Integrate[ArcCos[a*x]^3/x^3,x]","\frac{1}{2} \left(-3 i a^2 \text{Li}_2\left(-e^{2 i \cos ^{-1}(a x)}\right)+\frac{3 a \left(\sqrt{1-a^2 x^2}-i a x\right) \cos ^{-1}(a x)^2}{x}+6 a^2 \cos ^{-1}(a x) \log \left(1+e^{2 i \cos ^{-1}(a x)}\right)-\frac{\cos ^{-1}(a x)^3}{x^2}\right)","-\frac{3}{2} i a^2 \text{Li}_2\left(-e^{2 i \cos ^{-1}(a x)}\right)+\frac{3 a \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x}-\frac{3}{2} i a^2 \cos ^{-1}(a x)^2+3 a^2 \cos ^{-1}(a x) \log \left(1+e^{2 i \cos ^{-1}(a x)}\right)-\frac{\cos ^{-1}(a x)^3}{2 x^2}",1,"((3*a*((-I)*a*x + Sqrt[1 - a^2*x^2])*ArcCos[a*x]^2)/x - ArcCos[a*x]^3/x^2 + 6*a^2*ArcCos[a*x]*Log[1 + E^((2*I)*ArcCos[a*x])] - (3*I)*a^2*PolyLog[2, -E^((2*I)*ArcCos[a*x])])/2","A",1
30,1,165,192,0.8319935,"\int \frac{\cos ^{-1}(a x)^3}{x^4} \, dx","Integrate[ArcCos[a*x]^3/x^4,x]","-\frac{\cos ^{-1}(a x) \left(12 a^2 x^2+4 \cos ^{-1}(a x)^2-3 \cos ^{-1}(a x) \sin \left(2 \cos ^{-1}(a x)\right)\right)}{12 x^3}+a^3 \left(\tanh ^{-1}\left(\sqrt{1-a^2 x^2}\right)+i \cos ^{-1}(a x) \text{Li}_2\left(-i e^{i \cos ^{-1}(a x)}\right)-i \cos ^{-1}(a x) \text{Li}_2\left(i e^{i \cos ^{-1}(a x)}\right)-\text{Li}_3\left(-i e^{i \cos ^{-1}(a x)}\right)+\text{Li}_3\left(i e^{i \cos ^{-1}(a x)}\right)-i \cos ^{-1}(a x)^2 \tan ^{-1}\left(e^{i \cos ^{-1}(a x)}\right)\right)","i a^3 \cos ^{-1}(a x) \text{Li}_2\left(-i e^{i \cos ^{-1}(a x)}\right)-i a^3 \cos ^{-1}(a x) \text{Li}_2\left(i e^{i \cos ^{-1}(a x)}\right)-a^3 \text{Li}_3\left(-i e^{i \cos ^{-1}(a x)}\right)+a^3 \text{Li}_3\left(i e^{i \cos ^{-1}(a x)}\right)-i a^3 \cos ^{-1}(a x)^2 \tan ^{-1}\left(e^{i \cos ^{-1}(a x)}\right)+\frac{a \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x^2}-\frac{a^2 \cos ^{-1}(a x)}{x}+a^3 \tanh ^{-1}\left(\sqrt{1-a^2 x^2}\right)-\frac{\cos ^{-1}(a x)^3}{3 x^3}",1,"a^3*((-I)*ArcCos[a*x]^2*ArcTan[E^(I*ArcCos[a*x])] + ArcTanh[Sqrt[1 - a^2*x^2]] + I*ArcCos[a*x]*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] - I*ArcCos[a*x]*PolyLog[2, I*E^(I*ArcCos[a*x])] - PolyLog[3, (-I)*E^(I*ArcCos[a*x])] + PolyLog[3, I*E^(I*ArcCos[a*x])]) - (ArcCos[a*x]*(12*a^2*x^2 + 4*ArcCos[a*x]^2 - 3*ArcCos[a*x]*Sin[2*ArcCos[a*x]]))/(12*x^3)","A",1
31,1,149,169,0.5276347,"\int \frac{\cos ^{-1}(a x)^3}{x^5} \, dx","Integrate[ArcCos[a*x]^3/x^5,x]","-\frac{\cos ^{-1}(a x)^3}{4 x^4}+\frac{1}{4} a^4 \sqrt{1-a^2 x^2} \left(-\frac{2 i \text{Li}_2\left(-e^{2 i \cos ^{-1}(a x)}\right)}{\sqrt{1-a^2 x^2}}+\frac{\frac{\cos ^{-1}(a x)^2}{a^2 x^2}+2 \cos ^{-1}(a x)^2+1}{a x}+\frac{\cos ^{-1}(a x) \left(-\frac{1}{a^2 x^2}-2 i \cos ^{-1}(a x)+4 \log \left(1+e^{2 i \cos ^{-1}(a x)}\right)\right)}{\sqrt{1-a^2 x^2}}\right)","-\frac{1}{2} i a^4 \text{Li}_2\left(-e^{2 i \cos ^{-1}(a x)}\right)-\frac{1}{2} i a^4 \cos ^{-1}(a x)^2+a^4 \cos ^{-1}(a x) \log \left(1+e^{2 i \cos ^{-1}(a x)}\right)-\frac{a^2 \cos ^{-1}(a x)}{4 x^2}+\frac{a \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{4 x^3}+\frac{a^3 \sqrt{1-a^2 x^2}}{4 x}+\frac{a^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x}-\frac{\cos ^{-1}(a x)^3}{4 x^4}",1,"-1/4*ArcCos[a*x]^3/x^4 + (a^4*Sqrt[1 - a^2*x^2]*((1 + 2*ArcCos[a*x]^2 + ArcCos[a*x]^2/(a^2*x^2))/(a*x) + (ArcCos[a*x]*(-(1/(a^2*x^2)) - (2*I)*ArcCos[a*x] + 4*Log[1 + E^((2*I)*ArcCos[a*x])]))/Sqrt[1 - a^2*x^2] - ((2*I)*PolyLog[2, -E^((2*I)*ArcCos[a*x])])/Sqrt[1 - a^2*x^2]))/4","A",1
32,1,167,282,0.0938669,"\int x^5 \cos ^{-1}(a x)^4 \, dx","Integrate[x^5*ArcCos[a*x]^4,x]","\frac{108 \left(16 a^6 x^6-5\right) \cos ^{-1}(a x)^4+a^2 x^2 \left(32 a^4 x^4+195 a^2 x^2+2205\right)-144 a x \sqrt{1-a^2 x^2} \left(8 a^4 x^4+10 a^2 x^2+15\right) \cos ^{-1}(a x)^3+6 a x \sqrt{1-a^2 x^2} \left(32 a^4 x^4+130 a^2 x^2+735\right) \cos ^{-1}(a x)-9 \left(64 a^6 x^6+120 a^4 x^4+360 a^2 x^2-245\right) \cos ^{-1}(a x)^2}{10368 a^6}","-\frac{5 \cos ^{-1}(a x)^4}{96 a^6}+\frac{245 \cos ^{-1}(a x)^2}{1152 a^6}+\frac{245 x^2}{1152 a^4}-\frac{5 x^2 \cos ^{-1}(a x)^2}{16 a^4}+\frac{65 x^4}{3456 a^2}-\frac{5 x^4 \cos ^{-1}(a x)^2}{48 a^2}-\frac{x^5 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}+\frac{x^5 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{54 a}-\frac{5 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{24 a^5}+\frac{245 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{576 a^5}-\frac{5 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{36 a^3}+\frac{65 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{864 a^3}+\frac{1}{6} x^6 \cos ^{-1}(a x)^4-\frac{1}{18} x^6 \cos ^{-1}(a x)^2+\frac{x^6}{324}",1,"(a^2*x^2*(2205 + 195*a^2*x^2 + 32*a^4*x^4) + 6*a*x*Sqrt[1 - a^2*x^2]*(735 + 130*a^2*x^2 + 32*a^4*x^4)*ArcCos[a*x] - 9*(-245 + 360*a^2*x^2 + 120*a^4*x^4 + 64*a^6*x^6)*ArcCos[a*x]^2 - 144*a*x*Sqrt[1 - a^2*x^2]*(15 + 10*a^2*x^2 + 8*a^4*x^4)*ArcCos[a*x]^3 + 108*(-5 + 16*a^6*x^6)*ArcCos[a*x]^4)/(10368*a^6)","A",1
33,1,150,250,0.0868032,"\int x^4 \cos ^{-1}(a x)^4 \, dx","Integrate[x^4*ArcCos[a*x]^4,x]","\frac{16875 a^5 x^5 \cos ^{-1}(a x)^4+8 a x \left(81 a^4 x^4+680 a^2 x^2+31080\right)-900 a x \left(9 a^4 x^4+20 a^2 x^2+120\right) \cos ^{-1}(a x)^2-4500 \sqrt{1-a^2 x^2} \left(3 a^4 x^4+4 a^2 x^2+8\right) \cos ^{-1}(a x)^3+120 \sqrt{1-a^2 x^2} \left(27 a^4 x^4+136 a^2 x^2+2072\right) \cos ^{-1}(a x)}{84375 a^5}","\frac{16576 x}{5625 a^4}-\frac{32 x \cos ^{-1}(a x)^2}{25 a^4}+\frac{1088 x^3}{16875 a^2}-\frac{16 x^3 \cos ^{-1}(a x)^2}{75 a^2}-\frac{4 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{25 a}+\frac{24 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{625 a}-\frac{32 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{75 a^5}+\frac{16576 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{5625 a^5}-\frac{16 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{75 a^3}+\frac{1088 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{5625 a^3}+\frac{1}{5} x^5 \cos ^{-1}(a x)^4-\frac{12}{125} x^5 \cos ^{-1}(a x)^2+\frac{24 x^5}{3125}",1,"(8*a*x*(31080 + 680*a^2*x^2 + 81*a^4*x^4) + 120*Sqrt[1 - a^2*x^2]*(2072 + 136*a^2*x^2 + 27*a^4*x^4)*ArcCos[a*x] - 900*a*x*(120 + 20*a^2*x^2 + 9*a^4*x^4)*ArcCos[a*x]^2 - 4500*Sqrt[1 - a^2*x^2]*(8 + 4*a^2*x^2 + 3*a^4*x^4)*ArcCos[a*x]^3 + 16875*a^5*x^5*ArcCos[a*x]^4)/(84375*a^5)","A",1
34,1,135,198,0.0783453,"\int x^3 \cos ^{-1}(a x)^4 \, dx","Integrate[x^3*ArcCos[a*x]^4,x]","\frac{4 \left(8 a^4 x^4-3\right) \cos ^{-1}(a x)^4+3 a^2 x^2 \left(a^2 x^2+15\right)-16 a x \sqrt{1-a^2 x^2} \left(2 a^2 x^2+3\right) \cos ^{-1}(a x)^3+6 a x \sqrt{1-a^2 x^2} \left(2 a^2 x^2+15\right) \cos ^{-1}(a x)-3 \left(8 a^4 x^4+24 a^2 x^2-15\right) \cos ^{-1}(a x)^2}{128 a^4}","-\frac{3 \cos ^{-1}(a x)^4}{32 a^4}+\frac{45 \cos ^{-1}(a x)^2}{128 a^4}+\frac{45 x^2}{128 a^2}-\frac{9 x^2 \cos ^{-1}(a x)^2}{16 a^2}-\frac{x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{4 a}+\frac{3 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{32 a}-\frac{3 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{8 a^3}+\frac{45 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{64 a^3}+\frac{1}{4} x^4 \cos ^{-1}(a x)^4-\frac{3}{16} x^4 \cos ^{-1}(a x)^2+\frac{3 x^4}{128}",1,"(3*a^2*x^2*(15 + a^2*x^2) + 6*a*x*Sqrt[1 - a^2*x^2]*(15 + 2*a^2*x^2)*ArcCos[a*x] - 3*(-15 + 24*a^2*x^2 + 8*a^4*x^4)*ArcCos[a*x]^2 - 16*a*x*Sqrt[1 - a^2*x^2]*(3 + 2*a^2*x^2)*ArcCos[a*x]^3 + 4*(-3 + 8*a^4*x^4)*ArcCos[a*x]^4)/(128*a^4)","A",1
35,1,114,166,0.0832618,"\int x^2 \cos ^{-1}(a x)^4 \, dx","Integrate[x^2*ArcCos[a*x]^4,x]","\frac{27 a^3 x^3 \cos ^{-1}(a x)^4+8 a x \left(a^2 x^2+60\right)-36 \sqrt{1-a^2 x^2} \left(a^2 x^2+2\right) \cos ^{-1}(a x)^3-36 a x \left(a^2 x^2+6\right) \cos ^{-1}(a x)^2+24 \sqrt{1-a^2 x^2} \left(a^2 x^2+20\right) \cos ^{-1}(a x)}{81 a^3}","-\frac{4 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}+\frac{8 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a}+\frac{160 x}{27 a^2}-\frac{8 x \cos ^{-1}(a x)^2}{3 a^2}-\frac{8 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a^3}+\frac{160 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a^3}+\frac{1}{3} x^3 \cos ^{-1}(a x)^4-\frac{4}{9} x^3 \cos ^{-1}(a x)^2+\frac{8 x^3}{81}",1,"(8*a*x*(60 + a^2*x^2) + 24*Sqrt[1 - a^2*x^2]*(20 + a^2*x^2)*ArcCos[a*x] - 36*a*x*(6 + a^2*x^2)*ArcCos[a*x]^2 - 36*Sqrt[1 - a^2*x^2]*(2 + a^2*x^2)*ArcCos[a*x]^3 + 27*a^3*x^3*ArcCos[a*x]^4)/(81*a^3)","A",1
36,1,96,112,0.0473197,"\int x \cos ^{-1}(a x)^4 \, dx","Integrate[x*ArcCos[a*x]^4,x]","\frac{3 a^2 x^2+\left(2 a^2 x^2-1\right) \cos ^{-1}(a x)^4-4 a x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3+\left(3-6 a^2 x^2\right) \cos ^{-1}(a x)^2+6 a x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{4 a^2}","-\frac{x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{a}+\frac{3 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{2 a}-\frac{\cos ^{-1}(a x)^4}{4 a^2}+\frac{3 \cos ^{-1}(a x)^2}{4 a^2}+\frac{1}{2} x^2 \cos ^{-1}(a x)^4-\frac{3}{2} x^2 \cos ^{-1}(a x)^2+\frac{3 x^2}{4}",1,"(3*a^2*x^2 + 6*a*x*Sqrt[1 - a^2*x^2]*ArcCos[a*x] + (3 - 6*a^2*x^2)*ArcCos[a*x]^2 - 4*a*x*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3 + (-1 + 2*a^2*x^2)*ArcCos[a*x]^4)/(4*a^2)","A",1
37,1,69,69,0.0252253,"\int \cos ^{-1}(a x)^4 \, dx","Integrate[ArcCos[a*x]^4,x]","-\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{a}+\frac{24 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{a}+x \cos ^{-1}(a x)^4-12 x \cos ^{-1}(a x)^2+24 x","-\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{a}+\frac{24 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{a}+x \cos ^{-1}(a x)^4-12 x \cos ^{-1}(a x)^2+24 x",1,"24*x + (24*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/a - 12*x*ArcCos[a*x]^2 - (4*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/a + x*ArcCos[a*x]^4","A",1
38,1,119,119,0.0225391,"\int \frac{\cos ^{-1}(a x)^4}{x} \, dx","Integrate[ArcCos[a*x]^4/x,x]","-2 i \cos ^{-1}(a x)^3 \text{Li}_2\left(-e^{2 i \cos ^{-1}(a x)}\right)+3 \cos ^{-1}(a x)^2 \text{Li}_3\left(-e^{2 i \cos ^{-1}(a x)}\right)+3 i \cos ^{-1}(a x) \text{Li}_4\left(-e^{2 i \cos ^{-1}(a x)}\right)-\frac{3}{2} \text{Li}_5\left(-e^{2 i \cos ^{-1}(a x)}\right)-\frac{1}{5} i \cos ^{-1}(a x)^5+\cos ^{-1}(a x)^4 \log \left(1+e^{2 i \cos ^{-1}(a x)}\right)","-2 i \cos ^{-1}(a x)^3 \text{Li}_2\left(-e^{2 i \cos ^{-1}(a x)}\right)+3 \cos ^{-1}(a x)^2 \text{Li}_3\left(-e^{2 i \cos ^{-1}(a x)}\right)+3 i \cos ^{-1}(a x) \text{Li}_4\left(-e^{2 i \cos ^{-1}(a x)}\right)-\frac{3}{2} \text{Li}_5\left(-e^{2 i \cos ^{-1}(a x)}\right)-\frac{1}{5} i \cos ^{-1}(a x)^5+\cos ^{-1}(a x)^4 \log \left(1+e^{2 i \cos ^{-1}(a x)}\right)",1,"(-1/5*I)*ArcCos[a*x]^5 + ArcCos[a*x]^4*Log[1 + E^((2*I)*ArcCos[a*x])] - (2*I)*ArcCos[a*x]^3*PolyLog[2, -E^((2*I)*ArcCos[a*x])] + 3*ArcCos[a*x]^2*PolyLog[3, -E^((2*I)*ArcCos[a*x])] + (3*I)*ArcCos[a*x]*PolyLog[4, -E^((2*I)*ArcCos[a*x])] - (3*PolyLog[5, -E^((2*I)*ArcCos[a*x])])/2","A",1
39,1,549,176,1.2008585,"\int \frac{\cos ^{-1}(a x)^4}{x^2} \, dx","Integrate[ArcCos[a*x]^4/x^2,x]","a \left(12 i \cos ^{-1}(a x)^2 \text{Li}_2\left(-i e^{-i \cos ^{-1}(a x)}\right)+12 i \cos ^{-1}(a x)^2 \text{Li}_2\left(-i e^{i \cos ^{-1}(a x)}\right)-12 i \pi  \cos ^{-1}(a x) \text{Li}_2\left(-i e^{i \cos ^{-1}(a x)}\right)+24 \cos ^{-1}(a x) \text{Li}_3\left(-i e^{-i \cos ^{-1}(a x)}\right)-24 \cos ^{-1}(a x) \text{Li}_3\left(-i e^{i \cos ^{-1}(a x)}\right)+3 i \pi  \left(\pi -4 \cos ^{-1}(a x)\right) \text{Li}_2\left(i e^{-i \cos ^{-1}(a x)}\right)+3 i \pi ^2 \text{Li}_2\left(-i e^{i \cos ^{-1}(a x)}\right)-12 \pi  \text{Li}_3\left(i e^{-i \cos ^{-1}(a x)}\right)+12 \pi  \text{Li}_3\left(-i e^{i \cos ^{-1}(a x)}\right)-24 i \text{Li}_4\left(-i e^{-i \cos ^{-1}(a x)}\right)-24 i \text{Li}_4\left(-i e^{i \cos ^{-1}(a x)}\right)-\frac{\cos ^{-1}(a x)^4}{a x}+i \cos ^{-1}(a x)^4-2 i \pi  \cos ^{-1}(a x)^3+\frac{3}{2} i \pi ^2 \cos ^{-1}(a x)^2-\frac{1}{2} i \pi ^3 \cos ^{-1}(a x)+4 \cos ^{-1}(a x)^3 \log \left(1+i e^{-i \cos ^{-1}(a x)}\right)-4 \cos ^{-1}(a x)^3 \log \left(1+i e^{i \cos ^{-1}(a x)}\right)-6 \pi  \cos ^{-1}(a x)^2 \log \left(1-i e^{-i \cos ^{-1}(a x)}\right)+6 \pi  \cos ^{-1}(a x)^2 \log \left(1+i e^{i \cos ^{-1}(a x)}\right)+3 \pi ^2 \cos ^{-1}(a x) \log \left(1-i e^{-i \cos ^{-1}(a x)}\right)-3 \pi ^2 \cos ^{-1}(a x) \log \left(1+i e^{i \cos ^{-1}(a x)}\right)-\frac{1}{2} \pi ^3 \log \left(1+i e^{-i \cos ^{-1}(a x)}\right)+\frac{1}{2} \pi ^3 \log \left(1+i e^{i \cos ^{-1}(a x)}\right)+\frac{1}{2} \pi ^3 \log \left(\tan \left(\frac{1}{4} \left(2 \cos ^{-1}(a x)+\pi \right)\right)\right)-\frac{7 i \pi ^4}{16}\right)","12 i a \cos ^{-1}(a x)^2 \text{Li}_2\left(-i e^{i \cos ^{-1}(a x)}\right)-12 i a \cos ^{-1}(a x)^2 \text{Li}_2\left(i e^{i \cos ^{-1}(a x)}\right)-24 a \cos ^{-1}(a x) \text{Li}_3\left(-i e^{i \cos ^{-1}(a x)}\right)+24 a \cos ^{-1}(a x) \text{Li}_3\left(i e^{i \cos ^{-1}(a x)}\right)-24 i a \text{Li}_4\left(-i e^{i \cos ^{-1}(a x)}\right)+24 i a \text{Li}_4\left(i e^{i \cos ^{-1}(a x)}\right)-\frac{\cos ^{-1}(a x)^4}{x}-8 i a \cos ^{-1}(a x)^3 \tan ^{-1}\left(e^{i \cos ^{-1}(a x)}\right)",1,"a*(((-7*I)/16)*Pi^4 - (I/2)*Pi^3*ArcCos[a*x] + ((3*I)/2)*Pi^2*ArcCos[a*x]^2 - (2*I)*Pi*ArcCos[a*x]^3 + I*ArcCos[a*x]^4 - ArcCos[a*x]^4/(a*x) + 3*Pi^2*ArcCos[a*x]*Log[1 - I/E^(I*ArcCos[a*x])] - 6*Pi*ArcCos[a*x]^2*Log[1 - I/E^(I*ArcCos[a*x])] - (Pi^3*Log[1 + I/E^(I*ArcCos[a*x])])/2 + 4*ArcCos[a*x]^3*Log[1 + I/E^(I*ArcCos[a*x])] + (Pi^3*Log[1 + I*E^(I*ArcCos[a*x])])/2 - 3*Pi^2*ArcCos[a*x]*Log[1 + I*E^(I*ArcCos[a*x])] + 6*Pi*ArcCos[a*x]^2*Log[1 + I*E^(I*ArcCos[a*x])] - 4*ArcCos[a*x]^3*Log[1 + I*E^(I*ArcCos[a*x])] + (Pi^3*Log[Tan[(Pi + 2*ArcCos[a*x])/4]])/2 + (12*I)*ArcCos[a*x]^2*PolyLog[2, (-I)/E^(I*ArcCos[a*x])] + (3*I)*Pi*(Pi - 4*ArcCos[a*x])*PolyLog[2, I/E^(I*ArcCos[a*x])] + (3*I)*Pi^2*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] - (12*I)*Pi*ArcCos[a*x]*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] + (12*I)*ArcCos[a*x]^2*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] + 24*ArcCos[a*x]*PolyLog[3, (-I)/E^(I*ArcCos[a*x])] - 12*Pi*PolyLog[3, I/E^(I*ArcCos[a*x])] + 12*Pi*PolyLog[3, (-I)*E^(I*ArcCos[a*x])] - 24*ArcCos[a*x]*PolyLog[3, (-I)*E^(I*ArcCos[a*x])] - (24*I)*PolyLog[4, (-I)/E^(I*ArcCos[a*x])] - (24*I)*PolyLog[4, (-I)*E^(I*ArcCos[a*x])])","B",0
40,1,115,121,0.413049,"\int \frac{\cos ^{-1}(a x)^4}{x^3} \, dx","Integrate[ArcCos[a*x]^4/x^3,x]","-\frac{\cos ^{-1}(a x)^4}{2 x^2}-a^2 \left(-2 \cos ^{-1}(a x)^2 \left(\frac{\sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{a x}-i \cos ^{-1}(a x)+3 \log \left(1+e^{2 i \cos ^{-1}(a x)}\right)\right)+6 i \cos ^{-1}(a x) \text{Li}_2\left(-e^{2 i \cos ^{-1}(a x)}\right)-3 \text{Li}_3\left(-e^{2 i \cos ^{-1}(a x)}\right)\right)","-6 i a^2 \cos ^{-1}(a x) \text{Li}_2\left(-e^{2 i \cos ^{-1}(a x)}\right)+3 a^2 \text{Li}_3\left(-e^{2 i \cos ^{-1}(a x)}\right)+\frac{2 a \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{x}-2 i a^2 \cos ^{-1}(a x)^3+6 a^2 \cos ^{-1}(a x)^2 \log \left(1+e^{2 i \cos ^{-1}(a x)}\right)-\frac{\cos ^{-1}(a x)^4}{2 x^2}",1,"-1/2*ArcCos[a*x]^4/x^2 - a^2*(-2*ArcCos[a*x]^2*((-I)*ArcCos[a*x] + (Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(a*x) + 3*Log[1 + E^((2*I)*ArcCos[a*x])]) + (6*I)*ArcCos[a*x]*PolyLog[2, -E^((2*I)*ArcCos[a*x])] - 3*PolyLog[3, -E^((2*I)*ArcCos[a*x])])","A",1
41,1,1475,304,12.075091,"\int \frac{\cos ^{-1}(a x)^4}{x^4} \, dx","Integrate[ArcCos[a*x]^4/x^4,x]","a^3 \left(-\frac{\sin \left(\frac{1}{2} \cos ^{-1}(a x)\right) \cos ^{-1}(a x)^4}{6 \left(\cos \left(\frac{1}{2} \cos ^{-1}(a x)\right)-\sin \left(\frac{1}{2} \cos ^{-1}(a x)\right)\right)^3}+\frac{\sin \left(\frac{1}{2} \cos ^{-1}(a x)\right) \cos ^{-1}(a x)^4}{6 \left(\cos \left(\frac{1}{2} \cos ^{-1}(a x)\right)+\sin \left(\frac{1}{2} \cos ^{-1}(a x)\right)\right)^3}-\frac{1}{6} \left(\cos ^{-1}(a x)^2+12\right) \cos ^{-1}(a x)^2+4 \left(\cos ^{-1}(a x) \left(\log \left(1-i e^{i \cos ^{-1}(a x)}\right)-\log \left(1+i e^{i \cos ^{-1}(a x)}\right)\right)+i \left(\text{Li}_2\left(-i e^{i \cos ^{-1}(a x)}\right)-\text{Li}_2\left(i e^{i \cos ^{-1}(a x)}\right)\right)\right)+\frac{2}{3} \left(\frac{1}{8} \pi ^3 \log \left(\cot \left(\frac{1}{2} \left(\frac{\pi }{2}-\cos ^{-1}(a x)\right)\right)\right)+\frac{3}{4} \pi ^2 \left(\left(\frac{\pi }{2}-\cos ^{-1}(a x)\right) \left(\log \left(1-e^{i \left(\frac{\pi }{2}-\cos ^{-1}(a x)\right)}\right)-\log \left(1+e^{i \left(\frac{\pi }{2}-\cos ^{-1}(a x)\right)}\right)\right)+i \left(\text{Li}_2\left(-e^{i \left(\frac{\pi }{2}-\cos ^{-1}(a x)\right)}\right)-\text{Li}_2\left(e^{i \left(\frac{\pi }{2}-\cos ^{-1}(a x)\right)}\right)\right)\right)-\frac{3}{2} \pi  \left(\left(\log \left(1-e^{i \left(\frac{\pi }{2}-\cos ^{-1}(a x)\right)}\right)-\log \left(1+e^{i \left(\frac{\pi }{2}-\cos ^{-1}(a x)\right)}\right)\right) \left(\frac{\pi }{2}-\cos ^{-1}(a x)\right)^2+2 i \left(\text{Li}_2\left(-e^{i \left(\frac{\pi }{2}-\cos ^{-1}(a x)\right)}\right)-\text{Li}_2\left(e^{i \left(\frac{\pi }{2}-\cos ^{-1}(a x)\right)}\right)\right) \left(\frac{\pi }{2}-\cos ^{-1}(a x)\right)+2 \left(\text{Li}_3\left(e^{i \left(\frac{\pi }{2}-\cos ^{-1}(a x)\right)}\right)-\text{Li}_3\left(-e^{i \left(\frac{\pi }{2}-\cos ^{-1}(a x)\right)}\right)\right)\right)+8 \left(\frac{1}{64} i \left(\frac{\pi }{2}-\cos ^{-1}(a x)\right)^4-\frac{1}{8} \log \left(1+e^{i \left(\frac{\pi }{2}-\cos ^{-1}(a x)\right)}\right) \left(\frac{\pi }{2}-\cos ^{-1}(a x)\right)^3+\frac{3}{8} i \text{Li}_2\left(-e^{i \left(\frac{\pi }{2}-\cos ^{-1}(a x)\right)}\right) \left(\frac{\pi }{2}-\cos ^{-1}(a x)\right)^2-\frac{3}{4} \text{Li}_3\left(-e^{i \left(\frac{\pi }{2}-\cos ^{-1}(a x)\right)}\right) \left(\frac{\pi }{2}-\cos ^{-1}(a x)\right)+\frac{1}{4} i \left(\frac{1}{2} \left(\cos ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)^4-\frac{1}{8} \pi ^3 \left(i \left(\frac{1}{2} \left(\cos ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)-\log \left(1+e^{2 i \left(\frac{1}{2} \left(\cos ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right)\right)-\left(\frac{1}{2} \left(\cos ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)^3 \log \left(1+e^{2 i \left(\frac{1}{2} \left(\cos ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right)+\frac{3}{4} \pi ^2 \left(\frac{1}{2} i \left(\frac{1}{2} \left(\cos ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)^2-\log \left(1+e^{2 i \left(\frac{1}{2} \left(\cos ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right) \left(\frac{1}{2} \left(\cos ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)+\frac{1}{2} i \text{Li}_2\left(-e^{2 i \left(\frac{1}{2} \left(\cos ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right)\right)+\frac{3}{2} i \left(\frac{1}{2} \left(\cos ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)^2 \text{Li}_2\left(-e^{2 i \left(\frac{1}{2} \left(\cos ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right)-\frac{3}{2} \pi  \left(\frac{1}{3} i \left(\frac{1}{2} \left(\cos ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)^3-\log \left(1+e^{2 i \left(\frac{1}{2} \left(\cos ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right) \left(\frac{1}{2} \left(\cos ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)^2+i \text{Li}_2\left(-e^{2 i \left(\frac{1}{2} \left(\cos ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right) \left(\frac{1}{2} \left(\cos ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)-\frac{1}{2} \text{Li}_3\left(-e^{2 i \left(\frac{1}{2} \left(\cos ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right)\right)-\frac{3}{2} \left(\frac{1}{2} \left(\cos ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right) \text{Li}_3\left(-e^{2 i \left(\frac{1}{2} \left(\cos ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right)-\frac{3}{4} i \text{Li}_4\left(-e^{i \left(\frac{\pi }{2}-\cos ^{-1}(a x)\right)}\right)-\frac{3}{4} i \text{Li}_4\left(-e^{2 i \left(\frac{1}{2} \left(\cos ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right)\right)\right)-\frac{-\sin \left(\frac{1}{2} \cos ^{-1}(a x)\right) \cos ^{-1}(a x)^4-12 \sin \left(\frac{1}{2} \cos ^{-1}(a x)\right) \cos ^{-1}(a x)^2}{6 \left(\cos \left(\frac{1}{2} \cos ^{-1}(a x)\right)+\sin \left(\frac{1}{2} \cos ^{-1}(a x)\right)\right)}-\frac{\sin \left(\frac{1}{2} \cos ^{-1}(a x)\right) \cos ^{-1}(a x)^4+12 \sin \left(\frac{1}{2} \cos ^{-1}(a x)\right) \cos ^{-1}(a x)^2}{6 \left(\cos \left(\frac{1}{2} \cos ^{-1}(a x)\right)-\sin \left(\frac{1}{2} \cos ^{-1}(a x)\right)\right)}-\frac{\cos ^{-1}(a x)^4-4 \cos ^{-1}(a x)^3}{12 \left(\cos \left(\frac{1}{2} \cos ^{-1}(a x)\right)-\sin \left(\frac{1}{2} \cos ^{-1}(a x)\right)\right)^2}-\frac{\cos ^{-1}(a x)^4+4 \cos ^{-1}(a x)^3}{12 \left(\cos \left(\frac{1}{2} \cos ^{-1}(a x)\right)+\sin \left(\frac{1}{2} \cos ^{-1}(a x)\right)\right)^2}\right)","2 i a^3 \cos ^{-1}(a x)^2 \text{Li}_2\left(-i e^{i \cos ^{-1}(a x)}\right)-2 i a^3 \cos ^{-1}(a x)^2 \text{Li}_2\left(i e^{i \cos ^{-1}(a x)}\right)-4 a^3 \cos ^{-1}(a x) \text{Li}_3\left(-i e^{i \cos ^{-1}(a x)}\right)+4 a^3 \cos ^{-1}(a x) \text{Li}_3\left(i e^{i \cos ^{-1}(a x)}\right)+4 i a^3 \text{Li}_2\left(-i e^{i \cos ^{-1}(a x)}\right)-4 i a^3 \text{Li}_2\left(i e^{i \cos ^{-1}(a x)}\right)-4 i a^3 \text{Li}_4\left(-i e^{i \cos ^{-1}(a x)}\right)+4 i a^3 \text{Li}_4\left(i e^{i \cos ^{-1}(a x)}\right)-\frac{4}{3} i a^3 \cos ^{-1}(a x)^3 \tan ^{-1}\left(e^{i \cos ^{-1}(a x)}\right)-8 i a^3 \cos ^{-1}(a x) \tan ^{-1}\left(e^{i \cos ^{-1}(a x)}\right)+\frac{2 a \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{3 x^2}-\frac{2 a^2 \cos ^{-1}(a x)^2}{x}-\frac{\cos ^{-1}(a x)^4}{3 x^3}",1,"a^3*(-1/6*(ArcCos[a*x]^2*(12 + ArcCos[a*x]^2)) + 4*(ArcCos[a*x]*(Log[1 - I*E^(I*ArcCos[a*x])] - Log[1 + I*E^(I*ArcCos[a*x])]) + I*(PolyLog[2, (-I)*E^(I*ArcCos[a*x])] - PolyLog[2, I*E^(I*ArcCos[a*x])])) + (2*((Pi^3*Log[Cot[(Pi/2 - ArcCos[a*x])/2]])/8 + (3*Pi^2*((Pi/2 - ArcCos[a*x])*(Log[1 - E^(I*(Pi/2 - ArcCos[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcCos[a*x]))]) + I*(PolyLog[2, -E^(I*(Pi/2 - ArcCos[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcCos[a*x]))])))/4 - (3*Pi*((Pi/2 - ArcCos[a*x])^2*(Log[1 - E^(I*(Pi/2 - ArcCos[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcCos[a*x]))]) + (2*I)*(Pi/2 - ArcCos[a*x])*(PolyLog[2, -E^(I*(Pi/2 - ArcCos[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcCos[a*x]))]) + 2*(-PolyLog[3, -E^(I*(Pi/2 - ArcCos[a*x]))] + PolyLog[3, E^(I*(Pi/2 - ArcCos[a*x]))])))/2 + 8*((I/64)*(Pi/2 - ArcCos[a*x])^4 + (I/4)*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2)^4 - ((Pi/2 - ArcCos[a*x])^3*Log[1 + E^(I*(Pi/2 - ArcCos[a*x]))])/8 - (Pi^3*(I*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2) - Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2))]))/8 - (Pi/2 + (-1/2*Pi + ArcCos[a*x])/2)^3*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2))] + ((3*I)/8)*(Pi/2 - ArcCos[a*x])^2*PolyLog[2, -E^(I*(Pi/2 - ArcCos[a*x]))] + (3*Pi^2*((I/2)*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2)^2 - (Pi/2 + (-1/2*Pi + ArcCos[a*x])/2)*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2))] + (I/2)*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2))]))/4 + ((3*I)/2)*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2)^2*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2))] - (3*(Pi/2 - ArcCos[a*x])*PolyLog[3, -E^(I*(Pi/2 - ArcCos[a*x]))])/4 - (3*Pi*((I/3)*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2)^3 - (Pi/2 + (-1/2*Pi + ArcCos[a*x])/2)^2*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2))] + I*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2)*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2))] - PolyLog[3, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2))]/2))/2 - (3*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2)*PolyLog[3, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2))])/2 - ((3*I)/4)*PolyLog[4, -E^(I*(Pi/2 - ArcCos[a*x]))] - ((3*I)/4)*PolyLog[4, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcCos[a*x])/2))])))/3 - (-4*ArcCos[a*x]^3 + ArcCos[a*x]^4)/(12*(Cos[ArcCos[a*x]/2] - Sin[ArcCos[a*x]/2])^2) - (ArcCos[a*x]^4*Sin[ArcCos[a*x]/2])/(6*(Cos[ArcCos[a*x]/2] - Sin[ArcCos[a*x]/2])^3) + (ArcCos[a*x]^4*Sin[ArcCos[a*x]/2])/(6*(Cos[ArcCos[a*x]/2] + Sin[ArcCos[a*x]/2])^3) - (4*ArcCos[a*x]^3 + ArcCos[a*x]^4)/(12*(Cos[ArcCos[a*x]/2] + Sin[ArcCos[a*x]/2])^2) - (-12*ArcCos[a*x]^2*Sin[ArcCos[a*x]/2] - ArcCos[a*x]^4*Sin[ArcCos[a*x]/2])/(6*(Cos[ArcCos[a*x]/2] + Sin[ArcCos[a*x]/2])) - (12*ArcCos[a*x]^2*Sin[ArcCos[a*x]/2] + ArcCos[a*x]^4*Sin[ArcCos[a*x]/2])/(6*(Cos[ArcCos[a*x]/2] - Sin[ArcCos[a*x]/2])))","B",0
42,1,40,55,0.1214293,"\int \frac{x^6}{\cos ^{-1}(a x)} \, dx","Integrate[x^6/ArcCos[a*x],x]","-\frac{5 \text{Si}\left(\cos ^{-1}(a x)\right)+9 \text{Si}\left(3 \cos ^{-1}(a x)\right)+5 \text{Si}\left(5 \cos ^{-1}(a x)\right)+\text{Si}\left(7 \cos ^{-1}(a x)\right)}{64 a^7}","-\frac{5 \text{Si}\left(\cos ^{-1}(a x)\right)}{64 a^7}-\frac{9 \text{Si}\left(3 \cos ^{-1}(a x)\right)}{64 a^7}-\frac{5 \text{Si}\left(5 \cos ^{-1}(a x)\right)}{64 a^7}-\frac{\text{Si}\left(7 \cos ^{-1}(a x)\right)}{64 a^7}",1,"-1/64*(5*SinIntegral[ArcCos[a*x]] + 9*SinIntegral[3*ArcCos[a*x]] + 5*SinIntegral[5*ArcCos[a*x]] + SinIntegral[7*ArcCos[a*x]])/a^7","A",1
43,1,33,43,0.098951,"\int \frac{x^5}{\cos ^{-1}(a x)} \, dx","Integrate[x^5/ArcCos[a*x],x]","-\frac{5 \text{Si}\left(2 \cos ^{-1}(a x)\right)+4 \text{Si}\left(4 \cos ^{-1}(a x)\right)+\text{Si}\left(6 \cos ^{-1}(a x)\right)}{32 a^6}","-\frac{5 \text{Si}\left(2 \cos ^{-1}(a x)\right)}{32 a^6}-\frac{\text{Si}\left(4 \cos ^{-1}(a x)\right)}{8 a^6}-\frac{\text{Si}\left(6 \cos ^{-1}(a x)\right)}{32 a^6}",1,"-1/32*(5*SinIntegral[2*ArcCos[a*x]] + 4*SinIntegral[4*ArcCos[a*x]] + SinIntegral[6*ArcCos[a*x]])/a^6","A",1
44,1,31,41,0.0893381,"\int \frac{x^4}{\cos ^{-1}(a x)} \, dx","Integrate[x^4/ArcCos[a*x],x]","-\frac{2 \text{Si}\left(\cos ^{-1}(a x)\right)+3 \text{Si}\left(3 \cos ^{-1}(a x)\right)+\text{Si}\left(5 \cos ^{-1}(a x)\right)}{16 a^5}","-\frac{\text{Si}\left(\cos ^{-1}(a x)\right)}{8 a^5}-\frac{3 \text{Si}\left(3 \cos ^{-1}(a x)\right)}{16 a^5}-\frac{\text{Si}\left(5 \cos ^{-1}(a x)\right)}{16 a^5}",1,"-1/16*(2*SinIntegral[ArcCos[a*x]] + 3*SinIntegral[3*ArcCos[a*x]] + SinIntegral[5*ArcCos[a*x]])/a^5","A",1
45,1,24,29,0.0715332,"\int \frac{x^3}{\cos ^{-1}(a x)} \, dx","Integrate[x^3/ArcCos[a*x],x]","-\frac{2 \text{Si}\left(2 \cos ^{-1}(a x)\right)+\text{Si}\left(4 \cos ^{-1}(a x)\right)}{8 a^4}","-\frac{\text{Si}\left(2 \cos ^{-1}(a x)\right)}{4 a^4}-\frac{\text{Si}\left(4 \cos ^{-1}(a x)\right)}{8 a^4}",1,"-1/8*(2*SinIntegral[2*ArcCos[a*x]] + SinIntegral[4*ArcCos[a*x]])/a^4","A",1
46,1,20,27,0.0620836,"\int \frac{x^2}{\cos ^{-1}(a x)} \, dx","Integrate[x^2/ArcCos[a*x],x]","-\frac{\text{Si}\left(\cos ^{-1}(a x)\right)+\text{Si}\left(3 \cos ^{-1}(a x)\right)}{4 a^3}","-\frac{\text{Si}\left(\cos ^{-1}(a x)\right)}{4 a^3}-\frac{\text{Si}\left(3 \cos ^{-1}(a x)\right)}{4 a^3}",1,"-1/4*(SinIntegral[ArcCos[a*x]] + SinIntegral[3*ArcCos[a*x]])/a^3","A",1
47,1,14,14,0.0221507,"\int \frac{x}{\cos ^{-1}(a x)} \, dx","Integrate[x/ArcCos[a*x],x]","-\frac{\text{Si}\left(2 \cos ^{-1}(a x)\right)}{2 a^2}","-\frac{\text{Si}\left(2 \cos ^{-1}(a x)\right)}{2 a^2}",1,"-1/2*SinIntegral[2*ArcCos[a*x]]/a^2","A",1
48,1,10,10,0.0256011,"\int \frac{1}{\cos ^{-1}(a x)} \, dx","Integrate[ArcCos[a*x]^(-1),x]","-\frac{\text{Si}\left(\cos ^{-1}(a x)\right)}{a}","-\frac{\text{Si}\left(\cos ^{-1}(a x)\right)}{a}",1,"-(SinIntegral[ArcCos[a*x]]/a)","A",1
49,0,0,13,0.2524231,"\int \frac{1}{x \cos ^{-1}(a x)} \, dx","Integrate[1/(x*ArcCos[a*x]),x]","\int \frac{1}{x \cos ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{x \cos ^{-1}(a x)},x\right)",0,"Integrate[1/(x*ArcCos[a*x]), x]","A",-1
50,0,0,13,0.9285167,"\int \frac{1}{x^2 \cos ^{-1}(a x)} \, dx","Integrate[1/(x^2*ArcCos[a*x]),x]","\int \frac{1}{x^2 \cos ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{x^2 \cos ^{-1}(a x)},x\right)",0,"Integrate[1/(x^2*ArcCos[a*x]), x]","A",-1
51,1,86,82,0.1664936,"\int \frac{x^6}{\cos ^{-1}(a x)^2} \, dx","Integrate[x^6/ArcCos[a*x]^2,x]","-\frac{-64 a^6 x^6 \sqrt{1-a^2 x^2}+5 \cos ^{-1}(a x) \text{Ci}\left(\cos ^{-1}(a x)\right)+27 \cos ^{-1}(a x) \text{Ci}\left(3 \cos ^{-1}(a x)\right)+25 \cos ^{-1}(a x) \text{Ci}\left(5 \cos ^{-1}(a x)\right)+7 \cos ^{-1}(a x) \text{Ci}\left(7 \cos ^{-1}(a x)\right)}{64 a^7 \cos ^{-1}(a x)}","-\frac{5 \text{Ci}\left(\cos ^{-1}(a x)\right)}{64 a^7}-\frac{27 \text{Ci}\left(3 \cos ^{-1}(a x)\right)}{64 a^7}-\frac{25 \text{Ci}\left(5 \cos ^{-1}(a x)\right)}{64 a^7}-\frac{7 \text{Ci}\left(7 \cos ^{-1}(a x)\right)}{64 a^7}+\frac{x^6 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}",1,"-1/64*(-64*a^6*x^6*Sqrt[1 - a^2*x^2] + 5*ArcCos[a*x]*CosIntegral[ArcCos[a*x]] + 27*ArcCos[a*x]*CosIntegral[3*ArcCos[a*x]] + 25*ArcCos[a*x]*CosIntegral[5*ArcCos[a*x]] + 7*ArcCos[a*x]*CosIntegral[7*ArcCos[a*x]])/(a^7*ArcCos[a*x])","A",1
52,1,63,70,0.1712978,"\int \frac{x^5}{\cos ^{-1}(a x)^2} \, dx","Integrate[x^5/ArcCos[a*x]^2,x]","-\frac{-\frac{16 a^5 x^5 \sqrt{1-a^2 x^2}}{\cos ^{-1}(a x)}+5 \text{Ci}\left(2 \cos ^{-1}(a x)\right)+8 \text{Ci}\left(4 \cos ^{-1}(a x)\right)+3 \text{Ci}\left(6 \cos ^{-1}(a x)\right)}{16 a^6}","-\frac{5 \text{Ci}\left(2 \cos ^{-1}(a x)\right)}{16 a^6}-\frac{\text{Ci}\left(4 \cos ^{-1}(a x)\right)}{2 a^6}-\frac{3 \text{Ci}\left(6 \cos ^{-1}(a x)\right)}{16 a^6}+\frac{x^5 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}",1,"-1/16*((-16*a^5*x^5*Sqrt[1 - a^2*x^2])/ArcCos[a*x] + 5*CosIntegral[2*ArcCos[a*x]] + 8*CosIntegral[4*ArcCos[a*x]] + 3*CosIntegral[6*ArcCos[a*x]])/a^6","A",1
53,1,61,68,0.171556,"\int \frac{x^4}{\cos ^{-1}(a x)^2} \, dx","Integrate[x^4/ArcCos[a*x]^2,x]","-\frac{-\frac{16 a^4 x^4 \sqrt{1-a^2 x^2}}{\cos ^{-1}(a x)}+2 \text{Ci}\left(\cos ^{-1}(a x)\right)+9 \text{Ci}\left(3 \cos ^{-1}(a x)\right)+5 \text{Ci}\left(5 \cos ^{-1}(a x)\right)}{16 a^5}","-\frac{\text{Ci}\left(\cos ^{-1}(a x)\right)}{8 a^5}-\frac{9 \text{Ci}\left(3 \cos ^{-1}(a x)\right)}{16 a^5}-\frac{5 \text{Ci}\left(5 \cos ^{-1}(a x)\right)}{16 a^5}+\frac{x^4 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}",1,"-1/16*((-16*a^4*x^4*Sqrt[1 - a^2*x^2])/ArcCos[a*x] + 2*CosIntegral[ArcCos[a*x]] + 9*CosIntegral[3*ArcCos[a*x]] + 5*CosIntegral[5*ArcCos[a*x]])/a^5","A",1
54,1,50,56,0.1438938,"\int \frac{x^3}{\cos ^{-1}(a x)^2} \, dx","Integrate[x^3/ArcCos[a*x]^2,x]","-\frac{-\frac{2 a^3 x^3 \sqrt{1-a^2 x^2}}{\cos ^{-1}(a x)}+\text{Ci}\left(2 \cos ^{-1}(a x)\right)+\text{Ci}\left(4 \cos ^{-1}(a x)\right)}{2 a^4}","-\frac{\text{Ci}\left(2 \cos ^{-1}(a x)\right)}{2 a^4}-\frac{\text{Ci}\left(4 \cos ^{-1}(a x)\right)}{2 a^4}+\frac{x^3 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}",1,"-1/2*((-2*a^3*x^3*Sqrt[1 - a^2*x^2])/ArcCos[a*x] + CosIntegral[2*ArcCos[a*x]] + CosIntegral[4*ArcCos[a*x]])/a^4","A",1
55,1,50,54,0.1332882,"\int \frac{x^2}{\cos ^{-1}(a x)^2} \, dx","Integrate[x^2/ArcCos[a*x]^2,x]","-\frac{-\frac{4 a^2 x^2 \sqrt{1-a^2 x^2}}{\cos ^{-1}(a x)}+\text{Ci}\left(\cos ^{-1}(a x)\right)+3 \text{Ci}\left(3 \cos ^{-1}(a x)\right)}{4 a^3}","-\frac{\text{Ci}\left(\cos ^{-1}(a x)\right)}{4 a^3}-\frac{3 \text{Ci}\left(3 \cos ^{-1}(a x)\right)}{4 a^3}+\frac{x^2 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}",1,"-1/4*((-4*a^2*x^2*Sqrt[1 - a^2*x^2])/ArcCos[a*x] + CosIntegral[ArcCos[a*x]] + 3*CosIntegral[3*ArcCos[a*x]])/a^3","A",1
56,1,37,38,0.0968814,"\int \frac{x}{\cos ^{-1}(a x)^2} \, dx","Integrate[x/ArcCos[a*x]^2,x]","\frac{\frac{a x \sqrt{1-a^2 x^2}}{\cos ^{-1}(a x)}-\text{Ci}\left(2 \cos ^{-1}(a x)\right)}{a^2}","\frac{x \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}-\frac{\text{Ci}\left(2 \cos ^{-1}(a x)\right)}{a^2}",1,"((a*x*Sqrt[1 - a^2*x^2])/ArcCos[a*x] - CosIntegral[2*ArcCos[a*x]])/a^2","A",1
57,1,35,35,0.0422617,"\int \frac{1}{\cos ^{-1}(a x)^2} \, dx","Integrate[ArcCos[a*x]^(-2),x]","\frac{\sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}-\frac{\text{Ci}\left(\cos ^{-1}(a x)\right)}{a}","\frac{\sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}-\frac{\text{Ci}\left(\cos ^{-1}(a x)\right)}{a}",1,"Sqrt[1 - a^2*x^2]/(a*ArcCos[a*x]) - CosIntegral[ArcCos[a*x]]/a","A",1
58,0,0,13,1.034109,"\int \frac{1}{x \cos ^{-1}(a x)^2} \, dx","Integrate[1/(x*ArcCos[a*x]^2),x]","\int \frac{1}{x \cos ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{1}{x \cos ^{-1}(a x)^2},x\right)",0,"Integrate[1/(x*ArcCos[a*x]^2), x]","A",-1
59,0,0,13,18.2601318,"\int \frac{1}{x^2 \cos ^{-1}(a x)^2} \, dx","Integrate[1/(x^2*ArcCos[a*x]^2),x]","\int \frac{1}{x^2 \cos ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{1}{x^2 \cos ^{-1}(a x)^2},x\right)",0,"Integrate[1/(x^2*ArcCos[a*x]^2), x]","A",-1
60,1,103,98,0.1441116,"\int \frac{x^4}{\cos ^{-1}(a x)^3} \, dx","Integrate[x^4/ArcCos[a*x]^3,x]","\frac{80 a^5 x^5 \cos ^{-1}(a x)-64 a^3 x^3 \cos ^{-1}(a x)+16 a^4 x^4 \sqrt{1-a^2 x^2}+2 \cos ^{-1}(a x)^2 \text{Si}\left(\cos ^{-1}(a x)\right)+27 \cos ^{-1}(a x)^2 \text{Si}\left(3 \cos ^{-1}(a x)\right)+25 \cos ^{-1}(a x)^2 \text{Si}\left(5 \cos ^{-1}(a x)\right)}{32 a^5 \cos ^{-1}(a x)^2}","\frac{\text{Si}\left(\cos ^{-1}(a x)\right)}{16 a^5}+\frac{27 \text{Si}\left(3 \cos ^{-1}(a x)\right)}{32 a^5}+\frac{25 \text{Si}\left(5 \cos ^{-1}(a x)\right)}{32 a^5}-\frac{2 x^3}{a^2 \cos ^{-1}(a x)}+\frac{x^4 \sqrt{1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}+\frac{5 x^5}{2 \cos ^{-1}(a x)}",1,"(16*a^4*x^4*Sqrt[1 - a^2*x^2] - 64*a^3*x^3*ArcCos[a*x] + 80*a^5*x^5*ArcCos[a*x] + 2*ArcCos[a*x]^2*SinIntegral[ArcCos[a*x]] + 27*ArcCos[a*x]^2*SinIntegral[3*ArcCos[a*x]] + 25*ArcCos[a*x]^2*SinIntegral[5*ArcCos[a*x]])/(32*a^5*ArcCos[a*x]^2)","A",1
61,1,70,83,0.150482,"\int \frac{x^3}{\cos ^{-1}(a x)^3} \, dx","Integrate[x^3/ArcCos[a*x]^3,x]","\frac{\frac{a^2 x^2 \left(a x \sqrt{1-a^2 x^2}+\left(4 a^2 x^2-3\right) \cos ^{-1}(a x)\right)}{\cos ^{-1}(a x)^2}+\text{Si}\left(2 \cos ^{-1}(a x)\right)+2 \text{Si}\left(4 \cos ^{-1}(a x)\right)}{2 a^4}","\frac{\text{Si}\left(2 \cos ^{-1}(a x)\right)}{2 a^4}+\frac{\text{Si}\left(4 \cos ^{-1}(a x)\right)}{a^4}-\frac{3 x^2}{2 a^2 \cos ^{-1}(a x)}+\frac{x^3 \sqrt{1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}+\frac{2 x^4}{\cos ^{-1}(a x)}",1,"((a^2*x^2*(a*x*Sqrt[1 - a^2*x^2] + (-3 + 4*a^2*x^2)*ArcCos[a*x]))/ArcCos[a*x]^2 + SinIntegral[2*ArcCos[a*x]] + 2*SinIntegral[4*ArcCos[a*x]])/(2*a^4)","A",1
62,1,65,82,0.1393752,"\int \frac{x^2}{\cos ^{-1}(a x)^3} \, dx","Integrate[x^2/ArcCos[a*x]^3,x]","\frac{\frac{4 a x \left(a x \sqrt{1-a^2 x^2}+\left(3 a^2 x^2-2\right) \cos ^{-1}(a x)\right)}{\cos ^{-1}(a x)^2}+\text{Si}\left(\cos ^{-1}(a x)\right)+9 \text{Si}\left(3 \cos ^{-1}(a x)\right)}{8 a^3}","\frac{\text{Si}\left(\cos ^{-1}(a x)\right)}{8 a^3}+\frac{9 \text{Si}\left(3 \cos ^{-1}(a x)\right)}{8 a^3}+\frac{x^2 \sqrt{1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac{x}{a^2 \cos ^{-1}(a x)}+\frac{3 x^3}{2 \cos ^{-1}(a x)}",1,"((4*a*x*(a*x*Sqrt[1 - a^2*x^2] + (-2 + 3*a^2*x^2)*ArcCos[a*x]))/ArcCos[a*x]^2 + SinIntegral[ArcCos[a*x]] + 9*SinIntegral[3*ArcCos[a*x]])/(8*a^3)","A",1
63,1,63,63,0.0438248,"\int \frac{x}{\cos ^{-1}(a x)^3} \, dx","Integrate[x/ArcCos[a*x]^3,x]","\frac{\text{Si}\left(2 \cos ^{-1}(a x)\right)}{a^2}+\frac{x \sqrt{1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}+\frac{2 a^2 x^2-1}{2 a^2 \cos ^{-1}(a x)}","\frac{\text{Si}\left(2 \cos ^{-1}(a x)\right)}{a^2}+\frac{x \sqrt{1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}-\frac{1}{2 a^2 \cos ^{-1}(a x)}+\frac{x^2}{\cos ^{-1}(a x)}",1,"(x*Sqrt[1 - a^2*x^2])/(2*a*ArcCos[a*x]^2) + (-1 + 2*a^2*x^2)/(2*a^2*ArcCos[a*x]) + SinIntegral[2*ArcCos[a*x]]/a^2","A",1
64,1,47,51,0.0326941,"\int \frac{1}{\cos ^{-1}(a x)^3} \, dx","Integrate[ArcCos[a*x]^(-3),x]","\frac{\sqrt{1-a^2 x^2}+\cos ^{-1}(a x)^2 \text{Si}\left(\cos ^{-1}(a x)\right)+a x \cos ^{-1}(a x)}{2 a \cos ^{-1}(a x)^2}","\frac{\sqrt{1-a^2 x^2}}{2 a \cos ^{-1}(a x)^2}+\frac{\text{Si}\left(\cos ^{-1}(a x)\right)}{2 a}+\frac{x}{2 \cos ^{-1}(a x)}",1,"(Sqrt[1 - a^2*x^2] + a*x*ArcCos[a*x] + ArcCos[a*x]^2*SinIntegral[ArcCos[a*x]])/(2*a*ArcCos[a*x]^2)","A",1
65,0,0,13,0.7562993,"\int \frac{1}{x \cos ^{-1}(a x)^3} \, dx","Integrate[1/(x*ArcCos[a*x]^3),x]","\int \frac{1}{x \cos ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{1}{x \cos ^{-1}(a x)^3},x\right)",0,"Integrate[1/(x*ArcCos[a*x]^3), x]","A",-1
66,0,0,13,10.1947015,"\int \frac{1}{x^2 \cos ^{-1}(a x)^3} \, dx","Integrate[1/(x^2*ArcCos[a*x]^3),x]","\int \frac{1}{x^2 \cos ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{1}{x^2 \cos ^{-1}(a x)^3},x\right)",0,"Integrate[1/(x^2*ArcCos[a*x]^3), x]","A",-1
67,1,159,158,0.1847932,"\int \frac{x^4}{\cos ^{-1}(a x)^4} \, dx","Integrate[x^4/ArcCos[a*x]^4,x]","\frac{80 a^5 x^5 \cos ^{-1}(a x)-64 a^3 x^3 \cos ^{-1}(a x)+192 a^2 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2+32 a^4 x^4 \sqrt{1-a^2 x^2}-400 a^4 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2+2 \cos ^{-1}(a x)^3 \text{Ci}\left(\cos ^{-1}(a x)\right)+81 \cos ^{-1}(a x)^3 \text{Ci}\left(3 \cos ^{-1}(a x)\right)+125 \cos ^{-1}(a x)^3 \text{Ci}\left(5 \cos ^{-1}(a x)\right)}{96 a^5 \cos ^{-1}(a x)^3}","\frac{\text{Ci}\left(\cos ^{-1}(a x)\right)}{48 a^5}+\frac{27 \text{Ci}\left(3 \cos ^{-1}(a x)\right)}{32 a^5}+\frac{125 \text{Ci}\left(5 \cos ^{-1}(a x)\right)}{96 a^5}-\frac{2 x^3}{3 a^2 \cos ^{-1}(a x)^2}-\frac{25 x^4 \sqrt{1-a^2 x^2}}{6 a \cos ^{-1}(a x)}+\frac{x^4 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}+\frac{2 x^2 \sqrt{1-a^2 x^2}}{a^3 \cos ^{-1}(a x)}+\frac{5 x^5}{6 \cos ^{-1}(a x)^2}",1,"(32*a^4*x^4*Sqrt[1 - a^2*x^2] - 64*a^3*x^3*ArcCos[a*x] + 80*a^5*x^5*ArcCos[a*x] + 192*a^2*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2 - 400*a^4*x^4*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2 + 2*ArcCos[a*x]^3*CosIntegral[ArcCos[a*x]] + 81*ArcCos[a*x]^3*CosIntegral[3*ArcCos[a*x]] + 125*ArcCos[a*x]^3*CosIntegral[5*ArcCos[a*x]])/(96*a^5*ArcCos[a*x]^3)","A",1
68,1,107,143,0.3003532,"\int \frac{x^3}{\cos ^{-1}(a x)^4} \, dx","Integrate[x^3/ArcCos[a*x]^4,x]","\frac{\frac{a x \left(2 a^2 x^2 \sqrt{1-a^2 x^2}+a x \left(4 a^2 x^2-3\right) \cos ^{-1}(a x)-2 \sqrt{1-a^2 x^2} \left(8 a^2 x^2-3\right) \cos ^{-1}(a x)^2\right)}{\cos ^{-1}(a x)^3}+2 \text{Ci}\left(2 \cos ^{-1}(a x)\right)+8 \text{Ci}\left(4 \cos ^{-1}(a x)\right)}{6 a^4}","\frac{\text{Ci}\left(2 \cos ^{-1}(a x)\right)}{3 a^4}+\frac{4 \text{Ci}\left(4 \cos ^{-1}(a x)\right)}{3 a^4}-\frac{x^2}{2 a^2 \cos ^{-1}(a x)^2}-\frac{8 x^3 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)}+\frac{x^3 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}+\frac{x \sqrt{1-a^2 x^2}}{a^3 \cos ^{-1}(a x)}+\frac{2 x^4}{3 \cos ^{-1}(a x)^2}",1,"((a*x*(2*a^2*x^2*Sqrt[1 - a^2*x^2] + a*x*(-3 + 4*a^2*x^2)*ArcCos[a*x] - 2*Sqrt[1 - a^2*x^2]*(-3 + 8*a^2*x^2)*ArcCos[a*x]^2))/ArcCos[a*x]^3 + 2*CosIntegral[2*ArcCos[a*x]] + 8*CosIntegral[4*ArcCos[a*x]])/(6*a^4)","A",1
69,1,129,141,0.1534965,"\int \frac{x^2}{\cos ^{-1}(a x)^4} \, dx","Integrate[x^2/ArcCos[a*x]^4,x]","-\frac{10 \text{Ci}\left(\cos ^{-1}(a x)\right)}{3 a^3}-\frac{9 \left(-3 \text{Ci}\left(\cos ^{-1}(a x)\right)-\text{Ci}\left(3 \cos ^{-1}(a x)\right)\right)}{8 a^3}+\frac{3 a^2 x^3-2 x}{6 a^2 \cos ^{-1}(a x)^2}+\frac{x^2 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac{\sqrt{1-a^2 x^2} \left(9 a^2 x^2-2\right)}{6 a^3 \cos ^{-1}(a x)}","\frac{\text{Ci}\left(\cos ^{-1}(a x)\right)}{24 a^3}+\frac{9 \text{Ci}\left(3 \cos ^{-1}(a x)\right)}{8 a^3}-\frac{3 x^2 \sqrt{1-a^2 x^2}}{2 a \cos ^{-1}(a x)}+\frac{x^2 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac{x}{3 a^2 \cos ^{-1}(a x)^2}+\frac{\sqrt{1-a^2 x^2}}{3 a^3 \cos ^{-1}(a x)}+\frac{x^3}{2 \cos ^{-1}(a x)^2}",1,"(x^2*Sqrt[1 - a^2*x^2])/(3*a*ArcCos[a*x]^3) + (-2*x + 3*a^2*x^3)/(6*a^2*ArcCos[a*x]^2) - (Sqrt[1 - a^2*x^2]*(-2 + 9*a^2*x^2))/(6*a^3*ArcCos[a*x]) - (10*CosIntegral[ArcCos[a*x]])/(3*a^3) - (9*(-3*CosIntegral[ArcCos[a*x]] - CosIntegral[3*ArcCos[a*x]]))/(8*a^3)","A",1
70,1,86,97,0.119942,"\int \frac{x}{\cos ^{-1}(a x)^4} \, dx","Integrate[x/ArcCos[a*x]^4,x]","\frac{2 a x \sqrt{1-a^2 x^2}-4 a x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2+\left(2 a^2 x^2-1\right) \cos ^{-1}(a x)+4 \cos ^{-1}(a x)^3 \text{Ci}\left(2 \cos ^{-1}(a x)\right)}{6 a^2 \cos ^{-1}(a x)^3}","\frac{2 \text{Ci}\left(2 \cos ^{-1}(a x)\right)}{3 a^2}-\frac{2 x \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)}+\frac{x \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}-\frac{1}{6 a^2 \cos ^{-1}(a x)^2}+\frac{x^2}{3 \cos ^{-1}(a x)^2}",1,"(2*a*x*Sqrt[1 - a^2*x^2] + (-1 + 2*a^2*x^2)*ArcCos[a*x] - 4*a*x*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2 + 4*ArcCos[a*x]^3*CosIntegral[2*ArcCos[a*x]])/(6*a^2*ArcCos[a*x]^3)","A",1
71,1,71,78,0.0521445,"\int \frac{1}{\cos ^{-1}(a x)^4} \, dx","Integrate[ArcCos[a*x]^(-4),x]","\frac{2 \sqrt{1-a^2 x^2}-\sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2+\cos ^{-1}(a x)^3 \text{Ci}\left(\cos ^{-1}(a x)\right)+a x \cos ^{-1}(a x)}{6 a \cos ^{-1}(a x)^3}","-\frac{\sqrt{1-a^2 x^2}}{6 a \cos ^{-1}(a x)}+\frac{\sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^3}+\frac{\text{Ci}\left(\cos ^{-1}(a x)\right)}{6 a}+\frac{x}{6 \cos ^{-1}(a x)^2}",1,"(2*Sqrt[1 - a^2*x^2] + a*x*ArcCos[a*x] - Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2 + ArcCos[a*x]^3*CosIntegral[ArcCos[a*x]])/(6*a*ArcCos[a*x]^3)","A",1
72,0,0,13,3.048953,"\int \frac{1}{x \cos ^{-1}(a x)^4} \, dx","Integrate[1/(x*ArcCos[a*x]^4),x]","\int \frac{1}{x \cos ^{-1}(a x)^4} \, dx","\text{Int}\left(\frac{1}{x \cos ^{-1}(a x)^4},x\right)",0,"Integrate[1/(x*ArcCos[a*x]^4), x]","A",-1
73,0,0,13,29.9511568,"\int \frac{1}{x^2 \cos ^{-1}(a x)^4} \, dx","Integrate[1/(x^2*ArcCos[a*x]^4),x]","\int \frac{1}{x^2 \cos ^{-1}(a x)^4} \, dx","\text{Int}\left(\frac{1}{x^2 \cos ^{-1}(a x)^4},x\right)",0,"Integrate[1/(x^2*ArcCos[a*x]^4), x]","A",-1
74,1,212,121,0.3245593,"\int x^4 \sqrt{\cos ^{-1}(a x)} \, dx","Integrate[x^4*Sqrt[ArcCos[a*x]],x]","-\frac{25 \sqrt{3} \left(-i \cos ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{3}{2},-3 i \cos ^{-1}(a x)\right)+3 \sqrt{5} \left(-i \cos ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{3}{2},-5 i \cos ^{-1}(a x)\right)-150 \sqrt{\cos ^{-1}(a x)^2} \sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{3}{2},i \cos ^{-1}(a x)\right)-150 \sqrt{i \cos ^{-1}(a x)} \sqrt{\cos ^{-1}(a x)^2} \Gamma \left(\frac{3}{2},-i \cos ^{-1}(a x)\right)+25 \sqrt{3} \left(i \cos ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{3}{2},3 i \cos ^{-1}(a x)\right)+3 \sqrt{5} \left(i \cos ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{3}{2},5 i \cos ^{-1}(a x)\right)}{2400 a^5 \cos ^{-1}(a x)^{3/2}}","-\frac{\sqrt{\frac{\pi }{2}} C\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{8 a^5}-\frac{\sqrt{\frac{\pi }{6}} C\left(\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{16 a^5}-\frac{\sqrt{\frac{\pi }{10}} C\left(\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{80 a^5}+\frac{1}{5} x^5 \sqrt{\cos ^{-1}(a x)}",1,"-1/2400*(-150*Sqrt[I*ArcCos[a*x]]*Sqrt[ArcCos[a*x]^2]*Gamma[3/2, (-I)*ArcCos[a*x]] - 150*Sqrt[(-I)*ArcCos[a*x]]*Sqrt[ArcCos[a*x]^2]*Gamma[3/2, I*ArcCos[a*x]] + 25*Sqrt[3]*((-I)*ArcCos[a*x])^(3/2)*Gamma[3/2, (-3*I)*ArcCos[a*x]] + 25*Sqrt[3]*(I*ArcCos[a*x])^(3/2)*Gamma[3/2, (3*I)*ArcCos[a*x]] + 3*Sqrt[5]*((-I)*ArcCos[a*x])^(3/2)*Gamma[3/2, (-5*I)*ArcCos[a*x]] + 3*Sqrt[5]*(I*ArcCos[a*x])^(3/2)*Gamma[3/2, (5*I)*ArcCos[a*x]])/(a^5*ArcCos[a*x]^(3/2))","C",0
75,1,125,95,0.1347022,"\int x^3 \sqrt{\cos ^{-1}(a x)} \, dx","Integrate[x^3*Sqrt[ArcCos[a*x]],x]","-\frac{\sqrt{\cos ^{-1}(a x)} \left(4 i \sqrt{2} \cos ^{-1}(a x) \Gamma \left(\frac{3}{2},-2 i \cos ^{-1}(a x)\right)+4 \sqrt{2} \sqrt{\cos ^{-1}(a x)^2} \Gamma \left(\frac{3}{2},2 i \cos ^{-1}(a x)\right)+i \cos ^{-1}(a x) \Gamma \left(\frac{3}{2},-4 i \cos ^{-1}(a x)\right)+\sqrt{\cos ^{-1}(a x)^2} \Gamma \left(\frac{3}{2},4 i \cos ^{-1}(a x)\right)\right)}{128 a^4 \left(-i \cos ^{-1}(a x)\right)^{3/2}}","-\frac{\sqrt{\frac{\pi }{2}} C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{64 a^4}-\frac{\sqrt{\pi } C\left(\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right)}{16 a^4}-\frac{3 \sqrt{\cos ^{-1}(a x)}}{32 a^4}+\frac{1}{4} x^4 \sqrt{\cos ^{-1}(a x)}",1,"-1/128*(Sqrt[ArcCos[a*x]]*((4*I)*Sqrt[2]*ArcCos[a*x]*Gamma[3/2, (-2*I)*ArcCos[a*x]] + 4*Sqrt[2]*Sqrt[ArcCos[a*x]^2]*Gamma[3/2, (2*I)*ArcCos[a*x]] + I*ArcCos[a*x]*Gamma[3/2, (-4*I)*ArcCos[a*x]] + Sqrt[ArcCos[a*x]^2]*Gamma[3/2, (4*I)*ArcCos[a*x]]))/(a^4*((-I)*ArcCos[a*x])^(3/2))","C",0
76,1,122,86,0.2173808,"\int x^2 \sqrt{\cos ^{-1}(a x)} \, dx","Integrate[x^2*Sqrt[ArcCos[a*x]],x]","\frac{\sqrt{i \cos ^{-1}(a x)} \left(9 \sqrt{\cos ^{-1}(a x)^2} \Gamma \left(\frac{3}{2},-i \cos ^{-1}(a x)\right)-9 i \cos ^{-1}(a x) \Gamma \left(\frac{3}{2},i \cos ^{-1}(a x)\right)+\sqrt{3} \left(\sqrt{\cos ^{-1}(a x)^2} \Gamma \left(\frac{3}{2},-3 i \cos ^{-1}(a x)\right)-i \cos ^{-1}(a x) \Gamma \left(\frac{3}{2},3 i \cos ^{-1}(a x)\right)\right)\right)}{72 a^3 \cos ^{-1}(a x)^{3/2}}","-\frac{\sqrt{\frac{\pi }{2}} C\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{4 a^3}-\frac{\sqrt{\frac{\pi }{6}} C\left(\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{12 a^3}+\frac{1}{3} x^3 \sqrt{\cos ^{-1}(a x)}",1,"(Sqrt[I*ArcCos[a*x]]*(9*Sqrt[ArcCos[a*x]^2]*Gamma[3/2, (-I)*ArcCos[a*x]] - (9*I)*ArcCos[a*x]*Gamma[3/2, I*ArcCos[a*x]] + Sqrt[3]*(Sqrt[ArcCos[a*x]^2]*Gamma[3/2, (-3*I)*ArcCos[a*x]] - I*ArcCos[a*x]*Gamma[3/2, (3*I)*ArcCos[a*x]])))/(72*a^3*ArcCos[a*x]^(3/2))","C",0
77,1,49,59,0.0433343,"\int x \sqrt{\cos ^{-1}(a x)} \, dx","Integrate[x*Sqrt[ArcCos[a*x]],x]","\frac{\frac{1}{4} \sqrt{\cos ^{-1}(a x)} \cos \left(2 \cos ^{-1}(a x)\right)-\frac{1}{8} \sqrt{\pi } C\left(\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right)}{a^2}","-\frac{\sqrt{\pi } C\left(\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right)}{8 a^2}-\frac{\sqrt{\cos ^{-1}(a x)}}{4 a^2}+\frac{1}{2} x^2 \sqrt{\cos ^{-1}(a x)}",1,"((Sqrt[ArcCos[a*x]]*Cos[2*ArcCos[a*x]])/4 - (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])/8)/a^2","A",1
78,1,76,44,0.041332,"\int \sqrt{\cos ^{-1}(a x)} \, dx","Integrate[Sqrt[ArcCos[a*x]],x]","\frac{\sqrt{\cos ^{-1}(a x)} \left(\sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{3}{2},-i \cos ^{-1}(a x)\right)+\sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{3}{2},i \cos ^{-1}(a x)\right)\right)}{2 a \sqrt{\cos ^{-1}(a x)^2}}","x \sqrt{\cos ^{-1}(a x)}-\frac{\sqrt{\frac{\pi }{2}} C\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{a}",1,"(Sqrt[ArcCos[a*x]]*(Sqrt[I*ArcCos[a*x]]*Gamma[3/2, (-I)*ArcCos[a*x]] + Sqrt[(-I)*ArcCos[a*x]]*Gamma[3/2, I*ArcCos[a*x]]))/(2*a*Sqrt[ArcCos[a*x]^2])","C",0
79,0,0,15,0.4469165,"\int \frac{\sqrt{\cos ^{-1}(a x)}}{x} \, dx","Integrate[Sqrt[ArcCos[a*x]]/x,x]","\int \frac{\sqrt{\cos ^{-1}(a x)}}{x} \, dx","\text{Int}\left(\frac{\sqrt{\cos ^{-1}(a x)}}{x},x\right)",0,"Integrate[Sqrt[ArcCos[a*x]]/x, x]","A",-1
80,1,185,282,0.1391626,"\int x^4 \cos ^{-1}(a x)^{3/2} \, dx","Integrate[x^4*ArcCos[a*x]^(3/2),x]","-\frac{2250 \left(\sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{5}{2},-i \cos ^{-1}(a x)\right)+\sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{5}{2},i \cos ^{-1}(a x)\right)\right)+125 \sqrt{3} \left(\sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{5}{2},-3 i \cos ^{-1}(a x)\right)+\sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{5}{2},3 i \cos ^{-1}(a x)\right)\right)+9 \sqrt{5} \left(\sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{5}{2},-5 i \cos ^{-1}(a x)\right)+\sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{5}{2},5 i \cos ^{-1}(a x)\right)\right)}{36000 a^5 \sqrt{\cos ^{-1}(a x)}}","\frac{2 \sqrt{2 \pi } S\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{25 a^5}+\frac{11 \sqrt{\frac{\pi }{2}} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{400 a^5}+\frac{3 \sqrt{\frac{3 \pi }{2}} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{800 a^5}+\frac{\sqrt{\frac{\pi }{6}} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{50 a^5}+\frac{3 \sqrt{\frac{\pi }{10}} S\left(\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{800 a^5}-\frac{3 x^4 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{50 a}-\frac{4 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{25 a^5}-\frac{2 x^2 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{25 a^3}+\frac{1}{5} x^5 \cos ^{-1}(a x)^{3/2}",1,"-1/36000*(2250*(Sqrt[(-I)*ArcCos[a*x]]*Gamma[5/2, (-I)*ArcCos[a*x]] + Sqrt[I*ArcCos[a*x]]*Gamma[5/2, I*ArcCos[a*x]]) + 125*Sqrt[3]*(Sqrt[(-I)*ArcCos[a*x]]*Gamma[5/2, (-3*I)*ArcCos[a*x]] + Sqrt[I*ArcCos[a*x]]*Gamma[5/2, (3*I)*ArcCos[a*x]]) + 9*Sqrt[5]*(Sqrt[(-I)*ArcCos[a*x]]*Gamma[5/2, (-5*I)*ArcCos[a*x]] + Sqrt[I*ArcCos[a*x]]*Gamma[5/2, (5*I)*ArcCos[a*x]]))/(a^5*Sqrt[ArcCos[a*x]])","C",0
81,1,128,157,0.0894301,"\int x^3 \cos ^{-1}(a x)^{3/2} \, dx","Integrate[x^3*ArcCos[a*x]^(3/2),x]","-\frac{8 \sqrt{2} \sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{5}{2},-2 i \cos ^{-1}(a x)\right)+8 \sqrt{2} \sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{5}{2},2 i \cos ^{-1}(a x)\right)+\sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{5}{2},-4 i \cos ^{-1}(a x)\right)+\sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{5}{2},4 i \cos ^{-1}(a x)\right)}{512 a^4 \sqrt{\cos ^{-1}(a x)}}","\frac{3 \sqrt{\frac{\pi }{2}} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{512 a^4}+\frac{3 \sqrt{\pi } S\left(\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right)}{64 a^4}-\frac{3 \cos ^{-1}(a x)^{3/2}}{32 a^4}-\frac{3 x^3 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{32 a}-\frac{9 x \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{64 a^3}+\frac{1}{4} x^4 \cos ^{-1}(a x)^{3/2}",1,"-1/512*(8*Sqrt[2]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[5/2, (-2*I)*ArcCos[a*x]] + 8*Sqrt[2]*Sqrt[I*ArcCos[a*x]]*Gamma[5/2, (2*I)*ArcCos[a*x]] + Sqrt[(-I)*ArcCos[a*x]]*Gamma[5/2, (-4*I)*ArcCos[a*x]] + Sqrt[I*ArcCos[a*x]]*Gamma[5/2, (4*I)*ArcCos[a*x]])/(a^4*Sqrt[ArcCos[a*x]])","C",0
82,1,125,147,0.1009059,"\int x^2 \cos ^{-1}(a x)^{3/2} \, dx","Integrate[x^2*ArcCos[a*x]^(3/2),x]","-\frac{27 \sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{5}{2},-i \cos ^{-1}(a x)\right)+27 \sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{5}{2},i \cos ^{-1}(a x)\right)+\sqrt{3} \left(\sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{5}{2},-3 i \cos ^{-1}(a x)\right)+\sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{5}{2},3 i \cos ^{-1}(a x)\right)\right)}{216 a^3 \sqrt{\cos ^{-1}(a x)}}","\frac{3 \sqrt{\frac{\pi }{2}} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{8 a^3}+\frac{\sqrt{\frac{\pi }{6}} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{24 a^3}-\frac{x^2 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{6 a}-\frac{\sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{3 a^3}+\frac{1}{3} x^3 \cos ^{-1}(a x)^{3/2}",1,"-1/216*(27*Sqrt[(-I)*ArcCos[a*x]]*Gamma[5/2, (-I)*ArcCos[a*x]] + 27*Sqrt[I*ArcCos[a*x]]*Gamma[5/2, I*ArcCos[a*x]] + Sqrt[3]*(Sqrt[(-I)*ArcCos[a*x]]*Gamma[5/2, (-3*I)*ArcCos[a*x]] + Sqrt[I*ArcCos[a*x]]*Gamma[5/2, (3*I)*ArcCos[a*x]]))/(a^3*Sqrt[ArcCos[a*x]])","C",0
83,1,64,89,0.0731008,"\int x \cos ^{-1}(a x)^{3/2} \, dx","Integrate[x*ArcCos[a*x]^(3/2),x]","\frac{3 \sqrt{\pi } S\left(\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right)-2 \sqrt{\cos ^{-1}(a x)} \left(3 \sin \left(2 \cos ^{-1}(a x)\right)-4 \cos ^{-1}(a x) \cos \left(2 \cos ^{-1}(a x)\right)\right)}{32 a^2}","\frac{3 \sqrt{\pi } S\left(\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right)}{32 a^2}-\frac{3 x \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{8 a}-\frac{\cos ^{-1}(a x)^{3/2}}{4 a^2}+\frac{1}{2} x^2 \cos ^{-1}(a x)^{3/2}",1,"(3*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]] - 2*Sqrt[ArcCos[a*x]]*(-4*ArcCos[a*x]*Cos[2*ArcCos[a*x]] + 3*Sin[2*ArcCos[a*x]]))/(32*a^2)","A",1
84,1,66,75,0.0295927,"\int \cos ^{-1}(a x)^{3/2} \, dx","Integrate[ArcCos[a*x]^(3/2),x]","-\frac{\sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{5}{2},-i \cos ^{-1}(a x)\right)+\sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{5}{2},i \cos ^{-1}(a x)\right)}{2 a \sqrt{\cos ^{-1}(a x)}}","-\frac{3 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{2 a}+\frac{3 \sqrt{\frac{\pi }{2}} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{2 a}+x \cos ^{-1}(a x)^{3/2}",1,"-1/2*(Sqrt[(-I)*ArcCos[a*x]]*Gamma[5/2, (-I)*ArcCos[a*x]] + Sqrt[I*ArcCos[a*x]]*Gamma[5/2, I*ArcCos[a*x]])/(a*Sqrt[ArcCos[a*x]])","C",0
85,0,0,15,0.4534964,"\int \frac{\cos ^{-1}(a x)^{3/2}}{x} \, dx","Integrate[ArcCos[a*x]^(3/2)/x,x]","\int \frac{\cos ^{-1}(a x)^{3/2}}{x} \, dx","\text{Int}\left(\frac{\cos ^{-1}(a x)^{3/2}}{x},x\right)",0,"Integrate[ArcCos[a*x]^(3/2)/x, x]","A",-1
86,1,212,298,0.2290669,"\int x^4 \cos ^{-1}(a x)^{5/2} \, dx","Integrate[x^4*ArcCos[a*x]^(5/2),x]","-\frac{-625 \sqrt{3} \left(-i \cos ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{7}{2},-3 i \cos ^{-1}(a x)\right)-27 \sqrt{5} \left(-i \cos ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{7}{2},-5 i \cos ^{-1}(a x)\right)+33750 \sqrt{\cos ^{-1}(a x)^2} \sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{7}{2},i \cos ^{-1}(a x)\right)+33750 \sqrt{i \cos ^{-1}(a x)} \sqrt{\cos ^{-1}(a x)^2} \Gamma \left(\frac{7}{2},-i \cos ^{-1}(a x)\right)-625 \sqrt{3} \left(i \cos ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{7}{2},3 i \cos ^{-1}(a x)\right)-27 \sqrt{5} \left(i \cos ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{7}{2},5 i \cos ^{-1}(a x)\right)}{540000 a^5 \cos ^{-1}(a x)^{3/2}}","\frac{15 \sqrt{\frac{\pi }{2}} C\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{32 a^5}+\frac{\sqrt{\frac{3 \pi }{2}} C\left(\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{320 a^5}+\frac{\sqrt{\frac{\pi }{6}} C\left(\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{60 a^5}+\frac{3 \sqrt{\frac{\pi }{10}} C\left(\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{1600 a^5}-\frac{2 x \sqrt{\cos ^{-1}(a x)}}{5 a^4}-\frac{x^3 \sqrt{\cos ^{-1}(a x)}}{15 a^2}-\frac{x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{10 a}-\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{15 a^5}-\frac{2 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{15 a^3}+\frac{1}{5} x^5 \cos ^{-1}(a x)^{5/2}-\frac{3}{100} x^5 \sqrt{\cos ^{-1}(a x)}",1,"-1/540000*(33750*Sqrt[I*ArcCos[a*x]]*Sqrt[ArcCos[a*x]^2]*Gamma[7/2, (-I)*ArcCos[a*x]] + 33750*Sqrt[(-I)*ArcCos[a*x]]*Sqrt[ArcCos[a*x]^2]*Gamma[7/2, I*ArcCos[a*x]] - 625*Sqrt[3]*((-I)*ArcCos[a*x])^(3/2)*Gamma[7/2, (-3*I)*ArcCos[a*x]] - 625*Sqrt[3]*(I*ArcCos[a*x])^(3/2)*Gamma[7/2, (3*I)*ArcCos[a*x]] - 27*Sqrt[5]*((-I)*ArcCos[a*x])^(3/2)*Gamma[7/2, (-5*I)*ArcCos[a*x]] - 27*Sqrt[5]*(I*ArcCos[a*x])^(3/2)*Gamma[7/2, (5*I)*ArcCos[a*x]])/(a^5*ArcCos[a*x]^(3/2))","C",0
87,1,140,205,0.1548526,"\int x^3 \cos ^{-1}(a x)^{5/2} \, dx","Integrate[x^3*ArcCos[a*x]^(5/2),x]","-\frac{-16 \sqrt{2} \left(-i \cos ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{7}{2},-2 i \cos ^{-1}(a x)\right)-16 \sqrt{2} \left(i \cos ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{7}{2},2 i \cos ^{-1}(a x)\right)+\sqrt{\cos ^{-1}(a x)^2} \left(\sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{7}{2},-4 i \cos ^{-1}(a x)\right)+\sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{7}{2},4 i \cos ^{-1}(a x)\right)\right)}{2048 a^4 \cos ^{-1}(a x)^{3/2}}","\frac{15 \sqrt{\frac{\pi }{2}} C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{4096 a^4}+\frac{15 \sqrt{\pi } C\left(\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right)}{256 a^4}-\frac{3 \cos ^{-1}(a x)^{5/2}}{32 a^4}+\frac{225 \sqrt{\cos ^{-1}(a x)}}{2048 a^4}-\frac{45 x^2 \sqrt{\cos ^{-1}(a x)}}{256 a^2}-\frac{5 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{32 a}-\frac{15 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{64 a^3}+\frac{1}{4} x^4 \cos ^{-1}(a x)^{5/2}-\frac{15}{256} x^4 \sqrt{\cos ^{-1}(a x)}",1,"-1/2048*(-16*Sqrt[2]*((-I)*ArcCos[a*x])^(3/2)*Gamma[7/2, (-2*I)*ArcCos[a*x]] - 16*Sqrt[2]*(I*ArcCos[a*x])^(3/2)*Gamma[7/2, (2*I)*ArcCos[a*x]] + Sqrt[ArcCos[a*x]^2]*(Sqrt[I*ArcCos[a*x]]*Gamma[7/2, (-4*I)*ArcCos[a*x]] + Sqrt[(-I)*ArcCos[a*x]]*Gamma[7/2, (4*I)*ArcCos[a*x]]))/(a^4*ArcCos[a*x]^(3/2))","C",0
88,1,122,178,0.1345754,"\int x^2 \cos ^{-1}(a x)^{5/2} \, dx","Integrate[x^2*ArcCos[a*x]^(5/2),x]","-\frac{81 i \sqrt{\cos ^{-1}(a x)^2} \Gamma \left(\frac{7}{2},-i \cos ^{-1}(a x)\right)+81 \cos ^{-1}(a x) \Gamma \left(\frac{7}{2},i \cos ^{-1}(a x)\right)+\sqrt{3} \left(i \sqrt{\cos ^{-1}(a x)^2} \Gamma \left(\frac{7}{2},-3 i \cos ^{-1}(a x)\right)+\cos ^{-1}(a x) \Gamma \left(\frac{7}{2},3 i \cos ^{-1}(a x)\right)\right)}{648 a^3 \sqrt{i \cos ^{-1}(a x)} \sqrt{\cos ^{-1}(a x)}}","\frac{15 \sqrt{\frac{\pi }{2}} C\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{16 a^3}+\frac{5 \sqrt{\frac{\pi }{6}} C\left(\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{144 a^3}-\frac{5 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{18 a}-\frac{5 x \sqrt{\cos ^{-1}(a x)}}{6 a^2}-\frac{5 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{9 a^3}+\frac{1}{3} x^3 \cos ^{-1}(a x)^{5/2}-\frac{5}{36} x^3 \sqrt{\cos ^{-1}(a x)}",1,"-1/648*((81*I)*Sqrt[ArcCos[a*x]^2]*Gamma[7/2, (-I)*ArcCos[a*x]] + 81*ArcCos[a*x]*Gamma[7/2, I*ArcCos[a*x]] + Sqrt[3]*(I*Sqrt[ArcCos[a*x]^2]*Gamma[7/2, (-3*I)*ArcCos[a*x]] + ArcCos[a*x]*Gamma[7/2, (3*I)*ArcCos[a*x]]))/(a^3*Sqrt[I*ArcCos[a*x]]*Sqrt[ArcCos[a*x]])","C",0
89,1,73,119,0.0997533,"\int x \cos ^{-1}(a x)^{5/2} \, dx","Integrate[x*ArcCos[a*x]^(5/2),x]","\frac{15 \sqrt{\pi } C\left(\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right)-2 \sqrt{\cos ^{-1}(a x)} \left(\left(15-16 \cos ^{-1}(a x)^2\right) \cos \left(2 \cos ^{-1}(a x)\right)+20 \cos ^{-1}(a x) \sin \left(2 \cos ^{-1}(a x)\right)\right)}{128 a^2}","\frac{15 \sqrt{\pi } C\left(\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right)}{128 a^2}-\frac{5 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{8 a}-\frac{\cos ^{-1}(a x)^{5/2}}{4 a^2}+\frac{15 \sqrt{\cos ^{-1}(a x)}}{64 a^2}+\frac{1}{2} x^2 \cos ^{-1}(a x)^{5/2}-\frac{15}{32} x^2 \sqrt{\cos ^{-1}(a x)}",1,"(15*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]] - 2*Sqrt[ArcCos[a*x]]*((15 - 16*ArcCos[a*x]^2)*Cos[2*ArcCos[a*x]] + 20*ArcCos[a*x]*Sin[2*ArcCos[a*x]]))/(128*a^2)","A",1
90,1,76,88,0.0388777,"\int \cos ^{-1}(a x)^{5/2} \, dx","Integrate[ArcCos[a*x]^(5/2),x]","-\frac{\sqrt{\cos ^{-1}(a x)} \left(\sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{7}{2},-i \cos ^{-1}(a x)\right)+\sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{7}{2},i \cos ^{-1}(a x)\right)\right)}{2 a \sqrt{\cos ^{-1}(a x)^2}}","-\frac{5 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{2 a}+\frac{15 \sqrt{\frac{\pi }{2}} C\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{4 a}+x \cos ^{-1}(a x)^{5/2}-\frac{15}{4} x \sqrt{\cos ^{-1}(a x)}",1,"-1/2*(Sqrt[ArcCos[a*x]]*(Sqrt[I*ArcCos[a*x]]*Gamma[7/2, (-I)*ArcCos[a*x]] + Sqrt[(-I)*ArcCos[a*x]]*Gamma[7/2, I*ArcCos[a*x]]))/(a*Sqrt[ArcCos[a*x]^2])","C",0
91,0,0,15,0.4587321,"\int \frac{\cos ^{-1}(a x)^{5/2}}{x} \, dx","Integrate[ArcCos[a*x]^(5/2)/x,x]","\int \frac{\cos ^{-1}(a x)^{5/2}}{x} \, dx","\text{Int}\left(\frac{\cos ^{-1}(a x)^{5/2}}{x},x\right)",0,"Integrate[ArcCos[a*x]^(5/2)/x, x]","A",-1
92,1,192,106,0.1209767,"\int \frac{x^4}{\sqrt{\cos ^{-1}(a x)}} \, dx","Integrate[x^4/Sqrt[ArcCos[a*x]],x]","-\frac{-10 \sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \cos ^{-1}(a x)\right)-10 \sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \cos ^{-1}(a x)\right)-5 \sqrt{3} \sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \cos ^{-1}(a x)\right)-5 \sqrt{3} \sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \cos ^{-1}(a x)\right)-\sqrt{5} \sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},-5 i \cos ^{-1}(a x)\right)-\sqrt{5} \sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},5 i \cos ^{-1}(a x)\right)}{160 a^5 \sqrt{\cos ^{-1}(a x)}}","-\frac{\sqrt{\frac{\pi }{2}} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{4 a^5}-\frac{\sqrt{\frac{3 \pi }{2}} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{8 a^5}-\frac{\sqrt{\frac{\pi }{10}} S\left(\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{8 a^5}",1,"-1/160*(-10*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-I)*ArcCos[a*x]] - 10*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, I*ArcCos[a*x]] - 5*Sqrt[3]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-3*I)*ArcCos[a*x]] - 5*Sqrt[3]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (3*I)*ArcCos[a*x]] - Sqrt[5]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-5*I)*ArcCos[a*x]] - Sqrt[5]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (5*I)*ArcCos[a*x]])/(a^5*Sqrt[ArcCos[a*x]])","C",0
93,1,130,65,0.0905511,"\int \frac{x^3}{\sqrt{\cos ^{-1}(a x)}} \, dx","Integrate[x^3/Sqrt[ArcCos[a*x]],x]","-\frac{-2 \sqrt{2} \sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \cos ^{-1}(a x)\right)-2 \sqrt{2} \sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \cos ^{-1}(a x)\right)-\sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \cos ^{-1}(a x)\right)-\sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \cos ^{-1}(a x)\right)}{32 a^4 \sqrt{\cos ^{-1}(a x)}}","-\frac{\sqrt{\frac{\pi }{2}} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{8 a^4}-\frac{\sqrt{\pi } S\left(\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right)}{4 a^4}",1,"-1/32*(-2*Sqrt[2]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-2*I)*ArcCos[a*x]] - 2*Sqrt[2]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (2*I)*ArcCos[a*x]] - Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-4*I)*ArcCos[a*x]] - Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (4*I)*ArcCos[a*x]])/(a^4*Sqrt[ArcCos[a*x]])","C",0
94,1,126,71,0.0992847,"\int \frac{x^2}{\sqrt{\cos ^{-1}(a x)}} \, dx","Integrate[x^2/Sqrt[ArcCos[a*x]],x]","-\frac{-3 \sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \cos ^{-1}(a x)\right)-3 \sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \cos ^{-1}(a x)\right)-\sqrt{3} \left(\sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \cos ^{-1}(a x)\right)+\sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \cos ^{-1}(a x)\right)\right)}{24 a^3 \sqrt{\cos ^{-1}(a x)}}","-\frac{\sqrt{\frac{\pi }{2}} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{2 a^3}-\frac{\sqrt{\frac{\pi }{6}} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{2 a^3}",1,"-1/24*(-3*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-I)*ArcCos[a*x]] - 3*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, I*ArcCos[a*x]] - Sqrt[3]*(Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-3*I)*ArcCos[a*x]] + Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (3*I)*ArcCos[a*x]]))/(a^3*Sqrt[ArcCos[a*x]])","C",0
95,1,28,28,0.0288141,"\int \frac{x}{\sqrt{\cos ^{-1}(a x)}} \, dx","Integrate[x/Sqrt[ArcCos[a*x]],x]","-\frac{\sqrt{\pi } S\left(\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right)}{2 a^2}","-\frac{\sqrt{\pi } S\left(\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right)}{2 a^2}",1,"-1/2*(Sqrt[Pi]*FresnelS[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])/a^2","A",1
96,1,68,31,0.0315914,"\int \frac{1}{\sqrt{\cos ^{-1}(a x)}} \, dx","Integrate[1/Sqrt[ArcCos[a*x]],x]","-\frac{-\sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \cos ^{-1}(a x)\right)-\sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \cos ^{-1}(a x)\right)}{2 a \sqrt{\cos ^{-1}(a x)}}","-\frac{\sqrt{2 \pi } S\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{a}",1,"-1/2*(-(Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-I)*ArcCos[a*x]]) - Sqrt[I*ArcCos[a*x]]*Gamma[1/2, I*ArcCos[a*x]])/(a*Sqrt[ArcCos[a*x]])","C",0
97,0,0,15,0.3530342,"\int \frac{1}{x \sqrt{\cos ^{-1}(a x)}} \, dx","Integrate[1/(x*Sqrt[ArcCos[a*x]]),x]","\int \frac{1}{x \sqrt{\cos ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{1}{x \sqrt{\cos ^{-1}(a x)}},x\right)",0,"Integrate[1/(x*Sqrt[ArcCos[a*x]]), x]","A",-1
98,0,0,15,3.9762826,"\int \frac{1}{x^2 \sqrt{\cos ^{-1}(a x)}} \, dx","Integrate[1/(x^2*Sqrt[ArcCos[a*x]]),x]","\int \frac{1}{x^2 \sqrt{\cos ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{1}{x^2 \sqrt{\cos ^{-1}(a x)}},x\right)",0,"Integrate[1/(x^2*Sqrt[ArcCos[a*x]]), x]","A",-1
99,1,306,171,0.3493678,"\int \frac{x^6}{\cos ^{-1}(a x)^{3/2}} \, dx","Integrate[x^6/ArcCos[a*x]^(3/2),x]","\frac{i \left(-10 i \sqrt{1-a^2 x^2}-18 i \sin \left(3 \cos ^{-1}(a x)\right)-10 i \sin \left(5 \cos ^{-1}(a x)\right)-2 i \sin \left(7 \cos ^{-1}(a x)\right)+5 \sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \cos ^{-1}(a x)\right)-5 \sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \cos ^{-1}(a x)\right)+9 \sqrt{3} \sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \cos ^{-1}(a x)\right)-9 \sqrt{3} \sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \cos ^{-1}(a x)\right)+5 \sqrt{5} \sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},-5 i \cos ^{-1}(a x)\right)-5 \sqrt{5} \sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},5 i \cos ^{-1}(a x)\right)+\sqrt{7} \sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},-7 i \cos ^{-1}(a x)\right)-\sqrt{7} \sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},7 i \cos ^{-1}(a x)\right)\right)}{64 a^7 \sqrt{\cos ^{-1}(a x)}}","-\frac{5 \sqrt{\frac{\pi }{2}} C\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{16 a^7}-\frac{9 \sqrt{\frac{3 \pi }{2}} C\left(\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{16 a^7}-\frac{5 \sqrt{\frac{5 \pi }{2}} C\left(\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{16 a^7}-\frac{\sqrt{\frac{7 \pi }{2}} C\left(\sqrt{\frac{14}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{16 a^7}+\frac{2 x^6 \sqrt{1-a^2 x^2}}{a \sqrt{\cos ^{-1}(a x)}}",1,"((I/64)*((-10*I)*Sqrt[1 - a^2*x^2] + 5*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-I)*ArcCos[a*x]] - 5*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, I*ArcCos[a*x]] + 9*Sqrt[3]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-3*I)*ArcCos[a*x]] - 9*Sqrt[3]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (3*I)*ArcCos[a*x]] + 5*Sqrt[5]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-5*I)*ArcCos[a*x]] - 5*Sqrt[5]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (5*I)*ArcCos[a*x]] + Sqrt[7]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-7*I)*ArcCos[a*x]] - Sqrt[7]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (7*I)*ArcCos[a*x]] - (18*I)*Sin[3*ArcCos[a*x]] - (10*I)*Sin[5*ArcCos[a*x]] - (2*I)*Sin[7*ArcCos[a*x]]))/(a^7*Sqrt[ArcCos[a*x]])","C",0
100,1,226,127,0.5612939,"\int \frac{x^5}{\cos ^{-1}(a x)^{3/2}} \, dx","Integrate[x^5/ArcCos[a*x]^(3/2),x]","\frac{i \left(-10 i \sin \left(2 \cos ^{-1}(a x)\right)-8 i \sin \left(4 \cos ^{-1}(a x)\right)-2 i \sin \left(6 \cos ^{-1}(a x)\right)+5 \sqrt{2} \sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \cos ^{-1}(a x)\right)-5 \sqrt{2} \sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \cos ^{-1}(a x)\right)+8 \sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \cos ^{-1}(a x)\right)-8 \sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \cos ^{-1}(a x)\right)+\sqrt{6} \sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},-6 i \cos ^{-1}(a x)\right)-\sqrt{6} \sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},6 i \cos ^{-1}(a x)\right)\right)}{32 a^6 \sqrt{\cos ^{-1}(a x)}}","-\frac{\sqrt{\frac{\pi }{2}} C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{a^6}-\frac{\sqrt{3 \pi } C\left(2 \sqrt{\frac{3}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{8 a^6}-\frac{5 \sqrt{\pi } C\left(\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right)}{8 a^6}+\frac{2 x^5 \sqrt{1-a^2 x^2}}{a \sqrt{\cos ^{-1}(a x)}}",1,"((I/32)*(5*Sqrt[2]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-2*I)*ArcCos[a*x]] - 5*Sqrt[2]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (2*I)*ArcCos[a*x]] + 8*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-4*I)*ArcCos[a*x]] - 8*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (4*I)*ArcCos[a*x]] + Sqrt[6]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-6*I)*ArcCos[a*x]] - Sqrt[6]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (6*I)*ArcCos[a*x]] - (10*I)*Sin[2*ArcCos[a*x]] - (8*I)*Sin[4*ArcCos[a*x]] - (2*I)*Sin[6*ArcCos[a*x]]))/(a^6*Sqrt[ArcCos[a*x]])","C",0
101,1,233,136,0.2339486,"\int \frac{x^4}{\cos ^{-1}(a x)^{3/2}} \, dx","Integrate[x^4/ArcCos[a*x]^(3/2),x]","\frac{i \left(-4 i \sqrt{1-a^2 x^2}-6 i \sin \left(3 \cos ^{-1}(a x)\right)-2 i \sin \left(5 \cos ^{-1}(a x)\right)+2 \sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \cos ^{-1}(a x)\right)-2 \sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \cos ^{-1}(a x)\right)+3 \sqrt{3} \sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \cos ^{-1}(a x)\right)-3 \sqrt{3} \sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \cos ^{-1}(a x)\right)+\sqrt{5} \sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},-5 i \cos ^{-1}(a x)\right)-\sqrt{5} \sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},5 i \cos ^{-1}(a x)\right)\right)}{16 a^5 \sqrt{\cos ^{-1}(a x)}}","-\frac{\sqrt{\frac{\pi }{2}} C\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{2 a^5}-\frac{3 \sqrt{\frac{3 \pi }{2}} C\left(\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{4 a^5}-\frac{\sqrt{\frac{5 \pi }{2}} C\left(\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{4 a^5}+\frac{2 x^4 \sqrt{1-a^2 x^2}}{a \sqrt{\cos ^{-1}(a x)}}",1,"((I/16)*((-4*I)*Sqrt[1 - a^2*x^2] + 2*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-I)*ArcCos[a*x]] - 2*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, I*ArcCos[a*x]] + 3*Sqrt[3]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-3*I)*ArcCos[a*x]] - 3*Sqrt[3]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (3*I)*ArcCos[a*x]] + Sqrt[5]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-5*I)*ArcCos[a*x]] - Sqrt[5]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (5*I)*ArcCos[a*x]] - (6*I)*Sin[3*ArcCos[a*x]] - (2*I)*Sin[5*ArcCos[a*x]]))/(a^5*Sqrt[ArcCos[a*x]])","C",0
102,1,154,91,0.4773147,"\int \frac{x^3}{\cos ^{-1}(a x)^{3/2}} \, dx","Integrate[x^3/ArcCos[a*x]^(3/2),x]","\frac{2 \sin \left(2 \cos ^{-1}(a x)\right)+\sin \left(4 \cos ^{-1}(a x)\right)+i \sqrt{2} \sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \cos ^{-1}(a x)\right)-i \sqrt{2} \sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \cos ^{-1}(a x)\right)+i \sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \cos ^{-1}(a x)\right)-i \sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \cos ^{-1}(a x)\right)}{4 a^4 \sqrt{\cos ^{-1}(a x)}}","-\frac{\sqrt{\frac{\pi }{2}} C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{a^4}-\frac{\sqrt{\pi } C\left(\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right)}{a^4}+\frac{2 x^3 \sqrt{1-a^2 x^2}}{a \sqrt{\cos ^{-1}(a x)}}",1,"(I*Sqrt[2]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-2*I)*ArcCos[a*x]] - I*Sqrt[2]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (2*I)*ArcCos[a*x]] + I*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-4*I)*ArcCos[a*x]] - I*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (4*I)*ArcCos[a*x]] + 2*Sin[2*ArcCos[a*x]] + Sin[4*ArcCos[a*x]])/(4*a^4*Sqrt[ArcCos[a*x]])","C",0
103,1,159,97,0.1299467,"\int \frac{x^2}{\cos ^{-1}(a x)^{3/2}} \, dx","Integrate[x^2/ArcCos[a*x]^(3/2),x]","\frac{i \left(-2 i \sqrt{1-a^2 x^2}-2 i \sin \left(3 \cos ^{-1}(a x)\right)+\sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \cos ^{-1}(a x)\right)-\sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \cos ^{-1}(a x)\right)+\sqrt{3} \sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \cos ^{-1}(a x)\right)-\sqrt{3} \sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \cos ^{-1}(a x)\right)\right)}{4 a^3 \sqrt{\cos ^{-1}(a x)}}","-\frac{\sqrt{\frac{\pi }{2}} C\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{a^3}-\frac{\sqrt{\frac{3 \pi }{2}} C\left(\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{a^3}+\frac{2 x^2 \sqrt{1-a^2 x^2}}{a \sqrt{\cos ^{-1}(a x)}}",1,"((I/4)*((-2*I)*Sqrt[1 - a^2*x^2] + Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-I)*ArcCos[a*x]] - Sqrt[I*ArcCos[a*x]]*Gamma[1/2, I*ArcCos[a*x]] + Sqrt[3]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-3*I)*ArcCos[a*x]] - Sqrt[3]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (3*I)*ArcCos[a*x]] - (2*I)*Sin[3*ArcCos[a*x]]))/(a^3*Sqrt[ArcCos[a*x]])","C",0
104,1,44,55,0.0513925,"\int \frac{x}{\cos ^{-1}(a x)^{3/2}} \, dx","Integrate[x/ArcCos[a*x]^(3/2),x]","\frac{\frac{\sin \left(2 \cos ^{-1}(a x)\right)}{\sqrt{\cos ^{-1}(a x)}}-2 \sqrt{\pi } C\left(\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right)}{a^2}","\frac{2 x \sqrt{1-a^2 x^2}}{a \sqrt{\cos ^{-1}(a x)}}-\frac{2 \sqrt{\pi } C\left(\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right)}{a^2}",1,"(-2*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]] + Sin[2*ArcCos[a*x]]/Sqrt[ArcCos[a*x]])/a^2","A",1
105,1,86,59,0.0379041,"\int \frac{1}{\cos ^{-1}(a x)^{3/2}} \, dx","Integrate[ArcCos[a*x]^(-3/2),x]","-\frac{-2 \sqrt{1-a^2 x^2}-i \sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \cos ^{-1}(a x)\right)+i \sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \cos ^{-1}(a x)\right)}{a \sqrt{\cos ^{-1}(a x)}}","\frac{2 \sqrt{1-a^2 x^2}}{a \sqrt{\cos ^{-1}(a x)}}-\frac{2 \sqrt{2 \pi } C\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{a}",1,"-((-2*Sqrt[1 - a^2*x^2] - I*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-I)*ArcCos[a*x]] + I*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, I*ArcCos[a*x]])/(a*Sqrt[ArcCos[a*x]]))","C",0
106,0,0,15,0.513203,"\int \frac{1}{x \cos ^{-1}(a x)^{3/2}} \, dx","Integrate[1/(x*ArcCos[a*x]^(3/2)),x]","\int \frac{1}{x \cos ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{1}{x \cos ^{-1}(a x)^{3/2}},x\right)",0,"Integrate[1/(x*ArcCos[a*x]^(3/2)), x]","A",-1
107,1,322,235,1.8197307,"\int \frac{x^4}{\cos ^{-1}(a x)^{5/2}} \, dx","Integrate[x^4/ArcCos[a*x]^(5/2),x]","-\frac{2 \left(-\sqrt{1-a^2 x^2}-e^{-i \cos ^{-1}(a x)} \cos ^{-1}(a x)-e^{i \cos ^{-1}(a x)} \cos ^{-1}(a x)+\sqrt{-i \cos ^{-1}(a x)} \cos ^{-1}(a x) \Gamma \left(\frac{1}{2},-i \cos ^{-1}(a x)\right)+\sqrt{i \cos ^{-1}(a x)} \cos ^{-1}(a x) \Gamma \left(\frac{1}{2},i \cos ^{-1}(a x)\right)\right)-\sin \left(5 \cos ^{-1}(a x)\right)-5 \cos ^{-1}(a x) \left(e^{-5 i \cos ^{-1}(a x)}+e^{5 i \cos ^{-1}(a x)}-\sqrt{5} \sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},-5 i \cos ^{-1}(a x)\right)-\sqrt{5} \sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},5 i \cos ^{-1}(a x)\right)\right)-3 \left(\sin \left(3 \cos ^{-1}(a x)\right)+3 \cos ^{-1}(a x) \left(e^{-3 i \cos ^{-1}(a x)}+e^{3 i \cos ^{-1}(a x)}-\sqrt{3} \sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \cos ^{-1}(a x)\right)-\sqrt{3} \sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \cos ^{-1}(a x)\right)\right)\right)}{24 a^5 \cos ^{-1}(a x)^{3/2}}","-\frac{4 \sqrt{2 \pi } S\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{a^5}+\frac{25 \sqrt{\frac{\pi }{2}} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{3 a^5}-\frac{4 \sqrt{\frac{2 \pi }{3}} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{a^5}+\frac{25 \sqrt{\frac{\pi }{6}} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{2 a^5}+\frac{5 \sqrt{\frac{5 \pi }{2}} S\left(\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{6 a^5}-\frac{16 x^3}{3 a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{2 x^4 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}+\frac{20 x^5}{3 \sqrt{\cos ^{-1}(a x)}}",1,"-1/24*(2*(-Sqrt[1 - a^2*x^2] - ArcCos[a*x]/E^(I*ArcCos[a*x]) - E^(I*ArcCos[a*x])*ArcCos[a*x] + Sqrt[(-I)*ArcCos[a*x]]*ArcCos[a*x]*Gamma[1/2, (-I)*ArcCos[a*x]] + Sqrt[I*ArcCos[a*x]]*ArcCos[a*x]*Gamma[1/2, I*ArcCos[a*x]]) - 5*ArcCos[a*x]*(E^((-5*I)*ArcCos[a*x]) + E^((5*I)*ArcCos[a*x]) - Sqrt[5]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-5*I)*ArcCos[a*x]] - Sqrt[5]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (5*I)*ArcCos[a*x]]) - 3*(3*ArcCos[a*x]*(E^((-3*I)*ArcCos[a*x]) + E^((3*I)*ArcCos[a*x]) - Sqrt[3]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-3*I)*ArcCos[a*x]] - Sqrt[3]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (3*I)*ArcCos[a*x]]) + Sin[3*ArcCos[a*x]]) - Sin[5*ArcCos[a*x]])/(a^5*ArcCos[a*x]^(3/2))","C",0
108,1,203,126,0.9777503,"\int \frac{x^3}{\cos ^{-1}(a x)^{5/2}} \, dx","Integrate[x^3/ArcCos[a*x]^(5/2),x]","-\frac{-\sin \left(4 \cos ^{-1}(a x)\right)-4 \cos ^{-1}(a x) \left(e^{-4 i \cos ^{-1}(a x)}+e^{4 i \cos ^{-1}(a x)}-2 \sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \cos ^{-1}(a x)\right)-2 \sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \cos ^{-1}(a x)\right)\right)-2 \left(\sin \left(2 \cos ^{-1}(a x)\right)+2 \cos ^{-1}(a x) \left(e^{-2 i \cos ^{-1}(a x)}+e^{2 i \cos ^{-1}(a x)}-\sqrt{2} \sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \cos ^{-1}(a x)\right)-\sqrt{2} \sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \cos ^{-1}(a x)\right)\right)\right)}{12 a^4 \cos ^{-1}(a x)^{3/2}}","\frac{4 \sqrt{2 \pi } S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{3 a^4}+\frac{4 \sqrt{\pi } S\left(\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right)}{3 a^4}-\frac{4 x^2}{a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{2 x^3 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}+\frac{16 x^4}{3 \sqrt{\cos ^{-1}(a x)}}",1,"-1/12*(-4*ArcCos[a*x]*(E^((-4*I)*ArcCos[a*x]) + E^((4*I)*ArcCos[a*x]) - 2*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-4*I)*ArcCos[a*x]] - 2*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (4*I)*ArcCos[a*x]]) - 2*(2*ArcCos[a*x]*(E^((-2*I)*ArcCos[a*x]) + E^((2*I)*ArcCos[a*x]) - Sqrt[2]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-2*I)*ArcCos[a*x]] - Sqrt[2]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (2*I)*ArcCos[a*x]]) + Sin[2*ArcCos[a*x]]) - Sin[4*ArcCos[a*x]])/(a^4*ArcCos[a*x]^(3/2))","C",0
109,1,220,125,0.9071179,"\int \frac{x^2}{\cos ^{-1}(a x)^{5/2}} \, dx","Integrate[x^2/ArcCos[a*x]^(5/2),x]","-\frac{-\sqrt{1-a^2 x^2}-e^{-i \cos ^{-1}(a x)} \cos ^{-1}(a x)-e^{i \cos ^{-1}(a x)} \cos ^{-1}(a x)-\sin \left(3 \cos ^{-1}(a x)\right)+\sqrt{-i \cos ^{-1}(a x)} \cos ^{-1}(a x) \Gamma \left(\frac{1}{2},-i \cos ^{-1}(a x)\right)+\sqrt{i \cos ^{-1}(a x)} \cos ^{-1}(a x) \Gamma \left(\frac{1}{2},i \cos ^{-1}(a x)\right)-3 \cos ^{-1}(a x) \left(e^{-3 i \cos ^{-1}(a x)}+e^{3 i \cos ^{-1}(a x)}-\sqrt{3} \sqrt{-i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \cos ^{-1}(a x)\right)-\sqrt{3} \sqrt{i \cos ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \cos ^{-1}(a x)\right)\right)}{6 a^3 \cos ^{-1}(a x)^{3/2}}","\frac{\sqrt{2 \pi } S\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{3 a^3}+\frac{\sqrt{6 \pi } S\left(\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{a^3}+\frac{2 x^2 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{8 x}{3 a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{4 x^3}{\sqrt{\cos ^{-1}(a x)}}",1,"-1/6*(-Sqrt[1 - a^2*x^2] - ArcCos[a*x]/E^(I*ArcCos[a*x]) - E^(I*ArcCos[a*x])*ArcCos[a*x] + Sqrt[(-I)*ArcCos[a*x]]*ArcCos[a*x]*Gamma[1/2, (-I)*ArcCos[a*x]] + Sqrt[I*ArcCos[a*x]]*ArcCos[a*x]*Gamma[1/2, I*ArcCos[a*x]] - 3*ArcCos[a*x]*(E^((-3*I)*ArcCos[a*x]) + E^((3*I)*ArcCos[a*x]) - Sqrt[3]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-3*I)*ArcCos[a*x]] - Sqrt[3]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (3*I)*ArcCos[a*x]]) - Sin[3*ArcCos[a*x]])/(a^3*ArcCos[a*x]^(3/2))","C",0
110,1,61,89,0.0981817,"\int \frac{x}{\cos ^{-1}(a x)^{5/2}} \, dx","Integrate[x/ArcCos[a*x]^(5/2),x]","\frac{8 \sqrt{\pi } S\left(\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right)+\frac{4 \cos ^{-1}(a x) \cos \left(2 \cos ^{-1}(a x)\right)+\sin \left(2 \cos ^{-1}(a x)\right)}{\cos ^{-1}(a x)^{3/2}}}{3 a^2}","\frac{8 \sqrt{\pi } S\left(\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right)}{3 a^2}+\frac{2 x \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{4}{3 a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{8 x^2}{3 \sqrt{\cos ^{-1}(a x)}}",1,"(8*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]] + (4*ArcCos[a*x]*Cos[2*ArcCos[a*x]] + Sin[2*ArcCos[a*x]])/ArcCos[a*x]^(3/2))/(3*a^2)","A",1
111,1,122,76,0.2615225,"\int \frac{1}{\cos ^{-1}(a x)^{5/2}} \, dx","Integrate[ArcCos[a*x]^(-5/2),x]","-\frac{2 \left(-\sqrt{1-a^2 x^2}-e^{-i \cos ^{-1}(a x)} \cos ^{-1}(a x)-e^{i \cos ^{-1}(a x)} \cos ^{-1}(a x)+\sqrt{-i \cos ^{-1}(a x)} \cos ^{-1}(a x) \Gamma \left(\frac{1}{2},-i \cos ^{-1}(a x)\right)+\sqrt{i \cos ^{-1}(a x)} \cos ^{-1}(a x) \Gamma \left(\frac{1}{2},i \cos ^{-1}(a x)\right)\right)}{3 a \cos ^{-1}(a x)^{3/2}}","\frac{2 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}+\frac{4 \sqrt{2 \pi } S\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{3 a}+\frac{4 x}{3 \sqrt{\cos ^{-1}(a x)}}",1,"(-2*(-Sqrt[1 - a^2*x^2] - ArcCos[a*x]/E^(I*ArcCos[a*x]) - E^(I*ArcCos[a*x])*ArcCos[a*x] + Sqrt[(-I)*ArcCos[a*x]]*ArcCos[a*x]*Gamma[1/2, (-I)*ArcCos[a*x]] + Sqrt[I*ArcCos[a*x]]*ArcCos[a*x]*Gamma[1/2, I*ArcCos[a*x]]))/(3*a*ArcCos[a*x]^(3/2))","C",0
112,0,0,15,0.514671,"\int \frac{1}{x \cos ^{-1}(a x)^{5/2}} \, dx","Integrate[1/(x*ArcCos[a*x]^(5/2)),x]","\int \frac{1}{x \cos ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{1}{x \cos ^{-1}(a x)^{5/2}},x\right)",0,"Integrate[1/(x*ArcCos[a*x]^(5/2)), x]","A",-1
113,1,418,264,8.0617001,"\int \frac{x^4}{\cos ^{-1}(a x)^{7/2}} \, dx","Integrate[x^4/ArcCos[a*x]^(7/2),x]","-\frac{2 \left(-6 \sqrt{1-a^2 x^2}-2 i e^{i \cos ^{-1}(a x)} \cos ^{-1}(a x) \left(2 \cos ^{-1}(a x)-i\right)-4 \cos ^{-1}(a x) \left(-i \cos ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-i \cos ^{-1}(a x)\right)+e^{-i \cos ^{-1}(a x)} \cos ^{-1}(a x) \left(4 i \cos ^{-1}(a x)-4 e^{i \cos ^{-1}(a x)} \left(i \cos ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},i \cos ^{-1}(a x)\right)-2\right)\right)-6 \sin \left(5 \cos ^{-1}(a x)\right)-5 \cos ^{-1}(a x) \left(2 e^{5 i \cos ^{-1}(a x)} \left(1+10 i \cos ^{-1}(a x)\right)+20 \sqrt{5} \left(-i \cos ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-5 i \cos ^{-1}(a x)\right)+e^{-5 i \cos ^{-1}(a x)} \left(-20 i \cos ^{-1}(a x)+20 \sqrt{5} e^{5 i \cos ^{-1}(a x)} \left(i \cos ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},5 i \cos ^{-1}(a x)\right)+2\right)\right)+9 \left(-2 \sin \left(3 \cos ^{-1}(a x)\right)-2 \cos ^{-1}(a x) \left(e^{3 i \cos ^{-1}(a x)} \left(1+6 i \cos ^{-1}(a x)\right)+6 \sqrt{3} \left(-i \cos ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-3 i \cos ^{-1}(a x)\right)+e^{-3 i \cos ^{-1}(a x)} \left(-6 i \cos ^{-1}(a x)+6 \sqrt{3} e^{3 i \cos ^{-1}(a x)} \left(i \cos ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},3 i \cos ^{-1}(a x)\right)+1\right)\right)\right)}{240 a^5 \cos ^{-1}(a x)^{5/2}}","\frac{\sqrt{2 \pi } C\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{15 a^5}-\frac{8 \sqrt{6 \pi } C\left(\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{5 a^5}+\frac{5 \sqrt{\frac{3 \pi }{2}} C\left(\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{a^5}+\frac{5 \sqrt{\frac{5 \pi }{2}} C\left(\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{3 a^5}-\frac{16 x^3}{15 a^2 \cos ^{-1}(a x)^{3/2}}-\frac{40 x^4 \sqrt{1-a^2 x^2}}{3 a \sqrt{\cos ^{-1}(a x)}}+\frac{2 x^4 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}+\frac{32 x^2 \sqrt{1-a^2 x^2}}{5 a^3 \sqrt{\cos ^{-1}(a x)}}+\frac{4 x^5}{3 \cos ^{-1}(a x)^{3/2}}",1,"-1/240*(2*(-6*Sqrt[1 - a^2*x^2] - (2*I)*E^(I*ArcCos[a*x])*ArcCos[a*x]*(-I + 2*ArcCos[a*x]) - 4*((-I)*ArcCos[a*x])^(3/2)*ArcCos[a*x]*Gamma[1/2, (-I)*ArcCos[a*x]] + (ArcCos[a*x]*(-2 + (4*I)*ArcCos[a*x] - 4*E^(I*ArcCos[a*x])*(I*ArcCos[a*x])^(3/2)*Gamma[1/2, I*ArcCos[a*x]]))/E^(I*ArcCos[a*x])) - 5*ArcCos[a*x]*(2*E^((5*I)*ArcCos[a*x])*(1 + (10*I)*ArcCos[a*x]) + 20*Sqrt[5]*((-I)*ArcCos[a*x])^(3/2)*Gamma[1/2, (-5*I)*ArcCos[a*x]] + (2 - (20*I)*ArcCos[a*x] + 20*Sqrt[5]*E^((5*I)*ArcCos[a*x])*(I*ArcCos[a*x])^(3/2)*Gamma[1/2, (5*I)*ArcCos[a*x]])/E^((5*I)*ArcCos[a*x])) + 9*(-2*ArcCos[a*x]*(E^((3*I)*ArcCos[a*x])*(1 + (6*I)*ArcCos[a*x]) + 6*Sqrt[3]*((-I)*ArcCos[a*x])^(3/2)*Gamma[1/2, (-3*I)*ArcCos[a*x]] + (1 - (6*I)*ArcCos[a*x] + 6*Sqrt[3]*E^((3*I)*ArcCos[a*x])*(I*ArcCos[a*x])^(3/2)*Gamma[1/2, (3*I)*ArcCos[a*x]])/E^((3*I)*ArcCos[a*x])) - 2*Sin[3*ArcCos[a*x]]) - 6*Sin[5*ArcCos[a*x]])/(a^5*ArcCos[a*x]^(5/2))","C",0
114,1,264,190,4.3304162,"\int \frac{x^3}{\cos ^{-1}(a x)^{7/2}} \, dx","Integrate[x^3/ArcCos[a*x]^(7/2),x]","-\frac{-4 e^{-2 i \cos ^{-1}(a x)} \left(e^{4 i \cos ^{-1}(a x)} \left(1+4 i \cos ^{-1}(a x)\right)-4 i \cos ^{-1}(a x)+1\right) \cos ^{-1}(a x)-6 \sin \left(2 \cos ^{-1}(a x)\right)-3 \sin \left(4 \cos ^{-1}(a x)\right)+\frac{16 \sqrt{2} \cos ^{-1}(a x)^3 \Gamma \left(\frac{1}{2},-2 i \cos ^{-1}(a x)\right)}{\sqrt{-i \cos ^{-1}(a x)}}-2 \cos ^{-1}(a x) \left(2 e^{4 i \cos ^{-1}(a x)} \left(1+8 i \cos ^{-1}(a x)\right)+32 \left(-i \cos ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-4 i \cos ^{-1}(a x)\right)+2 e^{-4 i \cos ^{-1}(a x)} \left(-8 i \cos ^{-1}(a x)+16 e^{4 i \cos ^{-1}(a x)} \left(i \cos ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},4 i \cos ^{-1}(a x)\right)+1\right)\right)+16 i \sqrt{2} \left(i \cos ^{-1}(a x)\right)^{5/2} \Gamma \left(\frac{1}{2},2 i \cos ^{-1}(a x)\right)}{60 a^4 \cos ^{-1}(a x)^{5/2}}","\frac{32 \sqrt{2 \pi } C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{15 a^4}+\frac{16 \sqrt{\pi } C\left(\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right)}{15 a^4}-\frac{4 x^2}{5 a^2 \cos ^{-1}(a x)^{3/2}}-\frac{128 x^3 \sqrt{1-a^2 x^2}}{15 a \sqrt{\cos ^{-1}(a x)}}+\frac{2 x^3 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}+\frac{16 x \sqrt{1-a^2 x^2}}{5 a^3 \sqrt{\cos ^{-1}(a x)}}+\frac{16 x^4}{15 \cos ^{-1}(a x)^{3/2}}",1,"-1/60*((-4*(1 + E^((4*I)*ArcCos[a*x])*(1 + (4*I)*ArcCos[a*x]) - (4*I)*ArcCos[a*x])*ArcCos[a*x])/E^((2*I)*ArcCos[a*x]) + (16*Sqrt[2]*ArcCos[a*x]^3*Gamma[1/2, (-2*I)*ArcCos[a*x]])/Sqrt[(-I)*ArcCos[a*x]] + (16*I)*Sqrt[2]*(I*ArcCos[a*x])^(5/2)*Gamma[1/2, (2*I)*ArcCos[a*x]] - 2*ArcCos[a*x]*(2*E^((4*I)*ArcCos[a*x])*(1 + (8*I)*ArcCos[a*x]) + 32*((-I)*ArcCos[a*x])^(3/2)*Gamma[1/2, (-4*I)*ArcCos[a*x]] + (2*(1 - (8*I)*ArcCos[a*x] + 16*E^((4*I)*ArcCos[a*x])*(I*ArcCos[a*x])^(3/2)*Gamma[1/2, (4*I)*ArcCos[a*x]]))/E^((4*I)*ArcCos[a*x])) - 6*Sin[2*ArcCos[a*x]] - 3*Sin[4*ArcCos[a*x]])/(a^4*ArcCos[a*x]^(5/2))","C",0
115,1,281,191,2.8640652,"\int \frac{x^2}{\cos ^{-1}(a x)^{7/2}} \, dx","Integrate[x^2/ArcCos[a*x]^(7/2),x]","-\frac{-6 \sqrt{1-a^2 x^2}-2 i e^{i \cos ^{-1}(a x)} \cos ^{-1}(a x) \left(2 \cos ^{-1}(a x)-i\right)-6 \sin \left(3 \cos ^{-1}(a x)\right)-4 \cos ^{-1}(a x) \left(-i \cos ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-i \cos ^{-1}(a x)\right)+e^{-i \cos ^{-1}(a x)} \cos ^{-1}(a x) \left(4 i \cos ^{-1}(a x)-4 e^{i \cos ^{-1}(a x)} \left(i \cos ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},i \cos ^{-1}(a x)\right)-2\right)-6 \cos ^{-1}(a x) \left(e^{3 i \cos ^{-1}(a x)} \left(1+6 i \cos ^{-1}(a x)\right)+6 \sqrt{3} \left(-i \cos ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-3 i \cos ^{-1}(a x)\right)+e^{-3 i \cos ^{-1}(a x)} \left(-6 i \cos ^{-1}(a x)+6 \sqrt{3} e^{3 i \cos ^{-1}(a x)} \left(i \cos ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},3 i \cos ^{-1}(a x)\right)+1\right)\right)}{60 a^3 \cos ^{-1}(a x)^{5/2}}","\frac{2 \sqrt{2 \pi } C\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{15 a^3}+\frac{6 \sqrt{6 \pi } C\left(\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{5 a^3}-\frac{24 x^2 \sqrt{1-a^2 x^2}}{5 a \sqrt{\cos ^{-1}(a x)}}+\frac{2 x^2 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac{8 x}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac{16 \sqrt{1-a^2 x^2}}{15 a^3 \sqrt{\cos ^{-1}(a x)}}+\frac{4 x^3}{5 \cos ^{-1}(a x)^{3/2}}",1,"-1/60*(-6*Sqrt[1 - a^2*x^2] - (2*I)*E^(I*ArcCos[a*x])*ArcCos[a*x]*(-I + 2*ArcCos[a*x]) - 4*((-I)*ArcCos[a*x])^(3/2)*ArcCos[a*x]*Gamma[1/2, (-I)*ArcCos[a*x]] + (ArcCos[a*x]*(-2 + (4*I)*ArcCos[a*x] - 4*E^(I*ArcCos[a*x])*(I*ArcCos[a*x])^(3/2)*Gamma[1/2, I*ArcCos[a*x]]))/E^(I*ArcCos[a*x]) - 6*ArcCos[a*x]*(E^((3*I)*ArcCos[a*x])*(1 + (6*I)*ArcCos[a*x]) + 6*Sqrt[3]*((-I)*ArcCos[a*x])^(3/2)*Gamma[1/2, (-3*I)*ArcCos[a*x]] + (1 - (6*I)*ArcCos[a*x] + 6*Sqrt[3]*E^((3*I)*ArcCos[a*x])*(I*ArcCos[a*x])^(3/2)*Gamma[1/2, (3*I)*ArcCos[a*x]])/E^((3*I)*ArcCos[a*x])) - 6*Sin[3*ArcCos[a*x]])/(a^3*ArcCos[a*x]^(5/2))","C",0
116,1,75,119,0.1042159,"\int \frac{x}{\cos ^{-1}(a x)^{7/2}} \, dx","Integrate[x/ArcCos[a*x]^(7/2),x]","\frac{32 \sqrt{\pi } C\left(\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right)+\frac{4 \cos \left(2 \cos ^{-1}(a x)\right)}{\cos ^{-1}(a x)^{3/2}}-\frac{\left(16 \cos ^{-1}(a x)^2-3\right) \sin \left(2 \cos ^{-1}(a x)\right)}{\cos ^{-1}(a x)^{5/2}}}{15 a^2}","\frac{32 \sqrt{\pi } C\left(\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right)}{15 a^2}-\frac{32 x \sqrt{1-a^2 x^2}}{15 a \sqrt{\cos ^{-1}(a x)}}+\frac{2 x \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac{4}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac{8 x^2}{15 \cos ^{-1}(a x)^{3/2}}",1,"((4*Cos[2*ArcCos[a*x]])/ArcCos[a*x]^(3/2) + 32*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]] - ((-3 + 16*ArcCos[a*x]^2)*Sin[2*ArcCos[a*x]])/ArcCos[a*x]^(5/2))/(15*a^2)","A",1
117,1,151,105,1.1743726,"\int \frac{1}{\cos ^{-1}(a x)^{7/2}} \, dx","Integrate[ArcCos[a*x]^(-7/2),x]","-\frac{-6 \sqrt{1-a^2 x^2}-2 i e^{i \cos ^{-1}(a x)} \cos ^{-1}(a x) \left(2 \cos ^{-1}(a x)-i\right)-4 \cos ^{-1}(a x) \left(-i \cos ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-i \cos ^{-1}(a x)\right)+e^{-i \cos ^{-1}(a x)} \cos ^{-1}(a x) \left(4 i \cos ^{-1}(a x)-4 e^{i \cos ^{-1}(a x)} \left(i \cos ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},i \cos ^{-1}(a x)\right)-2\right)}{15 a \cos ^{-1}(a x)^{5/2}}","-\frac{8 \sqrt{1-a^2 x^2}}{15 a \sqrt{\cos ^{-1}(a x)}}+\frac{2 \sqrt{1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}+\frac{8 \sqrt{2 \pi } C\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right)}{15 a}+\frac{4 x}{15 \cos ^{-1}(a x)^{3/2}}",1,"-1/15*(-6*Sqrt[1 - a^2*x^2] - (2*I)*E^(I*ArcCos[a*x])*ArcCos[a*x]*(-I + 2*ArcCos[a*x]) - 4*((-I)*ArcCos[a*x])^(3/2)*ArcCos[a*x]*Gamma[1/2, (-I)*ArcCos[a*x]] + (ArcCos[a*x]*(-2 + (4*I)*ArcCos[a*x] - 4*E^(I*ArcCos[a*x])*(I*ArcCos[a*x])^(3/2)*Gamma[1/2, I*ArcCos[a*x]]))/E^(I*ArcCos[a*x]))/(a*ArcCos[a*x]^(5/2))","C",0
118,0,0,15,0.5319763,"\int \frac{1}{x \cos ^{-1}(a x)^{7/2}} \, dx","Integrate[1/(x*ArcCos[a*x]^(7/2)),x]","\int \frac{1}{x \cos ^{-1}(a x)^{7/2}} \, dx","\text{Int}\left(\frac{1}{x \cos ^{-1}(a x)^{7/2}},x\right)",0,"Integrate[1/(x*ArcCos[a*x]^(7/2)), x]","A",-1
119,0,0,65,1.3006965,"\int (b x)^m \cos ^{-1}(a x)^4 \, dx","Integrate[(b*x)^m*ArcCos[a*x]^4,x]","\int (b x)^m \cos ^{-1}(a x)^4 \, dx","\frac{4 a \text{Int}\left(\frac{\cos ^{-1}(a x)^3 (b x)^{m+1}}{\sqrt{1-a^2 x^2}},x\right)}{b (m+1)}+\frac{\cos ^{-1}(a x)^4 (b x)^{m+1}}{b (m+1)}",0,"Integrate[(b*x)^m*ArcCos[a*x]^4, x]","A",-1
120,0,0,65,1.18352,"\int (b x)^m \cos ^{-1}(a x)^3 \, dx","Integrate[(b*x)^m*ArcCos[a*x]^3,x]","\int (b x)^m \cos ^{-1}(a x)^3 \, dx","\frac{3 a \text{Int}\left(\frac{\cos ^{-1}(a x)^2 (b x)^{m+1}}{\sqrt{1-a^2 x^2}},x\right)}{b (m+1)}+\frac{\cos ^{-1}(a x)^3 (b x)^{m+1}}{b (m+1)}",0,"Integrate[(b*x)^m*ArcCos[a*x]^3, x]","A",-1
121,1,132,150,2.3796968,"\int (b x)^m \cos ^{-1}(a x)^2 \, dx","Integrate[(b*x)^m*ArcCos[a*x]^2,x]","\frac{x (b x)^m \left(a x \left(\sqrt{\pi } a 2^{-m} x \Gamma (m+2) \, _3\tilde{F}_2\left(1,\frac{m+3}{2},\frac{m+3}{2};\frac{m+4}{2},\frac{m+5}{2};a^2 x^2\right)+\frac{8 \sqrt{1-a^2 x^2} \cos ^{-1}(a x) \, _2F_1\left(1,\frac{m+3}{2};\frac{m+4}{2};a^2 x^2\right)}{m+2}\right)+4 \cos ^{-1}(a x)^2\right)}{4 (m+1)}","\frac{2 a^2 (b x)^{m+3} \, _3F_2\left(1,\frac{m}{2}+\frac{3}{2},\frac{m}{2}+\frac{3}{2};\frac{m}{2}+2,\frac{m}{2}+\frac{5}{2};a^2 x^2\right)}{b^3 (m+1) (m+2) (m+3)}+\frac{2 a \cos ^{-1}(a x) (b x)^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};a^2 x^2\right)}{b^2 (m+1) (m+2)}+\frac{\cos ^{-1}(a x)^2 (b x)^{m+1}}{b (m+1)}",1,"(x*(b*x)^m*(4*ArcCos[a*x]^2 + a*x*((8*Sqrt[1 - a^2*x^2]*ArcCos[a*x]*Hypergeometric2F1[1, (3 + m)/2, (4 + m)/2, a^2*x^2])/(2 + m) + (a*Sqrt[Pi]*x*Gamma[2 + m]*HypergeometricPFQRegularized[{1, (3 + m)/2, (3 + m)/2}, {(4 + m)/2, (5 + m)/2}, a^2*x^2])/2^m)))/(4*(1 + m))","C",0
122,1,54,68,0.0349518,"\int (b x)^m \cos ^{-1}(a x) \, dx","Integrate[(b*x)^m*ArcCos[a*x],x]","\frac{x (b x)^m \left(a x \, _2F_1\left(\frac{1}{2},\frac{m}{2}+1;\frac{m}{2}+2;a^2 x^2\right)+(m+2) \cos ^{-1}(a x)\right)}{(m+1) (m+2)}","\frac{a (b x)^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};a^2 x^2\right)}{b^2 (m+1) (m+2)}+\frac{\cos ^{-1}(a x) (b x)^{m+1}}{b (m+1)}",1,"(x*(b*x)^m*((2 + m)*ArcCos[a*x] + a*x*Hypergeometric2F1[1/2, 1 + m/2, 2 + m/2, a^2*x^2]))/((1 + m)*(2 + m))","A",1
123,0,0,15,0.7098044,"\int \frac{(b x)^m}{\cos ^{-1}(a x)} \, dx","Integrate[(b*x)^m/ArcCos[a*x],x]","\int \frac{(b x)^m}{\cos ^{-1}(a x)} \, dx","\text{Int}\left(\frac{(b x)^m}{\cos ^{-1}(a x)},x\right)",0,"Integrate[(b*x)^m/ArcCos[a*x], x]","A",-1
124,0,0,15,0.7541573,"\int \frac{(b x)^m}{\cos ^{-1}(a x)^2} \, dx","Integrate[(b*x)^m/ArcCos[a*x]^2,x]","\int \frac{(b x)^m}{\cos ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{(b x)^m}{\cos ^{-1}(a x)^2},x\right)",0,"Integrate[(b*x)^m/ArcCos[a*x]^2, x]","A",-1
125,0,0,17,6.4512974,"\int (b x)^m \cos ^{-1}(a x)^{3/2} \, dx","Integrate[(b*x)^m*ArcCos[a*x]^(3/2),x]","\int (b x)^m \cos ^{-1}(a x)^{3/2} \, dx","\text{Int}\left(\cos ^{-1}(a x)^{3/2} (b x)^m,x\right)",0,"Integrate[(b*x)^m*ArcCos[a*x]^(3/2), x]","A",-1
126,0,0,17,6.6815449,"\int (b x)^m \sqrt{\cos ^{-1}(a x)} \, dx","Integrate[(b*x)^m*Sqrt[ArcCos[a*x]],x]","\int (b x)^m \sqrt{\cos ^{-1}(a x)} \, dx","\text{Int}\left(\sqrt{\cos ^{-1}(a x)} (b x)^m,x\right)",0,"Integrate[(b*x)^m*Sqrt[ArcCos[a*x]], x]","A",-1
127,0,0,17,6.2606105,"\int \frac{(b x)^m}{\sqrt{\cos ^{-1}(a x)}} \, dx","Integrate[(b*x)^m/Sqrt[ArcCos[a*x]],x]","\int \frac{(b x)^m}{\sqrt{\cos ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{(b x)^m}{\sqrt{\cos ^{-1}(a x)}},x\right)",0,"Integrate[(b*x)^m/Sqrt[ArcCos[a*x]], x]","A",-1
128,0,0,17,3.7242811,"\int \frac{(b x)^m}{\cos ^{-1}(a x)^{3/2}} \, dx","Integrate[(b*x)^m/ArcCos[a*x]^(3/2),x]","\int \frac{(b x)^m}{\cos ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{(b x)^m}{\cos ^{-1}(a x)^{3/2}},x\right)",0,"Integrate[(b*x)^m/ArcCos[a*x]^(3/2), x]","A",-1
129,0,0,15,1.1238861,"\int (b x)^m \cos ^{-1}(a x)^n \, dx","Integrate[(b*x)^m*ArcCos[a*x]^n,x]","\int (b x)^m \cos ^{-1}(a x)^n \, dx","\text{Int}\left((b x)^m \cos ^{-1}(a x)^n,x\right)",0,"Integrate[(b*x)^m*ArcCos[a*x]^n, x]","A",-1
130,1,130,165,0.1272756,"\int x^3 \cos ^{-1}(a x)^n \, dx","Integrate[x^3*ArcCos[a*x]^n,x]","\frac{2^{-2 (n+3)} \cos ^{-1}(a x)^n \left(\cos ^{-1}(a x)^2\right)^{-n} \left(2^{n+2} \left(-i \cos ^{-1}(a x)\right)^n \Gamma \left(n+1,2 i \cos ^{-1}(a x)\right)+\left(-i \cos ^{-1}(a x)\right)^n \Gamma \left(n+1,4 i \cos ^{-1}(a x)\right)+2^{n+2} \left(i \cos ^{-1}(a x)\right)^n \Gamma \left(n+1,-2 i \cos ^{-1}(a x)\right)+\left(i \cos ^{-1}(a x)\right)^n \Gamma \left(n+1,-4 i \cos ^{-1}(a x)\right)\right)}{a^4}","\frac{2^{-n-4} \cos ^{-1}(a x)^n \left(-i \cos ^{-1}(a x)\right)^{-n} \Gamma \left(n+1,-2 i \cos ^{-1}(a x)\right)}{a^4}+\frac{2^{-2 (n+3)} \cos ^{-1}(a x)^n \left(-i \cos ^{-1}(a x)\right)^{-n} \Gamma \left(n+1,-4 i \cos ^{-1}(a x)\right)}{a^4}+\frac{2^{-n-4} \left(i \cos ^{-1}(a x)\right)^{-n} \cos ^{-1}(a x)^n \Gamma \left(n+1,2 i \cos ^{-1}(a x)\right)}{a^4}+\frac{2^{-2 (n+3)} \left(i \cos ^{-1}(a x)\right)^{-n} \cos ^{-1}(a x)^n \Gamma \left(n+1,4 i \cos ^{-1}(a x)\right)}{a^4}",1,"(ArcCos[a*x]^n*(2^(2 + n)*(I*ArcCos[a*x])^n*Gamma[1 + n, (-2*I)*ArcCos[a*x]] + 2^(2 + n)*((-I)*ArcCos[a*x])^n*Gamma[1 + n, (2*I)*ArcCos[a*x]] + (I*ArcCos[a*x])^n*Gamma[1 + n, (-4*I)*ArcCos[a*x]] + ((-I)*ArcCos[a*x])^n*Gamma[1 + n, (4*I)*ArcCos[a*x]]))/(2^(2*(3 + n))*a^4*(ArcCos[a*x]^2)^n)","A",1
131,1,152,163,0.2183195,"\int x^2 \cos ^{-1}(a x)^n \, dx","Integrate[x^2*ArcCos[a*x]^n,x]","\frac{\frac{1}{8} 3^{-n-1} \cos ^{-1}(a x)^n \left(\cos ^{-1}(a x)^2\right)^{-n} \left(\left(-i \cos ^{-1}(a x)\right)^n \Gamma \left(n+1,3 i \cos ^{-1}(a x)\right)+\left(i \cos ^{-1}(a x)\right)^n \Gamma \left(n+1,-3 i \cos ^{-1}(a x)\right)\right)+\frac{1}{4} \left(\frac{1}{2} \cos ^{-1}(a x)^n \left(-i \cos ^{-1}(a x)\right)^{-n} \Gamma \left(n+1,-i \cos ^{-1}(a x)\right)+\frac{1}{2} \left(i \cos ^{-1}(a x)\right)^{-n} \cos ^{-1}(a x)^n \Gamma \left(n+1,i \cos ^{-1}(a x)\right)\right)}{a^3}","\frac{\cos ^{-1}(a x)^n \left(-i \cos ^{-1}(a x)\right)^{-n} \Gamma \left(n+1,-i \cos ^{-1}(a x)\right)}{8 a^3}+\frac{3^{-n-1} \cos ^{-1}(a x)^n \left(-i \cos ^{-1}(a x)\right)^{-n} \Gamma \left(n+1,-3 i \cos ^{-1}(a x)\right)}{8 a^3}+\frac{\left(i \cos ^{-1}(a x)\right)^{-n} \cos ^{-1}(a x)^n \Gamma \left(n+1,i \cos ^{-1}(a x)\right)}{8 a^3}+\frac{3^{-n-1} \left(i \cos ^{-1}(a x)\right)^{-n} \cos ^{-1}(a x)^n \Gamma \left(n+1,3 i \cos ^{-1}(a x)\right)}{8 a^3}",1,"(((ArcCos[a*x]^n*Gamma[1 + n, (-I)*ArcCos[a*x]])/(2*((-I)*ArcCos[a*x])^n) + (ArcCos[a*x]^n*Gamma[1 + n, I*ArcCos[a*x]])/(2*(I*ArcCos[a*x])^n))/4 + (3^(-1 - n)*ArcCos[a*x]^n*((I*ArcCos[a*x])^n*Gamma[1 + n, (-3*I)*ArcCos[a*x]] + ((-I)*ArcCos[a*x])^n*Gamma[1 + n, (3*I)*ArcCos[a*x]]))/(8*(ArcCos[a*x]^2)^n))/a^3","A",1
132,1,74,83,0.0664356,"\int x \cos ^{-1}(a x)^n \, dx","Integrate[x*ArcCos[a*x]^n,x]","\frac{2^{-n-3} \cos ^{-1}(a x)^n \left(\cos ^{-1}(a x)^2\right)^{-n} \left(\left(-i \cos ^{-1}(a x)\right)^n \Gamma \left(n+1,2 i \cos ^{-1}(a x)\right)+\left(i \cos ^{-1}(a x)\right)^n \Gamma \left(n+1,-2 i \cos ^{-1}(a x)\right)\right)}{a^2}","\frac{2^{-n-3} \cos ^{-1}(a x)^n \left(-i \cos ^{-1}(a x)\right)^{-n} \Gamma \left(n+1,-2 i \cos ^{-1}(a x)\right)}{a^2}+\frac{2^{-n-3} \left(i \cos ^{-1}(a x)\right)^{-n} \cos ^{-1}(a x)^n \Gamma \left(n+1,2 i \cos ^{-1}(a x)\right)}{a^2}",1,"(2^(-3 - n)*ArcCos[a*x]^n*((I*ArcCos[a*x])^n*Gamma[1 + n, (-2*I)*ArcCos[a*x]] + ((-I)*ArcCos[a*x])^n*Gamma[1 + n, (2*I)*ArcCos[a*x]]))/(a^2*(ArcCos[a*x]^2)^n)","A",1
133,1,70,75,0.0399416,"\int \cos ^{-1}(a x)^n \, dx","Integrate[ArcCos[a*x]^n,x]","\frac{\cos ^{-1}(a x)^n \left(\cos ^{-1}(a x)^2\right)^{-n} \left(\left(-i \cos ^{-1}(a x)\right)^n \Gamma \left(n+1,i \cos ^{-1}(a x)\right)+\left(i \cos ^{-1}(a x)\right)^n \Gamma \left(n+1,-i \cos ^{-1}(a x)\right)\right)}{2 a}","\frac{\cos ^{-1}(a x)^n \left(-i \cos ^{-1}(a x)\right)^{-n} \Gamma \left(n+1,-i \cos ^{-1}(a x)\right)}{2 a}+\frac{\left(i \cos ^{-1}(a x)\right)^{-n} \cos ^{-1}(a x)^n \Gamma \left(n+1,i \cos ^{-1}(a x)\right)}{2 a}",1,"(ArcCos[a*x]^n*((I*ArcCos[a*x])^n*Gamma[1 + n, (-I)*ArcCos[a*x]] + ((-I)*ArcCos[a*x])^n*Gamma[1 + n, I*ArcCos[a*x]]))/(2*a*(ArcCos[a*x]^2)^n)","A",1
134,0,0,13,0.3605693,"\int \frac{\cos ^{-1}(a x)^n}{x} \, dx","Integrate[ArcCos[a*x]^n/x,x]","\int \frac{\cos ^{-1}(a x)^n}{x} \, dx","\text{Int}\left(\frac{\cos ^{-1}(a x)^n}{x},x\right)",0,"Integrate[ArcCos[a*x]^n/x, x]","A",-1
135,0,0,13,0.9529765,"\int \frac{\cos ^{-1}(a x)^n}{x^2} \, dx","Integrate[ArcCos[a*x]^n/x^2,x]","\int \frac{\cos ^{-1}(a x)^n}{x^2} \, dx","\text{Int}\left(\frac{\cos ^{-1}(a x)^n}{x^2},x\right)",0,"Integrate[ArcCos[a*x]^n/x^2, x]","A",-1
136,0,0,17,4.7196731,"\int (b x)^{3/2} \cos ^{-1}(a x)^n \, dx","Integrate[(b*x)^(3/2)*ArcCos[a*x]^n,x]","\int (b x)^{3/2} \cos ^{-1}(a x)^n \, dx","\text{Int}\left((b x)^{3/2} \cos ^{-1}(a x)^n,x\right)",0,"Integrate[(b*x)^(3/2)*ArcCos[a*x]^n, x]","A",-1
137,0,0,17,5.9510706,"\int \sqrt{b x} \cos ^{-1}(a x)^n \, dx","Integrate[Sqrt[b*x]*ArcCos[a*x]^n,x]","\int \sqrt{b x} \cos ^{-1}(a x)^n \, dx","\text{Int}\left(\sqrt{b x} \cos ^{-1}(a x)^n,x\right)",0,"Integrate[Sqrt[b*x]*ArcCos[a*x]^n, x]","A",-1
138,0,0,17,1.9191077,"\int \frac{\cos ^{-1}(a x)^n}{\sqrt{b x}} \, dx","Integrate[ArcCos[a*x]^n/Sqrt[b*x],x]","\int \frac{\cos ^{-1}(a x)^n}{\sqrt{b x}} \, dx","\text{Int}\left(\frac{\cos ^{-1}(a x)^n}{\sqrt{b x}},x\right)",0,"Integrate[ArcCos[a*x]^n/Sqrt[b*x], x]","A",-1
139,0,0,17,2.0054059,"\int \frac{\cos ^{-1}(a x)^n}{(b x)^{3/2}} \, dx","Integrate[ArcCos[a*x]^n/(b*x)^(3/2),x]","\int \frac{\cos ^{-1}(a x)^n}{(b x)^{3/2}} \, dx","\text{Int}\left(\frac{\cos ^{-1}(a x)^n}{(b x)^{3/2}},x\right)",0,"Integrate[ArcCos[a*x]^n/(b*x)^(3/2), x]","A",-1
140,1,68,76,0.0566202,"\int x^3 \left(a+b \cos ^{-1}(c x)\right) \, dx","Integrate[x^3*(a + b*ArcCos[c*x]),x]","\frac{a x^4}{4}+\frac{3 b \sin ^{-1}(c x)}{32 c^4}+b \sqrt{1-c^2 x^2} \left(-\frac{3 x}{32 c^3}-\frac{x^3}{16 c}\right)+\frac{1}{4} b x^4 \cos ^{-1}(c x)","\frac{1}{4} x^4 \left(a+b \cos ^{-1}(c x)\right)+\frac{3 b \sin ^{-1}(c x)}{32 c^4}-\frac{b x^3 \sqrt{1-c^2 x^2}}{16 c}-\frac{3 b x \sqrt{1-c^2 x^2}}{32 c^3}",1,"(a*x^4)/4 + b*Sqrt[1 - c^2*x^2]*((-3*x)/(32*c^3) - x^3/(16*c)) + (b*x^4*ArcCos[c*x])/4 + (3*b*ArcSin[c*x])/(32*c^4)","A",1
141,1,55,60,0.0499515,"\int x^2 \left(a+b \cos ^{-1}(c x)\right) \, dx","Integrate[x^2*(a + b*ArcCos[c*x]),x]","\frac{a x^3}{3}+b \left(-\frac{2}{9 c^3}-\frac{x^2}{9 c}\right) \sqrt{1-c^2 x^2}+\frac{1}{3} b x^3 \cos ^{-1}(c x)","\frac{1}{3} x^3 \left(a+b \cos ^{-1}(c x)\right)+\frac{b \left(1-c^2 x^2\right)^{3/2}}{9 c^3}-\frac{b \sqrt{1-c^2 x^2}}{3 c^3}",1,"(a*x^3)/3 + b*(-2/(9*c^3) - x^2/(9*c))*Sqrt[1 - c^2*x^2] + (b*x^3*ArcCos[c*x])/3","A",1
142,1,56,51,0.0414054,"\int x \left(a+b \cos ^{-1}(c x)\right) \, dx","Integrate[x*(a + b*ArcCos[c*x]),x]","\frac{a x^2}{2}-\frac{b x \sqrt{1-c^2 x^2}}{4 c}+\frac{b \sin ^{-1}(c x)}{4 c^2}+\frac{1}{2} b x^2 \cos ^{-1}(c x)","\frac{1}{2} x^2 \left(a+b \cos ^{-1}(c x)\right)-\frac{b x \sqrt{1-c^2 x^2}}{4 c}+\frac{b \sin ^{-1}(c x)}{4 c^2}",1,"(a*x^2)/2 - (b*x*Sqrt[1 - c^2*x^2])/(4*c) + (b*x^2*ArcCos[c*x])/2 + (b*ArcSin[c*x])/(4*c^2)","A",1
143,1,31,31,0.0116493,"\int \left(a+b \cos ^{-1}(c x)\right) \, dx","Integrate[a + b*ArcCos[c*x],x]","a x-\frac{b \sqrt{1-c^2 x^2}}{c}+b x \cos ^{-1}(c x)","a x-\frac{b \sqrt{1-c^2 x^2}}{c}+b x \cos ^{-1}(c x)",1,"a*x - (b*Sqrt[1 - c^2*x^2])/c + b*x*ArcCos[c*x]","A",1
144,1,58,63,0.0185669,"\int \frac{a+b \cos ^{-1}(c x)}{x} \, dx","Integrate[(a + b*ArcCos[c*x])/x,x]","a \log (x)-\frac{1}{2} i b \text{Li}_2\left(-e^{2 i \cos ^{-1}(c x)}\right)-\frac{1}{2} i b \cos ^{-1}(c x)^2+b \cos ^{-1}(c x) \log \left(1+e^{2 i \cos ^{-1}(c x)}\right)","-\frac{i \left(a+b \cos ^{-1}(c x)\right)^2}{2 b}+\log \left(1+e^{2 i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)-\frac{1}{2} i b \text{Li}_2\left(-e^{2 i \cos ^{-1}(c x)}\right)",1,"(-1/2*I)*b*ArcCos[c*x]^2 + b*ArcCos[c*x]*Log[1 + E^((2*I)*ArcCos[c*x])] + a*Log[x] - (I/2)*b*PolyLog[2, -E^((2*I)*ArcCos[c*x])]","A",0
145,1,43,32,0.0133488,"\int \frac{a+b \cos ^{-1}(c x)}{x^2} \, dx","Integrate[(a + b*ArcCos[c*x])/x^2,x]","-\frac{a}{x}+b c \log \left(\sqrt{1-c^2 x^2}+1\right)-b c \log (x)-\frac{b \cos ^{-1}(c x)}{x}","b c \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)-\frac{a+b \cos ^{-1}(c x)}{x}",1,"-(a/x) - (b*ArcCos[c*x])/x - b*c*Log[x] + b*c*Log[1 + Sqrt[1 - c^2*x^2]]","A",1
146,1,44,39,0.0247777,"\int \frac{a+b \cos ^{-1}(c x)}{x^3} \, dx","Integrate[(a + b*ArcCos[c*x])/x^3,x]","-\frac{a}{2 x^2}+\frac{b c \sqrt{1-c^2 x^2}}{2 x}-\frac{b \cos ^{-1}(c x)}{2 x^2}","\frac{b c \sqrt{1-c^2 x^2}}{2 x}-\frac{a+b \cos ^{-1}(c x)}{2 x^2}",1,"-1/2*a/x^2 + (b*c*Sqrt[1 - c^2*x^2])/(2*x) - (b*ArcCos[c*x])/(2*x^2)","A",1
147,1,79,62,0.0290386,"\int \frac{a+b \cos ^{-1}(c x)}{x^4} \, dx","Integrate[(a + b*ArcCos[c*x])/x^4,x]","-\frac{a}{3 x^3}-\frac{1}{6} b c^3 \log (x)+\frac{b c \sqrt{1-c^2 x^2}}{6 x^2}+\frac{1}{6} b c^3 \log \left(\sqrt{1-c^2 x^2}+1\right)-\frac{b \cos ^{-1}(c x)}{3 x^3}","-\frac{a+b \cos ^{-1}(c x)}{3 x^3}+\frac{b c \sqrt{1-c^2 x^2}}{6 x^2}+\frac{1}{6} b c^3 \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)",1,"-1/3*a/x^3 + (b*c*Sqrt[1 - c^2*x^2])/(6*x^2) - (b*ArcCos[c*x])/(3*x^3) - (b*c^3*Log[x])/6 + (b*c^3*Log[1 + Sqrt[1 - c^2*x^2]])/6","A",1
148,1,121,102,0.1569687,"\int x^2 \left(a+b \cos ^{-1}(c x)\right)^2 \, dx","Integrate[x^2*(a + b*ArcCos[c*x])^2,x]","\frac{9 a^2 c^3 x^3-6 a b \sqrt{1-c^2 x^2} \left(c^2 x^2+2\right)-6 b \cos ^{-1}(c x) \left(b \sqrt{1-c^2 x^2} \left(c^2 x^2+2\right)-3 a c^3 x^3\right)+9 b^2 c^3 x^3 \cos ^{-1}(c x)^2-2 b^2 c x \left(c^2 x^2+6\right)}{27 c^3}","-\frac{2 b x^2 \sqrt{1-c^2 x^2} \left(a+b \cos ^{-1}(c x)\right)}{9 c}-\frac{4 b \sqrt{1-c^2 x^2} \left(a+b \cos ^{-1}(c x)\right)}{9 c^3}+\frac{1}{3} x^3 \left(a+b \cos ^{-1}(c x)\right)^2-\frac{4 b^2 x}{9 c^2}-\frac{2}{27} b^2 x^3",1,"(9*a^2*c^3*x^3 - 6*a*b*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2) - 2*b^2*c*x*(6 + c^2*x^2) - 6*b*(-3*a*c^3*x^3 + b*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2))*ArcCos[c*x] + 9*b^2*c^3*x^3*ArcCos[c*x]^2)/(27*c^3)","A",1
149,1,104,76,0.1330638,"\int x \left(a+b \cos ^{-1}(c x)\right)^2 \, dx","Integrate[x*(a + b*ArcCos[c*x])^2,x]","\frac{c x \left(2 a^2 c x-2 a b \sqrt{1-c^2 x^2}-b^2 c x\right)+2 b c x \cos ^{-1}(c x) \left(2 a c x-b \sqrt{1-c^2 x^2}\right)+2 a b \sin ^{-1}(c x)+b^2 \left(2 c^2 x^2-1\right) \cos ^{-1}(c x)^2}{4 c^2}","-\frac{b x \sqrt{1-c^2 x^2} \left(a+b \cos ^{-1}(c x)\right)}{2 c}-\frac{\left(a+b \cos ^{-1}(c x)\right)^2}{4 c^2}+\frac{1}{2} x^2 \left(a+b \cos ^{-1}(c x)\right)^2-\frac{1}{4} b^2 x^2",1,"(c*x*(2*a^2*c*x - b^2*c*x - 2*a*b*Sqrt[1 - c^2*x^2]) + 2*b*c*x*(2*a*c*x - b*Sqrt[1 - c^2*x^2])*ArcCos[c*x] + b^2*(-1 + 2*c^2*x^2)*ArcCos[c*x]^2 + 2*a*b*ArcSin[c*x])/(4*c^2)","A",1
150,1,76,47,0.0716866,"\int \left(a+b \cos ^{-1}(c x)\right)^2 \, dx","Integrate[(a + b*ArcCos[c*x])^2,x]","x \left(a^2-2 b^2\right)-\frac{2 a b \sqrt{1-c^2 x^2}}{c}+\frac{2 b \cos ^{-1}(c x) \left(a c x-b \sqrt{1-c^2 x^2}\right)}{c}+b^2 x \cos ^{-1}(c x)^2","-\frac{2 b \sqrt{1-c^2 x^2} \left(a+b \cos ^{-1}(c x)\right)}{c}+x \left(a+b \cos ^{-1}(c x)\right)^2-2 b^2 x",1,"(a^2 - 2*b^2)*x - (2*a*b*Sqrt[1 - c^2*x^2])/c + (2*b*(a*c*x - b*Sqrt[1 - c^2*x^2])*ArcCos[c*x])/c + b^2*x*ArcCos[c*x]^2","A",1
151,1,128,92,0.1317841,"\int \frac{\left(a+b \cos ^{-1}(c x)\right)^2}{x} \, dx","Integrate[(a + b*ArcCos[c*x])^2/x,x]","a^2 \log (c x)-i b \text{Li}_2\left(-e^{2 i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)-i a b \cos ^{-1}(c x)^2+2 a b \cos ^{-1}(c x) \log \left(1+e^{2 i \cos ^{-1}(c x)}\right)+\frac{1}{2} b^2 \text{Li}_3\left(-e^{2 i \cos ^{-1}(c x)}\right)-\frac{1}{3} i b^2 \cos ^{-1}(c x)^3+b^2 \cos ^{-1}(c x)^2 \log \left(1+e^{2 i \cos ^{-1}(c x)}\right)","-i b \text{Li}_2\left(-e^{2 i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)-\frac{i \left(a+b \cos ^{-1}(c x)\right)^3}{3 b}+\log \left(1+e^{2 i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)^2+\frac{1}{2} b^2 \text{Li}_3\left(-e^{2 i \cos ^{-1}(c x)}\right)",1,"(-I)*a*b*ArcCos[c*x]^2 - (I/3)*b^2*ArcCos[c*x]^3 + 2*a*b*ArcCos[c*x]*Log[1 + E^((2*I)*ArcCos[c*x])] + b^2*ArcCos[c*x]^2*Log[1 + E^((2*I)*ArcCos[c*x])] + a^2*Log[c*x] - I*b*(a + b*ArcCos[c*x])*PolyLog[2, -E^((2*I)*ArcCos[c*x])] + (b^2*PolyLog[3, -E^((2*I)*ArcCos[c*x])])/2","A",0
152,1,134,89,0.2331096,"\int \frac{\left(a+b \cos ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[(a + b*ArcCos[c*x])^2/x^2,x]","-\frac{a^2+2 a b \left(\cos ^{-1}(c x)-c x \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)\right)+b^2 \left(\cos ^{-1}(c x)^2-2 c x \left(i \left(\text{Li}_2\left(-i e^{i \cos ^{-1}(c x)}\right)-\text{Li}_2\left(i e^{i \cos ^{-1}(c x)}\right)\right)+\cos ^{-1}(c x) \left(\log \left(1-i e^{i \cos ^{-1}(c x)}\right)-\log \left(1+i e^{i \cos ^{-1}(c x)}\right)\right)\right)\right)}{x}","-\frac{\left(a+b \cos ^{-1}(c x)\right)^2}{x}-4 i b c \tan ^{-1}\left(e^{i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)+2 i b^2 c \text{Li}_2\left(-i e^{i \cos ^{-1}(c x)}\right)-2 i b^2 c \text{Li}_2\left(i e^{i \cos ^{-1}(c x)}\right)",1,"-((a^2 + 2*a*b*(ArcCos[c*x] - c*x*ArcTanh[Sqrt[1 - c^2*x^2]]) + b^2*(ArcCos[c*x]^2 - 2*c*x*(ArcCos[c*x]*(Log[1 - I*E^(I*ArcCos[c*x])] - Log[1 + I*E^(I*ArcCos[c*x])]) + I*(PolyLog[2, (-I)*E^(I*ArcCos[c*x])] - PolyLog[2, I*E^(I*ArcCos[c*x])]))))/x)","A",0
153,1,218,178,0.2473991,"\int x^2 \left(a+b \cos ^{-1}(c x)\right)^3 \, dx","Integrate[x^2*(a + b*ArcCos[c*x])^3,x]","\frac{9 a^3 c^3 x^3-3 b \cos ^{-1}(c x) \left(-9 a^2 c^3 x^3+6 a b \sqrt{1-c^2 x^2} \left(c^2 x^2+2\right)+2 b^2 c x \left(c^2 x^2+6\right)\right)-9 a^2 b \sqrt{1-c^2 x^2} \left(c^2 x^2+2\right)-6 a b^2 c x \left(c^2 x^2+6\right)-9 b^2 \cos ^{-1}(c x)^2 \left(b \sqrt{1-c^2 x^2} \left(c^2 x^2+2\right)-3 a c^3 x^3\right)+9 b^3 c^3 x^3 \cos ^{-1}(c x)^3+2 b^3 \sqrt{1-c^2 x^2} \left(c^2 x^2+20\right)}{27 c^3}","-\frac{4 a b^2 x}{3 c^2}-\frac{2}{9} b^2 x^3 \left(a+b \cos ^{-1}(c x)\right)-\frac{b x^2 \sqrt{1-c^2 x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{3 c}-\frac{2 b \sqrt{1-c^2 x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{3 c^3}+\frac{1}{3} x^3 \left(a+b \cos ^{-1}(c x)\right)^3-\frac{4 b^3 x \cos ^{-1}(c x)}{3 c^2}-\frac{2 b^3 \left(1-c^2 x^2\right)^{3/2}}{27 c^3}+\frac{14 b^3 \sqrt{1-c^2 x^2}}{9 c^3}",1,"(9*a^3*c^3*x^3 - 9*a^2*b*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2) - 6*a*b^2*c*x*(6 + c^2*x^2) + 2*b^3*Sqrt[1 - c^2*x^2]*(20 + c^2*x^2) - 3*b*(-9*a^2*c^3*x^3 + 6*a*b*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2) + 2*b^2*c*x*(6 + c^2*x^2))*ArcCos[c*x] - 9*b^2*(-3*a*c^3*x^3 + b*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2))*ArcCos[c*x]^2 + 9*b^3*c^3*x^3*ArcCos[c*x]^3)/(27*c^3)","A",1
154,1,185,125,0.1801352,"\int x \left(a+b \cos ^{-1}(c x)\right)^3 \, dx","Integrate[x*(a + b*ArcCos[c*x])^3,x]","\frac{\left(6 a^2 b-3 b^3\right) \sin ^{-1}(c x)-6 b c x \cos ^{-1}(c x) \left(-2 a^2 c x+2 a b \sqrt{1-c^2 x^2}+b^2 c x\right)+c x \left(4 a^3 c x-6 a^2 b \sqrt{1-c^2 x^2}-6 a b^2 c x+3 b^3 \sqrt{1-c^2 x^2}\right)-6 b^2 \cos ^{-1}(c x)^2 \left(-2 a c^2 x^2+a+b c x \sqrt{1-c^2 x^2}\right)+2 b^3 \left(2 c^2 x^2-1\right) \cos ^{-1}(c x)^3}{8 c^2}","-\frac{3}{4} b^2 x^2 \left(a+b \cos ^{-1}(c x)\right)-\frac{3 b x \sqrt{1-c^2 x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{4 c}-\frac{\left(a+b \cos ^{-1}(c x)\right)^3}{4 c^2}+\frac{1}{2} x^2 \left(a+b \cos ^{-1}(c x)\right)^3+\frac{3 b^3 x \sqrt{1-c^2 x^2}}{8 c}-\frac{3 b^3 \sin ^{-1}(c x)}{8 c^2}",1,"(c*x*(4*a^3*c*x - 6*a*b^2*c*x - 6*a^2*b*Sqrt[1 - c^2*x^2] + 3*b^3*Sqrt[1 - c^2*x^2]) - 6*b*c*x*(-2*a^2*c*x + b^2*c*x + 2*a*b*Sqrt[1 - c^2*x^2])*ArcCos[c*x] - 6*b^2*(a - 2*a*c^2*x^2 + b*c*x*Sqrt[1 - c^2*x^2])*ArcCos[c*x]^2 + 2*b^3*(-1 + 2*c^2*x^2)*ArcCos[c*x]^3 + (6*a^2*b - 3*b^3)*ArcSin[c*x])/(8*c^2)","A",1
155,1,128,82,0.1220972,"\int \left(a+b \cos ^{-1}(c x)\right)^3 \, dx","Integrate[(a + b*ArcCos[c*x])^3,x]","\frac{-3 b \left(a^2-2 b^2\right) \sqrt{1-c^2 x^2}+3 b \cos ^{-1}(c x) \left(a^2 c x-2 a b \sqrt{1-c^2 x^2}-2 b^2 c x\right)+a c x \left(a^2-6 b^2\right)+3 b^2 \cos ^{-1}(c x)^2 \left(a c x-b \sqrt{1-c^2 x^2}\right)+b^3 c x \cos ^{-1}(c x)^3}{c}","-6 a b^2 x-\frac{3 b \sqrt{1-c^2 x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{c}+x \left(a+b \cos ^{-1}(c x)\right)^3+\frac{6 b^3 \sqrt{1-c^2 x^2}}{c}-6 b^3 x \cos ^{-1}(c x)",1,"(a*(a^2 - 6*b^2)*c*x - 3*b*(a^2 - 2*b^2)*Sqrt[1 - c^2*x^2] + 3*b*(a^2*c*x - 2*b^2*c*x - 2*a*b*Sqrt[1 - c^2*x^2])*ArcCos[c*x] + 3*b^2*(a*c*x - b*Sqrt[1 - c^2*x^2])*ArcCos[c*x]^2 + b^3*c*x*ArcCos[c*x]^3)/c","A",1
156,1,204,127,0.188148,"\int \frac{\left(a+b \cos ^{-1}(c x)\right)^3}{x} \, dx","Integrate[(a + b*ArcCos[c*x])^3/x,x]","\frac{1}{4} \left(4 a^3 \log (c x)-6 i a^2 b \cos ^{-1}(c x)^2+12 a^2 b \cos ^{-1}(c x) \log \left(1+e^{2 i \cos ^{-1}(c x)}\right)+6 b^2 \text{Li}_3\left(-e^{2 i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)-4 i a b^2 \cos ^{-1}(c x)^3+12 a b^2 \cos ^{-1}(c x)^2 \log \left(1+e^{2 i \cos ^{-1}(c x)}\right)-6 i b \text{Li}_2\left(-e^{2 i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)^2+3 i b^3 \text{Li}_4\left(-e^{2 i \cos ^{-1}(c x)}\right)-i b^3 \cos ^{-1}(c x)^4+4 b^3 \cos ^{-1}(c x)^3 \log \left(1+e^{2 i \cos ^{-1}(c x)}\right)\right)","\frac{3}{2} b^2 \text{Li}_3\left(-e^{2 i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)-\frac{3}{2} i b \text{Li}_2\left(-e^{2 i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)^2-\frac{i \left(a+b \cos ^{-1}(c x)\right)^4}{4 b}+\log \left(1+e^{2 i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)^3+\frac{3}{4} i b^3 \text{Li}_4\left(-e^{2 i \cos ^{-1}(c x)}\right)",1,"((-6*I)*a^2*b*ArcCos[c*x]^2 - (4*I)*a*b^2*ArcCos[c*x]^3 - I*b^3*ArcCos[c*x]^4 + 12*a^2*b*ArcCos[c*x]*Log[1 + E^((2*I)*ArcCos[c*x])] + 12*a*b^2*ArcCos[c*x]^2*Log[1 + E^((2*I)*ArcCos[c*x])] + 4*b^3*ArcCos[c*x]^3*Log[1 + E^((2*I)*ArcCos[c*x])] + 4*a^3*Log[c*x] - (6*I)*b*(a + b*ArcCos[c*x])^2*PolyLog[2, -E^((2*I)*ArcCos[c*x])] + 6*b^2*(a + b*ArcCos[c*x])*PolyLog[3, -E^((2*I)*ArcCos[c*x])] + (3*I)*b^3*PolyLog[4, -E^((2*I)*ArcCos[c*x])])/4","A",0
157,1,308,151,0.3196551,"\int \frac{\left(a+b \cos ^{-1}(c x)\right)^3}{x^2} \, dx","Integrate[(a + b*ArcCos[c*x])^3/x^2,x]","-\frac{a^3}{x}+3 a^2 b c \log \left(\sqrt{1-c^2 x^2}+1\right)-3 a^2 b c \log (x)-\frac{3 a^2 b \cos ^{-1}(c x)}{x}+3 a b^2 c \left(-\frac{\cos ^{-1}(c x)^2}{c x}+2 \left(i \left(\text{Li}_2\left(-i e^{i \cos ^{-1}(c x)}\right)-\text{Li}_2\left(i e^{i \cos ^{-1}(c x)}\right)\right)+\cos ^{-1}(c x) \left(\log \left(1-i e^{i \cos ^{-1}(c x)}\right)-\log \left(1+i e^{i \cos ^{-1}(c x)}\right)\right)\right)\right)+b^3 c \left(-\frac{\cos ^{-1}(c x)^3}{c x}+3 \left(2 i \cos ^{-1}(c x) \left(\text{Li}_2\left(-i e^{i \cos ^{-1}(c x)}\right)-\text{Li}_2\left(i e^{i \cos ^{-1}(c x)}\right)\right)-2 \left(\text{Li}_3\left(-i e^{i \cos ^{-1}(c x)}\right)-\text{Li}_3\left(i e^{i \cos ^{-1}(c x)}\right)\right)+\cos ^{-1}(c x)^2 \left(\log \left(1-i e^{i \cos ^{-1}(c x)}\right)-\log \left(1+i e^{i \cos ^{-1}(c x)}\right)\right)\right)\right)","6 i b^2 c \text{Li}_2\left(-i e^{i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)-6 i b^2 c \text{Li}_2\left(i e^{i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)-\frac{\left(a+b \cos ^{-1}(c x)\right)^3}{x}-6 i b c \tan ^{-1}\left(e^{i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)^2-6 b^3 c \text{Li}_3\left(-i e^{i \cos ^{-1}(c x)}\right)+6 b^3 c \text{Li}_3\left(i e^{i \cos ^{-1}(c x)}\right)",1,"-(a^3/x) - (3*a^2*b*ArcCos[c*x])/x - 3*a^2*b*c*Log[x] + 3*a^2*b*c*Log[1 + Sqrt[1 - c^2*x^2]] + 3*a*b^2*c*(-(ArcCos[c*x]^2/(c*x)) + 2*(ArcCos[c*x]*(Log[1 - I*E^(I*ArcCos[c*x])] - Log[1 + I*E^(I*ArcCos[c*x])]) + I*(PolyLog[2, (-I)*E^(I*ArcCos[c*x])] - PolyLog[2, I*E^(I*ArcCos[c*x])]))) + b^3*c*(-(ArcCos[c*x]^3/(c*x)) + 3*(ArcCos[c*x]^2*(Log[1 - I*E^(I*ArcCos[c*x])] - Log[1 + I*E^(I*ArcCos[c*x])]) + (2*I)*ArcCos[c*x]*(PolyLog[2, (-I)*E^(I*ArcCos[c*x])] - PolyLog[2, I*E^(I*ArcCos[c*x])]) - 2*(PolyLog[3, (-I)*E^(I*ArcCos[c*x])] - PolyLog[3, I*E^(I*ArcCos[c*x])])))","B",0
158,1,91,121,0.1931908,"\int \frac{x^2}{a+b \cos ^{-1}(c x)} \, dx","Integrate[x^2/(a + b*ArcCos[c*x]),x]","-\frac{\sin \left(\frac{a}{b}\right) \left(-\text{Ci}\left(\frac{a}{b}+\cos ^{-1}(c x)\right)\right)-\sin \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\cos ^{-1}(c x)\right)\right)+\cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\cos ^{-1}(c x)\right)+\cos \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\cos ^{-1}(c x)\right)\right)}{4 b c^3}","\frac{\sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \cos ^{-1}(c x)}{b}\right)}{4 b c^3}+\frac{\sin \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \cos ^{-1}(c x)\right)}{b}\right)}{4 b c^3}-\frac{\cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \cos ^{-1}(c x)}{b}\right)}{4 b c^3}-\frac{\cos \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \cos ^{-1}(c x)\right)}{b}\right)}{4 b c^3}",1,"-1/4*(-(CosIntegral[a/b + ArcCos[c*x]]*Sin[a/b]) - CosIntegral[3*(a/b + ArcCos[c*x])]*Sin[(3*a)/b] + Cos[a/b]*SinIntegral[a/b + ArcCos[c*x]] + Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcCos[c*x])])/(b*c^3)","A",1
159,1,56,63,0.0713647,"\int \frac{x}{a+b \cos ^{-1}(c x)} \, dx","Integrate[x/(a + b*ArcCos[c*x]),x]","-\frac{\cos \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 a}{b}+2 \cos ^{-1}(c x)\right)-\sin \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 a}{b}+2 \cos ^{-1}(c x)\right)}{2 b c^2}","\frac{\sin \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \cos ^{-1}(c x)\right)}{b}\right)}{2 b c^2}-\frac{\cos \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \cos ^{-1}(c x)\right)}{b}\right)}{2 b c^2}",1,"-1/2*(-(CosIntegral[(2*a)/b + 2*ArcCos[c*x]]*Sin[(2*a)/b]) + Cos[(2*a)/b]*SinIntegral[(2*a)/b + 2*ArcCos[c*x]])/(b*c^2)","A",1
160,1,46,54,0.0736898,"\int \frac{1}{a+b \cos ^{-1}(c x)} \, dx","Integrate[(a + b*ArcCos[c*x])^(-1),x]","-\frac{\cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\cos ^{-1}(c x)\right)-\sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\cos ^{-1}(c x)\right)}{b c}","\frac{\sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \cos ^{-1}(c x)}{b}\right)}{b c}-\frac{\cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \cos ^{-1}(c x)}{b}\right)}{b c}",1,"-((-(CosIntegral[a/b + ArcCos[c*x]]*Sin[a/b]) + Cos[a/b]*SinIntegral[a/b + ArcCos[c*x]])/(b*c))","A",1
161,0,0,17,0.3341572,"\int \frac{1}{x \left(a+b \cos ^{-1}(c x)\right)} \, dx","Integrate[1/(x*(a + b*ArcCos[c*x])),x]","\int \frac{1}{x \left(a+b \cos ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x \left(a+b \cos ^{-1}(c x)\right)},x\right)",0,"Integrate[1/(x*(a + b*ArcCos[c*x])), x]","A",-1
162,0,0,17,3.9417723,"\int \frac{1}{x^2 \left(a+b \cos ^{-1}(c x)\right)} \, dx","Integrate[1/(x^2*(a + b*ArcCos[c*x])),x]","\int \frac{1}{x^2 \left(a+b \cos ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x^2 \left(a+b \cos ^{-1}(c x)\right)},x\right)",0,"Integrate[1/(x^2*(a + b*ArcCos[c*x])), x]","A",-1
163,1,124,155,0.6131312,"\int \frac{x^2}{\left(a+b \cos ^{-1}(c x)\right)^2} \, dx","Integrate[x^2/(a + b*ArcCos[c*x])^2,x]","-\frac{-\frac{4 b c^2 x^2 \sqrt{1-c^2 x^2}}{a+b \cos ^{-1}(c x)}+\cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\cos ^{-1}(c x)\right)+3 \cos \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\cos ^{-1}(c x)\right)\right)+\sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\cos ^{-1}(c x)\right)+3 \sin \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\cos ^{-1}(c x)\right)\right)}{4 b^2 c^3}","-\frac{\cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \cos ^{-1}(c x)}{b}\right)}{4 b^2 c^3}-\frac{3 \cos \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \cos ^{-1}(c x)\right)}{b}\right)}{4 b^2 c^3}-\frac{\sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \cos ^{-1}(c x)}{b}\right)}{4 b^2 c^3}-\frac{3 \sin \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \cos ^{-1}(c x)\right)}{b}\right)}{4 b^2 c^3}+\frac{x^2 \sqrt{1-c^2 x^2}}{b c \left(a+b \cos ^{-1}(c x)\right)}",1,"-1/4*((-4*b*c^2*x^2*Sqrt[1 - c^2*x^2])/(a + b*ArcCos[c*x]) + Cos[a/b]*CosIntegral[a/b + ArcCos[c*x]] + 3*Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcCos[c*x])] + Sin[a/b]*SinIntegral[a/b + ArcCos[c*x]] + 3*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcCos[c*x])])/(b^2*c^3)","A",1
164,1,80,91,0.2681421,"\int \frac{x}{\left(a+b \cos ^{-1}(c x)\right)^2} \, dx","Integrate[x/(a + b*ArcCos[c*x])^2,x]","\frac{\frac{b c x \sqrt{1-c^2 x^2}}{a+b \cos ^{-1}(c x)}-\cos \left(\frac{2 a}{b}\right) \text{Ci}\left(2 \left(\frac{a}{b}+\cos ^{-1}(c x)\right)\right)-\sin \left(\frac{2 a}{b}\right) \text{Si}\left(2 \left(\frac{a}{b}+\cos ^{-1}(c x)\right)\right)}{b^2 c^2}","-\frac{\cos \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \cos ^{-1}(c x)\right)}{b}\right)}{b^2 c^2}-\frac{\sin \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \cos ^{-1}(c x)\right)}{b}\right)}{b^2 c^2}+\frac{x \sqrt{1-c^2 x^2}}{b c \left(a+b \cos ^{-1}(c x)\right)}",1,"((b*c*x*Sqrt[1 - c^2*x^2])/(a + b*ArcCos[c*x]) - Cos[(2*a)/b]*CosIntegral[2*(a/b + ArcCos[c*x])] - Sin[(2*a)/b]*SinIntegral[2*(a/b + ArcCos[c*x])])/(b^2*c^2)","A",1
165,1,72,86,0.1600701,"\int \frac{1}{\left(a+b \cos ^{-1}(c x)\right)^2} \, dx","Integrate[(a + b*ArcCos[c*x])^(-2),x]","\frac{\frac{b \sqrt{1-c^2 x^2}}{a+b \cos ^{-1}(c x)}-\cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\cos ^{-1}(c x)\right)-\sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\cos ^{-1}(c x)\right)}{b^2 c}","-\frac{\cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \cos ^{-1}(c x)}{b}\right)}{b^2 c}-\frac{\sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \cos ^{-1}(c x)}{b}\right)}{b^2 c}+\frac{\sqrt{1-c^2 x^2}}{b c \left(a+b \cos ^{-1}(c x)\right)}",1,"((b*Sqrt[1 - c^2*x^2])/(a + b*ArcCos[c*x]) - Cos[a/b]*CosIntegral[a/b + ArcCos[c*x]] - Sin[a/b]*SinIntegral[a/b + ArcCos[c*x]])/(b^2*c)","A",1
166,0,0,17,7.1204369,"\int \frac{1}{x \left(a+b \cos ^{-1}(c x)\right)^2} \, dx","Integrate[1/(x*(a + b*ArcCos[c*x])^2),x]","\int \frac{1}{x \left(a+b \cos ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{x \left(a+b \cos ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/(x*(a + b*ArcCos[c*x])^2), x]","A",-1
167,0,0,17,56.1201214,"\int \frac{1}{x^2 \left(a+b \cos ^{-1}(c x)\right)^2} \, dx","Integrate[1/(x^2*(a + b*ArcCos[c*x])^2),x]","\int \frac{1}{x^2 \left(a+b \cos ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{x^2 \left(a+b \cos ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/(x^2*(a + b*ArcCos[c*x])^2), x]","A",-1
168,1,169,197,0.4901797,"\int \frac{x^2}{\left(a+b \cos ^{-1}(c x)\right)^3} \, dx","Integrate[x^2/(a + b*ArcCos[c*x])^3,x]","\frac{\frac{4 b^2 x^2 \sqrt{1-c^2 x^2}}{c \left(a+b \cos ^{-1}(c x)\right)^2}-\frac{\sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\cos ^{-1}(c x)\right)}{c^3}-\frac{9 \sin \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\cos ^{-1}(c x)\right)\right)}{c^3}+\frac{\cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\cos ^{-1}(c x)\right)}{c^3}+\frac{9 \cos \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\cos ^{-1}(c x)\right)\right)}{c^3}-\frac{8 b x}{c^2 \left(a+b \cos ^{-1}(c x)\right)}+\frac{12 b x^3}{a+b \cos ^{-1}(c x)}}{8 b^3}","-\frac{\sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \cos ^{-1}(c x)}{b}\right)}{8 b^3 c^3}-\frac{9 \sin \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \cos ^{-1}(c x)\right)}{b}\right)}{8 b^3 c^3}+\frac{\cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \cos ^{-1}(c x)}{b}\right)}{8 b^3 c^3}+\frac{9 \cos \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \cos ^{-1}(c x)\right)}{b}\right)}{8 b^3 c^3}-\frac{x}{b^2 c^2 \left(a+b \cos ^{-1}(c x)\right)}+\frac{3 x^3}{2 b^2 \left(a+b \cos ^{-1}(c x)\right)}+\frac{x^2 \sqrt{1-c^2 x^2}}{2 b c \left(a+b \cos ^{-1}(c x)\right)^2}",1,"((4*b^2*x^2*Sqrt[1 - c^2*x^2])/(c*(a + b*ArcCos[c*x])^2) - (8*b*x)/(c^2*(a + b*ArcCos[c*x])) + (12*b*x^3)/(a + b*ArcCos[c*x]) - (CosIntegral[a/b + ArcCos[c*x]]*Sin[a/b])/c^3 - (9*CosIntegral[3*(a/b + ArcCos[c*x])]*Sin[(3*a)/b])/c^3 + (Cos[a/b]*SinIntegral[a/b + ArcCos[c*x]])/c^3 + (9*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcCos[c*x])])/c^3)/(8*b^3)","A",1
169,1,107,130,0.3054264,"\int \frac{x}{\left(a+b \cos ^{-1}(c x)\right)^3} \, dx","Integrate[x/(a + b*ArcCos[c*x])^3,x]","\frac{\frac{b^2 c x \sqrt{1-c^2 x^2}}{\left(a+b \cos ^{-1}(c x)\right)^2}+\frac{b \left(2 c^2 x^2-1\right)}{a+b \cos ^{-1}(c x)}-2 \sin \left(\frac{2 a}{b}\right) \text{Ci}\left(2 \left(\frac{a}{b}+\cos ^{-1}(c x)\right)\right)+2 \cos \left(\frac{2 a}{b}\right) \text{Si}\left(2 \left(\frac{a}{b}+\cos ^{-1}(c x)\right)\right)}{2 b^3 c^2}","-\frac{\sin \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \cos ^{-1}(c x)\right)}{b}\right)}{b^3 c^2}+\frac{\cos \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \cos ^{-1}(c x)\right)}{b}\right)}{b^3 c^2}-\frac{1}{2 b^2 c^2 \left(a+b \cos ^{-1}(c x)\right)}+\frac{x^2}{b^2 \left(a+b \cos ^{-1}(c x)\right)}+\frac{x \sqrt{1-c^2 x^2}}{2 b c \left(a+b \cos ^{-1}(c x)\right)^2}",1,"((b^2*c*x*Sqrt[1 - c^2*x^2])/(a + b*ArcCos[c*x])^2 + (b*(-1 + 2*c^2*x^2))/(a + b*ArcCos[c*x]) - 2*CosIntegral[2*(a/b + ArcCos[c*x])]*Sin[(2*a)/b] + 2*Cos[(2*a)/b]*SinIntegral[2*(a/b + ArcCos[c*x])])/(2*b^3*c^2)","A",1
170,1,89,111,0.2472344,"\int \frac{1}{\left(a+b \cos ^{-1}(c x)\right)^3} \, dx","Integrate[(a + b*ArcCos[c*x])^(-3),x]","\frac{\frac{b \left(a c x+b \sqrt{1-c^2 x^2}+b c x \cos ^{-1}(c x)\right)}{\left(a+b \cos ^{-1}(c x)\right)^2}-\sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\cos ^{-1}(c x)\right)+\cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\cos ^{-1}(c x)\right)}{2 b^3 c}","-\frac{\sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \cos ^{-1}(c x)}{b}\right)}{2 b^3 c}+\frac{\cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \cos ^{-1}(c x)}{b}\right)}{2 b^3 c}+\frac{x}{2 b^2 \left(a+b \cos ^{-1}(c x)\right)}+\frac{\sqrt{1-c^2 x^2}}{2 b c \left(a+b \cos ^{-1}(c x)\right)^2}",1,"((b*(a*c*x + b*Sqrt[1 - c^2*x^2] + b*c*x*ArcCos[c*x]))/(a + b*ArcCos[c*x])^2 - CosIntegral[a/b + ArcCos[c*x]]*Sin[a/b] + Cos[a/b]*SinIntegral[a/b + ArcCos[c*x]])/(2*b^3*c)","A",1
171,0,0,17,3.6679933,"\int \frac{1}{x \left(a+b \cos ^{-1}(c x)\right)^3} \, dx","Integrate[1/(x*(a + b*ArcCos[c*x])^3),x]","\int \frac{1}{x \left(a+b \cos ^{-1}(c x)\right)^3} \, dx","\text{Int}\left(\frac{1}{x \left(a+b \cos ^{-1}(c x)\right)^3},x\right)",0,"Integrate[1/(x*(a + b*ArcCos[c*x])^3), x]","A",-1
172,0,0,17,35.4427434,"\int \frac{1}{x^2 \left(a+b \cos ^{-1}(c x)\right)^3} \, dx","Integrate[1/(x^2*(a + b*ArcCos[c*x])^3),x]","\int \frac{1}{x^2 \left(a+b \cos ^{-1}(c x)\right)^3} \, dx","\text{Int}\left(\frac{1}{x^2 \left(a+b \cos ^{-1}(c x)\right)^3},x\right)",0,"Integrate[1/(x^2*(a + b*ArcCos[c*x])^3), x]","A",-1
173,1,243,242,0.534461,"\int x^2 \sqrt{a+b \cos ^{-1}(c x)} \, dx","Integrate[x^2*Sqrt[a + b*ArcCos[c*x]],x]","\frac{e^{-\frac{3 i a}{b}} \sqrt{a+b \cos ^{-1}(c x)} \left(9 e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)+9 e^{\frac{4 i a}{b}} \sqrt{-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)+\sqrt{3} \left(\sqrt{\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{3 i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)+e^{\frac{6 i a}{b}} \sqrt{-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{3 i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)\right)\right)}{72 c^3 \sqrt{\frac{\left(a+b \cos ^{-1}(c x)\right)^2}{b^2}}}","-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{4 c^3}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{b} \cos \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{12 c^3}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{4 c^3}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{b} \sin \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{12 c^3}+\frac{1}{3} x^3 \sqrt{a+b \cos ^{-1}(c x)}",1,"(Sqrt[a + b*ArcCos[c*x]]*(9*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcCos[c*x]))/b] + 9*E^(((4*I)*a)/b)*Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, (I*(a + b*ArcCos[c*x]))/b] + Sqrt[3]*(Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, ((-3*I)*(a + b*ArcCos[c*x]))/b] + E^(((6*I)*a)/b)*Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, ((3*I)*(a + b*ArcCos[c*x]))/b])))/(72*c^3*E^(((3*I)*a)/b)*Sqrt[(a + b*ArcCos[c*x])^2/b^2])","C",0
174,1,123,137,0.2432652,"\int x \sqrt{a+b \cos ^{-1}(c x)} \, dx","Integrate[x*Sqrt[a + b*ArcCos[c*x]],x]","-\frac{\sqrt{\pi } \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{\pi }}\right)+\sqrt{\pi } \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{\pi }}\right)-2 \sqrt{\frac{1}{b}} \cos \left(2 \cos ^{-1}(c x)\right) \sqrt{a+b \cos ^{-1}(c x)}}{8 \sqrt{\frac{1}{b}} c^2}","-\frac{\sqrt{\pi } \sqrt{b} \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right)}{8 c^2}-\frac{\sqrt{\pi } \sqrt{b} \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right)}{8 c^2}-\frac{\sqrt{a+b \cos ^{-1}(c x)}}{4 c^2}+\frac{1}{2} x^2 \sqrt{a+b \cos ^{-1}(c x)}",1,"-1/8*(-2*Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[c*x]]*Cos[2*ArcCos[c*x]] + Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[Pi]] + Sqrt[Pi]*FresnelS[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[Pi]]*Sin[(2*a)/b])/(Sqrt[b^(-1)]*c^2)","A",1
175,1,120,121,0.2465489,"\int \sqrt{a+b \cos ^{-1}(c x)} \, dx","Integrate[Sqrt[a + b*ArcCos[c*x]],x]","-\frac{e^{-\frac{i a}{b}} \sqrt{a+b \cos ^{-1}(c x)} \left(-\frac{\Gamma \left(\frac{3}{2},-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)}{\sqrt{-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}}}-\frac{e^{\frac{2 i a}{b}} \Gamma \left(\frac{3}{2},\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)}{\sqrt{\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}}}\right)}{2 c}","-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{c}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{c}+x \sqrt{a+b \cos ^{-1}(c x)}",1,"-1/2*(Sqrt[a + b*ArcCos[c*x]]*(-(Gamma[3/2, ((-I)*(a + b*ArcCos[c*x]))/b]/Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]) - (E^(((2*I)*a)/b)*Gamma[3/2, (I*(a + b*ArcCos[c*x]))/b])/Sqrt[(I*(a + b*ArcCos[c*x]))/b]))/(c*E^((I*a)/b))","C",0
176,0,0,19,4.2042414,"\int \frac{\sqrt{a+b \cos ^{-1}(c x)}}{x} \, dx","Integrate[Sqrt[a + b*ArcCos[c*x]]/x,x]","\int \frac{\sqrt{a+b \cos ^{-1}(c x)}}{x} \, dx","\text{Int}\left(\frac{\sqrt{a+b \cos ^{-1}(c x)}}{x},x\right)",0,"Integrate[Sqrt[a + b*ArcCos[c*x]]/x, x]","A",-1
177,0,0,19,11.5975064,"\int \frac{\sqrt{a+b \cos ^{-1}(c x)}}{x^2} \, dx","Integrate[Sqrt[a + b*ArcCos[c*x]]/x^2,x]","\int \frac{\sqrt{a+b \cos ^{-1}(c x)}}{x^2} \, dx","\text{Int}\left(\frac{\sqrt{a+b \cos ^{-1}(c x)}}{x^2},x\right)",0,"Integrate[Sqrt[a + b*ArcCos[c*x]]/x^2, x]","A",-1
178,1,589,313,10.1100477,"\int x^2 \left(a+b \cos ^{-1}(c x)\right)^{3/2} \, dx","Integrate[x^2*(a + b*ArcCos[c*x])^(3/2),x]","\frac{b \left(-18 \left(3 \sqrt{1-c^2 x^2}-2 c x \cos ^{-1}(c x)\right) \sqrt{a+b \cos ^{-1}(c x)}-9 \sqrt{2 \pi } \sqrt{\frac{1}{b}} \left(3 b \sin \left(\frac{a}{b}\right)-2 a \cos \left(\frac{a}{b}\right)\right) C\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}\right)-\sqrt{6 \pi } \sqrt{\frac{1}{b}} \left(b \sin \left(\frac{3 a}{b}\right)-2 a \cos \left(\frac{3 a}{b}\right)\right) C\left(\sqrt{\frac{1}{b}} \sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}\right)+9 \sqrt{2 \pi } \sqrt{\frac{1}{b}} \left(2 a \sin \left(\frac{a}{b}\right)+3 b \cos \left(\frac{a}{b}\right)\right) S\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}\right)+\sqrt{6 \pi } \sqrt{\frac{1}{b}} \left(2 a \sin \left(\frac{3 a}{b}\right)+b \cos \left(\frac{3 a}{b}\right)\right) S\left(\sqrt{\frac{1}{b}} \sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}\right)-6 \left(\sin \left(3 \cos ^{-1}(c x)\right)-2 \cos ^{-1}(c x) \cos \left(3 \cos ^{-1}(c x)\right)\right) \sqrt{a+b \cos ^{-1}(c x)}\right)}{144 c^3}+\frac{a e^{-\frac{3 i a}{b}} \sqrt{a+b \cos ^{-1}(c x)} \left(9 e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)+9 e^{\frac{4 i a}{b}} \sqrt{-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)+\sqrt{3} \left(\sqrt{\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{3 i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)+e^{\frac{6 i a}{b}} \sqrt{-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{3 i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)\right)\right)}{72 c^3 \sqrt{\frac{\left(a+b \cos ^{-1}(c x)\right)^2}{b^2}}}","-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{8 c^3}-\frac{\sqrt{\frac{\pi }{6}} b^{3/2} \sin \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{24 c^3}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{8 c^3}+\frac{\sqrt{\frac{\pi }{6}} b^{3/2} \cos \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{24 c^3}-\frac{b x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \cos ^{-1}(c x)}}{6 c}-\frac{b \sqrt{1-c^2 x^2} \sqrt{a+b \cos ^{-1}(c x)}}{3 c^3}+\frac{1}{3} x^3 \left(a+b \cos ^{-1}(c x)\right)^{3/2}",1,"(a*Sqrt[a + b*ArcCos[c*x]]*(9*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcCos[c*x]))/b] + 9*E^(((4*I)*a)/b)*Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, (I*(a + b*ArcCos[c*x]))/b] + Sqrt[3]*(Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, ((-3*I)*(a + b*ArcCos[c*x]))/b] + E^(((6*I)*a)/b)*Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, ((3*I)*(a + b*ArcCos[c*x]))/b])))/(72*c^3*E^(((3*I)*a)/b)*Sqrt[(a + b*ArcCos[c*x])^2/b^2]) + (b*(-18*Sqrt[a + b*ArcCos[c*x]]*(3*Sqrt[1 - c^2*x^2] - 2*c*x*ArcCos[c*x]) + 9*Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelS[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]]]*(3*b*Cos[a/b] + 2*a*Sin[a/b]) - 9*Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]]]*(-2*a*Cos[a/b] + 3*b*Sin[a/b]) + Sqrt[b^(-1)]*Sqrt[6*Pi]*FresnelS[Sqrt[b^(-1)]*Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]]]*(b*Cos[(3*a)/b] + 2*a*Sin[(3*a)/b]) - Sqrt[b^(-1)]*Sqrt[6*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]]]*(-2*a*Cos[(3*a)/b] + b*Sin[(3*a)/b]) - 6*Sqrt[a + b*ArcCos[c*x]]*(-2*ArcCos[c*x]*Cos[3*ArcCos[c*x]] + Sin[3*ArcCos[c*x]])))/(144*c^3)","C",0
179,1,155,172,0.9504581,"\int x \left(a+b \cos ^{-1}(c x)\right)^{3/2} \, dx","Integrate[x*(a + b*ArcCos[c*x])^(3/2),x]","\frac{-3 \sqrt{\pi } b \sin \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{\pi }}\right)+3 \sqrt{\pi } b \cos \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{\pi }}\right)+2 \sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}(c x)} \left(4 a \cos \left(2 \cos ^{-1}(c x)\right)+4 b \cos ^{-1}(c x) \cos \left(2 \cos ^{-1}(c x)\right)-3 b \sin \left(2 \cos ^{-1}(c x)\right)\right)}{32 \sqrt{\frac{1}{b}} c^2}","-\frac{3 \sqrt{\pi } b^{3/2} \sin \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right)}{32 c^2}+\frac{3 \sqrt{\pi } b^{3/2} \cos \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right)}{32 c^2}-\frac{3 b x \sqrt{1-c^2 x^2} \sqrt{a+b \cos ^{-1}(c x)}}{8 c}-\frac{\left(a+b \cos ^{-1}(c x)\right)^{3/2}}{4 c^2}+\frac{1}{2} x^2 \left(a+b \cos ^{-1}(c x)\right)^{3/2}",1,"(3*b*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[Pi]] - 3*b*Sqrt[Pi]*FresnelC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[Pi]]*Sin[(2*a)/b] + 2*Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[c*x]]*(4*a*Cos[2*ArcCos[c*x]] + 4*b*ArcCos[c*x]*Cos[2*ArcCos[c*x]] - 3*b*Sin[2*ArcCos[c*x]]))/(32*Sqrt[b^(-1)]*c^2)","A",1
180,1,295,159,2.6452203,"\int \left(a+b \cos ^{-1}(c x)\right)^{3/2} \, dx","Integrate[(a + b*ArcCos[c*x])^(3/2),x]","\frac{b \left(2 \left(2 c x \cos ^{-1}(c x)-3 \sqrt{1-c^2 x^2}\right) \sqrt{a+b \cos ^{-1}(c x)}-\sqrt{2 \pi } \sqrt{\frac{1}{b}} \left(3 b \sin \left(\frac{a}{b}\right)-2 a \cos \left(\frac{a}{b}\right)\right) C\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}\right)+\sqrt{2 \pi } \sqrt{\frac{1}{b}} \left(2 a \sin \left(\frac{a}{b}\right)+3 b \cos \left(\frac{a}{b}\right)\right) S\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}\right)\right)-2 a e^{-\frac{i a}{b}} \sqrt{a+b \cos ^{-1}(c x)} \left(-\frac{\Gamma \left(\frac{3}{2},-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)}{\sqrt{-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}}}-\frac{e^{\frac{2 i a}{b}} \Gamma \left(\frac{3}{2},\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)}{\sqrt{\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}}}\right)}{4 c}","-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{2 c}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{2 c}-\frac{3 b \sqrt{1-c^2 x^2} \sqrt{a+b \cos ^{-1}(c x)}}{2 c}+x \left(a+b \cos ^{-1}(c x)\right)^{3/2}",1,"((-2*a*Sqrt[a + b*ArcCos[c*x]]*(-(Gamma[3/2, ((-I)*(a + b*ArcCos[c*x]))/b]/Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]) - (E^(((2*I)*a)/b)*Gamma[3/2, (I*(a + b*ArcCos[c*x]))/b])/Sqrt[(I*(a + b*ArcCos[c*x]))/b]))/E^((I*a)/b) + b*(2*Sqrt[a + b*ArcCos[c*x]]*(-3*Sqrt[1 - c^2*x^2] + 2*c*x*ArcCos[c*x]) + Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelS[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]]]*(3*b*Cos[a/b] + 2*a*Sin[a/b]) - Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]]]*(-2*a*Cos[a/b] + 3*b*Sin[a/b])))/(4*c)","C",0
181,0,0,19,3.8569495,"\int \frac{\left(a+b \cos ^{-1}(c x)\right)^{3/2}}{x} \, dx","Integrate[(a + b*ArcCos[c*x])^(3/2)/x,x]","\int \frac{\left(a+b \cos ^{-1}(c x)\right)^{3/2}}{x} \, dx","\text{Int}\left(\frac{\left(a+b \cos ^{-1}(c x)\right)^{3/2}}{x},x\right)",0,"Integrate[(a + b*ArcCos[c*x])^(3/2)/x, x]","A",-1
182,0,0,19,11.1604789,"\int \frac{\left(a+b \cos ^{-1}(c x)\right)^{3/2}}{x^2} \, dx","Integrate[(a + b*ArcCos[c*x])^(3/2)/x^2,x]","\int \frac{\left(a+b \cos ^{-1}(c x)\right)^{3/2}}{x^2} \, dx","\text{Int}\left(\frac{\left(a+b \cos ^{-1}(c x)\right)^{3/2}}{x^2},x\right)",0,"Integrate[(a + b*ArcCos[c*x])^(3/2)/x^2, x]","A",-1
183,1,1002,358,17.2299918,"\int x^2 \left(a+b \cos ^{-1}(c x)\right)^{5/2} \, dx","Integrate[x^2*(a + b*ArcCos[c*x])^(5/2),x]","\frac{e^{-\frac{3 i a}{b}} \sqrt{a+b \cos ^{-1}(c x)} \left(9 e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)+9 e^{\frac{4 i a}{b}} \sqrt{-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)+\sqrt{3} \left(\sqrt{\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{3 i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)+e^{\frac{6 i a}{b}} \sqrt{-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{3 i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)\right)\right) a^2}{72 c^3 \sqrt{\frac{\left(a+b \cos ^{-1}(c x)\right)^2}{b^2}}}+\frac{b \left(-18 \sqrt{a+b \cos ^{-1}(c x)} \left(3 \sqrt{1-c^2 x^2}-2 c x \cos ^{-1}(c x)\right)+9 \sqrt{\frac{1}{b}} \sqrt{2 \pi } S\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}\right) \left(3 b \cos \left(\frac{a}{b}\right)+2 a \sin \left(\frac{a}{b}\right)\right)-9 \sqrt{\frac{1}{b}} \sqrt{2 \pi } C\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}\right) \left(3 b \sin \left(\frac{a}{b}\right)-2 a \cos \left(\frac{a}{b}\right)\right)+\sqrt{\frac{1}{b}} \sqrt{6 \pi } S\left(\sqrt{\frac{1}{b}} \sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}\right) \left(b \cos \left(\frac{3 a}{b}\right)+2 a \sin \left(\frac{3 a}{b}\right)\right)-\sqrt{\frac{1}{b}} \sqrt{6 \pi } C\left(\sqrt{\frac{1}{b}} \sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}\right) \left(b \sin \left(\frac{3 a}{b}\right)-2 a \cos \left(\frac{3 a}{b}\right)\right)-6 \sqrt{a+b \cos ^{-1}(c x)} \left(\sin \left(3 \cos ^{-1}(c x)\right)-2 \cos ^{-1}(c x) \cos \left(3 \cos ^{-1}(c x)\right)\right)\right) a}{72 c^3}-\frac{-\frac{54 \sqrt{a+b \cos ^{-1}(c x)} \left(2 \sqrt{1-c^2 x^2} \left(a-5 b \cos ^{-1}(c x)\right)+b c x \left(4 \cos ^{-1}(c x)^2-15\right)\right)}{\sqrt{\frac{1}{b}}}+27 \sqrt{2 \pi } C\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}\right) \left(\left(4 a^2-15 b^2\right) \cos \left(\frac{a}{b}\right)-12 a b \sin \left(\frac{a}{b}\right)\right)-27 \sqrt{2 \pi } S\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}\right) \left(\left(15 b^2-4 a^2\right) \sin \left(\frac{a}{b}\right)-12 a b \cos \left(\frac{a}{b}\right)\right)+\sqrt{6 \pi } C\left(\sqrt{\frac{1}{b}} \sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}\right) \left(\left(12 a^2-5 b^2\right) \cos \left(\frac{3 a}{b}\right)-12 a b \sin \left(\frac{3 a}{b}\right)\right)-\sqrt{6 \pi } S\left(\sqrt{\frac{1}{b}} \sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}\right) \left(\left(5 b^2-12 a^2\right) \sin \left(\frac{3 a}{b}\right)-12 a b \cos \left(\frac{3 a}{b}\right)\right)-\frac{6 \sqrt{a+b \cos ^{-1}(c x)} \left(b \left(12 \cos ^{-1}(c x)^2-5\right) \cos \left(3 \cos ^{-1}(c x)\right)+2 \left(a-5 b \cos ^{-1}(c x)\right) \sin \left(3 \cos ^{-1}(c x)\right)\right)}{\sqrt{\frac{1}{b}}}}{864 \sqrt{\frac{1}{b}} c^3}","\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{16 c^3}+\frac{5 \sqrt{\frac{\pi }{6}} b^{5/2} \cos \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{144 c^3}+\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{16 c^3}+\frac{5 \sqrt{\frac{\pi }{6}} b^{5/2} \sin \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{144 c^3}-\frac{5 b^2 x \sqrt{a+b \cos ^{-1}(c x)}}{6 c^2}-\frac{5}{36} b^2 x^3 \sqrt{a+b \cos ^{-1}(c x)}-\frac{5 b x^2 \sqrt{1-c^2 x^2} \left(a+b \cos ^{-1}(c x)\right)^{3/2}}{18 c}-\frac{5 b \sqrt{1-c^2 x^2} \left(a+b \cos ^{-1}(c x)\right)^{3/2}}{9 c^3}+\frac{1}{3} x^3 \left(a+b \cos ^{-1}(c x)\right)^{5/2}",1,"(a^2*Sqrt[a + b*ArcCos[c*x]]*(9*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcCos[c*x]))/b] + 9*E^(((4*I)*a)/b)*Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, (I*(a + b*ArcCos[c*x]))/b] + Sqrt[3]*(Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, ((-3*I)*(a + b*ArcCos[c*x]))/b] + E^(((6*I)*a)/b)*Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, ((3*I)*(a + b*ArcCos[c*x]))/b])))/(72*c^3*E^(((3*I)*a)/b)*Sqrt[(a + b*ArcCos[c*x])^2/b^2]) + (a*b*(-18*Sqrt[a + b*ArcCos[c*x]]*(3*Sqrt[1 - c^2*x^2] - 2*c*x*ArcCos[c*x]) + 9*Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelS[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]]]*(3*b*Cos[a/b] + 2*a*Sin[a/b]) - 9*Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]]]*(-2*a*Cos[a/b] + 3*b*Sin[a/b]) + Sqrt[b^(-1)]*Sqrt[6*Pi]*FresnelS[Sqrt[b^(-1)]*Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]]]*(b*Cos[(3*a)/b] + 2*a*Sin[(3*a)/b]) - Sqrt[b^(-1)]*Sqrt[6*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]]]*(-2*a*Cos[(3*a)/b] + b*Sin[(3*a)/b]) - 6*Sqrt[a + b*ArcCos[c*x]]*(-2*ArcCos[c*x]*Cos[3*ArcCos[c*x]] + Sin[3*ArcCos[c*x]])))/(72*c^3) - ((-54*Sqrt[a + b*ArcCos[c*x]]*(2*Sqrt[1 - c^2*x^2]*(a - 5*b*ArcCos[c*x]) + b*c*x*(-15 + 4*ArcCos[c*x]^2)))/Sqrt[b^(-1)] + 27*Sqrt[2*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]]]*((4*a^2 - 15*b^2)*Cos[a/b] - 12*a*b*Sin[a/b]) - 27*Sqrt[2*Pi]*FresnelS[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]]]*(-12*a*b*Cos[a/b] + (-4*a^2 + 15*b^2)*Sin[a/b]) + Sqrt[6*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]]]*((12*a^2 - 5*b^2)*Cos[(3*a)/b] - 12*a*b*Sin[(3*a)/b]) - Sqrt[6*Pi]*FresnelS[Sqrt[b^(-1)]*Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]]]*(-12*a*b*Cos[(3*a)/b] + (-12*a^2 + 5*b^2)*Sin[(3*a)/b]) - (6*Sqrt[a + b*ArcCos[c*x]]*(b*(-5 + 12*ArcCos[c*x]^2)*Cos[3*ArcCos[c*x]] + 2*(a - 5*b*ArcCos[c*x])*Sin[3*ArcCos[c*x]]))/Sqrt[b^(-1)])/(864*Sqrt[b^(-1)]*c^3)","C",0
184,1,201,216,2.1990291,"\int x \left(a+b \cos ^{-1}(c x)\right)^{5/2} \, dx","Integrate[x*(a + b*ArcCos[c*x])^(5/2),x]","\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}(c x)} \left(\left(16 a^2-15 b^2\right) \cos \left(2 \cos ^{-1}(c x)\right)+4 b \cos ^{-1}(c x) \left(8 a \cos \left(2 \cos ^{-1}(c x)\right)-5 b \sin \left(2 \cos ^{-1}(c x)\right)\right)-20 a b \sin \left(2 \cos ^{-1}(c x)\right)+16 b^2 \cos \left(2 \cos ^{-1}(c x)\right) \cos ^{-1}(c x)^2\right)+15 \sqrt{\pi } b^2 \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{\pi }}\right)+15 \sqrt{\pi } b^2 \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{\pi }}\right)}{128 \sqrt{\frac{1}{b}} c^2}","\frac{15 \sqrt{\pi } b^{5/2} \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right)}{128 c^2}+\frac{15 \sqrt{\pi } b^{5/2} \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right)}{128 c^2}+\frac{15 b^2 \sqrt{a+b \cos ^{-1}(c x)}}{64 c^2}-\frac{15}{32} b^2 x^2 \sqrt{a+b \cos ^{-1}(c x)}-\frac{5 b x \sqrt{1-c^2 x^2} \left(a+b \cos ^{-1}(c x)\right)^{3/2}}{8 c}-\frac{\left(a+b \cos ^{-1}(c x)\right)^{5/2}}{4 c^2}+\frac{1}{2} x^2 \left(a+b \cos ^{-1}(c x)\right)^{5/2}",1,"(15*b^2*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[Pi]] + 15*b^2*Sqrt[Pi]*FresnelS[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[Pi]]*Sin[(2*a)/b] + 2*Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[c*x]]*((16*a^2 - 15*b^2)*Cos[2*ArcCos[c*x]] + 16*b^2*ArcCos[c*x]^2*Cos[2*ArcCos[c*x]] - 20*a*b*Sin[2*ArcCos[c*x]] + 4*b*ArcCos[c*x]*(8*a*Cos[2*ArcCos[c*x]] - 5*b*Sin[2*ArcCos[c*x]])))/(128*Sqrt[b^(-1)]*c^2)","A",1
185,1,383,179,4.8514033,"\int \left(a+b \cos ^{-1}(c x)\right)^{5/2} \, dx","Integrate[(a + b*ArcCos[c*x])^(5/2),x]","\frac{b e^{-\frac{i a}{b}} \left(\sqrt{\frac{\pi }{2}} \sqrt{\frac{1}{b}} \left(4 a^2+15 b^2\right) \left(1+e^{\frac{2 i a}{b}}\right) \sqrt{a+b \cos ^{-1}(c x)} C\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}\right)-i \sqrt{\frac{\pi }{2}} \sqrt{\frac{1}{b}} \left(4 a^2+15 b^2\right) \left(-1+e^{\frac{2 i a}{b}}\right) \sqrt{a+b \cos ^{-1}(c x)} S\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}\right)+4 i a^2 \sqrt{-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)-4 i a^2 e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)-2 e^{\frac{i a}{b}} \left(a+b \cos ^{-1}(c x)\right) \left(5 \left(2 a \sqrt{1-c^2 x^2}+3 b c x\right)+\cos ^{-1}(c x) \left(10 b \sqrt{1-c^2 x^2}-8 a c x\right)-4 b c x \cos ^{-1}(c x)^2\right)\right)}{8 c \sqrt{a+b \cos ^{-1}(c x)}}","\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{4 c}+\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{4 c}-\frac{15}{4} b^2 x \sqrt{a+b \cos ^{-1}(c x)}-\frac{5 b \sqrt{1-c^2 x^2} \left(a+b \cos ^{-1}(c x)\right)^{3/2}}{2 c}+x \left(a+b \cos ^{-1}(c x)\right)^{5/2}",1,"(b*(-2*E^((I*a)/b)*(a + b*ArcCos[c*x])*(5*(3*b*c*x + 2*a*Sqrt[1 - c^2*x^2]) + (-8*a*c*x + 10*b*Sqrt[1 - c^2*x^2])*ArcCos[c*x] - 4*b*c*x*ArcCos[c*x]^2) + Sqrt[b^(-1)]*(4*a^2 + 15*b^2)*(1 + E^(((2*I)*a)/b))*Sqrt[Pi/2]*Sqrt[a + b*ArcCos[c*x]]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]]] - I*Sqrt[b^(-1)]*(4*a^2 + 15*b^2)*(-1 + E^(((2*I)*a)/b))*Sqrt[Pi/2]*Sqrt[a + b*ArcCos[c*x]]*FresnelS[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]]] + (4*I)*a^2*Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcCos[c*x]))/b] - (4*I)*a^2*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, (I*(a + b*ArcCos[c*x]))/b]))/(8*c*E^((I*a)/b)*Sqrt[a + b*ArcCos[c*x]])","C",0
186,0,0,19,3.8799386,"\int \frac{\left(a+b \cos ^{-1}(c x)\right)^{5/2}}{x} \, dx","Integrate[(a + b*ArcCos[c*x])^(5/2)/x,x]","\int \frac{\left(a+b \cos ^{-1}(c x)\right)^{5/2}}{x} \, dx","\text{Int}\left(\frac{\left(a+b \cos ^{-1}(c x)\right)^{5/2}}{x},x\right)",0,"Integrate[(a + b*ArcCos[c*x])^(5/2)/x, x]","A",-1
187,0,0,19,10.9885659,"\int \frac{\left(a+b \cos ^{-1}(c x)\right)^{5/2}}{x^2} \, dx","Integrate[(a + b*ArcCos[c*x])^(5/2)/x^2,x]","\int \frac{\left(a+b \cos ^{-1}(c x)\right)^{5/2}}{x^2} \, dx","\text{Int}\left(\frac{\left(a+b \cos ^{-1}(c x)\right)^{5/2}}{x^2},x\right)",0,"Integrate[(a + b*ArcCos[c*x])^(5/2)/x^2, x]","A",-1
188,1,225,223,0.462436,"\int \frac{x^2}{\sqrt{a+b \cos ^{-1}(c x)}} \, dx","Integrate[x^2/Sqrt[a + b*ArcCos[c*x]],x]","\frac{e^{-\frac{3 i a}{b}} \left(3 e^{\frac{2 i a}{b}} \sqrt{-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)+3 e^{\frac{4 i a}{b}} \sqrt{\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)+\sqrt{3} \left(\sqrt{-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)+e^{\frac{6 i a}{b}} \sqrt{\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{3 i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)\right)\right)}{24 c^3 \sqrt{a+b \cos ^{-1}(c x)}}","\frac{\sqrt{\frac{\pi }{2}} \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{2 \sqrt{b} c^3}+\frac{\sqrt{\frac{\pi }{6}} \sin \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{2 \sqrt{b} c^3}-\frac{\sqrt{\frac{\pi }{2}} \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{2 \sqrt{b} c^3}-\frac{\sqrt{\frac{\pi }{6}} \cos \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{2 \sqrt{b} c^3}",1,"(3*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcCos[c*x]))/b] + 3*E^(((4*I)*a)/b)*Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, (I*(a + b*ArcCos[c*x]))/b] + Sqrt[3]*(Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, ((-3*I)*(a + b*ArcCos[c*x]))/b] + E^(((6*I)*a)/b)*Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, ((3*I)*(a + b*ArcCos[c*x]))/b]))/(24*c^3*E^(((3*I)*a)/b)*Sqrt[a + b*ArcCos[c*x]])","C",0
189,1,91,99,0.1738986,"\int \frac{x}{\sqrt{a+b \cos ^{-1}(c x)}} \, dx","Integrate[x/Sqrt[a + b*ArcCos[c*x]],x]","-\frac{\sqrt{\pi } \sqrt{\frac{1}{b}} \left(\cos \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{\pi }}\right)-\sin \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{\pi }}\right)\right)}{2 c^2}","\frac{\sqrt{\pi } \sin \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right)}{2 \sqrt{b} c^2}-\frac{\sqrt{\pi } \cos \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right)}{2 \sqrt{b} c^2}",1,"-1/2*(Sqrt[b^(-1)]*Sqrt[Pi]*(Cos[(2*a)/b]*FresnelS[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[Pi]] - FresnelC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[Pi]]*Sin[(2*a)/b]))/c^2","A",1
190,1,118,102,0.0934231,"\int \frac{1}{\sqrt{a+b \cos ^{-1}(c x)}} \, dx","Integrate[1/Sqrt[a + b*ArcCos[c*x]],x]","\frac{e^{-\frac{i a}{b}} \left(\sqrt{-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)+e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)\right)}{2 c \sqrt{a+b \cos ^{-1}(c x)}}","\frac{\sqrt{2 \pi } \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{\sqrt{b} c}-\frac{\sqrt{2 \pi } \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{\sqrt{b} c}",1,"(Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcCos[c*x]))/b] + E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, (I*(a + b*ArcCos[c*x]))/b])/(2*c*E^((I*a)/b)*Sqrt[a + b*ArcCos[c*x]])","C",0
191,0,0,19,3.9953654,"\int \frac{1}{x \sqrt{a+b \cos ^{-1}(c x)}} \, dx","Integrate[1/(x*Sqrt[a + b*ArcCos[c*x]]),x]","\int \frac{1}{x \sqrt{a+b \cos ^{-1}(c x)}} \, dx","\text{Int}\left(\frac{1}{x \sqrt{a+b \cos ^{-1}(c x)}},x\right)",0,"Integrate[1/(x*Sqrt[a + b*ArcCos[c*x]]), x]","A",-1
192,0,0,19,13.6629076,"\int \frac{1}{x^2 \sqrt{a+b \cos ^{-1}(c x)}} \, dx","Integrate[1/(x^2*Sqrt[a + b*ArcCos[c*x]]),x]","\int \frac{1}{x^2 \sqrt{a+b \cos ^{-1}(c x)}} \, dx","\text{Int}\left(\frac{1}{x^2 \sqrt{a+b \cos ^{-1}(c x)}},x\right)",0,"Integrate[1/(x^2*Sqrt[a + b*ArcCos[c*x]]), x]","A",-1
193,1,273,252,0.5708932,"\int \frac{x^2}{\left(a+b \cos ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[x^2/(a + b*ArcCos[c*x])^(3/2),x]","\frac{e^{-\frac{3 i a}{b}} \left(8 c^2 x^2 e^{\frac{3 i a}{b}} \sqrt{1-c^2 x^2}+i e^{\frac{2 i a}{b}} \sqrt{-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)-i e^{\frac{4 i a}{b}} \sqrt{\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)+i \sqrt{3} \sqrt{-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)-i \sqrt{3} e^{\frac{6 i a}{b}} \sqrt{\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{3 i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)\right)}{4 b c^3 \sqrt{a+b \cos ^{-1}(c x)}}","-\frac{\sqrt{\frac{\pi }{2}} \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c^3}-\frac{\sqrt{\frac{3 \pi }{2}} \cos \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c^3}-\frac{\sqrt{\frac{\pi }{2}} \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c^3}-\frac{\sqrt{\frac{3 \pi }{2}} \sin \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c^3}+\frac{2 x^2 \sqrt{1-c^2 x^2}}{b c \sqrt{a+b \cos ^{-1}(c x)}}",1,"(8*c^2*E^(((3*I)*a)/b)*x^2*Sqrt[1 - c^2*x^2] + I*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcCos[c*x]))/b] - I*E^(((4*I)*a)/b)*Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, (I*(a + b*ArcCos[c*x]))/b] + I*Sqrt[3]*Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, ((-3*I)*(a + b*ArcCos[c*x]))/b] - I*Sqrt[3]*E^(((6*I)*a)/b)*Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, ((3*I)*(a + b*ArcCos[c*x]))/b])/(4*b*c^3*E^(((3*I)*a)/b)*Sqrt[a + b*ArcCos[c*x]])","C",0
194,1,124,130,0.3679492,"\int \frac{x}{\left(a+b \cos ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[x/(a + b*ArcCos[c*x])^(3/2),x]","\frac{-2 \sqrt{\pi } \left(\frac{1}{b}\right)^{3/2} \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{\pi }}\right)-2 \sqrt{\pi } \left(\frac{1}{b}\right)^{3/2} \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{\pi }}\right)+\frac{\sin \left(2 \cos ^{-1}(c x)\right)}{b \sqrt{a+b \cos ^{-1}(c x)}}}{c^2}","-\frac{2 \sqrt{\pi } \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right)}{b^{3/2} c^2}-\frac{2 \sqrt{\pi } \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right)}{b^{3/2} c^2}+\frac{2 x \sqrt{1-c^2 x^2}}{b c \sqrt{a+b \cos ^{-1}(c x)}}",1,"(-2*(b^(-1))^(3/2)*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[Pi]] - 2*(b^(-1))^(3/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[Pi]]*Sin[(2*a)/b] + Sin[2*ArcCos[c*x]]/(b*Sqrt[a + b*ArcCos[c*x]]))/c^2","A",1
195,1,150,137,0.1879734,"\int \frac{1}{\left(a+b \cos ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(a + b*ArcCos[c*x])^(-3/2),x]","-\frac{i e^{-\frac{i a}{b}} \left(2 i e^{\frac{i a}{b}} \sqrt{1-c^2 x^2}-\sqrt{-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)+e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)\right)}{b c \sqrt{a+b \cos ^{-1}(c x)}}","-\frac{2 \sqrt{2 \pi } \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c}-\frac{2 \sqrt{2 \pi } \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c}+\frac{2 \sqrt{1-c^2 x^2}}{b c \sqrt{a+b \cos ^{-1}(c x)}}",1,"((-I)*((2*I)*E^((I*a)/b)*Sqrt[1 - c^2*x^2] - Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcCos[c*x]))/b] + E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, (I*(a + b*ArcCos[c*x]))/b]))/(b*c*E^((I*a)/b)*Sqrt[a + b*ArcCos[c*x]])","C",0
196,0,0,19,4.7202551,"\int \frac{1}{x \left(a+b \cos ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[1/(x*(a + b*ArcCos[c*x])^(3/2)),x]","\int \frac{1}{x \left(a+b \cos ^{-1}(c x)\right)^{3/2}} \, dx","\text{Int}\left(\frac{1}{x \left(a+b \cos ^{-1}(c x)\right)^{3/2}},x\right)",0,"Integrate[1/(x*(a + b*ArcCos[c*x])^(3/2)), x]","A",-1
197,0,0,19,13.9432472,"\int \frac{1}{x^2 \left(a+b \cos ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[1/(x^2*(a + b*ArcCos[c*x])^(3/2)),x]","\int \frac{1}{x^2 \left(a+b \cos ^{-1}(c x)\right)^{3/2}} \, dx","\text{Int}\left(\frac{1}{x^2 \left(a+b \cos ^{-1}(c x)\right)^{3/2}},x\right)",0,"Integrate[1/(x^2*(a + b*ArcCos[c*x])^(3/2)), x]","A",-1
198,1,322,292,2.8831624,"\int \frac{x^2}{\left(a+b \cos ^{-1}(c x)\right)^{5/2}} \, dx","Integrate[x^2/(a + b*ArcCos[c*x])^(5/2),x]","-\frac{-\left(a+b \cos ^{-1}(c x)\right) \left(-e^{-\frac{i a}{b}} \sqrt{-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)-e^{\frac{i a}{b}} \sqrt{\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)+e^{-i \cos ^{-1}(c x)}+e^{i \cos ^{-1}(c x)}\right)-3 \left(a+b \cos ^{-1}(c x)\right) \left(-\sqrt{3} e^{-\frac{3 i a}{b}} \sqrt{-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)-\sqrt{3} e^{\frac{3 i a}{b}} \sqrt{\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{3 i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)+e^{-3 i \cos ^{-1}(c x)}+e^{3 i \cos ^{-1}(c x)}\right)-b \sqrt{1-c^2 x^2}+b \left(-\sin \left(3 \cos ^{-1}(c x)\right)\right)}{6 b^2 c^3 \left(a+b \cos ^{-1}(c x)\right)^{3/2}}","-\frac{\sqrt{2 \pi } \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{3 b^{5/2} c^3}-\frac{\sqrt{6 \pi } \sin \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{5/2} c^3}+\frac{\sqrt{2 \pi } \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{3 b^{5/2} c^3}+\frac{\sqrt{6 \pi } \cos \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{5/2} c^3}-\frac{8 x}{3 b^2 c^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{4 x^3}{b^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{2 x^2 \sqrt{1-c^2 x^2}}{3 b c \left(a+b \cos ^{-1}(c x)\right)^{3/2}}",1,"-1/6*(-(b*Sqrt[1 - c^2*x^2]) - (a + b*ArcCos[c*x])*(E^((-I)*ArcCos[c*x]) + E^(I*ArcCos[c*x]) - (Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcCos[c*x]))/b])/E^((I*a)/b) - E^((I*a)/b)*Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, (I*(a + b*ArcCos[c*x]))/b]) - 3*(a + b*ArcCos[c*x])*(E^((-3*I)*ArcCos[c*x]) + E^((3*I)*ArcCos[c*x]) - (Sqrt[3]*Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, ((-3*I)*(a + b*ArcCos[c*x]))/b])/E^(((3*I)*a)/b) - Sqrt[3]*E^(((3*I)*a)/b)*Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, ((3*I)*(a + b*ArcCos[c*x]))/b]) - b*Sin[3*ArcCos[c*x]])/(b^2*c^3*(a + b*ArcCos[c*x])^(3/2))","C",0
199,1,176,180,0.7644265,"\int \frac{x}{\left(a+b \cos ^{-1}(c x)\right)^{5/2}} \, dx","Integrate[x/(a + b*ArcCos[c*x])^(5/2),x]","\frac{-8 \sqrt{\pi } \sqrt{\frac{1}{b}} \sin \left(\frac{2 a}{b}\right) \left(a+b \cos ^{-1}(c x)\right)^{3/2} C\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{\pi }}\right)+8 \sqrt{\pi } \sqrt{\frac{1}{b}} \cos \left(\frac{2 a}{b}\right) \left(a+b \cos ^{-1}(c x)\right)^{3/2} S\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{\pi }}\right)+4 a \cos \left(2 \cos ^{-1}(c x)\right)+4 b \cos ^{-1}(c x) \cos \left(2 \cos ^{-1}(c x)\right)+b \sin \left(2 \cos ^{-1}(c x)\right)}{3 b^2 c^2 \left(a+b \cos ^{-1}(c x)\right)^{3/2}}","-\frac{8 \sqrt{\pi } \sin \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right)}{3 b^{5/2} c^2}+\frac{8 \sqrt{\pi } \cos \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right)}{3 b^{5/2} c^2}-\frac{4}{3 b^2 c^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{8 x^2}{3 b^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{2 x \sqrt{1-c^2 x^2}}{3 b c \left(a+b \cos ^{-1}(c x)\right)^{3/2}}",1,"(4*a*Cos[2*ArcCos[c*x]] + 4*b*ArcCos[c*x]*Cos[2*ArcCos[c*x]] + 8*Sqrt[b^(-1)]*Sqrt[Pi]*(a + b*ArcCos[c*x])^(3/2)*Cos[(2*a)/b]*FresnelS[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[Pi]] - 8*Sqrt[b^(-1)]*Sqrt[Pi]*(a + b*ArcCos[c*x])^(3/2)*FresnelC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[Pi]]*Sin[(2*a)/b] + b*Sin[2*ArcCos[c*x]])/(3*b^2*c^2*(a + b*ArcCos[c*x])^(3/2))","A",1
200,1,194,163,1.7254594,"\int \frac{1}{\left(a+b \cos ^{-1}(c x)\right)^{5/2}} \, dx","Integrate[(a + b*ArcCos[c*x])^(-5/2),x]","\frac{2 \left(e^{-\frac{i a}{b}} \left(a+b \cos ^{-1}(c x)\right) \left(e^{\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}}-\sqrt{-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)\right)-e^{-i \cos ^{-1}(c x)} \left(a+b \cos ^{-1}(c x)\right) \left(-1+e^{\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \sqrt{\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \cos ^{-1}(c x)\right)}{b}\right)\right)+b \sqrt{1-c^2 x^2}\right)}{3 b^2 c \left(a+b \cos ^{-1}(c x)\right)^{3/2}}","-\frac{4 \sqrt{2 \pi } \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{3 b^{5/2} c}+\frac{4 \sqrt{2 \pi } \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right)}{3 b^{5/2} c}+\frac{4 x}{3 b^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{2 \sqrt{1-c^2 x^2}}{3 b c \left(a+b \cos ^{-1}(c x)\right)^{3/2}}",1,"(2*(b*Sqrt[1 - c^2*x^2] + ((a + b*ArcCos[c*x])*(E^((I*(a + b*ArcCos[c*x]))/b) - Sqrt[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcCos[c*x]))/b]))/E^((I*a)/b) - ((a + b*ArcCos[c*x])*(-1 + E^((I*(a + b*ArcCos[c*x]))/b)*Sqrt[(I*(a + b*ArcCos[c*x]))/b]*Gamma[1/2, (I*(a + b*ArcCos[c*x]))/b]))/E^(I*ArcCos[c*x])))/(3*b^2*c*(a + b*ArcCos[c*x])^(3/2))","C",0
201,0,0,19,4.8887482,"\int \frac{1}{x \left(a+b \cos ^{-1}(c x)\right)^{5/2}} \, dx","Integrate[1/(x*(a + b*ArcCos[c*x])^(5/2)),x]","\int \frac{1}{x \left(a+b \cos ^{-1}(c x)\right)^{5/2}} \, dx","\text{Int}\left(\frac{1}{x \left(a+b \cos ^{-1}(c x)\right)^{5/2}},x\right)",0,"Integrate[1/(x*(a + b*ArcCos[c*x])^(5/2)), x]","A",-1
202,0,0,19,14.414935,"\int \frac{1}{x^2 \left(a+b \cos ^{-1}(c x)\right)^{5/2}} \, dx","Integrate[1/(x^2*(a + b*ArcCos[c*x])^(5/2)),x]","\int \frac{1}{x^2 \left(a+b \cos ^{-1}(c x)\right)^{5/2}} \, dx","\text{Int}\left(\frac{1}{x^2 \left(a+b \cos ^{-1}(c x)\right)^{5/2}},x\right)",0,"Integrate[1/(x^2*(a + b*ArcCos[c*x])^(5/2)), x]","A",-1
203,1,158,120,0.2861121,"\int (d x)^{5/2} \left(a+b \cos ^{-1}(c x)\right) \, dx","Integrate[(d*x)^(5/2)*(a + b*ArcCos[c*x]),x]","\frac{2 d^2 \sqrt{d x} \left(21 a c^3 x^3 \sqrt{1-c^2 x^2}+6 b c^4 x^4+4 b c^2 x^2+\frac{10 i b \sqrt{x} \sqrt{1-\frac{1}{c^2 x^2}} F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{-\frac{1}{c}}}{\sqrt{x}}\right)\right|-1\right)}{\sqrt{-\frac{1}{c}}}+21 b c^3 x^3 \sqrt{1-c^2 x^2} \cos ^{-1}(c x)-10 b\right)}{147 c^3 \sqrt{1-c^2 x^2}}","\frac{2 (d x)^{7/2} \left(a+b \cos ^{-1}(c x)\right)}{7 d}+\frac{20 b d^{5/2} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right)\right|-1\right)}{147 c^{7/2}}-\frac{4 b \sqrt{1-c^2 x^2} (d x)^{5/2}}{49 c}-\frac{20 b d^2 \sqrt{1-c^2 x^2} \sqrt{d x}}{147 c^3}",1,"(2*d^2*Sqrt[d*x]*(-10*b + 4*b*c^2*x^2 + 6*b*c^4*x^4 + 21*a*c^3*x^3*Sqrt[1 - c^2*x^2] + 21*b*c^3*x^3*Sqrt[1 - c^2*x^2]*ArcCos[c*x] + ((10*I)*b*Sqrt[1 - 1/(c^2*x^2)]*Sqrt[x]*EllipticF[I*ArcSinh[Sqrt[-c^(-1)]/Sqrt[x]], -1])/Sqrt[-c^(-1)]))/(147*c^3*Sqrt[1 - c^2*x^2])","C",1
204,1,66,124,0.0829325,"\int (d x)^{3/2} \left(a+b \cos ^{-1}(c x)\right) \, dx","Integrate[(d*x)^(3/2)*(a + b*ArcCos[c*x]),x]","\frac{2 (d x)^{3/2} \left(5 a c x+2 b \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};c^2 x^2\right)-2 b \sqrt{1-c^2 x^2}+5 b c x \cos ^{-1}(c x)\right)}{25 c}","\frac{2 (d x)^{5/2} \left(a+b \cos ^{-1}(c x)\right)}{5 d}-\frac{12 b d^{3/2} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right)\right|-1\right)}{25 c^{5/2}}+\frac{12 b d^{3/2} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right)\right|-1\right)}{25 c^{5/2}}-\frac{4 b \sqrt{1-c^2 x^2} (d x)^{3/2}}{25 c}",1,"(2*(d*x)^(3/2)*(5*a*c*x - 2*b*Sqrt[1 - c^2*x^2] + 5*b*c*x*ArcCos[c*x] + 2*b*Hypergeometric2F1[1/2, 3/4, 7/4, c^2*x^2]))/(25*c)","C",1
205,1,113,88,0.2119547,"\int \sqrt{d x} \left(a+b \cos ^{-1}(c x)\right) \, dx","Integrate[Sqrt[d*x]*(a + b*ArcCos[c*x]),x]","\frac{2}{9} \sqrt{d x} \left(3 a x-\frac{2 b \sqrt{1-c^2 x^2}}{c}-\frac{2 i b \sqrt{-\frac{1}{c}} \sqrt{x} \sqrt{1-\frac{1}{c^2 x^2}} F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{-\frac{1}{c}}}{\sqrt{x}}\right)\right|-1\right)}{\sqrt{1-c^2 x^2}}+3 b x \cos ^{-1}(c x)\right)","\frac{2 (d x)^{3/2} \left(a+b \cos ^{-1}(c x)\right)}{3 d}+\frac{4 b \sqrt{d} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right)\right|-1\right)}{9 c^{3/2}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{d x}}{9 c}",1,"(2*Sqrt[d*x]*(3*a*x - (2*b*Sqrt[1 - c^2*x^2])/c + 3*b*x*ArcCos[c*x] - ((2*I)*b*Sqrt[-c^(-1)]*Sqrt[1 - 1/(c^2*x^2)]*Sqrt[x]*EllipticF[I*ArcSinh[Sqrt[-c^(-1)]/Sqrt[x]], -1])/Sqrt[1 - c^2*x^2]))/9","C",1
206,1,45,89,0.0362734,"\int \frac{a+b \cos ^{-1}(c x)}{\sqrt{d x}} \, dx","Integrate[(a + b*ArcCos[c*x])/Sqrt[d*x],x]","\frac{2 x \left(3 \left(a+b \cos ^{-1}(c x)\right)+2 b c x \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};c^2 x^2\right)\right)}{3 \sqrt{d x}}","\frac{2 \sqrt{d x} \left(a+b \cos ^{-1}(c x)\right)}{d}-\frac{4 b F\left(\left.\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right)\right|-1\right)}{\sqrt{c} \sqrt{d}}+\frac{4 b E\left(\left.\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right)\right|-1\right)}{\sqrt{c} \sqrt{d}}",1,"(2*x*(3*(a + b*ArcCos[c*x]) + 2*b*c*x*Hypergeometric2F1[1/2, 3/4, 7/4, c^2*x^2]))/(3*Sqrt[d*x])","C",1
207,1,93,55,0.1651253,"\int \frac{a+b \cos ^{-1}(c x)}{(d x)^{3/2}} \, dx","Integrate[(a + b*ArcCos[c*x])/(d*x)^(3/2),x]","\frac{2 x \left(-a+\frac{2 i b \sqrt{-\frac{1}{c}} c^2 x^{3/2} \sqrt{1-\frac{1}{c^2 x^2}} F\left(\left.i \sinh ^{-1}\left(\frac{\sqrt{-\frac{1}{c}}}{\sqrt{x}}\right)\right|-1\right)}{\sqrt{1-c^2 x^2}}-b \cos ^{-1}(c x)\right)}{(d x)^{3/2}}","-\frac{2 \left(a+b \cos ^{-1}(c x)\right)}{d \sqrt{d x}}-\frac{4 b \sqrt{c} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right)\right|-1\right)}{d^{3/2}}",1,"(2*x*(-a - b*ArcCos[c*x] + ((2*I)*b*Sqrt[-c^(-1)]*c^2*Sqrt[1 - 1/(c^2*x^2)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c^(-1)]/Sqrt[x]], -1])/Sqrt[1 - c^2*x^2]))/(d*x)^(3/2)","C",1
208,1,68,125,0.0879312,"\int \frac{a+b \cos ^{-1}(c x)}{(d x)^{5/2}} \, dx","Integrate[(a + b*ArcCos[c*x])/(d*x)^(5/2),x]","\frac{2 x \left(2 b c^3 x^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};c^2 x^2\right)-3 \left(a-2 b c x \sqrt{1-c^2 x^2}+b \cos ^{-1}(c x)\right)\right)}{9 (d x)^{5/2}}","-\frac{2 \left(a+b \cos ^{-1}(c x)\right)}{3 d (d x)^{3/2}}-\frac{4 b c^{3/2} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right)\right|-1\right)}{3 d^{5/2}}+\frac{4 b c^{3/2} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{c} \sqrt{d x}}{\sqrt{d}}\right)\right|-1\right)}{3 d^{5/2}}+\frac{4 b c \sqrt{1-c^2 x^2}}{3 d^2 \sqrt{d x}}",1,"(2*x*(-3*(a - 2*b*c*x*Sqrt[1 - c^2*x^2] + b*ArcCos[c*x]) + 2*b*c^3*x^3*Hypergeometric2F1[1/2, 3/4, 7/4, c^2*x^2]))/(9*(d*x)^(5/2))","C",1
209,1,234,109,1.4534824,"\int (d x)^{5/2} \left(a+b \cos ^{-1}(c x)\right)^2 \, dx","Integrate[(d*x)^(5/2)*(a + b*ArcCos[c*x])^2,x]","\frac{(d x)^{5/2} \left(\frac{b^2 \left(\frac{105 \sqrt{2} \pi  c x \, _3F_2\left(\frac{3}{4},\frac{3}{4},1;\frac{5}{4},\frac{7}{4};c^2 x^2\right)}{\Gamma \left(\frac{5}{4}\right) \Gamma \left(\frac{7}{4}\right)}+882 c^3 x^3 \cos ^{-1}(c x)^2+840 \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{3}{4},1;\frac{5}{4};c^2 x^2\right) \cos ^{-1}(c x)-16 c x \left(9 c^2 x^2+35\right)-168 \sqrt{1-c^2 x^2} \left(3 c^2 x^2+5\right) \cos ^{-1}(c x)\right)}{c^3}+882 a^2 x^3+\frac{84 a b \left(21 c^3 x^3 \cos ^{-1}(c x)+10 \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};c^2 x^2\right)-2 \sqrt{1-c^2 x^2} \left(3 c^2 x^2+5\right)\right)}{c^3}\right)}{3087 x^2}","\frac{16 b^2 c^2 (d x)^{11/2} \, _3F_2\left(1,\frac{11}{4},\frac{11}{4};\frac{13}{4},\frac{15}{4};c^2 x^2\right)}{693 d^3}+\frac{8 b c (d x)^{9/2} \, _2F_1\left(\frac{1}{2},\frac{9}{4};\frac{13}{4};c^2 x^2\right) \left(a+b \cos ^{-1}(c x)\right)}{63 d^2}+\frac{2 (d x)^{7/2} \left(a+b \cos ^{-1}(c x)\right)^2}{7 d}",1,"((d*x)^(5/2)*(882*a^2*x^3 + (84*a*b*(-2*Sqrt[1 - c^2*x^2]*(5 + 3*c^2*x^2) + 21*c^3*x^3*ArcCos[c*x] + 10*Hypergeometric2F1[1/4, 1/2, 5/4, c^2*x^2]))/c^3 + (b^2*(-16*c*x*(35 + 9*c^2*x^2) - 168*Sqrt[1 - c^2*x^2]*(5 + 3*c^2*x^2)*ArcCos[c*x] + 882*c^3*x^3*ArcCos[c*x]^2 + 840*Sqrt[1 - c^2*x^2]*ArcCos[c*x]*Hypergeometric2F1[3/4, 1, 5/4, c^2*x^2] + (105*Sqrt[2]*c*Pi*x*HypergeometricPFQ[{3/4, 3/4, 1}, {5/4, 7/4}, c^2*x^2])/(Gamma[5/4]*Gamma[7/4])))/c^3))/(3087*x^2)","B",0
210,1,176,109,8.6174549,"\int (d x)^{3/2} \left(a+b \cos ^{-1}(c x)\right)^2 \, dx","Integrate[(d*x)^(3/2)*(a + b*ArcCos[c*x])^2,x]","\frac{(d x)^{3/2} \left(\frac{525 \sqrt{2} \pi  b^2 c^2 x^3 \, _3F_2\left(1,\frac{9}{4},\frac{9}{4};\frac{11}{4},\frac{13}{4};c^2 x^2\right)}{\Gamma \left(\frac{11}{4}\right) \Gamma \left(\frac{13}{4}\right)}+4480 a^2 x+\frac{128 b \left(28 a \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};c^2 x^2\right)-28 a \sqrt{1-c^2 x^2}+70 a c x \cos ^{-1}(c x)+20 b c^2 x^2 \sqrt{1-c^2 x^2} \, _2F_1\left(1,\frac{9}{4};\frac{11}{4};c^2 x^2\right) \cos ^{-1}(c x)+35 b c x \cos ^{-1}(c x)^2\right)}{c}\right)}{11200}","\frac{16 b^2 c^2 (d x)^{9/2} \, _3F_2\left(1,\frac{9}{4},\frac{9}{4};\frac{11}{4},\frac{13}{4};c^2 x^2\right)}{315 d^3}+\frac{8 b c (d x)^{7/2} \, _2F_1\left(\frac{1}{2},\frac{7}{4};\frac{11}{4};c^2 x^2\right) \left(a+b \cos ^{-1}(c x)\right)}{35 d^2}+\frac{2 (d x)^{5/2} \left(a+b \cos ^{-1}(c x)\right)^2}{5 d}",1,"((d*x)^(3/2)*(4480*a^2*x + (128*b*(-28*a*Sqrt[1 - c^2*x^2] + 70*a*c*x*ArcCos[c*x] + 35*b*c*x*ArcCos[c*x]^2 + 28*a*Hypergeometric2F1[1/2, 3/4, 7/4, c^2*x^2] + 20*b*c^2*x^2*Sqrt[1 - c^2*x^2]*ArcCos[c*x]*Hypergeometric2F1[1, 9/4, 11/4, c^2*x^2]))/c + (525*Sqrt[2]*b^2*c^2*Pi*x^3*HypergeometricPFQ[{1, 9/4, 9/4}, {11/4, 13/4}, c^2*x^2])/(Gamma[11/4]*Gamma[13/4])))/11200","A",0
211,1,202,109,0.622395,"\int \sqrt{d x} \left(a+b \cos ^{-1}(c x)\right)^2 \, dx","Integrate[Sqrt[d*x]*(a + b*ArcCos[c*x])^2,x]","\frac{1}{27} \sqrt{d x} \left(\frac{3 \sqrt{2} \pi  b^2 x \, _3F_2\left(\frac{3}{4},\frac{3}{4},1;\frac{5}{4},\frac{7}{4};c^2 x^2\right)}{\Gamma \left(\frac{5}{4}\right) \Gamma \left(\frac{7}{4}\right)}+\frac{2 \left(9 a^2 c x+12 a b \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};c^2 x^2\right)-12 a b \sqrt{1-c^2 x^2}+18 a b c x \cos ^{-1}(c x)+12 b^2 \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{3}{4},1;\frac{5}{4};c^2 x^2\right) \cos ^{-1}(c x)-12 b^2 \sqrt{1-c^2 x^2} \cos ^{-1}(c x)-8 b^2 c x+9 b^2 c x \cos ^{-1}(c x)^2\right)}{c}\right)","\frac{16 b^2 c^2 (d x)^{7/2} \, _3F_2\left(1,\frac{7}{4},\frac{7}{4};\frac{9}{4},\frac{11}{4};c^2 x^2\right)}{105 d^3}+\frac{8 b c (d x)^{5/2} \, _2F_1\left(\frac{1}{2},\frac{5}{4};\frac{9}{4};c^2 x^2\right) \left(a+b \cos ^{-1}(c x)\right)}{15 d^2}+\frac{2 (d x)^{3/2} \left(a+b \cos ^{-1}(c x)\right)^2}{3 d}",1,"(Sqrt[d*x]*((2*(9*a^2*c*x - 8*b^2*c*x - 12*a*b*Sqrt[1 - c^2*x^2] + 18*a*b*c*x*ArcCos[c*x] - 12*b^2*Sqrt[1 - c^2*x^2]*ArcCos[c*x] + 9*b^2*c*x*ArcCos[c*x]^2 + 12*a*b*Hypergeometric2F1[1/4, 1/2, 5/4, c^2*x^2] + 12*b^2*Sqrt[1 - c^2*x^2]*ArcCos[c*x]*Hypergeometric2F1[3/4, 1, 5/4, c^2*x^2]))/c + (3*Sqrt[2]*b^2*Pi*x*HypergeometricPFQ[{3/4, 3/4, 1}, {5/4, 7/4}, c^2*x^2])/(Gamma[5/4]*Gamma[7/4])))/27","A",0
212,1,142,107,1.4858265,"\int \frac{\left(a+b \cos ^{-1}(c x)\right)^2}{\sqrt{d x}} \, dx","Integrate[(a + b*ArcCos[c*x])^2/Sqrt[d*x],x]","\frac{3 \sqrt{2} \pi  b^2 c^2 x^3 \, _3F_2\left(1,\frac{5}{4},\frac{5}{4};\frac{7}{4},\frac{9}{4};c^2 x^2\right)+8 x \Gamma \left(\frac{7}{4}\right) \Gamma \left(\frac{9}{4}\right) \left(4 a b c x \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};c^2 x^2\right)+3 \left(a+b \cos ^{-1}(c x)\right)^2+2 b^2 \, _2F_1\left(1,\frac{5}{4};\frac{7}{4};c^2 x^2\right) \cos ^{-1}(c x) \sin \left(2 \cos ^{-1}(c x)\right)\right)}{12 \Gamma \left(\frac{7}{4}\right) \Gamma \left(\frac{9}{4}\right) \sqrt{d x}}","\frac{16 b^2 c^2 (d x)^{5/2} \, _3F_2\left(1,\frac{5}{4},\frac{5}{4};\frac{7}{4},\frac{9}{4};c^2 x^2\right)}{15 d^3}+\frac{8 b c (d x)^{3/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};c^2 x^2\right) \left(a+b \cos ^{-1}(c x)\right)}{3 d^2}+\frac{2 \sqrt{d x} \left(a+b \cos ^{-1}(c x)\right)^2}{d}",1,"(3*Sqrt[2]*b^2*c^2*Pi*x^3*HypergeometricPFQ[{1, 5/4, 5/4}, {7/4, 9/4}, c^2*x^2] + 8*x*Gamma[7/4]*Gamma[9/4]*(3*(a + b*ArcCos[c*x])^2 + 4*a*b*c*x*Hypergeometric2F1[1/2, 3/4, 7/4, c^2*x^2] + 2*b^2*ArcCos[c*x]*Hypergeometric2F1[1, 5/4, 7/4, c^2*x^2]*Sin[2*ArcCos[c*x]]))/(12*Sqrt[d*x]*Gamma[7/4]*Gamma[9/4])","A",0
213,1,129,105,0.5204653,"\int \frac{\left(a+b \cos ^{-1}(c x)\right)^2}{(d x)^{3/2}} \, dx","Integrate[(a + b*ArcCos[c*x])^2/(d*x)^(3/2),x]","\frac{x \left(-\frac{\sqrt{2} \pi  b^2 c^2 x^2 \, _3F_2\left(\frac{3}{4},\frac{3}{4},1;\frac{5}{4},\frac{7}{4};c^2 x^2\right)}{\Gamma \left(\frac{5}{4}\right) \Gamma \left(\frac{7}{4}\right)}-2 \left(4 a b c x \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};c^2 x^2\right)+\left(a+b \cos ^{-1}(c x)\right)^2+2 b^2 \, _2F_1\left(\frac{3}{4},1;\frac{5}{4};c^2 x^2\right) \cos ^{-1}(c x) \sin \left(2 \cos ^{-1}(c x)\right)\right)\right)}{(d x)^{3/2}}","-\frac{16 b^2 c^2 (d x)^{3/2} \, _3F_2\left(\frac{3}{4},\frac{3}{4},1;\frac{5}{4},\frac{7}{4};c^2 x^2\right)}{3 d^3}-\frac{8 b c \sqrt{d x} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};c^2 x^2\right) \left(a+b \cos ^{-1}(c x)\right)}{d^2}-\frac{2 \left(a+b \cos ^{-1}(c x)\right)^2}{d \sqrt{d x}}",1,"(x*(-((Sqrt[2]*b^2*c^2*Pi*x^2*HypergeometricPFQ[{3/4, 3/4, 1}, {5/4, 7/4}, c^2*x^2])/(Gamma[5/4]*Gamma[7/4])) - 2*((a + b*ArcCos[c*x])^2 + 4*a*b*c*x*Hypergeometric2F1[1/4, 1/2, 5/4, c^2*x^2] + 2*b^2*ArcCos[c*x]*Hypergeometric2F1[3/4, 1, 5/4, c^2*x^2]*Sin[2*ArcCos[c*x]])))/(d*x)^(3/2)","A",0
214,1,198,109,0.9328154,"\int \frac{\left(a+b \cos ^{-1}(c x)\right)^2}{(d x)^{5/2}} \, dx","Integrate[(a + b*ArcCos[c*x])^2/(d*x)^(5/2),x]","\frac{x \left(3 \sqrt{2} \pi  b^2 c^4 x^4 \, _3F_2\left(1,\frac{5}{4},\frac{5}{4};\frac{7}{4},\frac{9}{4};c^2 x^2\right)-8 \Gamma \left(\frac{7}{4}\right) \Gamma \left(\frac{9}{4}\right) \left(3 \left(a^2+2 b \cos ^{-1}(c x) \left(a-2 b c x \sqrt{1-c^2 x^2}\right)-8 b^2 c^2 x^2+b^2 \cos ^{-1}(c x)^2\right)-12 a b c x \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};c^2 x^2\right)-4 b^2 c^3 x^3 \sqrt{1-c^2 x^2} \, _2F_1\left(1,\frac{5}{4};\frac{7}{4};c^2 x^2\right) \cos ^{-1}(c x)\right)\right)}{36 \Gamma \left(\frac{7}{4}\right) \Gamma \left(\frac{9}{4}\right) (d x)^{5/2}}","\frac{16 b^2 c^2 \sqrt{d x} \, _3F_2\left(\frac{1}{4},\frac{1}{4},1;\frac{3}{4},\frac{5}{4};c^2 x^2\right)}{3 d^3}+\frac{8 b c \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};c^2 x^2\right) \left(a+b \cos ^{-1}(c x)\right)}{3 d^2 \sqrt{d x}}-\frac{2 \left(a+b \cos ^{-1}(c x)\right)^2}{3 d (d x)^{3/2}}",1,"(x*(-8*Gamma[7/4]*Gamma[9/4]*(3*(a^2 - 8*b^2*c^2*x^2 + 2*b*(a - 2*b*c*x*Sqrt[1 - c^2*x^2])*ArcCos[c*x] + b^2*ArcCos[c*x]^2) - 12*a*b*c*x*Hypergeometric2F1[-1/4, 1/2, 3/4, c^2*x^2] - 4*b^2*c^3*x^3*Sqrt[1 - c^2*x^2]*ArcCos[c*x]*Hypergeometric2F1[1, 5/4, 7/4, c^2*x^2]) + 3*Sqrt[2]*b^2*c^4*Pi*x^4*HypergeometricPFQ[{1, 5/4, 5/4}, {7/4, 9/4}, c^2*x^2]))/(36*(d*x)^(5/2)*Gamma[7/4]*Gamma[9/4])","A",0
215,0,0,69,34.9919533,"\int (d x)^{3/2} \left(a+b \cos ^{-1}(c x)\right)^3 \, dx","Integrate[(d*x)^(3/2)*(a + b*ArcCos[c*x])^3,x]","\int (d x)^{3/2} \left(a+b \cos ^{-1}(c x)\right)^3 \, dx","\frac{6 b c \text{Int}\left(\frac{(d x)^{5/2} \left(a+b \cos ^{-1}(c x)\right)^2}{\sqrt{1-c^2 x^2}},x\right)}{5 d}+\frac{2 (d x)^{5/2} \left(a+b \cos ^{-1}(c x)\right)^3}{5 d}",0,"Integrate[(d*x)^(3/2)*(a + b*ArcCos[c*x])^3, x]","A",-1
216,0,0,67,147.3276622,"\int \sqrt{d x} \left(a+b \cos ^{-1}(c x)\right)^3 \, dx","Integrate[Sqrt[d*x]*(a + b*ArcCos[c*x])^3,x]","\int \sqrt{d x} \left(a+b \cos ^{-1}(c x)\right)^3 \, dx","\frac{2 b c \text{Int}\left(\frac{(d x)^{3/2} \left(a+b \cos ^{-1}(c x)\right)^2}{\sqrt{1-c^2 x^2}},x\right)}{d}+\frac{2 (d x)^{3/2} \left(a+b \cos ^{-1}(c x)\right)^3}{3 d}",0,"Integrate[Sqrt[d*x]*(a + b*ArcCos[c*x])^3, x]","A",-1
217,0,0,65,11.3300743,"\int \frac{\left(a+b \cos ^{-1}(c x)\right)^3}{\sqrt{d x}} \, dx","Integrate[(a + b*ArcCos[c*x])^3/Sqrt[d*x],x]","\int \frac{\left(a+b \cos ^{-1}(c x)\right)^3}{\sqrt{d x}} \, dx","\frac{6 b c \text{Int}\left(\frac{\sqrt{d x} \left(a+b \cos ^{-1}(c x)\right)^2}{\sqrt{1-c^2 x^2}},x\right)}{d}+\frac{2 \sqrt{d x} \left(a+b \cos ^{-1}(c x)\right)^3}{d}",0,"Integrate[(a + b*ArcCos[c*x])^3/Sqrt[d*x], x]","A",-1
218,0,0,65,8.0640687,"\int \frac{\left(a+b \cos ^{-1}(c x)\right)^3}{(d x)^{3/2}} \, dx","Integrate[(a + b*ArcCos[c*x])^3/(d*x)^(3/2),x]","\int \frac{\left(a+b \cos ^{-1}(c x)\right)^3}{(d x)^{3/2}} \, dx","-\frac{6 b c \text{Int}\left(\frac{\left(a+b \cos ^{-1}(c x)\right)^2}{\sqrt{1-c^2 x^2} \sqrt{d x}},x\right)}{d}-\frac{2 \left(a+b \cos ^{-1}(c x)\right)^3}{d \sqrt{d x}}",0,"Integrate[(a + b*ArcCos[c*x])^3/(d*x)^(3/2), x]","A",-1
219,0,0,67,12.9642141,"\int \frac{\left(a+b \cos ^{-1}(c x)\right)^3}{(d x)^{5/2}} \, dx","Integrate[(a + b*ArcCos[c*x])^3/(d*x)^(5/2),x]","\int \frac{\left(a+b \cos ^{-1}(c x)\right)^3}{(d x)^{5/2}} \, dx","-\frac{2 b c \text{Int}\left(\frac{\left(a+b \cos ^{-1}(c x)\right)^2}{\sqrt{1-c^2 x^2} (d x)^{3/2}},x\right)}{d}-\frac{2 \left(a+b \cos ^{-1}(c x)\right)^3}{3 d (d x)^{3/2}}",0,"Integrate[(a + b*ArcCos[c*x])^3/(d*x)^(5/2), x]","A",-1
220,0,0,21,3.7831468,"\int \frac{(d x)^{3/2}}{a+b \cos ^{-1}(c x)} \, dx","Integrate[(d*x)^(3/2)/(a + b*ArcCos[c*x]),x]","\int \frac{(d x)^{3/2}}{a+b \cos ^{-1}(c x)} \, dx","\text{Int}\left(\frac{(d x)^{3/2}}{a+b \cos ^{-1}(c x)},x\right)",0,"Integrate[(d*x)^(3/2)/(a + b*ArcCos[c*x]), x]","A",-1
221,0,0,21,2.4575321,"\int \frac{\sqrt{d x}}{a+b \cos ^{-1}(c x)} \, dx","Integrate[Sqrt[d*x]/(a + b*ArcCos[c*x]),x]","\int \frac{\sqrt{d x}}{a+b \cos ^{-1}(c x)} \, dx","\text{Int}\left(\frac{\sqrt{d x}}{a+b \cos ^{-1}(c x)},x\right)",0,"Integrate[Sqrt[d*x]/(a + b*ArcCos[c*x]), x]","A",-1
222,0,0,21,0.9468762,"\int \frac{1}{\sqrt{d x} \left(a+b \cos ^{-1}(c x)\right)} \, dx","Integrate[1/(Sqrt[d*x]*(a + b*ArcCos[c*x])),x]","\int \frac{1}{\sqrt{d x} \left(a+b \cos ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{\sqrt{d x} \left(a+b \cos ^{-1}(c x)\right)},x\right)",0,"Integrate[1/(Sqrt[d*x]*(a + b*ArcCos[c*x])), x]","A",-1
223,0,0,21,3.4267986,"\int \frac{1}{(d x)^{3/2} \left(a+b \cos ^{-1}(c x)\right)} \, dx","Integrate[1/((d*x)^(3/2)*(a + b*ArcCos[c*x])),x]","\int \frac{1}{(d x)^{3/2} \left(a+b \cos ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{(d x)^{3/2} \left(a+b \cos ^{-1}(c x)\right)},x\right)",0,"Integrate[1/((d*x)^(3/2)*(a + b*ArcCos[c*x])), x]","A",-1
224,0,0,21,7.100824,"\int \frac{(d x)^{3/2}}{\left(a+b \cos ^{-1}(c x)\right)^2} \, dx","Integrate[(d*x)^(3/2)/(a + b*ArcCos[c*x])^2,x]","\int \frac{(d x)^{3/2}}{\left(a+b \cos ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{(d x)^{3/2}}{\left(a+b \cos ^{-1}(c x)\right)^2},x\right)",0,"Integrate[(d*x)^(3/2)/(a + b*ArcCos[c*x])^2, x]","A",-1
225,0,0,21,5.4407846,"\int \frac{\sqrt{d x}}{\left(a+b \cos ^{-1}(c x)\right)^2} \, dx","Integrate[Sqrt[d*x]/(a + b*ArcCos[c*x])^2,x]","\int \frac{\sqrt{d x}}{\left(a+b \cos ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{\sqrt{d x}}{\left(a+b \cos ^{-1}(c x)\right)^2},x\right)",0,"Integrate[Sqrt[d*x]/(a + b*ArcCos[c*x])^2, x]","A",-1
226,0,0,21,10.4308358,"\int \frac{1}{\sqrt{d x} \left(a+b \cos ^{-1}(c x)\right)^2} \, dx","Integrate[1/(Sqrt[d*x]*(a + b*ArcCos[c*x])^2),x]","\int \frac{1}{\sqrt{d x} \left(a+b \cos ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{\sqrt{d x} \left(a+b \cos ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/(Sqrt[d*x]*(a + b*ArcCos[c*x])^2), x]","A",-1
227,0,0,21,15.2119097,"\int \frac{1}{(d x)^{3/2} \left(a+b \cos ^{-1}(c x)\right)^2} \, dx","Integrate[1/((d*x)^(3/2)*(a + b*ArcCos[c*x])^2),x]","\int \frac{1}{(d x)^{3/2} \left(a+b \cos ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{(d x)^{3/2} \left(a+b \cos ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/((d*x)^(3/2)*(a + b*ArcCos[c*x])^2), x]","A",-1