1,1,56,62,0.04271,"\int \frac{\csc ^{-1}\left(a x^5\right)}{x} \, dx","Integrate[ArcCsc[a*x^5]/x,x]","\frac{1}{10} i \left(\text{Li}_2\left(e^{2 i \csc ^{-1}\left(a x^5\right)}\right)+\csc ^{-1}\left(a x^5\right) \left(\csc ^{-1}\left(a x^5\right)+2 i \log \left(1-e^{2 i \csc ^{-1}\left(a x^5\right)}\right)\right)\right)","\frac{1}{10} i \text{Li}_2\left(e^{2 i \csc ^{-1}\left(a x^5\right)}\right)+\frac{1}{10} i \csc ^{-1}\left(a x^5\right)^2-\frac{1}{5} \csc ^{-1}\left(a x^5\right) \log \left(1-e^{2 i \csc ^{-1}\left(a x^5\right)}\right)",1,"(I/10)*(ArcCsc[a*x^5]*(ArcCsc[a*x^5] + (2*I)*Log[1 - E^((2*I)*ArcCsc[a*x^5])]) + PolyLog[2, E^((2*I)*ArcCsc[a*x^5])])","A",1
2,1,40,58,0.0278018,"\int x^3 \csc ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[x^3*ArcCsc[Sqrt[x]],x]","\frac{1}{4} x^4 \csc ^{-1}\left(\sqrt{x}\right)+\frac{1}{140} \sqrt{x-1} \left(5 x^3+6 x^2+8 x+16\right)","\frac{1}{4} x^4 \csc ^{-1}\left(\sqrt{x}\right)+\frac{1}{28} (x-1)^{7/2}+\frac{3}{20} (x-1)^{5/2}+\frac{1}{4} (x-1)^{3/2}+\frac{\sqrt{x-1}}{4}",1,"(Sqrt[-1 + x]*(16 + 8*x + 6*x^2 + 5*x^3))/140 + (x^4*ArcCsc[Sqrt[x]])/4","A",1
3,1,35,47,0.0213415,"\int x^2 \csc ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[x^2*ArcCsc[Sqrt[x]],x]","\frac{1}{3} x^3 \csc ^{-1}\left(\sqrt{x}\right)+\frac{1}{45} \sqrt{x-1} \left(3 x^2+4 x+8\right)","\frac{1}{3} x^3 \csc ^{-1}\left(\sqrt{x}\right)+\frac{1}{15} (x-1)^{5/2}+\frac{2}{9} (x-1)^{3/2}+\frac{\sqrt{x-1}}{3}",1,"(Sqrt[-1 + x]*(8 + 4*x + 3*x^2))/45 + (x^3*ArcCsc[Sqrt[x]])/3","A",1
4,1,27,36,0.0216385,"\int x \csc ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[x*ArcCsc[Sqrt[x]],x]","\frac{1}{6} \left(3 x^2 \csc ^{-1}\left(\sqrt{x}\right)+\sqrt{x-1} (x+2)\right)","\frac{1}{2} x^2 \csc ^{-1}\left(\sqrt{x}\right)+\frac{1}{6} (x-1)^{3/2}+\frac{\sqrt{x-1}}{2}",1,"(Sqrt[-1 + x]*(2 + x) + 3*x^2*ArcCsc[Sqrt[x]])/6","A",1
5,1,16,16,0.0043532,"\int \csc ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[ArcCsc[Sqrt[x]],x]","\sqrt{x-1}+x \csc ^{-1}\left(\sqrt{x}\right)","\sqrt{x-1}+x \csc ^{-1}\left(\sqrt{x}\right)",1,"Sqrt[-1 + x] + x*ArcCsc[Sqrt[x]]","A",1
6,1,54,56,0.0320764,"\int \frac{\csc ^{-1}\left(\sqrt{x}\right)}{x} \, dx","Integrate[ArcCsc[Sqrt[x]]/x,x]","i \left(\text{Li}_2\left(e^{2 i \csc ^{-1}\left(\sqrt{x}\right)}\right)+\csc ^{-1}\left(\sqrt{x}\right) \left(\csc ^{-1}\left(\sqrt{x}\right)+2 i \log \left(1-e^{2 i \csc ^{-1}\left(\sqrt{x}\right)}\right)\right)\right)","i \text{Li}_2\left(e^{2 i \csc ^{-1}\left(\sqrt{x}\right)}\right)+i \csc ^{-1}\left(\sqrt{x}\right)^2-2 \csc ^{-1}\left(\sqrt{x}\right) \log \left(1-e^{2 i \csc ^{-1}\left(\sqrt{x}\right)}\right)",1,"I*(ArcCsc[Sqrt[x]]*(ArcCsc[Sqrt[x]] + (2*I)*Log[1 - E^((2*I)*ArcCsc[Sqrt[x]])]) + PolyLog[2, E^((2*I)*ArcCsc[Sqrt[x]])])","A",1
7,1,32,38,0.0248523,"\int \frac{\csc ^{-1}\left(\sqrt{x}\right)}{x^2} \, dx","Integrate[ArcCsc[Sqrt[x]]/x^2,x]","-\frac{\sqrt{x-1}-x \sin ^{-1}\left(\frac{1}{\sqrt{x}}\right)+2 \csc ^{-1}\left(\sqrt{x}\right)}{2 x}","-\frac{\sqrt{x-1}}{2 x}-\frac{1}{2} \tan ^{-1}\left(\sqrt{x-1}\right)-\frac{\csc ^{-1}\left(\sqrt{x}\right)}{x}",1,"-1/2*(Sqrt[-1 + x] + 2*ArcCsc[Sqrt[x]] - x*ArcSin[1/Sqrt[x]])/x","A",1
8,1,55,54,0.0475535,"\int \frac{\csc ^{-1}\left(\sqrt{x}\right)}{x^3} \, dx","Integrate[ArcCsc[Sqrt[x]]/x^3,x]","\sqrt{\frac{x-1}{x}} \left(-\frac{1}{8 x^{3/2}}-\frac{3}{16 \sqrt{x}}\right)-\frac{\csc ^{-1}\left(\sqrt{x}\right)}{2 x^2}+\frac{3}{16} \sin ^{-1}\left(\frac{1}{\sqrt{x}}\right)","-\frac{\sqrt{x-1}}{8 x^2}-\frac{\csc ^{-1}\left(\sqrt{x}\right)}{2 x^2}-\frac{3 \sqrt{x-1}}{16 x}-\frac{3}{16} \tan ^{-1}\left(\sqrt{x-1}\right)",1,"(-1/8*1/x^(3/2) - 3/(16*Sqrt[x]))*Sqrt[(-1 + x)/x] - ArcCsc[Sqrt[x]]/(2*x^2) + (3*ArcSin[1/Sqrt[x]])/16","A",1
9,1,42,56,0.0346184,"\int x^2 \csc ^{-1}\left(\frac{a}{x}\right) \, dx","Integrate[x^2*ArcCsc[a/x],x]","\frac{1}{9} a \left(2 a^2+x^2\right) \sqrt{1-\frac{x^2}{a^2}}+\frac{1}{3} x^3 \csc ^{-1}\left(\frac{a}{x}\right)","-\frac{1}{9} a^3 \left(1-\frac{x^2}{a^2}\right)^{3/2}+\frac{1}{3} a^3 \sqrt{1-\frac{x^2}{a^2}}+\frac{1}{3} x^3 \sin ^{-1}\left(\frac{x}{a}\right)",1,"(a*(2*a^2 + x^2)*Sqrt[1 - x^2/a^2])/9 + (x^3*ArcCsc[a/x])/3","A",1
10,1,44,47,0.024647,"\int x \csc ^{-1}\left(\frac{a}{x}\right) \, dx","Integrate[x*ArcCsc[a/x],x]","\frac{1}{4} \left(a x \sqrt{1-\frac{x^2}{a^2}}+a^2 \left(-\sin ^{-1}\left(\frac{x}{a}\right)\right)+2 x^2 \csc ^{-1}\left(\frac{a}{x}\right)\right)","\frac{1}{4} a x \sqrt{1-\frac{x^2}{a^2}}-\frac{1}{4} a^2 \sin ^{-1}\left(\frac{x}{a}\right)+\frac{1}{2} x^2 \sin ^{-1}\left(\frac{x}{a}\right)",1,"(a*x*Sqrt[1 - x^2/a^2] + 2*x^2*ArcCsc[a/x] - a^2*ArcSin[x/a])/4","A",1
11,1,25,25,0.0121781,"\int \csc ^{-1}\left(\frac{a}{x}\right) \, dx","Integrate[ArcCsc[a/x],x]","a \sqrt{1-\frac{x^2}{a^2}}+x \csc ^{-1}\left(\frac{a}{x}\right)","a \sqrt{1-\frac{x^2}{a^2}}+x \sin ^{-1}\left(\frac{x}{a}\right)",1,"a*Sqrt[1 - x^2/a^2] + x*ArcCsc[a/x]","A",1
12,1,54,59,0.0355457,"\int \frac{\csc ^{-1}\left(\frac{a}{x}\right)}{x} \, dx","Integrate[ArcCsc[a/x]/x,x]","\csc ^{-1}\left(\frac{a}{x}\right) \log \left(1-e^{2 i \csc ^{-1}\left(\frac{a}{x}\right)}\right)-\frac{1}{2} i \left(\csc ^{-1}\left(\frac{a}{x}\right)^2+\text{Li}_2\left(e^{2 i \csc ^{-1}\left(\frac{a}{x}\right)}\right)\right)","-\frac{1}{2} i \text{Li}_2\left(e^{2 i \sin ^{-1}\left(\frac{x}{a}\right)}\right)-\frac{1}{2} i \sin ^{-1}\left(\frac{x}{a}\right)^2+\sin ^{-1}\left(\frac{x}{a}\right) \log \left(1-e^{2 i \sin ^{-1}\left(\frac{x}{a}\right)}\right)",1,"ArcCsc[a/x]*Log[1 - E^((2*I)*ArcCsc[a/x])] - (I/2)*(ArcCsc[a/x]^2 + PolyLog[2, E^((2*I)*ArcCsc[a/x])])","A",0
13,1,93,32,0.1522111,"\int \frac{\csc ^{-1}\left(\frac{a}{x}\right)}{x^2} \, dx","Integrate[ArcCsc[a/x]/x^2,x]","-\frac{x \sqrt{\frac{a^2}{x^2}-1} \left(\log \left(\frac{a}{x \sqrt{\frac{a^2}{x^2}-1}}+1\right)-\log \left(1-\frac{a}{x \sqrt{\frac{a^2}{x^2}-1}}\right)\right)}{2 a^2 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\csc ^{-1}\left(\frac{a}{x}\right)}{x}","-\frac{\tanh ^{-1}\left(\sqrt{1-\frac{x^2}{a^2}}\right)}{a}-\frac{\sin ^{-1}\left(\frac{x}{a}\right)}{x}",1,"-(ArcCsc[a/x]/x) - (Sqrt[-1 + a^2/x^2]*x*(-Log[1 - a/(Sqrt[-1 + a^2/x^2]*x)] + Log[1 + a/(Sqrt[-1 + a^2/x^2]*x)]))/(2*a^2*Sqrt[1 - x^2/a^2])","B",1
14,1,35,38,0.027623,"\int \frac{\csc ^{-1}\left(\frac{a}{x}\right)}{x^3} \, dx","Integrate[ArcCsc[a/x]/x^3,x]","-\frac{x \sqrt{1-\frac{x^2}{a^2}}+a \csc ^{-1}\left(\frac{a}{x}\right)}{2 a x^2}","-\frac{\sqrt{1-\frac{x^2}{a^2}}}{2 a x}-\frac{\sin ^{-1}\left(\frac{x}{a}\right)}{2 x^2}",1,"-1/2*(x*Sqrt[1 - x^2/a^2] + a*ArcCsc[a/x])/(a*x^2)","A",1
15,1,69,60,0.0566233,"\int \frac{\csc ^{-1}\left(\frac{a}{x}\right)}{x^4} \, dx","Integrate[ArcCsc[a/x]/x^4,x]","-\frac{2 a^3 \csc ^{-1}\left(\frac{a}{x}\right)+a^2 x \sqrt{1-\frac{x^2}{a^2}}+x^3 \log \left(\sqrt{1-\frac{x^2}{a^2}}+1\right)-x^3 \log (x)}{6 a^3 x^3}","-\frac{\sqrt{1-\frac{x^2}{a^2}}}{6 a x^2}-\frac{\tanh ^{-1}\left(\sqrt{1-\frac{x^2}{a^2}}\right)}{6 a^3}-\frac{\sin ^{-1}\left(\frac{x}{a}\right)}{3 x^3}",1,"-1/6*(a^2*x*Sqrt[1 - x^2/a^2] + 2*a^3*ArcCsc[a/x] - x^3*Log[x] + x^3*Log[1 + Sqrt[1 - x^2/a^2]])/(a^3*x^3)","A",1
16,1,63,69,0.094299,"\int \frac{\csc ^{-1}\left(a x^n\right)}{x} \, dx","Integrate[ArcCsc[a*x^n]/x,x]","\log (x) \left(\csc ^{-1}\left(a x^n\right)-\sin ^{-1}\left(\frac{x^{-n}}{a}\right)\right)-\frac{x^{-n} \, _3F_2\left(\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};\frac{x^{-2 n}}{a^2}\right)}{a n}","\frac{i \text{Li}_2\left(e^{2 i \csc ^{-1}\left(a x^n\right)}\right)}{2 n}+\frac{i \csc ^{-1}\left(a x^n\right)^2}{2 n}-\frac{\csc ^{-1}\left(a x^n\right) \log \left(1-e^{2 i \csc ^{-1}\left(a x^n\right)}\right)}{n}",1,"-(HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, 1/(a^2*x^(2*n))]/(a*n*x^n)) + (ArcCsc[a*x^n] - ArcSin[1/(a*x^n)])*Log[x]","C",1
17,1,174,197,0.2245168,"\int x^4 \csc ^{-1}(a+b x) \, dx","Integrate[x^4*ArcCsc[a + b*x],x]","\frac{24 a^5 \sin ^{-1}\left(\frac{1}{a+b x}\right)-\sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} \left(-9 \left(4 a^2+1\right) b^2 x^2+2 a \left(48 a^2+31\right) b x+a^2 \left(154 a^2+71\right)+16 a b^3 x^3-6 b^4 x^4\right)+3 \left(40 a^4+40 a^2+3\right) \log \left((a+b x) \left(\sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}}+1\right)\right)+24 b^5 x^5 \csc ^{-1}(a+b x)}{120 b^5}","\frac{a^5 \csc ^{-1}(a+b x)}{5 b^5}-\frac{\left(53 a^2+20\right) a (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{30 b^5}+\frac{\left(58 a^2+9\right) (a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}}}{120 b^5}+\frac{\left(40 a^4+40 a^2+3\right) \tanh ^{-1}\left(\sqrt{1-\frac{1}{(a+b x)^2}}\right)}{40 b^5}-\frac{11 a x^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{60 b^3}+\frac{x^3 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{20 b^2}+\frac{1}{5} x^5 \csc ^{-1}(a+b x)",1,"(-(Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2]*(a^2*(71 + 154*a^2) + 2*a*(31 + 48*a^2)*b*x - 9*(1 + 4*a^2)*b^2*x^2 + 16*a*b^3*x^3 - 6*b^4*x^4)) + 24*b^5*x^5*ArcCsc[a + b*x] + 24*a^5*ArcSin[(a + b*x)^(-1)] + 3*(3 + 40*a^2 + 40*a^4)*Log[(a + b*x)*(1 + Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2])])/(120*b^5)","A",1
18,1,149,155,0.2883196,"\int x^3 \csc ^{-1}(a+b x) \, dx","Integrate[x^3*ArcCsc[a + b*x],x]","\frac{-3 a^4 \sin ^{-1}\left(\frac{1}{a+b x}\right)-6 \left(2 a^2+1\right) a \log \left((a+b x) \left(\sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}}+1\right)\right)+\sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} \left(13 a^3+9 a^2 b x-3 a b^2 x^2+2 a+b^3 x^3+2 b x\right)+3 b^4 x^4 \csc ^{-1}(a+b x)}{12 b^4}","-\frac{a^4 \csc ^{-1}(a+b x)}{4 b^4}+\frac{\left(17 a^2+2\right) (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{12 b^4}-\frac{\left(2 a^2+1\right) a \tanh ^{-1}\left(\sqrt{1-\frac{1}{(a+b x)^2}}\right)}{2 b^4}-\frac{a (a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}}}{3 b^4}+\frac{x^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{12 b^2}+\frac{1}{4} x^4 \csc ^{-1}(a+b x)",1,"(Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2]*(2*a + 13*a^3 + 2*b*x + 9*a^2*b*x - 3*a*b^2*x^2 + b^3*x^3) + 3*b^4*x^4*ArcCsc[a + b*x] - 3*a^4*ArcSin[(a + b*x)^(-1)] - 6*a*(1 + 2*a^2)*Log[(a + b*x)*(1 + Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2])])/(12*b^4)","A",1
19,1,129,116,0.1906225,"\int x^2 \csc ^{-1}(a+b x) \, dx","Integrate[x^2*ArcCsc[a + b*x],x]","\frac{2 a^3 \sin ^{-1}\left(\frac{1}{a+b x}\right)+\left(-5 a^2-4 a b x+b^2 x^2\right) \sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}}+\left(6 a^2+1\right) \log \left((a+b x) \left(\sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}}+1\right)\right)+2 b^3 x^3 \csc ^{-1}(a+b x)}{6 b^3}","\frac{a^3 \csc ^{-1}(a+b x)}{3 b^3}+\frac{\left(6 a^2+1\right) \tanh ^{-1}\left(\sqrt{1-\frac{1}{(a+b x)^2}}\right)}{6 b^3}-\frac{5 a (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{6 b^3}+\frac{x (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{6 b^2}+\frac{1}{3} x^3 \csc ^{-1}(a+b x)",1,"((-5*a^2 - 4*a*b*x + b^2*x^2)*Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2] + 2*b^3*x^3*ArcCsc[a + b*x] + 2*a^3*ArcSin[(a + b*x)^(-1)] + (1 + 6*a^2)*Log[(a + b*x)*(1 + Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2])])/(6*b^3)","A",1
20,1,110,79,0.1314264,"\int x \csc ^{-1}(a+b x) \, dx","Integrate[x*ArcCsc[a + b*x],x]","\frac{(a+b x) \sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}}-2 a \log \left((a+b x) \left(\sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}}+1\right)\right)+a^2 \left(-\sin ^{-1}\left(\frac{1}{a+b x}\right)\right)+b^2 x^2 \csc ^{-1}(a+b x)}{2 b^2}","-\frac{a^2 \csc ^{-1}(a+b x)}{2 b^2}+\frac{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{2 b^2}-\frac{a \tanh ^{-1}\left(\sqrt{1-\frac{1}{(a+b x)^2}}\right)}{b^2}+\frac{1}{2} x^2 \csc ^{-1}(a+b x)",1,"((a + b*x)*Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2] + b^2*x^2*ArcCsc[a + b*x] - a^2*ArcSin[(a + b*x)^(-1)] - 2*a*Log[(a + b*x)*(1 + Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2])])/(2*b^2)","A",1
21,1,114,36,0.1158837,"\int \csc ^{-1}(a+b x) \, dx","Integrate[ArcCsc[a + b*x],x]","\frac{(a+b x) \sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} \left(\tanh ^{-1}\left(\frac{a+b x}{\sqrt{a^2+2 a b x+b^2 x^2-1}}\right)-a \tan ^{-1}\left(\sqrt{(a+b x)^2-1}\right)\right)}{b \sqrt{a^2+2 a b x+b^2 x^2-1}}+x \csc ^{-1}(a+b x)","\frac{(a+b x) \csc ^{-1}(a+b x)}{b}+\frac{\tanh ^{-1}\left(\sqrt{1-\frac{1}{(a+b x)^2}}\right)}{b}",1,"x*ArcCsc[a + b*x] + ((a + b*x)*Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2]*(-(a*ArcTan[Sqrt[-1 + (a + b*x)^2]]) + ArcTanh[(a + b*x)/Sqrt[-1 + a^2 + 2*a*b*x + b^2*x^2]]))/(b*Sqrt[-1 + a^2 + 2*a*b*x + b^2*x^2])","B",1
22,1,375,210,0.4395669,"\int \frac{\csc ^{-1}(a+b x)}{x} \, dx","Integrate[ArcCsc[a + b*x]/x,x]","\frac{1}{8} \left(8 i \left(\text{Li}_2\left(-\frac{i \left(\sqrt{1-a^2}-1\right) e^{-i \csc ^{-1}(a+b x)}}{a}\right)+\text{Li}_2\left(\frac{i \left(\sqrt{1-a^2}+1\right) e^{-i \csc ^{-1}(a+b x)}}{a}\right)\right)-4 \log \left(1+\frac{i \left(\sqrt{1-a^2}-1\right) e^{-i \csc ^{-1}(a+b x)}}{a}\right) \left(-2 \csc ^{-1}(a+b x)+4 \sin ^{-1}\left(\frac{\sqrt{\frac{a-1}{a}}}{\sqrt{2}}\right)+\pi \right)-4 \log \left(1-\frac{i \left(\sqrt{1-a^2}+1\right) e^{-i \csc ^{-1}(a+b x)}}{a}\right) \left(-2 \csc ^{-1}(a+b x)-4 \sin ^{-1}\left(\frac{\sqrt{\frac{a-1}{a}}}{\sqrt{2}}\right)+\pi \right)-32 i \sin ^{-1}\left(\frac{\sqrt{\frac{a-1}{a}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{(a+1) \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(a+b x)+\pi \right)\right)}{\sqrt{1-a^2}}\right)+4 i \left(\csc ^{-1}(a+b x)^2+\text{Li}_2\left(e^{2 i \csc ^{-1}(a+b x)}\right)\right)+i \left(\pi -2 \csc ^{-1}(a+b x)\right)^2+4 \log \left(\frac{b x}{a+b x}\right) \left(\pi -2 \csc ^{-1}(a+b x)\right)-8 \csc ^{-1}(a+b x) \log \left(1-e^{2 i \csc ^{-1}(a+b x)}\right)+8 \log \left(\frac{b x}{a+b x}\right) \csc ^{-1}(a+b x)\right)","-i \text{Li}_2\left(-\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right)-i \text{Li}_2\left(-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)+\csc ^{-1}(a+b x) \log \left(1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right)+\csc ^{-1}(a+b x) \log \left(1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)+\frac{1}{2} i \text{Li}_2\left(e^{2 i \csc ^{-1}(a+b x)}\right)-\csc ^{-1}(a+b x) \log \left(1-e^{2 i \csc ^{-1}(a+b x)}\right)",1,"(I*(Pi - 2*ArcCsc[a + b*x])^2 - (32*I)*ArcSin[Sqrt[(-1 + a)/a]/Sqrt[2]]*ArcTan[((1 + a)*Cot[(Pi + 2*ArcCsc[a + b*x])/4])/Sqrt[1 - a^2]] - 4*(Pi - 2*ArcCsc[a + b*x] + 4*ArcSin[Sqrt[(-1 + a)/a]/Sqrt[2]])*Log[1 + (I*(-1 + Sqrt[1 - a^2]))/(a*E^(I*ArcCsc[a + b*x]))] - 4*(Pi - 2*ArcCsc[a + b*x] - 4*ArcSin[Sqrt[(-1 + a)/a]/Sqrt[2]])*Log[1 - (I*(1 + Sqrt[1 - a^2]))/(a*E^(I*ArcCsc[a + b*x]))] - 8*ArcCsc[a + b*x]*Log[1 - E^((2*I)*ArcCsc[a + b*x])] + 4*(Pi - 2*ArcCsc[a + b*x])*Log[(b*x)/(a + b*x)] + 8*ArcCsc[a + b*x]*Log[(b*x)/(a + b*x)] + (8*I)*(PolyLog[2, ((-I)*(-1 + Sqrt[1 - a^2]))/(a*E^(I*ArcCsc[a + b*x]))] + PolyLog[2, (I*(1 + Sqrt[1 - a^2]))/(a*E^(I*ArcCsc[a + b*x]))]) + (4*I)*(ArcCsc[a + b*x]^2 + PolyLog[2, E^((2*I)*ArcCsc[a + b*x])]))/8","A",0
23,1,115,69,0.4155312,"\int \frac{\csc ^{-1}(a+b x)}{x^2} \, dx","Integrate[ArcCsc[a + b*x]/x^2,x]","-\frac{\csc ^{-1}(a+b x)}{x}+\frac{b \left(-\sin ^{-1}\left(\frac{1}{a+b x}\right)+\frac{i \log \left(\frac{2 \left(-a \sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} (a+b x)-\frac{i a \left(a^2+a b x-1\right)}{\sqrt{1-a^2}}\right)}{b x}\right)}{\sqrt{1-a^2}}\right)}{a}","-\frac{2 b \tan ^{-1}\left(\frac{a-\tan \left(\frac{1}{2} \csc ^{-1}(a+b x)\right)}{\sqrt{1-a^2}}\right)}{a \sqrt{1-a^2}}-\frac{b \csc ^{-1}(a+b x)}{a}-\frac{\csc ^{-1}(a+b x)}{x}",1,"-(ArcCsc[a + b*x]/x) + (b*(-ArcSin[(a + b*x)^(-1)] + (I*Log[(2*(((-I)*a*(-1 + a^2 + a*b*x))/Sqrt[1 - a^2] - a*(a + b*x)*Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2]))/(b*x)])/Sqrt[1 - a^2]))/a","C",1
24,1,199,123,0.7696326,"\int \frac{\csc ^{-1}(a+b x)}{x^3} \, dx","Integrate[ArcCsc[a + b*x]/x^3,x]","\frac{\frac{b x (a+b x) \sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}}}{a \left(a^2-1\right)}+\frac{i \left(2 a^2-1\right) b^2 x^2 \log \left(\frac{4 (a-1) a^2 (a+1) \left(\sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} (a+b x)+\frac{i \left(a^2+a b x-1\right)}{\sqrt{1-a^2}}\right)}{\left(2 a^2-1\right) b^2 x}\right)}{a^2 \left(1-a^2\right)^{3/2}}+\frac{b^2 x^2 \sin ^{-1}\left(\frac{1}{a+b x}\right)}{a^2}-\csc ^{-1}(a+b x)}{2 x^2}","\frac{b^2 \csc ^{-1}(a+b x)}{2 a^2}+\frac{\left(1-2 a^2\right) b^2 \tan ^{-1}\left(\frac{a-\tan \left(\frac{1}{2} \csc ^{-1}(a+b x)\right)}{\sqrt{1-a^2}}\right)}{a^2 \left(1-a^2\right)^{3/2}}-\frac{b (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{2 a \left(1-a^2\right) x}-\frac{\csc ^{-1}(a+b x)}{2 x^2}",1,"((b*x*(a + b*x)*Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2])/(a*(-1 + a^2)) - ArcCsc[a + b*x] + (b^2*x^2*ArcSin[(a + b*x)^(-1)])/a^2 + (I*(-1 + 2*a^2)*b^2*x^2*Log[(4*(-1 + a)*a^2*(1 + a)*((I*(-1 + a^2 + a*b*x))/Sqrt[1 - a^2] + (a + b*x)*Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2]))/((-1 + 2*a^2)*b^2*x)])/(a^2*(1 - a^2)^(3/2)))/(2*x^2)","C",1
25,1,241,180,0.5231287,"\int \frac{\csc ^{-1}(a+b x)}{x^4} \, dx","Integrate[ArcCsc[a + b*x]/x^4,x]","\frac{1}{6} \left(-\frac{2 b^3 \sin ^{-1}\left(\frac{1}{a+b x}\right)}{a^3}+\frac{b \sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} \left(a^4-4 a^3 b x-a^2 \left(5 b^2 x^2+1\right)+a b x+2 b^2 x^2\right)}{a^2 \left(a^2-1\right)^2 x^2}+\frac{i \left(6 a^4-5 a^2+2\right) b^3 \log \left(\frac{12 a^3 \left(a^2-1\right)^2 \left(-\sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} (a+b x)-\frac{i \left(a^2+a b x-1\right)}{\sqrt{1-a^2}}\right)}{\left(6 a^4-5 a^2+2\right) b^3 x}\right)}{a^3 \left(1-a^2\right)^{5/2}}-\frac{2 \csc ^{-1}(a+b x)}{x^3}\right)","-\frac{b^3 \csc ^{-1}(a+b x)}{3 a^3}+\frac{\left(2-5 a^2\right) b^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{6 a^2 \left(1-a^2\right)^2 x}-\frac{b (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{6 a \left(1-a^2\right) x^2}-\frac{\left(6 a^4-5 a^2+2\right) b^3 \tan ^{-1}\left(\frac{a-\tan \left(\frac{1}{2} \csc ^{-1}(a+b x)\right)}{\sqrt{1-a^2}}\right)}{3 a^3 \left(1-a^2\right)^{5/2}}-\frac{\csc ^{-1}(a+b x)}{3 x^3}",1,"((b*Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2]*(a^4 + a*b*x - 4*a^3*b*x + 2*b^2*x^2 - a^2*(1 + 5*b^2*x^2)))/(a^2*(-1 + a^2)^2*x^2) - (2*ArcCsc[a + b*x])/x^3 - (2*b^3*ArcSin[(a + b*x)^(-1)])/a^3 + (I*(2 - 5*a^2 + 6*a^4)*b^3*Log[(12*a^3*(-1 + a^2)^2*(((-I)*(-1 + a^2 + a*b*x))/Sqrt[1 - a^2] - (a + b*x)*Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2]))/((2 - 5*a^2 + 6*a^4)*b^3*x)])/(a^3*(1 - a^2)^(5/2)))/6","C",1
26,1,307,239,0.5364519,"\int \frac{\csc ^{-1}(a+b x)}{x^5} \, dx","Integrate[ArcCsc[a + b*x]/x^5,x]","\frac{1}{8} \left(\frac{2 b^4 \sin ^{-1}\left(\frac{1}{a+b x}\right)}{a^4}+\frac{i \left(8 a^6-8 a^4+7 a^2-2\right) b^4 \log \left(\frac{16 a^4 \left(a^2-1\right)^3 \left(\sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} (a+b x)+\frac{i \left(a^2+a b x-1\right)}{\sqrt{1-a^2}}\right)}{\left(8 a^6-8 a^4+7 a^2-2\right) b^4 x}\right)}{a^4 \left(1-a^2\right)^{7/2}}+\frac{b \sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} \left(2 a^7-6 a^6 b x+2 a^5 \left(9 b^2 x^2-2\right)+a^4 b x \left(26 b^2 x^2+7\right)+a^3 \left(2-6 b^2 x^2\right)-a^2 \left(17 b^3 x^3+b x\right)+3 a b^2 x^2+6 b^3 x^3\right)}{3 a^3 \left(a^2-1\right)^3 x^3}-\frac{2 \csc ^{-1}(a+b x)}{x^4}\right)","\frac{b^4 \csc ^{-1}(a+b x)}{4 a^4}+\frac{\left(3-8 a^2\right) b^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{24 a^2 \left(1-a^2\right)^2 x^2}-\frac{b (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{12 a \left(1-a^2\right) x^3}+\frac{\left(-8 a^6+8 a^4-7 a^2+2\right) b^4 \tan ^{-1}\left(\frac{a-\tan \left(\frac{1}{2} \csc ^{-1}(a+b x)\right)}{\sqrt{1-a^2}}\right)}{4 a^4 \left(1-a^2\right)^{7/2}}-\frac{\left(26 a^4-17 a^2+6\right) b^3 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{24 a^3 \left(1-a^2\right)^3 x}-\frac{\csc ^{-1}(a+b x)}{4 x^4}",1,"((b*Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2]*(2*a^7 - 6*a^6*b*x + 3*a*b^2*x^2 + 6*b^3*x^3 + a^3*(2 - 6*b^2*x^2) + 2*a^5*(-2 + 9*b^2*x^2) + a^4*b*x*(7 + 26*b^2*x^2) - a^2*(b*x + 17*b^3*x^3)))/(3*a^3*(-1 + a^2)^3*x^3) - (2*ArcCsc[a + b*x])/x^4 + (2*b^4*ArcSin[(a + b*x)^(-1)])/a^4 + (I*(-2 + 7*a^2 - 8*a^4 + 8*a^6)*b^4*Log[(16*a^4*(-1 + a^2)^3*((I*(-1 + a^2 + a*b*x))/Sqrt[1 - a^2] + (a + b*x)*Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2]))/((-2 + 7*a^2 - 8*a^4 + 8*a^6)*b^4*x)])/(a^4*(1 - a^2)^(7/2)))/8","C",1
27,1,420,366,5.3264035,"\int x^3 \csc ^{-1}(a+b x)^2 \, dx","Integrate[x^3*ArcCsc[a + b*x]^2,x]","\frac{192 \left(2 a^3+a\right) \left(i \left(\text{Li}_2\left(-e^{i \csc ^{-1}(a+b x)}\right)-\text{Li}_2\left(e^{i \csc ^{-1}(a+b x)}\right)\right)+\csc ^{-1}(a+b x) \left(\log \left(1-e^{i \csc ^{-1}(a+b x)}\right)-\log \left(1+e^{i \csc ^{-1}(a+b x)}\right)\right)\right)-64 \left(9 a^2+1\right) \log \left(\frac{1}{a+b x}\right)+2 \left(\left(36 a^2+3\right) \csc ^{-1}(a+b x)^2-24 a \csc ^{-1}(a+b x)+2\right) \csc ^2\left(\frac{1}{2} \csc ^{-1}(a+b x)\right)+2 \left(\left(36 a^2+3\right) \csc ^{-1}(a+b x)^2+24 a \csc ^{-1}(a+b x)+2\right) \sec ^2\left(\frac{1}{2} \csc ^{-1}(a+b x)\right)-16 \left(3 \left(2 a^3+a\right) \csc ^{-1}(a+b x)^2-2 \left(9 a^2+1\right) \csc ^{-1}(a+b x)+6 a\right) \cot \left(\frac{1}{2} \csc ^{-1}(a+b x)\right)-16 \left(3 \left(2 a^3+a\right) \csc ^{-1}(a+b x)^2+2 \left(9 a^2+1\right) \csc ^{-1}(a+b x)+6 a\right) \tan \left(\frac{1}{2} \csc ^{-1}(a+b x)\right)+3 \csc ^{-1}(a+b x)^2 \csc ^4\left(\frac{1}{2} \csc ^{-1}(a+b x)\right)-\frac{2 \csc ^{-1}(a+b x) \left(6 a \csc ^{-1}(a+b x)-1\right) \csc ^4\left(\frac{1}{2} \csc ^{-1}(a+b x)\right)}{a+b x}-32 (a+b x)^3 \csc ^{-1}(a+b x) \left(6 a \csc ^{-1}(a+b x)+1\right) \sin ^4\left(\frac{1}{2} \csc ^{-1}(a+b x)\right)+3 \csc ^{-1}(a+b x)^2 \sec ^4\left(\frac{1}{2} \csc ^{-1}(a+b x)\right)}{192 b^4}","-\frac{a^4 \csc ^{-1}(a+b x)^2}{4 b^4}+\frac{2 i a^3 \text{Li}_2\left(-e^{i \csc ^{-1}(a+b x)}\right)}{b^4}-\frac{2 i a^3 \text{Li}_2\left(e^{i \csc ^{-1}(a+b x)}\right)}{b^4}-\frac{4 a^3 \csc ^{-1}(a+b x) \tanh ^{-1}\left(e^{i \csc ^{-1}(a+b x)}\right)}{b^4}+\frac{3 a^2 \log (a+b x)}{b^4}+\frac{3 a^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^4}+\frac{i a \text{Li}_2\left(-e^{i \csc ^{-1}(a+b x)}\right)}{b^4}-\frac{i a \text{Li}_2\left(e^{i \csc ^{-1}(a+b x)}\right)}{b^4}+\frac{(a+b x)^2}{12 b^4}+\frac{\log (a+b x)}{3 b^4}-\frac{a (a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^4}+\frac{(a+b x)^3 \sqrt{1-\frac{1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{6 b^4}+\frac{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^4}-\frac{2 a \csc ^{-1}(a+b x) \tanh ^{-1}\left(e^{i \csc ^{-1}(a+b x)}\right)}{b^4}-\frac{a x}{b^3}+\frac{1}{4} x^4 \csc ^{-1}(a+b x)^2",1,"(-16*(6*a - 2*(1 + 9*a^2)*ArcCsc[a + b*x] + 3*(a + 2*a^3)*ArcCsc[a + b*x]^2)*Cot[ArcCsc[a + b*x]/2] + 2*(2 - 24*a*ArcCsc[a + b*x] + (3 + 36*a^2)*ArcCsc[a + b*x]^2)*Csc[ArcCsc[a + b*x]/2]^2 + 3*ArcCsc[a + b*x]^2*Csc[ArcCsc[a + b*x]/2]^4 - (2*ArcCsc[a + b*x]*(-1 + 6*a*ArcCsc[a + b*x])*Csc[ArcCsc[a + b*x]/2]^4)/(a + b*x) - 64*(1 + 9*a^2)*Log[(a + b*x)^(-1)] + 192*(a + 2*a^3)*(ArcCsc[a + b*x]*(Log[1 - E^(I*ArcCsc[a + b*x])] - Log[1 + E^(I*ArcCsc[a + b*x])]) + I*(PolyLog[2, -E^(I*ArcCsc[a + b*x])] - PolyLog[2, E^(I*ArcCsc[a + b*x])])) + 2*(2 + 24*a*ArcCsc[a + b*x] + (3 + 36*a^2)*ArcCsc[a + b*x]^2)*Sec[ArcCsc[a + b*x]/2]^2 + 3*ArcCsc[a + b*x]^2*Sec[ArcCsc[a + b*x]/2]^4 - 32*(a + b*x)^3*ArcCsc[a + b*x]*(1 + 6*a*ArcCsc[a + b*x])*Sin[ArcCsc[a + b*x]/2]^4 - 16*(6*a + 2*(1 + 9*a^2)*ArcCsc[a + b*x] + 3*(a + 2*a^3)*ArcCsc[a + b*x]^2)*Tan[ArcCsc[a + b*x]/2])/(192*b^4)","A",0
28,1,314,272,4.7409344,"\int x^2 \csc ^{-1}(a+b x)^2 \, dx","Integrate[x^2*ArcCsc[a + b*x]^2,x]","-\frac{8 \left(6 a^2+1\right) \left(i \left(\text{Li}_2\left(-e^{i \csc ^{-1}(a+b x)}\right)-\text{Li}_2\left(e^{i \csc ^{-1}(a+b x)}\right)\right)+\csc ^{-1}(a+b x) \left(\log \left(1-e^{i \csc ^{-1}(a+b x)}\right)-\log \left(1+e^{i \csc ^{-1}(a+b x)}\right)\right)\right)-2 \left(\left(6 a^2+1\right) \csc ^{-1}(a+b x)^2-12 a \csc ^{-1}(a+b x)+2\right) \cot \left(\frac{1}{2} \csc ^{-1}(a+b x)\right)-2 \left(\left(6 a^2+1\right) \csc ^{-1}(a+b x)^2+12 a \csc ^{-1}(a+b x)+2\right) \tan \left(\frac{1}{2} \csc ^{-1}(a+b x)\right)-48 a \log \left(\frac{1}{a+b x}\right)-\frac{\csc ^{-1}(a+b x)^2 \csc ^4\left(\frac{1}{2} \csc ^{-1}(a+b x)\right)}{2 (a+b x)}+2 \csc ^{-1}(a+b x) \left(3 a \csc ^{-1}(a+b x)-1\right) \csc ^2\left(\frac{1}{2} \csc ^{-1}(a+b x)\right)-8 (a+b x)^3 \csc ^{-1}(a+b x)^2 \sin ^4\left(\frac{1}{2} \csc ^{-1}(a+b x)\right)+2 \csc ^{-1}(a+b x) \left(3 a \csc ^{-1}(a+b x)+1\right) \sec ^2\left(\frac{1}{2} \csc ^{-1}(a+b x)\right)}{24 b^3}","\frac{a^3 \csc ^{-1}(a+b x)^2}{3 b^3}-\frac{2 i a^2 \text{Li}_2\left(-e^{i \csc ^{-1}(a+b x)}\right)}{b^3}+\frac{2 i a^2 \text{Li}_2\left(e^{i \csc ^{-1}(a+b x)}\right)}{b^3}+\frac{4 a^2 \csc ^{-1}(a+b x) \tanh ^{-1}\left(e^{i \csc ^{-1}(a+b x)}\right)}{b^3}-\frac{i \text{Li}_2\left(-e^{i \csc ^{-1}(a+b x)}\right)}{3 b^3}+\frac{i \text{Li}_2\left(e^{i \csc ^{-1}(a+b x)}\right)}{3 b^3}-\frac{2 a \log (a+b x)}{b^3}-\frac{2 a (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^3}+\frac{(a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^3}+\frac{2 \csc ^{-1}(a+b x) \tanh ^{-1}\left(e^{i \csc ^{-1}(a+b x)}\right)}{3 b^3}+\frac{1}{3} x^3 \csc ^{-1}(a+b x)^2+\frac{x}{3 b^2}",1,"-1/24*(-2*(2 - 12*a*ArcCsc[a + b*x] + (1 + 6*a^2)*ArcCsc[a + b*x]^2)*Cot[ArcCsc[a + b*x]/2] + 2*ArcCsc[a + b*x]*(-1 + 3*a*ArcCsc[a + b*x])*Csc[ArcCsc[a + b*x]/2]^2 - (ArcCsc[a + b*x]^2*Csc[ArcCsc[a + b*x]/2]^4)/(2*(a + b*x)) - 48*a*Log[(a + b*x)^(-1)] + 8*(1 + 6*a^2)*(ArcCsc[a + b*x]*(Log[1 - E^(I*ArcCsc[a + b*x])] - Log[1 + E^(I*ArcCsc[a + b*x])]) + I*(PolyLog[2, -E^(I*ArcCsc[a + b*x])] - PolyLog[2, E^(I*ArcCsc[a + b*x])])) + 2*ArcCsc[a + b*x]*(1 + 3*a*ArcCsc[a + b*x])*Sec[ArcCsc[a + b*x]/2]^2 - 8*(a + b*x)^3*ArcCsc[a + b*x]^2*Sin[ArcCsc[a + b*x]/2]^4 - 2*(2 + 12*a*ArcCsc[a + b*x] + (1 + 6*a^2)*ArcCsc[a + b*x]^2)*Tan[ArcCsc[a + b*x]/2])/b^3","A",0
29,1,213,145,0.8151124,"\int x \csc ^{-1}(a+b x)^2 \, dx","Integrate[x*ArcCsc[a + b*x]^2,x]","\frac{2 a \sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} \csc ^{-1}(a+b x)+2 b x \sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} \csc ^{-1}(a+b x)-a^2 \csc ^{-1}(a+b x)^2+b^2 x^2 \csc ^{-1}(a+b x)^2+4 i a \text{Li}_2\left(-e^{i \csc ^{-1}(a+b x)}\right)-4 i a \text{Li}_2\left(e^{i \csc ^{-1}(a+b x)}\right)-2 \log \left(\frac{1}{a+b x}\right)+4 a \csc ^{-1}(a+b x) \log \left(1-e^{i \csc ^{-1}(a+b x)}\right)-4 a \csc ^{-1}(a+b x) \log \left(1+e^{i \csc ^{-1}(a+b x)}\right)}{2 b^2}","-\frac{a^2 \csc ^{-1}(a+b x)^2}{2 b^2}+\frac{2 i a \text{Li}_2\left(-e^{i \csc ^{-1}(a+b x)}\right)}{b^2}-\frac{2 i a \text{Li}_2\left(e^{i \csc ^{-1}(a+b x)}\right)}{b^2}+\frac{\log (a+b x)}{b^2}+\frac{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^2}-\frac{4 a \csc ^{-1}(a+b x) \tanh ^{-1}\left(e^{i \csc ^{-1}(a+b x)}\right)}{b^2}+\frac{1}{2} x^2 \csc ^{-1}(a+b x)^2",1,"(2*a*Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2]*ArcCsc[a + b*x] + 2*b*x*Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2]*ArcCsc[a + b*x] - a^2*ArcCsc[a + b*x]^2 + b^2*x^2*ArcCsc[a + b*x]^2 + 4*a*ArcCsc[a + b*x]*Log[1 - E^(I*ArcCsc[a + b*x])] - 4*a*ArcCsc[a + b*x]*Log[1 + E^(I*ArcCsc[a + b*x])] - 2*Log[(a + b*x)^(-1)] + (4*I)*a*PolyLog[2, -E^(I*ArcCsc[a + b*x])] - (4*I)*a*PolyLog[2, E^(I*ArcCsc[a + b*x])])/(2*b^2)","A",0
30,1,99,86,0.1323248,"\int \csc ^{-1}(a+b x)^2 \, dx","Integrate[ArcCsc[a + b*x]^2,x]","\frac{-2 i \text{Li}_2\left(-e^{i \csc ^{-1}(a+b x)}\right)+2 i \text{Li}_2\left(e^{i \csc ^{-1}(a+b x)}\right)+\csc ^{-1}(a+b x) \left((a+b x) \csc ^{-1}(a+b x)-2 \log \left(1-e^{i \csc ^{-1}(a+b x)}\right)+2 \log \left(1+e^{i \csc ^{-1}(a+b x)}\right)\right)}{b}","-\frac{2 i \text{Li}_2\left(-e^{i \csc ^{-1}(a+b x)}\right)}{b}+\frac{2 i \text{Li}_2\left(e^{i \csc ^{-1}(a+b x)}\right)}{b}+\frac{(a+b x) \csc ^{-1}(a+b x)^2}{b}+\frac{4 \csc ^{-1}(a+b x) \tanh ^{-1}\left(e^{i \csc ^{-1}(a+b x)}\right)}{b}",1,"(ArcCsc[a + b*x]*((a + b*x)*ArcCsc[a + b*x] - 2*Log[1 - E^(I*ArcCsc[a + b*x])] + 2*Log[1 + E^(I*ArcCsc[a + b*x])]) - (2*I)*PolyLog[2, -E^(I*ArcCsc[a + b*x])] + (2*I)*PolyLog[2, E^(I*ArcCsc[a + b*x])])/b","A",0
31,1,408,324,0.3459604,"\int \frac{\csc ^{-1}(a+b x)^2}{x} \, dx","Integrate[ArcCsc[a + b*x]^2/x,x]","-2 i \csc ^{-1}(a+b x) \text{Li}_2\left(\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}-1}\right)-2 i \csc ^{-1}(a+b x) \text{Li}_2\left(-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)+2 \text{Li}_3\left(\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}-1}\right)+2 \text{Li}_3\left(-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)+\csc ^{-1}(a+b x)^2 \log \left(1-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}-1}\right)+\csc ^{-1}(a+b x)^2 \log \left(1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)-2 i \csc ^{-1}(a+b x) \text{Li}_2\left(e^{-i \csc ^{-1}(a+b x)}\right)+2 i \csc ^{-1}(a+b x) \text{Li}_2\left(-e^{i \csc ^{-1}(a+b x)}\right)-2 \text{Li}_3\left(e^{-i \csc ^{-1}(a+b x)}\right)-2 \text{Li}_3\left(-e^{i \csc ^{-1}(a+b x)}\right)-\frac{1}{3} i \csc ^{-1}(a+b x)^3-\csc ^{-1}(a+b x)^2 \log \left(1-e^{-i \csc ^{-1}(a+b x)}\right)-\csc ^{-1}(a+b x)^2 \log \left(1+e^{i \csc ^{-1}(a+b x)}\right)+\frac{i \pi ^3}{6}","-2 i \csc ^{-1}(a+b x) \text{Li}_2\left(-\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right)-2 i \csc ^{-1}(a+b x) \text{Li}_2\left(-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)+2 \text{Li}_3\left(-\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right)+2 \text{Li}_3\left(-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)+\csc ^{-1}(a+b x)^2 \log \left(1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right)+\csc ^{-1}(a+b x)^2 \log \left(1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)+i \csc ^{-1}(a+b x) \text{Li}_2\left(e^{2 i \csc ^{-1}(a+b x)}\right)-\frac{1}{2} \text{Li}_3\left(e^{2 i \csc ^{-1}(a+b x)}\right)-\csc ^{-1}(a+b x)^2 \log \left(1-e^{2 i \csc ^{-1}(a+b x)}\right)",1,"(I/6)*Pi^3 - (I/3)*ArcCsc[a + b*x]^3 - ArcCsc[a + b*x]^2*Log[1 - E^((-I)*ArcCsc[a + b*x])] - ArcCsc[a + b*x]^2*Log[1 + E^(I*ArcCsc[a + b*x])] + ArcCsc[a + b*x]^2*Log[1 - (I*a*E^(I*ArcCsc[a + b*x]))/(-1 + Sqrt[1 - a^2])] + ArcCsc[a + b*x]^2*Log[1 + (I*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2])] - (2*I)*ArcCsc[a + b*x]*PolyLog[2, E^((-I)*ArcCsc[a + b*x])] + (2*I)*ArcCsc[a + b*x]*PolyLog[2, -E^(I*ArcCsc[a + b*x])] - (2*I)*ArcCsc[a + b*x]*PolyLog[2, (I*a*E^(I*ArcCsc[a + b*x]))/(-1 + Sqrt[1 - a^2])] - (2*I)*ArcCsc[a + b*x]*PolyLog[2, ((-I)*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2])] - 2*PolyLog[3, E^((-I)*ArcCsc[a + b*x])] - 2*PolyLog[3, -E^(I*ArcCsc[a + b*x])] + 2*PolyLog[3, (I*a*E^(I*ArcCsc[a + b*x]))/(-1 + Sqrt[1 - a^2])] + 2*PolyLog[3, ((-I)*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2])]","A",0
32,1,802,254,3.1825061,"\int \frac{\csc ^{-1}(a+b x)^2}{x^2} \, dx","Integrate[ArcCsc[a + b*x]^2/x^2,x]","-\frac{b \left(\frac{(a+b x) \csc ^{-1}(a+b x)^2}{b x}+\frac{2 \pi  \tan ^{-1}\left(\frac{a-\tan \left(\frac{1}{2} \csc ^{-1}(a+b x)\right)}{\sqrt{1-a^2}}\right)}{\sqrt{1-a^2}}+\frac{2 \left(-2 \cos ^{-1}\left(\frac{1}{a}\right) \tanh ^{-1}\left(\frac{(a+1) \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(a+b x)+\pi \right)\right)}{\sqrt{a^2-1}}\right)+\left(\pi -2 \csc ^{-1}(a+b x)\right) \tanh ^{-1}\left(\frac{(a-1) \tan \left(\frac{1}{4} \left(2 \csc ^{-1}(a+b x)+\pi \right)\right)}{\sqrt{a^2-1}}\right)+\left(\cos ^{-1}\left(\frac{1}{a}\right)+2 i \left(\tanh ^{-1}\left(\frac{(a-1) \tan \left(\frac{1}{4} \left(2 \csc ^{-1}(a+b x)+\pi \right)\right)}{\sqrt{a^2-1}}\right)-\tanh ^{-1}\left(\frac{(a+1) \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(a+b x)+\pi \right)\right)}{\sqrt{a^2-1}}\right)\right)\right) \log \left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{a^2-1} e^{-\frac{1}{2} i \csc ^{-1}(a+b x)}}{\sqrt{a} \sqrt{-\frac{b x}{a+b x}}}\right)+\left(\cos ^{-1}\left(\frac{1}{a}\right)+2 i \tanh ^{-1}\left(\frac{(a+1) \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(a+b x)+\pi \right)\right)}{\sqrt{a^2-1}}\right)-2 i \tanh ^{-1}\left(\frac{(a-1) \tan \left(\frac{1}{4} \left(2 \csc ^{-1}(a+b x)+\pi \right)\right)}{\sqrt{a^2-1}}\right)\right) \log \left(\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \sqrt{a^2-1} e^{\frac{1}{2} i \csc ^{-1}(a+b x)}}{\sqrt{a} \sqrt{-\frac{b x}{a+b x}}}\right)-\left(\cos ^{-1}\left(\frac{1}{a}\right)-2 i \tanh ^{-1}\left(\frac{(a+1) \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(a+b x)+\pi \right)\right)}{\sqrt{a^2-1}}\right)\right) \log \left(\frac{(a-1) \left(i a+\sqrt{a^2-1}+i\right) \left(\cot \left(\frac{1}{4} \left(2 \csc ^{-1}(a+b x)+\pi \right)\right)-i\right)}{a \left(a+\sqrt{a^2-1} \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(a+b x)+\pi \right)\right)-1\right)}\right)-\left(\cos ^{-1}\left(\frac{1}{a}\right)+2 i \tanh ^{-1}\left(\frac{(a+1) \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(a+b x)+\pi \right)\right)}{\sqrt{a^2-1}}\right)\right) \log \left(\frac{(a-1) \left(-i a+\sqrt{a^2-1}-i\right) \left(\cot \left(\frac{1}{4} \left(2 \csc ^{-1}(a+b x)+\pi \right)\right)+i\right)}{a \left(a+\sqrt{a^2-1} \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(a+b x)+\pi \right)\right)-1\right)}\right)+i \left(\text{Li}_2\left(\frac{\left(i \sqrt{a^2-1}+1\right) \left(-a+\sqrt{a^2-1} \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(a+b x)+\pi \right)\right)+1\right)}{a \left(a+\sqrt{a^2-1} \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(a+b x)+\pi \right)\right)-1\right)}\right)-\text{Li}_2\left(\frac{\left(1-i \sqrt{a^2-1}\right) \left(-a+\sqrt{a^2-1} \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(a+b x)+\pi \right)\right)+1\right)}{a \left(a+\sqrt{a^2-1} \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(a+b x)+\pi \right)\right)-1\right)}\right)\right)\right)}{\sqrt{a^2-1}}\right)}{a}","-\frac{2 b \text{Li}_2\left(-\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right)}{a \sqrt{1-a^2}}+\frac{2 b \text{Li}_2\left(-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)}{a \sqrt{1-a^2}}-\frac{2 i b \csc ^{-1}(a+b x) \log \left(1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right)}{a \sqrt{1-a^2}}+\frac{2 i b \csc ^{-1}(a+b x) \log \left(1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)}{a \sqrt{1-a^2}}-\frac{b \csc ^{-1}(a+b x)^2}{a}-\frac{\csc ^{-1}(a+b x)^2}{x}",1,"-((b*(((a + b*x)*ArcCsc[a + b*x]^2)/(b*x) + (2*Pi*ArcTan[(a - Tan[ArcCsc[a + b*x]/2])/Sqrt[1 - a^2]])/Sqrt[1 - a^2] + (2*(-2*ArcCos[a^(-1)]*ArcTanh[((1 + a)*Cot[(Pi + 2*ArcCsc[a + b*x])/4])/Sqrt[-1 + a^2]] + (Pi - 2*ArcCsc[a + b*x])*ArcTanh[((-1 + a)*Tan[(Pi + 2*ArcCsc[a + b*x])/4])/Sqrt[-1 + a^2]] + (ArcCos[a^(-1)] + (2*I)*(-ArcTanh[((1 + a)*Cot[(Pi + 2*ArcCsc[a + b*x])/4])/Sqrt[-1 + a^2]] + ArcTanh[((-1 + a)*Tan[(Pi + 2*ArcCsc[a + b*x])/4])/Sqrt[-1 + a^2]]))*Log[((1/2 + I/2)*Sqrt[-1 + a^2])/(Sqrt[a]*E^((I/2)*ArcCsc[a + b*x])*Sqrt[-((b*x)/(a + b*x))])] + (ArcCos[a^(-1)] + (2*I)*ArcTanh[((1 + a)*Cot[(Pi + 2*ArcCsc[a + b*x])/4])/Sqrt[-1 + a^2]] - (2*I)*ArcTanh[((-1 + a)*Tan[(Pi + 2*ArcCsc[a + b*x])/4])/Sqrt[-1 + a^2]])*Log[((1/2 - I/2)*Sqrt[-1 + a^2]*E^((I/2)*ArcCsc[a + b*x]))/(Sqrt[a]*Sqrt[-((b*x)/(a + b*x))])] - (ArcCos[a^(-1)] - (2*I)*ArcTanh[((1 + a)*Cot[(Pi + 2*ArcCsc[a + b*x])/4])/Sqrt[-1 + a^2]])*Log[((-1 + a)*(I + I*a + Sqrt[-1 + a^2])*(-I + Cot[(Pi + 2*ArcCsc[a + b*x])/4]))/(a*(-1 + a + Sqrt[-1 + a^2]*Cot[(Pi + 2*ArcCsc[a + b*x])/4]))] - (ArcCos[a^(-1)] + (2*I)*ArcTanh[((1 + a)*Cot[(Pi + 2*ArcCsc[a + b*x])/4])/Sqrt[-1 + a^2]])*Log[((-1 + a)*(-I - I*a + Sqrt[-1 + a^2])*(I + Cot[(Pi + 2*ArcCsc[a + b*x])/4]))/(a*(-1 + a + Sqrt[-1 + a^2]*Cot[(Pi + 2*ArcCsc[a + b*x])/4]))] + I*(-PolyLog[2, ((1 - I*Sqrt[-1 + a^2])*(1 - a + Sqrt[-1 + a^2]*Cot[(Pi + 2*ArcCsc[a + b*x])/4]))/(a*(-1 + a + Sqrt[-1 + a^2]*Cot[(Pi + 2*ArcCsc[a + b*x])/4]))] + PolyLog[2, ((1 + I*Sqrt[-1 + a^2])*(1 - a + Sqrt[-1 + a^2]*Cot[(Pi + 2*ArcCsc[a + b*x])/4]))/(a*(-1 + a + Sqrt[-1 + a^2]*Cot[(Pi + 2*ArcCsc[a + b*x])/4]))])))/Sqrt[-1 + a^2]))/a)","B",1
33,1,656,464,8.8883175,"\int x^2 \csc ^{-1}(a+b x)^3 \, dx","Integrate[x^2*ArcCsc[a + b*x]^3,x]","-\frac{24 i \left(6 a^2+1\right) \csc ^{-1}(a+b x) \text{Li}_2\left(-e^{i \csc ^{-1}(a+b x)}\right)-24 i \left(6 a^2+1\right) \csc ^{-1}(a+b x) \text{Li}_2\left(e^{i \csc ^{-1}(a+b x)}\right)-144 a^2 \text{Li}_3\left(-e^{i \csc ^{-1}(a+b x)}\right)+144 a^2 \text{Li}_3\left(e^{i \csc ^{-1}(a+b x)}\right)-12 a^2 \csc ^{-1}(a+b x)^3 \cot \left(\frac{1}{2} \csc ^{-1}(a+b x)\right)+72 a^2 \csc ^{-1}(a+b x)^2 \log \left(1-e^{i \csc ^{-1}(a+b x)}\right)-72 a^2 \csc ^{-1}(a+b x)^2 \log \left(1+e^{i \csc ^{-1}(a+b x)}\right)-12 a^2 \csc ^{-1}(a+b x)^3 \tan \left(\frac{1}{2} \csc ^{-1}(a+b x)\right)+72 i a \text{Li}_2\left(e^{2 i \csc ^{-1}(a+b x)}\right)-24 \text{Li}_3\left(-e^{i \csc ^{-1}(a+b x)}\right)+24 \text{Li}_3\left(e^{i \csc ^{-1}(a+b x)}\right)+72 i a \csc ^{-1}(a+b x)^2-\frac{\csc ^{-1}(a+b x)^3 \csc ^4\left(\frac{1}{2} \csc ^{-1}(a+b x)\right)}{2 (a+b x)}+6 a \csc ^{-1}(a+b x)^3 \csc ^2\left(\frac{1}{2} \csc ^{-1}(a+b x)\right)-3 \csc ^{-1}(a+b x)^2 \csc ^2\left(\frac{1}{2} \csc ^{-1}(a+b x)\right)-2 \csc ^{-1}(a+b x)^3 \cot \left(\frac{1}{2} \csc ^{-1}(a+b x)\right)+36 a \csc ^{-1}(a+b x)^2 \cot \left(\frac{1}{2} \csc ^{-1}(a+b x)\right)-12 \csc ^{-1}(a+b x) \cot \left(\frac{1}{2} \csc ^{-1}(a+b x)\right)+12 \csc ^{-1}(a+b x)^2 \log \left(1-e^{i \csc ^{-1}(a+b x)}\right)-12 \csc ^{-1}(a+b x)^2 \log \left(1+e^{i \csc ^{-1}(a+b x)}\right)-144 a \csc ^{-1}(a+b x) \log \left(1-e^{2 i \csc ^{-1}(a+b x)}\right)-8 (a+b x)^3 \csc ^{-1}(a+b x)^3 \sin ^4\left(\frac{1}{2} \csc ^{-1}(a+b x)\right)-2 \csc ^{-1}(a+b x)^3 \tan \left(\frac{1}{2} \csc ^{-1}(a+b x)\right)-36 a \csc ^{-1}(a+b x)^2 \tan \left(\frac{1}{2} \csc ^{-1}(a+b x)\right)-12 \csc ^{-1}(a+b x) \tan \left(\frac{1}{2} \csc ^{-1}(a+b x)\right)+6 a \csc ^{-1}(a+b x)^3 \sec ^2\left(\frac{1}{2} \csc ^{-1}(a+b x)\right)+3 \csc ^{-1}(a+b x)^2 \sec ^2\left(\frac{1}{2} \csc ^{-1}(a+b x)\right)+24 \log \left(\tan \left(\frac{1}{2} \csc ^{-1}(a+b x)\right)\right)}{24 b^3}","\frac{a^3 \csc ^{-1}(a+b x)^3}{3 b^3}-\frac{6 i a^2 \csc ^{-1}(a+b x) \text{Li}_2\left(-e^{i \csc ^{-1}(a+b x)}\right)}{b^3}+\frac{6 i a^2 \csc ^{-1}(a+b x) \text{Li}_2\left(e^{i \csc ^{-1}(a+b x)}\right)}{b^3}+\frac{6 a^2 \text{Li}_3\left(-e^{i \csc ^{-1}(a+b x)}\right)}{b^3}-\frac{6 a^2 \text{Li}_3\left(e^{i \csc ^{-1}(a+b x)}\right)}{b^3}+\frac{6 a^2 \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left(e^{i \csc ^{-1}(a+b x)}\right)}{b^3}-\frac{i \csc ^{-1}(a+b x) \text{Li}_2\left(-e^{i \csc ^{-1}(a+b x)}\right)}{b^3}+\frac{i \csc ^{-1}(a+b x) \text{Li}_2\left(e^{i \csc ^{-1}(a+b x)}\right)}{b^3}-\frac{3 i a \text{Li}_2\left(e^{2 i \csc ^{-1}(a+b x)}\right)}{b^3}+\frac{\text{Li}_3\left(-e^{i \csc ^{-1}(a+b x)}\right)}{b^3}-\frac{\text{Li}_3\left(e^{i \csc ^{-1}(a+b x)}\right)}{b^3}+\frac{(a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{2 b^3}-\frac{3 a (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{b^3}-\frac{3 i a \csc ^{-1}(a+b x)^2}{b^3}+\frac{(a+b x) \csc ^{-1}(a+b x)}{b^3}+\frac{\tanh ^{-1}\left(\sqrt{1-\frac{1}{(a+b x)^2}}\right)}{b^3}+\frac{6 a \csc ^{-1}(a+b x) \log \left(1-e^{2 i \csc ^{-1}(a+b x)}\right)}{b^3}+\frac{\csc ^{-1}(a+b x)^2 \tanh ^{-1}\left(e^{i \csc ^{-1}(a+b x)}\right)}{b^3}+\frac{1}{3} x^3 \csc ^{-1}(a+b x)^3",1,"-1/24*((72*I)*a*ArcCsc[a + b*x]^2 - 12*ArcCsc[a + b*x]*Cot[ArcCsc[a + b*x]/2] + 36*a*ArcCsc[a + b*x]^2*Cot[ArcCsc[a + b*x]/2] - 2*ArcCsc[a + b*x]^3*Cot[ArcCsc[a + b*x]/2] - 12*a^2*ArcCsc[a + b*x]^3*Cot[ArcCsc[a + b*x]/2] - 3*ArcCsc[a + b*x]^2*Csc[ArcCsc[a + b*x]/2]^2 + 6*a*ArcCsc[a + b*x]^3*Csc[ArcCsc[a + b*x]/2]^2 - (ArcCsc[a + b*x]^3*Csc[ArcCsc[a + b*x]/2]^4)/(2*(a + b*x)) + 12*ArcCsc[a + b*x]^2*Log[1 - E^(I*ArcCsc[a + b*x])] + 72*a^2*ArcCsc[a + b*x]^2*Log[1 - E^(I*ArcCsc[a + b*x])] - 12*ArcCsc[a + b*x]^2*Log[1 + E^(I*ArcCsc[a + b*x])] - 72*a^2*ArcCsc[a + b*x]^2*Log[1 + E^(I*ArcCsc[a + b*x])] - 144*a*ArcCsc[a + b*x]*Log[1 - E^((2*I)*ArcCsc[a + b*x])] + 24*Log[Tan[ArcCsc[a + b*x]/2]] + (24*I)*(1 + 6*a^2)*ArcCsc[a + b*x]*PolyLog[2, -E^(I*ArcCsc[a + b*x])] - (24*I)*(1 + 6*a^2)*ArcCsc[a + b*x]*PolyLog[2, E^(I*ArcCsc[a + b*x])] + (72*I)*a*PolyLog[2, E^((2*I)*ArcCsc[a + b*x])] - 24*PolyLog[3, -E^(I*ArcCsc[a + b*x])] - 144*a^2*PolyLog[3, -E^(I*ArcCsc[a + b*x])] + 24*PolyLog[3, E^(I*ArcCsc[a + b*x])] + 144*a^2*PolyLog[3, E^(I*ArcCsc[a + b*x])] + 3*ArcCsc[a + b*x]^2*Sec[ArcCsc[a + b*x]/2]^2 + 6*a*ArcCsc[a + b*x]^3*Sec[ArcCsc[a + b*x]/2]^2 - 8*(a + b*x)^3*ArcCsc[a + b*x]^3*Sin[ArcCsc[a + b*x]/2]^4 - 12*ArcCsc[a + b*x]*Tan[ArcCsc[a + b*x]/2] - 36*a*ArcCsc[a + b*x]^2*Tan[ArcCsc[a + b*x]/2] - 2*ArcCsc[a + b*x]^3*Tan[ArcCsc[a + b*x]/2] - 12*a^2*ArcCsc[a + b*x]^3*Tan[ArcCsc[a + b*x]/2])/b^3","A",0
34,1,314,264,0.8019691,"\int x \csc ^{-1}(a+b x)^3 \, dx","Integrate[x*ArcCsc[a + b*x]^3,x]","\frac{3 a \sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2+3 b x \sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2-a^2 \csc ^{-1}(a+b x)^3+b^2 x^2 \csc ^{-1}(a+b x)^3+12 i a \csc ^{-1}(a+b x) \text{Li}_2\left(-e^{i \csc ^{-1}(a+b x)}\right)-12 i a \csc ^{-1}(a+b x) \text{Li}_2\left(e^{i \csc ^{-1}(a+b x)}\right)+3 i \text{Li}_2\left(e^{2 i \csc ^{-1}(a+b x)}\right)-12 a \text{Li}_3\left(-e^{i \csc ^{-1}(a+b x)}\right)+12 a \text{Li}_3\left(e^{i \csc ^{-1}(a+b x)}\right)+3 i \csc ^{-1}(a+b x)^2+6 a \csc ^{-1}(a+b x)^2 \log \left(1-e^{i \csc ^{-1}(a+b x)}\right)-6 a \csc ^{-1}(a+b x)^2 \log \left(1+e^{i \csc ^{-1}(a+b x)}\right)-6 \csc ^{-1}(a+b x) \log \left(1-e^{2 i \csc ^{-1}(a+b x)}\right)}{2 b^2}","-\frac{a^2 \csc ^{-1}(a+b x)^3}{2 b^2}+\frac{6 i a \csc ^{-1}(a+b x) \text{Li}_2\left(-e^{i \csc ^{-1}(a+b x)}\right)}{b^2}-\frac{6 i a \csc ^{-1}(a+b x) \text{Li}_2\left(e^{i \csc ^{-1}(a+b x)}\right)}{b^2}+\frac{3 i \text{Li}_2\left(e^{2 i \csc ^{-1}(a+b x)}\right)}{2 b^2}-\frac{6 a \text{Li}_3\left(-e^{i \csc ^{-1}(a+b x)}\right)}{b^2}+\frac{6 a \text{Li}_3\left(e^{i \csc ^{-1}(a+b x)}\right)}{b^2}+\frac{3 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{2 b^2}+\frac{3 i \csc ^{-1}(a+b x)^2}{2 b^2}-\frac{3 \csc ^{-1}(a+b x) \log \left(1-e^{2 i \csc ^{-1}(a+b x)}\right)}{b^2}-\frac{6 a \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left(e^{i \csc ^{-1}(a+b x)}\right)}{b^2}+\frac{1}{2} x^2 \csc ^{-1}(a+b x)^3",1,"((3*I)*ArcCsc[a + b*x]^2 + 3*a*Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2]*ArcCsc[a + b*x]^2 + 3*b*x*Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2]*ArcCsc[a + b*x]^2 - a^2*ArcCsc[a + b*x]^3 + b^2*x^2*ArcCsc[a + b*x]^3 + 6*a*ArcCsc[a + b*x]^2*Log[1 - E^(I*ArcCsc[a + b*x])] - 6*a*ArcCsc[a + b*x]^2*Log[1 + E^(I*ArcCsc[a + b*x])] - 6*ArcCsc[a + b*x]*Log[1 - E^((2*I)*ArcCsc[a + b*x])] + (12*I)*a*ArcCsc[a + b*x]*PolyLog[2, -E^(I*ArcCsc[a + b*x])] - (12*I)*a*ArcCsc[a + b*x]*PolyLog[2, E^(I*ArcCsc[a + b*x])] + (3*I)*PolyLog[2, E^((2*I)*ArcCsc[a + b*x])] - 12*a*PolyLog[3, -E^(I*ArcCsc[a + b*x])] + 12*a*PolyLog[3, E^(I*ArcCsc[a + b*x])])/(2*b^2)","A",0
35,1,162,140,0.1090402,"\int \csc ^{-1}(a+b x)^3 \, dx","Integrate[ArcCsc[a + b*x]^3,x]","\frac{-6 i \csc ^{-1}(a+b x) \text{Li}_2\left(-e^{i \csc ^{-1}(a+b x)}\right)+6 i \csc ^{-1}(a+b x) \text{Li}_2\left(e^{i \csc ^{-1}(a+b x)}\right)+6 \text{Li}_3\left(-e^{i \csc ^{-1}(a+b x)}\right)-6 \text{Li}_3\left(e^{i \csc ^{-1}(a+b x)}\right)+a \csc ^{-1}(a+b x)^3+b x \csc ^{-1}(a+b x)^3-3 \csc ^{-1}(a+b x)^2 \log \left(1-e^{i \csc ^{-1}(a+b x)}\right)+3 \csc ^{-1}(a+b x)^2 \log \left(1+e^{i \csc ^{-1}(a+b x)}\right)}{b}","-\frac{6 i \csc ^{-1}(a+b x) \text{Li}_2\left(-e^{i \csc ^{-1}(a+b x)}\right)}{b}+\frac{6 i \csc ^{-1}(a+b x) \text{Li}_2\left(e^{i \csc ^{-1}(a+b x)}\right)}{b}+\frac{6 \text{Li}_3\left(-e^{i \csc ^{-1}(a+b x)}\right)}{b}-\frac{6 \text{Li}_3\left(e^{i \csc ^{-1}(a+b x)}\right)}{b}+\frac{(a+b x) \csc ^{-1}(a+b x)^3}{b}+\frac{6 \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left(e^{i \csc ^{-1}(a+b x)}\right)}{b}",1,"(a*ArcCsc[a + b*x]^3 + b*x*ArcCsc[a + b*x]^3 - 3*ArcCsc[a + b*x]^2*Log[1 - E^(I*ArcCsc[a + b*x])] + 3*ArcCsc[a + b*x]^2*Log[1 + E^(I*ArcCsc[a + b*x])] - (6*I)*ArcCsc[a + b*x]*PolyLog[2, -E^(I*ArcCsc[a + b*x])] + (6*I)*ArcCsc[a + b*x]*PolyLog[2, E^(I*ArcCsc[a + b*x])] + 6*PolyLog[3, -E^(I*ArcCsc[a + b*x])] - 6*PolyLog[3, E^(I*ArcCsc[a + b*x])])/b","A",0
36,1,554,448,0.3278774,"\int \frac{\csc ^{-1}(a+b x)^3}{x} \, dx","Integrate[ArcCsc[a + b*x]^3/x,x]","-3 i \csc ^{-1}(a+b x)^2 \text{Li}_2\left(\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}-1}\right)-3 i \csc ^{-1}(a+b x)^2 \text{Li}_2\left(-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)+6 \csc ^{-1}(a+b x) \text{Li}_3\left(\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}-1}\right)+6 \csc ^{-1}(a+b x) \text{Li}_3\left(-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)+6 i \text{Li}_4\left(\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}-1}\right)+6 i \text{Li}_4\left(-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)+\csc ^{-1}(a+b x)^3 \log \left(1-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}-1}\right)+\csc ^{-1}(a+b x)^3 \log \left(1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)-3 i \csc ^{-1}(a+b x)^2 \text{Li}_2\left(e^{-i \csc ^{-1}(a+b x)}\right)+3 i \csc ^{-1}(a+b x)^2 \text{Li}_2\left(-e^{i \csc ^{-1}(a+b x)}\right)-6 \csc ^{-1}(a+b x) \text{Li}_3\left(e^{-i \csc ^{-1}(a+b x)}\right)-6 \csc ^{-1}(a+b x) \text{Li}_3\left(-e^{i \csc ^{-1}(a+b x)}\right)+6 i \text{Li}_4\left(e^{-i \csc ^{-1}(a+b x)}\right)-6 i \text{Li}_4\left(-e^{i \csc ^{-1}(a+b x)}\right)-\frac{1}{4} i \csc ^{-1}(a+b x)^4-\csc ^{-1}(a+b x)^3 \log \left(1-e^{-i \csc ^{-1}(a+b x)}\right)-\csc ^{-1}(a+b x)^3 \log \left(1+e^{i \csc ^{-1}(a+b x)}\right)+\frac{i \pi ^4}{8}","-3 i \csc ^{-1}(a+b x)^2 \text{Li}_2\left(-\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right)-3 i \csc ^{-1}(a+b x)^2 \text{Li}_2\left(-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)+6 \csc ^{-1}(a+b x) \text{Li}_3\left(-\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right)+6 \csc ^{-1}(a+b x) \text{Li}_3\left(-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)+6 i \text{Li}_4\left(-\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right)+6 i \text{Li}_4\left(-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)+\csc ^{-1}(a+b x)^3 \log \left(1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right)+\csc ^{-1}(a+b x)^3 \log \left(1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)+\frac{3}{2} i \csc ^{-1}(a+b x)^2 \text{Li}_2\left(e^{2 i \csc ^{-1}(a+b x)}\right)-\frac{3}{2} \csc ^{-1}(a+b x) \text{Li}_3\left(e^{2 i \csc ^{-1}(a+b x)}\right)-\frac{3}{4} i \text{Li}_4\left(e^{2 i \csc ^{-1}(a+b x)}\right)-\csc ^{-1}(a+b x)^3 \log \left(1-e^{2 i \csc ^{-1}(a+b x)}\right)",1,"(I/8)*Pi^4 - (I/4)*ArcCsc[a + b*x]^4 - ArcCsc[a + b*x]^3*Log[1 - E^((-I)*ArcCsc[a + b*x])] - ArcCsc[a + b*x]^3*Log[1 + E^(I*ArcCsc[a + b*x])] + ArcCsc[a + b*x]^3*Log[1 - (I*a*E^(I*ArcCsc[a + b*x]))/(-1 + Sqrt[1 - a^2])] + ArcCsc[a + b*x]^3*Log[1 + (I*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2])] - (3*I)*ArcCsc[a + b*x]^2*PolyLog[2, E^((-I)*ArcCsc[a + b*x])] + (3*I)*ArcCsc[a + b*x]^2*PolyLog[2, -E^(I*ArcCsc[a + b*x])] - (3*I)*ArcCsc[a + b*x]^2*PolyLog[2, (I*a*E^(I*ArcCsc[a + b*x]))/(-1 + Sqrt[1 - a^2])] - (3*I)*ArcCsc[a + b*x]^2*PolyLog[2, ((-I)*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2])] - 6*ArcCsc[a + b*x]*PolyLog[3, E^((-I)*ArcCsc[a + b*x])] - 6*ArcCsc[a + b*x]*PolyLog[3, -E^(I*ArcCsc[a + b*x])] + 6*ArcCsc[a + b*x]*PolyLog[3, (I*a*E^(I*ArcCsc[a + b*x]))/(-1 + Sqrt[1 - a^2])] + 6*ArcCsc[a + b*x]*PolyLog[3, ((-I)*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2])] + (6*I)*PolyLog[4, E^((-I)*ArcCsc[a + b*x])] - (6*I)*PolyLog[4, -E^(I*ArcCsc[a + b*x])] + (6*I)*PolyLog[4, (I*a*E^(I*ArcCsc[a + b*x]))/(-1 + Sqrt[1 - a^2])] + (6*I)*PolyLog[4, ((-I)*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2])]","A",0
37,1,289,378,0.5103165,"\int \frac{\csc ^{-1}(a+b x)^3}{x^2} \, dx","Integrate[ArcCsc[a + b*x]^3/x^2,x]","-\frac{\frac{(a+b x) \csc ^{-1}(a+b x)^3}{x}+\frac{3 i b \left(-2 i \csc ^{-1}(a+b x) \text{Li}_2\left(\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}-1}\right)+2 i \csc ^{-1}(a+b x) \text{Li}_2\left(-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)+2 \text{Li}_3\left(\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}-1}\right)-2 \text{Li}_3\left(-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)+\csc ^{-1}(a+b x)^2 \log \left(1-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}-1}\right)-\csc ^{-1}(a+b x)^2 \log \left(1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)\right)}{\sqrt{1-a^2}}}{a}","-\frac{6 b \csc ^{-1}(a+b x) \text{Li}_2\left(-\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right)}{a \sqrt{1-a^2}}+\frac{6 b \csc ^{-1}(a+b x) \text{Li}_2\left(-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)}{a \sqrt{1-a^2}}-\frac{6 i b \text{Li}_3\left(-\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right)}{a \sqrt{1-a^2}}+\frac{6 i b \text{Li}_3\left(-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)}{a \sqrt{1-a^2}}-\frac{3 i b \csc ^{-1}(a+b x)^2 \log \left(1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right)}{a \sqrt{1-a^2}}+\frac{3 i b \csc ^{-1}(a+b x)^2 \log \left(1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)}{a \sqrt{1-a^2}}-\frac{b \csc ^{-1}(a+b x)^3}{a}-\frac{\csc ^{-1}(a+b x)^3}{x}",1,"-((((a + b*x)*ArcCsc[a + b*x]^3)/x + ((3*I)*b*(ArcCsc[a + b*x]^2*Log[1 - (I*a*E^(I*ArcCsc[a + b*x]))/(-1 + Sqrt[1 - a^2])] - ArcCsc[a + b*x]^2*Log[1 + (I*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2])] - (2*I)*ArcCsc[a + b*x]*PolyLog[2, (I*a*E^(I*ArcCsc[a + b*x]))/(-1 + Sqrt[1 - a^2])] + (2*I)*ArcCsc[a + b*x]*PolyLog[2, ((-I)*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2])] + 2*PolyLog[3, (I*a*E^(I*ArcCsc[a + b*x]))/(-1 + Sqrt[1 - a^2])] - 2*PolyLog[3, ((-I)*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2])]))/Sqrt[1 - a^2])/a)","A",1
38,1,127,48,0.2456315,"\int x^3 \csc ^{-1}\left(a+b x^4\right) \, dx","Integrate[x^3*ArcCsc[a + b*x^4],x]","\frac{\sqrt{\left(a+b x^4\right)^2-1} \left(\log \left(\frac{a+b x^4}{\sqrt{\left(a+b x^4\right)^2-1}}+1\right)-\log \left(1-\frac{a+b x^4}{\sqrt{\left(a+b x^4\right)^2-1}}\right)\right)}{8 b \left(a+b x^4\right) \sqrt{1-\frac{1}{\left(a+b x^4\right)^2}}}+\frac{\left(a+b x^4\right) \csc ^{-1}\left(a+b x^4\right)}{4 b}","\frac{\left(a+b x^4\right) \csc ^{-1}\left(a+b x^4\right)}{4 b}+\frac{\tanh ^{-1}\left(\sqrt{1-\frac{1}{\left(a+b x^4\right)^2}}\right)}{4 b}",1,"((a + b*x^4)*ArcCsc[a + b*x^4])/(4*b) + (Sqrt[-1 + (a + b*x^4)^2]*(-Log[1 - (a + b*x^4)/Sqrt[-1 + (a + b*x^4)^2]] + Log[1 + (a + b*x^4)/Sqrt[-1 + (a + b*x^4)^2]]))/(8*b*(a + b*x^4)*Sqrt[1 - (a + b*x^4)^(-2)])","B",1
39,1,130,48,0.2882186,"\int x^{-1+n} \csc ^{-1}\left(a+b x^n\right) \, dx","Integrate[x^(-1 + n)*ArcCsc[a + b*x^n],x]","\frac{\sqrt{\left(a+b x^n\right)^2-1} \left(\log \left(\frac{a+b x^n}{\sqrt{\left(a+b x^n\right)^2-1}}+1\right)-\log \left(1-\frac{a+b x^n}{\sqrt{\left(a+b x^n\right)^2-1}}\right)\right)}{2 b n \left(a+b x^n\right) \sqrt{1-\frac{1}{\left(a+b x^n\right)^2}}}+\frac{\left(a+b x^n\right) \csc ^{-1}\left(a+b x^n\right)}{b n}","\frac{\left(a+b x^n\right) \csc ^{-1}\left(a+b x^n\right)}{b n}+\frac{\tanh ^{-1}\left(\sqrt{1-\frac{1}{\left(a+b x^n\right)^2}}\right)}{b n}",1,"((a + b*x^n)*ArcCsc[a + b*x^n])/(b*n) + (Sqrt[-1 + (a + b*x^n)^2]*(-Log[1 - (a + b*x^n)/Sqrt[-1 + (a + b*x^n)^2]] + Log[1 + (a + b*x^n)/Sqrt[-1 + (a + b*x^n)^2]]))/(2*b*n*(a + b*x^n)*Sqrt[1 - (a + b*x^n)^(-2)])","B",1
40,1,280,85,0.7255833,"\int \csc ^{-1}\left(c e^{a+b x}\right) \, dx","Integrate[ArcCsc[c*E^(a + b*x)],x]","\frac{e^{-a-b x} \left(-4 \sqrt{1-c^2 e^{2 (a+b x)}} \text{Li}_2\left(\frac{1}{2} \left(1-\sqrt{1-c^2 e^{2 (a+b x)}}\right)\right)+\sqrt{1-c^2 e^{2 (a+b x)}} \left(\log ^2\left(c^2 e^{2 (a+b x)}\right)+2 \log ^2\left(\frac{1}{2} \left(\sqrt{1-c^2 e^{2 (a+b x)}}+1\right)\right)-4 \log \left(\frac{1}{2} \left(\sqrt{1-c^2 e^{2 (a+b x)}}+1\right)\right) \log \left(c^2 e^{2 (a+b x)}\right)\right)+4 \sqrt{c^2 e^{2 (a+b x)}-1} \left(2 b x-\log \left(c^2 e^{2 (a+b x)}\right)\right) \tan ^{-1}\left(\sqrt{c^2 e^{2 (a+b x)}-1}\right)\right)}{8 b c \sqrt{1-\frac{e^{-2 (a+b x)}}{c^2}}}+x \csc ^{-1}\left(c e^{a+b x}\right)","\frac{i \text{Li}_2\left(e^{2 i \csc ^{-1}\left(c e^{a+b x}\right)}\right)}{2 b}+\frac{i \csc ^{-1}\left(c e^{a+b x}\right)^2}{2 b}-\frac{\csc ^{-1}\left(c e^{a+b x}\right) \log \left(1-e^{2 i \csc ^{-1}\left(c e^{a+b x}\right)}\right)}{b}",1,"x*ArcCsc[c*E^(a + b*x)] + (E^(-a - b*x)*(4*Sqrt[-1 + c^2*E^(2*(a + b*x))]*ArcTan[Sqrt[-1 + c^2*E^(2*(a + b*x))]]*(2*b*x - Log[c^2*E^(2*(a + b*x))]) + Sqrt[1 - c^2*E^(2*(a + b*x))]*(Log[c^2*E^(2*(a + b*x))]^2 - 4*Log[c^2*E^(2*(a + b*x))]*Log[(1 + Sqrt[1 - c^2*E^(2*(a + b*x))])/2] + 2*Log[(1 + Sqrt[1 - c^2*E^(2*(a + b*x))])/2]^2) - 4*Sqrt[1 - c^2*E^(2*(a + b*x))]*PolyLog[2, (1 - Sqrt[1 - c^2*E^(2*(a + b*x))])/2]))/(8*b*c*Sqrt[1 - 1/(c^2*E^(2*(a + b*x)))])","B",1
41,1,79,95,0.4543748,"\int e^{\csc ^{-1}(a x)} x^2 \, dx","Integrate[E^ArcCsc[a*x]*x^2,x]","\frac{e^{\csc ^{-1}(a x)} \left(a^3 x^3 \left(-\cos \left(2 \csc ^{-1}(a x)\right)+\sin \left(2 \csc ^{-1}(a x)\right)+5\right)+(4+4 i) e^{i \csc ^{-1}(a x)} \, _2F_1\left(\frac{1}{2}-\frac{i}{2},1;\frac{3}{2}-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right)\right)}{12 a^3}","\frac{\left(\frac{4}{5}-\frac{12 i}{5}\right) e^{(1+3 i) \csc ^{-1}(a x)} \, _2F_1\left(\frac{3}{2}-\frac{i}{2},3;\frac{5}{2}-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right)}{a^3}-\frac{\left(\frac{8}{5}-\frac{24 i}{5}\right) e^{(1+3 i) \csc ^{-1}(a x)} \, _2F_1\left(\frac{3}{2}-\frac{i}{2},4;\frac{5}{2}-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right)}{a^3}",1,"(E^ArcCsc[a*x]*((4 + 4*I)*E^(I*ArcCsc[a*x])*Hypergeometric2F1[1/2 - I/2, 1, 3/2 - I/2, E^((2*I)*ArcCsc[a*x])] + a^3*x^3*(5 - Cos[2*ArcCsc[a*x]] + Sin[2*ArcCsc[a*x]])))/(12*a^3)","A",0
42,1,101,87,0.2626107,"\int e^{\csc ^{-1}(a x)} x \, dx","Integrate[E^ArcCsc[a*x]*x,x]","\frac{\left(\frac{1}{5}+\frac{i}{10}\right) e^{\csc ^{-1}(a x)} \left((2-i) a x \left(\sqrt{1-\frac{1}{a^2 x^2}}+a x\right)+(1+2 i) \, _2F_1\left(-\frac{i}{2},1;1-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right)+e^{2 i \csc ^{-1}(a x)} \, _2F_1\left(1,1-\frac{i}{2};2-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right)\right)}{a^2}","\frac{\left(\frac{8}{5}+\frac{4 i}{5}\right) e^{(1+2 i) \csc ^{-1}(a x)} \, _2F_1\left(1-\frac{i}{2},2;2-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right)}{a^2}-\frac{\left(\frac{16}{5}+\frac{8 i}{5}\right) e^{(1+2 i) \csc ^{-1}(a x)} \, _2F_1\left(1-\frac{i}{2},3;2-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right)}{a^2}",1,"((1/5 + I/10)*E^ArcCsc[a*x]*((2 - I)*a*x*(Sqrt[1 - 1/(a^2*x^2)] + a*x) + (1 + 2*I)*Hypergeometric2F1[-1/2*I, 1, 1 - I/2, E^((2*I)*ArcCsc[a*x])] + E^((2*I)*ArcCsc[a*x])*Hypergeometric2F1[1, 1 - I/2, 2 - I/2, E^((2*I)*ArcCsc[a*x])]))/a^2","A",0
43,1,54,87,0.0922056,"\int e^{\csc ^{-1}(a x)} \, dx","Integrate[E^ArcCsc[a*x],x]","\frac{e^{\csc ^{-1}(a x)} \left(a x+(1+i) e^{i \csc ^{-1}(a x)} \, _2F_1\left(\frac{1}{2}-\frac{i}{2},1;\frac{3}{2}-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right)\right)}{a}","\frac{(2-2 i) e^{(1+i) \csc ^{-1}(a x)} \, _2F_1\left(\frac{1}{2}-\frac{i}{2},2;\frac{3}{2}-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right)}{a}-\frac{(1-i) e^{(1+i) \csc ^{-1}(a x)} \, _2F_1\left(\frac{1}{2}-\frac{i}{2},1;\frac{3}{2}-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right)}{a}",1,"(E^ArcCsc[a*x]*(a*x + (1 + I)*E^(I*ArcCsc[a*x])*Hypergeometric2F1[1/2 - I/2, 1, 3/2 - I/2, E^((2*I)*ArcCsc[a*x])]))/a","A",0
44,1,75,43,0.0520054,"\int \frac{e^{\csc ^{-1}(a x)}}{x} \, dx","Integrate[E^ArcCsc[a*x]/x,x]","-i \left(-e^{\csc ^{-1}(a x)} \, _2F_1\left(-\frac{i}{2},1;1-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right)-\left(\frac{1}{5}-\frac{2 i}{5}\right) e^{(1+2 i) \csc ^{-1}(a x)} \, _2F_1\left(1,1-\frac{i}{2};2-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right)\right)","2 i e^{\csc ^{-1}(a x)} \, _2F_1\left(-\frac{i}{2},1;1-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right)-i e^{\csc ^{-1}(a x)}",1,"(-I)*(-(E^ArcCsc[a*x]*Hypergeometric2F1[-1/2*I, 1, 1 - I/2, E^((2*I)*ArcCsc[a*x])]) - (1/5 - (2*I)/5)*E^((1 + 2*I)*ArcCsc[a*x])*Hypergeometric2F1[1, 1 - I/2, 2 - I/2, E^((2*I)*ArcCsc[a*x])])","A",0
45,1,33,39,0.0384353,"\int \frac{e^{\csc ^{-1}(a x)}}{x^2} \, dx","Integrate[E^ArcCsc[a*x]/x^2,x]","-\frac{1}{2} a \left(\sqrt{1-\frac{1}{a^2 x^2}}+\frac{1}{a x}\right) e^{\csc ^{-1}(a x)}","-\frac{1}{2} a \sqrt{1-\frac{1}{a^2 x^2}} e^{\csc ^{-1}(a x)}-\frac{e^{\csc ^{-1}(a x)}}{2 x}",1,"-1/2*(a*E^ArcCsc[a*x]*(Sqrt[1 - 1/(a^2*x^2)] + 1/(a*x)))","A",1
46,1,30,41,0.0453315,"\int \frac{e^{\csc ^{-1}(a x)}}{x^3} \, dx","Integrate[E^ArcCsc[a*x]/x^3,x]","-\frac{1}{10} a^2 e^{\csc ^{-1}(a x)} \left(\sin \left(2 \csc ^{-1}(a x)\right)-2 \cos \left(2 \csc ^{-1}(a x)\right)\right)","\frac{1}{5} a^2 e^{\csc ^{-1}(a x)} \cos \left(2 \csc ^{-1}(a x)\right)-\frac{1}{10} a^2 e^{\csc ^{-1}(a x)} \sin \left(2 \csc ^{-1}(a x)\right)",1,"-1/10*(a^2*E^ArcCsc[a*x]*(-2*Cos[2*ArcCsc[a*x]] + Sin[2*ArcCsc[a*x]]))","A",1
47,1,54,84,0.1350101,"\int \frac{e^{\csc ^{-1}(a x)}}{x^4} \, dx","Integrate[E^ArcCsc[a*x]/x^4,x]","\frac{1}{40} a^3 e^{\csc ^{-1}(a x)} \left(-5 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{5}{a x}+\cos \left(3 \csc ^{-1}(a x)\right)+3 \sin \left(3 \csc ^{-1}(a x)\right)\right)","\frac{1}{40} a^3 e^{\csc ^{-1}(a x)} \cos \left(3 \csc ^{-1}(a x)\right)+\frac{3}{40} a^3 e^{\csc ^{-1}(a x)} \sin \left(3 \csc ^{-1}(a x)\right)-\frac{a^2 e^{\csc ^{-1}(a x)}}{8 x}-\frac{1}{8} a^3 \sqrt{1-\frac{1}{a^2 x^2}} e^{\csc ^{-1}(a x)}",1,"(a^3*E^ArcCsc[a*x]*(-5*Sqrt[1 - 1/(a^2*x^2)] - 5/(a*x) + Cos[3*ArcCsc[a*x]] + 3*Sin[3*ArcCsc[a*x]]))/40","A",1
48,1,50,81,0.1240858,"\int \frac{e^{\csc ^{-1}(a x)}}{x^5} \, dx","Integrate[E^ArcCsc[a*x]/x^5,x]","-\frac{1}{680} a^4 e^{\csc ^{-1}(a x)} \left(-68 \cos \left(2 \csc ^{-1}(a x)\right)+20 \cos \left(4 \csc ^{-1}(a x)\right)+34 \sin \left(2 \csc ^{-1}(a x)\right)-5 \sin \left(4 \csc ^{-1}(a x)\right)\right)","\frac{1}{10} a^4 e^{\csc ^{-1}(a x)} \cos \left(2 \csc ^{-1}(a x)\right)-\frac{1}{34} a^4 e^{\csc ^{-1}(a x)} \cos \left(4 \csc ^{-1}(a x)\right)-\frac{1}{20} a^4 e^{\csc ^{-1}(a x)} \sin \left(2 \csc ^{-1}(a x)\right)+\frac{1}{136} a^4 e^{\csc ^{-1}(a x)} \sin \left(4 \csc ^{-1}(a x)\right)",1,"-1/680*(a^4*E^ArcCsc[a*x]*(-68*Cos[2*ArcCsc[a*x]] + 20*Cos[4*ArcCsc[a*x]] + 34*Sin[2*ArcCsc[a*x]] - 5*Sin[4*ArcCsc[a*x]]))","A",1
49,1,59,69,0.0549405,"\int \frac{\csc ^{-1}(a+b x)}{\frac{a d}{b}+d x} \, dx","Integrate[ArcCsc[a + b*x]/((a*d)/b + d*x),x]","\frac{\frac{1}{2} i \left(\csc ^{-1}(a+b x)^2+\text{Li}_2\left(e^{2 i \csc ^{-1}(a+b x)}\right)\right)-\csc ^{-1}(a+b x) \log \left(1-e^{2 i \csc ^{-1}(a+b x)}\right)}{d}","\frac{i \text{Li}_2\left(e^{2 i \csc ^{-1}(a+b x)}\right)}{2 d}+\frac{i \csc ^{-1}(a+b x)^2}{2 d}-\frac{\csc ^{-1}(a+b x) \log \left(1-e^{2 i \csc ^{-1}(a+b x)}\right)}{d}",1,"(-(ArcCsc[a + b*x]*Log[1 - E^((2*I)*ArcCsc[a + b*x])]) + (I/2)*(ArcCsc[a + b*x]^2 + PolyLog[2, E^((2*I)*ArcCsc[a + b*x])]))/d","A",1