1,1,56,62,0.0376678,"\int \frac{\sec ^{-1}\left(a x^5\right)}{x} \, dx","Integrate[ArcSec[a*x^5]/x,x]","\frac{1}{10} i \left(\text{Li}_2\left(-e^{2 i \sec ^{-1}\left(a x^5\right)}\right)+\sec ^{-1}\left(a x^5\right) \left(\sec ^{-1}\left(a x^5\right)+2 i \log \left(1+e^{2 i \sec ^{-1}\left(a x^5\right)}\right)\right)\right)","\frac{1}{10} i \text{Li}_2\left(-e^{2 i \sec ^{-1}\left(a x^5\right)}\right)+\frac{1}{10} i \sec ^{-1}\left(a x^5\right)^2-\frac{1}{5} \sec ^{-1}\left(a x^5\right) \log \left(1+e^{2 i \sec ^{-1}\left(a x^5\right)}\right)",1,"(I/10)*(ArcSec[a*x^5]*(ArcSec[a*x^5] + (2*I)*Log[1 + E^((2*I)*ArcSec[a*x^5])]) + PolyLog[2, -E^((2*I)*ArcSec[a*x^5])])","A",1
2,1,40,58,0.0279398,"\int x^3 \sec ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[x^3*ArcSec[Sqrt[x]],x]","\frac{1}{4} x^4 \sec ^{-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 \sec ^{-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,"-1/140*(Sqrt[-1 + x]*(16 + 8*x + 6*x^2 + 5*x^3)) + (x^4*ArcSec[Sqrt[x]])/4","A",1
3,1,35,47,0.0224523,"\int x^2 \sec ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[x^2*ArcSec[Sqrt[x]],x]","\frac{1}{3} x^3 \sec ^{-1}\left(\sqrt{x}\right)-\frac{1}{45} \sqrt{x-1} \left(3 x^2+4 x+8\right)","\frac{1}{3} x^3 \sec ^{-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,"-1/45*(Sqrt[-1 + x]*(8 + 4*x + 3*x^2)) + (x^3*ArcSec[Sqrt[x]])/3","A",1
4,1,28,36,0.0180353,"\int x \sec ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[x*ArcSec[Sqrt[x]],x]","\frac{1}{2} x^2 \sec ^{-1}\left(\sqrt{x}\right)-\frac{1}{6} \sqrt{x-1} (x+2)","\frac{1}{2} x^2 \sec ^{-1}\left(\sqrt{x}\right)-\frac{1}{6} (x-1)^{3/2}-\frac{\sqrt{x-1}}{2}",1,"-1/6*(Sqrt[-1 + x]*(2 + x)) + (x^2*ArcSec[Sqrt[x]])/2","A",1
5,1,18,18,0.0059038,"\int \sec ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[ArcSec[Sqrt[x]],x]","x \sec ^{-1}\left(\sqrt{x}\right)-\sqrt{x-1}","x \sec ^{-1}\left(\sqrt{x}\right)-\sqrt{x-1}",1,"-Sqrt[-1 + x] + x*ArcSec[Sqrt[x]]","A",1
6,1,54,56,0.0287917,"\int \frac{\sec ^{-1}\left(\sqrt{x}\right)}{x} \, dx","Integrate[ArcSec[Sqrt[x]]/x,x]","i \left(\text{Li}_2\left(-e^{2 i \sec ^{-1}\left(\sqrt{x}\right)}\right)+\sec ^{-1}\left(\sqrt{x}\right) \left(\sec ^{-1}\left(\sqrt{x}\right)+2 i \log \left(1+e^{2 i \sec ^{-1}\left(\sqrt{x}\right)}\right)\right)\right)","i \text{Li}_2\left(-e^{2 i \sec ^{-1}\left(\sqrt{x}\right)}\right)+i \sec ^{-1}\left(\sqrt{x}\right)^2-2 \sec ^{-1}\left(\sqrt{x}\right) \log \left(1+e^{2 i \sec ^{-1}\left(\sqrt{x}\right)}\right)",1,"I*(ArcSec[Sqrt[x]]*(ArcSec[Sqrt[x]] + (2*I)*Log[1 + E^((2*I)*ArcSec[Sqrt[x]])]) + PolyLog[2, -E^((2*I)*ArcSec[Sqrt[x]])])","A",1
7,1,32,38,0.0215234,"\int \frac{\sec ^{-1}\left(\sqrt{x}\right)}{x^2} \, dx","Integrate[ArcSec[Sqrt[x]]/x^2,x]","\frac{\sqrt{x-1}-x \sin ^{-1}\left(\frac{1}{\sqrt{x}}\right)-2 \sec ^{-1}\left(\sqrt{x}\right)}{2 x}","\frac{\sqrt{x-1}}{2 x}+\frac{1}{2} \tan ^{-1}\left(\sqrt{x-1}\right)-\frac{\sec ^{-1}\left(\sqrt{x}\right)}{x}",1,"(Sqrt[-1 + x] - 2*ArcSec[Sqrt[x]] - x*ArcSin[1/Sqrt[x]])/(2*x)","A",1
8,1,55,54,0.033652,"\int \frac{\sec ^{-1}\left(\sqrt{x}\right)}{x^3} \, dx","Integrate[ArcSec[Sqrt[x]]/x^3,x]","\sqrt{\frac{x-1}{x}} \left(\frac{1}{8 x^{3/2}}+\frac{3}{16 \sqrt{x}}\right)-\frac{\sec ^{-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{\sec ^{-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*x^(3/2)) + 3/(16*Sqrt[x]))*Sqrt[(-1 + x)/x] - ArcSec[Sqrt[x]]/(2*x^2) - (3*ArcSin[1/Sqrt[x]])/16","A",1
9,1,45,68,0.0545139,"\int \frac{\sec ^{-1}\left(\sqrt{x}\right)}{x^4} \, dx","Integrate[ArcSec[Sqrt[x]]/x^4,x]","\frac{-15 x^3 \sin ^{-1}\left(\frac{1}{\sqrt{x}}\right)+\sqrt{x-1} \left(15 x^2+10 x+8\right)-48 \sec ^{-1}\left(\sqrt{x}\right)}{144 x^3}","\frac{\sqrt{x-1}}{18 x^3}-\frac{\sec ^{-1}\left(\sqrt{x}\right)}{3 x^3}+\frac{5 \sqrt{x-1}}{72 x^2}+\frac{5 \sqrt{x-1}}{48 x}+\frac{5}{48} \tan ^{-1}\left(\sqrt{x-1}\right)",1,"(Sqrt[-1 + x]*(8 + 10*x + 15*x^2) - 48*ArcSec[Sqrt[x]] - 15*x^3*ArcSin[1/Sqrt[x]])/(144*x^3)","A",1
10,1,42,56,0.0351505,"\int x^2 \sec ^{-1}\left(\frac{a}{x}\right) \, dx","Integrate[x^2*ArcSec[a/x],x]","\frac{1}{3} x^3 \sec ^{-1}\left(\frac{a}{x}\right)-\frac{1}{9} a \left(2 a^2+x^2\right) \sqrt{1-\frac{x^2}{a^2}}","\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 \cos ^{-1}\left(\frac{x}{a}\right)",1,"-1/9*(a*(2*a^2 + x^2)*Sqrt[1 - x^2/a^2]) + (x^3*ArcSec[a/x])/3","A",1
11,1,44,47,0.0249854,"\int x \sec ^{-1}\left(\frac{a}{x}\right) \, dx","Integrate[x*ArcSec[a/x],x]","\frac{1}{4} \left(-a x \sqrt{1-\frac{x^2}{a^2}}+a^2 \sin ^{-1}\left(\frac{x}{a}\right)+2 x^2 \sec ^{-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 \cos ^{-1}\left(\frac{x}{a}\right)",1,"(-(a*x*Sqrt[1 - x^2/a^2]) + 2*x^2*ArcSec[a/x] + a^2*ArcSin[x/a])/4","A",1
12,1,26,26,0.0124632,"\int \sec ^{-1}\left(\frac{a}{x}\right) \, dx","Integrate[ArcSec[a/x],x]","x \sec ^{-1}\left(\frac{a}{x}\right)-a \sqrt{1-\frac{x^2}{a^2}}","x \cos ^{-1}\left(\frac{x}{a}\right)-a \sqrt{1-\frac{x^2}{a^2}}",1,"-(a*Sqrt[1 - x^2/a^2]) + x*ArcSec[a/x]","A",1
13,1,59,59,0.0235222,"\int \frac{\sec ^{-1}\left(\frac{a}{x}\right)}{x} \, dx","Integrate[ArcSec[a/x]/x,x]","-\frac{1}{2} i \text{Li}_2\left(-e^{2 i \sec ^{-1}\left(\frac{a}{x}\right)}\right)-\frac{1}{2} i \sec ^{-1}\left(\frac{a}{x}\right)^2+\sec ^{-1}\left(\frac{a}{x}\right) \log \left(1+e^{2 i \sec ^{-1}\left(\frac{a}{x}\right)}\right)","-\frac{1}{2} i \text{Li}_2\left(-e^{2 i \cos ^{-1}\left(\frac{x}{a}\right)}\right)-\frac{1}{2} i \cos ^{-1}\left(\frac{x}{a}\right)^2+\cos ^{-1}\left(\frac{x}{a}\right) \log \left(1+e^{2 i \cos ^{-1}\left(\frac{x}{a}\right)}\right)",1,"(-1/2*I)*ArcSec[a/x]^2 + ArcSec[a/x]*Log[1 + E^((2*I)*ArcSec[a/x])] - (I/2)*PolyLog[2, -E^((2*I)*ArcSec[a/x])]","A",0
14,1,93,31,0.1421427,"\int \frac{\sec ^{-1}\left(\frac{a}{x}\right)}{x^2} \, dx","Integrate[ArcSec[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{\sec ^{-1}\left(\frac{a}{x}\right)}{x}","\frac{\tanh ^{-1}\left(\sqrt{1-\frac{x^2}{a^2}}\right)}{a}-\frac{\cos ^{-1}\left(\frac{x}{a}\right)}{x}",1,"-(ArcSec[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
15,1,36,38,0.0219786,"\int \frac{\sec ^{-1}\left(\frac{a}{x}\right)}{x^3} \, dx","Integrate[ArcSec[a/x]/x^3,x]","\frac{x \sqrt{1-\frac{x^2}{a^2}}-a \sec ^{-1}\left(\frac{a}{x}\right)}{2 a x^2}","\frac{\sqrt{1-\frac{x^2}{a^2}}}{2 a x}-\frac{\cos ^{-1}\left(\frac{x}{a}\right)}{2 x^2}",1,"(x*Sqrt[1 - x^2/a^2] - a*ArcSec[a/x])/(2*a*x^2)","A",1
16,1,69,60,0.044096,"\int \frac{\sec ^{-1}\left(\frac{a}{x}\right)}{x^4} \, dx","Integrate[ArcSec[a/x]/x^4,x]","\frac{-2 a^3 \sec ^{-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{\cos ^{-1}\left(\frac{x}{a}\right)}{3 x^3}",1,"(a^2*x*Sqrt[1 - x^2/a^2] - 2*a^3*ArcSec[a/x] - x^3*Log[x] + x^3*Log[1 + Sqrt[1 - x^2/a^2]])/(6*a^3*x^3)","A",1
17,1,60,69,0.0929838,"\int \frac{\sec ^{-1}\left(a x^n\right)}{x} \, dx","Integrate[ArcSec[a*x^n]/x,x]","\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}+\log (x) \left(\sin ^{-1}\left(\frac{x^{-n}}{a}\right)+\sec ^{-1}\left(a x^n\right)\right)","\frac{i \text{Li}_2\left(-e^{2 i \sec ^{-1}\left(a x^n\right)}\right)}{2 n}+\frac{i \sec ^{-1}\left(a x^n\right)^2}{2 n}-\frac{\sec ^{-1}\left(a x^n\right) \log \left(1+e^{2 i \sec ^{-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) + (ArcSec[a*x^n] + ArcSin[1/(a*x^n)])*Log[x]","C",1
18,1,173,197,0.1860689,"\int x^4 \sec ^{-1}(a+b x) \, dx","Integrate[x^4*ArcSec[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 \sec ^{-1}(a+b x)}{120 b^5}","\frac{a^5 \sec ^{-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 \sec ^{-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*ArcSec[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
19,1,150,155,0.286774,"\int x^3 \sec ^{-1}(a+b x) \, dx","Integrate[x^3*ArcSec[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 \sec ^{-1}(a+b x)}{12 b^4}","-\frac{a^4 \sec ^{-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 \sec ^{-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*ArcSec[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
20,1,131,116,0.1779311,"\int x^2 \sec ^{-1}(a+b x) \, dx","Integrate[x^2*ArcSec[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 \sec ^{-1}(a+b x)}{6 b^3}","\frac{a^3 \sec ^{-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 \sec ^{-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*ArcSec[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
21,1,110,78,0.1234488,"\int x \sec ^{-1}(a+b x) \, dx","Integrate[x*ArcSec[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 \sin ^{-1}\left(\frac{1}{a+b x}\right)+b^2 x^2 \sec ^{-1}(a+b x)}{2 b^2}","-\frac{a^2 \sec ^{-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 \sec ^{-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*ArcSec[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
22,1,115,37,0.1439825,"\int \sec ^{-1}(a+b x) \, dx","Integrate[ArcSec[a + b*x],x]","x \sec ^{-1}(a+b 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}}","\frac{(a+b x) \sec ^{-1}(a+b x)}{b}-\frac{\tanh ^{-1}\left(\sqrt{1-\frac{1}{(a+b x)^2}}\right)}{b}",1,"x*ArcSec[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
23,1,284,200,0.322965,"\int \frac{\sec ^{-1}(a+b x)}{x} \, dx","Integrate[ArcSec[a + b*x]/x,x]","-i \left(\text{Li}_2\left(-\frac{\left(\sqrt{1-a^2}-1\right) e^{i \sec ^{-1}(a+b x)}}{a}\right)+\text{Li}_2\left(\frac{\left(\sqrt{1-a^2}+1\right) e^{i \sec ^{-1}(a+b x)}}{a}\right)\right)+\log \left(1+\frac{\left(\sqrt{1-a^2}-1\right) e^{i \sec ^{-1}(a+b x)}}{a}\right) \left(\sec ^{-1}(a+b x)-2 \sin ^{-1}\left(\frac{\sqrt{\frac{a-1}{a}}}{\sqrt{2}}\right)\right)+\log \left(1-\frac{\left(\sqrt{1-a^2}+1\right) e^{i \sec ^{-1}(a+b x)}}{a}\right) \left(\sec ^{-1}(a+b x)+2 \sin ^{-1}\left(\frac{\sqrt{\frac{a-1}{a}}}{\sqrt{2}}\right)\right)-4 i \sin ^{-1}\left(\frac{\sqrt{\frac{a-1}{a}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{(a+1) \tan \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)}{\sqrt{1-a^2}}\right)+\frac{1}{2} i \text{Li}_2\left(-e^{2 i \sec ^{-1}(a+b x)}\right)-\sec ^{-1}(a+b x) \log \left(1+e^{2 i \sec ^{-1}(a+b x)}\right)","-i \text{Li}_2\left(\frac{a e^{i \sec ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right)-i \text{Li}_2\left(\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)+\sec ^{-1}(a+b x) \log \left(1-\frac{a e^{i \sec ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right)+\sec ^{-1}(a+b x) \log \left(1-\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)+\frac{1}{2} i \text{Li}_2\left(-e^{2 i \sec ^{-1}(a+b x)}\right)-\sec ^{-1}(a+b x) \log \left(1+e^{2 i \sec ^{-1}(a+b x)}\right)",1,"(-4*I)*ArcSin[Sqrt[(-1 + a)/a]/Sqrt[2]]*ArcTan[((1 + a)*Tan[ArcSec[a + b*x]/2])/Sqrt[1 - a^2]] + (ArcSec[a + b*x] - 2*ArcSin[Sqrt[(-1 + a)/a]/Sqrt[2]])*Log[1 + ((-1 + Sqrt[1 - a^2])*E^(I*ArcSec[a + b*x]))/a] + (ArcSec[a + b*x] + 2*ArcSin[Sqrt[(-1 + a)/a]/Sqrt[2]])*Log[1 - ((1 + Sqrt[1 - a^2])*E^(I*ArcSec[a + b*x]))/a] - ArcSec[a + b*x]*Log[1 + E^((2*I)*ArcSec[a + b*x])] - I*(PolyLog[2, -(((-1 + Sqrt[1 - a^2])*E^(I*ArcSec[a + b*x]))/a)] + PolyLog[2, ((1 + Sqrt[1 - a^2])*E^(I*ArcSec[a + b*x]))/a]) + (I/2)*PolyLog[2, -E^((2*I)*ArcSec[a + b*x])]","A",0
24,1,112,70,0.3144679,"\int \frac{\sec ^{-1}(a+b x)}{x^2} \, dx","Integrate[ArcSec[a + b*x]/x^2,x]","-\frac{\sec ^{-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{\sqrt{a+1} \tan \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)}{\sqrt{1-a}}\right)}{a \sqrt{1-a^2}}-\frac{b \sec ^{-1}(a+b x)}{a}-\frac{\sec ^{-1}(a+b x)}{x}",1,"-(ArcSec[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
25,1,198,125,1.0522386,"\int \frac{\sec ^{-1}(a+b x)}{x^3} \, dx","Integrate[ArcSec[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}+\sec ^{-1}(a+b x)}{2 x^2}","\frac{b^2 \sec ^{-1}(a+b x)}{2 a^2}-\frac{\left(1-2 a^2\right) b^2 \tan ^{-1}\left(\frac{\sqrt{a+1} \tan \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)}{\sqrt{1-a}}\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{\sec ^{-1}(a+b x)}{2 x^2}",1,"-1/2*((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)) + ArcSec[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)))/x^2","C",1
26,1,241,181,0.4310532,"\int \frac{\sec ^{-1}(a+b x)}{x^4} \, dx","Integrate[ArcSec[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 \sec ^{-1}(a+b x)}{x^3}\right)","-\frac{b^3 \sec ^{-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{\sqrt{a+1} \tan \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)}{\sqrt{1-a}}\right)}{3 a^3 \left(1-a^2\right)^{5/2}}-\frac{\sec ^{-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*ArcSec[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
27,1,667,381,9.4179988,"\int x^3 \sec ^{-1}(a+b x)^2 \, dx","Integrate[x^3*ArcSec[a + b*x]^2,x]","\frac{\left(1-\frac{a}{a+b x}\right)^3 \left(-24 a \left(2 a^2+1\right) \left(2 i \left(\text{Li}_2\left(-i e^{-i \sec ^{-1}(a+b x)}\right)-\text{Li}_2\left(i e^{-i \sec ^{-1}(a+b x)}\right)\right)+\left(\pi -2 \sec ^{-1}(a+b x)\right) \left(\log \left(1-i e^{-i \sec ^{-1}(a+b x)}\right)-\log \left(1+i e^{-i \sec ^{-1}(a+b x)}\right)\right)-\pi  \log \left(\cot \left(\frac{1}{4} \left(2 \sec ^{-1}(a+b x)+\pi \right)\right)\right)\right)+16 \left(9 a^2+1\right) \log \left(\frac{1}{a+b x}\right)+24 a \left(\left(2 a^2+1\right) \sec ^{-1}(a+b x)^2+2\right)+\frac{3 \left(12 a^2-4 a+1\right) \sec ^{-1}(a+b x)^2+(24 a-2) \sec ^{-1}(a+b x)+2}{\sqrt{1-\frac{1}{(a+b x)^2}}-1}-\frac{3 \left(12 a^2-4 a+1\right) \sec ^{-1}(a+b x)^2+(2-24 a) \sec ^{-1}(a+b x)+2}{\left(\cos \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)+\sin \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)\right)^2}+\frac{8 \left(6 a^3 \sec ^{-1}(a+b x)^2+18 a^2 \sec ^{-1}(a+b x)+3 a \left(\sec ^{-1}(a+b x)^2+2\right)+2 \sec ^{-1}(a+b x)\right) \sin \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)}{\cos \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)-\sin \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)}-\frac{8 \left(6 a^3 \sec ^{-1}(a+b x)^2-18 a^2 \sec ^{-1}(a+b x)+3 a \left(\sec ^{-1}(a+b x)^2+2\right)-2 \sec ^{-1}(a+b x)\right) \sin \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)}{\cos \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)+\sin \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)}-\frac{3 \sec ^{-1}(a+b x)^2}{\left(\cos \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)-\sin \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)\right)^4}-\frac{3 \sec ^{-1}(a+b x)^2}{\left(\cos \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)+\sin \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)\right)^4}+\frac{4 \left(6 a \sec ^{-1}(a+b x)+1\right) \sec ^{-1}(a+b x) \sin \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)}{\left(\cos \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)-\sin \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)\right)^3}+\frac{4 \left(1-6 a \sec ^{-1}(a+b x)\right) \sec ^{-1}(a+b x) \sin \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)}{\left(\cos \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)+\sin \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)\right)^3}\right)}{48 b^4 \left(\frac{a}{a+b x}-1\right)^3}","-\frac{a^4 \sec ^{-1}(a+b x)^2}{4 b^4}+\frac{2 i a^3 \text{Li}_2\left(-i e^{i \sec ^{-1}(a+b x)}\right)}{b^4}-\frac{2 i a^3 \text{Li}_2\left(i e^{i \sec ^{-1}(a+b x)}\right)}{b^4}-\frac{4 i a^3 \sec ^{-1}(a+b x) \tan ^{-1}\left(e^{i \sec ^{-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}} \sec ^{-1}(a+b x)}{b^4}+\frac{i a \text{Li}_2\left(-i e^{i \sec ^{-1}(a+b x)}\right)}{b^4}-\frac{i a \text{Li}_2\left(i e^{i \sec ^{-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}} \sec ^{-1}(a+b x)}{b^4}-\frac{(a+b x)^3 \sqrt{1-\frac{1}{(a+b x)^2}} \sec ^{-1}(a+b x)}{6 b^4}-\frac{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \sec ^{-1}(a+b x)}{3 b^4}-\frac{2 i a \sec ^{-1}(a+b x) \tan ^{-1}\left(e^{i \sec ^{-1}(a+b x)}\right)}{b^4}-\frac{a x}{b^3}+\frac{1}{4} x^4 \sec ^{-1}(a+b x)^2",1,"((1 - a/(a + b*x))^3*(24*a*(2 + (1 + 2*a^2)*ArcSec[a + b*x]^2) + (2 + (-2 + 24*a)*ArcSec[a + b*x] + 3*(1 - 4*a + 12*a^2)*ArcSec[a + b*x]^2)/(-1 + Sqrt[1 - (a + b*x)^(-2)]) + 16*(1 + 9*a^2)*Log[(a + b*x)^(-1)] - 24*a*(1 + 2*a^2)*((Pi - 2*ArcSec[a + b*x])*(Log[1 - I/E^(I*ArcSec[a + b*x])] - Log[1 + I/E^(I*ArcSec[a + b*x])]) - Pi*Log[Cot[(Pi + 2*ArcSec[a + b*x])/4]] + (2*I)*(PolyLog[2, (-I)/E^(I*ArcSec[a + b*x])] - PolyLog[2, I/E^(I*ArcSec[a + b*x])])) - (3*ArcSec[a + b*x]^2)/(Cos[ArcSec[a + b*x]/2] - Sin[ArcSec[a + b*x]/2])^4 + (4*ArcSec[a + b*x]*(1 + 6*a*ArcSec[a + b*x])*Sin[ArcSec[a + b*x]/2])/(Cos[ArcSec[a + b*x]/2] - Sin[ArcSec[a + b*x]/2])^3 + (8*(2*ArcSec[a + b*x] + 18*a^2*ArcSec[a + b*x] + 6*a^3*ArcSec[a + b*x]^2 + 3*a*(2 + ArcSec[a + b*x]^2))*Sin[ArcSec[a + b*x]/2])/(Cos[ArcSec[a + b*x]/2] - Sin[ArcSec[a + b*x]/2]) - (3*ArcSec[a + b*x]^2)/(Cos[ArcSec[a + b*x]/2] + Sin[ArcSec[a + b*x]/2])^4 + (4*ArcSec[a + b*x]*(1 - 6*a*ArcSec[a + b*x])*Sin[ArcSec[a + b*x]/2])/(Cos[ArcSec[a + b*x]/2] + Sin[ArcSec[a + b*x]/2])^3 - (2 + (2 - 24*a)*ArcSec[a + b*x] + 3*(1 - 4*a + 12*a^2)*ArcSec[a + b*x]^2)/(Cos[ArcSec[a + b*x]/2] + Sin[ArcSec[a + b*x]/2])^2 - (8*(-2*ArcSec[a + b*x] - 18*a^2*ArcSec[a + b*x] + 6*a^3*ArcSec[a + b*x]^2 + 3*a*(2 + ArcSec[a + b*x]^2))*Sin[ArcSec[a + b*x]/2])/(Cos[ArcSec[a + b*x]/2] + Sin[ArcSec[a + b*x]/2])))/(48*b^4*(-1 + a/(a + b*x))^3)","A",0
28,1,473,288,6.5820499,"\int x^2 \sec ^{-1}(a+b x)^2 \, dx","Integrate[x^2*ArcSec[a + b*x]^2,x]","\frac{2 \left(-6 a^2-1\right) \left(2 i \left(\text{Li}_2\left(-i e^{-i \sec ^{-1}(a+b x)}\right)-\text{Li}_2\left(i e^{-i \sec ^{-1}(a+b x)}\right)\right)+\left(\pi -2 \sec ^{-1}(a+b x)\right) \left(\log \left(1-i e^{-i \sec ^{-1}(a+b x)}\right)-\log \left(1+i e^{-i \sec ^{-1}(a+b x)}\right)\right)-\pi  \log \left(\cot \left(\frac{1}{4} \left(2 \sec ^{-1}(a+b x)+\pi \right)\right)\right)\right)+2 \left(6 a^2+1\right) \sec ^{-1}(a+b x)^2+\frac{2 \left(\left(6 a^2+1\right) \sec ^{-1}(a+b x)^2+12 a \sec ^{-1}(a+b x)+2\right) \sin \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)}{\cos \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)-\sin \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)}-\frac{2 \left(\left(6 a^2+1\right) \sec ^{-1}(a+b x)^2-12 a \sec ^{-1}(a+b x)+2\right) \sin \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)}{\cos \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)+\sin \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)}+24 a \log \left(\frac{1}{a+b x}\right)+\frac{\left((6 a-1) \sec ^{-1}(a+b x)+2\right) \sec ^{-1}(a+b x)}{\sqrt{1-\frac{1}{(a+b x)^2}}-1}+\frac{2 \sec ^{-1}(a+b x)^2 \sin \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)}{\left(\cos \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)-\sin \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)\right)^3}-\frac{2 \sec ^{-1}(a+b x)^2 \sin \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)}{\left(\cos \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)+\sin \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)\right)^3}+\frac{\left((1-6 a) \sec ^{-1}(a+b x)+2\right) \sec ^{-1}(a+b x)}{\left(\cos \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)+\sin \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)\right)^2}+4}{12 b^3}","\frac{a^3 \sec ^{-1}(a+b x)^2}{3 b^3}-\frac{2 i a^2 \text{Li}_2\left(-i e^{i \sec ^{-1}(a+b x)}\right)}{b^3}+\frac{2 i a^2 \text{Li}_2\left(i e^{i \sec ^{-1}(a+b x)}\right)}{b^3}+\frac{4 i a^2 \sec ^{-1}(a+b x) \tan ^{-1}\left(e^{i \sec ^{-1}(a+b x)}\right)}{b^3}-\frac{i \text{Li}_2\left(-i e^{i \sec ^{-1}(a+b x)}\right)}{3 b^3}+\frac{i \text{Li}_2\left(i e^{i \sec ^{-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}} \sec ^{-1}(a+b x)}{b^3}-\frac{(a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}} \sec ^{-1}(a+b x)}{3 b^3}+\frac{2 i \sec ^{-1}(a+b x) \tan ^{-1}\left(e^{i \sec ^{-1}(a+b x)}\right)}{3 b^3}+\frac{1}{3} x^3 \sec ^{-1}(a+b x)^2+\frac{x}{3 b^2}",1,"(4 + 2*(1 + 6*a^2)*ArcSec[a + b*x]^2 + (ArcSec[a + b*x]*(2 + (-1 + 6*a)*ArcSec[a + b*x]))/(-1 + Sqrt[1 - (a + b*x)^(-2)]) + 24*a*Log[(a + b*x)^(-1)] + 2*(-1 - 6*a^2)*((Pi - 2*ArcSec[a + b*x])*(Log[1 - I/E^(I*ArcSec[a + b*x])] - Log[1 + I/E^(I*ArcSec[a + b*x])]) - Pi*Log[Cot[(Pi + 2*ArcSec[a + b*x])/4]] + (2*I)*(PolyLog[2, (-I)/E^(I*ArcSec[a + b*x])] - PolyLog[2, I/E^(I*ArcSec[a + b*x])])) + (2*ArcSec[a + b*x]^2*Sin[ArcSec[a + b*x]/2])/(Cos[ArcSec[a + b*x]/2] - Sin[ArcSec[a + b*x]/2])^3 + (2*(2 + 12*a*ArcSec[a + b*x] + (1 + 6*a^2)*ArcSec[a + b*x]^2)*Sin[ArcSec[a + b*x]/2])/(Cos[ArcSec[a + b*x]/2] - Sin[ArcSec[a + b*x]/2]) - (2*ArcSec[a + b*x]^2*Sin[ArcSec[a + b*x]/2])/(Cos[ArcSec[a + b*x]/2] + Sin[ArcSec[a + b*x]/2])^3 + (ArcSec[a + b*x]*(2 + (1 - 6*a)*ArcSec[a + b*x]))/(Cos[ArcSec[a + b*x]/2] + Sin[ArcSec[a + b*x]/2])^2 - (2*(2 - 12*a*ArcSec[a + b*x] + (1 + 6*a^2)*ArcSec[a + b*x]^2)*Sin[ArcSec[a + b*x]/2])/(Cos[ArcSec[a + b*x]/2] + Sin[ArcSec[a + b*x]/2]))/(12*b^3)","A",0
29,1,142,154,0.1472326,"\int x \sec ^{-1}(a+b x)^2 \, dx","Integrate[x*ArcSec[a + b*x]^2,x]","\frac{-\frac{1}{2} a^2 \sec ^{-1}(a+b x)^2+\frac{1}{2} b^2 x^2 \sec ^{-1}(a+b x)^2+2 i a \text{Li}_2\left(-i e^{i \sec ^{-1}(a+b x)}\right)-2 i a \text{Li}_2\left(i e^{i \sec ^{-1}(a+b x)}\right)+\log (a+b x)-(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \sec ^{-1}(a+b x)-4 i a \sec ^{-1}(a+b x) \tan ^{-1}\left(e^{i \sec ^{-1}(a+b x)}\right)}{b^2}","-\frac{a^2 \sec ^{-1}(a+b x)^2}{2 b^2}+\frac{2 i a \text{Li}_2\left(-i e^{i \sec ^{-1}(a+b x)}\right)}{b^2}-\frac{2 i a \text{Li}_2\left(i e^{i \sec ^{-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}} \sec ^{-1}(a+b x)}{b^2}-\frac{4 i a \sec ^{-1}(a+b x) \tan ^{-1}\left(e^{i \sec ^{-1}(a+b x)}\right)}{b^2}+\frac{1}{2} x^2 \sec ^{-1}(a+b x)^2",1,"(-((a + b*x)*Sqrt[1 - (a + b*x)^(-2)]*ArcSec[a + b*x]) - (a^2*ArcSec[a + b*x]^2)/2 + (b^2*x^2*ArcSec[a + b*x]^2)/2 - (4*I)*a*ArcSec[a + b*x]*ArcTan[E^(I*ArcSec[a + b*x])] + Log[a + b*x] + (2*I)*a*PolyLog[2, (-I)*E^(I*ArcSec[a + b*x])] - (2*I)*a*PolyLog[2, I*E^(I*ArcSec[a + b*x])])/b^2","A",1
30,1,111,94,0.1134973,"\int \sec ^{-1}(a+b x)^2 \, dx","Integrate[ArcSec[a + b*x]^2,x]","\frac{-2 i \text{Li}_2\left(-i e^{i \sec ^{-1}(a+b x)}\right)+2 i \text{Li}_2\left(i e^{i \sec ^{-1}(a+b x)}\right)+\sec ^{-1}(a+b x) \left((a+b x) \sec ^{-1}(a+b x)-2 \log \left(1-i e^{i \sec ^{-1}(a+b x)}\right)+2 \log \left(1+i e^{i \sec ^{-1}(a+b x)}\right)\right)}{b}","-\frac{2 i \text{Li}_2\left(-i e^{i \sec ^{-1}(a+b x)}\right)}{b}+\frac{2 i \text{Li}_2\left(i e^{i \sec ^{-1}(a+b x)}\right)}{b}+\frac{(a+b x) \sec ^{-1}(a+b x)^2}{b}+\frac{4 i \sec ^{-1}(a+b x) \tan ^{-1}\left(e^{i \sec ^{-1}(a+b x)}\right)}{b}",1,"(ArcSec[a + b*x]*((a + b*x)*ArcSec[a + b*x] - 2*Log[1 - I*E^(I*ArcSec[a + b*x])] + 2*Log[1 + I*E^(I*ArcSec[a + b*x])]) - (2*I)*PolyLog[2, (-I)*E^(I*ArcSec[a + b*x])] + (2*I)*PolyLog[2, I*E^(I*ArcSec[a + b*x])])/b","A",0
31,1,813,310,2.3778734,"\int \frac{\sec ^{-1}(a+b x)^2}{x} \, dx","Integrate[ArcSec[a + b*x]^2/x,x]","\log \left(\frac{e^{i \sec ^{-1}(a+b x)} a}{\sqrt{1-a^2}-1}+1\right) \sec ^{-1}(a+b x)^2+\log \left(\frac{e^{i \sec ^{-1}(a+b x)} \left(\sqrt{1-a^2}-1\right)}{a}+1\right) \sec ^{-1}(a+b x)^2+\log \left(1-\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right) \sec ^{-1}(a+b x)^2+\log \left(1-\frac{\left(\sqrt{1-a^2}+1\right) e^{i \sec ^{-1}(a+b x)}}{a}\right) \sec ^{-1}(a+b x)^2-2 \log \left(1+e^{2 i \sec ^{-1}(a+b x)}\right) \sec ^{-1}(a+b x)^2+\log \left(\frac{2 \left(i \sqrt{1-\frac{1}{(a+b x)^2}}+\frac{1}{a+b x}\right)}{a+b x}\right) \sec ^{-1}(a+b x)^2-\log \left(\frac{\left(\sqrt{1-a^2}-1\right) \left(i \sqrt{1-\frac{1}{(a+b x)^2}}+\frac{1}{a+b x}\right)}{a}+1\right) \sec ^{-1}(a+b x)^2-\log \left(1-\frac{\left(\sqrt{1-a^2}+1\right) \left(i \sqrt{1-\frac{1}{(a+b x)^2}}+\frac{1}{a+b x}\right)}{a}\right) \sec ^{-1}(a+b x)^2-4 \sin ^{-1}\left(\frac{\sqrt{\frac{a-1}{a}}}{\sqrt{2}}\right) \log \left(\frac{e^{i \sec ^{-1}(a+b x)} \left(\sqrt{1-a^2}-1\right)}{a}+1\right) \sec ^{-1}(a+b x)+4 \sin ^{-1}\left(\frac{\sqrt{\frac{a-1}{a}}}{\sqrt{2}}\right) \log \left(1-\frac{\left(\sqrt{1-a^2}+1\right) e^{i \sec ^{-1}(a+b x)}}{a}\right) \sec ^{-1}(a+b x)+4 \sin ^{-1}\left(\frac{\sqrt{\frac{a-1}{a}}}{\sqrt{2}}\right) \log \left(\frac{\left(\sqrt{1-a^2}-1\right) \left(i \sqrt{1-\frac{1}{(a+b x)^2}}+\frac{1}{a+b x}\right)}{a}+1\right) \sec ^{-1}(a+b x)-4 \sin ^{-1}\left(\frac{\sqrt{\frac{a-1}{a}}}{\sqrt{2}}\right) \log \left(1-\frac{\left(\sqrt{1-a^2}+1\right) \left(i \sqrt{1-\frac{1}{(a+b x)^2}}+\frac{1}{a+b x}\right)}{a}\right) \sec ^{-1}(a+b x)-2 i \text{Li}_2\left(-\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}-1}\right) \sec ^{-1}(a+b x)-2 i \text{Li}_2\left(\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right) \sec ^{-1}(a+b x)+i \text{Li}_2\left(-e^{2 i \sec ^{-1}(a+b x)}\right) \sec ^{-1}(a+b x)+2 \text{Li}_3\left(-\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}-1}\right)+2 \text{Li}_3\left(\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)-\frac{1}{2} \text{Li}_3\left(-e^{2 i \sec ^{-1}(a+b x)}\right)","-2 i \sec ^{-1}(a+b x) \text{Li}_2\left(\frac{a e^{i \sec ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right)-2 i \sec ^{-1}(a+b x) \text{Li}_2\left(\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)+2 \text{Li}_3\left(\frac{a e^{i \sec ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right)+2 \text{Li}_3\left(\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)+\sec ^{-1}(a+b x)^2 \log \left(1-\frac{a e^{i \sec ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right)+\sec ^{-1}(a+b x)^2 \log \left(1-\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)+i \sec ^{-1}(a+b x) \text{Li}_2\left(-e^{2 i \sec ^{-1}(a+b x)}\right)-\frac{1}{2} \text{Li}_3\left(-e^{2 i \sec ^{-1}(a+b x)}\right)-\sec ^{-1}(a+b x)^2 \log \left(1+e^{2 i \sec ^{-1}(a+b x)}\right)",1,"ArcSec[a + b*x]^2*Log[1 + (a*E^(I*ArcSec[a + b*x]))/(-1 + Sqrt[1 - a^2])] + ArcSec[a + b*x]^2*Log[1 + ((-1 + Sqrt[1 - a^2])*E^(I*ArcSec[a + b*x]))/a] - 4*ArcSec[a + b*x]*ArcSin[Sqrt[(-1 + a)/a]/Sqrt[2]]*Log[1 + ((-1 + Sqrt[1 - a^2])*E^(I*ArcSec[a + b*x]))/a] + ArcSec[a + b*x]^2*Log[1 - (a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])] + ArcSec[a + b*x]^2*Log[1 - ((1 + Sqrt[1 - a^2])*E^(I*ArcSec[a + b*x]))/a] + 4*ArcSec[a + b*x]*ArcSin[Sqrt[(-1 + a)/a]/Sqrt[2]]*Log[1 - ((1 + Sqrt[1 - a^2])*E^(I*ArcSec[a + b*x]))/a] - 2*ArcSec[a + b*x]^2*Log[1 + E^((2*I)*ArcSec[a + b*x])] + ArcSec[a + b*x]^2*Log[(2*((a + b*x)^(-1) + I*Sqrt[1 - (a + b*x)^(-2)]))/(a + b*x)] - ArcSec[a + b*x]^2*Log[1 + ((-1 + Sqrt[1 - a^2])*((a + b*x)^(-1) + I*Sqrt[1 - (a + b*x)^(-2)]))/a] + 4*ArcSec[a + b*x]*ArcSin[Sqrt[(-1 + a)/a]/Sqrt[2]]*Log[1 + ((-1 + Sqrt[1 - a^2])*((a + b*x)^(-1) + I*Sqrt[1 - (a + b*x)^(-2)]))/a] - ArcSec[a + b*x]^2*Log[1 - ((1 + Sqrt[1 - a^2])*((a + b*x)^(-1) + I*Sqrt[1 - (a + b*x)^(-2)]))/a] - 4*ArcSec[a + b*x]*ArcSin[Sqrt[(-1 + a)/a]/Sqrt[2]]*Log[1 - ((1 + Sqrt[1 - a^2])*((a + b*x)^(-1) + I*Sqrt[1 - (a + b*x)^(-2)]))/a] - (2*I)*ArcSec[a + b*x]*PolyLog[2, -((a*E^(I*ArcSec[a + b*x]))/(-1 + Sqrt[1 - a^2]))] - (2*I)*ArcSec[a + b*x]*PolyLog[2, (a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])] + I*ArcSec[a + b*x]*PolyLog[2, -E^((2*I)*ArcSec[a + b*x])] + 2*PolyLog[3, -((a*E^(I*ArcSec[a + b*x]))/(-1 + Sqrt[1 - a^2]))] + 2*PolyLog[3, (a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])] - PolyLog[3, -E^((2*I)*ArcSec[a + b*x])]/2","B",0
32,1,686,244,2.293039,"\int \frac{\sec ^{-1}(a+b x)^2}{x^2} \, dx","Integrate[ArcSec[a + b*x]^2/x^2,x]","-\frac{\frac{(a+b x) \sec ^{-1}(a+b x)^2}{x}+\frac{2 b \left(i \left(\text{Li}_2\left(\frac{\left(i \sqrt{a^2-1}+1\right) \left(-a+\sqrt{a^2-1} \tan \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)+1\right)}{a \left(a+\sqrt{a^2-1} \tan \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)-1\right)}\right)-\text{Li}_2\left(\frac{\left(1-i \sqrt{a^2-1}\right) \left(-a+\sqrt{a^2-1} \tan \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)+1\right)}{a \left(a+\sqrt{a^2-1} \tan \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)-1\right)}\right)\right)+2 \sec ^{-1}(a+b x) \tanh ^{-1}\left(\frac{(a-1) \cot \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)}{\sqrt{a^2-1}}\right)-2 \cos ^{-1}\left(\frac{1}{a}\right) \tanh ^{-1}\left(\frac{(a+1) \tan \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)}{\sqrt{a^2-1}}\right)-\log \left(\frac{(a-1) \left(\sqrt{a^2-1}+i a+i\right) \left(\tan \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)-i\right)}{a \left(\sqrt{a^2-1} \tan \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)+a-1\right)}\right) \left(\cos ^{-1}\left(\frac{1}{a}\right)-2 i \tanh ^{-1}\left(\frac{(a+1) \tan \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)}{\sqrt{a^2-1}}\right)\right)-\log \left(\frac{(a-1) \left(\sqrt{a^2-1}-i a-i\right) \left(\tan \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)+i\right)}{a \left(\sqrt{a^2-1} \tan \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)+a-1\right)}\right) \left(\cos ^{-1}\left(\frac{1}{a}\right)+2 i \tanh ^{-1}\left(\frac{(a+1) \tan \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)}{\sqrt{a^2-1}}\right)\right)+\log \left(\frac{\sqrt{a^2-1} e^{-\frac{1}{2} i \sec ^{-1}(a+b x)}}{\sqrt{2} \sqrt{a} \sqrt{-\frac{b x}{a+b x}}}\right) \left(-2 i \tanh ^{-1}\left(\frac{(a-1) \cot \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)}{\sqrt{a^2-1}}\right)+2 i \tanh ^{-1}\left(\frac{(a+1) \tan \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)}{\sqrt{a^2-1}}\right)+\cos ^{-1}\left(\frac{1}{a}\right)\right)+\log \left(\frac{\sqrt{a^2-1} e^{\frac{1}{2} i \sec ^{-1}(a+b x)}}{\sqrt{2} \sqrt{a} \sqrt{-\frac{b x}{a+b x}}}\right) \left(\cos ^{-1}\left(\frac{1}{a}\right)+2 i \left(\tanh ^{-1}\left(\frac{(a-1) \cot \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)}{\sqrt{a^2-1}}\right)-\tanh ^{-1}\left(\frac{(a+1) \tan \left(\frac{1}{2} \sec ^{-1}(a+b x)\right)}{\sqrt{a^2-1}}\right)\right)\right)\right)}{\sqrt{a^2-1}}}{a}","-\frac{2 b \text{Li}_2\left(\frac{a e^{i \sec ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right)}{a \sqrt{1-a^2}}+\frac{2 b \text{Li}_2\left(\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)}{a \sqrt{1-a^2}}-\frac{2 i b \sec ^{-1}(a+b x) \log \left(1-\frac{a e^{i \sec ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right)}{a \sqrt{1-a^2}}+\frac{2 i b \sec ^{-1}(a+b x) \log \left(1-\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)}{a \sqrt{1-a^2}}-\frac{b \sec ^{-1}(a+b x)^2}{a}-\frac{\sec ^{-1}(a+b x)^2}{x}",1,"-((((a + b*x)*ArcSec[a + b*x]^2)/x + (2*b*(2*ArcSec[a + b*x]*ArcTanh[((-1 + a)*Cot[ArcSec[a + b*x]/2])/Sqrt[-1 + a^2]] - 2*ArcCos[a^(-1)]*ArcTanh[((1 + a)*Tan[ArcSec[a + b*x]/2])/Sqrt[-1 + a^2]] + (ArcCos[a^(-1)] - (2*I)*ArcTanh[((-1 + a)*Cot[ArcSec[a + b*x]/2])/Sqrt[-1 + a^2]] + (2*I)*ArcTanh[((1 + a)*Tan[ArcSec[a + b*x]/2])/Sqrt[-1 + a^2]])*Log[Sqrt[-1 + a^2]/(Sqrt[2]*Sqrt[a]*E^((I/2)*ArcSec[a + b*x])*Sqrt[-((b*x)/(a + b*x))])] + (ArcCos[a^(-1)] + (2*I)*(ArcTanh[((-1 + a)*Cot[ArcSec[a + b*x]/2])/Sqrt[-1 + a^2]] - ArcTanh[((1 + a)*Tan[ArcSec[a + b*x]/2])/Sqrt[-1 + a^2]]))*Log[(Sqrt[-1 + a^2]*E^((I/2)*ArcSec[a + b*x]))/(Sqrt[2]*Sqrt[a]*Sqrt[-((b*x)/(a + b*x))])] - (ArcCos[a^(-1)] - (2*I)*ArcTanh[((1 + a)*Tan[ArcSec[a + b*x]/2])/Sqrt[-1 + a^2]])*Log[((-1 + a)*(I + I*a + Sqrt[-1 + a^2])*(-I + Tan[ArcSec[a + b*x]/2]))/(a*(-1 + a + Sqrt[-1 + a^2]*Tan[ArcSec[a + b*x]/2]))] - (ArcCos[a^(-1)] + (2*I)*ArcTanh[((1 + a)*Tan[ArcSec[a + b*x]/2])/Sqrt[-1 + a^2]])*Log[((-1 + a)*(-I - I*a + Sqrt[-1 + a^2])*(I + Tan[ArcSec[a + b*x]/2]))/(a*(-1 + a + Sqrt[-1 + a^2]*Tan[ArcSec[a + b*x]/2]))] + I*(-PolyLog[2, ((1 - I*Sqrt[-1 + a^2])*(1 - a + Sqrt[-1 + a^2]*Tan[ArcSec[a + b*x]/2]))/(a*(-1 + a + Sqrt[-1 + a^2]*Tan[ArcSec[a + b*x]/2]))] + PolyLog[2, ((1 + I*Sqrt[-1 + a^2])*(1 - a + Sqrt[-1 + a^2]*Tan[ArcSec[a + b*x]/2]))/(a*(-1 + a + Sqrt[-1 + a^2]*Tan[ArcSec[a + b*x]/2]))])))/Sqrt[-1 + a^2])/a)","B",1
33,1,442,494,0.5254793,"\int x^2 \sec ^{-1}(a+b x)^3 \, dx","Integrate[x^2*ArcSec[a + b*x]^3,x]","\frac{\frac{1}{3} a^3 \sec ^{-1}(a+b x)^3+6 a^2 \left(\text{Li}_3\left(-i e^{i \sec ^{-1}(a+b x)}\right)-i \sec ^{-1}(a+b x) \text{Li}_2\left(-i e^{i \sec ^{-1}(a+b x)}\right)\right)+6 i a^2 \left(\sec ^{-1}(a+b x) \text{Li}_2\left(i e^{i \sec ^{-1}(a+b x)}\right)+i \text{Li}_3\left(i e^{i \sec ^{-1}(a+b x)}\right)\right)+6 i a^2 \sec ^{-1}(a+b x)^2 \tan ^{-1}\left(e^{i \sec ^{-1}(a+b x)}\right)+\frac{1}{3} b^3 x^3 \sec ^{-1}(a+b x)^3-i \sec ^{-1}(a+b x) \text{Li}_2\left(-i e^{i \sec ^{-1}(a+b x)}\right)+i \sec ^{-1}(a+b x) \text{Li}_2\left(i e^{i \sec ^{-1}(a+b x)}\right)+\text{Li}_3\left(-i e^{i \sec ^{-1}(a+b x)}\right)-\text{Li}_3\left(i e^{i \sec ^{-1}(a+b x)}\right)-3 i a \left(\text{Li}_2\left(-e^{2 i \sec ^{-1}(a+b x)}\right)+\sec ^{-1}(a+b x) \left(\sec ^{-1}(a+b x)+2 i \log \left(1+e^{2 i \sec ^{-1}(a+b x)}\right)\right)\right)-\frac{1}{2} (a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2+3 a (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2+(a+b x) \sec ^{-1}(a+b x)-\tanh ^{-1}\left(\sqrt{1-\frac{1}{(a+b x)^2}}\right)+i \sec ^{-1}(a+b x)^2 \tan ^{-1}\left(e^{i \sec ^{-1}(a+b x)}\right)}{b^3}","\frac{a^3 \sec ^{-1}(a+b x)^3}{3 b^3}-\frac{6 i a^2 \sec ^{-1}(a+b x) \text{Li}_2\left(-i e^{i \sec ^{-1}(a+b x)}\right)}{b^3}+\frac{6 i a^2 \sec ^{-1}(a+b x) \text{Li}_2\left(i e^{i \sec ^{-1}(a+b x)}\right)}{b^3}+\frac{6 a^2 \text{Li}_3\left(-i e^{i \sec ^{-1}(a+b x)}\right)}{b^3}-\frac{6 a^2 \text{Li}_3\left(i e^{i \sec ^{-1}(a+b x)}\right)}{b^3}+\frac{6 i a^2 \sec ^{-1}(a+b x)^2 \tan ^{-1}\left(e^{i \sec ^{-1}(a+b x)}\right)}{b^3}-\frac{i \sec ^{-1}(a+b x) \text{Li}_2\left(-i e^{i \sec ^{-1}(a+b x)}\right)}{b^3}+\frac{i \sec ^{-1}(a+b x) \text{Li}_2\left(i e^{i \sec ^{-1}(a+b x)}\right)}{b^3}-\frac{3 i a \text{Li}_2\left(-e^{2 i \sec ^{-1}(a+b x)}\right)}{b^3}+\frac{\text{Li}_3\left(-i e^{i \sec ^{-1}(a+b x)}\right)}{b^3}-\frac{\text{Li}_3\left(i e^{i \sec ^{-1}(a+b x)}\right)}{b^3}-\frac{(a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2}{2 b^3}+\frac{3 a (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2}{b^3}-\frac{3 i a \sec ^{-1}(a+b x)^2}{b^3}+\frac{(a+b x) \sec ^{-1}(a+b x)}{b^3}-\frac{\tanh ^{-1}\left(\sqrt{1-\frac{1}{(a+b x)^2}}\right)}{b^3}+\frac{6 a \sec ^{-1}(a+b x) \log \left(1+e^{2 i \sec ^{-1}(a+b x)}\right)}{b^3}+\frac{i \sec ^{-1}(a+b x)^2 \tan ^{-1}\left(e^{i \sec ^{-1}(a+b x)}\right)}{b^3}+\frac{1}{3} x^3 \sec ^{-1}(a+b x)^3",1,"((a + b*x)*ArcSec[a + b*x] + 3*a*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)]*ArcSec[a + b*x]^2 - ((a + b*x)^2*Sqrt[1 - (a + b*x)^(-2)]*ArcSec[a + b*x]^2)/2 + (a^3*ArcSec[a + b*x]^3)/3 + (b^3*x^3*ArcSec[a + b*x]^3)/3 + I*ArcSec[a + b*x]^2*ArcTan[E^(I*ArcSec[a + b*x])] + (6*I)*a^2*ArcSec[a + b*x]^2*ArcTan[E^(I*ArcSec[a + b*x])] - ArcTanh[Sqrt[1 - (a + b*x)^(-2)]] - I*ArcSec[a + b*x]*PolyLog[2, (-I)*E^(I*ArcSec[a + b*x])] + I*ArcSec[a + b*x]*PolyLog[2, I*E^(I*ArcSec[a + b*x])] - (3*I)*a*(ArcSec[a + b*x]*(ArcSec[a + b*x] + (2*I)*Log[1 + E^((2*I)*ArcSec[a + b*x])]) + PolyLog[2, -E^((2*I)*ArcSec[a + b*x])]) + PolyLog[3, (-I)*E^(I*ArcSec[a + b*x])] + 6*a^2*((-I)*ArcSec[a + b*x]*PolyLog[2, (-I)*E^(I*ArcSec[a + b*x])] + PolyLog[3, (-I)*E^(I*ArcSec[a + b*x])]) + (6*I)*a^2*(ArcSec[a + b*x]*PolyLog[2, I*E^(I*ArcSec[a + b*x])] + I*PolyLog[3, I*E^(I*ArcSec[a + b*x])]) - PolyLog[3, I*E^(I*ArcSec[a + b*x])])/b^3","A",1
34,1,248,278,0.4380363,"\int x \sec ^{-1}(a+b x)^3 \, dx","Integrate[x*ArcSec[a + b*x]^3,x]","\frac{1}{2} \left(x^2 \sec ^{-1}(a+b x)^3-\frac{3 \left(\frac{1}{3} a^2 \sec ^{-1}(a+b x)^3+4 i a \sec ^{-1}(a+b x) \text{Li}_2\left(i e^{i \sec ^{-1}(a+b x)}\right)-i \text{Li}_2\left(-e^{2 i \sec ^{-1}(a+b x)}\right)+4 a \left(\text{Li}_3\left(-i e^{i \sec ^{-1}(a+b x)}\right)-i \sec ^{-1}(a+b x) \text{Li}_2\left(-i e^{i \sec ^{-1}(a+b x)}\right)\right)-4 a \text{Li}_3\left(i e^{i \sec ^{-1}(a+b x)}\right)+(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2-i \sec ^{-1}(a+b x)^2+2 \sec ^{-1}(a+b x) \log \left(1+e^{2 i \sec ^{-1}(a+b x)}\right)+4 i a \sec ^{-1}(a+b x)^2 \tan ^{-1}\left(e^{i \sec ^{-1}(a+b x)}\right)\right)}{b^2}\right)","-\frac{a^2 \sec ^{-1}(a+b x)^3}{2 b^2}+\frac{6 i a \sec ^{-1}(a+b x) \text{Li}_2\left(-i e^{i \sec ^{-1}(a+b x)}\right)}{b^2}-\frac{6 i a \sec ^{-1}(a+b x) \text{Li}_2\left(i e^{i \sec ^{-1}(a+b x)}\right)}{b^2}+\frac{3 i \text{Li}_2\left(-e^{2 i \sec ^{-1}(a+b x)}\right)}{2 b^2}-\frac{6 a \text{Li}_3\left(-i e^{i \sec ^{-1}(a+b x)}\right)}{b^2}+\frac{6 a \text{Li}_3\left(i e^{i \sec ^{-1}(a+b x)}\right)}{b^2}-\frac{3 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \sec ^{-1}(a+b x)^2}{2 b^2}+\frac{3 i \sec ^{-1}(a+b x)^2}{2 b^2}-\frac{3 \sec ^{-1}(a+b x) \log \left(1+e^{2 i \sec ^{-1}(a+b x)}\right)}{b^2}-\frac{6 i a \sec ^{-1}(a+b x)^2 \tan ^{-1}\left(e^{i \sec ^{-1}(a+b x)}\right)}{b^2}+\frac{1}{2} x^2 \sec ^{-1}(a+b x)^3",1,"(x^2*ArcSec[a + b*x]^3 - (3*((-I)*ArcSec[a + b*x]^2 + (a + b*x)*Sqrt[1 - (a + b*x)^(-2)]*ArcSec[a + b*x]^2 + (a^2*ArcSec[a + b*x]^3)/3 + (4*I)*a*ArcSec[a + b*x]^2*ArcTan[E^(I*ArcSec[a + b*x])] + 2*ArcSec[a + b*x]*Log[1 + E^((2*I)*ArcSec[a + b*x])] + (4*I)*a*ArcSec[a + b*x]*PolyLog[2, I*E^(I*ArcSec[a + b*x])] - I*PolyLog[2, -E^((2*I)*ArcSec[a + b*x])] + 4*a*((-I)*ArcSec[a + b*x]*PolyLog[2, (-I)*E^(I*ArcSec[a + b*x])] + PolyLog[3, (-I)*E^(I*ArcSec[a + b*x])]) - 4*a*PolyLog[3, I*E^(I*ArcSec[a + b*x])]))/b^2)/2","A",1
35,1,160,154,0.0987383,"\int \sec ^{-1}(a+b x)^3 \, dx","Integrate[ArcSec[a + b*x]^3,x]","\frac{-6 i \sec ^{-1}(a+b x) \left(\text{Li}_2\left(-i e^{i \sec ^{-1}(a+b x)}\right)-\text{Li}_2\left(i e^{i \sec ^{-1}(a+b x)}\right)\right)+6 \left(\text{Li}_3\left(-i e^{i \sec ^{-1}(a+b x)}\right)-\text{Li}_3\left(i e^{i \sec ^{-1}(a+b x)}\right)\right)+(a+b x) \sec ^{-1}(a+b x)^3-3 \sec ^{-1}(a+b x)^2 \left(\log \left(1-i e^{i \sec ^{-1}(a+b x)}\right)-\log \left(1+i e^{i \sec ^{-1}(a+b x)}\right)\right)}{b}","-\frac{6 i \sec ^{-1}(a+b x) \text{Li}_2\left(-i e^{i \sec ^{-1}(a+b x)}\right)}{b}+\frac{6 i \sec ^{-1}(a+b x) \text{Li}_2\left(i e^{i \sec ^{-1}(a+b x)}\right)}{b}+\frac{6 \text{Li}_3\left(-i e^{i \sec ^{-1}(a+b x)}\right)}{b}-\frac{6 \text{Li}_3\left(i e^{i \sec ^{-1}(a+b x)}\right)}{b}+\frac{(a+b x) \sec ^{-1}(a+b x)^3}{b}+\frac{6 i \sec ^{-1}(a+b x)^2 \tan ^{-1}\left(e^{i \sec ^{-1}(a+b x)}\right)}{b}",1,"((a + b*x)*ArcSec[a + b*x]^3 - 3*ArcSec[a + b*x]^2*(Log[1 - I*E^(I*ArcSec[a + b*x])] - Log[1 + I*E^(I*ArcSec[a + b*x])]) - (6*I)*ArcSec[a + b*x]*(PolyLog[2, (-I)*E^(I*ArcSec[a + b*x])] - PolyLog[2, I*E^(I*ArcSec[a + b*x])]) + 6*(PolyLog[3, (-I)*E^(I*ArcSec[a + b*x])] - PolyLog[3, I*E^(I*ArcSec[a + b*x])]))/b","A",0
36,1,1058,430,4.0551523,"\int \frac{\sec ^{-1}(a+b x)^3}{x} \, dx","Integrate[ArcSec[a + b*x]^3/x,x]","2 \log \left(\frac{e^{i \sec ^{-1}(a+b x)} a}{\sqrt{1-a^2}-1}+1\right) \sec ^{-1}(a+b x)^3+\log \left(\frac{e^{i \sec ^{-1}(a+b x)} \left(\sqrt{1-a^2}-1\right)}{a}+1\right) \sec ^{-1}(a+b x)^3+2 \log \left(1-\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right) \sec ^{-1}(a+b x)^3+\log \left(1-\frac{\left(\sqrt{1-a^2}+1\right) e^{i \sec ^{-1}(a+b x)}}{a}\right) \sec ^{-1}(a+b x)^3-3 \log \left(1+e^{2 i \sec ^{-1}(a+b x)}\right) \sec ^{-1}(a+b x)^3+2 \log \left(\frac{2 \left(i \sqrt{1-\frac{1}{(a+b x)^2}}+\frac{1}{a+b x}\right)}{a+b x}\right) \sec ^{-1}(a+b x)^3-\log \left(\frac{a \left(i \sqrt{1-\frac{1}{(a+b x)^2}}+\frac{1}{a+b x}\right)}{\sqrt{1-a^2}-1}+1\right) \sec ^{-1}(a+b x)^3-\log \left(\frac{\left(\sqrt{1-a^2}-1\right) \left(i \sqrt{1-\frac{1}{(a+b x)^2}}+\frac{1}{a+b x}\right)}{a}+1\right) \sec ^{-1}(a+b x)^3-\log \left(1-\frac{a \left(i \sqrt{1-\frac{1}{(a+b x)^2}}+\frac{1}{a+b x}\right)}{\sqrt{1-a^2}+1}\right) \sec ^{-1}(a+b x)^3-\log \left(1-\frac{\left(\sqrt{1-a^2}+1\right) \left(i \sqrt{1-\frac{1}{(a+b x)^2}}+\frac{1}{a+b x}\right)}{a}\right) \sec ^{-1}(a+b x)^3-6 \sin ^{-1}\left(\frac{\sqrt{\frac{a-1}{a}}}{\sqrt{2}}\right) \log \left(\frac{e^{i \sec ^{-1}(a+b x)} \left(\sqrt{1-a^2}-1\right)}{a}+1\right) \sec ^{-1}(a+b x)^2+6 \sin ^{-1}\left(\frac{\sqrt{\frac{a-1}{a}}}{\sqrt{2}}\right) \log \left(1-\frac{\left(\sqrt{1-a^2}+1\right) e^{i \sec ^{-1}(a+b x)}}{a}\right) \sec ^{-1}(a+b x)^2+6 \sin ^{-1}\left(\frac{\sqrt{\frac{a-1}{a}}}{\sqrt{2}}\right) \log \left(\frac{\left(\sqrt{1-a^2}-1\right) \left(i \sqrt{1-\frac{1}{(a+b x)^2}}+\frac{1}{a+b x}\right)}{a}+1\right) \sec ^{-1}(a+b x)^2-6 \sin ^{-1}\left(\frac{\sqrt{\frac{a-1}{a}}}{\sqrt{2}}\right) \log \left(1-\frac{\left(\sqrt{1-a^2}+1\right) \left(i \sqrt{1-\frac{1}{(a+b x)^2}}+\frac{1}{a+b x}\right)}{a}\right) \sec ^{-1}(a+b x)^2-3 i \text{Li}_2\left(-\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}-1}\right) \sec ^{-1}(a+b x)^2-3 i \text{Li}_2\left(\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right) \sec ^{-1}(a+b x)^2+\frac{3}{2} i \text{Li}_2\left(-e^{2 i \sec ^{-1}(a+b x)}\right) \sec ^{-1}(a+b x)^2+6 \text{Li}_3\left(-\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}-1}\right) \sec ^{-1}(a+b x)+6 \text{Li}_3\left(\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right) \sec ^{-1}(a+b x)-\frac{3}{2} \text{Li}_3\left(-e^{2 i \sec ^{-1}(a+b x)}\right) \sec ^{-1}(a+b x)+6 i \text{Li}_4\left(-\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}-1}\right)+6 i \text{Li}_4\left(\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)-\frac{3}{4} i \text{Li}_4\left(-e^{2 i \sec ^{-1}(a+b x)}\right)","-3 i \sec ^{-1}(a+b x)^2 \text{Li}_2\left(\frac{a e^{i \sec ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right)-3 i \sec ^{-1}(a+b x)^2 \text{Li}_2\left(\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)+6 \sec ^{-1}(a+b x) \text{Li}_3\left(\frac{a e^{i \sec ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right)+6 \sec ^{-1}(a+b x) \text{Li}_3\left(\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)+6 i \text{Li}_4\left(\frac{a e^{i \sec ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right)+6 i \text{Li}_4\left(\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)+\sec ^{-1}(a+b x)^3 \log \left(1-\frac{a e^{i \sec ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right)+\sec ^{-1}(a+b x)^3 \log \left(1-\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)+\frac{3}{2} i \sec ^{-1}(a+b x)^2 \text{Li}_2\left(-e^{2 i \sec ^{-1}(a+b x)}\right)-\frac{3}{2} \sec ^{-1}(a+b x) \text{Li}_3\left(-e^{2 i \sec ^{-1}(a+b x)}\right)-\frac{3}{4} i \text{Li}_4\left(-e^{2 i \sec ^{-1}(a+b x)}\right)-\sec ^{-1}(a+b x)^3 \log \left(1+e^{2 i \sec ^{-1}(a+b x)}\right)",1,"2*ArcSec[a + b*x]^3*Log[1 + (a*E^(I*ArcSec[a + b*x]))/(-1 + Sqrt[1 - a^2])] + ArcSec[a + b*x]^3*Log[1 + ((-1 + Sqrt[1 - a^2])*E^(I*ArcSec[a + b*x]))/a] - 6*ArcSec[a + b*x]^2*ArcSin[Sqrt[(-1 + a)/a]/Sqrt[2]]*Log[1 + ((-1 + Sqrt[1 - a^2])*E^(I*ArcSec[a + b*x]))/a] + 2*ArcSec[a + b*x]^3*Log[1 - (a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])] + ArcSec[a + b*x]^3*Log[1 - ((1 + Sqrt[1 - a^2])*E^(I*ArcSec[a + b*x]))/a] + 6*ArcSec[a + b*x]^2*ArcSin[Sqrt[(-1 + a)/a]/Sqrt[2]]*Log[1 - ((1 + Sqrt[1 - a^2])*E^(I*ArcSec[a + b*x]))/a] - 3*ArcSec[a + b*x]^3*Log[1 + E^((2*I)*ArcSec[a + b*x])] + 2*ArcSec[a + b*x]^3*Log[(2*((a + b*x)^(-1) + I*Sqrt[1 - (a + b*x)^(-2)]))/(a + b*x)] - ArcSec[a + b*x]^3*Log[1 + (a*((a + b*x)^(-1) + I*Sqrt[1 - (a + b*x)^(-2)]))/(-1 + Sqrt[1 - a^2])] - ArcSec[a + b*x]^3*Log[1 + ((-1 + Sqrt[1 - a^2])*((a + b*x)^(-1) + I*Sqrt[1 - (a + b*x)^(-2)]))/a] + 6*ArcSec[a + b*x]^2*ArcSin[Sqrt[(-1 + a)/a]/Sqrt[2]]*Log[1 + ((-1 + Sqrt[1 - a^2])*((a + b*x)^(-1) + I*Sqrt[1 - (a + b*x)^(-2)]))/a] - ArcSec[a + b*x]^3*Log[1 - (a*((a + b*x)^(-1) + I*Sqrt[1 - (a + b*x)^(-2)]))/(1 + Sqrt[1 - a^2])] - ArcSec[a + b*x]^3*Log[1 - ((1 + Sqrt[1 - a^2])*((a + b*x)^(-1) + I*Sqrt[1 - (a + b*x)^(-2)]))/a] - 6*ArcSec[a + b*x]^2*ArcSin[Sqrt[(-1 + a)/a]/Sqrt[2]]*Log[1 - ((1 + Sqrt[1 - a^2])*((a + b*x)^(-1) + I*Sqrt[1 - (a + b*x)^(-2)]))/a] - (3*I)*ArcSec[a + b*x]^2*PolyLog[2, -((a*E^(I*ArcSec[a + b*x]))/(-1 + Sqrt[1 - a^2]))] - (3*I)*ArcSec[a + b*x]^2*PolyLog[2, (a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])] + ((3*I)/2)*ArcSec[a + b*x]^2*PolyLog[2, -E^((2*I)*ArcSec[a + b*x])] + 6*ArcSec[a + b*x]*PolyLog[3, -((a*E^(I*ArcSec[a + b*x]))/(-1 + Sqrt[1 - a^2]))] + 6*ArcSec[a + b*x]*PolyLog[3, (a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])] - (3*ArcSec[a + b*x]*PolyLog[3, -E^((2*I)*ArcSec[a + b*x])])/2 + (6*I)*PolyLog[4, -((a*E^(I*ArcSec[a + b*x]))/(-1 + Sqrt[1 - a^2]))] + (6*I)*PolyLog[4, (a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])] - ((3*I)/4)*PolyLog[4, -E^((2*I)*ArcSec[a + b*x])]","B",0
37,-1,0,362,180.0016589,"\int \frac{\sec ^{-1}(a+b x)^3}{x^2} \, dx","Integrate[ArcSec[a + b*x]^3/x^2,x]","\text{\$Aborted}","-\frac{6 b \sec ^{-1}(a+b x) \text{Li}_2\left(\frac{a e^{i \sec ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right)}{a \sqrt{1-a^2}}+\frac{6 b \sec ^{-1}(a+b x) \text{Li}_2\left(\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)}{a \sqrt{1-a^2}}-\frac{6 i b \text{Li}_3\left(\frac{a e^{i \sec ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right)}{a \sqrt{1-a^2}}+\frac{6 i b \text{Li}_3\left(\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)}{a \sqrt{1-a^2}}-\frac{3 i b \sec ^{-1}(a+b x)^2 \log \left(1-\frac{a e^{i \sec ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right)}{a \sqrt{1-a^2}}+\frac{3 i b \sec ^{-1}(a+b x)^2 \log \left(1-\frac{a e^{i \sec ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right)}{a \sqrt{1-a^2}}-\frac{b \sec ^{-1}(a+b x)^3}{a}-\frac{\sec ^{-1}(a+b x)^3}{x}",1,"$Aborted","F",-1
38,1,148,58,0.2373487,"\int x \left(a+b \sec ^{-1}\left(c+d x^2\right)\right) \, dx","Integrate[x*(a + b*ArcSec[c + d*x^2]),x]","\frac{a x^2}{2}-\frac{b \left(c+d x^2\right) \sqrt{\frac{c^2+2 c d x^2+d^2 x^4-1}{\left(c+d x^2\right)^2}} \left(\tanh ^{-1}\left(\frac{c+d x^2}{\sqrt{c^2+2 c d x^2+d^2 x^4-1}}\right)-c \tan ^{-1}\left(\sqrt{\left(c+d x^2\right)^2-1}\right)\right)}{2 d \sqrt{c^2+2 c d x^2+d^2 x^4-1}}+\frac{1}{2} b x^2 \sec ^{-1}\left(c+d x^2\right)","\frac{a x^2}{2}+\frac{b \left(c+d x^2\right) \sec ^{-1}\left(c+d x^2\right)}{2 d}-\frac{b \tanh ^{-1}\left(\sqrt{1-\frac{1}{\left(c+d x^2\right)^2}}\right)}{2 d}",1,"(a*x^2)/2 + (b*x^2*ArcSec[c + d*x^2])/2 - (b*(c + d*x^2)*Sqrt[(-1 + c^2 + 2*c*d*x^2 + d^2*x^4)/(c + d*x^2)^2]*(-(c*ArcTan[Sqrt[-1 + (c + d*x^2)^2]]) + ArcTanh[(c + d*x^2)/Sqrt[-1 + c^2 + 2*c*d*x^2 + d^2*x^4]]))/(2*d*Sqrt[-1 + c^2 + 2*c*d*x^2 + d^2*x^4])","B",1
39,1,148,58,0.15544,"\int x^2 \left(a+b \sec ^{-1}\left(c+d x^3\right)\right) \, dx","Integrate[x^2*(a + b*ArcSec[c + d*x^3]),x]","\frac{a x^3}{3}-\frac{b \left(c+d x^3\right) \sqrt{\frac{c^2+2 c d x^3+d^2 x^6-1}{\left(c+d x^3\right)^2}} \left(\tanh ^{-1}\left(\frac{c+d x^3}{\sqrt{c^2+2 c d x^3+d^2 x^6-1}}\right)-c \tan ^{-1}\left(\sqrt{\left(c+d x^3\right)^2-1}\right)\right)}{3 d \sqrt{c^2+2 c d x^3+d^2 x^6-1}}+\frac{1}{3} b x^3 \sec ^{-1}\left(c+d x^3\right)","\frac{a x^3}{3}+\frac{b \left(c+d x^3\right) \sec ^{-1}\left(c+d x^3\right)}{3 d}-\frac{b \tanh ^{-1}\left(\sqrt{1-\frac{1}{\left(c+d x^3\right)^2}}\right)}{3 d}",1,"(a*x^3)/3 + (b*x^3*ArcSec[c + d*x^3])/3 - (b*(c + d*x^3)*Sqrt[(-1 + c^2 + 2*c*d*x^3 + d^2*x^6)/(c + d*x^3)^2]*(-(c*ArcTan[Sqrt[-1 + (c + d*x^3)^2]]) + ArcTanh[(c + d*x^3)/Sqrt[-1 + c^2 + 2*c*d*x^3 + d^2*x^6]]))/(3*d*Sqrt[-1 + c^2 + 2*c*d*x^3 + d^2*x^6])","B",1
40,1,137,58,0.3828775,"\int x^3 \left(a+b \sec ^{-1}\left(c+d x^4\right)\right) \, dx","Integrate[x^3*(a + b*ArcSec[c + d*x^4]),x]","\frac{a x^4}{4}-\frac{b \sqrt{\left(c+d x^4\right)^2-1} \left(\log \left(\frac{c+d x^4}{\sqrt{\left(c+d x^4\right)^2-1}}+1\right)-\log \left(1-\frac{c+d x^4}{\sqrt{\left(c+d x^4\right)^2-1}}\right)\right)}{8 d \left(c+d x^4\right) \sqrt{1-\frac{1}{\left(c+d x^4\right)^2}}}+\frac{b \left(c+d x^4\right) \sec ^{-1}\left(c+d x^4\right)}{4 d}","\frac{a x^4}{4}+\frac{b \left(c+d x^4\right) \sec ^{-1}\left(c+d x^4\right)}{4 d}-\frac{b \tanh ^{-1}\left(\sqrt{1-\frac{1}{\left(c+d x^4\right)^2}}\right)}{4 d}",1,"(a*x^4)/4 + (b*(c + d*x^4)*ArcSec[c + d*x^4])/(4*d) - (b*Sqrt[-1 + (c + d*x^4)^2]*(-Log[1 - (c + d*x^4)/Sqrt[-1 + (c + d*x^4)^2]] + Log[1 + (c + d*x^4)/Sqrt[-1 + (c + d*x^4)^2]]))/(8*d*(c + d*x^4)*Sqrt[1 - (c + d*x^4)^(-2)])","B",1
41,1,130,49,0.3395374,"\int x^{-1+n} \sec ^{-1}\left(a+b x^n\right) \, dx","Integrate[x^(-1 + n)*ArcSec[a + b*x^n],x]","\frac{\left(a+b x^n\right) \sec ^{-1}\left(a+b x^n\right)}{b n}-\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) \sec ^{-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)*ArcSec[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
42,1,280,85,0.8553918,"\int \sec ^{-1}\left(c e^{a+b x}\right) \, dx","Integrate[ArcSec[c*E^(a + b*x)],x]","x \sec ^{-1}\left(c e^{a+b x}\right)-\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}}}","\frac{i \text{Li}_2\left(-e^{2 i \sec ^{-1}\left(c e^{a+b x}\right)}\right)}{2 b}+\frac{i \sec ^{-1}\left(c e^{a+b x}\right)^2}{2 b}-\frac{\sec ^{-1}\left(c e^{a+b x}\right) \log \left(1+e^{2 i \sec ^{-1}\left(c e^{a+b x}\right)}\right)}{b}",1,"x*ArcSec[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
43,1,95,99,0.3121855,"\int e^{\sec ^{-1}(a x)} x^2 \, dx","Integrate[E^ArcSec[a*x]*x^2,x]","\frac{e^{\sec ^{-1}(a x)} \left(a^4 x^4 \left(\cos \left(2 \sec ^{-1}(a x)\right)-\sin \left(2 \sec ^{-1}(a x)\right)+5\right)-(4+4 i) \left(a x \sqrt{1-\frac{1}{a^2 x^2}}-i\right) \, _2F_1\left(\frac{1}{2}-\frac{i}{2},1;\frac{3}{2}-\frac{i}{2};-e^{2 i \sec ^{-1}(a x)}\right)\right)}{12 a^4 x}","\frac{\left(\frac{24}{5}+\frac{8 i}{5}\right) e^{(1+3 i) \sec ^{-1}(a x)} \, _2F_1\left(\frac{3}{2}-\frac{i}{2},4;\frac{5}{2}-\frac{i}{2};-e^{2 i \sec ^{-1}(a x)}\right)}{a^3}-\frac{\left(\frac{12}{5}+\frac{4 i}{5}\right) e^{(1+3 i) \sec ^{-1}(a x)} \, _2F_1\left(\frac{3}{2}-\frac{i}{2},3;\frac{5}{2}-\frac{i}{2};-e^{2 i \sec ^{-1}(a x)}\right)}{a^3}",1,"(E^ArcSec[a*x]*((-4 - 4*I)*(-I + a*Sqrt[1 - 1/(a^2*x^2)]*x)*Hypergeometric2F1[1/2 - I/2, 1, 3/2 - I/2, -E^((2*I)*ArcSec[a*x])] + a^4*x^4*(5 + Cos[2*ArcSec[a*x]] - Sin[2*ArcSec[a*x]])))/(12*a^4*x)","A",0
44,1,107,91,0.1972489,"\int e^{\sec ^{-1}(a x)} x \, dx","Integrate[E^ArcSec[a*x]*x,x]","\frac{\left(\frac{1}{5}+\frac{i}{10}\right) e^{\sec ^{-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 \sec ^{-1}(a x)}\right)-e^{2 i \sec ^{-1}(a x)} \, _2F_1\left(1,1-\frac{i}{2};2-\frac{i}{2};-e^{2 i \sec ^{-1}(a x)}\right)\right)}{a^2}","\frac{\left(\frac{16}{5}+\frac{8 i}{5}\right) e^{(1+2 i) \sec ^{-1}(a x)} \, _2F_1\left(1-\frac{i}{2},3;2-\frac{i}{2};-e^{2 i \sec ^{-1}(a x)}\right)}{a^2}-\frac{\left(\frac{8}{5}+\frac{4 i}{5}\right) e^{(1+2 i) \sec ^{-1}(a x)} \, _2F_1\left(1-\frac{i}{2},2;2-\frac{i}{2};-e^{2 i \sec ^{-1}(a x)}\right)}{a^2}",1,"((1/5 + I/10)*E^ArcSec[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)*ArcSec[a*x])] - E^((2*I)*ArcSec[a*x])*Hypergeometric2F1[1, 1 - I/2, 2 - I/2, -E^((2*I)*ArcSec[a*x])]))/a^2","A",0
45,1,54,91,0.0603377,"\int e^{\sec ^{-1}(a x)} \, dx","Integrate[E^ArcSec[a*x],x]","x e^{\sec ^{-1}(a x)}-\frac{(1-i) e^{(1+i) \sec ^{-1}(a x)} \, _2F_1\left(\frac{1}{2}-\frac{i}{2},1;\frac{3}{2}-\frac{i}{2};-e^{2 i \sec ^{-1}(a x)}\right)}{a}","\frac{(2+2 i) e^{(1+i) \sec ^{-1}(a x)} \, _2F_1\left(\frac{1}{2}-\frac{i}{2},2;\frac{3}{2}-\frac{i}{2};-e^{2 i \sec ^{-1}(a x)}\right)}{a}-\frac{(1+i) e^{(1+i) \sec ^{-1}(a x)} \, _2F_1\left(\frac{1}{2}-\frac{i}{2},1;\frac{3}{2}-\frac{i}{2};-e^{2 i \sec ^{-1}(a x)}\right)}{a}",1,"E^ArcSec[a*x]*x - ((1 - I)*E^((1 + I)*ArcSec[a*x])*Hypergeometric2F1[1/2 - I/2, 1, 3/2 - I/2, -E^((2*I)*ArcSec[a*x])])/a","A",0
46,1,79,45,0.0542362,"\int \frac{e^{\sec ^{-1}(a x)}}{x} \, dx","Integrate[E^ArcSec[a*x]/x,x]","-i \left(\left(\frac{1}{5}-\frac{2 i}{5}\right) e^{(1+2 i) \sec ^{-1}(a x)} \, _2F_1\left(1,1-\frac{i}{2};2-\frac{i}{2};-e^{2 i \sec ^{-1}(a x)}\right)-e^{\sec ^{-1}(a x)} \, _2F_1\left(-\frac{i}{2},1;1-\frac{i}{2};-e^{2 i \sec ^{-1}(a x)}\right)\right)","2 i e^{\sec ^{-1}(a x)} \, _2F_1\left(-\frac{i}{2},1;1-\frac{i}{2};-e^{2 i \sec ^{-1}(a x)}\right)-i e^{\sec ^{-1}(a x)}",1,"(-I)*(-(E^ArcSec[a*x]*Hypergeometric2F1[-1/2*I, 1, 1 - I/2, -E^((2*I)*ArcSec[a*x])]) + (1/5 - (2*I)/5)*E^((1 + 2*I)*ArcSec[a*x])*Hypergeometric2F1[1, 1 - I/2, 2 - I/2, -E^((2*I)*ArcSec[a*x])])","A",0
47,1,34,39,0.0426816,"\int \frac{e^{\sec ^{-1}(a x)}}{x^2} \, dx","Integrate[E^ArcSec[a*x]/x^2,x]","\frac{1}{2} a \left(\sqrt{1-\frac{1}{a^2 x^2}}-\frac{1}{a x}\right) e^{\sec ^{-1}(a x)}","\frac{1}{2} a \sqrt{1-\frac{1}{a^2 x^2}} e^{\sec ^{-1}(a x)}-\frac{e^{\sec ^{-1}(a x)}}{2 x}",1,"(a*E^ArcSec[a*x]*(Sqrt[1 - 1/(a^2*x^2)] - 1/(a*x)))/2","A",1
48,1,30,41,0.0445296,"\int \frac{e^{\sec ^{-1}(a x)}}{x^3} \, dx","Integrate[E^ArcSec[a*x]/x^3,x]","\frac{1}{10} a^2 e^{\sec ^{-1}(a x)} \left(\sin \left(2 \sec ^{-1}(a x)\right)-2 \cos \left(2 \sec ^{-1}(a x)\right)\right)","\frac{1}{10} a^2 e^{\sec ^{-1}(a x)} \sin \left(2 \sec ^{-1}(a x)\right)-\frac{1}{5} a^2 e^{\sec ^{-1}(a x)} \cos \left(2 \sec ^{-1}(a x)\right)",1,"(a^2*E^ArcSec[a*x]*(-2*Cos[2*ArcSec[a*x]] + Sin[2*ArcSec[a*x]]))/10","A",1
49,1,54,84,0.1389298,"\int \frac{e^{\sec ^{-1}(a x)}}{x^4} \, dx","Integrate[E^ArcSec[a*x]/x^4,x]","\frac{1}{40} a^3 e^{\sec ^{-1}(a x)} \left(5 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{5}{a x}-3 \cos \left(3 \sec ^{-1}(a x)\right)+\sin \left(3 \sec ^{-1}(a x)\right)\right)","-\frac{3}{40} a^3 e^{\sec ^{-1}(a x)} \cos \left(3 \sec ^{-1}(a x)\right)+\frac{1}{40} a^3 e^{\sec ^{-1}(a x)} \sin \left(3 \sec ^{-1}(a x)\right)-\frac{a^2 e^{\sec ^{-1}(a x)}}{8 x}+\frac{1}{8} a^3 \sqrt{1-\frac{1}{a^2 x^2}} e^{\sec ^{-1}(a x)}",1,"(a^3*E^ArcSec[a*x]*(5*Sqrt[1 - 1/(a^2*x^2)] - 5/(a*x) - 3*Cos[3*ArcSec[a*x]] + Sin[3*ArcSec[a*x]]))/40","A",1
50,1,59,69,0.0502475,"\int \frac{\sec ^{-1}(a+b x)}{\frac{a d}{b}+d x} \, dx","Integrate[ArcSec[a + b*x]/((a*d)/b + d*x),x]","\frac{i \left(\text{Li}_2\left(-e^{2 i \sec ^{-1}(a+b x)}\right)+\sec ^{-1}(a+b x) \left(\sec ^{-1}(a+b x)+2 i \log \left(1+e^{2 i \sec ^{-1}(a+b x)}\right)\right)\right)}{2 d}","\frac{i \text{Li}_2\left(-e^{2 i \sec ^{-1}(a+b x)}\right)}{2 d}+\frac{i \sec ^{-1}(a+b x)^2}{2 d}-\frac{\sec ^{-1}(a+b x) \log \left(1+e^{2 i \sec ^{-1}(a+b x)}\right)}{d}",1,"((I/2)*(ArcSec[a + b*x]*(ArcSec[a + b*x] + (2*I)*Log[1 + E^((2*I)*ArcSec[a + b*x])]) + PolyLog[2, -E^((2*I)*ArcSec[a + b*x])]))/d","A",1