1,1,182,164,0.4638444,"\int x^4 \text{sech}^{-1}(a x)^2 \, dx","Integrate[x^4*ArcSech[a*x]^2,x]","\frac{12 a^5 x^5 \text{sech}^{-1}(a x)^2-2 a^3 x^3-6 a^3 x^3 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)+9 i \text{Li}_2\left(-i e^{-\text{sech}^{-1}(a x)}\right)-9 i \text{Li}_2\left(i e^{-\text{sech}^{-1}(a x)}\right)-9 a x-9 a x \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)+9 i \text{sech}^{-1}(a x) \log \left(1-i e^{-\text{sech}^{-1}(a x)}\right)-9 i \text{sech}^{-1}(a x) \log \left(1+i e^{-\text{sech}^{-1}(a x)}\right)}{60 a^5}","\frac{3 i \text{Li}_2\left(-i e^{\text{sech}^{-1}(a x)}\right)}{20 a^5}-\frac{3 i \text{Li}_2\left(i e^{\text{sech}^{-1}(a x)}\right)}{20 a^5}-\frac{3 \text{sech}^{-1}(a x) \tan ^{-1}\left(e^{\text{sech}^{-1}(a x)}\right)}{10 a^5}-\frac{3 x}{20 a^4}-\frac{3 x \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)}{20 a^4}-\frac{x^3}{30 a^2}-\frac{x^3 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)}{10 a^2}+\frac{1}{5} x^5 \text{sech}^{-1}(a x)^2",1,"(-9*a*x - 2*a^3*x^3 - 9*a*x*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x] - 6*a^3*x^3*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x] + 12*a^5*x^5*ArcSech[a*x]^2 + (9*I)*ArcSech[a*x]*Log[1 - I/E^ArcSech[a*x]] - (9*I)*ArcSech[a*x]*Log[1 + I/E^ArcSech[a*x]] + (9*I)*PolyLog[2, (-I)/E^ArcSech[a*x]] - (9*I)*PolyLog[2, I/E^ArcSech[a*x]])/(60*a^5)","A",0
2,1,77,104,0.1175944,"\int x^3 \text{sech}^{-1}(a x)^2 \, dx","Integrate[x^3*ArcSech[a*x]^2,x]","-\frac{-3 a^4 x^4 \text{sech}^{-1}(a x)^2+a^2 x^2+2 \sqrt{\frac{1-a x}{a x+1}} \left(a^3 x^3+a^2 x^2+2 a x+2\right) \text{sech}^{-1}(a x)+4 \log (x)}{12 a^4}","-\frac{\log (x)}{3 a^4}-\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)}{3 a^4}-\frac{x^2}{12 a^2}-\frac{x^2 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)}{6 a^2}+\frac{1}{4} x^4 \text{sech}^{-1}(a x)^2",1,"-1/12*(a^2*x^2 + 2*Sqrt[(1 - a*x)/(1 + a*x)]*(2 + 2*a*x + a^2*x^2 + a^3*x^3)*ArcSech[a*x] - 3*a^4*x^4*ArcSech[a*x]^2 + 4*Log[x])/a^4","A",1
3,1,138,117,0.2539322,"\int x^2 \text{sech}^{-1}(a x)^2 \, dx","Integrate[x^2*ArcSech[a*x]^2,x]","\frac{a^3 x^3 \text{sech}^{-1}(a x)^2+i \text{Li}_2\left(-i e^{-\text{sech}^{-1}(a x)}\right)-i \text{Li}_2\left(i e^{-\text{sech}^{-1}(a x)}\right)-a x-a x \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)+i \text{sech}^{-1}(a x) \log \left(1-i e^{-\text{sech}^{-1}(a x)}\right)-i \text{sech}^{-1}(a x) \log \left(1+i e^{-\text{sech}^{-1}(a x)}\right)}{3 a^3}","\frac{i \text{Li}_2\left(-i e^{\text{sech}^{-1}(a x)}\right)}{3 a^3}-\frac{i \text{Li}_2\left(i e^{\text{sech}^{-1}(a x)}\right)}{3 a^3}-\frac{2 \text{sech}^{-1}(a x) \tan ^{-1}\left(e^{\text{sech}^{-1}(a x)}\right)}{3 a^3}-\frac{x}{3 a^2}-\frac{x \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)}{3 a^2}+\frac{1}{3} x^3 \text{sech}^{-1}(a x)^2",1,"(-(a*x) - a*x*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x] + a^3*x^3*ArcSech[a*x]^2 + I*ArcSech[a*x]*Log[1 - I/E^ArcSech[a*x]] - I*ArcSech[a*x]*Log[1 + I/E^ArcSech[a*x]] + I*PolyLog[2, (-I)/E^ArcSech[a*x]] - I*PolyLog[2, I/E^ArcSech[a*x]])/(3*a^3)","A",0
4,1,53,53,0.0645313,"\int x \text{sech}^{-1}(a x)^2 \, dx","Integrate[x*ArcSech[a*x]^2,x]","-\frac{\log (x)}{a^2}-\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)}{a^2}+\frac{1}{2} x^2 \text{sech}^{-1}(a x)^2","-\frac{\log (x)}{a^2}-\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)}{a^2}+\frac{1}{2} x^2 \text{sech}^{-1}(a x)^2",1,"-((Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x])/a^2) + (x^2*ArcSech[a*x]^2)/2 - Log[x]/a^2","A",1
5,1,90,63,0.1999562,"\int \text{sech}^{-1}(a x)^2 \, dx","Integrate[ArcSech[a*x]^2,x]","\frac{i \left(2 \text{Li}_2\left(-i e^{-\text{sech}^{-1}(a x)}\right)-2 \text{Li}_2\left(i e^{-\text{sech}^{-1}(a x)}\right)+\text{sech}^{-1}(a x) \left(-i a x \text{sech}^{-1}(a x)+2 \log \left(1-i e^{-\text{sech}^{-1}(a x)}\right)-2 \log \left(1+i e^{-\text{sech}^{-1}(a x)}\right)\right)\right)}{a}","\frac{2 i \text{Li}_2\left(-i e^{\text{sech}^{-1}(a x)}\right)}{a}-\frac{2 i \text{Li}_2\left(i e^{\text{sech}^{-1}(a x)}\right)}{a}+x \text{sech}^{-1}(a x)^2-\frac{4 \text{sech}^{-1}(a x) \tan ^{-1}\left(e^{\text{sech}^{-1}(a x)}\right)}{a}",1,"(I*(ArcSech[a*x]*((-I)*a*x*ArcSech[a*x] + 2*Log[1 - I/E^ArcSech[a*x]] - 2*Log[1 + I/E^ArcSech[a*x]]) + 2*PolyLog[2, (-I)/E^ArcSech[a*x]] - 2*PolyLog[2, I/E^ArcSech[a*x]]))/a","A",0
6,1,63,64,0.0355148,"\int \frac{\text{sech}^{-1}(a x)^2}{x} \, dx","Integrate[ArcSech[a*x]^2/x,x]","\text{sech}^{-1}(a x) \text{Li}_2\left(-e^{-2 \text{sech}^{-1}(a x)}\right)+\frac{1}{2} \text{Li}_3\left(-e^{-2 \text{sech}^{-1}(a x)}\right)-\frac{1}{3} \text{sech}^{-1}(a x)^3-\text{sech}^{-1}(a x)^2 \log \left(e^{-2 \text{sech}^{-1}(a x)}+1\right)","-\text{sech}^{-1}(a x) \text{Li}_2\left(-e^{2 \text{sech}^{-1}(a x)}\right)+\frac{1}{2} \text{Li}_3\left(-e^{2 \text{sech}^{-1}(a x)}\right)+\frac{1}{3} \text{sech}^{-1}(a x)^3-\text{sech}^{-1}(a x)^2 \log \left(e^{2 \text{sech}^{-1}(a x)}+1\right)",1,"-1/3*ArcSech[a*x]^3 - ArcSech[a*x]^2*Log[1 + E^(-2*ArcSech[a*x])] + ArcSech[a*x]*PolyLog[2, -E^(-2*ArcSech[a*x])] + PolyLog[3, -E^(-2*ArcSech[a*x])]/2","A",0
7,1,42,49,0.100313,"\int \frac{\text{sech}^{-1}(a x)^2}{x^2} \, dx","Integrate[ArcSech[a*x]^2/x^2,x]","-\frac{\text{sech}^{-1}(a x)^2-2 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)+2}{x}","-\frac{\text{sech}^{-1}(a x)^2}{x}+\frac{2 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)}{x}-\frac{2}{x}",1,"-((2 - 2*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x] + ArcSech[a*x]^2)/x)","A",1
8,1,54,90,0.0439721,"\int \frac{\text{sech}^{-1}(a x)^2}{x^3} \, dx","Integrate[ArcSech[a*x]^2/x^3,x]","\frac{\left(a^2 x^2-2\right) \text{sech}^{-1}(a x)^2+2 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)-1}{4 x^2}","-\frac{1}{4} a^2 \text{sech}^{-1}(a x)^2-\frac{(1-a x) (a x+1)}{4 x^2}-\frac{(1-a x) (a x+1) \text{sech}^{-1}(a x)^2}{2 x^2}+\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)}{2 x^2}",1,"(-1 + 2*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x] + (-2 + a^2*x^2)*ArcSech[a*x]^2)/(4*x^2)","A",1
9,1,73,102,0.1091788,"\int \frac{\text{sech}^{-1}(a x)^2}{x^4} \, dx","Integrate[ArcSech[a*x]^2/x^4,x]","\frac{-2 \left(6 a^2 x^2+1\right)+6 \sqrt{\frac{1-a x}{a x+1}} \left(2 a^3 x^3+2 a^2 x^2+a x+1\right) \text{sech}^{-1}(a x)-9 \text{sech}^{-1}(a x)^2}{27 x^3}","-\frac{4 a^2}{9 x}+\frac{4 a^2 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)}{9 x}-\frac{\text{sech}^{-1}(a x)^2}{3 x^3}+\frac{2 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)}{9 x^3}-\frac{2}{27 x^3}",1,"(-2*(1 + 6*a^2*x^2) + 6*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x + 2*a^2*x^2 + 2*a^3*x^3)*ArcSech[a*x] - 9*ArcSech[a*x]^2)/(27*x^3)","A",1
10,1,281,297,0.6388956,"\int x^4 \text{sech}^{-1}(a x)^3 \, dx","Integrate[x^4*ArcSech[a*x]^3,x]","\frac{8 a^5 x^5 \text{sech}^{-1}(a x)^3-6 a^3 x^3 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)^2-4 a^3 x^3 \text{sech}^{-1}(a x)+18 i \text{sech}^{-1}(a x) \text{Li}_2\left(-i e^{-\text{sech}^{-1}(a x)}\right)-18 i \text{sech}^{-1}(a x) \text{Li}_2\left(i e^{-\text{sech}^{-1}(a x)}\right)+18 i \text{Li}_3\left(-i e^{-\text{sech}^{-1}(a x)}\right)-18 i \text{Li}_3\left(i e^{-\text{sech}^{-1}(a x)}\right)+2 a x \sqrt{\frac{1-a x}{a x+1}} (a x+1)-9 a x \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)^2-18 a x \text{sech}^{-1}(a x)+9 i \text{sech}^{-1}(a x)^2 \log \left(1-i e^{-\text{sech}^{-1}(a x)}\right)-9 i \text{sech}^{-1}(a x)^2 \log \left(1+i e^{-\text{sech}^{-1}(a x)}\right)+40 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \text{sech}^{-1}(a x)\right)\right)}{40 a^5}","\frac{9 i \text{sech}^{-1}(a x) \text{Li}_2\left(-i e^{\text{sech}^{-1}(a x)}\right)}{20 a^5}-\frac{9 i \text{sech}^{-1}(a x) \text{Li}_2\left(i e^{\text{sech}^{-1}(a x)}\right)}{20 a^5}-\frac{9 i \text{Li}_3\left(-i e^{\text{sech}^{-1}(a x)}\right)}{20 a^5}+\frac{9 i \text{Li}_3\left(i e^{\text{sech}^{-1}(a x)}\right)}{20 a^5}+\frac{\tan ^{-1}\left(\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1)}{a x}\right)}{2 a^5}-\frac{9 \text{sech}^{-1}(a x)^2 \tan ^{-1}\left(e^{\text{sech}^{-1}(a x)}\right)}{20 a^5}+\frac{x \sqrt{\frac{1-a x}{a x+1}} (a x+1)}{20 a^4}-\frac{9 x \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)^2}{40 a^4}-\frac{9 x \text{sech}^{-1}(a x)}{20 a^4}-\frac{3 x^3 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)^2}{20 a^2}-\frac{x^3 \text{sech}^{-1}(a x)}{10 a^2}+\frac{1}{5} x^5 \text{sech}^{-1}(a x)^3",1,"(2*a*x*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x) - 18*a*x*ArcSech[a*x] - 4*a^3*x^3*ArcSech[a*x] - 9*a*x*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x]^2 - 6*a^3*x^3*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x]^2 + 8*a^5*x^5*ArcSech[a*x]^3 + 40*ArcTan[Tanh[ArcSech[a*x]/2]] + (9*I)*ArcSech[a*x]^2*Log[1 - I/E^ArcSech[a*x]] - (9*I)*ArcSech[a*x]^2*Log[1 + I/E^ArcSech[a*x]] + (18*I)*ArcSech[a*x]*PolyLog[2, (-I)/E^ArcSech[a*x]] - (18*I)*ArcSech[a*x]*PolyLog[2, I/E^ArcSech[a*x]] + (18*I)*PolyLog[3, (-I)/E^ArcSech[a*x]] - (18*I)*PolyLog[3, I/E^ArcSech[a*x]])/(40*a^5)","A",0
11,1,188,184,0.6362533,"\int x^3 \text{sech}^{-1}(a x)^3 \, dx","Integrate[x^3*ArcSech[a*x]^3,x]","\frac{a^4 x^4 \text{sech}^{-1}(a x)^3+\text{sech}^{-1}(a x) \left(4 \log \left(e^{-2 \text{sech}^{-1}(a x)}+1\right)-a^2 x^2\right)-\left(a^3 x^3 \sqrt{\frac{1-a x}{a x+1}}+a^2 x^2 \sqrt{\frac{1-a x}{a x+1}}+2 a x \sqrt{\frac{1-a x}{a x+1}}+2 \sqrt{\frac{1-a x}{a x+1}}-2\right) \text{sech}^{-1}(a x)^2-2 \text{Li}_2\left(-e^{-2 \text{sech}^{-1}(a x)}\right)+\sqrt{\frac{1-a x}{a x+1}} (a x+1)}{4 a^4}","\frac{\text{Li}_2\left(-e^{2 \text{sech}^{-1}(a x)}\right)}{2 a^4}+\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1)}{4 a^4}-\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)^2}{2 a^4}-\frac{\text{sech}^{-1}(a x)^2}{2 a^4}+\frac{\text{sech}^{-1}(a x) \log \left(e^{2 \text{sech}^{-1}(a x)}+1\right)}{a^4}-\frac{x^2 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)^2}{4 a^2}-\frac{x^2 \text{sech}^{-1}(a x)}{4 a^2}+\frac{1}{4} x^4 \text{sech}^{-1}(a x)^3",1,"(Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x) - (-2 + 2*Sqrt[(1 - a*x)/(1 + a*x)] + 2*a*x*Sqrt[(1 - a*x)/(1 + a*x)] + a^2*x^2*Sqrt[(1 - a*x)/(1 + a*x)] + a^3*x^3*Sqrt[(1 - a*x)/(1 + a*x)])*ArcSech[a*x]^2 + a^4*x^4*ArcSech[a*x]^3 + ArcSech[a*x]*(-(a^2*x^2) + 4*Log[1 + E^(-2*ArcSech[a*x])]) - 2*PolyLog[2, -E^(-2*ArcSech[a*x])])/(4*a^4)","A",0
12,1,199,198,0.5018507,"\int x^2 \text{sech}^{-1}(a x)^3 \, dx","Integrate[x^2*ArcSech[a*x]^3,x]","\frac{2 a^3 x^3 \text{sech}^{-1}(a x)^3+3 i \left(2 \text{sech}^{-1}(a x) \text{Li}_2\left(-i e^{-\text{sech}^{-1}(a x)}\right)-2 \text{sech}^{-1}(a x) \text{Li}_2\left(i e^{-\text{sech}^{-1}(a x)}\right)+2 \text{Li}_3\left(-i e^{-\text{sech}^{-1}(a x)}\right)-2 \text{Li}_3\left(i e^{-\text{sech}^{-1}(a x)}\right)+\text{sech}^{-1}(a x)^2 \log \left(1-i e^{-\text{sech}^{-1}(a x)}\right)-\text{sech}^{-1}(a x)^2 \log \left(1+i e^{-\text{sech}^{-1}(a x)}\right)-4 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \text{sech}^{-1}(a x)\right)\right)\right)-3 a x \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)^2-6 a x \text{sech}^{-1}(a x)}{6 a^3}","\frac{i \text{sech}^{-1}(a x) \text{Li}_2\left(-i e^{\text{sech}^{-1}(a x)}\right)}{a^3}-\frac{i \text{sech}^{-1}(a x) \text{Li}_2\left(i e^{\text{sech}^{-1}(a x)}\right)}{a^3}-\frac{i \text{Li}_3\left(-i e^{\text{sech}^{-1}(a x)}\right)}{a^3}+\frac{i \text{Li}_3\left(i e^{\text{sech}^{-1}(a x)}\right)}{a^3}+\frac{\tan ^{-1}\left(\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1)}{a x}\right)}{a^3}-\frac{\text{sech}^{-1}(a x)^2 \tan ^{-1}\left(e^{\text{sech}^{-1}(a x)}\right)}{a^3}-\frac{x \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)^2}{2 a^2}-\frac{x \text{sech}^{-1}(a x)}{a^2}+\frac{1}{3} x^3 \text{sech}^{-1}(a x)^3",1,"(-6*a*x*ArcSech[a*x] - 3*a*x*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x]^2 + 2*a^3*x^3*ArcSech[a*x]^3 + (3*I)*((-4*I)*ArcTan[Tanh[ArcSech[a*x]/2]] + ArcSech[a*x]^2*Log[1 - I/E^ArcSech[a*x]] - ArcSech[a*x]^2*Log[1 + I/E^ArcSech[a*x]] + 2*ArcSech[a*x]*PolyLog[2, (-I)/E^ArcSech[a*x]] - 2*ArcSech[a*x]*PolyLog[2, I/E^ArcSech[a*x]] + 2*PolyLog[3, (-I)/E^ArcSech[a*x]] - 2*PolyLog[3, I/E^ArcSech[a*x]]))/(6*a^3)","A",0
13,1,101,102,0.3966262,"\int x \text{sech}^{-1}(a x)^3 \, dx","Integrate[x*ArcSech[a*x]^3,x]","\frac{\text{sech}^{-1}(a x) \left(a^2 x^2 \text{sech}^{-1}(a x)^2-3 \left(a x \sqrt{\frac{1-a x}{a x+1}}+\sqrt{\frac{1-a x}{a x+1}}-1\right) \text{sech}^{-1}(a x)+6 \log \left(e^{-2 \text{sech}^{-1}(a x)}+1\right)\right)-3 \text{Li}_2\left(-e^{-2 \text{sech}^{-1}(a x)}\right)}{2 a^2}","\frac{3 \text{Li}_2\left(-e^{2 \text{sech}^{-1}(a x)}\right)}{2 a^2}-\frac{3 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)^2}{2 a^2}-\frac{3 \text{sech}^{-1}(a x)^2}{2 a^2}+\frac{3 \text{sech}^{-1}(a x) \log \left(e^{2 \text{sech}^{-1}(a x)}+1\right)}{a^2}+\frac{1}{2} x^2 \text{sech}^{-1}(a x)^3",1,"(ArcSech[a*x]*(-3*(-1 + Sqrt[(1 - a*x)/(1 + a*x)] + a*x*Sqrt[(1 - a*x)/(1 + a*x)])*ArcSech[a*x] + a^2*x^2*ArcSech[a*x]^2 + 6*Log[1 + E^(-2*ArcSech[a*x])]) - 3*PolyLog[2, -E^(-2*ArcSech[a*x])])/(2*a^2)","A",0
14,1,128,111,0.1297494,"\int \text{sech}^{-1}(a x)^3 \, dx","Integrate[ArcSech[a*x]^3,x]","x \text{sech}^{-1}(a x)^3-\frac{3 i \left(-2 \text{sech}^{-1}(a x) \left(\text{Li}_2\left(-i e^{-\text{sech}^{-1}(a x)}\right)-\text{Li}_2\left(i e^{-\text{sech}^{-1}(a x)}\right)\right)-2 \left(\text{Li}_3\left(-i e^{-\text{sech}^{-1}(a x)}\right)-\text{Li}_3\left(i e^{-\text{sech}^{-1}(a x)}\right)\right)-\left(\text{sech}^{-1}(a x)^2 \left(\log \left(1-i e^{-\text{sech}^{-1}(a x)}\right)-\log \left(1+i e^{-\text{sech}^{-1}(a x)}\right)\right)\right)\right)}{a}","\frac{6 i \text{sech}^{-1}(a x) \text{Li}_2\left(-i e^{\text{sech}^{-1}(a x)}\right)}{a}-\frac{6 i \text{sech}^{-1}(a x) \text{Li}_2\left(i e^{\text{sech}^{-1}(a x)}\right)}{a}-\frac{6 i \text{Li}_3\left(-i e^{\text{sech}^{-1}(a x)}\right)}{a}+\frac{6 i \text{Li}_3\left(i e^{\text{sech}^{-1}(a x)}\right)}{a}+x \text{sech}^{-1}(a x)^3-\frac{6 \text{sech}^{-1}(a x)^2 \tan ^{-1}\left(e^{\text{sech}^{-1}(a x)}\right)}{a}",1,"x*ArcSech[a*x]^3 - ((3*I)*(-(ArcSech[a*x]^2*(Log[1 - I/E^ArcSech[a*x]] - Log[1 + I/E^ArcSech[a*x]])) - 2*ArcSech[a*x]*(PolyLog[2, (-I)/E^ArcSech[a*x]] - PolyLog[2, I/E^ArcSech[a*x]]) - 2*(PolyLog[3, (-I)/E^ArcSech[a*x]] - PolyLog[3, I/E^ArcSech[a*x]])))/a","A",0
15,1,84,88,0.0531714,"\int \frac{\text{sech}^{-1}(a x)^3}{x} \, dx","Integrate[ArcSech[a*x]^3/x,x]","\frac{1}{4} \left(6 \text{sech}^{-1}(a x)^2 \text{Li}_2\left(-e^{-2 \text{sech}^{-1}(a x)}\right)+6 \text{sech}^{-1}(a x) \text{Li}_3\left(-e^{-2 \text{sech}^{-1}(a x)}\right)+3 \text{Li}_4\left(-e^{-2 \text{sech}^{-1}(a x)}\right)-\text{sech}^{-1}(a x)^4-4 \text{sech}^{-1}(a x)^3 \log \left(e^{-2 \text{sech}^{-1}(a x)}+1\right)\right)","-\frac{3}{2} \text{sech}^{-1}(a x)^2 \text{Li}_2\left(-e^{2 \text{sech}^{-1}(a x)}\right)+\frac{3}{2} \text{sech}^{-1}(a x) \text{Li}_3\left(-e^{2 \text{sech}^{-1}(a x)}\right)-\frac{3}{4} \text{Li}_4\left(-e^{2 \text{sech}^{-1}(a x)}\right)+\frac{1}{4} \text{sech}^{-1}(a x)^4-\text{sech}^{-1}(a x)^3 \log \left(e^{2 \text{sech}^{-1}(a x)}+1\right)",1,"(-ArcSech[a*x]^4 - 4*ArcSech[a*x]^3*Log[1 + E^(-2*ArcSech[a*x])] + 6*ArcSech[a*x]^2*PolyLog[2, -E^(-2*ArcSech[a*x])] + 6*ArcSech[a*x]*PolyLog[3, -E^(-2*ArcSech[a*x])] + 3*PolyLog[4, -E^(-2*ArcSech[a*x])])/4","A",0
16,1,75,83,0.0875631,"\int \frac{\text{sech}^{-1}(a x)^3}{x^2} \, dx","Integrate[ArcSech[a*x]^3/x^2,x]","\frac{6 \sqrt{\frac{1-a x}{a x+1}} (a x+1)-\text{sech}^{-1}(a x)^3+3 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)^2-6 \text{sech}^{-1}(a x)}{x}","\frac{6 \sqrt{\frac{1-a x}{a x+1}} (a x+1)}{x}-\frac{\text{sech}^{-1}(a x)^3}{x}+\frac{3 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)^2}{x}-\frac{6 \text{sech}^{-1}(a x)}{x}",1,"(6*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x) - 6*ArcSech[a*x] + 3*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x]^2 - ArcSech[a*x]^3)/x","A",1
17,1,147,137,0.1530208,"\int \frac{\text{sech}^{-1}(a x)^3}{x^3} \, dx","Integrate[ArcSech[a*x]^3/x^3,x]","\frac{-3 a^2 x^2 \log (x)+3 a^2 x^2 \log \left(a x \sqrt{\frac{1-a x}{a x+1}}+\sqrt{\frac{1-a x}{a x+1}}+1\right)+2 \left(a^2 x^2-2\right) \text{sech}^{-1}(a x)^3+3 \sqrt{\frac{1-a x}{a x+1}} (a x+1)+6 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)^2-6 \text{sech}^{-1}(a x)}{8 x^2}","-\frac{1}{4} a^2 \text{sech}^{-1}(a x)^3-\frac{3}{8} a^2 \text{sech}^{-1}(a x)+\frac{3 \sqrt{\frac{1-a x}{a x+1}} (a x+1)}{8 x^2}-\frac{(1-a x) (a x+1) \text{sech}^{-1}(a x)^3}{2 x^2}+\frac{3 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)^2}{4 x^2}-\frac{3 (1-a x) (a x+1) \text{sech}^{-1}(a x)}{4 x^2}",1,"(3*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x) - 6*ArcSech[a*x] + 6*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x]^2 + 2*(-2 + a^2*x^2)*ArcSech[a*x]^3 - 3*a^2*x^2*Log[x] + 3*a^2*x^2*Log[1 + Sqrt[(1 - a*x)/(1 + a*x)] + a*x*Sqrt[(1 - a*x)/(1 + a*x)]])/(8*x^2)","A",1
18,1,120,179,0.1276468,"\int \frac{\text{sech}^{-1}(a x)^3}{x^4} \, dx","Integrate[ArcSech[a*x]^3/x^4,x]","\frac{-6 \left(6 a^2 x^2+1\right) \text{sech}^{-1}(a x)+2 \sqrt{\frac{1-a x}{a x+1}} \left(20 a^3 x^3+20 a^2 x^2+a x+1\right)+9 \sqrt{\frac{1-a x}{a x+1}} \left(2 a^3 x^3+2 a^2 x^2+a x+1\right) \text{sech}^{-1}(a x)^2-9 \text{sech}^{-1}(a x)^3}{27 x^3}","\frac{14 a^2 \sqrt{\frac{1-a x}{a x+1}} (a x+1)}{9 x}+\frac{2 a^2 \sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)^2}{3 x}-\frac{4 a^2 \text{sech}^{-1}(a x)}{3 x}+\frac{2 \left(\frac{1-a x}{a x+1}\right)^{3/2} (a x+1)^3}{27 x^3}+\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1) \text{sech}^{-1}(a x)^2}{3 x^3}-\frac{\text{sech}^{-1}(a x)^3}{3 x^3}-\frac{2 \text{sech}^{-1}(a x)}{9 x^3}",1,"(2*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x + 20*a^2*x^2 + 20*a^3*x^3) - 6*(1 + 6*a^2*x^2)*ArcSech[a*x] + 9*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x + 2*a^2*x^2 + 2*a^3*x^3)*ArcSech[a*x]^2 - 9*ArcSech[a*x]^3)/(27*x^3)","A",1
19,1,143,142,0.2221491,"\int x^6 \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[x^6*(a + b*ArcSech[c*x]),x]","\frac{a x^7}{7}+\frac{5 i b \log \left(2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)-2 i c x\right)}{112 c^7}+b \sqrt{\frac{1-c x}{c x+1}} \left(-\frac{5 x}{112 c^6}-\frac{5 x^2}{112 c^5}-\frac{5 x^3}{168 c^4}-\frac{5 x^4}{168 c^3}-\frac{x^5}{42 c^2}-\frac{x^6}{42 c}\right)+\frac{1}{7} b x^7 \text{sech}^{-1}(c x)","\frac{1}{7} x^7 \left(a+b \text{sech}^{-1}(c x)\right)+\frac{5 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sin ^{-1}(c x)}{112 c^7}-\frac{5 b x \sqrt{1-c x}}{112 c^6 \sqrt{\frac{1}{c x+1}}}-\frac{5 b x^3 \sqrt{1-c x}}{168 c^4 \sqrt{\frac{1}{c x+1}}}-\frac{b x^5 \sqrt{1-c x}}{42 c^2 \sqrt{\frac{1}{c x+1}}}",1,"(a*x^7)/7 + b*Sqrt[(1 - c*x)/(1 + c*x)]*((-5*x)/(112*c^6) - (5*x^2)/(112*c^5) - (5*x^3)/(168*c^4) - (5*x^4)/(168*c^3) - x^5/(42*c^2) - x^6/(42*c)) + (b*x^7*ArcSech[c*x])/7 + (((5*I)/112)*b*Log[(-2*I)*c*x + 2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)])/c^7","C",1
20,1,97,109,0.1098782,"\int x^5 \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[x^5*(a + b*ArcSech[c*x]),x]","\frac{a x^6}{6}+b \sqrt{\frac{1-c x}{c x+1}} \left(-\frac{4}{45 c^6}-\frac{4 x}{45 c^5}-\frac{2 x^2}{45 c^4}-\frac{2 x^3}{45 c^3}-\frac{x^4}{30 c^2}-\frac{x^5}{30 c}\right)+\frac{1}{6} b x^6 \text{sech}^{-1}(c x)","\frac{1}{6} x^6 \left(a+b \text{sech}^{-1}(c x)\right)-\frac{4 b \sqrt{1-c x}}{45 c^6 \sqrt{\frac{1}{c x+1}}}-\frac{2 b x^2 \sqrt{1-c x}}{45 c^4 \sqrt{\frac{1}{c x+1}}}-\frac{b x^4 \sqrt{1-c x}}{30 c^2 \sqrt{\frac{1}{c x+1}}}",1,"(a*x^6)/6 + b*Sqrt[(1 - c*x)/(1 + c*x)]*(-4/(45*c^6) - (4*x)/(45*c^5) - (2*x^2)/(45*c^4) - (2*x^3)/(45*c^3) - x^4/(30*c^2) - x^5/(30*c)) + (b*x^6*ArcSech[c*x])/6","A",1
21,1,123,110,0.1360418,"\int x^4 \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[x^4*(a + b*ArcSech[c*x]),x]","\frac{a x^5}{5}+\frac{3 i b \log \left(2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)-2 i c x\right)}{40 c^5}+b \sqrt{\frac{1-c x}{c x+1}} \left(-\frac{3 x}{40 c^4}-\frac{3 x^2}{40 c^3}-\frac{x^3}{20 c^2}-\frac{x^4}{20 c}\right)+\frac{1}{5} b x^5 \text{sech}^{-1}(c x)","\frac{1}{5} x^5 \left(a+b \text{sech}^{-1}(c x)\right)+\frac{3 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sin ^{-1}(c x)}{40 c^5}-\frac{3 b x \sqrt{1-c x}}{40 c^4 \sqrt{\frac{1}{c x+1}}}-\frac{b x^3 \sqrt{1-c x}}{20 c^2 \sqrt{\frac{1}{c x+1}}}",1,"(a*x^5)/5 + b*Sqrt[(1 - c*x)/(1 + c*x)]*((-3*x)/(40*c^4) - (3*x^2)/(40*c^3) - x^3/(20*c^2) - x^4/(20*c)) + (b*x^5*ArcSech[c*x])/5 + (((3*I)/40)*b*Log[(-2*I)*c*x + 2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)])/c^5","C",1
22,1,77,77,0.0793466,"\int x^3 \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[x^3*(a + b*ArcSech[c*x]),x]","\frac{a x^4}{4}+b \sqrt{\frac{1-c x}{c x+1}} \left(-\frac{1}{6 c^4}-\frac{x}{6 c^3}-\frac{x^2}{12 c^2}-\frac{x^3}{12 c}\right)+\frac{1}{4} b x^4 \text{sech}^{-1}(c x)","\frac{1}{4} x^4 \left(a+b \text{sech}^{-1}(c x)\right)-\frac{b \sqrt{1-c x}}{6 c^4 \sqrt{\frac{1}{c x+1}}}-\frac{b x^2 \sqrt{1-c x}}{12 c^2 \sqrt{\frac{1}{c x+1}}}",1,"(a*x^4)/4 + b*Sqrt[(1 - c*x)/(1 + c*x)]*(-1/6*1/c^4 - x/(6*c^3) - x^2/(12*c^2) - x^3/(12*c)) + (b*x^4*ArcSech[c*x])/4","A",1
23,1,103,78,0.0903088,"\int x^2 \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[x^2*(a + b*ArcSech[c*x]),x]","\frac{a x^3}{3}+\frac{i b \log \left(2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)-2 i c x\right)}{6 c^3}+b \sqrt{\frac{1-c x}{c x+1}} \left(-\frac{x}{6 c^2}-\frac{x^2}{6 c}\right)+\frac{1}{3} b x^3 \text{sech}^{-1}(c x)","\frac{1}{3} x^3 \left(a+b \text{sech}^{-1}(c x)\right)+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sin ^{-1}(c x)}{6 c^3}-\frac{b x \sqrt{1-c x}}{6 c^2 \sqrt{\frac{1}{c x+1}}}",1,"(a*x^3)/3 + b*Sqrt[(1 - c*x)/(1 + c*x)]*(-1/6*x/c^2 - x^2/(6*c)) + (b*x^3*ArcSech[c*x])/3 + ((I/6)*b*Log[(-2*I)*c*x + 2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)])/c^3","C",1
24,1,57,45,0.0539697,"\int x \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[x*(a + b*ArcSech[c*x]),x]","\frac{a x^2}{2}+b \left(-\frac{1}{2 c^2}-\frac{x}{2 c}\right) \sqrt{\frac{1-c x}{c x+1}}+\frac{1}{2} b x^2 \text{sech}^{-1}(c x)","\frac{1}{2} x^2 \left(a+b \text{sech}^{-1}(c x)\right)-\frac{b \sqrt{1-c x}}{2 c^2 \sqrt{\frac{1}{c x+1}}}",1,"(a*x^2)/2 + b*(-1/2*1/c^2 - x/(2*c))*Sqrt[(1 - c*x)/(1 + c*x)] + (b*x^2*ArcSech[c*x])/2","A",1
25,1,60,40,0.1071689,"\int \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[a + b*ArcSech[c*x],x]","a x-\frac{b \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c (c x-1)}+b x \text{sech}^{-1}(c x)","a x+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sin ^{-1}(c x)}{c}+b x \text{sech}^{-1}(c x)",1,"a*x + b*x*ArcSech[c*x] - (b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c*(-1 + c*x))","A",1
26,1,47,56,0.0478365,"\int \frac{a+b \text{sech}^{-1}(c x)}{x} \, dx","Integrate[(a + b*ArcSech[c*x])/x,x]","a \log (x)+\frac{1}{2} b \left(\text{Li}_2\left(-e^{-2 \text{sech}^{-1}(c x)}\right)-\text{sech}^{-1}(c x) \left(\text{sech}^{-1}(c x)+2 \log \left(e^{-2 \text{sech}^{-1}(c x)}+1\right)\right)\right)","-\frac{\left(a+b \text{sech}^{-1}(c x)\right)^2}{2 b}-\log \left(e^{-2 \text{sech}^{-1}(c x)}+1\right) \left(a+b \text{sech}^{-1}(c x)\right)+\frac{1}{2} b \text{Li}_2\left(-e^{-2 \text{sech}^{-1}(c x)}\right)",1,"a*Log[x] + (b*(-(ArcSech[c*x]*(ArcSech[c*x] + 2*Log[1 + E^(-2*ArcSech[c*x])])) + PolyLog[2, -E^(-2*ArcSech[c*x])]))/2","A",0
27,1,42,40,0.0569707,"\int \frac{a+b \text{sech}^{-1}(c x)}{x^2} \, dx","Integrate[(a + b*ArcSech[c*x])/x^2,x]","-\frac{a}{x}+b \left(c+\frac{1}{x}\right) \sqrt{\frac{1-c x}{c x+1}}-\frac{b \text{sech}^{-1}(c x)}{x}","\frac{b \sqrt{1-c x}}{x \sqrt{\frac{1}{c x+1}}}-\frac{a+b \text{sech}^{-1}(c x)}{x}",1,"-(a/x) + b*(c + x^(-1))*Sqrt[(1 - c*x)/(1 + c*x)] - (b*ArcSech[c*x])/x","A",1
28,1,117,94,0.0765807,"\int \frac{a+b \text{sech}^{-1}(c x)}{x^3} \, dx","Integrate[(a + b*ArcSech[c*x])/x^3,x]","-\frac{a}{2 x^2}-\frac{1}{4} b c^2 \log (x)+\frac{1}{4} b c^2 \log \left(c x \sqrt{\frac{1-c x}{c x+1}}+\sqrt{\frac{1-c x}{c x+1}}+1\right)+b \left(\frac{c}{4 x}+\frac{1}{4 x^2}\right) \sqrt{\frac{1-c x}{c x+1}}-\frac{b \text{sech}^{-1}(c x)}{2 x^2}","-\frac{a+b \text{sech}^{-1}(c x)}{2 x^2}+\frac{1}{4} b c^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left(\sqrt{1-c x} \sqrt{c x+1}\right)+\frac{b \sqrt{1-c x}}{4 x^2 \sqrt{\frac{1}{c x+1}}}",1,"-1/2*a/x^2 + b*(1/(4*x^2) + c/(4*x))*Sqrt[(1 - c*x)/(1 + c*x)] - (b*ArcSech[c*x])/(2*x^2) - (b*c^2*Log[x])/4 + (b*c^2*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]])/4","A",1
29,1,74,77,0.0692034,"\int \frac{a+b \text{sech}^{-1}(c x)}{x^4} \, dx","Integrate[(a + b*ArcSech[c*x])/x^4,x]","-\frac{a}{3 x^3}+b \left(\frac{2 c^3}{9}+\frac{2 c^2}{9 x}+\frac{c}{9 x^2}+\frac{1}{9 x^3}\right) \sqrt{\frac{1-c x}{c x+1}}-\frac{b \text{sech}^{-1}(c x)}{3 x^3}","-\frac{a+b \text{sech}^{-1}(c x)}{3 x^3}+\frac{2 b c^2 \sqrt{1-c x}}{9 x \sqrt{\frac{1}{c x+1}}}+\frac{b \sqrt{1-c x}}{9 x^3 \sqrt{\frac{1}{c x+1}}}",1,"-1/3*a/x^3 + b*((2*c^3)/9 + 1/(9*x^3) + c/(9*x^2) + (2*c^2)/(9*x))*Sqrt[(1 - c*x)/(1 + c*x)] - (b*ArcSech[c*x])/(3*x^3)","A",1
30,1,137,126,0.1128579,"\int \frac{a+b \text{sech}^{-1}(c x)}{x^5} \, dx","Integrate[(a + b*ArcSech[c*x])/x^5,x]","-\frac{a}{4 x^4}-\frac{3}{32} b c^4 \log (x)+\frac{3}{32} b c^4 \log \left(c x \sqrt{\frac{1-c x}{c x+1}}+\sqrt{\frac{1-c x}{c x+1}}+1\right)+b \left(\frac{3 c^3}{32 x}+\frac{3 c^2}{32 x^2}+\frac{c}{16 x^3}+\frac{1}{16 x^4}\right) \sqrt{\frac{1-c x}{c x+1}}-\frac{b \text{sech}^{-1}(c x)}{4 x^4}","-\frac{a+b \text{sech}^{-1}(c x)}{4 x^4}+\frac{3}{32} b c^4 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left(\sqrt{1-c x} \sqrt{c x+1}\right)+\frac{3 b c^2 \sqrt{1-c x}}{32 x^2 \sqrt{\frac{1}{c x+1}}}+\frac{b \sqrt{1-c x}}{16 x^4 \sqrt{\frac{1}{c x+1}}}",1,"-1/4*a/x^4 + b*(1/(16*x^4) + c/(16*x^3) + (3*c^2)/(32*x^2) + (3*c^3)/(32*x))*Sqrt[(1 - c*x)/(1 + c*x)] - (b*ArcSech[c*x])/(4*x^4) - (3*b*c^4*Log[x])/32 + (3*b*c^4*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]])/32","A",1
31,1,94,109,0.0927483,"\int \frac{a+b \text{sech}^{-1}(c x)}{x^6} \, dx","Integrate[(a + b*ArcSech[c*x])/x^6,x]","-\frac{a}{5 x^5}+b \left(\frac{8 c^5}{75}+\frac{8 c^4}{75 x}+\frac{4 c^3}{75 x^2}+\frac{4 c^2}{75 x^3}+\frac{c}{25 x^4}+\frac{1}{25 x^5}\right) \sqrt{\frac{1-c x}{c x+1}}-\frac{b \text{sech}^{-1}(c x)}{5 x^5}","-\frac{a+b \text{sech}^{-1}(c x)}{5 x^5}+\frac{8 b c^4 \sqrt{1-c x}}{75 x \sqrt{\frac{1}{c x+1}}}+\frac{4 b c^2 \sqrt{1-c x}}{75 x^3 \sqrt{\frac{1}{c x+1}}}+\frac{b \sqrt{1-c x}}{25 x^5 \sqrt{\frac{1}{c x+1}}}",1,"-1/5*a/x^5 + b*((8*c^5)/75 + 1/(25*x^5) + c/(25*x^4) + (4*c^2)/(75*x^3) + (4*c^3)/(75*x^2) + (8*c^4)/(75*x))*Sqrt[(1 - c*x)/(1 + c*x)] - (b*ArcSech[c*x])/(5*x^5)","A",1
32,1,157,158,0.1580098,"\int \frac{a+b \text{sech}^{-1}(c x)}{x^7} \, dx","Integrate[(a + b*ArcSech[c*x])/x^7,x]","-\frac{a}{6 x^6}-\frac{5}{96} b c^6 \log (x)+\frac{5}{96} b c^6 \log \left(c x \sqrt{\frac{1-c x}{c x+1}}+\sqrt{\frac{1-c x}{c x+1}}+1\right)+b \left(\frac{5 c^5}{96 x}+\frac{5 c^4}{96 x^2}+\frac{5 c^3}{144 x^3}+\frac{5 c^2}{144 x^4}+\frac{c}{36 x^5}+\frac{1}{36 x^6}\right) \sqrt{\frac{1-c x}{c x+1}}-\frac{b \text{sech}^{-1}(c x)}{6 x^6}","-\frac{a+b \text{sech}^{-1}(c x)}{6 x^6}+\frac{5}{96} b c^6 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left(\sqrt{1-c x} \sqrt{c x+1}\right)+\frac{5 b c^4 \sqrt{1-c x}}{96 x^2 \sqrt{\frac{1}{c x+1}}}+\frac{5 b c^2 \sqrt{1-c x}}{144 x^4 \sqrt{\frac{1}{c x+1}}}+\frac{b \sqrt{1-c x}}{36 x^6 \sqrt{\frac{1}{c x+1}}}",1,"-1/6*a/x^6 + b*(1/(36*x^6) + c/(36*x^5) + (5*c^2)/(144*x^4) + (5*c^3)/(144*x^3) + (5*c^4)/(96*x^2) + (5*c^5)/(96*x))*Sqrt[(1 - c*x)/(1 + c*x)] - (b*ArcSech[c*x])/(6*x^6) - (5*b*c^6*Log[x])/96 + (5*b*c^6*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]])/96","A",1
33,1,212,124,0.348647,"\int x^3 \left(a+b \text{sech}^{-1}(c x)\right)^2 \, dx","Integrate[x^3*(a + b*ArcSech[c*x])^2,x]","-\frac{-3 a^2 c^4 x^4+2 a b c^3 x^3 \sqrt{\frac{1-c x}{c x+1}}+2 a b c^2 x^2 \sqrt{\frac{1-c x}{c x+1}}+2 b \text{sech}^{-1}(c x) \left(b \sqrt{\frac{1-c x}{c x+1}} \left(c^3 x^3+c^2 x^2+2 c x+2\right)-3 a c^4 x^4\right)+4 a b c x \sqrt{\frac{1-c x}{c x+1}}+4 a b \sqrt{\frac{1-c x}{c x+1}}-3 b^2 c^4 x^4 \text{sech}^{-1}(c x)^2+b^2 c^2 x^2+4 b^2 \log (x)}{12 c^4}","-\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(a+b \text{sech}^{-1}(c x)\right)}{3 c^4}-\frac{b x^2 \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(a+b \text{sech}^{-1}(c x)\right)}{6 c^2}+\frac{1}{4} x^4 \left(a+b \text{sech}^{-1}(c x)\right)^2-\frac{b^2 \log (x)}{3 c^4}-\frac{b^2 x^2}{12 c^2}",1,"-1/12*(b^2*c^2*x^2 - 3*a^2*c^4*x^4 + 4*a*b*Sqrt[(1 - c*x)/(1 + c*x)] + 4*a*b*c*x*Sqrt[(1 - c*x)/(1 + c*x)] + 2*a*b*c^2*x^2*Sqrt[(1 - c*x)/(1 + c*x)] + 2*a*b*c^3*x^3*Sqrt[(1 - c*x)/(1 + c*x)] + 2*b*(-3*a*c^4*x^4 + b*Sqrt[(1 - c*x)/(1 + c*x)]*(2 + 2*c*x + c^2*x^2 + c^3*x^3))*ArcSech[c*x] - 3*b^2*c^4*x^4*ArcSech[c*x]^2 + 4*b^2*Log[x])/c^4","A",1
34,1,224,140,1.2749109,"\int x^2 \left(a+b \text{sech}^{-1}(c x)\right)^2 \, dx","Integrate[x^2*(a + b*ArcSech[c*x])^2,x]","\frac{1}{3} \left(a^2 x^3+a b \left(2 x^3 \text{sech}^{-1}(c x)-\frac{\sqrt{\frac{1-c x}{c x+1}} \left(c^3 x^3+\sqrt{1-c^2 x^2} \sin ^{-1}(c x)-c x\right)}{c^3 (c x-1)}\right)+\frac{b^2 \left(c^3 x^3 \text{sech}^{-1}(c x)^2+i \text{Li}_2\left(-i e^{-\text{sech}^{-1}(c x)}\right)-i \text{Li}_2\left(i e^{-\text{sech}^{-1}(c x)}\right)-c x-c x \sqrt{\frac{1-c x}{c x+1}} (c x+1) \text{sech}^{-1}(c x)+i \text{sech}^{-1}(c x) \log \left(1-i e^{-\text{sech}^{-1}(c x)}\right)-i \text{sech}^{-1}(c x) \log \left(1+i e^{-\text{sech}^{-1}(c x)}\right)\right)}{c^3}\right)","-\frac{2 b \tan ^{-1}\left(e^{\text{sech}^{-1}(c x)}\right) \left(a+b \text{sech}^{-1}(c x)\right)}{3 c^3}-\frac{b x \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(a+b \text{sech}^{-1}(c x)\right)}{3 c^2}+\frac{1}{3} x^3 \left(a+b \text{sech}^{-1}(c x)\right)^2+\frac{i b^2 \text{Li}_2\left(-i e^{\text{sech}^{-1}(c x)}\right)}{3 c^3}-\frac{i b^2 \text{Li}_2\left(i e^{\text{sech}^{-1}(c x)}\right)}{3 c^3}-\frac{b^2 x}{3 c^2}",1,"(a^2*x^3 + a*b*(2*x^3*ArcSech[c*x] - (Sqrt[(1 - c*x)/(1 + c*x)]*(-(c*x) + c^3*x^3 + Sqrt[1 - c^2*x^2]*ArcSin[c*x]))/(c^3*(-1 + c*x))) + (b^2*(-(c*x) - c*x*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*ArcSech[c*x] + c^3*x^3*ArcSech[c*x]^2 + I*ArcSech[c*x]*Log[1 - I/E^ArcSech[c*x]] - I*ArcSech[c*x]*Log[1 + I/E^ArcSech[c*x]] + I*PolyLog[2, (-I)/E^ArcSech[c*x]] - I*PolyLog[2, I/E^ArcSech[c*x]]))/c^3)/3","A",0
35,1,112,65,0.2530582,"\int x \left(a+b \text{sech}^{-1}(c x)\right)^2 \, dx","Integrate[x*(a + b*ArcSech[c*x])^2,x]","\frac{a \left(a c^2 x^2-2 b \sqrt{\frac{1-c x}{c x+1}} (c x+1)\right)-2 b \text{sech}^{-1}(c x) \left(b \sqrt{\frac{1-c x}{c x+1}} (c x+1)-a c^2 x^2\right)+b^2 c^2 x^2 \text{sech}^{-1}(c x)^2-2 b^2 \log (c x)}{2 c^2}","-\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(a+b \text{sech}^{-1}(c x)\right)}{c^2}+\frac{1}{2} x^2 \left(a+b \text{sech}^{-1}(c x)\right)^2-\frac{b^2 \log (x)}{c^2}",1,"(a*(a*c^2*x^2 - 2*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)) - 2*b*(-(a*c^2*x^2) + b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))*ArcSech[c*x] + b^2*c^2*x^2*ArcSech[c*x]^2 - 2*b^2*Log[c*x])/(2*c^2)","A",1
36,1,126,78,0.2661823,"\int \left(a+b \text{sech}^{-1}(c x)\right)^2 \, dx","Integrate[(a + b*ArcSech[c*x])^2,x]","a^2 x+\frac{2 a b \left(c x \text{sech}^{-1}(c x)-2 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \text{sech}^{-1}(c x)\right)\right)\right)}{c}+\frac{i b^2 \left(2 \text{Li}_2\left(-i e^{-\text{sech}^{-1}(c x)}\right)-2 \text{Li}_2\left(i e^{-\text{sech}^{-1}(c x)}\right)+\text{sech}^{-1}(c x) \left(-i c x \text{sech}^{-1}(c x)+2 \log \left(1-i e^{-\text{sech}^{-1}(c x)}\right)-2 \log \left(1+i e^{-\text{sech}^{-1}(c x)}\right)\right)\right)}{c}","x \left(a+b \text{sech}^{-1}(c x)\right)^2-\frac{4 b \tan ^{-1}\left(e^{\text{sech}^{-1}(c x)}\right) \left(a+b \text{sech}^{-1}(c x)\right)}{c}+\frac{2 i b^2 \text{Li}_2\left(-i e^{\text{sech}^{-1}(c x)}\right)}{c}-\frac{2 i b^2 \text{Li}_2\left(i e^{\text{sech}^{-1}(c x)}\right)}{c}",1,"a^2*x + (2*a*b*(c*x*ArcSech[c*x] - 2*ArcTan[Tanh[ArcSech[c*x]/2]]))/c + (I*b^2*(ArcSech[c*x]*((-I)*c*x*ArcSech[c*x] + 2*Log[1 - I/E^ArcSech[c*x]] - 2*Log[1 + I/E^ArcSech[c*x]]) + 2*PolyLog[2, (-I)/E^ArcSech[c*x]] - 2*PolyLog[2, I/E^ArcSech[c*x]]))/c","A",0
37,1,116,83,0.1737002,"\int \frac{\left(a+b \text{sech}^{-1}(c x)\right)^2}{x} \, dx","Integrate[(a + b*ArcSech[c*x])^2/x,x]","a^2 \log (c x)+a b \left(\text{Li}_2\left(-e^{-2 \text{sech}^{-1}(c x)}\right)-\text{sech}^{-1}(c x) \left(\text{sech}^{-1}(c x)+2 \log \left(e^{-2 \text{sech}^{-1}(c x)}+1\right)\right)\right)+b^2 \left(\text{sech}^{-1}(c x) \text{Li}_2\left(-e^{-2 \text{sech}^{-1}(c x)}\right)+\frac{1}{2} \text{Li}_3\left(-e^{-2 \text{sech}^{-1}(c x)}\right)-\frac{1}{3} \text{sech}^{-1}(c x)^3-\text{sech}^{-1}(c x)^2 \log \left(e^{-2 \text{sech}^{-1}(c x)}+1\right)\right)","-b \text{Li}_2\left(-e^{2 \text{sech}^{-1}(c x)}\right) \left(a+b \text{sech}^{-1}(c x)\right)+\frac{\left(a+b \text{sech}^{-1}(c x)\right)^3}{3 b}-\log \left(e^{2 \text{sech}^{-1}(c x)}+1\right) \left(a+b \text{sech}^{-1}(c x)\right)^2+\frac{1}{2} b^2 \text{Li}_3\left(-e^{2 \text{sech}^{-1}(c x)}\right)",1,"a^2*Log[c*x] + a*b*(-(ArcSech[c*x]*(ArcSech[c*x] + 2*Log[1 + E^(-2*ArcSech[c*x])])) + PolyLog[2, -E^(-2*ArcSech[c*x])]) + b^2*(-1/3*ArcSech[c*x]^3 - ArcSech[c*x]^2*Log[1 + E^(-2*ArcSech[c*x])] + ArcSech[c*x]*PolyLog[2, -E^(-2*ArcSech[c*x])] + PolyLog[3, -E^(-2*ArcSech[c*x])]/2)","A",0
38,1,87,61,0.2429474,"\int \frac{\left(a+b \text{sech}^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[(a + b*ArcSech[c*x])^2/x^2,x]","-\frac{a^2-2 a b \sqrt{\frac{1-c x}{c x+1}} (c x+1)-2 b \text{sech}^{-1}(c x) \left(b \sqrt{\frac{1-c x}{c x+1}} (c x+1)-a\right)+b^2 \text{sech}^{-1}(c x)^2+2 b^2}{x}","\frac{2 b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(a+b \text{sech}^{-1}(c x)\right)}{x}-\frac{\left(a+b \text{sech}^{-1}(c x)\right)^2}{x}-\frac{2 b^2}{x}",1,"-((a^2 + 2*b^2 - 2*a*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) - 2*b*(-a + b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))*ArcSech[c*x] + b^2*ArcSech[c*x]^2)/x)","A",1
39,1,183,118,0.1793512,"\int \frac{\left(a+b \text{sech}^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[(a + b*ArcSech[c*x])^2/x^3,x]","\frac{-2 a^2-2 a b c^2 x^2 \log (x)+2 a b c^2 x^2 \log \left(c x \sqrt{\frac{1-c x}{c x+1}}+\sqrt{\frac{1-c x}{c x+1}}+1\right)+2 a b \sqrt{\frac{1-c x}{c x+1}}+2 a b c x \sqrt{\frac{1-c x}{c x+1}}+2 b \text{sech}^{-1}(c x) \left(b \sqrt{\frac{1-c x}{c x+1}} (c x+1)-2 a\right)+b^2 \left(c^2 x^2-2\right) \text{sech}^{-1}(c x)^2-b^2}{4 x^2}","-\frac{1}{2} a b c^2 \text{sech}^{-1}(c x)+\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(a+b \text{sech}^{-1}(c x)\right)}{2 x^2}-\frac{(1-c x) (c x+1) \left(a+b \text{sech}^{-1}(c x)\right)^2}{2 x^2}-\frac{1}{4} b^2 c^2 \text{sech}^{-1}(c x)^2-\frac{b^2 (1-c x) (c x+1)}{4 x^2}",1,"(-2*a^2 - b^2 + 2*a*b*Sqrt[(1 - c*x)/(1 + c*x)] + 2*a*b*c*x*Sqrt[(1 - c*x)/(1 + c*x)] + 2*b*(-2*a + b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))*ArcSech[c*x] + b^2*(-2 + c^2*x^2)*ArcSech[c*x]^2 - 2*a*b*c^2*x^2*Log[x] + 2*a*b*c^2*x^2*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]])/(4*x^2)","A",1
40,1,134,122,0.2902893,"\int \frac{\left(a+b \text{sech}^{-1}(c x)\right)^2}{x^4} \, dx","Integrate[(a + b*ArcSech[c*x])^2/x^4,x]","\frac{-9 a^2+6 a b \sqrt{\frac{1-c x}{c x+1}} \left(2 c^3 x^3+2 c^2 x^2+c x+1\right)+6 b \text{sech}^{-1}(c x) \left(b \sqrt{\frac{1-c x}{c x+1}} \left(2 c^3 x^3+2 c^2 x^2+c x+1\right)-3 a\right)-2 b^2 \left(6 c^2 x^2+1\right)-9 b^2 \text{sech}^{-1}(c x)^2}{27 x^3}","\frac{4 b c^2 \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(a+b \text{sech}^{-1}(c x)\right)}{9 x}+\frac{2 b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(a+b \text{sech}^{-1}(c x)\right)}{9 x^3}-\frac{\left(a+b \text{sech}^{-1}(c x)\right)^2}{3 x^3}-\frac{4 b^2 c^2}{9 x}-\frac{2 b^2}{27 x^3}",1,"(-9*a^2 - 2*b^2*(1 + 6*c^2*x^2) + 6*a*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x + 2*c^2*x^2 + 2*c^3*x^3) + 6*b*(-3*a + b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x + 2*c^2*x^2 + 2*c^3*x^3))*ArcSech[c*x] - 9*b^2*ArcSech[c*x]^2)/(27*x^3)","A",1
41,1,268,151,0.2824239,"\int \frac{\left(a+b \text{sech}^{-1}(c x)\right)^2}{x^5} \, dx","Integrate[(a + b*ArcSech[c*x])^2/x^5,x]","\frac{-8 a^2-6 a b c^4 x^4 \log (x)+6 a b c^4 x^4 \log \left(c x \sqrt{\frac{1-c x}{c x+1}}+\sqrt{\frac{1-c x}{c x+1}}+1\right)+6 a b c^3 x^3 \sqrt{\frac{1-c x}{c x+1}}+6 a b c^2 x^2 \sqrt{\frac{1-c x}{c x+1}}+2 b \text{sech}^{-1}(c x) \left(b \sqrt{\frac{1-c x}{c x+1}} \left(3 c^3 x^3+3 c^2 x^2+2 c x+2\right)-8 a\right)+4 a b c x \sqrt{\frac{1-c x}{c x+1}}+4 a b \sqrt{\frac{1-c x}{c x+1}}+b^2 \left(3 c^4 x^4-8\right) \text{sech}^{-1}(c x)^2-3 b^2 c^2 x^2-b^2}{32 x^4}","\frac{3}{16} a b c^4 \text{sech}^{-1}(c x)+\frac{3 b c^2 \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(a+b \text{sech}^{-1}(c x)\right)}{16 x^2}-\frac{\left(a+b \text{sech}^{-1}(c x)\right)^2}{4 x^4}+\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(a+b \text{sech}^{-1}(c x)\right)}{8 x^4}+\frac{3}{32} b^2 c^4 \text{sech}^{-1}(c x)^2-\frac{3 b^2 c^2}{32 x^2}-\frac{b^2}{32 x^4}",1,"(-8*a^2 - b^2 - 3*b^2*c^2*x^2 + 4*a*b*Sqrt[(1 - c*x)/(1 + c*x)] + 4*a*b*c*x*Sqrt[(1 - c*x)/(1 + c*x)] + 6*a*b*c^2*x^2*Sqrt[(1 - c*x)/(1 + c*x)] + 6*a*b*c^3*x^3*Sqrt[(1 - c*x)/(1 + c*x)] + 2*b*(-8*a + b*Sqrt[(1 - c*x)/(1 + c*x)]*(2 + 2*c*x + 3*c^2*x^2 + 3*c^3*x^3))*ArcSech[c*x] + b^2*(-8 + 3*c^4*x^4)*ArcSech[c*x]^2 - 6*a*b*c^4*x^4*Log[x] + 6*a*b*c^4*x^4*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]])/(32*x^4)","A",1
42,1,337,223,2.1462606,"\int x^3 \left(a+b \text{sech}^{-1}(c x)\right)^3 \, dx","Integrate[x^3*(a + b*ArcSech[c*x])^3,x]","\frac{1}{4} \left(a^3 x^4+a^2 b \left(3 x^4 \text{sech}^{-1}(c x)-\frac{\sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(c^2 x^2+2\right)}{c^4}\right)+\frac{a b^2 \left(3 c^4 x^4 \text{sech}^{-1}(c x)^2-c^2 x^2-2 \sqrt{\frac{1-c x}{c x+1}} \left(c^3 x^3+c^2 x^2+2 c x+2\right) \text{sech}^{-1}(c x)+4 \log \left(\frac{1}{c x}\right)\right)}{c^4}-\frac{b^3 \left(\text{sech}^{-1}(c x) \left(c^2 x^2-4 \log \left(e^{-2 \text{sech}^{-1}(c x)}+1\right)\right)+\left(c^3 x^3 \sqrt{\frac{1-c x}{c x+1}}+c^2 x^2 \sqrt{\frac{1-c x}{c x+1}}+2 c x \sqrt{\frac{1-c x}{c x+1}}+2 \sqrt{\frac{1-c x}{c x+1}}-2\right) \text{sech}^{-1}(c x)^2+2 \text{Li}_2\left(-e^{-2 \text{sech}^{-1}(c x)}\right)-\sqrt{\frac{1-c x}{c x+1}} (c x+1)\right)}{c^4}+b^3 x^4 \text{sech}^{-1}(c x)^3\right)","\frac{b^2 \log \left(e^{2 \text{sech}^{-1}(c x)}+1\right) \left(a+b \text{sech}^{-1}(c x)\right)}{c^4}-\frac{b^2 x^2 \left(a+b \text{sech}^{-1}(c x)\right)}{4 c^2}-\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(a+b \text{sech}^{-1}(c x)\right)^2}{2 c^4}-\frac{b \left(a+b \text{sech}^{-1}(c x)\right)^2}{2 c^4}-\frac{b x^2 \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(a+b \text{sech}^{-1}(c x)\right)^2}{4 c^2}+\frac{1}{4} x^4 \left(a+b \text{sech}^{-1}(c x)\right)^3+\frac{b^3 \text{Li}_2\left(-e^{2 \text{sech}^{-1}(c x)}\right)}{2 c^4}+\frac{b^3 \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{4 c^4}",1,"(a^3*x^4 + b^3*x^4*ArcSech[c*x]^3 + a^2*b*(-((Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(2 + c^2*x^2))/c^4) + 3*x^4*ArcSech[c*x]) + (a*b^2*(-(c^2*x^2) - 2*Sqrt[(1 - c*x)/(1 + c*x)]*(2 + 2*c*x + c^2*x^2 + c^3*x^3)*ArcSech[c*x] + 3*c^4*x^4*ArcSech[c*x]^2 + 4*Log[1/(c*x)]))/c^4 - (b^3*(-(Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)) + (-2 + 2*Sqrt[(1 - c*x)/(1 + c*x)] + 2*c*x*Sqrt[(1 - c*x)/(1 + c*x)] + c^2*x^2*Sqrt[(1 - c*x)/(1 + c*x)] + c^3*x^3*Sqrt[(1 - c*x)/(1 + c*x)])*ArcSech[c*x]^2 + ArcSech[c*x]*(c^2*x^2 - 4*Log[1 + E^(-2*ArcSech[c*x])]) + 2*PolyLog[2, -E^(-2*ArcSech[c*x])]))/c^4)/4","A",0
43,1,440,242,1.1128792,"\int x^2 \left(a+b \text{sech}^{-1}(c x)\right)^3 \, dx","Integrate[x^2*(a + b*ArcSech[c*x])^3,x]","\frac{2 a^3 c^3 x^3+6 a^2 b c^3 x^3 \text{sech}^{-1}(c x)-3 a^2 b c x \sqrt{\frac{1-c x}{c x+1}} (c x+1)+3 i a^2 b \log \left(2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)-2 i c x\right)-6 a b^2 \left(-c^3 x^3 \text{sech}^{-1}(c x)^2-i \text{Li}_2\left(-i e^{-\text{sech}^{-1}(c x)}\right)+i \text{Li}_2\left(i e^{-\text{sech}^{-1}(c x)}\right)+c x+c x \sqrt{\frac{1-c x}{c x+1}} (c x+1) \text{sech}^{-1}(c x)-i \text{sech}^{-1}(c x) \log \left(1-i e^{-\text{sech}^{-1}(c x)}\right)+i \text{sech}^{-1}(c x) \log \left(1+i e^{-\text{sech}^{-1}(c x)}\right)\right)-\left(b^3 \left(-2 c^3 x^3 \text{sech}^{-1}(c x)^3-3 i \left(2 \text{sech}^{-1}(c x) \text{Li}_2\left(-i e^{-\text{sech}^{-1}(c x)}\right)-2 \text{sech}^{-1}(c x) \text{Li}_2\left(i e^{-\text{sech}^{-1}(c x)}\right)+2 \text{Li}_3\left(-i e^{-\text{sech}^{-1}(c x)}\right)-2 \text{Li}_3\left(i e^{-\text{sech}^{-1}(c x)}\right)+\text{sech}^{-1}(c x)^2 \log \left(1-i e^{-\text{sech}^{-1}(c x)}\right)-\text{sech}^{-1}(c x)^2 \log \left(1+i e^{-\text{sech}^{-1}(c x)}\right)-4 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \text{sech}^{-1}(c x)\right)\right)\right)+3 c x \sqrt{\frac{1-c x}{c x+1}} (c x+1) \text{sech}^{-1}(c x)^2+6 c x \text{sech}^{-1}(c x)\right)\right)}{6 c^3}","\frac{i b^2 \text{Li}_2\left(-i e^{\text{sech}^{-1}(c x)}\right) \left(a+b \text{sech}^{-1}(c x)\right)}{c^3}-\frac{i b^2 \text{Li}_2\left(i e^{\text{sech}^{-1}(c x)}\right) \left(a+b \text{sech}^{-1}(c x)\right)}{c^3}-\frac{b^2 x \left(a+b \text{sech}^{-1}(c x)\right)}{c^2}-\frac{b \tan ^{-1}\left(e^{\text{sech}^{-1}(c x)}\right) \left(a+b \text{sech}^{-1}(c x)\right)^2}{c^3}-\frac{b x \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(a+b \text{sech}^{-1}(c x)\right)^2}{2 c^2}+\frac{1}{3} x^3 \left(a+b \text{sech}^{-1}(c x)\right)^3-\frac{i b^3 \text{Li}_3\left(-i e^{\text{sech}^{-1}(c x)}\right)}{c^3}+\frac{i b^3 \text{Li}_3\left(i e^{\text{sech}^{-1}(c x)}\right)}{c^3}+\frac{b^3 \tan ^{-1}\left(\frac{\sqrt{\frac{1-c x}{c x+1}} (c x+1)}{c x}\right)}{c^3}",1,"(2*a^3*c^3*x^3 - 3*a^2*b*c*x*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + 6*a^2*b*c^3*x^3*ArcSech[c*x] + (3*I)*a^2*b*Log[(-2*I)*c*x + 2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)] - 6*a*b^2*(c*x + c*x*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*ArcSech[c*x] - c^3*x^3*ArcSech[c*x]^2 - I*ArcSech[c*x]*Log[1 - I/E^ArcSech[c*x]] + I*ArcSech[c*x]*Log[1 + I/E^ArcSech[c*x]] - I*PolyLog[2, (-I)/E^ArcSech[c*x]] + I*PolyLog[2, I/E^ArcSech[c*x]]) - b^3*(6*c*x*ArcSech[c*x] + 3*c*x*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*ArcSech[c*x]^2 - 2*c^3*x^3*ArcSech[c*x]^3 - (3*I)*((-4*I)*ArcTan[Tanh[ArcSech[c*x]/2]] + ArcSech[c*x]^2*Log[1 - I/E^ArcSech[c*x]] - ArcSech[c*x]^2*Log[1 + I/E^ArcSech[c*x]] + 2*ArcSech[c*x]*PolyLog[2, (-I)/E^ArcSech[c*x]] - 2*ArcSech[c*x]*PolyLog[2, I/E^ArcSech[c*x]] + 2*PolyLog[3, (-I)/E^ArcSech[c*x]] - 2*PolyLog[3, I/E^ArcSech[c*x]])))/(6*c^3)","A",0
44,1,219,126,0.9543652,"\int x \left(a+b \text{sech}^{-1}(c x)\right)^3 \, dx","Integrate[x*(a + b*ArcSech[c*x])^3,x]","\frac{a \left(a \left(a c^2 x^2-3 b \sqrt{\frac{1-c x}{c x+1}} (c x+1)\right)+6 b^2 \log \left(\frac{1}{c x}\right)\right)-3 b^2 \text{sech}^{-1}(c x)^2 \left(b \left(c x \sqrt{\frac{1-c x}{c x+1}}+\sqrt{\frac{1-c x}{c x+1}}-1\right)-a c^2 x^2\right)+3 b \text{sech}^{-1}(c x) \left(a \left(a c^2 x^2-2 b \sqrt{\frac{1-c x}{c x+1}} (c x+1)\right)+2 b^2 \log \left(e^{-2 \text{sech}^{-1}(c x)}+1\right)\right)+b^3 c^2 x^2 \text{sech}^{-1}(c x)^3-3 b^3 \text{Li}_2\left(-e^{-2 \text{sech}^{-1}(c x)}\right)}{2 c^2}","\frac{3 b^2 \log \left(e^{2 \text{sech}^{-1}(c x)}+1\right) \left(a+b \text{sech}^{-1}(c x)\right)}{c^2}-\frac{3 b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(a+b \text{sech}^{-1}(c x)\right)^2}{2 c^2}-\frac{3 b \left(a+b \text{sech}^{-1}(c x)\right)^2}{2 c^2}+\frac{1}{2} x^2 \left(a+b \text{sech}^{-1}(c x)\right)^3+\frac{3 b^3 \text{Li}_2\left(-e^{2 \text{sech}^{-1}(c x)}\right)}{2 c^2}",1,"(-3*b^2*(-(a*c^2*x^2) + b*(-1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]))*ArcSech[c*x]^2 + b^3*c^2*x^2*ArcSech[c*x]^3 + 3*b*ArcSech[c*x]*(a*(a*c^2*x^2 - 2*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)) + 2*b^2*Log[1 + E^(-2*ArcSech[c*x])]) + a*(a*(a*c^2*x^2 - 3*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)) + 6*b^2*Log[1/(c*x)]) - 3*b^3*PolyLog[2, -E^(-2*ArcSech[c*x])])/(2*c^2)","A",0
45,1,282,140,0.4415803,"\int \left(a+b \text{sech}^{-1}(c x)\right)^3 \, dx","Integrate[(a + b*ArcSech[c*x])^3,x]","a^3 x-\frac{3 a^2 b \tan ^{-1}\left(\frac{c x \sqrt{\frac{1-c x}{c x+1}}}{c x-1}\right)}{c}+3 a^2 b x \text{sech}^{-1}(c x)+\frac{3 i a b^2 \left(2 \text{Li}_2\left(-i e^{-\text{sech}^{-1}(c x)}\right)-2 \text{Li}_2\left(i e^{-\text{sech}^{-1}(c x)}\right)+\text{sech}^{-1}(c x) \left(-i c x \text{sech}^{-1}(c x)+2 \log \left(1-i e^{-\text{sech}^{-1}(c x)}\right)-2 \log \left(1+i e^{-\text{sech}^{-1}(c x)}\right)\right)\right)}{c}+\frac{b^3 \left(c x \text{sech}^{-1}(c x)^3-3 i \left(-2 \text{sech}^{-1}(c x) \left(\text{Li}_2\left(-i e^{-\text{sech}^{-1}(c x)}\right)-\text{Li}_2\left(i e^{-\text{sech}^{-1}(c x)}\right)\right)-2 \left(\text{Li}_3\left(-i e^{-\text{sech}^{-1}(c x)}\right)-\text{Li}_3\left(i e^{-\text{sech}^{-1}(c x)}\right)\right)-\left(\text{sech}^{-1}(c x)^2 \left(\log \left(1-i e^{-\text{sech}^{-1}(c x)}\right)-\log \left(1+i e^{-\text{sech}^{-1}(c x)}\right)\right)\right)\right)\right)}{c}","\frac{6 i b^2 \text{Li}_2\left(-i e^{\text{sech}^{-1}(c x)}\right) \left(a+b \text{sech}^{-1}(c x)\right)}{c}-\frac{6 i b^2 \text{Li}_2\left(i e^{\text{sech}^{-1}(c x)}\right) \left(a+b \text{sech}^{-1}(c x)\right)}{c}+x \left(a+b \text{sech}^{-1}(c x)\right)^3-\frac{6 b \tan ^{-1}\left(e^{\text{sech}^{-1}(c x)}\right) \left(a+b \text{sech}^{-1}(c x)\right)^2}{c}-\frac{6 i b^3 \text{Li}_3\left(-i e^{\text{sech}^{-1}(c x)}\right)}{c}+\frac{6 i b^3 \text{Li}_3\left(i e^{\text{sech}^{-1}(c x)}\right)}{c}",1,"a^3*x + 3*a^2*b*x*ArcSech[c*x] - (3*a^2*b*ArcTan[(c*x*Sqrt[(1 - c*x)/(1 + c*x)])/(-1 + c*x)])/c + ((3*I)*a*b^2*(ArcSech[c*x]*((-I)*c*x*ArcSech[c*x] + 2*Log[1 - I/E^ArcSech[c*x]] - 2*Log[1 + I/E^ArcSech[c*x]]) + 2*PolyLog[2, (-I)/E^ArcSech[c*x]] - 2*PolyLog[2, I/E^ArcSech[c*x]]))/c + (b^3*(c*x*ArcSech[c*x]^3 - (3*I)*(-(ArcSech[c*x]^2*(Log[1 - I/E^ArcSech[c*x]] - Log[1 + I/E^ArcSech[c*x]])) - 2*ArcSech[c*x]*(PolyLog[2, (-I)/E^ArcSech[c*x]] - PolyLog[2, I/E^ArcSech[c*x]]) - 2*(PolyLog[3, (-I)/E^ArcSech[c*x]] - PolyLog[3, I/E^ArcSech[c*x]]))))/c","B",0
46,1,182,114,0.2491263,"\int \frac{\left(a+b \text{sech}^{-1}(c x)\right)^3}{x} \, dx","Integrate[(a + b*ArcSech[c*x])^3/x,x]","\frac{1}{4} \left(4 a^3 \log (c x)-6 a^2 b \text{sech}^{-1}(c x)^2-12 a^2 b \text{sech}^{-1}(c x) \log \left(e^{-2 \text{sech}^{-1}(c x)}+1\right)+6 b^2 \text{Li}_3\left(-e^{-2 \text{sech}^{-1}(c x)}\right) \left(a+b \text{sech}^{-1}(c x)\right)-4 a b^2 \text{sech}^{-1}(c x)^3-12 a b^2 \text{sech}^{-1}(c x)^2 \log \left(e^{-2 \text{sech}^{-1}(c x)}+1\right)+6 b \text{Li}_2\left(-e^{-2 \text{sech}^{-1}(c x)}\right) \left(a+b \text{sech}^{-1}(c x)\right)^2+3 b^3 \text{Li}_4\left(-e^{-2 \text{sech}^{-1}(c x)}\right)-b^3 \text{sech}^{-1}(c x)^4-4 b^3 \text{sech}^{-1}(c x)^3 \log \left(e^{-2 \text{sech}^{-1}(c x)}+1\right)\right)","\frac{3}{2} b^2 \text{Li}_3\left(-e^{2 \text{sech}^{-1}(c x)}\right) \left(a+b \text{sech}^{-1}(c x)\right)-\frac{3}{2} b \text{Li}_2\left(-e^{2 \text{sech}^{-1}(c x)}\right) \left(a+b \text{sech}^{-1}(c x)\right)^2+\frac{\left(a+b \text{sech}^{-1}(c x)\right)^4}{4 b}-\log \left(e^{2 \text{sech}^{-1}(c x)}+1\right) \left(a+b \text{sech}^{-1}(c x)\right)^3-\frac{3}{4} b^3 \text{Li}_4\left(-e^{2 \text{sech}^{-1}(c x)}\right)",1,"(-6*a^2*b*ArcSech[c*x]^2 - 4*a*b^2*ArcSech[c*x]^3 - b^3*ArcSech[c*x]^4 - 12*a^2*b*ArcSech[c*x]*Log[1 + E^(-2*ArcSech[c*x])] - 12*a*b^2*ArcSech[c*x]^2*Log[1 + E^(-2*ArcSech[c*x])] - 4*b^3*ArcSech[c*x]^3*Log[1 + E^(-2*ArcSech[c*x])] + 4*a^3*Log[c*x] + 6*b*(a + b*ArcSech[c*x])^2*PolyLog[2, -E^(-2*ArcSech[c*x])] + 6*b^2*(a + b*ArcSech[c*x])*PolyLog[3, -E^(-2*ArcSech[c*x])] + 3*b^3*PolyLog[4, -E^(-2*ArcSech[c*x])])/4","A",0
47,1,165,102,0.359044,"\int \frac{\left(a+b \text{sech}^{-1}(c x)\right)^3}{x^2} \, dx","Integrate[(a + b*ArcSech[c*x])^3/x^2,x]","-\frac{a^3+3 b \text{sech}^{-1}(c x) \left(a^2-2 a b \sqrt{\frac{1-c x}{c x+1}} (c x+1)+2 b^2\right)-3 a^2 b \sqrt{\frac{1-c x}{c x+1}} (c x+1)-3 b^2 \text{sech}^{-1}(c x)^2 \left(b \sqrt{\frac{1-c x}{c x+1}} (c x+1)-a\right)+6 a b^2-6 b^3 \sqrt{\frac{1-c x}{c x+1}} (c x+1)+b^3 \text{sech}^{-1}(c x)^3}{x}","-\frac{6 b^2 \left(a+b \text{sech}^{-1}(c x)\right)}{x}+\frac{3 b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(a+b \text{sech}^{-1}(c x)\right)^2}{x}-\frac{\left(a+b \text{sech}^{-1}(c x)\right)^3}{x}+\frac{6 b^3 \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{x}",1,"-((a^3 + 6*a*b^2 - 3*a^2*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) - 6*b^3*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + 3*b*(a^2 + 2*b^2 - 2*a*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))*ArcSech[c*x] - 3*b^2*(-a + b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))*ArcSech[c*x]^2 + b^3*ArcSech[c*x]^3)/x)","A",1
48,1,245,163,0.478285,"\int \frac{\left(a+b \text{sech}^{-1}(c x)\right)^3}{x^3} \, dx","Integrate[(a + b*ArcSech[c*x])^3/x^3,x]","\frac{-4 a^3-3 b c^2 x^2 \left(2 a^2+b^2\right) \log (x)+3 b c^2 x^2 \left(2 a^2+b^2\right) \log \left(c x \sqrt{\frac{1-c x}{c x+1}}+\sqrt{\frac{1-c x}{c x+1}}+1\right)+3 b \left(2 a^2+b^2\right) \sqrt{\frac{1-c x}{c x+1}} (c x+1)-6 b \text{sech}^{-1}(c x) \left(2 a^2-2 a b \sqrt{\frac{1-c x}{c x+1}} (c x+1)+b^2\right)+6 b^2 \text{sech}^{-1}(c x)^2 \left(a \left(c^2 x^2-2\right)+b \sqrt{\frac{1-c x}{c x+1}} (c x+1)\right)-6 a b^2+2 b^3 \left(c^2 x^2-2\right) \text{sech}^{-1}(c x)^3}{8 x^2}","-\frac{3 b^2 (1-c x) (c x+1) \left(a+b \text{sech}^{-1}(c x)\right)}{4 x^2}-\frac{1}{4} c^2 \left(a+b \text{sech}^{-1}(c x)\right)^3+\frac{3 b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(a+b \text{sech}^{-1}(c x)\right)^2}{4 x^2}-\frac{(1-c x) (c x+1) \left(a+b \text{sech}^{-1}(c x)\right)^3}{2 x^2}-\frac{3}{8} b^3 c^2 \text{sech}^{-1}(c x)+\frac{3 b^3 \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{8 x^2}",1,"(-4*a^3 - 6*a*b^2 + 3*b*(2*a^2 + b^2)*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) - 6*b*(2*a^2 + b^2 - 2*a*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))*ArcSech[c*x] + 6*b^2*(b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + a*(-2 + c^2*x^2))*ArcSech[c*x]^2 + 2*b^3*(-2 + c^2*x^2)*ArcSech[c*x]^3 - 3*b*(2*a^2 + b^2)*c^2*x^2*Log[x] + 3*b*(2*a^2 + b^2)*c^2*x^2*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]])/(8*x^2)","A",1
49,1,256,213,0.4203887,"\int \frac{\left(a+b \text{sech}^{-1}(c x)\right)^3}{x^4} \, dx","Integrate[(a + b*ArcSech[c*x])^3/x^4,x]","\frac{-9 a^3-3 b \text{sech}^{-1}(c x) \left(9 a^2-6 a b \sqrt{\frac{1-c x}{c x+1}} \left(2 c^3 x^3+2 c^2 x^2+c x+1\right)+2 b^2 \left(6 c^2 x^2+1\right)\right)+9 a^2 b \sqrt{\frac{1-c x}{c x+1}} \left(2 c^3 x^3+2 c^2 x^2+c x+1\right)-6 a b^2 \left(6 c^2 x^2+1\right)+9 b^2 \text{sech}^{-1}(c x)^2 \left(b \sqrt{\frac{1-c x}{c x+1}} \left(2 c^3 x^3+2 c^2 x^2+c x+1\right)-3 a\right)+2 b^3 \sqrt{\frac{1-c x}{c x+1}} \left(20 c^3 x^3+20 c^2 x^2+c x+1\right)-9 b^3 \text{sech}^{-1}(c x)^3}{27 x^3}","-\frac{4 b^2 c^2 \left(a+b \text{sech}^{-1}(c x)\right)}{3 x}-\frac{2 b^2 \left(a+b \text{sech}^{-1}(c x)\right)}{9 x^3}+\frac{2 b c^2 \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(a+b \text{sech}^{-1}(c x)\right)^2}{3 x}+\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(a+b \text{sech}^{-1}(c x)\right)^2}{3 x^3}-\frac{\left(a+b \text{sech}^{-1}(c x)\right)^3}{3 x^3}+\frac{14 b^3 c^2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{9 x}+\frac{2 b^3 \left(\frac{1-c x}{c x+1}\right)^{3/2} (c x+1)^3}{27 x^3}",1,"(-9*a^3 - 6*a*b^2*(1 + 6*c^2*x^2) + 9*a^2*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x + 2*c^2*x^2 + 2*c^3*x^3) + 2*b^3*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x + 20*c^2*x^2 + 20*c^3*x^3) - 3*b*(9*a^2 + 2*b^2*(1 + 6*c^2*x^2) - 6*a*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x + 2*c^2*x^2 + 2*c^3*x^3))*ArcSech[c*x] + 9*b^2*(-3*a + b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x + 2*c^2*x^2 + 2*c^3*x^3))*ArcSech[c*x]^2 - 9*b^3*ArcSech[c*x]^3)/(27*x^3)","A",1
50,1,332,242,0.7866347,"\int \frac{\left(a+b \text{sech}^{-1}(c x)\right)^3}{x^5} \, dx","Integrate[(a + b*ArcSech[c*x])^3/x^5,x]","\frac{-9 b c^4 x^4 \left(8 a^2+5 b^2\right) \log (x)+9 b c^4 x^4 \left(8 a^2+5 b^2\right) \log \left(c x \sqrt{\frac{1-c x}{c x+1}}+\sqrt{\frac{1-c x}{c x+1}}+1\right)+3 b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(8 a^2 \left(3 c^2 x^2+2\right)+b^2 \left(15 c^2 x^2+2\right)\right)-24 b \text{sech}^{-1}(c x) \left(8 a^2-2 a b \sqrt{\frac{1-c x}{c x+1}} \left(3 c^3 x^3+3 c^2 x^2+2 c x+2\right)+b^2 \left(3 c^2 x^2+1\right)\right)-8 a \left(8 a^2+3 b^2\right)-72 a b^2 c^2 x^2+24 b^2 \text{sech}^{-1}(c x)^2 \left(a \left(3 c^4 x^4-8\right)+b \sqrt{\frac{1-c x}{c x+1}} \left(3 c^3 x^3+3 c^2 x^2+2 c x+2\right)\right)+8 b^3 \left(3 c^4 x^4-8\right) \text{sech}^{-1}(c x)^3}{256 x^4}","-\frac{9 b^2 c^2 \left(a+b \text{sech}^{-1}(c x)\right)}{32 x^2}-\frac{3 b^2 \left(a+b \text{sech}^{-1}(c x)\right)}{32 x^4}+\frac{3}{32} c^4 \left(a+b \text{sech}^{-1}(c x)\right)^3+\frac{9 b c^2 \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(a+b \text{sech}^{-1}(c x)\right)^2}{32 x^2}-\frac{\left(a+b \text{sech}^{-1}(c x)\right)^3}{4 x^4}+\frac{3 b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(a+b \text{sech}^{-1}(c x)\right)^2}{16 x^4}+\frac{45}{256} b^3 c^4 \text{sech}^{-1}(c x)+\frac{45 b^3 c^2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{256 x^2}+\frac{3 b^3 \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{128 x^4}",1,"(-8*a*(8*a^2 + 3*b^2) - 72*a*b^2*c^2*x^2 + 3*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(8*a^2*(2 + 3*c^2*x^2) + b^2*(2 + 15*c^2*x^2)) - 24*b*(8*a^2 + b^2*(1 + 3*c^2*x^2) - 2*a*b*Sqrt[(1 - c*x)/(1 + c*x)]*(2 + 2*c*x + 3*c^2*x^2 + 3*c^3*x^3))*ArcSech[c*x] + 24*b^2*(b*Sqrt[(1 - c*x)/(1 + c*x)]*(2 + 2*c*x + 3*c^2*x^2 + 3*c^3*x^3) + a*(-8 + 3*c^4*x^4))*ArcSech[c*x]^2 + 8*b^3*(-8 + 3*c^4*x^4)*ArcSech[c*x]^3 - 9*b*(8*a^2 + 5*b^2)*c^4*x^4*Log[x] + 9*b*(8*a^2 + 5*b^2)*c^4*x^4*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]])/(256*x^4)","A",1
51,0,0,15,2.3745712,"\int \frac{x}{a+b \text{sech}^{-1}(c x)} \, dx","Integrate[x/(a + b*ArcSech[c*x]),x]","\int \frac{x}{a+b \text{sech}^{-1}(c x)} \, dx","\text{Int}\left(\frac{x}{a+b \text{sech}^{-1}(c x)},x\right)",0,"Integrate[x/(a + b*ArcSech[c*x]), x]","A",-1
52,0,0,13,0.0338132,"\int \frac{1}{a+b \text{sech}^{-1}(c x)} \, dx","Integrate[(a + b*ArcSech[c*x])^(-1),x]","\int \frac{1}{a+b \text{sech}^{-1}(c x)} \, dx","\text{Int}\left(\frac{1}{a+b \text{sech}^{-1}(c x)},x\right)",0,"Integrate[(a + b*ArcSech[c*x])^(-1), x]","A",-1
53,0,0,17,0.3253506,"\int \frac{1}{x \left(a+b \text{sech}^{-1}(c x)\right)} \, dx","Integrate[1/(x*(a + b*ArcSech[c*x])),x]","\int \frac{1}{x \left(a+b \text{sech}^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x \left(a+b \text{sech}^{-1}(c x)\right)},x\right)",0,"Integrate[1/(x*(a + b*ArcSech[c*x])), x]","A",-1
54,1,43,46,0.0812708,"\int \frac{1}{x^2 \left(a+b \text{sech}^{-1}(c x)\right)} \, dx","Integrate[1/(x^2*(a + b*ArcSech[c*x])),x]","\frac{c \left(\sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)-\cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)\right)}{b}","\frac{c \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)}{b}-\frac{c \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)}{b}",1,"(c*(CoshIntegral[a/b + ArcSech[c*x]]*Sinh[a/b] - Cosh[a/b]*SinhIntegral[a/b + ArcSech[c*x]]))/b","A",1
55,1,56,63,0.0849077,"\int \frac{1}{x^3 \left(a+b \text{sech}^{-1}(c x)\right)} \, dx","Integrate[1/(x^3*(a + b*ArcSech[c*x])),x]","\frac{c^2 \left(\sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 a}{b}+2 \text{sech}^{-1}(c x)\right)-\cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 a}{b}+2 \text{sech}^{-1}(c x)\right)\right)}{2 b}","\frac{c^2 \sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 a}{b}+2 \text{sech}^{-1}(c x)\right)}{2 b}-\frac{c^2 \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 a}{b}+2 \text{sech}^{-1}(c x)\right)}{2 b}",1,"(c^2*(CoshIntegral[(2*a)/b + 2*ArcSech[c*x]]*Sinh[(2*a)/b] - Cosh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcSech[c*x]]))/(2*b)","A",1
56,1,91,117,0.1657776,"\int \frac{1}{x^4 \left(a+b \text{sech}^{-1}(c x)\right)} \, dx","Integrate[1/(x^4*(a + b*ArcSech[c*x])),x]","-\frac{c^3 \left(\sinh \left(\frac{a}{b}\right) \left(-\text{Chi}\left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)\right)-\sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)\right)+\cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)+\cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)\right)\right)}{4 b}","\frac{c^3 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)}{4 b}+\frac{c^3 \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 a}{b}+3 \text{sech}^{-1}(c x)\right)}{4 b}-\frac{c^3 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)}{4 b}-\frac{c^3 \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 a}{b}+3 \text{sech}^{-1}(c x)\right)}{4 b}",1,"-1/4*(c^3*(-(CoshIntegral[a/b + ArcSech[c*x]]*Sinh[a/b]) - CoshIntegral[3*(a/b + ArcSech[c*x])]*Sinh[(3*a)/b] + Cosh[a/b]*SinhIntegral[a/b + ArcSech[c*x]] + Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSech[c*x])]))/b","A",1
57,0,0,15,17.7565578,"\int \frac{x}{\left(a+b \text{sech}^{-1}(c x)\right)^2} \, dx","Integrate[x/(a + b*ArcSech[c*x])^2,x]","\int \frac{x}{\left(a+b \text{sech}^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{x}{\left(a+b \text{sech}^{-1}(c x)\right)^2},x\right)",0,"Integrate[x/(a + b*ArcSech[c*x])^2, x]","A",-1
58,0,0,13,9.7142517,"\int \frac{1}{\left(a+b \text{sech}^{-1}(c x)\right)^2} \, dx","Integrate[(a + b*ArcSech[c*x])^(-2),x]","\int \frac{1}{\left(a+b \text{sech}^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{\left(a+b \text{sech}^{-1}(c x)\right)^2},x\right)",0,"Integrate[(a + b*ArcSech[c*x])^(-2), x]","A",-1
59,0,0,17,5.4310909,"\int \frac{1}{x \left(a+b \text{sech}^{-1}(c x)\right)^2} \, dx","Integrate[1/(x*(a + b*ArcSech[c*x])^2),x]","\int \frac{1}{x \left(a+b \text{sech}^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{x \left(a+b \text{sech}^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/(x*(a + b*ArcSech[c*x])^2), x]","A",-1
60,1,82,86,0.3245361,"\int \frac{1}{x^2 \left(a+b \text{sech}^{-1}(c x)\right)^2} \, dx","Integrate[1/(x^2*(a + b*ArcSech[c*x])^2),x]","\frac{-c \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)+c \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)+\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{x \left(a+b \text{sech}^{-1}(c x)\right)}}{b^2}","-\frac{c \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)}{b^2}+\frac{c \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)}{b^2}+\frac{\sqrt{\frac{1-c x}{c x+1}} (c x+1)}{b x \left(a+b \text{sech}^{-1}(c x)\right)}",1,"((b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(x*(a + b*ArcSech[c*x])) - c*Cosh[a/b]*CoshIntegral[a/b + ArcSech[c*x]] + c*Sinh[a/b]*SinhIntegral[a/b + ArcSech[c*x]])/b^2","A",1
61,1,92,85,0.3617465,"\int \frac{1}{x^3 \left(a+b \text{sech}^{-1}(c x)\right)^2} \, dx","Integrate[1/(x^3*(a + b*ArcSech[c*x])^2),x]","\frac{c^2 \left(-\cosh \left(\frac{2 a}{b}\right)\right) \text{Chi}\left(2 \left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)\right)+c^2 \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)\right)+\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{x^2 \left(a+b \text{sech}^{-1}(c x)\right)}}{b^2}","-\frac{c^2 \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 a}{b}+2 \text{sech}^{-1}(c x)\right)}{b^2}+\frac{c^2 \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 a}{b}+2 \text{sech}^{-1}(c x)\right)}{b^2}+\frac{c^2 \sinh \left(2 \text{sech}^{-1}(c x)\right)}{2 b \left(a+b \text{sech}^{-1}(c x)\right)}",1,"((b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(x^2*(a + b*ArcSech[c*x])) - c^2*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcSech[c*x])] + c^2*Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSech[c*x])])/b^2","A",1
62,1,250,190,0.5425895,"\int \frac{1}{x^4 \left(a+b \text{sech}^{-1}(c x)\right)^2} \, dx","Integrate[1/(x^4*(a + b*ArcSech[c*x])^2),x]","\frac{-c^3 x^3 \cosh \left(\frac{a}{b}\right) \left(a+b \text{sech}^{-1}(c x)\right) \text{Chi}\left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)-3 c^3 x^3 \cosh \left(\frac{3 a}{b}\right) \left(a+b \text{sech}^{-1}(c x)\right) \text{Chi}\left(3 \left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)\right)+a c^3 x^3 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)+b c^3 x^3 \sinh \left(\frac{a}{b}\right) \text{sech}^{-1}(c x) \text{Shi}\left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)+3 a c^3 x^3 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)\right)+3 b c^3 x^3 \sinh \left(\frac{3 a}{b}\right) \text{sech}^{-1}(c x) \text{Shi}\left(3 \left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)\right)+4 b c x \sqrt{\frac{1-c x}{c x+1}}+4 b \sqrt{\frac{1-c x}{c x+1}}}{4 b^2 x^3 \left(a+b \text{sech}^{-1}(c x)\right)}","-\frac{c^3 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)}{4 b^2}-\frac{3 c^3 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 a}{b}+3 \text{sech}^{-1}(c x)\right)}{4 b^2}+\frac{c^3 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)}{4 b^2}+\frac{3 c^3 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 a}{b}+3 \text{sech}^{-1}(c x)\right)}{4 b^2}+\frac{c^3 \sinh \left(3 \text{sech}^{-1}(c x)\right)}{4 b \left(a+b \text{sech}^{-1}(c x)\right)}+\frac{c^2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{4 b x \left(a+b \text{sech}^{-1}(c x)\right)}",1,"(4*b*Sqrt[(1 - c*x)/(1 + c*x)] + 4*b*c*x*Sqrt[(1 - c*x)/(1 + c*x)] - c^3*x^3*(a + b*ArcSech[c*x])*Cosh[a/b]*CoshIntegral[a/b + ArcSech[c*x]] - 3*c^3*x^3*(a + b*ArcSech[c*x])*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcSech[c*x])] + a*c^3*x^3*Sinh[a/b]*SinhIntegral[a/b + ArcSech[c*x]] + b*c^3*x^3*ArcSech[c*x]*Sinh[a/b]*SinhIntegral[a/b + ArcSech[c*x]] + 3*a*c^3*x^3*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSech[c*x])] + 3*b*c^3*x^3*ArcSech[c*x]*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSech[c*x])])/(4*b^2*x^3*(a + b*ArcSech[c*x]))","A",1
63,0,0,15,6.282119,"\int \frac{x}{\left(a+b \text{sech}^{-1}(c x)\right)^3} \, dx","Integrate[x/(a + b*ArcSech[c*x])^3,x]","\int \frac{x}{\left(a+b \text{sech}^{-1}(c x)\right)^3} \, dx","\text{Int}\left(\frac{x}{\left(a+b \text{sech}^{-1}(c x)\right)^3},x\right)",0,"Integrate[x/(a + b*ArcSech[c*x])^3, x]","A",-1
64,0,0,13,3.918859,"\int \frac{1}{\left(a+b \text{sech}^{-1}(c x)\right)^3} \, dx","Integrate[(a + b*ArcSech[c*x])^(-3),x]","\int \frac{1}{\left(a+b \text{sech}^{-1}(c x)\right)^3} \, dx","\text{Int}\left(\frac{1}{\left(a+b \text{sech}^{-1}(c x)\right)^3},x\right)",0,"Integrate[(a + b*ArcSech[c*x])^(-3), x]","A",-1
65,0,0,17,2.6914965,"\int \frac{1}{x \left(a+b \text{sech}^{-1}(c x)\right)^3} \, dx","Integrate[1/(x*(a + b*ArcSech[c*x])^3),x]","\int \frac{1}{x \left(a+b \text{sech}^{-1}(c x)\right)^3} \, dx","\text{Int}\left(\frac{1}{x \left(a+b \text{sech}^{-1}(c x)\right)^3},x\right)",0,"Integrate[1/(x*(a + b*ArcSech[c*x])^3), x]","A",-1
66,1,103,114,0.3073379,"\int \frac{1}{x^2 \left(a+b \text{sech}^{-1}(c x)\right)^3} \, dx","Integrate[1/(x^2*(a + b*ArcSech[c*x])^3),x]","\frac{\frac{b^2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{x \left(a+b \text{sech}^{-1}(c x)\right)^2}+c \left(\sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)-\cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)\right)+\frac{b}{a x+b x \text{sech}^{-1}(c x)}}{2 b^3}","\frac{c \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)}{2 b^3}-\frac{c \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)}{2 b^3}+\frac{1}{2 b^2 x \left(a+b \text{sech}^{-1}(c x)\right)}+\frac{\sqrt{\frac{1-c x}{c x+1}} (c x+1)}{2 b x \left(a+b \text{sech}^{-1}(c x)\right)^2}",1,"((b^2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(x*(a + b*ArcSech[c*x])^2) + b/(a*x + b*x*ArcSech[c*x]) + c*(CoshIntegral[a/b + ArcSech[c*x]]*Sinh[a/b] - Cosh[a/b]*SinhIntegral[a/b + ArcSech[c*x]]))/(2*b^3)","A",1
67,1,122,112,0.4305125,"\int \frac{1}{x^3 \left(a+b \text{sech}^{-1}(c x)\right)^3} \, dx","Integrate[1/(x^3*(a + b*ArcSech[c*x])^3),x]","\frac{\frac{b^2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{x^2 \left(a+b \text{sech}^{-1}(c x)\right)^2}+2 c^2 \left(\sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)\right)-\cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)\right)\right)+\frac{b \left(2-c^2 x^2\right)}{x^2 \left(a+b \text{sech}^{-1}(c x)\right)}}{2 b^3}","\frac{c^2 \sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 a}{b}+2 \text{sech}^{-1}(c x)\right)}{b^3}-\frac{c^2 \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 a}{b}+2 \text{sech}^{-1}(c x)\right)}{b^3}+\frac{c^2 \cosh \left(2 \text{sech}^{-1}(c x)\right)}{2 b^2 \left(a+b \text{sech}^{-1}(c x)\right)}+\frac{c^2 \sinh \left(2 \text{sech}^{-1}(c x)\right)}{4 b \left(a+b \text{sech}^{-1}(c x)\right)^2}",1,"((b^2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(x^2*(a + b*ArcSech[c*x])^2) + (b*(2 - c^2*x^2))/(x^2*(a + b*ArcSech[c*x])) + 2*c^2*(CoshIntegral[2*(a/b + ArcSech[c*x])]*Sinh[(2*a)/b] - Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSech[c*x])]))/(2*b^3)","A",1
68,1,204,240,0.627651,"\int \frac{1}{x^4 \left(a+b \text{sech}^{-1}(c x)\right)^3} \, dx","Integrate[1/(x^4*(a + b*ArcSech[c*x])^3),x]","\frac{\frac{4 b^2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{x^3 \left(a+b \text{sech}^{-1}(c x)\right)^2}-8 c^3 \left(\sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)-\cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)\right)+9 c^3 \left(\sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)+\sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)\right)-\cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)-\cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)\right)\right)+\frac{4 b \left(3-2 c^2 x^2\right)}{x^3 \left(a+b \text{sech}^{-1}(c x)\right)}}{8 b^3}","\frac{c^3 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)}{8 b^3}+\frac{9 c^3 \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 a}{b}+3 \text{sech}^{-1}(c x)\right)}{8 b^3}-\frac{c^3 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\text{sech}^{-1}(c x)\right)}{8 b^3}-\frac{9 c^3 \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 a}{b}+3 \text{sech}^{-1}(c x)\right)}{8 b^3}+\frac{3 c^3 \cosh \left(3 \text{sech}^{-1}(c x)\right)}{8 b^2 \left(a+b \text{sech}^{-1}(c x)\right)}+\frac{c^2}{8 b^2 x \left(a+b \text{sech}^{-1}(c x)\right)}+\frac{c^3 \sinh \left(3 \text{sech}^{-1}(c x)\right)}{8 b \left(a+b \text{sech}^{-1}(c x)\right)^2}+\frac{c^2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{8 b x \left(a+b \text{sech}^{-1}(c x)\right)^2}",1,"((4*b^2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(x^3*(a + b*ArcSech[c*x])^2) + (4*b*(3 - 2*c^2*x^2))/(x^3*(a + b*ArcSech[c*x])) - 8*c^3*(CoshIntegral[a/b + ArcSech[c*x]]*Sinh[a/b] - Cosh[a/b]*SinhIntegral[a/b + ArcSech[c*x]]) + 9*c^3*(CoshIntegral[a/b + ArcSech[c*x]]*Sinh[a/b] + CoshIntegral[3*(a/b + ArcSech[c*x])]*Sinh[(3*a)/b] - Cosh[a/b]*SinhIntegral[a/b + ArcSech[c*x]] - Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSech[c*x])]))/(8*b^3)","A",0
69,0,0,19,5.6070464,"\int (d x)^m \left(a+b \text{sech}^{-1}(c x)\right)^3 \, dx","Integrate[(d*x)^m*(a + b*ArcSech[c*x])^3,x]","\int (d x)^m \left(a+b \text{sech}^{-1}(c x)\right)^3 \, dx","\text{Int}\left((d x)^m \left(a+b \text{sech}^{-1}(c x)\right)^3,x\right)",0,"Integrate[(d*x)^m*(a + b*ArcSech[c*x])^3, x]","A",-1
70,0,0,19,3.6127156,"\int (d x)^m \left(a+b \text{sech}^{-1}(c x)\right)^2 \, dx","Integrate[(d*x)^m*(a + b*ArcSech[c*x])^2,x]","\int (d x)^m \left(a+b \text{sech}^{-1}(c x)\right)^2 \, dx","\text{Int}\left((d x)^m \left(a+b \text{sech}^{-1}(c x)\right)^2,x\right)",0,"Integrate[(d*x)^m*(a + b*ArcSech[c*x])^2, x]","A",-1
71,1,97,87,0.1492604,"\int (d x)^m \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[(d*x)^m*(a + b*ArcSech[c*x]),x]","\frac{x (d x)^m \left((m+1) (c x-1) \left(a+b \text{sech}^{-1}(c x)\right)-b \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right)\right)}{(m+1)^2 (c x-1)}","\frac{(d x)^{m+1} \left(a+b \text{sech}^{-1}(c x)\right)}{d (m+1)}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} (d x)^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right)}{d (m+1)^2}",1,"(x*(d*x)^m*((1 + m)*(-1 + c*x)*(a + b*ArcSech[c*x]) - b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2]))/((1 + m)^2*(-1 + c*x))","A",1
72,0,0,19,0.4145864,"\int \frac{(d x)^m}{a+b \text{sech}^{-1}(c x)} \, dx","Integrate[(d*x)^m/(a + b*ArcSech[c*x]),x]","\int \frac{(d x)^m}{a+b \text{sech}^{-1}(c x)} \, dx","\text{Int}\left(\frac{(d x)^m}{a+b \text{sech}^{-1}(c x)},x\right)",0,"Integrate[(d*x)^m/(a + b*ArcSech[c*x]), x]","A",-1
73,0,0,19,0.8235161,"\int \frac{(d x)^m}{\left(a+b \text{sech}^{-1}(c x)\right)^2} \, dx","Integrate[(d*x)^m/(a + b*ArcSech[c*x])^2,x]","\int \frac{(d x)^m}{\left(a+b \text{sech}^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{(d x)^m}{\left(a+b \text{sech}^{-1}(c x)\right)^2},x\right)",0,"Integrate[(d*x)^m/(a + b*ArcSech[c*x])^2, x]","A",-1
74,1,190,264,0.4246612,"\int (d+e x)^3 \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^3*(a + b*ArcSech[c*x]),x]","\frac{1}{4} \left(4 a d^3 x+6 a d^2 e x^2+4 a d e^2 x^3+a e^3 x^4-\frac{b e \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(c^2 \left(18 d^2+6 d e x+e^2 x^2\right)+2 e^2\right)}{3 c^4}+\frac{2 i b d \left(2 c^2 d^2+e^2\right) \log \left(2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)-2 i c x\right)}{c^3}+b x \text{sech}^{-1}(c x) \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)\right)","\frac{(d+e x)^4 \left(a+b \text{sech}^{-1}(c x)\right)}{4 e}-\frac{b d^4 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{4 e}-\frac{b d e^2 x \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{2 c^2}-\frac{b e^3 x^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{12 c^2}-\frac{b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(9 c^2 d^2+e^2\right)}{6 c^4}+\frac{b d \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(2 c^2 d^2+e^2\right) \sin ^{-1}(c x)}{2 c^3}",1,"(4*a*d^3*x + 6*a*d^2*e*x^2 + 4*a*d*e^2*x^3 + a*e^3*x^4 - (b*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(2*e^2 + c^2*(18*d^2 + 6*d*e*x + e^2*x^2)))/(3*c^4) + b*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)*ArcSech[c*x] + ((2*I)*b*d*(2*c^2*d^2 + e^2)*Log[(-2*I)*c*x + 2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)])/c^3)/4","C",1
75,1,147,201,0.2462422,"\int (d+e x)^2 \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^2*(a + b*ArcSech[c*x]),x]","\frac{2 a c^3 x \left(3 d^2+3 d e x+e^2 x^2\right)+2 b c^3 x \text{sech}^{-1}(c x) \left(3 d^2+3 d e x+e^2 x^2\right)+i b \left(6 c^2 d^2+e^2\right) \log \left(2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)-2 i c x\right)-b c e \sqrt{\frac{1-c x}{c x+1}} (c x+1) (6 d+e x)}{6 c^3}","\frac{(d+e x)^3 \left(a+b \text{sech}^{-1}(c x)\right)}{3 e}-\frac{b d^3 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{3 e}-\frac{b d e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{c^2}-\frac{b e^2 x \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{6 c^2}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(6 c^2 d^2+e^2\right) \sin ^{-1}(c x)}{6 c^3}",1,"(-(b*c*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(6*d + e*x)) + 2*a*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2) + 2*b*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2)*ArcSech[c*x] + I*b*(6*c^2*d^2 + e^2)*Log[(-2*I)*c*x + 2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)])/(6*c^3)","C",1
76,1,122,142,0.3768686,"\int (d+e x) \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[(d + e*x)*(a + b*ArcSech[c*x]),x]","a d x+\frac{1}{2} a e x^2-\frac{b d \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c (c x-1)}+b e \left(-\frac{1}{2 c^2}-\frac{x}{2 c}\right) \sqrt{\frac{1-c x}{c x+1}}+b d x \text{sech}^{-1}(c x)+\frac{1}{2} b e x^2 \text{sech}^{-1}(c x)","\frac{(d+e x)^2 \left(a+b \text{sech}^{-1}(c x)\right)}{2 e}-\frac{b d^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{2 e}-\frac{b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{2 c^2}+\frac{b d \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sin ^{-1}(c x)}{c}",1,"a*d*x + (a*e*x^2)/2 + b*e*(-1/2*1/c^2 - x/(2*c))*Sqrt[(1 - c*x)/(1 + c*x)] + b*d*x*ArcSech[c*x] + (b*e*x^2*ArcSech[c*x])/2 - (b*d*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c*(-1 + c*x))","A",1
77,1,60,40,0.0774956,"\int \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[a + b*ArcSech[c*x],x]","a x-\frac{b \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c (c x-1)}+b x \text{sech}^{-1}(c x)","a x+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sin ^{-1}(c x)}{c}+b x \text{sech}^{-1}(c x)",1,"a*x + b*x*ArcSech[c*x] - (b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c*(-1 + c*x))","A",1
78,1,393,229,0.5800979,"\int \frac{a+b \text{sech}^{-1}(c x)}{d+e x} \, dx","Integrate[(a + b*ArcSech[c*x])/(d + e*x),x]","\frac{a \log (d+e x)}{e}+\frac{b \left(\text{Li}_2\left(-e^{-2 \text{sech}^{-1}(c x)}\right)-2 \left(\text{Li}_2\left(\frac{\left(\sqrt{e^2-c^2 d^2}-e\right) e^{-\text{sech}^{-1}(c x)}}{c d}\right)+\text{Li}_2\left(-\frac{\left(e+\sqrt{e^2-c^2 d^2}\right) e^{-\text{sech}^{-1}(c x)}}{c d}\right)-\text{sech}^{-1}(c x) \log \left(\frac{\left(e-\sqrt{e^2-c^2 d^2}\right) e^{-\text{sech}^{-1}(c x)}}{c d}+1\right)-\text{sech}^{-1}(c x) \log \left(\frac{\left(\sqrt{e^2-c^2 d^2}+e\right) e^{-\text{sech}^{-1}(c x)}}{c d}+1\right)+2 i \sin ^{-1}\left(\frac{\sqrt{\frac{e}{c d}+1}}{\sqrt{2}}\right) \log \left(\frac{\left(e-\sqrt{e^2-c^2 d^2}\right) e^{-\text{sech}^{-1}(c x)}}{c d}+1\right)-2 i \sin ^{-1}\left(\frac{\sqrt{\frac{e}{c d}+1}}{\sqrt{2}}\right) \log \left(\frac{\left(\sqrt{e^2-c^2 d^2}+e\right) e^{-\text{sech}^{-1}(c x)}}{c d}+1\right)-4 i \sin ^{-1}\left(\frac{\sqrt{\frac{e}{c d}+1}}{\sqrt{2}}\right) \tanh ^{-1}\left(\frac{(e-c d) \tanh \left(\frac{1}{2} \text{sech}^{-1}(c x)\right)}{\sqrt{e^2-c^2 d^2}}\right)+\text{sech}^{-1}(c x) \log \left(e^{-2 \text{sech}^{-1}(c x)}+1\right)\right)\right)}{2 e}","\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{\left(e-\sqrt{e^2-c^2 d^2}\right) e^{-\text{sech}^{-1}(c x)}}{c d}+1\right)}{e}+\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{\left(\sqrt{e^2-c^2 d^2}+e\right) e^{-\text{sech}^{-1}(c x)}}{c d}+1\right)}{e}-\frac{\log \left(e^{-2 \text{sech}^{-1}(c x)}+1\right) \left(a+b \text{sech}^{-1}(c x)\right)}{e}-\frac{b \text{Li}_2\left(-\frac{\left(e-\sqrt{e^2-c^2 d^2}\right) e^{-\text{sech}^{-1}(c x)}}{c d}\right)}{e}-\frac{b \text{Li}_2\left(-\frac{\left(e+\sqrt{e^2-c^2 d^2}\right) e^{-\text{sech}^{-1}(c x)}}{c d}\right)}{e}+\frac{b \text{Li}_2\left(-e^{-2 \text{sech}^{-1}(c x)}\right)}{2 e}",1,"(a*Log[d + e*x])/e + (b*(PolyLog[2, -E^(-2*ArcSech[c*x])] - 2*((-4*I)*ArcSin[Sqrt[1 + e/(c*d)]/Sqrt[2]]*ArcTanh[((-(c*d) + e)*Tanh[ArcSech[c*x]/2])/Sqrt[-(c^2*d^2) + e^2]] + ArcSech[c*x]*Log[1 + E^(-2*ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (e - Sqrt[-(c^2*d^2) + e^2])/(c*d*E^ArcSech[c*x])] + (2*I)*ArcSin[Sqrt[1 + e/(c*d)]/Sqrt[2]]*Log[1 + (e - Sqrt[-(c^2*d^2) + e^2])/(c*d*E^ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (e + Sqrt[-(c^2*d^2) + e^2])/(c*d*E^ArcSech[c*x])] - (2*I)*ArcSin[Sqrt[1 + e/(c*d)]/Sqrt[2]]*Log[1 + (e + Sqrt[-(c^2*d^2) + e^2])/(c*d*E^ArcSech[c*x])] + PolyLog[2, (-e + Sqrt[-(c^2*d^2) + e^2])/(c*d*E^ArcSech[c*x])] + PolyLog[2, -((e + Sqrt[-(c^2*d^2) + e^2])/(c*d*E^ArcSech[c*x]))])))/(2*e)","C",0
79,1,222,147,0.2434769,"\int \frac{a+b \text{sech}^{-1}(c x)}{(d+e x)^2} \, dx","Integrate[(a + b*ArcSech[c*x])/(d + e*x)^2,x]","-\frac{a}{e (d+e x)}+\frac{b \log (d+e x)}{d \sqrt{e^2-c^2 d^2}}-\frac{b \log \left(c x \sqrt{\frac{1-c x}{c x+1}} \sqrt{e^2-c^2 d^2}+\sqrt{\frac{1-c x}{c x+1}} \sqrt{e^2-c^2 d^2}+c^2 d x+e\right)}{d \sqrt{e^2-c^2 d^2}}+\frac{b \log \left(c x \sqrt{\frac{1-c x}{c x+1}}+\sqrt{\frac{1-c x}{c x+1}}+1\right)}{d e}-\frac{b \text{sech}^{-1}(c x)}{e (d+e x)}-\frac{b \log (x)}{d e}","-\frac{a+b \text{sech}^{-1}(c x)}{e (d+e x)}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tan ^{-1}\left(\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right)}{d \sqrt{c^2 d^2-e^2}}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{d e}",1,"-(a/(e*(d + e*x))) - (b*ArcSech[c*x])/(e*(d + e*x)) - (b*Log[x])/(d*e) + (b*Log[d + e*x])/(d*Sqrt[-(c^2*d^2) + e^2]) + (b*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]])/(d*e) - (b*Log[e + c^2*d*x + Sqrt[-(c^2*d^2) + e^2]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[-(c^2*d^2) + e^2]*x*Sqrt[(1 - c*x)/(1 + c*x)]])/(d*Sqrt[-(c^2*d^2) + e^2])","A",1
80,1,342,306,0.6950842,"\int \frac{a+b \text{sech}^{-1}(c x)}{(d+e x)^3} \, dx","Integrate[(a + b*ArcSech[c*x])/(d + e*x)^3,x]","\frac{1}{2} \left(-\frac{a}{e (d+e x)^2}-\frac{i b \left(2 c^2 d^2-e^2\right) \log \left(\frac{4 d^2 e \sqrt{c^2 d^2-e^2} \left(c x \sqrt{\frac{1-c x}{c x+1}} \sqrt{c^2 d^2-e^2}+\sqrt{\frac{1-c x}{c x+1}} \sqrt{c^2 d^2-e^2}+i c^2 d x+i e\right)}{b \left(2 c^2 d^2-e^2\right) (d+e x)}\right)}{d^2 (c d-e) (c d+e) \sqrt{c^2 d^2-e^2}}+\frac{b \log \left(c x \sqrt{\frac{1-c x}{c x+1}}+\sqrt{\frac{1-c x}{c x+1}}+1\right)}{d^2 e}+\frac{b \sqrt{\frac{1-c x}{c x+1}} (c e x+e)}{d (c d-e) (c d+e) (d+e x)}-\frac{b \text{sech}^{-1}(c x)}{e (d+e x)^2}-\frac{b \log (x)}{d^2 e}\right)","-\frac{a+b \text{sech}^{-1}(c x)}{2 e (d+e x)^2}+\frac{b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{2 d \left(c^2 d^2-e^2\right) (d+e x)}+\frac{b c^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tan ^{-1}\left(\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right)}{2 \left(c^2 d^2-e^2\right)^{3/2}}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tan ^{-1}\left(\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right)}{2 d^2 \sqrt{c^2 d^2-e^2}}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{2 d^2 e}",1,"(-(a/(e*(d + e*x)^2)) + (b*Sqrt[(1 - c*x)/(1 + c*x)]*(e + c*e*x))/(d*(c*d - e)*(c*d + e)*(d + e*x)) - (b*ArcSech[c*x])/(e*(d + e*x)^2) - (b*Log[x])/(d^2*e) + (b*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]])/(d^2*e) - (I*b*(2*c^2*d^2 - e^2)*Log[(4*d^2*e*Sqrt[c^2*d^2 - e^2]*(I*e + I*c^2*d*x + Sqrt[c^2*d^2 - e^2]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[c^2*d^2 - e^2]*x*Sqrt[(1 - c*x)/(1 + c*x)]))/(b*(2*c^2*d^2 - e^2)*(d + e*x))])/(d^2*(c*d - e)*(c*d + e)*Sqrt[c^2*d^2 - e^2]))/2","C",1
81,1,2653,343,10.7793051,"\int (d+e x)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^(3/2)*(a + b*ArcSech[c*x]),x]","\text{Result too large to show}","\frac{2 (d+e x)^{5/2} \left(a+b \text{sech}^{-1}(c x)\right)}{5 e}-\frac{4 b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \sqrt{d+e x}}{15 c^2}-\frac{4 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(2 c^2 d^2+e^2\right) \sqrt{\frac{c (d+e x)}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{15 c^3 \sqrt{d+e x}}-\frac{4 b d^3 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{5 e \sqrt{d+e x}}-\frac{28 b d \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{15 c \sqrt{\frac{c (d+e x)}{c d+e}}}",1,"Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[d + e*x]*((-4*b*e)/(15*c^2) - (4*b*e*x)/(15*c)) + Sqrt[d + e*x]*((2*a*d^2)/(5*e) + (4*a*d*x)/5 + (2*a*e*x^2)/5) + (2*b*(d + e*x)^(5/2)*ArcSech[c*x])/(5*e) - (4*b*(7*c*d*e*Sqrt[(1 - c*x)/(1 + c*x)]*(e - (e*(1 - c*x))/(1 + c*x) + c*d*(1 + (1 - c*x)/(1 + c*x))) + ((7*I)*c^2*d^2*e*(c*d + e)*Sqrt[1 + (1 - c*x)/(1 + c*x)]*Sqrt[(e - (e*(1 - c*x))/(1 + c*x) + c*d*(1 + (1 - c*x)/(1 + c*x)))/(c*d + e)]*(EllipticE[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], (c*d - e)/(c*d + e)] - EllipticF[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], (c*d - e)/(c*d + e)]))/(c*d - e) - ((7*I)*c*d*e^2*(c*d + e)*Sqrt[1 + (1 - c*x)/(1 + c*x)]*Sqrt[(e - (e*(1 - c*x))/(1 + c*x) + c*d*(1 + (1 - c*x)/(1 + c*x)))/(c*d + e)]*(EllipticE[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], (c*d - e)/(c*d + e)] - EllipticF[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], (c*d - e)/(c*d + e)]))/(c*d - e) + (3*I)*c^3*d^3*Sqrt[1 + (1 - c*x)/(1 + c*x)]*Sqrt[(e - (e*(1 - c*x))/(1 + c*x) + c*d*(1 + (1 - c*x)/(1 + c*x)))/(c*d + e)]*EllipticF[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], (c*d - e)/(c*d + e)] - (2*I)*c^2*d^2*e*Sqrt[1 + (1 - c*x)/(1 + c*x)]*Sqrt[(e - (e*(1 - c*x))/(1 + c*x) + c*d*(1 + (1 - c*x)/(1 + c*x)))/(c*d + e)]*EllipticF[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], (c*d - e)/(c*d + e)] - I*e^3*Sqrt[1 + (1 - c*x)/(1 + c*x)]*Sqrt[(e - (e*(1 - c*x))/(1 + c*x) + c*d*(1 + (1 - c*x)/(1 + c*x)))/(c*d + e)]*EllipticF[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], (c*d - e)/(c*d + e)] + ((3 + 3*I)*c^3*d^3*(-I + Sqrt[(1 - c*x)/(1 + c*x)])*(I + Sqrt[(1 - c*x)/(1 + c*x)])*Sqrt[((-I)*(Sqrt[-(c*d) - e]*Sqrt[c*d - e] + c*d*Sqrt[(1 - c*x)/(1 + c*x)] - e*Sqrt[(1 - c*x)/(1 + c*x)]))/(((-I)*c*d + Sqrt[-(c*d) - e]*Sqrt[c*d - e] + I*e)*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[((-I)*(Sqrt[-(c*d) - e]*Sqrt[c*d - e] - c*d*Sqrt[(1 - c*x)/(1 + c*x)] + e*Sqrt[(1 - c*x)/(1 + c*x)]))/((I*c*d + Sqrt[-(c*d) - e]*Sqrt[c*d - e] - I*e)*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*(EllipticF[ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2] - (1 - I)*EllipticPi[(I*Sqrt[-(c*d) - e] - Sqrt[c*d - e])/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e]), ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2]))/Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))] + ((3 + 3*I)*c^3*d^3*(1 + I*Sqrt[(1 - c*x)/(1 + c*x)])*(I + Sqrt[(1 - c*x)/(1 + c*x)])*Sqrt[((-I)*(Sqrt[-(c*d) - e]*Sqrt[c*d - e] + c*d*Sqrt[(1 - c*x)/(1 + c*x)] - e*Sqrt[(1 - c*x)/(1 + c*x)]))/(((-I)*c*d + Sqrt[-(c*d) - e]*Sqrt[c*d - e] + I*e)*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[((-I)*(Sqrt[-(c*d) - e]*Sqrt[c*d - e] - c*d*Sqrt[(1 - c*x)/(1 + c*x)] + e*Sqrt[(1 - c*x)/(1 + c*x)]))/((I*c*d + Sqrt[-(c*d) - e]*Sqrt[c*d - e] - I*e)*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*(EllipticF[ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2] - (1 + I)*EllipticPi[((-I)*Sqrt[-(c*d) - e] + Sqrt[c*d - e])/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e]), ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2]))/Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]))/(15*c^3*e*(1 + (1 - c*x)/(1 + c*x))*Sqrt[(c*d + e + (c*d*(1 - c*x))/(1 + c*x) - (e*(1 - c*x))/(1 + c*x))/(c + (c*(1 - c*x))/(1 + c*x))])","C",0
82,1,2938,279,13.5592644,"\int \sqrt{d+e x} \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[Sqrt[d + e*x]*(a + b*ArcSech[c*x]),x]","\text{Result too large to show}","\frac{2 (d+e x)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right)}{3 e}-\frac{4 b d^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 e \sqrt{d+e x}}-\frac{4 b d \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{c (d+e x)}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 c \sqrt{d+e x}}-\frac{4 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 c \sqrt{\frac{c (d+e x)}{c d+e}}}",1,"((2*a*d)/(3*e) + (2*a*x)/3)*Sqrt[d + e*x] + (2*b*(d + e*x)^(3/2)*ArcSech[c*x])/(3*e) + (4*b*(-((e*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[c*(1 + (1 - c*x)/(1 + c*x))]*Sqrt[c + (c*(1 - c*x))/(1 + c*x)]*Sqrt[(c*d + e + (c*d*(1 - c*x))/(1 + c*x) - (e*(1 - c*x))/(1 + c*x))/(c + (c*(1 - c*x))/(1 + c*x))])/(c*(1 + (1 - c*x)/(1 + c*x)))) + (Sqrt[c*(1 + (1 - c*x)/(1 + c*x))]*Sqrt[c + (c*(1 - c*x))/(1 + c*x)]*Sqrt[c*(1 + (1 - c*x)/(1 + c*x))*(c*d + e + (c*d*(1 - c*x))/(1 + c*x) - (e*(1 - c*x))/(1 + c*x))]*Sqrt[(c*d + e + (c*d*(1 - c*x))/(1 + c*x) - (e*(1 - c*x))/(1 + c*x))/(c + (c*(1 - c*x))/(1 + c*x))]*((I*c*d*(-(c*d) - e)*e*Sqrt[1 + (1 - c*x)/(1 + c*x)]*Sqrt[1 - ((c*d - e)*(1 - c*x))/((-(c*d) - e)*(1 + c*x))]*(EllipticE[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], -((c*d - e)/(-(c*d) - e))] - EllipticF[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], -((c*d - e)/(-(c*d) - e))]))/((c*d - e)*Sqrt[c*(1 + (1 - c*x)/(1 + c*x))*(c*d + e + ((c*d - e)*(1 - c*x))/(1 + c*x))]) - (I*(-(c*d) - e)*e^2*Sqrt[1 + (1 - c*x)/(1 + c*x)]*Sqrt[1 - ((c*d - e)*(1 - c*x))/((-(c*d) - e)*(1 + c*x))]*(EllipticE[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], -((c*d - e)/(-(c*d) - e))] - EllipticF[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], -((c*d - e)/(-(c*d) - e))]))/((c*d - e)*Sqrt[c*(1 + (1 - c*x)/(1 + c*x))*(c*d + e + ((c*d - e)*(1 - c*x))/(1 + c*x))]) - (I*c^2*d^2*Sqrt[1 + (1 - c*x)/(1 + c*x)]*Sqrt[1 - ((c*d - e)*(1 - c*x))/((-(c*d) - e)*(1 + c*x))]*EllipticF[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], -((c*d - e)/(-(c*d) - e))])/Sqrt[c*(1 + (1 - c*x)/(1 + c*x))*(c*d + e + ((c*d - e)*(1 - c*x))/(1 + c*x))] + (I*c*d*e*Sqrt[1 + (1 - c*x)/(1 + c*x)]*Sqrt[1 - ((c*d - e)*(1 - c*x))/((-(c*d) - e)*(1 + c*x))]*EllipticF[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], -((c*d - e)/(-(c*d) - e))])/Sqrt[c*(1 + (1 - c*x)/(1 + c*x))*(c*d + e + ((c*d - e)*(1 - c*x))/(1 + c*x))] + (I*c^2*d^2*(I + Sqrt[-(c*d) - e]/Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)])^2*Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[(I*(-(Sqrt[-(c*d) - e]/Sqrt[c*d - e]) + Sqrt[(1 - c*x)/(1 + c*x)]))/((I + Sqrt[-(c*d) - e]/Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[(I*(Sqrt[-(c*d) - e]/Sqrt[c*d - e] + Sqrt[(1 - c*x)/(1 + c*x)]))/((I - Sqrt[-(c*d) - e]/Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*((1 + I)*EllipticF[ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2] - (2*I)*EllipticPi[((-I)*(I + Sqrt[-(c*d) - e]/Sqrt[c*d - e]))/(-I + Sqrt[-(c*d) - e]/Sqrt[c*d - e]), ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2]))/((I - Sqrt[-(c*d) - e]/Sqrt[c*d - e])*Sqrt[c*(1 + (1 - c*x)/(1 + c*x))*(c*d + e + ((c*d - e)*(1 - c*x))/(1 + c*x))]) - (I*c^2*d^2*(I + Sqrt[-(c*d) - e]/Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)])^2*Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[(I*(-(Sqrt[-(c*d) - e]/Sqrt[c*d - e]) + Sqrt[(1 - c*x)/(1 + c*x)]))/((I + Sqrt[-(c*d) - e]/Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[(I*(Sqrt[-(c*d) - e]/Sqrt[c*d - e] + Sqrt[(1 - c*x)/(1 + c*x)]))/((I - Sqrt[-(c*d) - e]/Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*((-1 + I)*EllipticF[ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2] - (2*I)*EllipticPi[(I*(I + Sqrt[-(c*d) - e]/Sqrt[c*d - e]))/(-I + Sqrt[-(c*d) - e]/Sqrt[c*d - e]), ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2]))/((I - Sqrt[-(c*d) - e]/Sqrt[c*d - e])*Sqrt[c*(1 + (1 - c*x)/(1 + c*x))*(c*d + e + ((c*d - e)*(1 - c*x))/(1 + c*x))])))/(c*(1 + (1 - c*x)/(1 + c*x))*(c*d + e + (c*d*(1 - c*x))/(1 + c*x) - (e*(1 - c*x))/(1 + c*x)))))/(3*c*e)","C",0
83,1,1707,187,11.747619,"\int \frac{a+b \text{sech}^{-1}(c x)}{\sqrt{d+e x}} \, dx","Integrate[(a + b*ArcSech[c*x])/Sqrt[d + e*x],x]","\frac{2 \sqrt{d+e x} a}{e}+\frac{2 b \sqrt{d+e x} \text{sech}^{-1}(c x)}{e}-\frac{4 i b \sqrt{\frac{c d+\frac{c (1-c x) d}{c x+1}+e-\frac{e (1-c x)}{c x+1}}{\frac{(1-c x) c}{c x+1}+c}} \left(2 c d \sqrt{-\frac{i \left(c \sqrt{\frac{1-c x}{c x+1}} d+\sqrt{-c d-e} \sqrt{c d-e}-e \sqrt{\frac{1-c x}{c x+1}}\right)}{\left(-i c d+i e+\sqrt{-c d-e} \sqrt{c d-e}\right) \left(\sqrt{\frac{1-c x}{c x+1}}-i\right)}} \sqrt{-\frac{i \left(-c \sqrt{\frac{1-c x}{c x+1}} d+\sqrt{-c d-e} \sqrt{c d-e}+e \sqrt{\frac{1-c x}{c x+1}}\right)}{\left(i c d-i e+\sqrt{-c d-e} \sqrt{c d-e}\right) \left(\sqrt{\frac{1-c x}{c x+1}}-i\right)}} \left(\frac{1-c x}{c x+1}+1\right) F\left(\sin ^{-1}\left(\sqrt{\frac{\left(\sqrt{-c d-e}-i \sqrt{c d-e}\right) \left(\sqrt{\frac{1-c x}{c x+1}}+i\right)}{\left(\sqrt{-c d-e}+i \sqrt{c d-e}\right) \left(\sqrt{\frac{1-c x}{c x+1}}-i\right)}}\right)|\frac{\left(\sqrt{-c d-e}+i \sqrt{c d-e}\right)^2}{\left(\sqrt{-c d-e}-i \sqrt{c d-e}\right)^2}\right)+(c d-e) \sqrt{\frac{\left(\sqrt{-c d-e}-i \sqrt{c d-e}\right) \left(\sqrt{\frac{1-c x}{c x+1}}+i\right)}{\left(\sqrt{-c d-e}+i \sqrt{c d-e}\right) \left(\sqrt{\frac{1-c x}{c x+1}}-i\right)}} \sqrt{\frac{1-c x}{c x+1}+1} \sqrt{\frac{-\frac{(1-c x) e}{c x+1}+e+c d \left(\frac{1-c x}{c x+1}+1\right)}{c d+e}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{1-c x}{c x+1}}\right)|\frac{c d-e}{c d+e}\right)+2 i c d \sqrt{-\frac{i \left(c \sqrt{\frac{1-c x}{c x+1}} d+\sqrt{-c d-e} \sqrt{c d-e}-e \sqrt{\frac{1-c x}{c x+1}}\right)}{\left(-i c d+i e+\sqrt{-c d-e} \sqrt{c d-e}\right) \left(\sqrt{\frac{1-c x}{c x+1}}-i\right)}} \sqrt{-\frac{i \left(-c \sqrt{\frac{1-c x}{c x+1}} d+\sqrt{-c d-e} \sqrt{c d-e}+e \sqrt{\frac{1-c x}{c x+1}}\right)}{\left(i c d-i e+\sqrt{-c d-e} \sqrt{c d-e}\right) \left(\sqrt{\frac{1-c x}{c x+1}}-i\right)}} \left(\frac{1-c x}{c x+1}+1\right) \left(\Pi \left(\frac{i \sqrt{-c d-e}-\sqrt{c d-e}}{\sqrt{-c d-e}-i \sqrt{c d-e}};\sin ^{-1}\left(\sqrt{\frac{\left(\sqrt{-c d-e}-i \sqrt{c d-e}\right) \left(\sqrt{\frac{1-c x}{c x+1}}+i\right)}{\left(\sqrt{-c d-e}+i \sqrt{c d-e}\right) \left(\sqrt{\frac{1-c x}{c x+1}}-i\right)}}\right)|\frac{\left(\sqrt{-c d-e}+i \sqrt{c d-e}\right)^2}{\left(\sqrt{-c d-e}-i \sqrt{c d-e}\right)^2}\right)-\Pi \left(\frac{\sqrt{c d-e}-i \sqrt{-c d-e}}{\sqrt{-c d-e}-i \sqrt{c d-e}};\sin ^{-1}\left(\sqrt{\frac{\left(\sqrt{-c d-e}-i \sqrt{c d-e}\right) \left(\sqrt{\frac{1-c x}{c x+1}}+i\right)}{\left(\sqrt{-c d-e}+i \sqrt{c d-e}\right) \left(\sqrt{\frac{1-c x}{c x+1}}-i\right)}}\right)|\frac{\left(\sqrt{-c d-e}+i \sqrt{c d-e}\right)^2}{\left(\sqrt{-c d-e}-i \sqrt{c d-e}\right)^2}\right)\right)\right)}{e \sqrt{\frac{\left(\sqrt{-c d-e}-i \sqrt{c d-e}\right) \left(\sqrt{\frac{1-c x}{c x+1}}+i\right)}{\left(\sqrt{-c d-e}+i \sqrt{c d-e}\right) \left(\sqrt{\frac{1-c x}{c x+1}}-i\right)}} \left(-\frac{(1-c x) e}{c x+1}+e+c d \left(\frac{1-c x}{c x+1}+1\right)\right)}","\frac{2 \sqrt{d+e x} \left(a+b \text{sech}^{-1}(c x)\right)}{e}-\frac{4 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{c (d+e x)}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{c \sqrt{d+e x}}-\frac{4 b d \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{e \sqrt{d+e x}}",1,"(2*a*Sqrt[d + e*x])/e + (2*b*Sqrt[d + e*x]*ArcSech[c*x])/e - ((4*I)*b*Sqrt[(c*d + e + (c*d*(1 - c*x))/(1 + c*x) - (e*(1 - c*x))/(1 + c*x))/(c + (c*(1 - c*x))/(1 + c*x))]*(2*c*d*Sqrt[((-I)*(Sqrt[-(c*d) - e]*Sqrt[c*d - e] + c*d*Sqrt[(1 - c*x)/(1 + c*x)] - e*Sqrt[(1 - c*x)/(1 + c*x)]))/(((-I)*c*d + Sqrt[-(c*d) - e]*Sqrt[c*d - e] + I*e)*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[((-I)*(Sqrt[-(c*d) - e]*Sqrt[c*d - e] - c*d*Sqrt[(1 - c*x)/(1 + c*x)] + e*Sqrt[(1 - c*x)/(1 + c*x)]))/((I*c*d + Sqrt[-(c*d) - e]*Sqrt[c*d - e] - I*e)*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*(1 + (1 - c*x)/(1 + c*x))*EllipticF[ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2] + (c*d - e)*Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[1 + (1 - c*x)/(1 + c*x)]*Sqrt[(e - (e*(1 - c*x))/(1 + c*x) + c*d*(1 + (1 - c*x)/(1 + c*x)))/(c*d + e)]*EllipticF[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], (c*d - e)/(c*d + e)] + (2*I)*c*d*Sqrt[((-I)*(Sqrt[-(c*d) - e]*Sqrt[c*d - e] + c*d*Sqrt[(1 - c*x)/(1 + c*x)] - e*Sqrt[(1 - c*x)/(1 + c*x)]))/(((-I)*c*d + Sqrt[-(c*d) - e]*Sqrt[c*d - e] + I*e)*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[((-I)*(Sqrt[-(c*d) - e]*Sqrt[c*d - e] - c*d*Sqrt[(1 - c*x)/(1 + c*x)] + e*Sqrt[(1 - c*x)/(1 + c*x)]))/((I*c*d + Sqrt[-(c*d) - e]*Sqrt[c*d - e] - I*e)*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*(1 + (1 - c*x)/(1 + c*x))*(EllipticPi[(I*Sqrt[-(c*d) - e] - Sqrt[c*d - e])/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e]), ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2] - EllipticPi[((-I)*Sqrt[-(c*d) - e] + Sqrt[c*d - e])/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e]), ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2])))/(e*Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*(e - (e*(1 - c*x))/(1 + c*x) + c*d*(1 + (1 - c*x)/(1 + c*x))))","C",0
84,1,1675,105,11.3390798,"\int \frac{a+b \text{sech}^{-1}(c x)}{(d+e x)^{3/2}} \, dx","Integrate[(a + b*ArcSech[c*x])/(d + e*x)^(3/2),x]","-\frac{2 a}{e \sqrt{d+e x}}-\frac{2 b \text{sech}^{-1}(c x)}{e \sqrt{d+e x}}+\frac{4 i b \left(2 \sqrt{-\frac{i \left(c \sqrt{\frac{1-c x}{c x+1}} d+\sqrt{-c d-e} \sqrt{c d-e}-e \sqrt{\frac{1-c x}{c x+1}}\right)}{\left(-i c d+i e+\sqrt{-c d-e} \sqrt{c d-e}\right) \left(\sqrt{\frac{1-c x}{c x+1}}-i\right)}} \sqrt{-\frac{i \left(-c \sqrt{\frac{1-c x}{c x+1}} d+\sqrt{-c d-e} \sqrt{c d-e}+e \sqrt{\frac{1-c x}{c x+1}}\right)}{\left(i c d-i e+\sqrt{-c d-e} \sqrt{c d-e}\right) \left(\sqrt{\frac{1-c x}{c x+1}}-i\right)}} \left(\frac{1-c x}{c x+1}+1\right) F\left(\sin ^{-1}\left(\sqrt{\frac{\left(\sqrt{-c d-e}-i \sqrt{c d-e}\right) \left(\sqrt{\frac{1-c x}{c x+1}}+i\right)}{\left(\sqrt{-c d-e}+i \sqrt{c d-e}\right) \left(\sqrt{\frac{1-c x}{c x+1}}-i\right)}}\right)|\frac{\left(\sqrt{-c d-e}+i \sqrt{c d-e}\right)^2}{\left(\sqrt{-c d-e}-i \sqrt{c d-e}\right)^2}\right)+\sqrt{\frac{\left(\sqrt{-c d-e}-i \sqrt{c d-e}\right) \left(\sqrt{\frac{1-c x}{c x+1}}+i\right)}{\left(\sqrt{-c d-e}+i \sqrt{c d-e}\right) \left(\sqrt{\frac{1-c x}{c x+1}}-i\right)}} \sqrt{\frac{1-c x}{c x+1}+1} \sqrt{\frac{-\frac{(1-c x) e}{c x+1}+e+c d \left(\frac{1-c x}{c x+1}+1\right)}{c d+e}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{1-c x}{c x+1}}\right)|\frac{c d-e}{c d+e}\right)+2 i \sqrt{-\frac{i \left(c \sqrt{\frac{1-c x}{c x+1}} d+\sqrt{-c d-e} \sqrt{c d-e}-e \sqrt{\frac{1-c x}{c x+1}}\right)}{\left(-i c d+i e+\sqrt{-c d-e} \sqrt{c d-e}\right) \left(\sqrt{\frac{1-c x}{c x+1}}-i\right)}} \sqrt{-\frac{i \left(-c \sqrt{\frac{1-c x}{c x+1}} d+\sqrt{-c d-e} \sqrt{c d-e}+e \sqrt{\frac{1-c x}{c x+1}}\right)}{\left(i c d-i e+\sqrt{-c d-e} \sqrt{c d-e}\right) \left(\sqrt{\frac{1-c x}{c x+1}}-i\right)}} \left(\frac{1-c x}{c x+1}+1\right) \left(\Pi \left(\frac{i \sqrt{-c d-e}-\sqrt{c d-e}}{\sqrt{-c d-e}-i \sqrt{c d-e}};\sin ^{-1}\left(\sqrt{\frac{\left(\sqrt{-c d-e}-i \sqrt{c d-e}\right) \left(\sqrt{\frac{1-c x}{c x+1}}+i\right)}{\left(\sqrt{-c d-e}+i \sqrt{c d-e}\right) \left(\sqrt{\frac{1-c x}{c x+1}}-i\right)}}\right)|\frac{\left(\sqrt{-c d-e}+i \sqrt{c d-e}\right)^2}{\left(\sqrt{-c d-e}-i \sqrt{c d-e}\right)^2}\right)-\Pi \left(\frac{\sqrt{c d-e}-i \sqrt{-c d-e}}{\sqrt{-c d-e}-i \sqrt{c d-e}};\sin ^{-1}\left(\sqrt{\frac{\left(\sqrt{-c d-e}-i \sqrt{c d-e}\right) \left(\sqrt{\frac{1-c x}{c x+1}}+i\right)}{\left(\sqrt{-c d-e}+i \sqrt{c d-e}\right) \left(\sqrt{\frac{1-c x}{c x+1}}-i\right)}}\right)|\frac{\left(\sqrt{-c d-e}+i \sqrt{c d-e}\right)^2}{\left(\sqrt{-c d-e}-i \sqrt{c d-e}\right)^2}\right)\right)\right)}{e \sqrt{\frac{\left(\sqrt{-c d-e}-i \sqrt{c d-e}\right) \left(\sqrt{\frac{1-c x}{c x+1}}+i\right)}{\left(\sqrt{-c d-e}+i \sqrt{c d-e}\right) \left(\sqrt{\frac{1-c x}{c x+1}}-i\right)}} \left(\frac{1-c x}{c x+1}+1\right) \sqrt{\frac{c d+\frac{c (1-c x) d}{c x+1}+e-\frac{e (1-c x)}{c x+1}}{\frac{(1-c x) c}{c x+1}+c}}}","\frac{4 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{e \sqrt{d+e x}}-\frac{2 \left(a+b \text{sech}^{-1}(c x)\right)}{e \sqrt{d+e x}}",1,"(-2*a)/(e*Sqrt[d + e*x]) - (2*b*ArcSech[c*x])/(e*Sqrt[d + e*x]) + ((4*I)*b*(2*Sqrt[((-I)*(Sqrt[-(c*d) - e]*Sqrt[c*d - e] + c*d*Sqrt[(1 - c*x)/(1 + c*x)] - e*Sqrt[(1 - c*x)/(1 + c*x)]))/(((-I)*c*d + Sqrt[-(c*d) - e]*Sqrt[c*d - e] + I*e)*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[((-I)*(Sqrt[-(c*d) - e]*Sqrt[c*d - e] - c*d*Sqrt[(1 - c*x)/(1 + c*x)] + e*Sqrt[(1 - c*x)/(1 + c*x)]))/((I*c*d + Sqrt[-(c*d) - e]*Sqrt[c*d - e] - I*e)*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*(1 + (1 - c*x)/(1 + c*x))*EllipticF[ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2] + Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[1 + (1 - c*x)/(1 + c*x)]*Sqrt[(e - (e*(1 - c*x))/(1 + c*x) + c*d*(1 + (1 - c*x)/(1 + c*x)))/(c*d + e)]*EllipticF[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], (c*d - e)/(c*d + e)] + (2*I)*Sqrt[((-I)*(Sqrt[-(c*d) - e]*Sqrt[c*d - e] + c*d*Sqrt[(1 - c*x)/(1 + c*x)] - e*Sqrt[(1 - c*x)/(1 + c*x)]))/(((-I)*c*d + Sqrt[-(c*d) - e]*Sqrt[c*d - e] + I*e)*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[((-I)*(Sqrt[-(c*d) - e]*Sqrt[c*d - e] - c*d*Sqrt[(1 - c*x)/(1 + c*x)] + e*Sqrt[(1 - c*x)/(1 + c*x)]))/((I*c*d + Sqrt[-(c*d) - e]*Sqrt[c*d - e] - I*e)*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*(1 + (1 - c*x)/(1 + c*x))*(EllipticPi[(I*Sqrt[-(c*d) - e] - Sqrt[c*d - e])/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e]), ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2] - EllipticPi[((-I)*Sqrt[-(c*d) - e] + Sqrt[c*d - e])/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e]), ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2])))/(e*Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*(1 + (1 - c*x)/(1 + c*x))*Sqrt[(c*d + e + (c*d*(1 - c*x))/(1 + c*x) - (e*(1 - c*x))/(1 + c*x))/(c + (c*(1 - c*x))/(1 + c*x))])","C",0
85,1,4527,278,13.74053,"\int \frac{a+b \text{sech}^{-1}(c x)}{(d+e x)^{5/2}} \, dx","Integrate[(a + b*ArcSech[c*x])/(d + e*x)^(5/2),x]","\text{Result too large to show}","-\frac{2 \left(a+b \text{sech}^{-1}(c x)\right)}{3 e (d+e x)^{3/2}}+\frac{4 b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{3 d \left(c^2 d^2-e^2\right) \sqrt{d+e x}}-\frac{4 b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 d \left(c^2 d^2-e^2\right) \sqrt{\frac{c (d+e x)}{c d+e}}}+\frac{4 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 d e \sqrt{d+e x}}",1,"(-2*a)/(3*e*(d + e*x)^(3/2)) + Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[d + e*x]*((4*b*c)/(3*d*(c^2*d^2 - e^2)) - (4*b)/(3*d*(c*d + e)*(d + e*x))) - (2*b*ArcSech[c*x])/(3*e*(d + e*x)^(3/2)) - (4*b*((e*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[c*(1 + (1 - c*x)/(1 + c*x))]*(c*d + e + (c*d*(1 - c*x))/(1 + c*x) - (e*(1 - c*x))/(1 + c*x)))/((1 + (1 - c*x)/(1 + c*x))*Sqrt[c + (c*(1 - c*x))/(1 + c*x)]*Sqrt[(c*d + e + (c*d*(1 - c*x))/(1 + c*x) - (e*(1 - c*x))/(1 + c*x))/(c + (c*(1 - c*x))/(1 + c*x))]) - ((c*d - e)*Sqrt[c*(1 + (1 - c*x)/(1 + c*x))]*Sqrt[c*(1 + (1 - c*x)/(1 + c*x))*(c*d + e + (c*d*(1 - c*x))/(1 + c*x) - (e*(1 - c*x))/(1 + c*x))]*((I*(-(c*d) - e)*e*Sqrt[1 + (1 - c*x)/(1 + c*x)]*Sqrt[1 - ((c*d - e)*(1 - c*x))/((-(c*d) - e)*(1 + c*x))]*(EllipticE[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], -((c*d - e)/(-(c*d) - e))] - EllipticF[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], -((c*d - e)/(-(c*d) - e))]))/((c*d - e)*Sqrt[c*(1 + (1 - c*x)/(1 + c*x))*(c*d + e + ((c*d - e)*(1 - c*x))/(1 + c*x))]) + (I*c*d*Sqrt[1 + (1 - c*x)/(1 + c*x)]*Sqrt[1 - ((c*d - e)*(1 - c*x))/((-(c*d) - e)*(1 + c*x))]*EllipticF[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], -((c*d - e)/(-(c*d) - e))])/Sqrt[c*(1 + (1 - c*x)/(1 + c*x))*(c*d + e + ((c*d - e)*(1 - c*x))/(1 + c*x))] + (I*e*Sqrt[1 + (1 - c*x)/(1 + c*x)]*Sqrt[1 - ((c*d - e)*(1 - c*x))/((-(c*d) - e)*(1 + c*x))]*EllipticF[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], -((c*d - e)/(-(c*d) - e))])/Sqrt[c*(1 + (1 - c*x)/(1 + c*x))*(c*d + e + ((c*d - e)*(1 - c*x))/(1 + c*x))] - (I*c*d*(I + Sqrt[-(c*d) - e]/Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)])^2*Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[(I*(-(Sqrt[-(c*d) - e]/Sqrt[c*d - e]) + Sqrt[(1 - c*x)/(1 + c*x)]))/((I + Sqrt[-(c*d) - e]/Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[(I*(Sqrt[-(c*d) - e]/Sqrt[c*d - e] + Sqrt[(1 - c*x)/(1 + c*x)]))/((I - Sqrt[-(c*d) - e]/Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*((1 + I)*EllipticF[ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2] - (2*I)*EllipticPi[((-I)*(I + Sqrt[-(c*d) - e]/Sqrt[c*d - e]))/(-I + Sqrt[-(c*d) - e]/Sqrt[c*d - e]), ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2]))/((I - Sqrt[-(c*d) - e]/Sqrt[c*d - e])*Sqrt[c*(1 + (1 - c*x)/(1 + c*x))*(c*d + e + ((c*d - e)*(1 - c*x))/(1 + c*x))]) - (I*(I + Sqrt[-(c*d) - e]/Sqrt[c*d - e])*e*(-I + Sqrt[(1 - c*x)/(1 + c*x)])^2*Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[(I*(-(Sqrt[-(c*d) - e]/Sqrt[c*d - e]) + Sqrt[(1 - c*x)/(1 + c*x)]))/((I + Sqrt[-(c*d) - e]/Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[(I*(Sqrt[-(c*d) - e]/Sqrt[c*d - e] + Sqrt[(1 - c*x)/(1 + c*x)]))/((I - Sqrt[-(c*d) - e]/Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*((1 + I)*EllipticF[ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2] - (2*I)*EllipticPi[((-I)*(I + Sqrt[-(c*d) - e]/Sqrt[c*d - e]))/(-I + Sqrt[-(c*d) - e]/Sqrt[c*d - e]), ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2]))/((I - Sqrt[-(c*d) - e]/Sqrt[c*d - e])*Sqrt[c*(1 + (1 - c*x)/(1 + c*x))*(c*d + e + ((c*d - e)*(1 - c*x))/(1 + c*x))]) + (I*c*d*(I + Sqrt[-(c*d) - e]/Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)])^2*Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[(I*(-(Sqrt[-(c*d) - e]/Sqrt[c*d - e]) + Sqrt[(1 - c*x)/(1 + c*x)]))/((I + Sqrt[-(c*d) - e]/Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[(I*(Sqrt[-(c*d) - e]/Sqrt[c*d - e] + Sqrt[(1 - c*x)/(1 + c*x)]))/((I - Sqrt[-(c*d) - e]/Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*((-1 + I)*EllipticF[ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2] - (2*I)*EllipticPi[(I*(I + Sqrt[-(c*d) - e]/Sqrt[c*d - e]))/(-I + Sqrt[-(c*d) - e]/Sqrt[c*d - e]), ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2]))/((I - Sqrt[-(c*d) - e]/Sqrt[c*d - e])*Sqrt[c*(1 + (1 - c*x)/(1 + c*x))*(c*d + e + ((c*d - e)*(1 - c*x))/(1 + c*x))]) + (I*(I + Sqrt[-(c*d) - e]/Sqrt[c*d - e])*e*(-I + Sqrt[(1 - c*x)/(1 + c*x)])^2*Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[(I*(-(Sqrt[-(c*d) - e]/Sqrt[c*d - e]) + Sqrt[(1 - c*x)/(1 + c*x)]))/((I + Sqrt[-(c*d) - e]/Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[(I*(Sqrt[-(c*d) - e]/Sqrt[c*d - e] + Sqrt[(1 - c*x)/(1 + c*x)]))/((I - Sqrt[-(c*d) - e]/Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*((-1 + I)*EllipticF[ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2] - (2*I)*EllipticPi[(I*(I + Sqrt[-(c*d) - e]/Sqrt[c*d - e]))/(-I + Sqrt[-(c*d) - e]/Sqrt[c*d - e]), ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2]))/((I - Sqrt[-(c*d) - e]/Sqrt[c*d - e])*Sqrt[c*(1 + (1 - c*x)/(1 + c*x))*(c*d + e + ((c*d - e)*(1 - c*x))/(1 + c*x))])))/((1 + (1 - c*x)/(1 + c*x))*Sqrt[c + (c*(1 - c*x))/(1 + c*x)]*Sqrt[(c*d + e + (c*d*(1 - c*x))/(1 + c*x) - (e*(1 - c*x))/(1 + c*x))/(c + (c*(1 - c*x))/(1 + c*x))])))/(3*d*e*(c^2*d^2 - e^2))","C",0
86,1,8675,609,14.9649336,"\int \frac{a+b \text{sech}^{-1}(c x)}{(d+e x)^{7/2}} \, dx","Integrate[(a + b*ArcSech[c*x])/(d + e*x)^(7/2),x]","\text{Result too large to show}","-\frac{2 \left(a+b \text{sech}^{-1}(c x)\right)}{5 e (d+e x)^{5/2}}+\frac{16 b c^2 e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{15 \left(c^2 d^2-e^2\right)^2 \sqrt{d+e x}}+\frac{4 b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{5 d^2 \left(c^2 d^2-e^2\right) \sqrt{d+e x}}+\frac{4 b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{15 d \left(c^2 d^2-e^2\right) (d+e x)^{3/2}}+\frac{4 b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{c (d+e x)}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{15 d \left(c^2 d^2-e^2\right) \sqrt{d+e x}}-\frac{4 b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{5 d^2 \left(c^2 d^2-e^2\right) \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{16 b c^3 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{15 \left(c^2 d^2-e^2\right)^2 \sqrt{\frac{c (d+e x)}{c d+e}}}+\frac{4 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{5 d^2 e \sqrt{d+e x}}",1,"Result too large to show","C",0
87,0,0,87,1.9551915,"\int (d+e x)^m \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^m*(a + b*ArcSech[c*x]),x]","\int (d+e x)^m \left(a+b \text{sech}^{-1}(c x)\right) \, dx","\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \text{Int}\left(\frac{(d+e x)^{m+1}}{x \sqrt{1-c^2 x^2}},x\right)}{e (m+1)}+\frac{(d+e x)^{m+1} \left(a+b \text{sech}^{-1}(c x)\right)}{e (m+1)}",0,"Integrate[(d + e*x)^m*(a + b*ArcSech[c*x]), x]","A",-1
88,1,162,229,0.3437772,"\int x^4 \left(d+e x^2\right) \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[x^4*(d + e*x^2)*(a + b*ArcSech[c*x]),x]","\frac{48 a c^7 x^5 \left(7 d+5 e x^2\right)+48 b c^7 x^5 \text{sech}^{-1}(c x) \left(7 d+5 e x^2\right)+3 i b \left(42 c^2 d+25 e\right) \log \left(2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)-2 i c x\right)-b c x \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(c^4 \left(84 d x^2+40 e x^4\right)+2 c^2 \left(63 d+25 e x^2\right)+75 e\right)}{1680 c^7}","\frac{1}{5} d x^5 \left(a+b \text{sech}^{-1}(c x)\right)+\frac{1}{7} e x^7 \left(a+b \text{sech}^{-1}(c x)\right)-\frac{b e x^5 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{42 c^2}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(42 c^2 d+25 e\right) \sin ^{-1}(c x)}{560 c^7}-\frac{b x \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(42 c^2 d+25 e\right)}{560 c^6}-\frac{b x^3 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(42 c^2 d+25 e\right)}{840 c^4}",1,"(48*a*c^7*x^5*(7*d + 5*e*x^2) - b*c*x*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(75*e + 2*c^2*(63*d + 25*e*x^2) + c^4*(84*d*x^2 + 40*e*x^4)) + 48*b*c^7*x^5*(7*d + 5*e*x^2)*ArcSech[c*x] + (3*I)*b*(42*c^2*d + 25*e)*Log[(-2*I)*c*x + 2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)])/(1680*c^7)","C",1
89,1,144,174,0.2241093,"\int x^2 \left(d+e x^2\right) \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[x^2*(d + e*x^2)*(a + b*ArcSech[c*x]),x]","\frac{8 a c^5 x^3 \left(5 d+3 e x^2\right)+8 b c^5 x^3 \text{sech}^{-1}(c x) \left(5 d+3 e x^2\right)-b c x \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(c^2 \left(20 d+6 e x^2\right)+9 e\right)+i b \left(20 c^2 d+9 e\right) \log \left(2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)-2 i c x\right)}{120 c^5}","\frac{1}{3} d x^3 \left(a+b \text{sech}^{-1}(c x)\right)+\frac{1}{5} e x^5 \left(a+b \text{sech}^{-1}(c x)\right)-\frac{b e x^3 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{20 c^2}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(20 c^2 d+9 e\right) \sin ^{-1}(c x)}{120 c^5}-\frac{b x \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(20 c^2 d+9 e\right)}{120 c^4}",1,"(8*a*c^5*x^3*(5*d + 3*e*x^2) - b*c*x*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(9*e + c^2*(20*d + 6*e*x^2)) + 8*b*c^5*x^3*(5*d + 3*e*x^2)*ArcSech[c*x] + I*b*(20*c^2*d + 9*e)*Log[(-2*I)*c*x + 2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)])/(120*c^5)","C",1
90,1,169,112,0.3701429,"\int \left(d+e x^2\right) \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[(d + e*x^2)*(a + b*ArcSech[c*x]),x]","a d x+\frac{1}{3} a e x^3+\frac{i b e \log \left(2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)-2 i c x\right)}{6 c^3}-\frac{b d \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c (c x-1)}+b e \sqrt{\frac{1-c x}{c x+1}} \left(-\frac{x}{6 c^2}-\frac{x^2}{6 c}\right)+b d x \text{sech}^{-1}(c x)+\frac{1}{3} b e x^3 \text{sech}^{-1}(c x)","d x \left(a+b \text{sech}^{-1}(c x)\right)+\frac{1}{3} e x^3 \left(a+b \text{sech}^{-1}(c x)\right)-\frac{b e x \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{6 c^2}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(6 c^2 d+e\right) \sin ^{-1}(c x)}{6 c^3}",1,"a*d*x + (a*e*x^3)/3 + b*e*Sqrt[(1 - c*x)/(1 + c*x)]*(-1/6*x/c^2 - x^2/(6*c)) + b*d*x*ArcSech[c*x] + (b*e*x^3*ArcSech[c*x])/3 - (b*d*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c*(-1 + c*x)) + ((I/6)*b*e*Log[(-2*I)*c*x + 2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)])/c^3","C",1
91,1,107,96,0.2633018,"\int \frac{\left(d+e x^2\right) \left(a+b \text{sech}^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + e*x^2)*(a + b*ArcSech[c*x]))/x^2,x]","-\frac{a d}{x}+a e x-\frac{b e \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c (c x-1)}+b d \left(c+\frac{1}{x}\right) \sqrt{\frac{1-c x}{c x+1}}-\frac{b d \text{sech}^{-1}(c x)}{x}+b e x \text{sech}^{-1}(c x)","-\frac{d \left(a+b \text{sech}^{-1}(c x)\right)}{x}+e x \left(a+b \text{sech}^{-1}(c x)\right)+\frac{b d \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{x}+\frac{b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sin ^{-1}(c x)}{c}",1,"-((a*d)/x) + a*e*x + b*d*(c + x^(-1))*Sqrt[(1 - c*x)/(1 + c*x)] - (b*d*ArcSech[c*x])/x + b*e*x*ArcSech[c*x] - (b*e*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c*(-1 + c*x))","A",1
92,1,76,126,0.1186724,"\int \frac{\left(d+e x^2\right) \left(a+b \text{sech}^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + e*x^2)*(a + b*ArcSech[c*x]))/x^4,x]","\frac{-3 a \left(d+3 e x^2\right)+b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(2 c^2 d x^2+d+9 e x^2\right)-3 b \text{sech}^{-1}(c x) \left(d+3 e x^2\right)}{9 x^3}","-\frac{d \left(a+b \text{sech}^{-1}(c x)\right)}{3 x^3}-\frac{e \left(a+b \text{sech}^{-1}(c x)\right)}{x}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(2 c^2 d+9 e\right)}{9 x}+\frac{b d \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{9 x^3}",1,"(-3*a*(d + 3*e*x^2) + b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(d + 2*c^2*d*x^2 + 9*e*x^2) - 3*b*(d + 3*e*x^2)*ArcSech[c*x])/(9*x^3)","A",1
93,1,101,183,0.1758632,"\int \frac{\left(d+e x^2\right) \left(a+b \text{sech}^{-1}(c x)\right)}{x^6} \, dx","Integrate[((d + e*x^2)*(a + b*ArcSech[c*x]))/x^6,x]","\frac{-15 a \left(3 d+5 e x^2\right)+b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(25 e x^2 \left(2 c^2 x^2+1\right)+3 d \left(8 c^4 x^4+4 c^2 x^2+3\right)\right)-15 b \text{sech}^{-1}(c x) \left(3 d+5 e x^2\right)}{225 x^5}","-\frac{d \left(a+b \text{sech}^{-1}(c x)\right)}{5 x^5}-\frac{e \left(a+b \text{sech}^{-1}(c x)\right)}{3 x^3}+\frac{2 b c^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(12 c^2 d+25 e\right)}{225 x}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(12 c^2 d+25 e\right)}{225 x^3}+\frac{b d \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{25 x^5}",1,"(-15*a*(3*d + 5*e*x^2) + b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(25*e*x^2*(1 + 2*c^2*x^2) + 3*d*(3 + 4*c^2*x^2 + 8*c^4*x^4)) - 15*b*(3*d + 5*e*x^2)*ArcSech[c*x])/(225*x^5)","A",1
94,1,117,238,0.2223838,"\int \frac{\left(d+e x^2\right) \left(a+b \text{sech}^{-1}(c x)\right)}{x^8} \, dx","Integrate[((d + e*x^2)*(a + b*ArcSech[c*x]))/x^8,x]","\frac{-105 a \left(5 d+7 e x^2\right)+b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(49 e x^2 \left(8 c^4 x^4+4 c^2 x^2+3\right)+15 d \left(16 c^6 x^6+8 c^4 x^4+6 c^2 x^2+5\right)\right)-105 b \text{sech}^{-1}(c x) \left(5 d+7 e x^2\right)}{3675 x^7}","-\frac{d \left(a+b \text{sech}^{-1}(c x)\right)}{7 x^7}-\frac{e \left(a+b \text{sech}^{-1}(c x)\right)}{5 x^5}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(30 c^2 d+49 e\right)}{1225 x^5}+\frac{4 b c^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(30 c^2 d+49 e\right)}{3675 x^3}+\frac{b d \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{49 x^7}+\frac{8 b c^4 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(30 c^2 d+49 e\right)}{3675 x}",1,"(-105*a*(5*d + 7*e*x^2) + b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(49*e*x^2*(3 + 4*c^2*x^2 + 8*c^4*x^4) + 15*d*(5 + 6*c^2*x^2 + 8*c^4*x^4 + 16*c^6*x^6)) - 105*b*(5*d + 7*e*x^2)*ArcSech[c*x])/(3675*x^7)","A",1
95,1,126,232,0.2175651,"\int x^5 \left(d+e x^2\right) \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[x^5*(d + e*x^2)*(a + b*ArcSech[c*x]),x]","\frac{1}{24} a x^6 \left(4 d+3 e x^2\right)-\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(c^6 \left(84 d x^4+45 e x^6\right)+2 c^4 \left(56 d x^2+27 e x^4\right)+8 c^2 \left(28 d+9 e x^2\right)+144 e\right)}{2520 c^8}+\frac{1}{24} b x^6 \text{sech}^{-1}(c x) \left(4 d+3 e x^2\right)","\frac{1}{6} d x^6 \left(a+b \text{sech}^{-1}(c x)\right)+\frac{1}{8} e x^8 \left(a+b \text{sech}^{-1}(c x)\right)-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(1-c^2 x^2\right)^{5/2} \left(4 c^2 d+9 e\right)}{120 c^8}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(1-c^2 x^2\right)^{3/2} \left(8 c^2 d+9 e\right)}{72 c^8}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(4 c^2 d+3 e\right)}{24 c^8}+\frac{b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(1-c^2 x^2\right)^{7/2}}{56 c^8}",1,"(a*x^6*(4*d + 3*e*x^2))/24 - (b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(144*e + 8*c^2*(28*d + 9*e*x^2) + 2*c^4*(56*d*x^2 + 27*e*x^4) + c^6*(84*d*x^4 + 45*e*x^6)))/(2520*c^8) + (b*x^6*(4*d + 3*e*x^2)*ArcSech[c*x])/24","A",1
96,1,106,180,0.1875813,"\int x^3 \left(d+e x^2\right) \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[x^3*(d + e*x^2)*(a + b*ArcSech[c*x]),x]","\frac{1}{180} \left(15 a x^4 \left(3 d+2 e x^2\right)-\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(3 c^4 \left(5 d x^2+2 e x^4\right)+c^2 \left(30 d+8 e x^2\right)+16 e\right)}{c^6}+15 b x^4 \text{sech}^{-1}(c x) \left(3 d+2 e x^2\right)\right)","\frac{1}{4} d x^4 \left(a+b \text{sech}^{-1}(c x)\right)+\frac{1}{6} e x^6 \left(a+b \text{sech}^{-1}(c x)\right)+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(1-c^2 x^2\right)^{3/2} \left(3 c^2 d+4 e\right)}{36 c^6}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(3 c^2 d+2 e\right)}{12 c^6}-\frac{b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(1-c^2 x^2\right)^{5/2}}{30 c^6}",1,"(15*a*x^4*(3*d + 2*e*x^2) - (b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(16*e + c^2*(30*d + 8*e*x^2) + 3*c^4*(5*d*x^2 + 2*e*x^4)))/c^6 + 15*b*x^4*(3*d + 2*e*x^2)*ArcSech[c*x])/180","A",1
97,1,85,164,0.1352826,"\int x \left(d+e x^2\right) \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[x*(d + e*x^2)*(a + b*ArcSech[c*x]),x]","\frac{1}{12} \left(3 a x^2 \left(2 d+e x^2\right)-\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(c^2 \left(6 d+e x^2\right)+2 e\right)}{c^4}+3 b x^2 \text{sech}^{-1}(c x) \left(2 d+e x^2\right)\right)","\frac{\left(d+e x^2\right)^2 \left(a+b \text{sech}^{-1}(c x)\right)}{4 e}-\frac{b d^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{4 e}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(2 c^2 d+e\right)}{4 c^4}+\frac{b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(1-c^2 x^2\right)^{3/2}}{12 c^4}",1,"(3*a*x^2*(2*d + e*x^2) - (b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(2*e + c^2*(6*d + e*x^2)))/c^4 + 3*b*x^2*(2*d + e*x^2)*ArcSech[c*x])/12","A",1
98,1,98,296,0.3143241,"\int \frac{\left(d+e x^2\right) \left(a+b \text{sech}^{-1}(c x)\right)}{x} \, dx","Integrate[((d + e*x^2)*(a + b*ArcSech[c*x]))/x,x]","\frac{1}{2} \left(2 a d \log (x)+a e x^2-\frac{b e \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{c^2}+b d \text{Li}_2\left(-e^{-2 \text{sech}^{-1}(c x)}\right)-b d \text{sech}^{-1}(c x) \left(\text{sech}^{-1}(c x)+2 \log \left(e^{-2 \text{sech}^{-1}(c x)}+1\right)\right)+b e x^2 \text{sech}^{-1}(c x)\right)","-d \log \left(\frac{1}{x}\right) \left(a+b \text{sech}^{-1}(c x)\right)+\frac{1}{2} e x^2 \left(a+b \text{sech}^{-1}(c x)\right)+\frac{i b d \sqrt{1-\frac{1}{c^2 x^2}} \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)}{2 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{i b d \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{b d \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left(1-e^{2 i \csc ^{-1}(c x)}\right)}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{b d \sqrt{1-\frac{1}{c^2 x^2}} \log \left(\frac{1}{x}\right) \csc ^{-1}(c x)}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{b e x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}{2 c}",1,"(a*e*x^2 - (b*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/c^2 + b*e*x^2*ArcSech[c*x] - b*d*ArcSech[c*x]*(ArcSech[c*x] + 2*Log[1 + E^(-2*ArcSech[c*x])]) + 2*a*d*Log[x] + b*d*PolyLog[2, -E^(-2*ArcSech[c*x])])/2","A",0
99,1,149,309,0.61441,"\int \frac{\left(d+e x^2\right) \left(a+b \text{sech}^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + e*x^2)*(a + b*ArcSech[c*x]))/x^3,x]","\frac{1}{4} \left(-\frac{2 a d}{x^2}+4 a e \log (x)-\frac{b d \sqrt{\frac{1-c x}{c x+1}} \left(-c^2 x^2+c^2 x^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)+1\right)}{x^2 (c x-1)}-\frac{2 b d \text{sech}^{-1}(c x)}{x^2}+2 b e \text{Li}_2\left(-e^{-2 \text{sech}^{-1}(c x)}\right)-2 b e \text{sech}^{-1}(c x) \left(\text{sech}^{-1}(c x)+2 \log \left(e^{-2 \text{sech}^{-1}(c x)}+1\right)\right)\right)","-\frac{d \left(a+b \text{sech}^{-1}(c x)\right)}{2 x^2}-e \log \left(\frac{1}{x}\right) \left(a+b \text{sech}^{-1}(c x)\right)+\frac{1}{4} b c^2 d \text{sech}^{-1}(c x)+\frac{i b e \sqrt{1-\frac{1}{c^2 x^2}} \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)}{2 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{i b e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{b e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left(1-e^{2 i \csc ^{-1}(c x)}\right)}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{b e \sqrt{1-\frac{1}{c^2 x^2}} \log \left(\frac{1}{x}\right) \csc ^{-1}(c x)}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{b c d \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}{4 x}",1,"((-2*a*d)/x^2 - (2*b*d*ArcSech[c*x])/x^2 - (b*d*Sqrt[(1 - c*x)/(1 + c*x)]*(1 - c^2*x^2 + c^2*x^2*Sqrt[1 - c^2*x^2]*ArcTanh[Sqrt[1 - c^2*x^2]]))/(x^2*(-1 + c*x)) - 2*b*e*ArcSech[c*x]*(ArcSech[c*x] + 2*Log[1 + E^(-2*ArcSech[c*x])]) + 4*a*e*Log[x] + 2*b*e*PolyLog[2, -E^(-2*ArcSech[c*x])])/4","A",0
100,1,207,275,0.4990266,"\int x^2 \left(d+e x^2\right)^2 \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[x^2*(d + e*x^2)^2*(a + b*ArcSech[c*x]),x]","\frac{16 a c^7 x^3 \left(35 d^2+42 d e x^2+15 e^2 x^4\right)+16 b c^7 x^3 \text{sech}^{-1}(c x) \left(35 d^2+42 d e x^2+15 e^2 x^4\right)-b c x \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(8 c^4 \left(35 d^2+21 d e x^2+5 e^2 x^4\right)+2 c^2 e \left(126 d+25 e x^2\right)+75 e^2\right)+i b \left(280 c^4 d^2+252 c^2 d e+75 e^2\right) \log \left(2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)-2 i c x\right)}{1680 c^7}","\frac{1}{3} d^2 x^3 \left(a+b \text{sech}^{-1}(c x)\right)+\frac{2}{5} d e x^5 \left(a+b \text{sech}^{-1}(c x)\right)+\frac{1}{7} e^2 x^7 \left(a+b \text{sech}^{-1}(c x)\right)-\frac{b e^2 x^5 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{42 c^2}-\frac{b e x^3 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(84 c^2 d+25 e\right)}{840 c^4}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(280 c^4 d^2+252 c^2 d e+75 e^2\right) \sin ^{-1}(c x)}{1680 c^7}-\frac{b x \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(280 c^4 d^2+252 c^2 d e+75 e^2\right)}{1680 c^6}",1,"(16*a*c^7*x^3*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4) - b*c*x*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(75*e^2 + 2*c^2*e*(126*d + 25*e*x^2) + 8*c^4*(35*d^2 + 21*d*e*x^2 + 5*e^2*x^4)) + 16*b*c^7*x^3*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4)*ArcSech[c*x] + I*b*(280*c^4*d^2 + 252*c^2*d*e + 75*e^2)*Log[(-2*I)*c*x + 2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)])/(1680*c^7)","C",1
101,1,174,204,0.3451322,"\int \left(d+e x^2\right)^2 \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[(d + e*x^2)^2*(a + b*ArcSech[c*x]),x]","\frac{8 a c^5 x \left(15 d^2+10 d e x^2+3 e^2 x^4\right)+8 b c^5 x \text{sech}^{-1}(c x) \left(15 d^2+10 d e x^2+3 e^2 x^4\right)-b c e x \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(c^2 \left(40 d+6 e x^2\right)+9 e\right)+i b \left(120 c^4 d^2+40 c^2 d e+9 e^2\right) \log \left(2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)-2 i c x\right)}{120 c^5}","d^2 x \left(a+b \text{sech}^{-1}(c x)\right)+\frac{2}{3} d e x^3 \left(a+b \text{sech}^{-1}(c x)\right)+\frac{1}{5} e^2 x^5 \left(a+b \text{sech}^{-1}(c x)\right)-\frac{b e^2 x^3 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{20 c^2}-\frac{b e x \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(40 c^2 d+9 e\right)}{120 c^4}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(120 c^4 d^2+40 c^2 d e+9 e^2\right) \sin ^{-1}(c x)}{120 c^5}",1,"(8*a*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) - b*c*e*x*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(9*e + c^2*(40*d + 6*e*x^2)) + 8*b*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4)*ArcSech[c*x] + I*b*(120*c^4*d^2 + 40*c^2*d*e + 9*e^2)*Log[(-2*I)*c*x + 2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)])/(120*c^5)","C",1
102,1,158,177,0.3151913,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \text{sech}^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcSech[c*x]))/x^2,x]","\frac{2 a c^3 \left(-3 d^2+6 d e x^2+e^2 x^4\right)+2 b c^3 \text{sech}^{-1}(c x) \left(-3 d^2+6 d e x^2+e^2 x^4\right)-b c \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(e^2 x^2-6 c^2 d^2\right)+i b e x \left(12 c^2 d+e\right) \log \left(2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)-2 i c x\right)}{6 c^3 x}","-\frac{d^2 \left(a+b \text{sech}^{-1}(c x)\right)}{x}+2 d e x \left(a+b \text{sech}^{-1}(c x)\right)+\frac{1}{3} e^2 x^3 \left(a+b \text{sech}^{-1}(c x)\right)+\frac{b d^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{x}-\frac{b e^2 x \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{6 c^2}+\frac{b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(12 c^2 d+e\right) \sin ^{-1}(c x)}{6 c^3}",1,"(-(b*c*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(-6*c^2*d^2 + e^2*x^2)) + 2*a*c^3*(-3*d^2 + 6*d*e*x^2 + e^2*x^4) + 2*b*c^3*(-3*d^2 + 6*d*e*x^2 + e^2*x^4)*ArcSech[c*x] + I*b*e*(12*c^2*d + e)*x*Log[(-2*I)*c*x + 2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)])/(6*c^3*x)","C",1
103,1,149,176,0.3082885,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \text{sech}^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcSech[c*x]))/x^4,x]","\frac{-3 a c \left(d^2+6 d e x^2-3 e^2 x^4\right)+b c d \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(2 c^2 d x^2+d+18 e x^2\right)-3 b c \text{sech}^{-1}(c x) \left(d^2+6 d e x^2-3 e^2 x^4\right)+9 i b e^2 x^3 \log \left(2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)-2 i c x\right)}{9 c x^3}","-\frac{d^2 \left(a+b \text{sech}^{-1}(c x)\right)}{3 x^3}-\frac{2 d e \left(a+b \text{sech}^{-1}(c x)\right)}{x}+e^2 x \left(a+b \text{sech}^{-1}(c x)\right)+\frac{b d^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{9 x^3}+\frac{2 b d \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(c^2 d+9 e\right)}{9 x}+\frac{b e^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sin ^{-1}(c x)}{c}",1,"(b*c*d*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(d + 2*c^2*d*x^2 + 18*e*x^2) - 3*a*c*(d^2 + 6*d*e*x^2 - 3*e^2*x^4) - 3*b*c*(d^2 + 6*d*e*x^2 - 3*e^2*x^4)*ArcSech[c*x] + (9*I)*b*e^2*x^3*Log[(-2*I)*c*x + 2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)])/(9*c*x^3)","C",1
104,1,134,213,0.289868,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \text{sech}^{-1}(c x)\right)}{x^6} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcSech[c*x]))/x^6,x]","\frac{-15 a \left(3 d^2+10 d e x^2+15 e^2 x^4\right)+b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(50 d e x^2 \left(2 c^2 x^2+1\right)+3 d^2 \left(8 c^4 x^4+4 c^2 x^2+3\right)+225 e^2 x^4\right)-15 b \text{sech}^{-1}(c x) \left(3 d^2+10 d e x^2+15 e^2 x^4\right)}{225 x^5}","-\frac{d^2 \left(a+b \text{sech}^{-1}(c x)\right)}{5 x^5}-\frac{2 d e \left(a+b \text{sech}^{-1}(c x)\right)}{3 x^3}-\frac{e^2 \left(a+b \text{sech}^{-1}(c x)\right)}{x}+\frac{b d^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{25 x^5}+\frac{2 b d \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(6 c^2 d+25 e\right)}{225 x^3}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(24 c^4 d^2+100 c^2 d e+225 e^2\right)}{225 x}",1,"(-15*a*(3*d^2 + 10*d*e*x^2 + 15*e^2*x^4) + b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(225*e^2*x^4 + 50*d*e*x^2*(1 + 2*c^2*x^2) + 3*d^2*(3 + 4*c^2*x^2 + 8*c^4*x^4)) - 15*b*(3*d^2 + 10*d*e*x^2 + 15*e^2*x^4)*ArcSech[c*x])/(225*x^5)","A",1
105,1,160,281,0.3983849,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \text{sech}^{-1}(c x)\right)}{x^8} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcSech[c*x]))/x^8,x]","\frac{-105 a \left(15 d^2+42 d e x^2+35 e^2 x^4\right)+b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(1225 e^2 x^4 \left(2 c^2 x^2+1\right)+294 d e x^2 \left(8 c^4 x^4+4 c^2 x^2+3\right)+45 d^2 \left(16 c^6 x^6+8 c^4 x^4+6 c^2 x^2+5\right)\right)-105 b \text{sech}^{-1}(c x) \left(15 d^2+42 d e x^2+35 e^2 x^4\right)}{11025 x^7}","-\frac{d^2 \left(a+b \text{sech}^{-1}(c x)\right)}{7 x^7}-\frac{2 d e \left(a+b \text{sech}^{-1}(c x)\right)}{5 x^5}-\frac{e^2 \left(a+b \text{sech}^{-1}(c x)\right)}{3 x^3}+\frac{b d^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{49 x^7}+\frac{2 b d \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(15 c^2 d+49 e\right)}{1225 x^5}+\frac{2 b c^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(360 c^4 d^2+1176 c^2 d e+1225 e^2\right)}{11025 x}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(360 c^4 d^2+1176 c^2 d e+1225 e^2\right)}{11025 x^3}",1,"(-105*a*(15*d^2 + 42*d*e*x^2 + 35*e^2*x^4) + b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(1225*e^2*x^4*(1 + 2*c^2*x^2) + 294*d*e*x^2*(3 + 4*c^2*x^2 + 8*c^4*x^4) + 45*d^2*(5 + 6*c^2*x^2 + 8*c^4*x^4 + 16*c^6*x^6)) - 105*b*(15*d^2 + 42*d*e*x^2 + 35*e^2*x^4)*ArcSech[c*x])/(11025*x^7)","A",1
106,1,168,278,0.3103298,"\int x^3 \left(d+e x^2\right)^2 \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[x^3*(d + e*x^2)^2*(a + b*ArcSech[c*x]),x]","\frac{1}{24} \left(6 a d^2 x^4+8 a d e x^6+3 a e^2 x^8-\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(3 c^6 \left(70 d^2 x^2+56 d e x^4+15 e^2 x^6\right)+c^4 \left(420 d^2+224 d e x^2+54 e^2 x^4\right)+8 c^2 e \left(56 d+9 e x^2\right)+144 e^2\right)}{105 c^8}+b x^4 \text{sech}^{-1}(c x) \left(6 d^2+8 d e x^2+3 e^2 x^4\right)\right)","\frac{1}{4} d^2 x^4 \left(a+b \text{sech}^{-1}(c x)\right)+\frac{1}{3} d e x^6 \left(a+b \text{sech}^{-1}(c x)\right)+\frac{1}{8} e^2 x^8 \left(a+b \text{sech}^{-1}(c x)\right)-\frac{b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(1-c^2 x^2\right)^{5/2} \left(8 c^2 d+9 e\right)}{120 c^8}+\frac{b e^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(1-c^2 x^2\right)^{7/2}}{56 c^8}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(1-c^2 x^2\right)^{3/2} \left(6 c^4 d^2+16 c^2 d e+9 e^2\right)}{72 c^8}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(6 c^4 d^2+8 c^2 d e+3 e^2\right)}{24 c^8}",1,"(6*a*d^2*x^4 + 8*a*d*e*x^6 + 3*a*e^2*x^8 - (b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(144*e^2 + 8*c^2*e*(56*d + 9*e*x^2) + c^4*(420*d^2 + 224*d*e*x^2 + 54*e^2*x^4) + 3*c^6*(70*d^2*x^2 + 56*d*e*x^4 + 15*e^2*x^6)))/(105*c^8) + b*x^4*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4)*ArcSech[c*x])/24","A",1
107,1,139,230,0.3240032,"\int x \left(d+e x^2\right)^2 \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[x*(d + e*x^2)^2*(a + b*ArcSech[c*x]),x]","\frac{1}{6} a x^2 \left(3 d^2+3 d e x^2+e^2 x^4\right)-\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(3 c^4 \left(15 d^2+5 d e x^2+e^2 x^4\right)+2 c^2 e \left(15 d+2 e x^2\right)+8 e^2\right)}{90 c^6}+\frac{1}{6} b x^2 \text{sech}^{-1}(c x) \left(3 d^2+3 d e x^2+e^2 x^4\right)","\frac{\left(d+e x^2\right)^3 \left(a+b \text{sech}^{-1}(c x)\right)}{6 e}-\frac{b d^3 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{6 e}+\frac{b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(1-c^2 x^2\right)^{3/2} \left(3 c^2 d+2 e\right)}{18 c^6}-\frac{b e^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(1-c^2 x^2\right)^{5/2}}{30 c^6}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(3 c^4 d^2+3 c^2 d e+e^2\right)}{6 c^6}",1,"(a*x^2*(3*d^2 + 3*d*e*x^2 + e^2*x^4))/6 - (b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(8*e^2 + 2*c^2*e*(15*d + 2*e*x^2) + 3*c^4*(15*d^2 + 5*d*e*x^2 + e^2*x^4)))/(90*c^6) + (b*x^2*(3*d^2 + 3*d*e*x^2 + e^2*x^4)*ArcSech[c*x])/6","A",1
108,1,176,370,0.4438009,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \text{sech}^{-1}(c x)\right)}{x} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcSech[c*x]))/x,x]","a d^2 \log (x)+a d e x^2+\frac{1}{4} a e^2 x^4-\frac{b d e \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{c^2}-\frac{b e^2 \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(c^2 x^2+2\right)}{12 c^4}+\frac{1}{2} b d^2 \text{Li}_2\left(-e^{-2 \text{sech}^{-1}(c x)}\right)-\frac{1}{2} b d^2 \text{sech}^{-1}(c x) \left(\text{sech}^{-1}(c x)+2 \log \left(e^{-2 \text{sech}^{-1}(c x)}+1\right)\right)+b d e x^2 \text{sech}^{-1}(c x)+\frac{1}{4} b e^2 x^4 \text{sech}^{-1}(c x)","-d^2 \log \left(\frac{1}{x}\right) \left(a+b \text{sech}^{-1}(c x)\right)+d e x^2 \left(a+b \text{sech}^{-1}(c x)\right)+\frac{1}{4} e^2 x^4 \left(a+b \text{sech}^{-1}(c x)\right)+\frac{i b d^2 \sqrt{1-\frac{1}{c^2 x^2}} \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)}{2 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{i b d^2 \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{b d^2 \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left(1-e^{2 i \csc ^{-1}(c x)}\right)}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{b d^2 \sqrt{1-\frac{1}{c^2 x^2}} \log \left(\frac{1}{x}\right) \csc ^{-1}(c x)}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{b e x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1} \left(6 c^2 d+e\right)}{6 c^3}-\frac{b e^2 x^3 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}{12 c}",1,"a*d*e*x^2 + (a*e^2*x^4)/4 - (b*d*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/c^2 - (b*e^2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(2 + c^2*x^2))/(12*c^4) + b*d*e*x^2*ArcSech[c*x] + (b*e^2*x^4*ArcSech[c*x])/4 - (b*d^2*ArcSech[c*x]*(ArcSech[c*x] + 2*Log[1 + E^(-2*ArcSech[c*x])]))/2 + a*d^2*Log[x] + (b*d^2*PolyLog[2, -E^(-2*ArcSech[c*x])])/2","A",0
109,1,212,373,0.8389001,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \text{sech}^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcSech[c*x]))/x^3,x]","\frac{1}{4} \left(-\frac{2 a d^2}{x^2}+8 a d e \log (x)+2 a e^2 x^2-\frac{b d^2 \sqrt{\frac{1-c x}{c x+1}} \left(-c^2 x^2+c^2 x^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)+1\right)}{x^2 (c x-1)}-\frac{2 b e^2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{c^2}-\frac{2 b d^2 \text{sech}^{-1}(c x)}{x^2}+4 b d e \text{Li}_2\left(-e^{-2 \text{sech}^{-1}(c x)}\right)-4 b d e \text{sech}^{-1}(c x) \left(\text{sech}^{-1}(c x)+2 \log \left(e^{-2 \text{sech}^{-1}(c x)}+1\right)\right)+2 b e^2 x^2 \text{sech}^{-1}(c x)\right)","-\frac{d^2 \left(a+b \text{sech}^{-1}(c x)\right)}{2 x^2}-2 d e \log \left(\frac{1}{x}\right) \left(a+b \text{sech}^{-1}(c x)\right)+\frac{1}{2} e^2 x^2 \left(a+b \text{sech}^{-1}(c x)\right)+\frac{1}{4} b c^2 d^2 \text{sech}^{-1}(c x)+\frac{i b d e \sqrt{1-\frac{1}{c^2 x^2}} \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{i b d e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x)^2}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{2 b d e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left(1-e^{2 i \csc ^{-1}(c x)}\right)}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{2 b d e \sqrt{1-\frac{1}{c^2 x^2}} \log \left(\frac{1}{x}\right) \csc ^{-1}(c x)}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{b c d^2 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}{4 x}-\frac{b e^2 x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}{2 c}",1,"((-2*a*d^2)/x^2 + 2*a*e^2*x^2 - (2*b*e^2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/c^2 - (2*b*d^2*ArcSech[c*x])/x^2 + 2*b*e^2*x^2*ArcSech[c*x] - (b*d^2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 - c^2*x^2 + c^2*x^2*Sqrt[1 - c^2*x^2]*ArcTanh[Sqrt[1 - c^2*x^2]]))/(x^2*(-1 + c*x)) - 4*b*d*e*ArcSech[c*x]*(ArcSech[c*x] + 2*Log[1 + E^(-2*ArcSech[c*x])]) + 8*a*d*e*Log[x] + 4*b*d*e*PolyLog[2, -E^(-2*ArcSech[c*x])])/4","A",0
110,1,921,519,1.7549231,"\int \frac{x^2 \left(a+b \text{sech}^{-1}(c x)\right)}{d+e x^2} \, dx","Integrate[(x^2*(a + b*ArcSech[c*x]))/(d + e*x^2),x]","\frac{4 a c \sqrt{e} x-4 a c \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)+b \left(4 \sqrt{e} \left(c x \text{sech}^{-1}(c x)-2 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \text{sech}^{-1}(c x)\right)\right)\right)-2 i c \sqrt{d} \left(-4 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tanh ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \tanh \left(\frac{1}{2} \text{sech}^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)+\text{sech}^{-1}(c x) \log \left(1+e^{-2 \text{sech}^{-1}(c x)}\right)-\text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+2 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-\text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-2 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+\text{Li}_2\left(\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)+\text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)\right)+2 i c \sqrt{d} \left(-4 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tanh ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \tanh \left(\frac{1}{2} \text{sech}^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)+\text{sech}^{-1}(c x) \log \left(1+e^{-2 \text{sech}^{-1}(c x)}\right)-\text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+2 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-\text{sech}^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)-2 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)+\text{Li}_2\left(-\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)+\text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)\right)\right)}{4 c e^{3/2}}","\frac{\sqrt{-d} \left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 e^{3/2}}-\frac{\sqrt{-d} \left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}+1\right)}{2 e^{3/2}}+\frac{\sqrt{-d} \left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 e^{3/2}}-\frac{\sqrt{-d} \left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}+1\right)}{2 e^{3/2}}+\frac{x \left(a+b \text{sech}^{-1}(c x)\right)}{e}-\frac{b \sqrt{-d} \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 e^{3/2}}+\frac{b \sqrt{-d} \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 e^{3/2}}-\frac{b \sqrt{-d} \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 e^{3/2}}+\frac{b \sqrt{-d} \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 e^{3/2}}-\frac{b \tan ^{-1}\left(\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}\right)}{c e}",1,"(4*a*c*Sqrt[e]*x - 4*a*c*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + b*(4*Sqrt[e]*(c*x*ArcSech[c*x] - 2*ArcTan[Tanh[ArcSech[c*x]/2]]) - (2*I)*c*Sqrt[d]*((-4*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[((I*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] + ArcSech[c*x]*Log[1 + E^(-2*ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (2*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (2*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])]) + (2*I)*c*Sqrt[d]*((-4*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[(((-I)*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] + ArcSech[c*x]*Log[1 + E^(-2*ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (2*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - ArcSech[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (2*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])))/(4*c*e^(3/2))","C",0
111,1,860,459,0.4064171,"\int \frac{x \left(a+b \text{sech}^{-1}(c x)\right)}{d+e x^2} \, dx","Integrate[(x*(a + b*ArcSech[c*x]))/(d + e*x^2),x]","\frac{4 i b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tanh ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \tanh \left(\frac{1}{2} \text{sech}^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)+4 i b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tanh ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \tanh \left(\frac{1}{2} \text{sech}^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)-2 b \text{sech}^{-1}(c x) \log \left(1+e^{-2 \text{sech}^{-1}(c x)}\right)+b \text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-2 i b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+b \text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-2 i b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+b \text{sech}^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)+2 i b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)+b \text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+2 i b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+a \log \left(e x^2+d\right)+b \text{Li}_2\left(-e^{-2 \text{sech}^{-1}(c x)}\right)-b \text{Li}_2\left(-\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)-b \text{Li}_2\left(\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)-b \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)-b \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)}{2 e}","\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 e}+\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}+1\right)}{2 e}+\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 e}+\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}+1\right)}{2 e}-\frac{\left(a+b \text{sech}^{-1}(c x)\right)^2}{b e}-\frac{\log \left(e^{-2 \text{sech}^{-1}(c x)}+1\right) \left(a+b \text{sech}^{-1}(c x)\right)}{e}+\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 e}+\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 e}+\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 e}+\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 e}+\frac{b \text{Li}_2\left(-e^{-2 \text{sech}^{-1}(c x)}\right)}{2 e}",1,"((4*I)*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[(((-I)*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] + (4*I)*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[((I*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] - 2*b*ArcSech[c*x]*Log[1 + E^(-2*ArcSech[c*x])] + b*ArcSech[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (2*I)*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + b*ArcSech[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (2*I)*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + b*ArcSech[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (2*I)*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + b*ArcSech[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (2*I)*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + a*Log[d + e*x^2] + b*PolyLog[2, -E^(-2*ArcSech[c*x])] - b*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - b*PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - b*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - b*PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/(2*e)","C",0
112,1,849,469,0.3914512,"\int \frac{a+b \text{sech}^{-1}(c x)}{d+e x^2} \, dx","Integrate[(a + b*ArcSech[c*x])/(d + e*x^2),x]","\frac{2 a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)-4 b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tanh ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \tanh \left(\frac{1}{2} \text{sech}^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)+4 b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tanh ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \tanh \left(\frac{1}{2} \text{sech}^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)-i b \text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-2 b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+i b \text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+2 b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+i b \text{sech}^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)-2 b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)-i b \text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+2 b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-i b \text{Li}_2\left(-\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)+i b \text{Li}_2\left(\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)+i b \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)-i b \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)}{2 \sqrt{d} \sqrt{e}}","\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}+1\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}+1\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 \sqrt{-d} \sqrt{e}}",1,"(2*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]] - 4*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[(((-I)*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] + 4*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[((I*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] - I*b*ArcSech[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - 2*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + I*b*ArcSech[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + 2*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + I*b*ArcSech[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - 2*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - I*b*ArcSech[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + 2*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - I*b*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + I*b*PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + I*b*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - I*b*PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/(2*Sqrt[d]*Sqrt[e])","C",0
113,1,386,417,0.9346568,"\int \frac{a+b \text{sech}^{-1}(c x)}{x \left(d+e x^2\right)} \, dx","Integrate[(a + b*ArcSech[c*x])/(x*(d + e*x^2)),x]","\frac{-2 a \log \left(d+e x^2\right)+4 a \log (x)+b \left(\text{Li}_2\left(-\frac{\left(d c^2+2 e-2 \sqrt{e \left(d c^2+e\right)}\right) e^{-2 \text{sech}^{-1}(c x)}}{c^2 d}\right)+\text{Li}_2\left(-\frac{\left(d c^2+2 \left(e+\sqrt{e \left(d c^2+e\right)}\right)\right) e^{-2 \text{sech}^{-1}(c x)}}{c^2 d}\right)-2 \left(\text{sech}^{-1}(c x) \left(\log \left(\frac{e^{-2 \text{sech}^{-1}(c x)} \left(-2 \sqrt{e \left(c^2 d+e\right)}+c^2 d \left(e^{2 \text{sech}^{-1}(c x)}+1\right)+2 e\right)}{c^2 d}\right)+\log \left(\frac{e^{-2 \text{sech}^{-1}(c x)} \left(2 \left(\sqrt{e \left(c^2 d+e\right)}+e\right)+c^2 d \left(e^{2 \text{sech}^{-1}(c x)}+1\right)\right)}{c^2 d}\right)\right)+i \sin ^{-1}\left(\sqrt{\frac{e}{c^2 d}+1}\right) \left(2 \tanh ^{-1}\left(\frac{e \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{\sqrt{e \left(c^2 d+e\right)}}\right)-\log \left(\frac{e^{-2 \text{sech}^{-1}(c x)} \left(-2 \sqrt{e \left(c^2 d+e\right)}+c^2 d \left(e^{2 \text{sech}^{-1}(c x)}+1\right)+2 e\right)}{c^2 d}\right)+\log \left(\frac{e^{-2 \text{sech}^{-1}(c x)} \left(2 \left(\sqrt{e \left(c^2 d+e\right)}+e\right)+c^2 d \left(e^{2 \text{sech}^{-1}(c x)}+1\right)\right)}{c^2 d}\right)\right)+\text{sech}^{-1}(c x)^2\right)\right)}{4 d}","-\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 d}-\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}+1\right)}{2 d}-\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 d}-\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}+1\right)}{2 d}+\frac{\left(a+b \text{sech}^{-1}(c x)\right)^2}{2 b d}-\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 d}-\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 d}-\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 d}-\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 d}",1,"(4*a*Log[x] - 2*a*Log[d + e*x^2] + b*(-2*(ArcSech[c*x]^2 + I*ArcSin[Sqrt[1 + e/(c^2*d)]]*(2*ArcTanh[(e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/Sqrt[e*(c^2*d + e)]] - Log[(2*e - 2*Sqrt[e*(c^2*d + e)] + c^2*d*(1 + E^(2*ArcSech[c*x])))/(c^2*d*E^(2*ArcSech[c*x]))] + Log[(2*(e + Sqrt[e*(c^2*d + e)]) + c^2*d*(1 + E^(2*ArcSech[c*x])))/(c^2*d*E^(2*ArcSech[c*x]))]) + ArcSech[c*x]*(Log[(2*e - 2*Sqrt[e*(c^2*d + e)] + c^2*d*(1 + E^(2*ArcSech[c*x])))/(c^2*d*E^(2*ArcSech[c*x]))] + Log[(2*(e + Sqrt[e*(c^2*d + e)]) + c^2*d*(1 + E^(2*ArcSech[c*x])))/(c^2*d*E^(2*ArcSech[c*x]))])) + PolyLog[2, -((c^2*d + 2*e - 2*Sqrt[e*(c^2*d + e)])/(c^2*d*E^(2*ArcSech[c*x])))] + PolyLog[2, -((c^2*d + 2*(e + Sqrt[e*(c^2*d + e)]))/(c^2*d*E^(2*ArcSech[c*x])))]))/(4*d)","C",0
114,1,933,523,1.8617998,"\int \frac{a+b \text{sech}^{-1}(c x)}{x^2 \left(d+e x^2\right)} \, dx","Integrate[(a + b*ArcSech[c*x])/(x^2*(d + e*x^2)),x]","\frac{-4 \sqrt{e} x \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) a-4 \sqrt{d} a+b \left(4 \sqrt{d} \sqrt{\frac{1-c x}{c x+1}} (c x+1)-4 \sqrt{d} \text{sech}^{-1}(c x)-2 i \sqrt{e} x \left(-4 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tanh ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \tanh \left(\frac{1}{2} \text{sech}^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)+\text{sech}^{-1}(c x) \log \left(1+e^{-2 \text{sech}^{-1}(c x)}\right)-\text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+2 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-\text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-2 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+\text{Li}_2\left(\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)+\text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)\right)+2 i \sqrt{e} x \left(-4 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tanh ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \tanh \left(\frac{1}{2} \text{sech}^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)+\text{sech}^{-1}(c x) \log \left(1+e^{-2 \text{sech}^{-1}(c x)}\right)-\text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+2 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-\text{sech}^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)-2 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)+\text{Li}_2\left(-\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)+\text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)\right)\right)}{4 d^{3/2} x}","\frac{\sqrt{e} \left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 (-d)^{3/2}}-\frac{\sqrt{e} \left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}+1\right)}{2 (-d)^{3/2}}+\frac{\sqrt{e} \left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 (-d)^{3/2}}-\frac{\sqrt{e} \left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}+1\right)}{2 (-d)^{3/2}}-\frac{a}{d x}-\frac{b \sqrt{e} \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 (-d)^{3/2}}+\frac{b \sqrt{e} \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 (-d)^{3/2}}-\frac{b \sqrt{e} \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 (-d)^{3/2}}+\frac{b \sqrt{e} \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 (-d)^{3/2}}+\frac{b c \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}{d}-\frac{b \text{sech}^{-1}(c x)}{d x}",1,"(-4*a*Sqrt[d] - 4*a*Sqrt[e]*x*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + b*(4*Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) - 4*Sqrt[d]*ArcSech[c*x] - (2*I)*Sqrt[e]*x*((-4*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[((I*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] + ArcSech[c*x]*Log[1 + E^(-2*ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (2*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (2*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])]) + (2*I)*Sqrt[e]*x*((-4*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[(((-I)*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] + ArcSech[c*x]*Log[1 + E^(-2*ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (2*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - ArcSech[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (2*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])))/(4*d^(3/2)*x)","C",0
115,1,1278,631,5.6178242,"\int \frac{x^5 \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^5*(a + b*ArcSech[c*x]))/(d + e*x^2)^2,x]","-\frac{\frac{2 a d^2}{e x^2+d}+4 a \log \left(e x^2+d\right) d-2 a e x^2+b \left(-2 e \text{sech}^{-1}(c x) x^2+\frac{2 e \sqrt{\frac{1-c x}{c x+1}} x}{c}+\frac{d^{3/2} \text{sech}^{-1}(c x)}{\sqrt{d}-i \sqrt{e} x}+\frac{d^{3/2} \text{sech}^{-1}(c x)}{i \sqrt{e} x+\sqrt{d}}+16 i d \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tanh ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \tanh \left(\frac{1}{2} \text{sech}^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)+16 i d \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tanh ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \tanh \left(\frac{1}{2} \text{sech}^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)-8 d \text{sech}^{-1}(c x) \log \left(1+e^{-2 \text{sech}^{-1}(c x)}\right)+4 d \text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-8 i d \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+4 d \text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-8 i d \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+4 d \text{sech}^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)+8 i d \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)+4 d \text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+8 i d \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+2 d \log (x)-2 d \log \left(c \sqrt{\frac{1-c x}{c x+1}} x+\sqrt{\frac{1-c x}{c x+1}}+1\right)+\frac{d \sqrt{e} \log \left(\frac{2 i \sqrt{e} \left(\sqrt{d} \sqrt{\frac{1-c x}{c x+1}} (c x+1)+\frac{i d x c^2+\sqrt{d} \sqrt{e}}{\sqrt{d c^2+e}}\right)}{\sqrt{e} x+i \sqrt{d}}\right)}{\sqrt{d c^2+e}}+\frac{d \sqrt{e} \log \left(\frac{2 \sqrt{e} \left(i \sqrt{d} \sqrt{\frac{1-c x}{c x+1}} (c x+1)+\frac{d x c^2+i \sqrt{d} \sqrt{e}}{\sqrt{d c^2+e}}\right)}{\sqrt{e} x-i \sqrt{d}}\right)}{\sqrt{d c^2+e}}+4 d \text{Li}_2\left(-e^{-2 \text{sech}^{-1}(c x)}\right)-4 d \text{Li}_2\left(-\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)-4 d \text{Li}_2\left(\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)-4 d \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)-4 d \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)+\frac{2 e \sqrt{\frac{1-c x}{c x+1}}}{c^2}\right)}{4 e^3}","-\frac{d \left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{e^3}-\frac{d \left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}+1\right)}{e^3}-\frac{d \left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{e^3}-\frac{d \left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}+1\right)}{e^3}+\frac{2 d \left(a+b \text{sech}^{-1}(c x)\right)^2}{b e^3}+\frac{2 d \log \left(e^{-2 \text{sech}^{-1}(c x)}+1\right) \left(a+b \text{sech}^{-1}(c x)\right)}{e^3}+\frac{d \left(a+b \text{sech}^{-1}(c x)\right)}{2 e^2 \left(\frac{d}{x^2}+e\right)}+\frac{x^2 \left(a+b \text{sech}^{-1}(c x)\right)}{2 e^2}-\frac{b d \sqrt{\frac{1}{c^2 x^2}-1} \tanh ^{-1}\left(\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{\frac{1}{c^2 x^2}-1}}\right)}{2 e^{5/2} \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1} \sqrt{c^2 d+e}}-\frac{b d \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{e^3}-\frac{b d \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{e^3}-\frac{b d \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{e^3}-\frac{b d \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{e^3}-\frac{b d \text{Li}_2\left(-e^{-2 \text{sech}^{-1}(c x)}\right)}{e^3}-\frac{b x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}{2 c e^2}",1,"-1/4*(-2*a*e*x^2 + (2*a*d^2)/(d + e*x^2) + 4*a*d*Log[d + e*x^2] + b*((2*e*Sqrt[(1 - c*x)/(1 + c*x)])/c^2 + (2*e*x*Sqrt[(1 - c*x)/(1 + c*x)])/c - 2*e*x^2*ArcSech[c*x] + (d^(3/2)*ArcSech[c*x])/(Sqrt[d] - I*Sqrt[e]*x) + (d^(3/2)*ArcSech[c*x])/(Sqrt[d] + I*Sqrt[e]*x) + (16*I)*d*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[(((-I)*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] + (16*I)*d*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[((I*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] - 8*d*ArcSech[c*x]*Log[1 + E^(-2*ArcSech[c*x])] + 4*d*ArcSech[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (8*I)*d*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + 4*d*ArcSech[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (8*I)*d*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + 4*d*ArcSech[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (8*I)*d*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + 4*d*ArcSech[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (8*I)*d*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + 2*d*Log[x] - 2*d*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]] + (d*Sqrt[e]*Log[((2*I)*Sqrt[e]*(Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + (Sqrt[d]*Sqrt[e] + I*c^2*d*x)/Sqrt[c^2*d + e]))/(I*Sqrt[d] + Sqrt[e]*x)])/Sqrt[c^2*d + e] + (d*Sqrt[e]*Log[(2*Sqrt[e]*(I*Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + (I*Sqrt[d]*Sqrt[e] + c^2*d*x)/Sqrt[c^2*d + e]))/((-I)*Sqrt[d] + Sqrt[e]*x)])/Sqrt[c^2*d + e] + 4*d*PolyLog[2, -E^(-2*ArcSech[c*x])] - 4*d*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - 4*d*PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - 4*d*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - 4*d*PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])]))/e^3","C",0
116,1,1208,580,1.2901458,"\int \frac{x^3 \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^3*(a + b*ArcSech[c*x]))/(d + e*x^2)^2,x]","\frac{\frac{2 a d}{e x^2+d}+\frac{b \text{sech}^{-1}(c x) \sqrt{d}}{\sqrt{d}-i \sqrt{e} x}+\frac{b \text{sech}^{-1}(c x) \sqrt{d}}{i \sqrt{e} x+\sqrt{d}}+8 i b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tanh ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \tanh \left(\frac{1}{2} \text{sech}^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)+8 i b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tanh ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \tanh \left(\frac{1}{2} \text{sech}^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)-4 b \text{sech}^{-1}(c x) \log \left(1+e^{-2 \text{sech}^{-1}(c x)}\right)+2 b \text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-4 i b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+2 b \text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-4 i b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+2 b \text{sech}^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)+4 i b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)+2 b \text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+4 i b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+2 b \log (x)+2 a \log \left(e x^2+d\right)-2 b \log \left(c \sqrt{\frac{1-c x}{c x+1}} x+\sqrt{\frac{1-c x}{c x+1}}+1\right)+\frac{b \sqrt{e} \log \left(\frac{2 i \sqrt{e} \left(\sqrt{d} \sqrt{\frac{1-c x}{c x+1}} (c x+1)+\frac{i d x c^2+\sqrt{d} \sqrt{e}}{\sqrt{d c^2+e}}\right)}{\sqrt{e} x+i \sqrt{d}}\right)}{\sqrt{d c^2+e}}+\frac{b \sqrt{e} \log \left(\frac{2 \sqrt{e} \left(i \sqrt{d} \sqrt{\frac{1-c x}{c x+1}} (c x+1)+\frac{d x c^2+i \sqrt{d} \sqrt{e}}{\sqrt{d c^2+e}}\right)}{\sqrt{e} x-i \sqrt{d}}\right)}{\sqrt{d c^2+e}}+2 b \text{Li}_2\left(-e^{-2 \text{sech}^{-1}(c x)}\right)-2 b \text{Li}_2\left(-\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)-2 b \text{Li}_2\left(\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)-2 b \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)-2 b \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)}{4 e^2}","\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 e^2}+\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}+1\right)}{2 e^2}+\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 e^2}+\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}+1\right)}{2 e^2}-\frac{a+b \text{sech}^{-1}(c x)}{2 e \left(\frac{d}{x^2}+e\right)}-\frac{\left(a+b \text{sech}^{-1}(c x)\right)^2}{b e^2}-\frac{\log \left(e^{-2 \text{sech}^{-1}(c x)}+1\right) \left(a+b \text{sech}^{-1}(c x)\right)}{e^2}+\frac{b \sqrt{\frac{1}{c^2 x^2}-1} \tanh ^{-1}\left(\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{\frac{1}{c^2 x^2}-1}}\right)}{2 e^{3/2} \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1} \sqrt{c^2 d+e}}+\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 e^2}+\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 e^2}+\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 e^2}+\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 e^2}+\frac{b \text{Li}_2\left(-e^{-2 \text{sech}^{-1}(c x)}\right)}{2 e^2}",1,"((2*a*d)/(d + e*x^2) + (b*Sqrt[d]*ArcSech[c*x])/(Sqrt[d] - I*Sqrt[e]*x) + (b*Sqrt[d]*ArcSech[c*x])/(Sqrt[d] + I*Sqrt[e]*x) + (8*I)*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[(((-I)*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] + (8*I)*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[((I*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] - 4*b*ArcSech[c*x]*Log[1 + E^(-2*ArcSech[c*x])] + 2*b*ArcSech[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (4*I)*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + 2*b*ArcSech[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (4*I)*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + 2*b*ArcSech[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (4*I)*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + 2*b*ArcSech[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (4*I)*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + 2*b*Log[x] + 2*a*Log[d + e*x^2] - 2*b*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]] + (b*Sqrt[e]*Log[((2*I)*Sqrt[e]*(Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + (Sqrt[d]*Sqrt[e] + I*c^2*d*x)/Sqrt[c^2*d + e]))/(I*Sqrt[d] + Sqrt[e]*x)])/Sqrt[c^2*d + e] + (b*Sqrt[e]*Log[(2*Sqrt[e]*(I*Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + (I*Sqrt[d]*Sqrt[e] + c^2*d*x)/Sqrt[c^2*d + e]))/((-I)*Sqrt[d] + Sqrt[e]*x)])/Sqrt[c^2*d + e] + 2*b*PolyLog[2, -E^(-2*ArcSech[c*x])] - 2*b*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - 2*b*PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - 2*b*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - 2*b*PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/(4*e^2)","C",0
117,1,345,147,1.052856,"\int \frac{x \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x*(a + b*ArcSech[c*x]))/(d + e*x^2)^2,x]","-\frac{\frac{2 a}{d+e x^2}+\frac{b \sqrt{e} \log \left(\frac{4 \left(\frac{c^2 d^{3/2} \sqrt{e} x+i d e}{\sqrt{c^2 d+e} \left(\sqrt{d}+i \sqrt{e} x\right)}+\frac{d e \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{e x-i \sqrt{d} \sqrt{e}}\right)}{b}\right)}{d \sqrt{c^2 d+e}}+\frac{b \sqrt{e} \log \left(\frac{4 \left(\frac{d e+i c^2 d^{3/2} \sqrt{e} x}{\sqrt{c^2 d+e} \left(\sqrt{e} x+i \sqrt{d}\right)}+\frac{d e \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{e x+i \sqrt{d} \sqrt{e}}\right)}{b}\right)}{d \sqrt{c^2 d+e}}+\frac{2 b \text{sech}^{-1}(c x)}{d+e x^2}-\frac{2 b \log \left(c x \sqrt{\frac{1-c x}{c x+1}}+\sqrt{\frac{1-c x}{c x+1}}+1\right)}{d}+\frac{2 b \log (x)}{d}}{4 e}","-\frac{a+b \text{sech}^{-1}(c x)}{2 e \left(d+e x^2\right)}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{2 d e}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{\sqrt{c^2 d+e}}\right)}{2 d \sqrt{e} \sqrt{c^2 d+e}}",1,"-1/4*((2*a)/(d + e*x^2) + (2*b*ArcSech[c*x])/(d + e*x^2) + (2*b*Log[x])/d - (2*b*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]])/d + (b*Sqrt[e]*Log[(4*((I*d*e + c^2*d^(3/2)*Sqrt[e]*x)/(Sqrt[c^2*d + e]*(Sqrt[d] + I*Sqrt[e]*x)) + (d*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/((-I)*Sqrt[d]*Sqrt[e] + e*x)))/b])/(d*Sqrt[c^2*d + e]) + (b*Sqrt[e]*Log[(4*((d*e + I*c^2*d^(3/2)*Sqrt[e]*x)/(Sqrt[c^2*d + e]*(I*Sqrt[d] + Sqrt[e]*x)) + (d*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(I*Sqrt[d]*Sqrt[e] + e*x)))/b])/(d*Sqrt[c^2*d + e]))/e","C",1
118,0,0,542,42.089396,"\int \frac{a+b \text{sech}^{-1}(c x)}{x \left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcSech[c*x])/(x*(d + e*x^2)^2),x]","\int \frac{a+b \text{sech}^{-1}(c x)}{x \left(d+e x^2\right)^2} \, dx","-\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 d^2}-\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}+1\right)}{2 d^2}-\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 d^2}-\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}+1\right)}{2 d^2}-\frac{e \left(a+b \text{sech}^{-1}(c x)\right)}{2 d^2 \left(\frac{d}{x^2}+e\right)}+\frac{\left(a+b \text{sech}^{-1}(c x)\right)^2}{2 b d^2}-\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 d^2}-\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 d^2}-\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 d^2}-\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 d^2}+\frac{b \sqrt{e} \sqrt{\frac{1}{c^2 x^2}-1} \tanh ^{-1}\left(\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{\frac{1}{c^2 x^2}-1}}\right)}{2 d^2 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1} \sqrt{c^2 d+e}}",1,"Integrate[(a + b*ArcSech[c*x])/(x*(d + e*x^2)^2), x]","F",-1
119,1,1270,840,1.7716428,"\int \frac{x^4 \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^4*(a + b*ArcSech[c*x]))/(d + e*x^2)^2,x]","\frac{4 b \sqrt{e} \text{sech}^{-1}(c x) x+4 a \sqrt{e} x+\frac{2 a d \sqrt{e} x}{e x^2+d}+\frac{b d \text{sech}^{-1}(c x)}{\sqrt{e} x-i \sqrt{d}}+\frac{b d \text{sech}^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}-6 a \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)-\frac{8 b \sqrt{e} \tan ^{-1}\left(\tanh \left(\frac{1}{2} \text{sech}^{-1}(c x)\right)\right)}{c}+12 b \sqrt{d} \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tanh ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \tanh \left(\frac{1}{2} \text{sech}^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)-12 b \sqrt{d} \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tanh ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \tanh \left(\frac{1}{2} \text{sech}^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)+3 i b \sqrt{d} \text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+6 b \sqrt{d} \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-3 i b \sqrt{d} \text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-6 b \sqrt{d} \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-3 i b \sqrt{d} \text{sech}^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)+6 b \sqrt{d} \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)+3 i b \sqrt{d} \text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-6 b \sqrt{d} \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-\frac{i b \sqrt{d} \sqrt{e} \log \left(\frac{2 i \sqrt{e} \left(\sqrt{d} \sqrt{\frac{1-c x}{c x+1}} (c x+1)+\frac{i d x c^2+\sqrt{d} \sqrt{e}}{\sqrt{d c^2+e}}\right)}{\sqrt{e} x+i \sqrt{d}}\right)}{\sqrt{d c^2+e}}+\frac{i b \sqrt{d} \sqrt{e} \log \left(\frac{2 \sqrt{e} \left(i \sqrt{d} \sqrt{\frac{1-c x}{c x+1}} (c x+1)+\frac{d x c^2+i \sqrt{d} \sqrt{e}}{\sqrt{d c^2+e}}\right)}{\sqrt{e} x-i \sqrt{d}}\right)}{\sqrt{d c^2+e}}+3 i b \sqrt{d} \text{Li}_2\left(-\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)-3 i b \sqrt{d} \text{Li}_2\left(\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)-3 i b \sqrt{d} \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)+3 i b \sqrt{d} \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)}{4 e^{5/2}}","\frac{x \left(a+b \text{sech}^{-1}(c x)\right)}{e^2}+\frac{3 \sqrt{-d} \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right) \left(a+b \text{sech}^{-1}(c x)\right)}{4 e^{5/2}}-\frac{3 \sqrt{-d} \log \left(\frac{\sqrt{-d} e^{\text{sech}^{-1}(c x)} c}{\sqrt{e}-\sqrt{d c^2+e}}+1\right) \left(a+b \text{sech}^{-1}(c x)\right)}{4 e^{5/2}}+\frac{3 \sqrt{-d} \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right) \left(a+b \text{sech}^{-1}(c x)\right)}{4 e^{5/2}}-\frac{3 \sqrt{-d} \log \left(\frac{\sqrt{-d} e^{\text{sech}^{-1}(c x)} c}{\sqrt{e}+\sqrt{d c^2+e}}+1\right) \left(a+b \text{sech}^{-1}(c x)\right)}{4 e^{5/2}}-\frac{d \left(a+b \text{sech}^{-1}(c x)\right)}{4 e^2 \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}+\frac{d \left(a+b \text{sech}^{-1}(c x)\right)}{4 e^2 \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)}+\frac{b d \tan ^{-1}\left(\frac{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{\frac{1}{c x}-1}}\right)}{2 \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}} e^2}+\frac{b d \tan ^{-1}\left(\frac{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{\frac{1}{c x}-1}}\right)}{2 \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}} e^2}-\frac{b \tan ^{-1}\left(\sqrt{\frac{1}{c x}-1} \sqrt{1+\frac{1}{c x}}\right)}{c e^2}-\frac{3 b \sqrt{-d} \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{4 e^{5/2}}+\frac{3 b \sqrt{-d} \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{4 e^{5/2}}-\frac{3 b \sqrt{-d} \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{4 e^{5/2}}+\frac{3 b \sqrt{-d} \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{4 e^{5/2}}",1,"(4*a*Sqrt[e]*x + (2*a*d*Sqrt[e]*x)/(d + e*x^2) + 4*b*Sqrt[e]*x*ArcSech[c*x] + (b*d*ArcSech[c*x])/((-I)*Sqrt[d] + Sqrt[e]*x) + (b*d*ArcSech[c*x])/(I*Sqrt[d] + Sqrt[e]*x) - 6*a*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] - (8*b*Sqrt[e]*ArcTan[Tanh[ArcSech[c*x]/2]])/c + 12*b*Sqrt[d]*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[(((-I)*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] - 12*b*Sqrt[d]*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[((I*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] + (3*I)*b*Sqrt[d]*ArcSech[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + 6*b*Sqrt[d]*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (3*I)*b*Sqrt[d]*ArcSech[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - 6*b*Sqrt[d]*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (3*I)*b*Sqrt[d]*ArcSech[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + 6*b*Sqrt[d]*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (3*I)*b*Sqrt[d]*ArcSech[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - 6*b*Sqrt[d]*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (I*b*Sqrt[d]*Sqrt[e]*Log[((2*I)*Sqrt[e]*(Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + (Sqrt[d]*Sqrt[e] + I*c^2*d*x)/Sqrt[c^2*d + e]))/(I*Sqrt[d] + Sqrt[e]*x)])/Sqrt[c^2*d + e] + (I*b*Sqrt[d]*Sqrt[e]*Log[(2*Sqrt[e]*(I*Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + (I*Sqrt[d]*Sqrt[e] + c^2*d*x)/Sqrt[c^2*d + e]))/((-I)*Sqrt[d] + Sqrt[e]*x)])/Sqrt[c^2*d + e] + (3*I)*b*Sqrt[d]*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (3*I)*b*Sqrt[d]*PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (3*I)*b*Sqrt[d]*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (3*I)*b*Sqrt[d]*PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/(4*e^(5/2))","C",0
120,1,1226,786,1.7466989,"\int \frac{x^2 \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^2*(a + b*ArcSech[c*x]))/(d + e*x^2)^2,x]","\frac{-\frac{2 a \sqrt{e} x}{e x^2+d}+\frac{b \text{sech}^{-1}(c x)}{i \sqrt{d}-\sqrt{e} x}-\frac{b \text{sech}^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}+\frac{2 a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{d}}-\frac{4 b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tanh ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \tanh \left(\frac{1}{2} \text{sech}^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)}{\sqrt{d}}+\frac{4 b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tanh ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \tanh \left(\frac{1}{2} \text{sech}^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)}{\sqrt{d}}-\frac{i b \text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{d}}-\frac{2 b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{d}}+\frac{i b \text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{d}}+\frac{2 b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{d}}+\frac{i b \text{sech}^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{d}}-\frac{2 b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{d}}-\frac{i b \text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{d}}+\frac{2 b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{d}}+\frac{i b \sqrt{e} \log \left(\frac{2 i \sqrt{e} \left(\sqrt{d} \sqrt{\frac{1-c x}{c x+1}} (c x+1)+\frac{i d x c^2+\sqrt{d} \sqrt{e}}{\sqrt{d c^2+e}}\right)}{\sqrt{e} x+i \sqrt{d}}\right)}{\sqrt{d} \sqrt{d c^2+e}}-\frac{i b \sqrt{e} \log \left(\frac{2 \sqrt{e} \left(i \sqrt{d} \sqrt{\frac{1-c x}{c x+1}} (c x+1)+\frac{d x c^2+i \sqrt{d} \sqrt{e}}{\sqrt{d c^2+e}}\right)}{\sqrt{e} x-i \sqrt{d}}\right)}{\sqrt{d} \sqrt{d c^2+e}}-\frac{i b \text{Li}_2\left(-\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{d}}+\frac{i b \text{Li}_2\left(\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{d}}+\frac{i b \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{d}}-\frac{i b \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{d}}}{4 e^{3/2}}","\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}+1\right)}{4 \sqrt{-d} e^{3/2}}+\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}+1\right)}{4 \sqrt{-d} e^{3/2}}+\frac{a+b \text{sech}^{-1}(c x)}{4 e \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}-\frac{a+b \text{sech}^{-1}(c x)}{4 e \left(\sqrt{-d} \sqrt{e}+\frac{d}{x}\right)}-\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{4 \sqrt{-d} e^{3/2}}+\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{4 \sqrt{-d} e^{3/2}}+\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{b \tan ^{-1}\left(\frac{\sqrt{\frac{1}{c x}+1} \sqrt{c d-\sqrt{-d} \sqrt{e}}}{\sqrt{\frac{1}{c x}-1} \sqrt{c d+\sqrt{-d} \sqrt{e}}}\right)}{2 e \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}-\frac{b \tan ^{-1}\left(\frac{\sqrt{\frac{1}{c x}+1} \sqrt{c d+\sqrt{-d} \sqrt{e}}}{\sqrt{\frac{1}{c x}-1} \sqrt{c d-\sqrt{-d} \sqrt{e}}}\right)}{2 e \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}",1,"((-2*a*Sqrt[e]*x)/(d + e*x^2) + (b*ArcSech[c*x])/(I*Sqrt[d] - Sqrt[e]*x) - (b*ArcSech[c*x])/(I*Sqrt[d] + Sqrt[e]*x) + (2*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] - (4*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[(((-I)*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]])/Sqrt[d] + (4*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[((I*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]])/Sqrt[d] - (I*b*ArcSech[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/Sqrt[d] - (2*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/Sqrt[d] + (I*b*ArcSech[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/Sqrt[d] + (2*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/Sqrt[d] + (I*b*ArcSech[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/Sqrt[d] - (2*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/Sqrt[d] - (I*b*ArcSech[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/Sqrt[d] + (2*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/Sqrt[d] + (I*b*Sqrt[e]*Log[((2*I)*Sqrt[e]*(Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + (Sqrt[d]*Sqrt[e] + I*c^2*d*x)/Sqrt[c^2*d + e]))/(I*Sqrt[d] + Sqrt[e]*x)])/(Sqrt[d]*Sqrt[c^2*d + e]) - (I*b*Sqrt[e]*Log[(2*Sqrt[e]*(I*Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + (I*Sqrt[d]*Sqrt[e] + c^2*d*x)/Sqrt[c^2*d + e]))/((-I)*Sqrt[d] + Sqrt[e]*x)])/(Sqrt[d]*Sqrt[c^2*d + e]) - (I*b*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/Sqrt[d] + (I*b*PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/Sqrt[d] + (I*b*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/Sqrt[d] - (I*b*PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/Sqrt[d])/(4*e^(3/2))","C",0
121,1,1216,786,1.789741,"\int \frac{a+b \text{sech}^{-1}(c x)}{\left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcSech[c*x])/(d + e*x^2)^2,x]","\frac{\frac{2 a \sqrt{d} x}{e x^2+d}+\frac{b \sqrt{d} \text{sech}^{-1}(c x)}{e x-i \sqrt{d} \sqrt{e}}+\frac{b \sqrt{d} \text{sech}^{-1}(c x)}{e x+i \sqrt{d} \sqrt{e}}+\frac{2 a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{e}}-\frac{4 b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tanh ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \tanh \left(\frac{1}{2} \text{sech}^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)}{\sqrt{e}}+\frac{4 b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tanh ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \tanh \left(\frac{1}{2} \text{sech}^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)}{\sqrt{e}}-\frac{i b \text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{e}}-\frac{2 b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{e}}+\frac{i b \text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{e}}+\frac{2 b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{e}}+\frac{i b \text{sech}^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{e}}-\frac{2 b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{e}}-\frac{i b \text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{e}}+\frac{2 b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{e}}-\frac{i b \log \left(\frac{2 i \sqrt{e} \left(\sqrt{d} \sqrt{\frac{1-c x}{c x+1}} (c x+1)+\frac{i d x c^2+\sqrt{d} \sqrt{e}}{\sqrt{d c^2+e}}\right)}{\sqrt{e} x+i \sqrt{d}}\right)}{\sqrt{d c^2+e}}+\frac{i b \log \left(\frac{2 \sqrt{e} \left(i \sqrt{d} \sqrt{\frac{1-c x}{c x+1}} (c x+1)+\frac{d x c^2+i \sqrt{d} \sqrt{e}}{\sqrt{d c^2+e}}\right)}{\sqrt{e} x-i \sqrt{d}}\right)}{\sqrt{d c^2+e}}-\frac{i b \text{Li}_2\left(-\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{e}}+\frac{i b \text{Li}_2\left(\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{e}}+\frac{i b \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{e}}-\frac{i b \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{e}}}{4 d^{3/2}}","-\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}+1\right)}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}+1\right)}{4 (-d)^{3/2} \sqrt{e}}-\frac{a+b \text{sech}^{-1}(c x)}{4 d \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}+\frac{a+b \text{sech}^{-1}(c x)}{4 d \left(\sqrt{-d} \sqrt{e}+\frac{d}{x}\right)}+\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \tan ^{-1}\left(\frac{\sqrt{\frac{1}{c x}+1} \sqrt{c d-\sqrt{-d} \sqrt{e}}}{\sqrt{\frac{1}{c x}-1} \sqrt{c d+\sqrt{-d} \sqrt{e}}}\right)}{2 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}+\frac{b \tan ^{-1}\left(\frac{\sqrt{\frac{1}{c x}+1} \sqrt{c d+\sqrt{-d} \sqrt{e}}}{\sqrt{\frac{1}{c x}-1} \sqrt{c d-\sqrt{-d} \sqrt{e}}}\right)}{2 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}",1,"((2*a*Sqrt[d]*x)/(d + e*x^2) + (b*Sqrt[d]*ArcSech[c*x])/((-I)*Sqrt[d]*Sqrt[e] + e*x) + (b*Sqrt[d]*ArcSech[c*x])/(I*Sqrt[d]*Sqrt[e] + e*x) + (2*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (4*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[(((-I)*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]])/Sqrt[e] + (4*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[((I*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]])/Sqrt[e] - (I*b*ArcSech[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/Sqrt[e] - (2*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/Sqrt[e] + (I*b*ArcSech[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/Sqrt[e] + (2*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/Sqrt[e] + (I*b*ArcSech[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/Sqrt[e] - (2*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/Sqrt[e] - (I*b*ArcSech[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/Sqrt[e] + (2*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/Sqrt[e] - (I*b*Log[((2*I)*Sqrt[e]*(Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + (Sqrt[d]*Sqrt[e] + I*c^2*d*x)/Sqrt[c^2*d + e]))/(I*Sqrt[d] + Sqrt[e]*x)])/Sqrt[c^2*d + e] + (I*b*Log[(2*Sqrt[e]*(I*Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + (I*Sqrt[d]*Sqrt[e] + c^2*d*x)/Sqrt[c^2*d + e]))/((-I)*Sqrt[d] + Sqrt[e]*x)])/Sqrt[c^2*d + e] - (I*b*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/Sqrt[e] + (I*b*PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/Sqrt[e] + (I*b*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/Sqrt[e] - (I*b*PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/Sqrt[e])/(4*d^(3/2))","C",0
122,1,1305,844,1.3929777,"\int \frac{a+b \text{sech}^{-1}(c x)}{x^2 \left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcSech[c*x])/(x^2*(d + e*x^2)^2),x]","\frac{-6 \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) a-\frac{4 \sqrt{d} a}{x}-\frac{2 \sqrt{d} e x a}{e x^2+d}-\frac{4 b \sqrt{d} \text{sech}^{-1}(c x)}{x}-\frac{b \sqrt{d} e \text{sech}^{-1}(c x)}{e x-i \sqrt{d} \sqrt{e}}-\frac{b \sqrt{d} e \text{sech}^{-1}(c x)}{e x+i \sqrt{d} \sqrt{e}}+12 b \sqrt{e} \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tanh ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \tanh \left(\frac{1}{2} \text{sech}^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)-12 b \sqrt{e} \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tanh ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \tanh \left(\frac{1}{2} \text{sech}^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)+3 i b \sqrt{e} \text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+6 b \sqrt{e} \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-3 i b \sqrt{e} \text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-6 b \sqrt{e} \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-3 i b \sqrt{e} \text{sech}^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)+6 b \sqrt{e} \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)+3 i b \sqrt{e} \text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-6 b \sqrt{e} \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+\frac{i b e \log \left(\frac{2 i \sqrt{e} \left(\sqrt{d} \sqrt{\frac{1-c x}{c x+1}} (c x+1)+\frac{i d x c^2+\sqrt{d} \sqrt{e}}{\sqrt{d c^2+e}}\right)}{\sqrt{e} x+i \sqrt{d}}\right)}{\sqrt{d c^2+e}}-\frac{i b e \log \left(\frac{2 \sqrt{e} \left(i \sqrt{d} \sqrt{\frac{1-c x}{c x+1}} (c x+1)+\frac{d x c^2+i \sqrt{d} \sqrt{e}}{\sqrt{d c^2+e}}\right)}{\sqrt{e} x-i \sqrt{d}}\right)}{\sqrt{d c^2+e}}+3 i b \sqrt{e} \text{Li}_2\left(-\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)-3 i b \sqrt{e} \text{Li}_2\left(\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)-3 i b \sqrt{e} \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)+3 i b \sqrt{e} \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)+4 b c \sqrt{d} \sqrt{\frac{1-c x}{c x+1}}+\frac{4 b \sqrt{d} \sqrt{\frac{1-c x}{c x+1}}}{x}}{4 d^{5/2}}","-\frac{a}{d^2 x}-\frac{b \text{sech}^{-1}(c x)}{d^2 x}+\frac{e \left(a+b \text{sech}^{-1}(c x)\right)}{4 d^2 \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}-\frac{e \left(a+b \text{sech}^{-1}(c x)\right)}{4 d^2 \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)}-\frac{b e \tan ^{-1}\left(\frac{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{\frac{1}{c x}-1}}\right)}{2 d^2 \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}-\frac{b e \tan ^{-1}\left(\frac{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{\frac{1}{c x}-1}}\right)}{2 d^2 \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}-\frac{3 \sqrt{e} \left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{\sqrt{-d} e^{\text{sech}^{-1}(c x)} c}{\sqrt{e}-\sqrt{d c^2+e}}+1\right)}{4 (-d)^{5/2}}-\frac{3 \sqrt{e} \left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{\sqrt{-d} e^{\text{sech}^{-1}(c x)} c}{\sqrt{e}+\sqrt{d c^2+e}}+1\right)}{4 (-d)^{5/2}}+\frac{3 b \sqrt{e} \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{4 (-d)^{5/2}}-\frac{3 b \sqrt{e} \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{4 (-d)^{5/2}}+\frac{3 b \sqrt{e} \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{4 (-d)^{5/2}}-\frac{3 b \sqrt{e} \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{4 (-d)^{5/2}}+\frac{b c \sqrt{\frac{1}{c x}-1} \sqrt{1+\frac{1}{c x}}}{d^2}",1,"((-4*a*Sqrt[d])/x + 4*b*c*Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)] + (4*b*Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)])/x - (2*a*Sqrt[d]*e*x)/(d + e*x^2) - (4*b*Sqrt[d]*ArcSech[c*x])/x - (b*Sqrt[d]*e*ArcSech[c*x])/((-I)*Sqrt[d]*Sqrt[e] + e*x) - (b*Sqrt[d]*e*ArcSech[c*x])/(I*Sqrt[d]*Sqrt[e] + e*x) - 6*a*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + 12*b*Sqrt[e]*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[(((-I)*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] - 12*b*Sqrt[e]*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[((I*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] + (3*I)*b*Sqrt[e]*ArcSech[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + 6*b*Sqrt[e]*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (3*I)*b*Sqrt[e]*ArcSech[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - 6*b*Sqrt[e]*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (3*I)*b*Sqrt[e]*ArcSech[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + 6*b*Sqrt[e]*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (3*I)*b*Sqrt[e]*ArcSech[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - 6*b*Sqrt[e]*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (I*b*e*Log[((2*I)*Sqrt[e]*(Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + (Sqrt[d]*Sqrt[e] + I*c^2*d*x)/Sqrt[c^2*d + e]))/(I*Sqrt[d] + Sqrt[e]*x)])/Sqrt[c^2*d + e] - (I*b*e*Log[(2*Sqrt[e]*(I*Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + (I*Sqrt[d]*Sqrt[e] + c^2*d*x)/Sqrt[c^2*d + e]))/((-I)*Sqrt[d] + Sqrt[e]*x)])/Sqrt[c^2*d + e] + (3*I)*b*Sqrt[e]*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (3*I)*b*Sqrt[e]*PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (3*I)*b*Sqrt[e]*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (3*I)*b*Sqrt[e]*PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/(4*d^(5/2))","C",0
123,1,2000,778,7.7397574,"\int \frac{x^5 \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^5*(a + b*ArcSech[c*x]))/(d + e*x^2)^3,x]","-\frac{a d^2}{4 e^3 \left(e x^2+d\right)^2}+\frac{a d}{e^3 \left(e x^2+d\right)}+\frac{a \log \left(e x^2+d\right)}{2 e^3}+b \left(-\frac{d \left(-\frac{i \sqrt{e} \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{\sqrt{d} \left(d c^2+e\right) \left(\sqrt{e} x-i \sqrt{d}\right)}-\frac{\text{sech}^{-1}(c x)}{\sqrt{e} \left(\sqrt{e} x-i \sqrt{d}\right)^2}+\frac{\log (x)}{d \sqrt{e}}-\frac{\log \left(c \sqrt{\frac{1-c x}{c x+1}} x+\sqrt{\frac{1-c x}{c x+1}}+1\right)}{d \sqrt{e}}+\frac{\left(2 d c^2+e\right) \log \left(-\frac{4 d \sqrt{e} \sqrt{d c^2+e} \left(-i \sqrt{d} x c^2+\sqrt{d c^2+e} x \sqrt{\frac{1-c x}{c x+1}} c+\sqrt{e}+\sqrt{d c^2+e} \sqrt{\frac{1-c x}{c x+1}}\right)}{\left(2 d c^2+e\right) \left(\sqrt{e} x-i \sqrt{d}\right)}\right)}{d \left(d c^2+e\right)^{3/2}}\right)}{16 e^{5/2}}-\frac{d \left(\frac{i \sqrt{e} \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{\sqrt{d} \left(d c^2+e\right) \left(\sqrt{e} x+i \sqrt{d}\right)}-\frac{\text{sech}^{-1}(c x)}{\sqrt{e} \left(\sqrt{e} x+i \sqrt{d}\right)^2}+\frac{\log (x)}{d \sqrt{e}}-\frac{\log \left(c \sqrt{\frac{1-c x}{c x+1}} x+\sqrt{\frac{1-c x}{c x+1}}+1\right)}{d \sqrt{e}}+\frac{\left(2 d c^2+e\right) \log \left(-\frac{4 d \sqrt{e} \sqrt{d c^2+e} \left(i \sqrt{d} x c^2+\sqrt{d c^2+e} x \sqrt{\frac{1-c x}{c x+1}} c+\sqrt{e}+\sqrt{d c^2+e} \sqrt{\frac{1-c x}{c x+1}}\right)}{\left(2 d c^2+e\right) \left(\sqrt{e} x+i \sqrt{d}\right)}\right)}{d \left(d c^2+e\right)^{3/2}}\right)}{16 e^{5/2}}-\frac{7 i \sqrt{d} \left(\frac{i \left(\frac{\log (x)}{\sqrt{e}}-\frac{\log \left(c \sqrt{\frac{1-c x}{c x+1}} x+\sqrt{\frac{1-c x}{c x+1}}+1\right)}{\sqrt{e}}+\frac{\log \left(\frac{2 i \sqrt{e} \left(\sqrt{d} \sqrt{\frac{1-c x}{c x+1}} (c x+1)+\frac{i d x c^2+\sqrt{d} \sqrt{e}}{\sqrt{d c^2+e}}\right)}{\sqrt{e} x+i \sqrt{d}}\right)}{\sqrt{d c^2+e}}\right)}{\sqrt{d}}-\frac{\text{sech}^{-1}(c x)}{e x+i \sqrt{d} \sqrt{e}}\right)}{16 e^{5/2}}+\frac{7 i \sqrt{d} \left(-\frac{\text{sech}^{-1}(c x)}{e x-i \sqrt{d} \sqrt{e}}-\frac{i \left(\frac{\log (x)}{\sqrt{e}}-\frac{\log \left(c \sqrt{\frac{1-c x}{c x+1}} x+\sqrt{\frac{1-c x}{c x+1}}+1\right)}{\sqrt{e}}+\frac{\log \left(\frac{2 \sqrt{e} \left(i \sqrt{d} \sqrt{\frac{1-c x}{c x+1}} (c x+1)+\frac{d x c^2+i \sqrt{d} \sqrt{e}}{\sqrt{d c^2+e}}\right)}{\sqrt{e} x-i \sqrt{d}}\right)}{\sqrt{d c^2+e}}\right)}{\sqrt{d}}\right)}{16 e^{5/2}}+\frac{\text{Li}_2\left(-e^{-2 \text{sech}^{-1}(c x)}\right)-2 \left(-4 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tanh ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \tanh \left(\frac{1}{2} \text{sech}^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)+\text{sech}^{-1}(c x) \log \left(1+e^{-2 \text{sech}^{-1}(c x)}\right)-\text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+2 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-\text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-2 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+\text{Li}_2\left(\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)+\text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)\right)}{4 e^3}-\frac{2 \left(-4 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tanh ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \tanh \left(\frac{1}{2} \text{sech}^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)+\text{sech}^{-1}(c x) \log \left(1+e^{-2 \text{sech}^{-1}(c x)}\right)-\text{sech}^{-1}(c x) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+2 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{-\text{sech}^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-\text{sech}^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)-2 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)+\text{Li}_2\left(-\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)+\text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-\text{sech}^{-1}(c x)}}{c \sqrt{d}}\right)\right)-\text{Li}_2\left(-e^{-2 \text{sech}^{-1}(c x)}\right)}{4 e^3}\right)","\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 e^3}+\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}+1\right)}{2 e^3}+\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 e^3}+\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}+1\right)}{2 e^3}-\frac{a+b \text{sech}^{-1}(c x)}{2 e^2 \left(\frac{d}{x^2}+e\right)}-\frac{a+b \text{sech}^{-1}(c x)}{4 e \left(\frac{d}{x^2}+e\right)^2}-\frac{\left(a+b \text{sech}^{-1}(c x)\right)^2}{b e^3}-\frac{\log \left(e^{-2 \text{sech}^{-1}(c x)}+1\right) \left(a+b \text{sech}^{-1}(c x)\right)}{e^3}+\frac{b \sqrt{\frac{1}{c^2 x^2}-1} \left(c^2 d+2 e\right) \tanh ^{-1}\left(\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{\frac{1}{c^2 x^2}-1}}\right)}{8 e^{5/2} \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1} \left(c^2 d+e\right)^{3/2}}+\frac{b \sqrt{\frac{1}{c^2 x^2}-1} \tanh ^{-1}\left(\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{\frac{1}{c^2 x^2}-1}}\right)}{2 e^{5/2} \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1} \sqrt{c^2 d+e}}+\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 e^3}+\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 e^3}+\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 e^3}+\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 e^3}+\frac{b d \left(c^2-\frac{1}{x^2}\right)}{8 c e^2 x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1} \left(c^2 d+e\right) \left(\frac{d}{x^2}+e\right)}+\frac{b \text{Li}_2\left(-e^{-2 \text{sech}^{-1}(c x)}\right)}{2 e^3}",1,"-1/4*(a*d^2)/(e^3*(d + e*x^2)^2) + (a*d)/(e^3*(d + e*x^2)) + (a*Log[d + e*x^2])/(2*e^3) + b*(-1/16*(d*(((-I)*Sqrt[e]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(Sqrt[d]*(c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - ArcSech[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) + Log[x]/(d*Sqrt[e]) - Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]/(d*Sqrt[e]) + ((2*c^2*d + e)*Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*(Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[c^2*d + e]*x*Sqrt[(1 - c*x)/(1 + c*x)]))/((2*c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/e^(5/2) - (d*((I*Sqrt[e]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(Sqrt[d]*(c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcSech[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) + Log[x]/(d*Sqrt[e]) - Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]/(d*Sqrt[e]) + ((2*c^2*d + e)*Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*(Sqrt[e] + I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[c^2*d + e]*x*Sqrt[(1 - c*x)/(1 + c*x)]))/((2*c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/(16*e^(5/2)) - (((7*I)/16)*Sqrt[d]*(-(ArcSech[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) + (I*(Log[x]/Sqrt[e] - Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]/Sqrt[e] + Log[((2*I)*Sqrt[e]*(Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + (Sqrt[d]*Sqrt[e] + I*c^2*d*x)/Sqrt[c^2*d + e]))/(I*Sqrt[d] + Sqrt[e]*x)]/Sqrt[c^2*d + e]))/Sqrt[d]))/e^(5/2) + (((7*I)/16)*Sqrt[d]*(-(ArcSech[c*x]/((-I)*Sqrt[d]*Sqrt[e] + e*x)) - (I*(Log[x]/Sqrt[e] - Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]/Sqrt[e] + Log[(2*Sqrt[e]*(I*Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + (I*Sqrt[d]*Sqrt[e] + c^2*d*x)/Sqrt[c^2*d + e]))/((-I)*Sqrt[d] + Sqrt[e]*x)]/Sqrt[c^2*d + e]))/Sqrt[d]))/e^(5/2) + (PolyLog[2, -E^(-2*ArcSech[c*x])] - 2*((-4*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[((I*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] + ArcSech[c*x]*Log[1 + E^(-2*ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (2*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (2*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])]))/(4*e^3) - (-PolyLog[2, -E^(-2*ArcSech[c*x])] + 2*((-4*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[(((-I)*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] + ArcSech[c*x]*Log[1 + E^(-2*ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (2*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - ArcSech[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (2*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])]))/(4*e^3))","C",0
124,1,486,173,1.5929529,"\int \frac{x^3 \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^3*(a + b*ArcSech[c*x]))/(d + e*x^2)^3,x]","-\frac{\frac{8 a}{d+e x^2}-\frac{4 a d}{\left(d+e x^2\right)^2}+\frac{b \sqrt{e} \left(c^2 d+2 e\right) \log \left(\frac{16 d e^{3/2} \sqrt{c^2 d+e} \left(c x \sqrt{\frac{1-c x}{c x+1}} \sqrt{c^2 d+e}+\sqrt{\frac{1-c x}{c x+1}} \sqrt{c^2 d+e}-i c^2 \sqrt{d} x+\sqrt{e}\right)}{b \left(c^2 d+2 e\right) \left(\sqrt{e} x-i \sqrt{d}\right)}\right)}{d \left(c^2 d+e\right)^{3/2}}+\frac{b \sqrt{e} \left(c^2 d+2 e\right) \log \left(\frac{16 d e^{3/2} \sqrt{c^2 d+e} \left(c x \sqrt{\frac{1-c x}{c x+1}} \sqrt{c^2 d+e}+\sqrt{\frac{1-c x}{c x+1}} \sqrt{c^2 d+e}+i c^2 \sqrt{d} x+\sqrt{e}\right)}{b \left(c^2 d+2 e\right) \left(\sqrt{e} x+i \sqrt{d}\right)}\right)}{d \left(c^2 d+e\right)^{3/2}}-\frac{2 e \sqrt{\frac{1-c x}{c x+1}} (b c x+b)}{\left(c^2 d+e\right) \left(d+e x^2\right)}+\frac{4 b \text{sech}^{-1}(c x) \left(d+2 e x^2\right)}{\left(d+e x^2\right)^2}-\frac{4 b \log \left(c x \sqrt{\frac{1-c x}{c x+1}}+\sqrt{\frac{1-c x}{c x+1}}+1\right)}{d}+\frac{4 b \log (x)}{d}}{16 e^2}","\frac{x^4 \left(a+b \text{sech}^{-1}(c x)\right)}{4 d \left(d+e x^2\right)^2}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(c^2 d+2 e\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{\sqrt{c^2 d+e}}\right)}{8 d e^{3/2} \left(c^2 d+e\right)^{3/2}}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{8 e \left(c^2 d+e\right) \left(d+e x^2\right)}",1,"-1/16*((-4*a*d)/(d + e*x^2)^2 + (8*a)/(d + e*x^2) - (2*e*Sqrt[(1 - c*x)/(1 + c*x)]*(b + b*c*x))/((c^2*d + e)*(d + e*x^2)) + (4*b*(d + 2*e*x^2)*ArcSech[c*x])/(d + e*x^2)^2 + (4*b*Log[x])/d - (4*b*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]])/d + (b*Sqrt[e]*(c^2*d + 2*e)*Log[(16*d*e^(3/2)*Sqrt[c^2*d + e]*(Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[c^2*d + e]*x*Sqrt[(1 - c*x)/(1 + c*x)]))/(b*(c^2*d + 2*e)*((-I)*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2)) + (b*Sqrt[e]*(c^2*d + 2*e)*Log[(16*d*e^(3/2)*Sqrt[c^2*d + e]*(Sqrt[e] + I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[c^2*d + e]*x*Sqrt[(1 - c*x)/(1 + c*x)]))/(b*(c^2*d + 2*e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2)))/e^2","C",1
125,1,486,217,1.0987146,"\int \frac{x \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x*(a + b*ArcSech[c*x]))/(d + e*x^2)^3,x]","\frac{1}{16} \left(-\frac{4 a}{e \left(d+e x^2\right)^2}-\frac{b \left(3 c^2 d+2 e\right) \log \left(\frac{16 d^2 \sqrt{e} \sqrt{c^2 d+e} \left(c x \sqrt{\frac{1-c x}{c x+1}} \sqrt{c^2 d+e}+\sqrt{\frac{1-c x}{c x+1}} \sqrt{c^2 d+e}-i c^2 \sqrt{d} x+\sqrt{e}\right)}{b \left(3 c^2 d+2 e\right) \left(\sqrt{e} x-i \sqrt{d}\right)}\right)}{d^2 \sqrt{e} \left(c^2 d+e\right)^{3/2}}-\frac{b \left(3 c^2 d+2 e\right) \log \left(\frac{16 d^2 \sqrt{e} \sqrt{c^2 d+e} \left(c x \sqrt{\frac{1-c x}{c x+1}} \sqrt{c^2 d+e}+\sqrt{\frac{1-c x}{c x+1}} \sqrt{c^2 d+e}+i c^2 \sqrt{d} x+\sqrt{e}\right)}{b \left(3 c^2 d+2 e\right) \left(\sqrt{e} x+i \sqrt{d}\right)}\right)}{d^2 \sqrt{e} \left(c^2 d+e\right)^{3/2}}-\frac{2 \sqrt{\frac{1-c x}{c x+1}} (b c x+b)}{d \left(c^2 d+e\right) \left(d+e x^2\right)}+\frac{4 b \log \left(c x \sqrt{\frac{1-c x}{c x+1}}+\sqrt{\frac{1-c x}{c x+1}}+1\right)}{d^2 e}-\frac{4 b \text{sech}^{-1}(c x)}{e \left(d+e x^2\right)^2}-\frac{4 b \log (x)}{d^2 e}\right)","-\frac{a+b \text{sech}^{-1}(c x)}{4 e \left(d+e x^2\right)^2}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{4 d^2 e}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(3 c^2 d+2 e\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{\sqrt{c^2 d+e}}\right)}{8 d^2 \sqrt{e} \left(c^2 d+e\right)^{3/2}}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{8 d \left(c^2 d+e\right) \left(d+e x^2\right)}",1,"((-4*a)/(e*(d + e*x^2)^2) - (2*Sqrt[(1 - c*x)/(1 + c*x)]*(b + b*c*x))/(d*(c^2*d + e)*(d + e*x^2)) - (4*b*ArcSech[c*x])/(e*(d + e*x^2)^2) - (4*b*Log[x])/(d^2*e) + (4*b*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]])/(d^2*e) - (b*(3*c^2*d + 2*e)*Log[(16*d^2*Sqrt[e]*Sqrt[c^2*d + e]*(Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[c^2*d + e]*x*Sqrt[(1 - c*x)/(1 + c*x)]))/(b*(3*c^2*d + 2*e)*((-I)*Sqrt[d] + Sqrt[e]*x))])/(d^2*Sqrt[e]*(c^2*d + e)^(3/2)) - (b*(3*c^2*d + 2*e)*Log[(16*d^2*Sqrt[e]*Sqrt[c^2*d + e]*(Sqrt[e] + I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[c^2*d + e]*x*Sqrt[(1 - c*x)/(1 + c*x)]))/(b*(3*c^2*d + 2*e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d^2*Sqrt[e]*(c^2*d + e)^(3/2)))/16","C",1
126,0,0,741,63.2240251,"\int \frac{a+b \text{sech}^{-1}(c x)}{x \left(d+e x^2\right)^3} \, dx","Integrate[(a + b*ArcSech[c*x])/(x*(d + e*x^2)^3),x]","\int \frac{a+b \text{sech}^{-1}(c x)}{x \left(d+e x^2\right)^3} \, dx","-\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 d^3}-\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}+1\right)}{2 d^3}-\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 d^3}-\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}+1\right)}{2 d^3}+\frac{e^2 \left(a+b \text{sech}^{-1}(c x)\right)}{4 d^3 \left(\frac{d}{x^2}+e\right)^2}-\frac{e \left(a+b \text{sech}^{-1}(c x)\right)}{d^3 \left(\frac{d}{x^2}+e\right)}+\frac{\left(a+b \text{sech}^{-1}(c x)\right)^2}{2 b d^3}-\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 d^3}-\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 d^3}-\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 d^3}-\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 d^3}-\frac{b \sqrt{e} \sqrt{\frac{1}{c^2 x^2}-1} \left(c^2 d+2 e\right) \tanh ^{-1}\left(\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{\frac{1}{c^2 x^2}-1}}\right)}{8 d^3 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1} \left(c^2 d+e\right)^{3/2}}+\frac{b \sqrt{e} \sqrt{\frac{1}{c^2 x^2}-1} \tanh ^{-1}\left(\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{\frac{1}{c^2 x^2}-1}}\right)}{d^3 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1} \sqrt{c^2 d+e}}-\frac{b e \left(c^2-\frac{1}{x^2}\right)}{8 c d^2 x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1} \left(c^2 d+e\right) \left(\frac{d}{x^2}+e\right)}",1,"Integrate[(a + b*ArcSech[c*x])/(x*(d + e*x^2)^3), x]","F",-1
127,1,2022,1272,6.2167541,"\int \frac{x^4 \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^4*(a + b*ArcSech[c*x]))/(d + e*x^2)^3,x]","\text{Result too large to show}","\frac{b \sqrt{-d} \sqrt{\frac{1}{c x}-1} \sqrt{1+\frac{1}{c x}} c}{16 e^{3/2} \left(d c^2+e\right) \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}+\frac{b \sqrt{-d} \sqrt{\frac{1}{c x}-1} \sqrt{1+\frac{1}{c x}} c}{16 e^{3/2} \left(d c^2+e\right) \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)}+\frac{3 \left(a+b \text{sech}^{-1}(c x)\right)}{16 e^2 \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}-\frac{3 \left(a+b \text{sech}^{-1}(c x)\right)}{16 e^2 \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)}+\frac{\sqrt{-d} \left(a+b \text{sech}^{-1}(c x)\right)}{16 e^{3/2} \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)^2}-\frac{\sqrt{-d} \left(a+b \text{sech}^{-1}(c x)\right)}{16 e^{3/2} \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)^2}-\frac{b d \tan ^{-1}\left(\frac{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{\frac{1}{c x}-1}}\right)}{8 \left(c d-\sqrt{-d} \sqrt{e}\right)^{3/2} \left(c d+\sqrt{-d} \sqrt{e}\right)^{3/2} e}-\frac{3 b \tan ^{-1}\left(\frac{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{\frac{1}{c x}-1}}\right)}{8 \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}} e^2}-\frac{b d \tan ^{-1}\left(\frac{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{\frac{1}{c x}-1}}\right)}{8 \left(c d-\sqrt{-d} \sqrt{e}\right)^{3/2} \left(c d+\sqrt{-d} \sqrt{e}\right)^{3/2} e}-\frac{3 b \tan ^{-1}\left(\frac{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{\frac{1}{c x}-1}}\right)}{8 \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}} e^2}+\frac{3 \left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{16 \sqrt{-d} e^{5/2}}-\frac{3 \left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{\sqrt{-d} e^{\text{sech}^{-1}(c x)} c}{\sqrt{e}-\sqrt{d c^2+e}}+1\right)}{16 \sqrt{-d} e^{5/2}}+\frac{3 \left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{16 \sqrt{-d} e^{5/2}}-\frac{3 \left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{\sqrt{-d} e^{\text{sech}^{-1}(c x)} c}{\sqrt{e}+\sqrt{d c^2+e}}+1\right)}{16 \sqrt{-d} e^{5/2}}-\frac{3 b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{16 \sqrt{-d} e^{5/2}}+\frac{3 b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{16 \sqrt{-d} e^{5/2}}-\frac{3 b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{16 \sqrt{-d} e^{5/2}}+\frac{3 b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{16 \sqrt{-d} e^{5/2}}",1,"(a*d*x)/(4*e^2*(d + e*x^2)^2) - (5*a*x)/(8*e^2*(d + e*x^2)) + (3*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*Sqrt[d]*e^(5/2)) + b*(((I/16)*Sqrt[d]*(((-I)*Sqrt[e]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(Sqrt[d]*(c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - ArcSech[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) + Log[x]/(d*Sqrt[e]) - Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]/(d*Sqrt[e]) + ((2*c^2*d + e)*Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*(Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[c^2*d + e]*x*Sqrt[(1 - c*x)/(1 + c*x)]))/((2*c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/e^2 - ((I/16)*Sqrt[d]*((I*Sqrt[e]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(Sqrt[d]*(c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcSech[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) + Log[x]/(d*Sqrt[e]) - Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]/(d*Sqrt[e]) + ((2*c^2*d + e)*Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*(Sqrt[e] + I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[c^2*d + e]*x*Sqrt[(1 - c*x)/(1 + c*x)]))/((2*c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/e^2 + (5*(-(ArcSech[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) + (I*(Log[x]/Sqrt[e] - Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]/Sqrt[e] + Log[((2*I)*Sqrt[e]*(Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + (Sqrt[d]*Sqrt[e] + I*c^2*d*x)/Sqrt[c^2*d + e]))/(I*Sqrt[d] + Sqrt[e]*x)]/Sqrt[c^2*d + e]))/Sqrt[d]))/(16*e^2) + (5*(-(ArcSech[c*x]/((-I)*Sqrt[d]*Sqrt[e] + e*x)) - (I*(Log[x]/Sqrt[e] - Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]/Sqrt[e] + Log[(2*Sqrt[e]*(I*Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + (I*Sqrt[d]*Sqrt[e] + c^2*d*x)/Sqrt[c^2*d + e]))/((-I)*Sqrt[d] + Sqrt[e]*x)]/Sqrt[c^2*d + e]))/Sqrt[d]))/(16*e^2) - (((3*I)/32)*(PolyLog[2, -E^(-2*ArcSech[c*x])] - 2*((-4*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[((I*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] + ArcSech[c*x]*Log[1 + E^(-2*ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (2*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (2*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])))/(Sqrt[d]*e^(5/2)) - (((3*I)/32)*(-PolyLog[2, -E^(-2*ArcSech[c*x])] + 2*((-4*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[(((-I)*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] + ArcSech[c*x]*Log[1 + E^(-2*ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (2*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - ArcSech[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (2*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])))/(Sqrt[d]*e^(5/2)))","C",0
128,1,2030,1276,6.1425282,"\int \frac{x^2 \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^2*(a + b*ArcSech[c*x]))/(d + e*x^2)^3,x]","\text{Result too large to show}","\frac{b \sqrt{\frac{1}{c x}-1} \sqrt{1+\frac{1}{c x}} c}{16 \sqrt{-d} \sqrt{e} \left(d c^2+e\right) \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}+\frac{b \sqrt{\frac{1}{c x}-1} \sqrt{1+\frac{1}{c x}} c}{16 \sqrt{-d} \sqrt{e} \left(d c^2+e\right) \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)}+\frac{a+b \text{sech}^{-1}(c x)}{16 d e \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}-\frac{a+b \text{sech}^{-1}(c x)}{16 d e \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)}+\frac{a+b \text{sech}^{-1}(c x)}{16 \sqrt{-d} \sqrt{e} \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)^2}-\frac{a+b \text{sech}^{-1}(c x)}{16 \sqrt{-d} \sqrt{e} \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)^2}-\frac{b \tan ^{-1}\left(\frac{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{\frac{1}{c x}-1}}\right)}{8 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}} e}+\frac{b \tan ^{-1}\left(\frac{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{\frac{1}{c x}-1}}\right)}{8 \left(c d-\sqrt{-d} \sqrt{e}\right)^{3/2} \left(c d+\sqrt{-d} \sqrt{e}\right)^{3/2}}-\frac{b \tan ^{-1}\left(\frac{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{\frac{1}{c x}-1}}\right)}{8 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}} e}+\frac{b \tan ^{-1}\left(\frac{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{\frac{1}{c x}-1}}\right)}{8 \left(c d-\sqrt{-d} \sqrt{e}\right)^{3/2} \left(c d+\sqrt{-d} \sqrt{e}\right)^{3/2}}-\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{16 (-d)^{3/2} e^{3/2}}+\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{\sqrt{-d} e^{\text{sech}^{-1}(c x)} c}{\sqrt{e}-\sqrt{d c^2+e}}+1\right)}{16 (-d)^{3/2} e^{3/2}}-\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{16 (-d)^{3/2} e^{3/2}}+\frac{\left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{\sqrt{-d} e^{\text{sech}^{-1}(c x)} c}{\sqrt{e}+\sqrt{d c^2+e}}+1\right)}{16 (-d)^{3/2} e^{3/2}}+\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{16 (-d)^{3/2} e^{3/2}}-\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{16 (-d)^{3/2} e^{3/2}}+\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{16 (-d)^{3/2} e^{3/2}}-\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{16 (-d)^{3/2} e^{3/2}}",1,"-1/4*(a*x)/(e*(d + e*x^2)^2) + (a*x)/(8*d*e*(d + e*x^2)) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(3/2)*e^(3/2)) + b*(((-1/16*I)*(((-I)*Sqrt[e]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(Sqrt[d]*(c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - ArcSech[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) + Log[x]/(d*Sqrt[e]) - Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]/(d*Sqrt[e]) + ((2*c^2*d + e)*Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*(Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[c^2*d + e]*x*Sqrt[(1 - c*x)/(1 + c*x)]))/((2*c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/(Sqrt[d]*e) + ((I/16)*((I*Sqrt[e]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(Sqrt[d]*(c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcSech[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) + Log[x]/(d*Sqrt[e]) - Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]/(d*Sqrt[e]) + ((2*c^2*d + e)*Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*(Sqrt[e] + I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[c^2*d + e]*x*Sqrt[(1 - c*x)/(1 + c*x)]))/((2*c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/(Sqrt[d]*e) - (-(ArcSech[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) + (I*(Log[x]/Sqrt[e] - Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]/Sqrt[e] + Log[((2*I)*Sqrt[e]*(Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + (Sqrt[d]*Sqrt[e] + I*c^2*d*x)/Sqrt[c^2*d + e]))/(I*Sqrt[d] + Sqrt[e]*x)]/Sqrt[c^2*d + e]))/Sqrt[d])/(16*d*e) - (-(ArcSech[c*x]/((-I)*Sqrt[d]*Sqrt[e] + e*x)) - (I*(Log[x]/Sqrt[e] - Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]/Sqrt[e] + Log[(2*Sqrt[e]*(I*Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + (I*Sqrt[d]*Sqrt[e] + c^2*d*x)/Sqrt[c^2*d + e]))/((-I)*Sqrt[d] + Sqrt[e]*x)]/Sqrt[c^2*d + e]))/Sqrt[d])/(16*d*e) - ((I/32)*(PolyLog[2, -E^(-2*ArcSech[c*x])] - 2*((-4*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[((I*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] + ArcSech[c*x]*Log[1 + E^(-2*ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (2*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (2*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])))/(d^(3/2)*e^(3/2)) - ((I/32)*(-PolyLog[2, -E^(-2*ArcSech[c*x])] + 2*((-4*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[(((-I)*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] + ArcSech[c*x]*Log[1 + E^(-2*ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (2*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - ArcSech[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (2*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])))/(d^(3/2)*e^(3/2)))","C",0
129,1,2015,1272,6.0726421,"\int \frac{a+b \text{sech}^{-1}(c x)}{\left(d+e x^2\right)^3} \, dx","Integrate[(a + b*ArcSech[c*x])/(d + e*x^2)^3,x]","\text{Result too large to show}","\frac{b \sqrt{e} \sqrt{\frac{1}{c x}-1} \sqrt{1+\frac{1}{c x}} c}{16 (-d)^{3/2} \left(d c^2+e\right) \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}+\frac{b \sqrt{e} \sqrt{\frac{1}{c x}-1} \sqrt{1+\frac{1}{c x}} c}{16 (-d)^{3/2} \left(d c^2+e\right) \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)}-\frac{5 \left(a+b \text{sech}^{-1}(c x)\right)}{16 d^2 \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}+\frac{5 \left(a+b \text{sech}^{-1}(c x)\right)}{16 d^2 \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)}+\frac{\sqrt{e} \left(a+b \text{sech}^{-1}(c x)\right)}{16 (-d)^{3/2} \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)^2}-\frac{\sqrt{e} \left(a+b \text{sech}^{-1}(c x)\right)}{16 (-d)^{3/2} \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)^2}-\frac{b e \tan ^{-1}\left(\frac{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{\frac{1}{c x}-1}}\right)}{8 d \left(c d-\sqrt{-d} \sqrt{e}\right)^{3/2} \left(c d+\sqrt{-d} \sqrt{e}\right)^{3/2}}+\frac{5 b \tan ^{-1}\left(\frac{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{\frac{1}{c x}-1}}\right)}{8 d^2 \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}-\frac{b e \tan ^{-1}\left(\frac{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{\frac{1}{c x}-1}}\right)}{8 d \left(c d-\sqrt{-d} \sqrt{e}\right)^{3/2} \left(c d+\sqrt{-d} \sqrt{e}\right)^{3/2}}+\frac{5 b \tan ^{-1}\left(\frac{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{\frac{1}{c x}-1}}\right)}{8 d^2 \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}+\frac{3 \left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{\sqrt{-d} e^{\text{sech}^{-1}(c x)} c}{\sqrt{e}-\sqrt{d c^2+e}}+1\right)}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 \left(a+b \text{sech}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \left(a+b \text{sech}^{-1}(c x)\right) \log \left(\frac{\sqrt{-d} e^{\text{sech}^{-1}(c x)} c}{\sqrt{e}+\sqrt{d c^2+e}}+1\right)}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{16 (-d)^{5/2} \sqrt{e}}",1,"(a*x)/(4*d*(d + e*x^2)^2) + (3*a*x)/(8*d^2*(d + e*x^2)) + (3*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*Sqrt[e]) + b*(((I/16)*(((-I)*Sqrt[e]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(Sqrt[d]*(c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - ArcSech[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) + Log[x]/(d*Sqrt[e]) - Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]/(d*Sqrt[e]) + ((2*c^2*d + e)*Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*(Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[c^2*d + e]*x*Sqrt[(1 - c*x)/(1 + c*x)]))/((2*c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/d^(3/2) - ((I/16)*((I*Sqrt[e]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(Sqrt[d]*(c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcSech[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) + Log[x]/(d*Sqrt[e]) - Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]/(d*Sqrt[e]) + ((2*c^2*d + e)*Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*(Sqrt[e] + I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[c^2*d + e]*x*Sqrt[(1 - c*x)/(1 + c*x)]))/((2*c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/d^(3/2) - (3*(-(ArcSech[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) + (I*(Log[x]/Sqrt[e] - Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]/Sqrt[e] + Log[((2*I)*Sqrt[e]*(Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + (Sqrt[d]*Sqrt[e] + I*c^2*d*x)/Sqrt[c^2*d + e]))/(I*Sqrt[d] + Sqrt[e]*x)]/Sqrt[c^2*d + e]))/Sqrt[d]))/(16*d^2) - (3*(-(ArcSech[c*x]/((-I)*Sqrt[d]*Sqrt[e] + e*x)) - (I*(Log[x]/Sqrt[e] - Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]/Sqrt[e] + Log[(2*Sqrt[e]*(I*Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + (I*Sqrt[d]*Sqrt[e] + c^2*d*x)/Sqrt[c^2*d + e]))/((-I)*Sqrt[d] + Sqrt[e]*x)]/Sqrt[c^2*d + e]))/Sqrt[d]))/(16*d^2) - (((3*I)/32)*(PolyLog[2, -E^(-2*ArcSech[c*x])] - 2*((-4*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[((I*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] + ArcSech[c*x]*Log[1 + E^(-2*ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (2*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (2*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])))/(d^(5/2)*Sqrt[e]) - (((3*I)/32)*(-PolyLog[2, -E^(-2*ArcSech[c*x])] + 2*((-4*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[(((-I)*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] + ArcSech[c*x]*Log[1 + E^(-2*ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (2*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - ArcSech[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (2*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])))/(d^(5/2)*Sqrt[e]))","C",0
130,1,409,447,3.1020141,"\int x^5 \sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[x^5*Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]),x]","\frac{\sqrt{d+e x^2} \left(16 a c^6 \left(8 d^3-4 d^2 e x^2+3 d e^2 x^4+15 e^3 x^6\right)+16 b c^6 \text{sech}^{-1}(c x) \left(8 d^3-4 d^2 e x^2+3 d e^2 x^4+15 e^3 x^6\right)-b e \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(c^4 \left(-41 d^2+22 d e x^2+40 e^2 x^4\right)+2 c^2 e \left(19 d+25 e x^2\right)+75 e^2\right)\right)}{1680 c^6 e^3}-\frac{b \sqrt{\frac{1-c x}{c x+1}} \sqrt{c^2 x^2-1} \left(128 c^9 d^{7/2} \sqrt{d+e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right)+\sqrt{c^2} \sqrt{e} \sqrt{c^2 d+e} \left(105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\right) \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2} \sqrt{c^2 d+e}}\right)\right)}{1680 c^9 e^3 (c x-1) \sqrt{d+e x^2}}","\frac{d^2 \left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right)}{3 e^3}-\frac{2 d \left(d+e x^2\right)^{5/2} \left(a+b \text{sech}^{-1}(c x)\right)}{5 e^3}+\frac{\left(d+e x^2\right)^{7/2} \left(a+b \text{sech}^{-1}(c x)\right)}{7 e^3}-\frac{8 b d^{7/2} \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{105 e^3}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(d+e x^2\right)^{5/2}}{42 c^2 e^2}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(29 c^2 d-25 e\right) \left(d+e x^2\right)^{3/2}}{840 c^4 e^2}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(23 c^4 d^2+12 c^2 d e-75 e^2\right) \sqrt{d+e x^2}}{1680 c^6 e^2}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\right) \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right)}{1680 c^7 e^{5/2}}",1,"(Sqrt[d + e*x^2]*(16*a*c^6*(8*d^3 - 4*d^2*e*x^2 + 3*d*e^2*x^4 + 15*e^3*x^6) - b*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(75*e^2 + 2*c^2*e*(19*d + 25*e*x^2) + c^4*(-41*d^2 + 22*d*e*x^2 + 40*e^2*x^4)) + 16*b*c^6*(8*d^3 - 4*d^2*e*x^2 + 3*d*e^2*x^4 + 15*e^3*x^6)*ArcSech[c*x]))/(1680*c^6*e^3) - (b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[-1 + c^2*x^2]*(Sqrt[c^2]*Sqrt[e]*Sqrt[c^2*d + e]*(105*c^6*d^3 - 35*c^4*d^2*e + 63*c^2*d*e^2 + 75*e^3)*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSinh[(c*Sqrt[e]*Sqrt[-1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d + e])] + 128*c^9*d^(7/2)*Sqrt[d + e*x^2]*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2]]))/(1680*c^9*e^3*(-1 + c*x)*Sqrt[d + e*x^2])","A",1
131,1,365,329,1.4945237,"\int x^3 \sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[x^3*Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]),x]","-\frac{\sqrt{d+e x^2} \left(8 a c^4 \left(2 d^2-d e x^2-3 e^2 x^4\right)+8 b c^4 \text{sech}^{-1}(c x) \left(2 d^2-d e x^2-3 e^2 x^4\right)+b e \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(c^2 \left(7 d+6 e x^2\right)+9 e\right)\right)}{120 c^4 e^2}-\frac{b \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^2 x^2} \left(16 c^7 d^{5/2} \sqrt{-d-e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{1-c^2 x^2}}{\sqrt{-d-e x^2}}\right)+\sqrt{-c^2} \sqrt{e} \sqrt{c^2 (-d)-e} \left(15 c^4 d^2-10 c^2 d e-9 e^2\right) \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sin ^{-1}\left(\frac{c \sqrt{e} \sqrt{1-c^2 x^2}}{\sqrt{-c^2} \sqrt{c^2 (-d)-e}}\right)\right)}{120 c^7 e^2 (c x-1) \sqrt{d+e x^2}}","-\frac{d \left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right)}{3 e^2}+\frac{\left(d+e x^2\right)^{5/2} \left(a+b \text{sech}^{-1}(c x)\right)}{5 e^2}+\frac{2 b d^{5/2} \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{15 e^2}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(d+e x^2\right)^{3/2}}{20 c^2 e}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(c^2 d+9 e\right) \sqrt{d+e x^2}}{120 c^4 e}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(15 c^4 d^2-10 c^2 d e-9 e^2\right) \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right)}{120 c^5 e^{3/2}}",1,"-1/120*(Sqrt[d + e*x^2]*(8*a*c^4*(2*d^2 - d*e*x^2 - 3*e^2*x^4) + b*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(9*e + c^2*(7*d + 6*e*x^2)) + 8*b*c^4*(2*d^2 - d*e*x^2 - 3*e^2*x^4)*ArcSech[c*x]))/(c^4*e^2) - (b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^2*x^2]*(Sqrt[-c^2]*Sqrt[-(c^2*d) - e]*Sqrt[e]*(15*c^4*d^2 - 10*c^2*d*e - 9*e^2)*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSin[(c*Sqrt[e]*Sqrt[1 - c^2*x^2])/(Sqrt[-c^2]*Sqrt[-(c^2*d) - e])] + 16*c^7*d^(5/2)*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 - c^2*x^2])/Sqrt[-d - e*x^2]]))/(120*c^7*e^2*(-1 + c*x)*Sqrt[d + e*x^2])","A",1
132,1,307,221,1.2784641,"\int x \sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[x*Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]),x]","\frac{\sqrt{d+e x^2} \left(2 a c^2 \left(d+e x^2\right)+2 b c^2 \text{sech}^{-1}(c x) \left(d+e x^2\right)-b e \sqrt{\frac{1-c x}{c x+1}} (c x+1)\right)}{6 c^2 e}+\frac{b \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^2 x^2} \left(\sqrt{-c^2} \sqrt{e} \sqrt{c^2 (-d)-e} \left(3 c^2 d+e\right) \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sin ^{-1}\left(\frac{c \sqrt{e} \sqrt{1-c^2 x^2}}{\sqrt{-c^2} \sqrt{c^2 (-d)-e}}\right)+2 c^5 d^{3/2} \sqrt{-d-e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{1-c^2 x^2}}{\sqrt{-d-e x^2}}\right)\right)}{6 c^5 e (c x-1) \sqrt{d+e x^2}}","\frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right)}{3 e}-\frac{b d^{3/2} \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{3 e}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{6 c^2}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(3 c^2 d+e\right) \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right)}{6 c^3 \sqrt{e}}",1,"(Sqrt[d + e*x^2]*(-(b*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)) + 2*a*c^2*(d + e*x^2) + 2*b*c^2*(d + e*x^2)*ArcSech[c*x]))/(6*c^2*e) + (b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^2*x^2]*(Sqrt[-c^2]*Sqrt[-(c^2*d) - e]*Sqrt[e]*(3*c^2*d + e)*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSin[(c*Sqrt[e]*Sqrt[1 - c^2*x^2])/(Sqrt[-c^2]*Sqrt[-(c^2*d) - e])] + 2*c^5*d^(3/2)*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 - c^2*x^2])/Sqrt[-d - e*x^2]]))/(6*c^5*e*(-1 + c*x)*Sqrt[d + e*x^2])","A",1
133,0,0,26,5.5249332,"\int \frac{\sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right)}{x} \, dx","Integrate[(Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]))/x,x]","\int \frac{\sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right)}{x} \, dx","\text{Int}\left(\frac{\sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right)}{x},x\right)",0,"Integrate[(Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]))/x, x]","A",-1
134,0,0,26,6.3412329,"\int \frac{\sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right)}{x^3} \, dx","Integrate[(Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]))/x^3,x]","\int \frac{\sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right)}{x^3} \, dx","\text{Int}\left(\frac{\sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right)}{x^3},x\right)",0,"Integrate[(Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]))/x^3, x]","A",-1
135,0,0,26,13.8974526,"\int x^2 \sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[x^2*Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]),x]","\int x^2 \sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right) \, dx","\text{Int}\left(x^2 \sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right),x\right)",0,"Integrate[x^2*Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]), x]","A",-1
136,0,0,23,3.1709234,"\int \sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]),x]","\int \sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right) \, dx","\text{Int}\left(\sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right),x\right)",0,"Integrate[Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]), x]","A",-1
137,0,0,26,1.9265705,"\int \frac{\sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right)}{x^2} \, dx","Integrate[(Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]))/x^2,x]","\int \frac{\sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right)}{x^2} \, dx","\text{Int}\left(\frac{\sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right)}{x^2},x\right)",0,"Integrate[(Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]))/x^2, x]","A",-1
138,1,576,312,4.2148381,"\int \frac{\sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right)}{x^4} \, dx","Integrate[(Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]))/x^4,x]","\frac{-\frac{3 a \left(d+e x^2\right)^2}{d x^3}+\frac{2 b \sqrt{\frac{1-c x}{c x+1}} \left(c^2 d+2 e\right) \left(d+e x^2\right)}{d x}-\frac{2 i b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(c \sqrt{d}-i \sqrt{e}\right)^2 \sqrt{\frac{c \left(\sqrt{d}-i \sqrt{e} x\right)}{(c x+1) \left(c \sqrt{d}-i \sqrt{e}\right)}} \sqrt{\frac{c \left(\sqrt{d}+i \sqrt{e} x\right)}{(c x+1) \left(c \sqrt{d}+i \sqrt{e}\right)}} \left(\sqrt{e} \left(-3 \sqrt{e}+2 i c \sqrt{d}\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{\left(d c^2+e\right) (1-c x)}{\left(\sqrt{d} c+i \sqrt{e}\right)^2 (c x+1)}}\right)|\frac{\left(\sqrt{d} c+i \sqrt{e}\right)^2}{\left(c \sqrt{d}-i \sqrt{e}\right)^2}\right)+\left(c^2 d+2 e\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{\left(d c^2+e\right) (1-c x)}{\left(\sqrt{d} c+i \sqrt{e}\right)^2 (c x+1)}}\right)|\frac{\left(\sqrt{d} c+i \sqrt{e}\right)^2}{\left(c \sqrt{d}-i \sqrt{e}\right)^2}\right)\right)}{c d \sqrt{-\frac{(c x-1) \left(c \sqrt{d}-i \sqrt{e}\right)}{(c x+1) \left(c \sqrt{d}+i \sqrt{e}\right)}}}+\frac{b c \sqrt{\frac{1-c x}{c x+1}} \left(d+e x^2\right)}{x^2}+\frac{b \sqrt{\frac{1-c x}{c x+1}} \left(d+e x^2\right)}{x^3}-\frac{3 b \text{sech}^{-1}(c x) \left(d+e x^2\right)^2}{d x^3}}{9 \sqrt{d+e x^2}}","-\frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right)}{3 d x^3}+\frac{2 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(c^2 d+2 e\right) \sqrt{d+e x^2}}{9 d x}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(c^2 d+e\right) \left(2 c^2 d+3 e\right) \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{9 c d \sqrt{d+e x^2}}+\frac{2 b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(c^2 d+2 e\right) \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{9 d \sqrt{\frac{e x^2}{d}+1}}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{9 x^3}",1,"((b*Sqrt[(1 - c*x)/(1 + c*x)]*(d + e*x^2))/x^3 + (b*c*Sqrt[(1 - c*x)/(1 + c*x)]*(d + e*x^2))/x^2 + (2*b*(c^2*d + 2*e)*Sqrt[(1 - c*x)/(1 + c*x)]*(d + e*x^2))/(d*x) - (3*a*(d + e*x^2)^2)/(d*x^3) - (3*b*(d + e*x^2)^2*ArcSech[c*x])/(d*x^3) - ((2*I)*b*(c*Sqrt[d] - I*Sqrt[e])^2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[(c*(Sqrt[d] - I*Sqrt[e]*x))/((c*Sqrt[d] - I*Sqrt[e])*(1 + c*x))]*Sqrt[(c*(Sqrt[d] + I*Sqrt[e]*x))/((c*Sqrt[d] + I*Sqrt[e])*(1 + c*x))]*((c^2*d + 2*e)*EllipticE[I*ArcSinh[Sqrt[((c^2*d + e)*(1 - c*x))/((c*Sqrt[d] + I*Sqrt[e])^2*(1 + c*x))]], (c*Sqrt[d] + I*Sqrt[e])^2/(c*Sqrt[d] - I*Sqrt[e])^2] + ((2*I)*c*Sqrt[d] - 3*Sqrt[e])*Sqrt[e]*EllipticF[I*ArcSinh[Sqrt[((c^2*d + e)*(1 - c*x))/((c*Sqrt[d] + I*Sqrt[e])^2*(1 + c*x))]], (c*Sqrt[d] + I*Sqrt[e])^2/(c*Sqrt[d] - I*Sqrt[e])^2]))/(c*d*Sqrt[-(((c*Sqrt[d] - I*Sqrt[e])*(-1 + c*x))/((c*Sqrt[d] + I*Sqrt[e])*(1 + c*x)))]))/(9*Sqrt[d + e*x^2])","C",1
139,1,641,446,6.3786656,"\int \frac{\sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right)}{x^6} \, dx","Integrate[(Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]))/x^6,x]","\frac{\frac{15 a \left(2 e x^2-3 d\right) \left(d+e x^2\right)^2}{x^5}+\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(d+e x^2\right) \left(d e x^2 \left(19 c^2 x^2+8\right)+3 d^2 \left(8 c^4 x^4+4 c^2 x^2+3\right)-31 e^2 x^4\right)}{x^5}+\frac{b \sqrt{\frac{1-c x}{c x+1}} \left(-\left(c^2 \left(24 c^4 d^2+19 c^2 d e-31 e^2\right) \left(d+e x^2\right)\right)-\frac{i (c x+1) \left(c \sqrt{d}-i \sqrt{e}\right)^2 \sqrt{\frac{c \left(\sqrt{d}-i \sqrt{e} x\right)}{(c x+1) \left(c \sqrt{d}-i \sqrt{e}\right)}} \sqrt{\frac{c \left(\sqrt{d}+i \sqrt{e} x\right)}{(c x+1) \left(c \sqrt{d}+i \sqrt{e}\right)}} \left(\left(24 c^4 d^2+19 c^2 d e-31 e^2\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{\left(d c^2+e\right) (1-c x)}{\left(\sqrt{d} c+i \sqrt{e}\right)^2 (c x+1)}}\right)|\frac{\left(\sqrt{d} c+i \sqrt{e}\right)^2}{\left(c \sqrt{d}-i \sqrt{e}\right)^2}\right)+2 \sqrt{e} \left(24 i c^3 d^{3/2}-36 c^2 d \sqrt{e}-29 i c \sqrt{d} e+30 e^{3/2}\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{\left(d c^2+e\right) (1-c x)}{\left(\sqrt{d} c+i \sqrt{e}\right)^2 (c x+1)}}\right)|\frac{\left(\sqrt{d} c+i \sqrt{e}\right)^2}{\left(c \sqrt{d}-i \sqrt{e}\right)^2}\right)\right)}{\sqrt{-\frac{(c x-1) \left(c \sqrt{d}-i \sqrt{e}\right)}{(c x+1) \left(c \sqrt{d}+i \sqrt{e}\right)}}}\right)}{c}+\frac{15 b \text{sech}^{-1}(c x) \left(2 e x^2-3 d\right) \left(d+e x^2\right)^2}{x^5}}{225 d^2 \sqrt{d+e x^2}}","\frac{2 e \left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right)}{15 d^2 x^3}-\frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right)}{5 d x^5}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(d+e x^2\right)^{3/2}}{25 d x^5}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(12 c^2 d-e\right) \sqrt{d+e x^2}}{225 d x^3}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(24 c^4 d^2+19 c^2 d e-31 e^2\right) \sqrt{d+e x^2}}{225 d^2 x}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(c^2 d+e\right) \left(24 c^4 d^2+7 c^2 d e-30 e^2\right) \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{225 c d^2 \sqrt{d+e x^2}}+\frac{b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(24 c^4 d^2+19 c^2 d e-31 e^2\right) \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{225 d^2 \sqrt{\frac{e x^2}{d}+1}}",1,"((15*a*(d + e*x^2)^2*(-3*d + 2*e*x^2))/x^5 + (b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(d + e*x^2)*(-31*e^2*x^4 + d*e*x^2*(8 + 19*c^2*x^2) + 3*d^2*(3 + 4*c^2*x^2 + 8*c^4*x^4)))/x^5 + (15*b*(d + e*x^2)^2*(-3*d + 2*e*x^2)*ArcSech[c*x])/x^5 + (b*Sqrt[(1 - c*x)/(1 + c*x)]*(-(c^2*(24*c^4*d^2 + 19*c^2*d*e - 31*e^2)*(d + e*x^2)) - (I*(c*Sqrt[d] - I*Sqrt[e])^2*(1 + c*x)*Sqrt[(c*(Sqrt[d] - I*Sqrt[e]*x))/((c*Sqrt[d] - I*Sqrt[e])*(1 + c*x))]*Sqrt[(c*(Sqrt[d] + I*Sqrt[e]*x))/((c*Sqrt[d] + I*Sqrt[e])*(1 + c*x))]*((24*c^4*d^2 + 19*c^2*d*e - 31*e^2)*EllipticE[I*ArcSinh[Sqrt[((c^2*d + e)*(1 - c*x))/((c*Sqrt[d] + I*Sqrt[e])^2*(1 + c*x))]], (c*Sqrt[d] + I*Sqrt[e])^2/(c*Sqrt[d] - I*Sqrt[e])^2] + 2*Sqrt[e]*((24*I)*c^3*d^(3/2) - 36*c^2*d*Sqrt[e] - (29*I)*c*Sqrt[d]*e + 30*e^(3/2))*EllipticF[I*ArcSinh[Sqrt[((c^2*d + e)*(1 - c*x))/((c*Sqrt[d] + I*Sqrt[e])^2*(1 + c*x))]], (c*Sqrt[d] + I*Sqrt[e])^2/(c*Sqrt[d] - I*Sqrt[e])^2]))/Sqrt[-(((c*Sqrt[d] - I*Sqrt[e])*(-1 + c*x))/((c*Sqrt[d] + I*Sqrt[e])*(1 + c*x)))]))/c)/(225*d^2*Sqrt[d + e*x^2])","C",1
140,1,382,418,2.9863245,"\int x^3 \left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[x^3*(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]),x]","\frac{b \sqrt{\frac{1-c x}{c x+1}} \sqrt{c^2 x^2-1} \left(32 c^9 d^{7/2} \sqrt{d+e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right)+\sqrt{c^2} \sqrt{e} \sqrt{c^2 d+e} \left(35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right) \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2} \sqrt{c^2 d+e}}\right)\right)}{560 c^9 e^2 (c x-1) \sqrt{d+e x^2}}-\frac{\sqrt{d+e x^2} \left(48 a c^6 \left(2 d-5 e x^2\right) \left(d+e x^2\right)^2+48 b c^6 \text{sech}^{-1}(c x) \left(2 d-5 e x^2\right) \left(d+e x^2\right)^2+b e \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(c^4 \left(57 d^2+106 d e x^2+40 e^2 x^4\right)+2 c^2 e \left(82 d+25 e x^2\right)+75 e^2\right)\right)}{1680 c^6 e^2}","-\frac{d \left(d+e x^2\right)^{5/2} \left(a+b \text{sech}^{-1}(c x)\right)}{5 e^2}+\frac{\left(d+e x^2\right)^{7/2} \left(a+b \text{sech}^{-1}(c x)\right)}{7 e^2}+\frac{2 b d^{7/2} \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{35 e^2}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(d+e x^2\right)^{5/2}}{42 c^2 e}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(13 c^2 d+25 e\right) \left(d+e x^2\right)^{3/2}}{840 c^4 e}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(3 c^4 d^2-38 c^2 d e-25 e^2\right) \sqrt{d+e x^2}}{560 c^6 e}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right) \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right)}{560 c^7 e^{3/2}}",1,"-1/1680*(Sqrt[d + e*x^2]*(48*a*c^6*(2*d - 5*e*x^2)*(d + e*x^2)^2 + b*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(75*e^2 + 2*c^2*e*(82*d + 25*e*x^2) + c^4*(57*d^2 + 106*d*e*x^2 + 40*e^2*x^4)) + 48*b*c^6*(2*d - 5*e*x^2)*(d + e*x^2)^2*ArcSech[c*x]))/(c^6*e^2) + (b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[-1 + c^2*x^2]*(Sqrt[c^2]*Sqrt[e]*Sqrt[c^2*d + e]*(35*c^6*d^3 - 35*c^4*d^2*e - 63*c^2*d*e^2 - 25*e^3)*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSinh[(c*Sqrt[e]*Sqrt[-1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d + e])] + 32*c^9*d^(7/2)*Sqrt[d + e*x^2]*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2]]))/(560*c^9*e^2*(-1 + c*x)*Sqrt[d + e*x^2])","A",1
141,1,342,297,1.5437833,"\int x \left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[x*(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]),x]","\frac{\sqrt{d+e x^2} \left(8 a c^4 \left(d+e x^2\right)^2+8 b c^4 \text{sech}^{-1}(c x) \left(d+e x^2\right)^2-b e \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(c^2 \left(9 d+2 e x^2\right)+3 e\right)\right)}{40 c^4 e}+\frac{b \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^2 x^2} \left(8 c^7 d^{5/2} \sqrt{-d-e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{1-c^2 x^2}}{\sqrt{-d-e x^2}}\right)+\sqrt{-c^2} \sqrt{e} \sqrt{c^2 (-d)-e} \left(15 c^4 d^2+10 c^2 d e+3 e^2\right) \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sin ^{-1}\left(\frac{c \sqrt{e} \sqrt{1-c^2 x^2}}{\sqrt{-c^2} \sqrt{c^2 (-d)-e}}\right)\right)}{40 c^7 e (c x-1) \sqrt{d+e x^2}}","\frac{\left(d+e x^2\right)^{5/2} \left(a+b \text{sech}^{-1}(c x)\right)}{5 e}-\frac{b d^{5/2} \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{5 e}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(d+e x^2\right)^{3/2}}{20 c^2}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(7 c^2 d+3 e\right) \sqrt{d+e x^2}}{40 c^4}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(15 c^4 d^2+10 c^2 d e+3 e^2\right) \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right)}{40 c^5 \sqrt{e}}",1,"(Sqrt[d + e*x^2]*(8*a*c^4*(d + e*x^2)^2 - b*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(3*e + c^2*(9*d + 2*e*x^2)) + 8*b*c^4*(d + e*x^2)^2*ArcSech[c*x]))/(40*c^4*e) + (b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^2*x^2]*(Sqrt[-c^2]*Sqrt[-(c^2*d) - e]*Sqrt[e]*(15*c^4*d^2 + 10*c^2*d*e + 3*e^2)*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSin[(c*Sqrt[e]*Sqrt[1 - c^2*x^2])/(Sqrt[-c^2]*Sqrt[-(c^2*d) - e])] + 8*c^7*d^(5/2)*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 - c^2*x^2])/Sqrt[-d - e*x^2]]))/(40*c^7*e*(-1 + c*x)*Sqrt[d + e*x^2])","A",1
142,0,0,26,6.3751264,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right)}{x} \, dx","Integrate[((d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/x,x]","\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right)}{x} \, dx","\text{Int}\left(\frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right)}{x},x\right)",0,"Integrate[((d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/x, x]","A",-1
143,0,0,26,6.2221472,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/x^3,x]","\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right)}{x^3} \, dx","\text{Int}\left(\frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right)}{x^3},x\right)",0,"Integrate[((d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/x^3, x]","A",-1
144,0,0,26,13.6487899,"\int x^2 \left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[x^2*(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]),x]","\int x^2 \left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right) \, dx","\text{Int}\left(x^2 \left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right),x\right)",0,"Integrate[x^2*(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]), x]","A",-1
145,0,0,23,4.4754142,"\int \left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]),x]","\int \left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right) \, dx","\text{Int}\left(\left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right),x\right)",0,"Integrate[(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]), x]","A",-1
146,0,0,26,6.8695187,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/x^2,x]","\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right)}{x^2} \, dx","\text{Int}\left(\frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right)}{x^2},x\right)",0,"Integrate[((d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/x^2, x]","A",-1
147,0,0,26,16.5793992,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/x^4,x]","\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right)}{x^4} \, dx","\text{Int}\left(\frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right)}{x^4},x\right)",0,"Integrate[((d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/x^4, x]","A",-1
148,1,620,409,6.3713911,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right)}{x^6} \, dx","Integrate[((d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/x^6,x]","\frac{-\frac{15 a \left(d+e x^2\right)^3}{x^5}+\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(d+e x^2\right) \left(d e x^2 \left(23 c^2 x^2+11\right)+d^2 \left(8 c^4 x^4+4 c^2 x^2+3\right)+23 e^2 x^4\right)}{x^5}+\frac{b \sqrt{\frac{1-c x}{c x+1}} \left(-\left(c^2 \left(8 c^4 d^2+23 c^2 d e+23 e^2\right) \left(d+e x^2\right)\right)-\frac{i (c x+1) \left(c \sqrt{d}-i \sqrt{e}\right)^2 \sqrt{\frac{c \left(\sqrt{d}-i \sqrt{e} x\right)}{(c x+1) \left(c \sqrt{d}-i \sqrt{e}\right)}} \sqrt{\frac{c \left(\sqrt{d}+i \sqrt{e} x\right)}{(c x+1) \left(c \sqrt{d}+i \sqrt{e}\right)}} \left(\left(8 c^4 d^2+23 c^2 d e+23 e^2\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{\left(d c^2+e\right) (1-c x)}{\left(\sqrt{d} c+i \sqrt{e}\right)^2 (c x+1)}}\right)|\frac{\left(\sqrt{d} c+i \sqrt{e}\right)^2}{\left(c \sqrt{d}-i \sqrt{e}\right)^2}\right)+2 \sqrt{e} \left(8 i c^3 d^{3/2}-12 c^2 d \sqrt{e}+7 i c \sqrt{d} e-15 e^{3/2}\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{\left(d c^2+e\right) (1-c x)}{\left(\sqrt{d} c+i \sqrt{e}\right)^2 (c x+1)}}\right)|\frac{\left(\sqrt{d} c+i \sqrt{e}\right)^2}{\left(c \sqrt{d}-i \sqrt{e}\right)^2}\right)\right)}{\sqrt{-\frac{(c x-1) \left(c \sqrt{d}-i \sqrt{e}\right)}{(c x+1) \left(c \sqrt{d}+i \sqrt{e}\right)}}}\right)}{c}-\frac{15 b \text{sech}^{-1}(c x) \left(d+e x^2\right)^3}{x^5}}{75 d \sqrt{d+e x^2}}","-\frac{\left(d+e x^2\right)^{5/2} \left(a+b \text{sech}^{-1}(c x)\right)}{5 d x^5}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(d+e x^2\right)^{3/2}}{25 x^5}+\frac{4 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(c^2 d+2 e\right) \sqrt{d+e x^2}}{75 x^3}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(8 c^4 d^2+23 c^2 d e+23 e^2\right) \sqrt{d+e x^2}}{75 d x}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(c^2 d+e\right) \left(8 c^4 d^2+19 c^2 d e+15 e^2\right) \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{75 c d \sqrt{d+e x^2}}+\frac{b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(8 c^4 d^2+23 c^2 d e+23 e^2\right) \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{75 d \sqrt{\frac{e x^2}{d}+1}}",1,"((-15*a*(d + e*x^2)^3)/x^5 + (b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(d + e*x^2)*(23*e^2*x^4 + d*e*x^2*(11 + 23*c^2*x^2) + d^2*(3 + 4*c^2*x^2 + 8*c^4*x^4)))/x^5 - (15*b*(d + e*x^2)^3*ArcSech[c*x])/x^5 + (b*Sqrt[(1 - c*x)/(1 + c*x)]*(-(c^2*(8*c^4*d^2 + 23*c^2*d*e + 23*e^2)*(d + e*x^2)) - (I*(c*Sqrt[d] - I*Sqrt[e])^2*(1 + c*x)*Sqrt[(c*(Sqrt[d] - I*Sqrt[e]*x))/((c*Sqrt[d] - I*Sqrt[e])*(1 + c*x))]*Sqrt[(c*(Sqrt[d] + I*Sqrt[e]*x))/((c*Sqrt[d] + I*Sqrt[e])*(1 + c*x))]*((8*c^4*d^2 + 23*c^2*d*e + 23*e^2)*EllipticE[I*ArcSinh[Sqrt[((c^2*d + e)*(1 - c*x))/((c*Sqrt[d] + I*Sqrt[e])^2*(1 + c*x))]], (c*Sqrt[d] + I*Sqrt[e])^2/(c*Sqrt[d] - I*Sqrt[e])^2] + 2*Sqrt[e]*((8*I)*c^3*d^(3/2) - 12*c^2*d*Sqrt[e] + (7*I)*c*Sqrt[d]*e - 15*e^(3/2))*EllipticF[I*ArcSinh[Sqrt[((c^2*d + e)*(1 - c*x))/((c*Sqrt[d] + I*Sqrt[e])^2*(1 + c*x))]], (c*Sqrt[d] + I*Sqrt[e])^2/(c*Sqrt[d] - I*Sqrt[e])^2]))/Sqrt[-(((c*Sqrt[d] - I*Sqrt[e])*(-1 + c*x))/((c*Sqrt[d] + I*Sqrt[e])*(1 + c*x)))]))/c)/(75*d*Sqrt[d + e*x^2])","C",1
149,1,728,556,7.8105782,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right)}{x^8} \, dx","Integrate[((d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/x^8,x]","\frac{\frac{105 a \left(2 e x^2-5 d\right) \left(d+e x^2\right)^3}{x^7}+\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(d+e x^2\right) \left(d e^2 x^4 \left(193 c^2 x^2+71\right)+3 d^2 e x^2 \left(176 c^4 x^4+83 c^2 x^2+61\right)+15 d^3 \left(16 c^6 x^6+8 c^4 x^4+6 c^2 x^2+5\right)-247 e^3 x^6\right)}{x^7}+\frac{b \sqrt{\frac{1-c x}{c x+1}} \left(-\left(c^2 \left(240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right) \left(d+e x^2\right)\right)-\frac{i (c x+1) \left(c \sqrt{d}-i \sqrt{e}\right)^2 \sqrt{\frac{c \left(\sqrt{d}-i \sqrt{e} x\right)}{(c x+1) \left(c \sqrt{d}-i \sqrt{e}\right)}} \sqrt{\frac{c \left(\sqrt{d}+i \sqrt{e} x\right)}{(c x+1) \left(c \sqrt{d}+i \sqrt{e}\right)}} \left(\left(240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{\left(d c^2+e\right) (1-c x)}{\left(\sqrt{d} c+i \sqrt{e}\right)^2 (c x+1)}}\right)|\frac{\left(\sqrt{d} c+i \sqrt{e}\right)^2}{\left(c \sqrt{d}-i \sqrt{e}\right)^2}\right)+2 \sqrt{e} \left(240 i c^5 d^{5/2}-360 c^4 d^2 \sqrt{e}+48 i c^3 d^{3/2} e-207 c^2 d e^{3/2}-173 i c \sqrt{d} e^2+210 e^{5/2}\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{\left(d c^2+e\right) (1-c x)}{\left(\sqrt{d} c+i \sqrt{e}\right)^2 (c x+1)}}\right)|\frac{\left(\sqrt{d} c+i \sqrt{e}\right)^2}{\left(c \sqrt{d}-i \sqrt{e}\right)^2}\right)\right)}{\sqrt{-\frac{(c x-1) \left(c \sqrt{d}-i \sqrt{e}\right)}{(c x+1) \left(c \sqrt{d}+i \sqrt{e}\right)}}}\right)}{c}+\frac{105 b \text{sech}^{-1}(c x) \left(2 e x^2-5 d\right) \left(d+e x^2\right)^3}{x^7}}{3675 d^2 \sqrt{d+e x^2}}","\frac{2 e \left(d+e x^2\right)^{5/2} \left(a+b \text{sech}^{-1}(c x)\right)}{35 d^2 x^5}-\frac{\left(d+e x^2\right)^{5/2} \left(a+b \text{sech}^{-1}(c x)\right)}{7 d x^7}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(d+e x^2\right)^{5/2}}{49 d x^7}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(30 c^2 d+11 e\right) \left(d+e x^2\right)^{3/2}}{1225 d x^5}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(120 c^4 d^2+159 c^2 d e-37 e^2\right) \sqrt{d+e x^2}}{3675 d x^3}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right) \sqrt{d+e x^2}}{3675 d^2 x}-\frac{2 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(c^2 d+e\right) \left(120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right) \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{3675 c d^2 \sqrt{d+e x^2}}+\frac{b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right) \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{3675 d^2 \sqrt{\frac{e x^2}{d}+1}}",1,"((105*a*(d + e*x^2)^3*(-5*d + 2*e*x^2))/x^7 + (b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(d + e*x^2)*(-247*e^3*x^6 + d*e^2*x^4*(71 + 193*c^2*x^2) + 3*d^2*e*x^2*(61 + 83*c^2*x^2 + 176*c^4*x^4) + 15*d^3*(5 + 6*c^2*x^2 + 8*c^4*x^4 + 16*c^6*x^6)))/x^7 + (105*b*(d + e*x^2)^3*(-5*d + 2*e*x^2)*ArcSech[c*x])/x^7 + (b*Sqrt[(1 - c*x)/(1 + c*x)]*(-(c^2*(240*c^6*d^3 + 528*c^4*d^2*e + 193*c^2*d*e^2 - 247*e^3)*(d + e*x^2)) - (I*(c*Sqrt[d] - I*Sqrt[e])^2*(1 + c*x)*Sqrt[(c*(Sqrt[d] - I*Sqrt[e]*x))/((c*Sqrt[d] - I*Sqrt[e])*(1 + c*x))]*Sqrt[(c*(Sqrt[d] + I*Sqrt[e]*x))/((c*Sqrt[d] + I*Sqrt[e])*(1 + c*x))]*((240*c^6*d^3 + 528*c^4*d^2*e + 193*c^2*d*e^2 - 247*e^3)*EllipticE[I*ArcSinh[Sqrt[((c^2*d + e)*(1 - c*x))/((c*Sqrt[d] + I*Sqrt[e])^2*(1 + c*x))]], (c*Sqrt[d] + I*Sqrt[e])^2/(c*Sqrt[d] - I*Sqrt[e])^2] + 2*Sqrt[e]*((240*I)*c^5*d^(5/2) - 360*c^4*d^2*Sqrt[e] + (48*I)*c^3*d^(3/2)*e - 207*c^2*d*e^(3/2) - (173*I)*c*Sqrt[d]*e^2 + 210*e^(5/2))*EllipticF[I*ArcSinh[Sqrt[((c^2*d + e)*(1 - c*x))/((c*Sqrt[d] + I*Sqrt[e])^2*(1 + c*x))]], (c*Sqrt[d] + I*Sqrt[e])^2/(c*Sqrt[d] - I*Sqrt[e])^2]))/Sqrt[-(((c*Sqrt[d] - I*Sqrt[e])*(-1 + c*x))/((c*Sqrt[d] + I*Sqrt[e])*(1 + c*x)))]))/c)/(3675*d^2*Sqrt[d + e*x^2])","C",1
150,1,366,356,1.6018283,"\int \frac{x^5 \left(a+b \text{sech}^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","Integrate[(x^5*(a + b*ArcSech[c*x]))/Sqrt[d + e*x^2],x]","\frac{\sqrt{d+e x^2} \left(8 a c^4 \left(8 d^2-4 d e x^2+3 e^2 x^4\right)+8 b c^4 \text{sech}^{-1}(c x) \left(8 d^2-4 d e x^2+3 e^2 x^4\right)-b e \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(c^2 \left(6 e x^2-13 d\right)+9 e\right)\right)}{120 c^4 e^3}+\frac{b \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^2 x^2} \left(64 c^7 d^{5/2} \sqrt{-d-e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{1-c^2 x^2}}{\sqrt{-d-e x^2}}\right)+\sqrt{-c^2} \sqrt{e} \sqrt{c^2 (-d)-e} \left(45 c^4 d^2-10 c^2 d e+9 e^2\right) \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sin ^{-1}\left(\frac{c \sqrt{e} \sqrt{1-c^2 x^2}}{\sqrt{-c^2} \sqrt{c^2 (-d)-e}}\right)\right)}{120 c^7 e^3 (c x-1) \sqrt{d+e x^2}}","\frac{d^2 \sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right)}{e^3}-\frac{2 d \left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right)}{3 e^3}+\frac{\left(d+e x^2\right)^{5/2} \left(a+b \text{sech}^{-1}(c x)\right)}{5 e^3}-\frac{8 b d^{5/2} \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{15 e^3}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(d+e x^2\right)^{3/2}}{20 c^2 e^2}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(19 c^2 d-9 e\right) \sqrt{d+e x^2}}{120 c^4 e^2}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(45 c^4 d^2-10 c^2 d e+9 e^2\right) \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right)}{120 c^5 e^{5/2}}",1,"(Sqrt[d + e*x^2]*(8*a*c^4*(8*d^2 - 4*d*e*x^2 + 3*e^2*x^4) - b*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(9*e + c^2*(-13*d + 6*e*x^2)) + 8*b*c^4*(8*d^2 - 4*d*e*x^2 + 3*e^2*x^4)*ArcSech[c*x]))/(120*c^4*e^3) + (b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^2*x^2]*(Sqrt[-c^2]*Sqrt[-(c^2*d) - e]*Sqrt[e]*(45*c^4*d^2 - 10*c^2*d*e + 9*e^2)*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSin[(c*Sqrt[e]*Sqrt[1 - c^2*x^2])/(Sqrt[-c^2]*Sqrt[-(c^2*d) - e])] + 64*c^7*d^(5/2)*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 - c^2*x^2])/Sqrt[-d - e*x^2]]))/(120*c^7*e^3*(-1 + c*x)*Sqrt[d + e*x^2])","A",1
151,1,406,251,1.316761,"\int \frac{x^3 \left(a+b \text{sech}^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","Integrate[(x^3*(a + b*ArcSech[c*x]))/Sqrt[d + e*x^2],x]","-\frac{\sqrt{d+e x^2} \left(2 a c^2 \left(2 d-e x^2\right)+2 b c^2 \text{sech}^{-1}(c x) \left(2 d-e x^2\right)+b e \sqrt{\frac{1-c x}{c x+1}} (c x+1)\right)}{6 c^2 e^2}-\frac{b \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^2 x^2} \left(\sqrt{-c^2} e^{3/2} \sqrt{c^2 (-d)-e} \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sin ^{-1}\left(\frac{\sqrt{-c^2} \sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{c^2 (-d)-e}}\right)-3 \left(-c^2\right)^{3/2} d \sqrt{e} \sqrt{c^2 (-d)-e} \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sin ^{-1}\left(\frac{c \sqrt{e} \sqrt{1-c^2 x^2}}{\sqrt{-c^2} \sqrt{c^2 (-d)-e}}\right)+4 c^5 d^{3/2} \sqrt{-d-e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{1-c^2 x^2}}{\sqrt{-d-e x^2}}\right)\right)}{6 c^5 e^2 (c x-1) \sqrt{d+e x^2}}","-\frac{d \sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right)}{e^2}+\frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right)}{3 e^2}+\frac{2 b d^{3/2} \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{3 e^2}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{6 c^2 e}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(3 c^2 d-e\right) \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right)}{6 c^3 e^{3/2}}",1,"-1/6*(Sqrt[d + e*x^2]*(b*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + 2*a*c^2*(2*d - e*x^2) + 2*b*c^2*(2*d - e*x^2)*ArcSech[c*x]))/(c^2*e^2) - (b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^2*x^2]*(-3*(-c^2)^(3/2)*d*Sqrt[-(c^2*d) - e]*Sqrt[e]*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSin[(c*Sqrt[e]*Sqrt[1 - c^2*x^2])/(Sqrt[-c^2]*Sqrt[-(c^2*d) - e])] + Sqrt[-c^2]*Sqrt[-(c^2*d) - e]*e^(3/2)*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSin[(Sqrt[-c^2]*Sqrt[e]*Sqrt[1 - c^2*x^2])/(c*Sqrt[-(c^2*d) - e])] + 4*c^5*d^(3/2)*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 - c^2*x^2])/Sqrt[-d - e*x^2]]))/(6*c^5*e^2*(-1 + c*x)*Sqrt[d + e*x^2])","A",1
152,1,239,153,0.5639173,"\int \frac{x \left(a+b \text{sech}^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","Integrate[(x*(a + b*ArcSech[c*x]))/Sqrt[d + e*x^2],x]","\frac{\sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right)}{e}+\frac{b \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^2 x^2} \left(\sqrt{-c^2} \sqrt{e} \sqrt{c^2 (-d)-e} \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sin ^{-1}\left(\frac{c \sqrt{e} \sqrt{1-c^2 x^2}}{\sqrt{-c^2} \sqrt{c^2 (-d)-e}}\right)+c^3 \sqrt{d} \sqrt{-d-e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{1-c^2 x^2}}{\sqrt{-d-e x^2}}\right)\right)}{c^3 e (c x-1) \sqrt{d+e x^2}}","\frac{\sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right)}{e}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right)}{c \sqrt{e}}-\frac{b \sqrt{d} \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{e}",1,"(Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]))/e + (b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^2*x^2]*(Sqrt[-c^2]*Sqrt[-(c^2*d) - e]*Sqrt[e]*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSin[(c*Sqrt[e]*Sqrt[1 - c^2*x^2])/(Sqrt[-c^2]*Sqrt[-(c^2*d) - e])] + c^3*Sqrt[d]*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 - c^2*x^2])/Sqrt[-d - e*x^2]]))/(c^3*e*(-1 + c*x)*Sqrt[d + e*x^2])","A",1
153,0,0,26,3.2537129,"\int \frac{a+b \text{sech}^{-1}(c x)}{x \sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcSech[c*x])/(x*Sqrt[d + e*x^2]),x]","\int \frac{a+b \text{sech}^{-1}(c x)}{x \sqrt{d+e x^2}} \, dx","\text{Int}\left(\frac{a+b \text{sech}^{-1}(c x)}{x \sqrt{d+e x^2}},x\right)",0,"Integrate[(a + b*ArcSech[c*x])/(x*Sqrt[d + e*x^2]), x]","A",-1
154,0,0,26,23.9584944,"\int \frac{a+b \text{sech}^{-1}(c x)}{x^3 \sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcSech[c*x])/(x^3*Sqrt[d + e*x^2]),x]","\int \frac{a+b \text{sech}^{-1}(c x)}{x^3 \sqrt{d+e x^2}} \, dx","\text{Int}\left(\frac{a+b \text{sech}^{-1}(c x)}{x^3 \sqrt{d+e x^2}},x\right)",0,"Integrate[(a + b*ArcSech[c*x])/(x^3*Sqrt[d + e*x^2]), x]","A",-1
155,0,0,26,7.1810298,"\int \frac{x^2 \left(a+b \text{sech}^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","Integrate[(x^2*(a + b*ArcSech[c*x]))/Sqrt[d + e*x^2],x]","\int \frac{x^2 \left(a+b \text{sech}^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","\text{Int}\left(\frac{x^2 \left(a+b \text{sech}^{-1}(c x)\right)}{\sqrt{d+e x^2}},x\right)",0,"Integrate[(x^2*(a + b*ArcSech[c*x]))/Sqrt[d + e*x^2], x]","A",-1
156,0,0,23,1.0661263,"\int \frac{a+b \text{sech}^{-1}(c x)}{\sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcSech[c*x])/Sqrt[d + e*x^2],x]","\int \frac{a+b \text{sech}^{-1}(c x)}{\sqrt{d+e x^2}} \, dx","\text{Int}\left(\frac{a+b \text{sech}^{-1}(c x)}{\sqrt{d+e x^2}},x\right)",0,"Integrate[(a + b*ArcSech[c*x])/Sqrt[d + e*x^2], x]","A",-1
157,1,501,221,4.3782738,"\int \frac{a+b \text{sech}^{-1}(c x)}{x^2 \sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcSech[c*x])/(x^2*Sqrt[d + e*x^2]),x]","-\frac{a \left(\frac{d}{x}+e x\right)+\frac{b \sqrt{\frac{1-c x}{c x+1}} \left(\sqrt{e} x+i \sqrt{d}\right) \sqrt{\frac{c \left(\sqrt{d}+i \sqrt{e} x\right)}{(c x+1) \left(c \sqrt{d}+i \sqrt{e}\right)}} \left(2 i \sqrt{e} F\left(i \sinh ^{-1}\left(\sqrt{\frac{\left(d c^2+e\right) (1-c x)}{\left(\sqrt{d} c+i \sqrt{e}\right)^2 (c x+1)}}\right)|\frac{\left(\sqrt{d} c+i \sqrt{e}\right)^2}{\left(c \sqrt{d}-i \sqrt{e}\right)^2}\right)+\left(c \sqrt{d}-i \sqrt{e}\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{\left(d c^2+e\right) (1-c x)}{\left(\sqrt{d} c+i \sqrt{e}\right)^2 (c x+1)}}\right)|\frac{\left(\sqrt{d} c+i \sqrt{e}\right)^2}{\left(c \sqrt{d}-i \sqrt{e}\right)^2}\right)\right)}{\sqrt{-\frac{(c x-1) \left(c \sqrt{d}-i \sqrt{e}\right)}{(c x+1) \left(c \sqrt{d}+i \sqrt{e}\right)}} \sqrt{\frac{c \left(\sqrt{d}-i \sqrt{e} x\right)}{(c x+1) \left(c \sqrt{d}-i \sqrt{e}\right)}}}-\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(d+e x^2\right)}{x}+b c \sqrt{\frac{1-c x}{c x+1}} \left(d+e x^2\right)+\frac{b \text{sech}^{-1}(c x) \left(d+e x^2\right)}{x}}{d \sqrt{d+e x^2}}","-\frac{\sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right)}{d x}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{d x}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(c^2 d+e\right) \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{c d \sqrt{d+e x^2}}+\frac{b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{d \sqrt{\frac{e x^2}{d}+1}}",1,"-((a*(d/x + e*x) + b*c*Sqrt[(1 - c*x)/(1 + c*x)]*(d + e*x^2) - (b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(d + e*x^2))/x + (b*(d + e*x^2)*ArcSech[c*x])/x + (b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[(c*(Sqrt[d] + I*Sqrt[e]*x))/((c*Sqrt[d] + I*Sqrt[e])*(1 + c*x))]*(I*Sqrt[d] + Sqrt[e]*x)*((c*Sqrt[d] - I*Sqrt[e])*EllipticE[I*ArcSinh[Sqrt[((c^2*d + e)*(1 - c*x))/((c*Sqrt[d] + I*Sqrt[e])^2*(1 + c*x))]], (c*Sqrt[d] + I*Sqrt[e])^2/(c*Sqrt[d] - I*Sqrt[e])^2] + (2*I)*Sqrt[e]*EllipticF[I*ArcSinh[Sqrt[((c^2*d + e)*(1 - c*x))/((c*Sqrt[d] + I*Sqrt[e])^2*(1 + c*x))]], (c*Sqrt[d] + I*Sqrt[e])^2/(c*Sqrt[d] - I*Sqrt[e])^2]))/(Sqrt[-(((c*Sqrt[d] - I*Sqrt[e])*(-1 + c*x))/((c*Sqrt[d] + I*Sqrt[e])*(1 + c*x)))]*Sqrt[(c*(Sqrt[d] - I*Sqrt[e]*x))/((c*Sqrt[d] - I*Sqrt[e])*(1 + c*x))]))/(d*Sqrt[d + e*x^2]))","C",1
158,1,612,346,5.0295551,"\int \frac{a+b \text{sech}^{-1}(c x)}{x^4 \sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcSech[c*x])/(x^4*Sqrt[d + e*x^2]),x]","\frac{-\frac{3 a \left(d-2 e x^2\right) \left(d+e x^2\right)}{x^3}+\frac{b \sqrt{\frac{1-c x}{c x+1}} \left(2 c^2 d-5 e\right) \left(d+e x^2\right)}{x}-\frac{b \sqrt{\frac{1-c x}{c x+1}} \left(\sqrt{e} x+i \sqrt{d}\right) \sqrt{\frac{c \left(\sqrt{d}+i \sqrt{e} x\right)}{(c x+1) \left(c \sqrt{d}+i \sqrt{e}\right)}} \left(2 \sqrt{e} \left(2 i c^2 d-c \sqrt{d} \sqrt{e}-6 i e\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{\left(d c^2+e\right) (1-c x)}{\left(\sqrt{d} c+i \sqrt{e}\right)^2 (c x+1)}}\right)|\frac{\left(\sqrt{d} c+i \sqrt{e}\right)^2}{\left(c \sqrt{d}-i \sqrt{e}\right)^2}\right)+\left(2 c^3 d^{3/2}-2 i c^2 d \sqrt{e}-5 c \sqrt{d} e+5 i e^{3/2}\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{\left(d c^2+e\right) (1-c x)}{\left(\sqrt{d} c+i \sqrt{e}\right)^2 (c x+1)}}\right)|\frac{\left(\sqrt{d} c+i \sqrt{e}\right)^2}{\left(c \sqrt{d}-i \sqrt{e}\right)^2}\right)\right)}{\sqrt{-\frac{(c x-1) \left(c \sqrt{d}-i \sqrt{e}\right)}{(c x+1) \left(c \sqrt{d}+i \sqrt{e}\right)}} \sqrt{\frac{c \left(\sqrt{d}-i \sqrt{e} x\right)}{(c x+1) \left(c \sqrt{d}-i \sqrt{e}\right)}}}+\frac{b c d \sqrt{\frac{1-c x}{c x+1}} \left(d+e x^2\right)}{x^2}+\frac{b d \sqrt{\frac{1-c x}{c x+1}} \left(d+e x^2\right)}{x^3}-\frac{3 b \text{sech}^{-1}(c x) \left(d-2 e x^2\right) \left(d+e x^2\right)}{x^3}}{9 d^2 \sqrt{d+e x^2}}","\frac{2 e \sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right)}{3 d^2 x}-\frac{\sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right)}{3 d x^3}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left(2 c^2 d-5 e\right) \sqrt{d+e x^2}}{9 d^2 x}-\frac{2 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(c^2 d-3 e\right) \left(c^2 d+e\right) \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{9 c d^2 \sqrt{d+e x^2}}+\frac{b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(2 c^2 d-5 e\right) \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{9 d^2 \sqrt{\frac{e x^2}{d}+1}}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{9 d x^3}",1,"((b*d*Sqrt[(1 - c*x)/(1 + c*x)]*(d + e*x^2))/x^3 + (b*c*d*Sqrt[(1 - c*x)/(1 + c*x)]*(d + e*x^2))/x^2 + (b*(2*c^2*d - 5*e)*Sqrt[(1 - c*x)/(1 + c*x)]*(d + e*x^2))/x - (3*a*(d - 2*e*x^2)*(d + e*x^2))/x^3 - (3*b*(d - 2*e*x^2)*(d + e*x^2)*ArcSech[c*x])/x^3 - (b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[(c*(Sqrt[d] + I*Sqrt[e]*x))/((c*Sqrt[d] + I*Sqrt[e])*(1 + c*x))]*(I*Sqrt[d] + Sqrt[e]*x)*((2*c^3*d^(3/2) - (2*I)*c^2*d*Sqrt[e] - 5*c*Sqrt[d]*e + (5*I)*e^(3/2))*EllipticE[I*ArcSinh[Sqrt[((c^2*d + e)*(1 - c*x))/((c*Sqrt[d] + I*Sqrt[e])^2*(1 + c*x))]], (c*Sqrt[d] + I*Sqrt[e])^2/(c*Sqrt[d] - I*Sqrt[e])^2] + 2*((2*I)*c^2*d - c*Sqrt[d]*Sqrt[e] - (6*I)*e)*Sqrt[e]*EllipticF[I*ArcSinh[Sqrt[((c^2*d + e)*(1 - c*x))/((c*Sqrt[d] + I*Sqrt[e])^2*(1 + c*x))]], (c*Sqrt[d] + I*Sqrt[e])^2/(c*Sqrt[d] - I*Sqrt[e])^2]))/(Sqrt[-(((c*Sqrt[d] - I*Sqrt[e])*(-1 + c*x))/((c*Sqrt[d] + I*Sqrt[e])*(1 + c*x)))]*Sqrt[(c*(Sqrt[d] - I*Sqrt[e]*x))/((c*Sqrt[d] - I*Sqrt[e])*(1 + c*x))]))/(9*d^2*Sqrt[d + e*x^2])","C",1
159,1,436,278,1.4979145,"\int \frac{x^5 \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(x^5*(a + b*ArcSech[c*x]))/(d + e*x^2)^(3/2),x]","\frac{-2 a c^2 \left(8 d^2+4 d e x^2-e^2 x^4\right)-2 b c^2 \text{sech}^{-1}(c x) \left(8 d^2+4 d e x^2-e^2 x^4\right)-b e \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(d+e x^2\right)}{6 c^2 e^3 \sqrt{d+e x^2}}-\frac{b \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^2 x^2} \left(\sqrt{-c^2} e^{3/2} \sqrt{c^2 (-d)-e} \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sin ^{-1}\left(\frac{\sqrt{-c^2} \sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{c^2 (-d)-e}}\right)-9 \left(-c^2\right)^{3/2} d \sqrt{e} \sqrt{c^2 (-d)-e} \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sin ^{-1}\left(\frac{c \sqrt{e} \sqrt{1-c^2 x^2}}{\sqrt{-c^2} \sqrt{c^2 (-d)-e}}\right)+16 c^5 d^{3/2} \sqrt{-d-e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{1-c^2 x^2}}{\sqrt{-d-e x^2}}\right)\right)}{6 c^5 e^3 (c x-1) \sqrt{d+e x^2}}","-\frac{d^2 \left(a+b \text{sech}^{-1}(c x)\right)}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right)}{e^3}+\frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right)}{3 e^3}+\frac{8 b d^{3/2} \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{3 e^3}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{6 c^2 e^2}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(9 c^2 d-e\right) \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right)}{6 c^3 e^{5/2}}",1,"(-(b*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(d + e*x^2)) - 2*a*c^2*(8*d^2 + 4*d*e*x^2 - e^2*x^4) - 2*b*c^2*(8*d^2 + 4*d*e*x^2 - e^2*x^4)*ArcSech[c*x])/(6*c^2*e^3*Sqrt[d + e*x^2]) - (b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^2*x^2]*(-9*(-c^2)^(3/2)*d*Sqrt[-(c^2*d) - e]*Sqrt[e]*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSin[(c*Sqrt[e]*Sqrt[1 - c^2*x^2])/(Sqrt[-c^2]*Sqrt[-(c^2*d) - e])] + Sqrt[-c^2]*Sqrt[-(c^2*d) - e]*e^(3/2)*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSin[(Sqrt[-c^2]*Sqrt[e]*Sqrt[1 - c^2*x^2])/(c*Sqrt[-(c^2*d) - e])] + 16*c^5*d^(3/2)*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 - c^2*x^2])/Sqrt[-d - e*x^2]]))/(6*c^5*e^3*(-1 + c*x)*Sqrt[d + e*x^2])","A",1
160,1,249,177,1.7763592,"\int \frac{x^3 \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(x^3*(a + b*ArcSech[c*x]))/(d + e*x^2)^(3/2),x]","\frac{\left(2 d+e x^2\right) \left(a+b \text{sech}^{-1}(c x)\right)}{e^2 \sqrt{d+e x^2}}+\frac{b \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^2 x^2} \left(\sqrt{-c^2} \sqrt{e} \sqrt{c^2 (-d)-e} \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sin ^{-1}\left(\frac{c \sqrt{e} \sqrt{1-c^2 x^2}}{\sqrt{-c^2} \sqrt{c^2 (-d)-e}}\right)+2 c^3 \sqrt{d} \sqrt{-d-e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{1-c^2 x^2}}{\sqrt{-d-e x^2}}\right)\right)}{c^3 e^2 (c x-1) \sqrt{d+e x^2}}","\frac{\sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right)}{e^2}+\frac{d \left(a+b \text{sech}^{-1}(c x)\right)}{e^2 \sqrt{d+e x^2}}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right)}{c e^{3/2}}-\frac{2 b \sqrt{d} \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{e^2}",1,"((2*d + e*x^2)*(a + b*ArcSech[c*x]))/(e^2*Sqrt[d + e*x^2]) + (b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^2*x^2]*(Sqrt[-c^2]*Sqrt[-(c^2*d) - e]*Sqrt[e]*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSin[(c*Sqrt[e]*Sqrt[1 - c^2*x^2])/(Sqrt[-c^2]*Sqrt[-(c^2*d) - e])] + 2*c^3*Sqrt[d]*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 - c^2*x^2])/Sqrt[-d - e*x^2]]))/(c^3*e^2*(-1 + c*x)*Sqrt[d + e*x^2])","A",1
161,1,135,87,0.6346478,"\int \frac{x \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(x*(a + b*ArcSech[c*x]))/(d + e*x^2)^(3/2),x]","-\frac{a+b \text{sech}^{-1}(c x)}{e \sqrt{d+e x^2}}-\frac{b \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^2 x^2} \sqrt{-d-e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{1-c^2 x^2}}{\sqrt{-d-e x^2}}\right)}{\sqrt{d} e (c x-1) \sqrt{d+e x^2}}","\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{\sqrt{d} e}-\frac{a+b \text{sech}^{-1}(c x)}{e \sqrt{d+e x^2}}",1,"-((a + b*ArcSech[c*x])/(e*Sqrt[d + e*x^2])) - (b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^2*x^2]*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 - c^2*x^2])/Sqrt[-d - e*x^2]])/(Sqrt[d]*e*(-1 + c*x)*Sqrt[d + e*x^2])","A",1
162,0,0,26,32.0216783,"\int \frac{a+b \text{sech}^{-1}(c x)}{x \left(d+e x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSech[c*x])/(x*(d + e*x^2)^(3/2)),x]","\int \frac{a+b \text{sech}^{-1}(c x)}{x \left(d+e x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{a+b \text{sech}^{-1}(c x)}{x \left(d+e x^2\right)^{3/2}},x\right)",0,"Integrate[(a + b*ArcSech[c*x])/(x*(d + e*x^2)^(3/2)), x]","A",-1
163,0,0,26,37.5419122,"\int \frac{a+b \text{sech}^{-1}(c x)}{x^3 \left(d+e x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSech[c*x])/(x^3*(d + e*x^2)^(3/2)),x]","\int \frac{a+b \text{sech}^{-1}(c x)}{x^3 \left(d+e x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{a+b \text{sech}^{-1}(c x)}{x^3 \left(d+e x^2\right)^{3/2}},x\right)",0,"Integrate[(a + b*ArcSech[c*x])/(x^3*(d + e*x^2)^(3/2)), x]","A",-1
164,0,0,26,9.906721,"\int \frac{x^4 \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(x^4*(a + b*ArcSech[c*x]))/(d + e*x^2)^(3/2),x]","\int \frac{x^4 \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{x^4 \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}},x\right)",0,"Integrate[(x^4*(a + b*ArcSech[c*x]))/(d + e*x^2)^(3/2), x]","A",-1
165,0,0,26,4.712452,"\int \frac{x^2 \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(x^2*(a + b*ArcSech[c*x]))/(d + e*x^2)^(3/2),x]","\int \frac{x^2 \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{x^2 \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}},x\right)",0,"Integrate[(x^2*(a + b*ArcSech[c*x]))/(d + e*x^2)^(3/2), x]","A",-1
166,1,334,92,1.390979,"\int \frac{a+b \text{sech}^{-1}(c x)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSech[c*x])/(d + e*x^2)^(3/2),x]","\frac{x \left(a+b \text{sech}^{-1}(c x)\right)}{d \sqrt{d+e x^2}}+\frac{2 i b \sqrt{\frac{1-c x}{c x+1}} \left(\sqrt{e} x-i \sqrt{d}\right) \sqrt{\frac{(c x+1) \left(c \sqrt{d}+i \sqrt{e}\right)}{(c x-1) \left(c \sqrt{d}-i \sqrt{e}\right)}} \sqrt{-\frac{c \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right)+\frac{i \sqrt{e} x}{\sqrt{d}}-1}{1-c x}} F\left(\sin ^{-1}\left(\sqrt{\frac{-x c+\frac{i \sqrt{d} c}{\sqrt{e}}+\frac{i \sqrt{e} x}{\sqrt{d}}+1}{2-2 c x}}\right)|-\frac{4 i c \sqrt{d} \sqrt{e}}{\left(c \sqrt{d}-i \sqrt{e}\right)^2}\right)}{d \left(c \sqrt{d}+i \sqrt{e}\right) \sqrt{d+e x^2} \sqrt{\frac{\frac{i c \sqrt{d}}{\sqrt{e}}+c (-x)+\frac{i \sqrt{e} x}{\sqrt{d}}+1}{1-c x}}}","\frac{x \left(a+b \text{sech}^{-1}(c x)\right)}{d \sqrt{d+e x^2}}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{c d \sqrt{d+e x^2}}",1,"(x*(a + b*ArcSech[c*x]))/(d*Sqrt[d + e*x^2]) + ((2*I)*b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[((c*Sqrt[d] + I*Sqrt[e])*(1 + c*x))/((c*Sqrt[d] - I*Sqrt[e])*(-1 + c*x))]*((-I)*Sqrt[d] + Sqrt[e]*x)*Sqrt[-((-1 + (I*Sqrt[e]*x)/Sqrt[d] + c*((I*Sqrt[d])/Sqrt[e] + x))/(1 - c*x))]*EllipticF[ArcSin[Sqrt[(1 + (I*c*Sqrt[d])/Sqrt[e] - c*x + (I*Sqrt[e]*x)/Sqrt[d])/(2 - 2*c*x)]], ((-4*I)*c*Sqrt[d]*Sqrt[e])/(c*Sqrt[d] - I*Sqrt[e])^2])/(d*(c*Sqrt[d] + I*Sqrt[e])*Sqrt[(1 + (I*c*Sqrt[d])/Sqrt[e] - c*x + (I*Sqrt[e]*x)/Sqrt[d])/(1 - c*x)]*Sqrt[d + e*x^2])","C",1
167,1,501,249,4.4901473,"\int \frac{a+b \text{sech}^{-1}(c x)}{x^2 \left(d+e x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSech[c*x])/(x^2*(d + e*x^2)^(3/2)),x]","\frac{-\frac{a \left(d+2 e x^2\right)}{x}+\frac{b \sqrt{\frac{1-c x}{c x+1}} \left(-c^2 \left(d+e x^2\right)+\frac{(c x+1) \sqrt{\frac{c \left(\sqrt{d}-i \sqrt{e} x\right)}{(c x+1) \left(c \sqrt{d}-i \sqrt{e}\right)}} \sqrt{\frac{c \left(\sqrt{d}+i \sqrt{e} x\right)}{(c x+1) \left(c \sqrt{d}+i \sqrt{e}\right)}} \left(2 \sqrt{e} \left(c \sqrt{d}-2 i \sqrt{e}\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{\left(d c^2+e\right) (1-c x)}{\left(\sqrt{d} c+i \sqrt{e}\right)^2 (c x+1)}}\right)|\frac{\left(\sqrt{d} c+i \sqrt{e}\right)^2}{\left(c \sqrt{d}-i \sqrt{e}\right)^2}\right)-i \left(c \sqrt{d}-i \sqrt{e}\right)^2 E\left(i \sinh ^{-1}\left(\sqrt{\frac{\left(d c^2+e\right) (1-c x)}{\left(\sqrt{d} c+i \sqrt{e}\right)^2 (c x+1)}}\right)|\frac{\left(\sqrt{d} c+i \sqrt{e}\right)^2}{\left(c \sqrt{d}-i \sqrt{e}\right)^2}\right)\right)}{\sqrt{-\frac{(c x-1) \left(c \sqrt{d}-i \sqrt{e}\right)}{(c x+1) \left(c \sqrt{d}+i \sqrt{e}\right)}}}\right)}{c}+\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(d+e x^2\right)}{x}-\frac{b \text{sech}^{-1}(c x) \left(d+2 e x^2\right)}{x}}{d^2 \sqrt{d+e x^2}}","-\frac{2 e x \left(a+b \text{sech}^{-1}(c x)\right)}{d^2 \sqrt{d+e x^2}}-\frac{a+b \text{sech}^{-1}(c x)}{d x \sqrt{d+e x^2}}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{d^2 x}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left(c^2 d+2 e\right) \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{c d^2 \sqrt{d+e x^2}}+\frac{b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{d^2 \sqrt{\frac{e x^2}{d}+1}}",1,"((b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(d + e*x^2))/x - (a*(d + 2*e*x^2))/x - (b*(d + 2*e*x^2)*ArcSech[c*x])/x + (b*Sqrt[(1 - c*x)/(1 + c*x)]*(-(c^2*(d + e*x^2)) + ((1 + c*x)*Sqrt[(c*(Sqrt[d] - I*Sqrt[e]*x))/((c*Sqrt[d] - I*Sqrt[e])*(1 + c*x))]*Sqrt[(c*(Sqrt[d] + I*Sqrt[e]*x))/((c*Sqrt[d] + I*Sqrt[e])*(1 + c*x))]*((-I)*(c*Sqrt[d] - I*Sqrt[e])^2*EllipticE[I*ArcSinh[Sqrt[((c^2*d + e)*(1 - c*x))/((c*Sqrt[d] + I*Sqrt[e])^2*(1 + c*x))]], (c*Sqrt[d] + I*Sqrt[e])^2/(c*Sqrt[d] - I*Sqrt[e])^2] + 2*(c*Sqrt[d] - (2*I)*Sqrt[e])*Sqrt[e]*EllipticF[I*ArcSinh[Sqrt[((c^2*d + e)*(1 - c*x))/((c*Sqrt[d] + I*Sqrt[e])^2*(1 + c*x))]], (c*Sqrt[d] + I*Sqrt[e])^2/(c*Sqrt[d] - I*Sqrt[e])^2]))/Sqrt[-(((c*Sqrt[d] - I*Sqrt[e])*(-1 + c*x))/((c*Sqrt[d] + I*Sqrt[e])*(1 + c*x)))]))/c)/(d^2*Sqrt[d + e*x^2])","C",1
168,1,348,272,1.9709064,"\int \frac{x^5 \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(x^5*(a + b*ArcSech[c*x]))/(d + e*x^2)^(5/2),x]","\frac{a \left(c^2 d+e\right) \left(8 d^2+12 d e x^2+3 e^2 x^4\right)+b \left(c^2 d+e\right) \text{sech}^{-1}(c x) \left(8 d^2+12 d e x^2+3 e^2 x^4\right)-b d e \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(d+e x^2\right)}{3 e^3 \left(c^2 d+e\right) \left(d+e x^2\right)^{3/2}}+\frac{b \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^2 x^2} \left(3 \sqrt{-c^2} \sqrt{e} \sqrt{c^2 (-d)-e} \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sin ^{-1}\left(\frac{c \sqrt{e} \sqrt{1-c^2 x^2}}{\sqrt{-c^2} \sqrt{c^2 (-d)-e}}\right)+8 c^3 \sqrt{d} \sqrt{-d-e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{1-c^2 x^2}}{\sqrt{-d-e x^2}}\right)\right)}{3 c^3 e^3 (c x-1) \sqrt{d+e x^2}}","-\frac{d^2 \left(a+b \text{sech}^{-1}(c x)\right)}{3 e^3 \left(d+e x^2\right)^{3/2}}+\frac{2 d \left(a+b \text{sech}^{-1}(c x)\right)}{e^3 \sqrt{d+e x^2}}+\frac{\sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right)}{e^3}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right)}{c e^{5/2}}-\frac{8 b \sqrt{d} \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{3 e^3}-\frac{b d \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{3 e^2 \left(c^2 d+e\right) \sqrt{d+e x^2}}",1,"(-(b*d*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(d + e*x^2)) + a*(c^2*d + e)*(8*d^2 + 12*d*e*x^2 + 3*e^2*x^4) + b*(c^2*d + e)*(8*d^2 + 12*d*e*x^2 + 3*e^2*x^4)*ArcSech[c*x])/(3*e^3*(c^2*d + e)*(d + e*x^2)^(3/2)) + (b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^2*x^2]*(3*Sqrt[-c^2]*Sqrt[-(c^2*d) - e]*Sqrt[e]*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSin[(c*Sqrt[e]*Sqrt[1 - c^2*x^2])/(Sqrt[-c^2]*Sqrt[-(c^2*d) - e])] + 8*c^3*Sqrt[d]*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 - c^2*x^2])/Sqrt[-d - e*x^2]]))/(3*c^3*e^3*(-1 + c*x)*Sqrt[d + e*x^2])","A",1
169,1,218,179,0.3770872,"\int \frac{x^3 \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(x^3*(a + b*ArcSech[c*x]))/(d + e*x^2)^(5/2),x]","\frac{-a \left(c^2 d+e\right) \left(2 d+3 e x^2\right)-b \left(c^2 d+e\right) \text{sech}^{-1}(c x) \left(2 d+3 e x^2\right)+b e \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(d+e x^2\right)}{3 e^2 \left(c^2 d+e\right) \left(d+e x^2\right)^{3/2}}-\frac{2 b \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^2 x^2} \sqrt{-d-e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{1-c^2 x^2}}{\sqrt{-d-e x^2}}\right)}{3 \sqrt{d} e^2 (c x-1) \sqrt{d+e x^2}}","-\frac{a+b \text{sech}^{-1}(c x)}{e^2 \sqrt{d+e x^2}}+\frac{d \left(a+b \text{sech}^{-1}(c x)\right)}{3 e^2 \left(d+e x^2\right)^{3/2}}+\frac{2 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{3 \sqrt{d} e^2}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{3 e \left(c^2 d+e\right) \sqrt{d+e x^2}}",1,"(b*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(d + e*x^2) - a*(c^2*d + e)*(2*d + 3*e*x^2) - b*(c^2*d + e)*(2*d + 3*e*x^2)*ArcSech[c*x])/(3*e^2*(c^2*d + e)*(d + e*x^2)^(3/2)) - (2*b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^2*x^2]*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 - c^2*x^2])/Sqrt[-d - e*x^2]])/(3*Sqrt[d]*e^2*(-1 + c*x)*Sqrt[d + e*x^2])","A",1
170,1,204,154,0.3157636,"\int \frac{x \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(x*(a + b*ArcSech[c*x]))/(d + e*x^2)^(5/2),x]","\frac{-a d \left(c^2 d+e\right)-b d \left(c^2 d+e\right) \text{sech}^{-1}(c x)-b e \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(d+e x^2\right)}{3 d e \left(c^2 d+e\right) \left(d+e x^2\right)^{3/2}}-\frac{b \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^2 x^2} \sqrt{-d-e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{1-c^2 x^2}}{\sqrt{-d-e x^2}}\right)}{3 d^{3/2} e (c x-1) \sqrt{d+e x^2}}","-\frac{a+b \text{sech}^{-1}(c x)}{3 e \left(d+e x^2\right)^{3/2}}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{3 d^{3/2} e}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{3 d \left(c^2 d+e\right) \sqrt{d+e x^2}}",1,"(-(a*d*(c^2*d + e)) - b*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(d + e*x^2) - b*d*(c^2*d + e)*ArcSech[c*x])/(3*d*e*(c^2*d + e)*(d + e*x^2)^(3/2)) - (b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^2*x^2]*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 - c^2*x^2])/Sqrt[-d - e*x^2]])/(3*d^(3/2)*e*(-1 + c*x)*Sqrt[d + e*x^2])","A",1
171,0,0,26,43.6758927,"\int \frac{a+b \text{sech}^{-1}(c x)}{x \left(d+e x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSech[c*x])/(x*(d + e*x^2)^(5/2)),x]","\int \frac{a+b \text{sech}^{-1}(c x)}{x \left(d+e x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{a+b \text{sech}^{-1}(c x)}{x \left(d+e x^2\right)^{5/2}},x\right)",0,"Integrate[(a + b*ArcSech[c*x])/(x*(d + e*x^2)^(5/2)), x]","A",-1
172,0,0,26,54.7484686,"\int \frac{a+b \text{sech}^{-1}(c x)}{x^3 \left(d+e x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSech[c*x])/(x^3*(d + e*x^2)^(5/2)),x]","\int \frac{a+b \text{sech}^{-1}(c x)}{x^3 \left(d+e x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{a+b \text{sech}^{-1}(c x)}{x^3 \left(d+e x^2\right)^{5/2}},x\right)",0,"Integrate[(a + b*ArcSech[c*x])/(x^3*(d + e*x^2)^(5/2)), x]","A",-1
173,0,0,26,14.8911467,"\int \frac{x^6 \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(x^6*(a + b*ArcSech[c*x]))/(d + e*x^2)^(5/2),x]","\int \frac{x^6 \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{x^6 \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}},x\right)",0,"Integrate[(x^6*(a + b*ArcSech[c*x]))/(d + e*x^2)^(5/2), x]","A",-1
174,0,0,26,13.5357431,"\int \frac{x^4 \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(x^4*(a + b*ArcSech[c*x]))/(d + e*x^2)^(5/2),x]","\int \frac{x^4 \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{x^4 \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}},x\right)",0,"Integrate[(x^4*(a + b*ArcSech[c*x]))/(d + e*x^2)^(5/2), x]","A",-1
175,1,488,246,2.6558335,"\int \frac{x^2 \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(x^2*(a + b*ArcSech[c*x]))/(d + e*x^2)^(5/2),x]","\frac{a x^3-\frac{b \sqrt{\frac{1-c x}{c x+1}} \left(d+e x^2\right) (e x-c d)}{e \left(c^2 d+e\right)}+\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(d+e x^2\right) \sqrt{\frac{c \left(\sqrt{d}+i \sqrt{e} x\right)}{(c x+1) \left(c \sqrt{d}+i \sqrt{e}\right)}} \sqrt{\frac{c \left(\sqrt{e} x+i \sqrt{d}\right)}{(c x+1) \left(\sqrt{e}+i c \sqrt{d}\right)}} \left(\left(\sqrt{e}+i c \sqrt{d}\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{\left(d c^2+e\right) (1-c x)}{\left(\sqrt{d} c+i \sqrt{e}\right)^2 (c x+1)}}\right)|\frac{\left(\sqrt{d} c+i \sqrt{e}\right)^2}{\left(c \sqrt{d}-i \sqrt{e}\right)^2}\right)-2 \sqrt{e} F\left(i \sinh ^{-1}\left(\sqrt{\frac{\left(d c^2+e\right) (1-c x)}{\left(\sqrt{d} c+i \sqrt{e}\right)^2 (c x+1)}}\right)|\frac{\left(\sqrt{d} c+i \sqrt{e}\right)^2}{\left(c \sqrt{d}-i \sqrt{e}\right)^2}\right)\right)}{c e \left(c \sqrt{d}+i \sqrt{e}\right) \sqrt{\frac{(c x-1) \left(\sqrt{e}+i c \sqrt{d}\right)}{(c x+1) \left(\sqrt{e}-i c \sqrt{d}\right)}}}+b x^3 \text{sech}^{-1}(c x)}{3 d \left(d+e x^2\right)^{3/2}}","\frac{x^3 \left(a+b \text{sech}^{-1}(c x)\right)}{3 d \left(d+e x^2\right)^{3/2}}-\frac{b x \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{3 d \left(c^2 d+e\right) \sqrt{d+e x^2}}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{3 c d e \sqrt{d+e x^2}}-\frac{b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{3 d e \left(c^2 d+e\right) \sqrt{\frac{e x^2}{d}+1}}",1,"(a*x^3 - (b*Sqrt[(1 - c*x)/(1 + c*x)]*(-(c*d) + e*x)*(d + e*x^2))/(e*(c^2*d + e)) + b*x^3*ArcSech[c*x] + (b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[(c*(Sqrt[d] + I*Sqrt[e]*x))/((c*Sqrt[d] + I*Sqrt[e])*(1 + c*x))]*Sqrt[(c*(I*Sqrt[d] + Sqrt[e]*x))/((I*c*Sqrt[d] + Sqrt[e])*(1 + c*x))]*(d + e*x^2)*((I*c*Sqrt[d] + Sqrt[e])*EllipticE[I*ArcSinh[Sqrt[((c^2*d + e)*(1 - c*x))/((c*Sqrt[d] + I*Sqrt[e])^2*(1 + c*x))]], (c*Sqrt[d] + I*Sqrt[e])^2/(c*Sqrt[d] - I*Sqrt[e])^2] - 2*Sqrt[e]*EllipticF[I*ArcSinh[Sqrt[((c^2*d + e)*(1 - c*x))/((c*Sqrt[d] + I*Sqrt[e])^2*(1 + c*x))]], (c*Sqrt[d] + I*Sqrt[e])^2/(c*Sqrt[d] - I*Sqrt[e])^2]))/(c*(c*Sqrt[d] + I*Sqrt[e])*e*Sqrt[((I*c*Sqrt[d] + Sqrt[e])*(-1 + c*x))/(((-I)*c*Sqrt[d] + Sqrt[e])*(1 + c*x))]))/(3*d*(d + e*x^2)^(3/2))","C",1
176,1,517,266,5.5046892,"\int \frac{a+b \text{sech}^{-1}(c x)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSech[c*x])/(d + e*x^2)^(5/2),x]","\frac{a x \left(3 d+2 e x^2\right)+\frac{b \sqrt{\frac{1-c x}{c x+1}} \left(d+e x^2\right) (e x-c d)}{c^2 d+e}-\frac{i b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left(d+e x^2\right) \sqrt{\frac{c \left(\sqrt{d}-i \sqrt{e} x\right)}{(c x+1) \left(c \sqrt{d}-i \sqrt{e}\right)}} \sqrt{\frac{c \left(\sqrt{d}+i \sqrt{e} x\right)}{(c x+1) \left(c \sqrt{d}+i \sqrt{e}\right)}} \left(\left(c \sqrt{d}-i \sqrt{e}\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{\left(d c^2+e\right) (1-c x)}{\left(\sqrt{d} c+i \sqrt{e}\right)^2 (c x+1)}}\right)|\frac{\left(\sqrt{d} c+i \sqrt{e}\right)^2}{\left(c \sqrt{d}-i \sqrt{e}\right)^2}\right)-2 \left(3 c \sqrt{d}+2 i \sqrt{e}\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{\left(d c^2+e\right) (1-c x)}{\left(\sqrt{d} c+i \sqrt{e}\right)^2 (c x+1)}}\right)|\frac{\left(\sqrt{d} c+i \sqrt{e}\right)^2}{\left(c \sqrt{d}-i \sqrt{e}\right)^2}\right)\right)}{c \left(c \sqrt{d}+i \sqrt{e}\right) \sqrt{-\frac{(c x-1) \left(c \sqrt{d}-i \sqrt{e}\right)}{(c x+1) \left(c \sqrt{d}+i \sqrt{e}\right)}}}+b x \text{sech}^{-1}(c x) \left(3 d+2 e x^2\right)}{3 d^2 \left(d+e x^2\right)^{3/2}}","\frac{2 x \left(a+b \text{sech}^{-1}(c x)\right)}{3 d^2 \sqrt{d+e x^2}}+\frac{x \left(a+b \text{sech}^{-1}(c x)\right)}{3 d \left(d+e x^2\right)^{3/2}}+\frac{b e x \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{3 d^2 \left(c^2 d+e\right) \sqrt{d+e x^2}}+\frac{2 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{3 c d^2 \sqrt{d+e x^2}}+\frac{b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{3 d^2 \left(c^2 d+e\right) \sqrt{\frac{e x^2}{d}+1}}",1,"((b*Sqrt[(1 - c*x)/(1 + c*x)]*(-(c*d) + e*x)*(d + e*x^2))/(c^2*d + e) + a*x*(3*d + 2*e*x^2) + b*x*(3*d + 2*e*x^2)*ArcSech[c*x] - (I*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[(c*(Sqrt[d] - I*Sqrt[e]*x))/((c*Sqrt[d] - I*Sqrt[e])*(1 + c*x))]*Sqrt[(c*(Sqrt[d] + I*Sqrt[e]*x))/((c*Sqrt[d] + I*Sqrt[e])*(1 + c*x))]*(d + e*x^2)*((c*Sqrt[d] - I*Sqrt[e])*EllipticE[I*ArcSinh[Sqrt[((c^2*d + e)*(1 - c*x))/((c*Sqrt[d] + I*Sqrt[e])^2*(1 + c*x))]], (c*Sqrt[d] + I*Sqrt[e])^2/(c*Sqrt[d] - I*Sqrt[e])^2] - 2*(3*c*Sqrt[d] + (2*I)*Sqrt[e])*EllipticF[I*ArcSinh[Sqrt[((c^2*d + e)*(1 - c*x))/((c*Sqrt[d] + I*Sqrt[e])^2*(1 + c*x))]], (c*Sqrt[d] + I*Sqrt[e])^2/(c*Sqrt[d] - I*Sqrt[e])^2]))/(c*(c*Sqrt[d] + I*Sqrt[e])*Sqrt[-(((c*Sqrt[d] - I*Sqrt[e])*(-1 + c*x))/((c*Sqrt[d] + I*Sqrt[e])*(1 + c*x)))]))/(3*d^2*(d + e*x^2)^(3/2))","C",1
177,0,0,596,0.2463809,"\int (f x)^m \left(d+e x^2\right)^3 \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d + e*x^2)^3*(a + b*ArcSech[c*x]),x]","\int (f x)^m \left(d+e x^2\right)^3 \left(a+b \text{sech}^{-1}(c x)\right) \, dx","\frac{d^3 (f x)^{m+1} \left(a+b \text{sech}^{-1}(c x)\right)}{f (m+1)}+\frac{3 d^2 e (f x)^{m+3} \left(a+b \text{sech}^{-1}(c x)\right)}{f^3 (m+3)}+\frac{3 d e^2 (f x)^{m+5} \left(a+b \text{sech}^{-1}(c x)\right)}{f^5 (m+5)}+\frac{e^3 (f x)^{m+7} \left(a+b \text{sech}^{-1}(c x)\right)}{f^7 (m+7)}-\frac{b e^3 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} (f x)^{m+5}}{c^2 f^5 (m+6) (m+7)}-\frac{b e^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} (f x)^{m+3} \left(3 c^2 d \left(m^2+13 m+42\right)+e (m+5)^2\right)}{c^4 f^3 (m+4) (m+5) (m+6) (m+7)}-\frac{b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} (f x)^{m+1} \left(3 c^4 d^2 \left(m^4+22 m^3+179 m^2+638 m+840\right)+3 c^2 d e (m+3)^2 \left(m^2+13 m+42\right)+e^2 \left(m^2+8 m+15\right)^2\right)}{c^6 f (m+2) (m+3) (m+4) (m+5) (m+6) (m+7)}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} (f x)^{m+1} \left(\frac{c^6 d^3 (m+2) (m+4) (m+6)}{m+1}+\frac{e (m+1) \left(3 c^4 d^2 \left(m^4+22 m^3+179 m^2+638 m+840\right)+3 c^2 d e (m+3)^2 \left(m^2+13 m+42\right)+e^2 \left(m^2+8 m+15\right)^2\right)}{(m+3) (m+5) (m+7)}\right) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right)}{c^6 f (m+1) (m+2) (m+4) (m+6)}",1,"Integrate[(f*x)^m*(d + e*x^2)^3*(a + b*ArcSech[c*x]), x]","F",-1
178,0,0,372,0.1605409,"\int (f x)^m \left(d+e x^2\right)^2 \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d + e*x^2)^2*(a + b*ArcSech[c*x]),x]","\int (f x)^m \left(d+e x^2\right)^2 \left(a+b \text{sech}^{-1}(c x)\right) \, dx","\frac{d^2 (f x)^{m+1} \left(a+b \text{sech}^{-1}(c x)\right)}{f (m+1)}+\frac{2 d e (f x)^{m+3} \left(a+b \text{sech}^{-1}(c x)\right)}{f^3 (m+3)}+\frac{e^2 (f x)^{m+5} \left(a+b \text{sech}^{-1}(c x)\right)}{f^5 (m+5)}-\frac{b e^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} (f x)^{m+3}}{c^2 f^3 (m+4) (m+5)}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} (f x)^{m+1} \left(c^4 d^2 (m+2) (m+3) (m+4) (m+5)+e (m+1)^2 \left(2 c^2 d \left(m^2+9 m+20\right)+e (m+3)^2\right)\right) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right)}{c^4 f (m+1)^2 (m+2) (m+3) (m+4) (m+5)}-\frac{b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} (f x)^{m+1} \left(2 c^2 d \left(m^2+9 m+20\right)+e (m+3)^2\right)}{c^4 f (m+2) (m+3) (m+4) (m+5)}",1,"Integrate[(f*x)^m*(d + e*x^2)^2*(a + b*ArcSech[c*x]), x]","F",-1
179,0,0,206,0.125647,"\int (f x)^m \left(d+e x^2\right) \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d + e*x^2)*(a + b*ArcSech[c*x]),x]","\int (f x)^m \left(d+e x^2\right) \left(a+b \text{sech}^{-1}(c x)\right) \, dx","\frac{d (f x)^{m+1} \left(a+b \text{sech}^{-1}(c x)\right)}{f (m+1)}+\frac{e (f x)^{m+3} \left(a+b \text{sech}^{-1}(c x)\right)}{f^3 (m+3)}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} (f x)^{m+1} \left(c^2 d (m+2) (m+3)+e (m+1)^2\right) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right)}{c^2 f (m+1)^2 (m+2) (m+3)}-\frac{b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} (f x)^{m+1}}{c^2 f (m+2) (m+3)}",1,"Integrate[(f*x)^m*(d + e*x^2)*(a + b*ArcSech[c*x]), x]","F",-1
180,0,0,26,2.4825723,"\int \frac{(f x)^m \left(a+b \text{sech}^{-1}(c x)\right)}{d+e x^2} \, dx","Integrate[((f*x)^m*(a + b*ArcSech[c*x]))/(d + e*x^2),x]","\int \frac{(f x)^m \left(a+b \text{sech}^{-1}(c x)\right)}{d+e x^2} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \text{sech}^{-1}(c x)\right)}{d+e x^2},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcSech[c*x]))/(d + e*x^2), x]","A",-1
181,0,0,26,7.5501449,"\int \frac{(f x)^m \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[((f*x)^m*(a + b*ArcSech[c*x]))/(d + e*x^2)^2,x]","\int \frac{(f x)^m \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^2},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcSech[c*x]))/(d + e*x^2)^2, x]","A",-1
182,0,0,28,1.0771177,"\int (f x)^m \left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]),x]","\int (f x)^m \left(d+e x^2\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right) \, dx","\text{Int}\left(\left(d+e x^2\right)^{3/2} (f x)^m \left(a+b \text{sech}^{-1}(c x)\right),x\right)",0,"Integrate[(f*x)^m*(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]), x]","A",-1
183,0,0,28,0.1221296,"\int (f x)^m \sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]),x]","\int (f x)^m \sqrt{d+e x^2} \left(a+b \text{sech}^{-1}(c x)\right) \, dx","\text{Int}\left(\sqrt{d+e x^2} (f x)^m \left(a+b \text{sech}^{-1}(c x)\right),x\right)",0,"Integrate[(f*x)^m*Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]), x]","A",-1
184,0,0,28,1.4203363,"\int \frac{(f x)^m \left(a+b \text{sech}^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","Integrate[((f*x)^m*(a + b*ArcSech[c*x]))/Sqrt[d + e*x^2],x]","\int \frac{(f x)^m \left(a+b \text{sech}^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \text{sech}^{-1}(c x)\right)}{\sqrt{d+e x^2}},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcSech[c*x]))/Sqrt[d + e*x^2], x]","A",-1
185,0,0,28,1.7294942,"\int \frac{(f x)^m \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[((f*x)^m*(a + b*ArcSech[c*x]))/(d + e*x^2)^(3/2),x]","\int \frac{(f x)^m \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \text{sech}^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcSech[c*x]))/(d + e*x^2)^(3/2), x]","A",-1
186,1,213,473,0.4425276,"\int \frac{x^{11} \left(a+b \text{sech}^{-1}(c x)\right)}{\sqrt{1-c^4 x^4}} \, dx","Integrate[(x^11*(a + b*ArcSech[c*x]))/Sqrt[1 - c^4*x^4],x]","\frac{-105 a \sqrt{1-c^4 x^4} \left(3 c^8 x^8+4 c^4 x^4+8\right)-840 b \log \left(-\sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^4 x^4}-c x+1\right)-105 b \sqrt{1-c^4 x^4} \left(3 c^8 x^8+4 c^4 x^4+8\right) \text{sech}^{-1}(c x)+\frac{b \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^4 x^4} \left(35 c^8 x^8+5 c^6 x^6+78 c^4 x^4+36 c^2 x^2+768\right)}{c x-1}+840 b \log (x (1-c x))}{3150 c^{12}}","-\frac{\left(1-c^4 x^4\right)^{5/2} \left(a+b \text{sech}^{-1}(c x)\right)}{10 c^{12}}+\frac{\left(1-c^4 x^4\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right)}{3 c^{12}}-\frac{\sqrt{1-c^4 x^4} \left(a+b \text{sech}^{-1}(c x)\right)}{2 c^{12}}-\frac{b \sqrt{1-c^2 x^2} \left(c^2 x^2+1\right)^{9/2}}{90 c^{13} x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{3 b \sqrt{1-c^2 x^2} \left(c^2 x^2+1\right)^{7/2}}{70 c^{13} x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{13 b \sqrt{1-c^2 x^2} \left(c^2 x^2+1\right)^{5/2}}{150 c^{13} x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{7 b \sqrt{1-c^2 x^2} \left(c^2 x^2+1\right)^{3/2}}{90 c^{13} x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{c^2 x^2+1}}{15 c^{13} x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{4 b \sqrt{1-c^2 x^2} \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)}{15 c^{13} x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}",1,"(-105*a*Sqrt[1 - c^4*x^4]*(8 + 4*c^4*x^4 + 3*c^8*x^8) + (b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^4*x^4]*(768 + 36*c^2*x^2 + 78*c^4*x^4 + 5*c^6*x^6 + 35*c^8*x^8))/(-1 + c*x) - 105*b*Sqrt[1 - c^4*x^4]*(8 + 4*c^4*x^4 + 3*c^8*x^8)*ArcSech[c*x] + 840*b*Log[x*(1 - c*x)] - 840*b*Log[1 - c*x - Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^4*x^4]])/(3150*c^12)","A",1
187,1,178,316,0.382681,"\int \frac{x^7 \left(a+b \text{sech}^{-1}(c x)\right)}{\sqrt{1-c^4 x^4}} \, dx","Integrate[(x^7*(a + b*ArcSech[c*x]))/Sqrt[1 - c^4*x^4],x]","\frac{-15 a \sqrt{1-c^4 x^4} \left(c^4 x^4+2\right)-30 b \log \left(-\sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^4 x^4}-c x+1\right)-15 b \sqrt{1-c^4 x^4} \left(c^4 x^4+2\right) \text{sech}^{-1}(c x)+\frac{b \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^4 x^4} \left(3 c^4 x^4+c^2 x^2+28\right)}{c x-1}+30 b \log (x (1-c x))}{90 c^8}","\frac{\left(1-c^4 x^4\right)^{3/2} \left(a+b \text{sech}^{-1}(c x)\right)}{6 c^8}-\frac{\sqrt{1-c^4 x^4} \left(a+b \text{sech}^{-1}(c x)\right)}{2 c^8}-\frac{b \sqrt{1-c^2 x^2} \left(c^2 x^2+1\right)^{5/2}}{30 c^9 x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{b \sqrt{1-c^2 x^2} \left(c^2 x^2+1\right)^{3/2}}{18 c^9 x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{b \sqrt{1-c^2 x^2} \sqrt{c^2 x^2+1}}{3 c^9 x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{b \sqrt{1-c^2 x^2} \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)}{3 c^9 x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}",1,"(-15*a*Sqrt[1 - c^4*x^4]*(2 + c^4*x^4) + (b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^4*x^4]*(28 + c^2*x^2 + 3*c^4*x^4))/(-1 + c*x) - 15*b*Sqrt[1 - c^4*x^4]*(2 + c^4*x^4)*ArcSech[c*x] + 30*b*Log[x*(1 - c*x)] - 30*b*Log[1 - c*x - Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^4*x^4]])/(90*c^8)","A",1
188,1,140,159,0.3527845,"\int \frac{x^3 \left(a+b \text{sech}^{-1}(c x)\right)}{\sqrt{1-c^4 x^4}} \, dx","Integrate[(x^3*(a + b*ArcSech[c*x]))/Sqrt[1 - c^4*x^4],x]","-\frac{a \sqrt{1-c^4 x^4}+\frac{b \sqrt{1-c^4 x^4}}{\sqrt{\frac{1-c x}{c x+1}} (c x+1)}+b \log \left(-\sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^4 x^4}-c x+1\right)+b \sqrt{1-c^4 x^4} \text{sech}^{-1}(c x)-b \log (x (1-c x))}{2 c^4}","-\frac{\sqrt{1-c^4 x^4} \left(a+b \text{sech}^{-1}(c x)\right)}{2 c^4}-\frac{b \sqrt{1-c^2 x^2} \sqrt{c^2 x^2+1}}{2 c^5 x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{b \sqrt{1-c^2 x^2} \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)}{2 c^5 x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}",1,"-1/2*(a*Sqrt[1 - c^4*x^4] + (b*Sqrt[1 - c^4*x^4])/(Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)) + b*Sqrt[1 - c^4*x^4]*ArcSech[c*x] - b*Log[x*(1 - c*x)] + b*Log[1 - c*x - Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^4*x^4]])/c^4","A",1
189,0,0,29,0.4144918,"\int \frac{a+b \text{sech}^{-1}(c x)}{x \sqrt{1-c^4 x^4}} \, dx","Integrate[(a + b*ArcSech[c*x])/(x*Sqrt[1 - c^4*x^4]),x]","\int \frac{a+b \text{sech}^{-1}(c x)}{x \sqrt{1-c^4 x^4}} \, dx","\text{Int}\left(\frac{a+b \text{sech}^{-1}(c x)}{x \sqrt{1-c^4 x^4}},x\right)",0,"Integrate[(a + b*ArcSech[c*x])/(x*Sqrt[1 - c^4*x^4]), x]","A",-1
190,0,0,29,3.5385206,"\int \frac{a+b \text{sech}^{-1}(c x)}{x^5 \sqrt{1-c^4 x^4}} \, dx","Integrate[(a + b*ArcSech[c*x])/(x^5*Sqrt[1 - c^4*x^4]),x]","\int \frac{a+b \text{sech}^{-1}(c x)}{x^5 \sqrt{1-c^4 x^4}} \, dx","\text{Int}\left(\frac{a+b \text{sech}^{-1}(c x)}{x^5 \sqrt{1-c^4 x^4}},x\right)",0,"Integrate[(a + b*ArcSech[c*x])/(x^5*Sqrt[1 - c^4*x^4]), x]","A",-1