3.1.10 \(\int x^4 \text {sech}^{-1}(a x)^3 \, dx\) [10]

3.1.10.1 Optimal result

Integrand size = 10, antiderivative size = 297 \[ \int x^4 \text {sech}^{-1}(a x)^3 \, dx=\frac {x \sqrt {\frac {1-a x}{1+a x}} (1+a x)}{20 a^4}-\frac {9 x \text {sech}^{-1}(a x)}{20 a^4}-\frac {x^3 \text {sech}^{-1}(a x)}{10 a^2}-\frac {9 x \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)^2}{40 a^4}-\frac {3 x^3 \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)^2}{20 a^2}+\frac {1}{5} x^5 \text {sech}^{-1}(a x)^3-\frac {9 \text {sech}^{-1}(a x)^2 \arctan \left (e^{\text {sech}^{-1}(a x)}\right )}{20 a^5}+\frac {\arctan \left (\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)}{a x}\right )}{2 a^5}+\frac {9 i \text {sech}^{-1}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {sech}^{-1}(a x)}\right )}{20 a^5}-\frac {9 i \text {sech}^{-1}(a x) \operatorname {PolyLog}\left (2,i e^{\text {sech}^{-1}(a x)}\right )}{20 a^5}-\frac {9 i \operatorname {PolyLog}\left (3,-i e^{\text {sech}^{-1}(a x)}\right )}{20 a^5}+\frac {9 i \operatorname {PolyLog}\left (3,i e^{\text {sech}^{-1}(a x)}\right )}{20 a^5} \]

output

-9/20*x*arcsech(a*x)/a^4-1/10*x^3*arcsech(a*x)/a^2+1/5*x^5*arcsech(a*x)^3- 
9/20*arcsech(a*x)^2*arctan(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))/a^5+1/2* 
arctan((a*x+1)*((-a*x+1)/(a*x+1))^(1/2)/a/x)/a^5+9/20*I*arcsech(a*x)*polyl 
og(2,-I*(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2)))/a^5-9/20*I*arcsech(a*x)*p 
olylog(2,I*(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2)))/a^5-9/20*I*polylog(3,- 
I*(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2)))/a^5+9/20*I*polylog(3,I*(1/a/x+( 
1/a/x-1)^(1/2)*(1+1/a/x)^(1/2)))/a^5+1/20*x*(a*x+1)*((-a*x+1)/(a*x+1))^(1/ 
2)/a^4-9/40*x*(a*x+1)*arcsech(a*x)^2*((-a*x+1)/(a*x+1))^(1/2)/a^4-3/20*x^3 
*(a*x+1)*arcsech(a*x)^2*((-a*x+1)/(a*x+1))^(1/2)/a^2
 

3.1.10.2 Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.95 \[ \int x^4 \text {sech}^{-1}(a x)^3 \, dx=\frac {2 a x \sqrt {\frac {1-a x}{1+a x}} (1+a x)-18 a x \text {sech}^{-1}(a x)-4 a^3 x^3 \text {sech}^{-1}(a x)-9 a x \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)^2-6 a^3 x^3 \sqrt {\frac {1-a x}{1+a x}} (1+a x) \text {sech}^{-1}(a x)^2+8 a^5 x^5 \text {sech}^{-1}(a x)^3+40 \arctan \left (\tanh \left (\frac {1}{2} \text {sech}^{-1}(a x)\right )\right )+9 i \text {sech}^{-1}(a x)^2 \log \left (1-i e^{-\text {sech}^{-1}(a x)}\right )-9 i \text {sech}^{-1}(a x)^2 \log \left (1+i e^{-\text {sech}^{-1}(a x)}\right )+18 i \text {sech}^{-1}(a x) \operatorname {PolyLog}\left (2,-i e^{-\text {sech}^{-1}(a x)}\right )-18 i \text {sech}^{-1}(a x) \operatorname {PolyLog}\left (2,i e^{-\text {sech}^{-1}(a x)}\right )+18 i \operatorname {PolyLog}\left (3,-i e^{-\text {sech}^{-1}(a x)}\right )-18 i \operatorname {PolyLog}\left (3,i e^{-\text {sech}^{-1}(a x)}\right )}{40 a^5} \]

input

Integrate[x^4*ArcSech[a*x]^3,x]
 

output

(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)
 

3.1.10.3 Rubi [A] (verified)

Time = 1.05 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.07, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.500, Rules used = {6839, 5941, 3042, 4674, 3042, 4255, 3042, 4257, 4674, 3042, 4257, 4668, 3011, 2720, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^4 \text {sech}^{-1}(a x)^3 \, dx\)

\(\Big \downarrow \) 6839

\(\displaystyle -\frac {\int a^5 x^5 \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^3d\text {sech}^{-1}(a x)}{a^5}\)

\(\Big \downarrow \) 5941

\(\displaystyle -\frac {\frac {3}{5} \int a^5 x^5 \text {sech}^{-1}(a x)^2d\text {sech}^{-1}(a x)-\frac {1}{5} a^5 x^5 \text {sech}^{-1}(a x)^3}{a^5}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {1}{5} a^5 x^5 \text {sech}^{-1}(a x)^3+\frac {3}{5} \int \text {sech}^{-1}(a x)^2 \csc \left (i \text {sech}^{-1}(a x)+\frac {\pi }{2}\right )^5d\text {sech}^{-1}(a x)}{a^5}\)

\(\Big \downarrow \) 4674

\(\displaystyle -\frac {\frac {3}{5} \left (-\frac {1}{6} \int a^3 x^3d\text {sech}^{-1}(a x)+\frac {3}{4} \int a^3 x^3 \text {sech}^{-1}(a x)^2d\text {sech}^{-1}(a x)+\frac {1}{4} a^3 x^3 \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2+\frac {1}{6} a^3 x^3 \text {sech}^{-1}(a x)\right )-\frac {1}{5} a^5 x^5 \text {sech}^{-1}(a x)^3}{a^5}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {1}{5} a^5 x^5 \text {sech}^{-1}(a x)^3+\frac {3}{5} \left (-\frac {1}{6} \int \csc \left (i \text {sech}^{-1}(a x)+\frac {\pi }{2}\right )^3d\text {sech}^{-1}(a x)+\frac {3}{4} \int \text {sech}^{-1}(a x)^2 \csc \left (i \text {sech}^{-1}(a x)+\frac {\pi }{2}\right )^3d\text {sech}^{-1}(a x)+\frac {1}{4} a^3 x^3 \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2+\frac {1}{6} a^3 x^3 \text {sech}^{-1}(a x)\right )}{a^5}\)

\(\Big \downarrow \) 4255

\(\displaystyle -\frac {-\frac {1}{5} a^5 x^5 \text {sech}^{-1}(a x)^3+\frac {3}{5} \left (\frac {1}{6} \left (-\frac {1}{2} \int a xd\text {sech}^{-1}(a x)-\frac {1}{2} a x \sqrt {\frac {1-a x}{a x+1}} (a x+1)\right )+\frac {3}{4} \int \text {sech}^{-1}(a x)^2 \csc \left (i \text {sech}^{-1}(a x)+\frac {\pi }{2}\right )^3d\text {sech}^{-1}(a x)+\frac {1}{4} a^3 x^3 \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2+\frac {1}{6} a^3 x^3 \text {sech}^{-1}(a x)\right )}{a^5}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {1}{5} a^5 x^5 \text {sech}^{-1}(a x)^3+\frac {3}{5} \left (\frac {1}{6} \left (-\frac {1}{2} a x \sqrt {\frac {1-a x}{a x+1}} (a x+1)-\frac {1}{2} \int \csc \left (i \text {sech}^{-1}(a x)+\frac {\pi }{2}\right )d\text {sech}^{-1}(a x)\right )+\frac {3}{4} \int \text {sech}^{-1}(a x)^2 \csc \left (i \text {sech}^{-1}(a x)+\frac {\pi }{2}\right )^3d\text {sech}^{-1}(a x)+\frac {1}{4} a^3 x^3 \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2+\frac {1}{6} a^3 x^3 \text {sech}^{-1}(a x)\right )}{a^5}\)

\(\Big \downarrow \) 4257

\(\displaystyle -\frac {-\frac {1}{5} a^5 x^5 \text {sech}^{-1}(a x)^3+\frac {3}{5} \left (\frac {3}{4} \int \text {sech}^{-1}(a x)^2 \csc \left (i \text {sech}^{-1}(a x)+\frac {\pi }{2}\right )^3d\text {sech}^{-1}(a x)+\frac {1}{4} a^3 x^3 \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2+\frac {1}{6} a^3 x^3 \text {sech}^{-1}(a x)+\frac {1}{6} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)}{a x}\right )-\frac {1}{2} a x \sqrt {\frac {1-a x}{a x+1}} (a x+1)\right )\right )}{a^5}\)

\(\Big \downarrow \) 4674

\(\displaystyle -\frac {\frac {3}{5} \left (\frac {3}{4} \left (-\int a xd\text {sech}^{-1}(a x)+\frac {1}{2} \int a x \text {sech}^{-1}(a x)^2d\text {sech}^{-1}(a x)+\frac {1}{2} a x \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2+a x \text {sech}^{-1}(a x)\right )+\frac {1}{4} a^3 x^3 \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2+\frac {1}{6} a^3 x^3 \text {sech}^{-1}(a x)+\frac {1}{6} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)}{a x}\right )-\frac {1}{2} a x \sqrt {\frac {1-a x}{a x+1}} (a x+1)\right )\right )-\frac {1}{5} a^5 x^5 \text {sech}^{-1}(a x)^3}{a^5}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {1}{5} a^5 x^5 \text {sech}^{-1}(a x)^3+\frac {3}{5} \left (\frac {3}{4} \left (-\int \csc \left (i \text {sech}^{-1}(a x)+\frac {\pi }{2}\right )d\text {sech}^{-1}(a x)+\frac {1}{2} \int \text {sech}^{-1}(a x)^2 \csc \left (i \text {sech}^{-1}(a x)+\frac {\pi }{2}\right )d\text {sech}^{-1}(a x)+\frac {1}{2} a x \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2+a x \text {sech}^{-1}(a x)\right )+\frac {1}{4} a^3 x^3 \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2+\frac {1}{6} a^3 x^3 \text {sech}^{-1}(a x)+\frac {1}{6} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)}{a x}\right )-\frac {1}{2} a x \sqrt {\frac {1-a x}{a x+1}} (a x+1)\right )\right )}{a^5}\)

\(\Big \downarrow \) 4257

\(\displaystyle -\frac {-\frac {1}{5} a^5 x^5 \text {sech}^{-1}(a x)^3+\frac {3}{5} \left (\frac {3}{4} \left (\frac {1}{2} \int \text {sech}^{-1}(a x)^2 \csc \left (i \text {sech}^{-1}(a x)+\frac {\pi }{2}\right )d\text {sech}^{-1}(a x)-\arctan \left (\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)}{a x}\right )+\frac {1}{2} a x \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2+a x \text {sech}^{-1}(a x)\right )+\frac {1}{4} a^3 x^3 \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2+\frac {1}{6} a^3 x^3 \text {sech}^{-1}(a x)+\frac {1}{6} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)}{a x}\right )-\frac {1}{2} a x \sqrt {\frac {1-a x}{a x+1}} (a x+1)\right )\right )}{a^5}\)

\(\Big \downarrow \) 4668

\(\displaystyle -\frac {-\frac {1}{5} a^5 x^5 \text {sech}^{-1}(a x)^3+\frac {3}{5} \left (\frac {3}{4} \left (\frac {1}{2} \left (-2 i \int \text {sech}^{-1}(a x) \log \left (1-i e^{\text {sech}^{-1}(a x)}\right )d\text {sech}^{-1}(a x)+2 i \int \text {sech}^{-1}(a x) \log \left (1+i e^{\text {sech}^{-1}(a x)}\right )d\text {sech}^{-1}(a x)+2 \text {sech}^{-1}(a x)^2 \arctan \left (e^{\text {sech}^{-1}(a x)}\right )\right )-\arctan \left (\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)}{a x}\right )+\frac {1}{2} a x \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2+a x \text {sech}^{-1}(a x)\right )+\frac {1}{4} a^3 x^3 \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2+\frac {1}{6} a^3 x^3 \text {sech}^{-1}(a x)+\frac {1}{6} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)}{a x}\right )-\frac {1}{2} a x \sqrt {\frac {1-a x}{a x+1}} (a x+1)\right )\right )}{a^5}\)

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {-\frac {1}{5} a^5 x^5 \text {sech}^{-1}(a x)^3+\frac {3}{5} \left (\frac {3}{4} \left (\frac {1}{2} \left (2 i \left (\int \operatorname {PolyLog}\left (2,-i e^{\text {sech}^{-1}(a x)}\right )d\text {sech}^{-1}(a x)-\text {sech}^{-1}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {sech}^{-1}(a x)}\right )\right )-2 i \left (\int \operatorname {PolyLog}\left (2,i e^{\text {sech}^{-1}(a x)}\right )d\text {sech}^{-1}(a x)-\text {sech}^{-1}(a x) \operatorname {PolyLog}\left (2,i e^{\text {sech}^{-1}(a x)}\right )\right )+2 \text {sech}^{-1}(a x)^2 \arctan \left (e^{\text {sech}^{-1}(a x)}\right )\right )-\arctan \left (\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)}{a x}\right )+\frac {1}{2} a x \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2+a x \text {sech}^{-1}(a x)\right )+\frac {1}{4} a^3 x^3 \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2+\frac {1}{6} a^3 x^3 \text {sech}^{-1}(a x)+\frac {1}{6} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)}{a x}\right )-\frac {1}{2} a x \sqrt {\frac {1-a x}{a x+1}} (a x+1)\right )\right )}{a^5}\)

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {-\frac {1}{5} a^5 x^5 \text {sech}^{-1}(a x)^3+\frac {3}{5} \left (\frac {3}{4} \left (\frac {1}{2} \left (2 i \left (\int e^{-\text {sech}^{-1}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {sech}^{-1}(a x)}\right )de^{\text {sech}^{-1}(a x)}-\text {sech}^{-1}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {sech}^{-1}(a x)}\right )\right )-2 i \left (\int e^{-\text {sech}^{-1}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {sech}^{-1}(a x)}\right )de^{\text {sech}^{-1}(a x)}-\text {sech}^{-1}(a x) \operatorname {PolyLog}\left (2,i e^{\text {sech}^{-1}(a x)}\right )\right )+2 \text {sech}^{-1}(a x)^2 \arctan \left (e^{\text {sech}^{-1}(a x)}\right )\right )-\arctan \left (\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)}{a x}\right )+\frac {1}{2} a x \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2+a x \text {sech}^{-1}(a x)\right )+\frac {1}{4} a^3 x^3 \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2+\frac {1}{6} a^3 x^3 \text {sech}^{-1}(a x)+\frac {1}{6} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)}{a x}\right )-\frac {1}{2} a x \sqrt {\frac {1-a x}{a x+1}} (a x+1)\right )\right )}{a^5}\)

\(\Big \downarrow \) 7143

\(\displaystyle -\frac {-\frac {1}{5} a^5 x^5 \text {sech}^{-1}(a x)^3+\frac {3}{5} \left (\frac {1}{4} a^3 x^3 \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2+\frac {1}{6} a^3 x^3 \text {sech}^{-1}(a x)+\frac {3}{4} \left (\frac {1}{2} \left (2 \text {sech}^{-1}(a x)^2 \arctan \left (e^{\text {sech}^{-1}(a x)}\right )+2 i \left (\operatorname {PolyLog}\left (3,-i e^{\text {sech}^{-1}(a x)}\right )-\text {sech}^{-1}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {sech}^{-1}(a x)}\right )\right )-2 i \left (\operatorname {PolyLog}\left (3,i e^{\text {sech}^{-1}(a x)}\right )-\text {sech}^{-1}(a x) \operatorname {PolyLog}\left (2,i e^{\text {sech}^{-1}(a x)}\right )\right )\right )-\arctan \left (\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)}{a x}\right )+\frac {1}{2} a x \sqrt {\frac {1-a x}{a x+1}} (a x+1) \text {sech}^{-1}(a x)^2+a x \text {sech}^{-1}(a x)\right )+\frac {1}{6} \left (-\frac {1}{2} \arctan \left (\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)}{a x}\right )-\frac {1}{2} a x \sqrt {\frac {1-a x}{a x+1}} (a x+1)\right )\right )}{a^5}\)

input

Int[x^4*ArcSech[a*x]^3,x]
 

output

-((-1/5*(a^5*x^5*ArcSech[a*x]^3) + (3*((a^3*x^3*ArcSech[a*x])/6 + (a^3*x^3 
*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x]^2)/4 + (-1/2*(a*x*Sqrt[( 
1 - a*x)/(1 + a*x)]*(1 + a*x)) - ArcTan[(Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a* 
x))/(a*x)]/2)/6 + (3*(a*x*ArcSech[a*x] + (a*x*Sqrt[(1 - a*x)/(1 + a*x)]*(1 
 + a*x)*ArcSech[a*x]^2)/2 - ArcTan[(Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x))/( 
a*x)] + (2*ArcSech[a*x]^2*ArcTan[E^ArcSech[a*x]] + (2*I)*(-(ArcSech[a*x]*P 
olyLog[2, (-I)*E^ArcSech[a*x]]) + PolyLog[3, (-I)*E^ArcSech[a*x]]) - (2*I) 
*(-(ArcSech[a*x]*PolyLog[2, I*E^ArcSech[a*x]]) + PolyLog[3, I*E^ArcSech[a* 
x]]))/2))/4))/5)/a^5)
 

3.1.10.3.1 Defintions of rubi rules used

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4255

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) 
  Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] 
&& IntegerQ[2*n]
 

rule 4257

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] 
 /; FreeQ[{c, d}, x]
 

rule 4668

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ 
))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( 
I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[ 
1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c 
+ d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c 
, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
 

rule 4674

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbo 
l] :> Simp[(-b^2)*(c + d*x)^m*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n 
- 1))), x] + (-Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^ 
2*(n - 1)*(n - 2))), x] + Simp[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))) 
Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Simp[b^2*((n - 2)/ 
(n - 1))   Int[(c + d*x)^m*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c 
, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]
 

rule 5941

Int[(x_)^(m_.)*Sech[(a_.) + (b_.)*(x_)^(n_.)]^(p_.)*Tanh[(a_.) + (b_.)*(x_) 
^(n_.)]^(q_.), x_Symbol] :> Simp[(-x^(m - n + 1))*(Sech[a + b*x^n]^p/(b*n*p 
)), x] + Simp[(m - n + 1)/(b*n*p)   Int[x^(m - n)*Sech[a + b*x^n]^p, x], x] 
 /; FreeQ[{a, b, p}, x] && RationalQ[m] && IntegerQ[n] && GeQ[m - n, 0] && 
EqQ[q, 1]
 

rule 6839

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
-(c^(m + 1))^(-1)   Subst[Int[(a + b*x)^n*Sech[x]^(m + 1)*Tanh[x], x], x, A 
rcSech[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (G 
tQ[n, 0] || LtQ[m, -1])
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

3.1.10.4 Maple [F]

\[\int x^{4} \operatorname {arcsech}\left (a x \right )^{3}d x\]

input

int(x^4*arcsech(a*x)^3,x)
 

output

int(x^4*arcsech(a*x)^3,x)
 

3.1.10.5 Fricas [F]

\[ \int x^4 \text {sech}^{-1}(a x)^3 \, dx=\int { x^{4} \operatorname {arsech}\left (a x\right )^{3} \,d x } \]

input

integrate(x^4*arcsech(a*x)^3,x, algorithm="fricas")
 

output

integral(x^4*arcsech(a*x)^3, x)
 

3.1.10.6 Sympy [F]

\[ \int x^4 \text {sech}^{-1}(a x)^3 \, dx=\int x^{4} \operatorname {asech}^{3}{\left (a x \right )}\, dx \]

input

integrate(x**4*asech(a*x)**3,x)
 

output

Integral(x**4*asech(a*x)**3, x)
 

3.1.10.7 Maxima [F]

\[ \int x^4 \text {sech}^{-1}(a x)^3 \, dx=\int { x^{4} \operatorname {arsech}\left (a x\right )^{3} \,d x } \]

input

integrate(x^4*arcsech(a*x)^3,x, algorithm="maxima")
 

output

integrate(x^4*arcsech(a*x)^3, x)
 

3.1.10.8 Giac [F]

\[ \int x^4 \text {sech}^{-1}(a x)^3 \, dx=\int { x^{4} \operatorname {arsech}\left (a x\right )^{3} \,d x } \]

input

integrate(x^4*arcsech(a*x)^3,x, algorithm="giac")
 

output

integrate(x^4*arcsech(a*x)^3, x)
 

3.1.10.9 Mupad [F(-1)]

Timed out. \[ \int x^4 \text {sech}^{-1}(a x)^3 \, dx=\int x^4\,{\mathrm {acosh}\left (\frac {1}{a\,x}\right )}^3 \,d x \]

input

int(x^4*acosh(1/(a*x))^3,x)
 

output

int(x^4*acosh(1/(a*x))^3, x)