\(\int \frac {\text {sech}^{-1}(a+b x)^3}{x} \, dx\) [17]

Optimal result

Integrand size = 12, antiderivative size = 378 \[ \int \frac {\text {sech}^{-1}(a+b x)^3}{x} \, dx=\text {sech}^{-1}(a+b x)^3 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\text {sech}^{-1}(a+b x)^3 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\text {sech}^{-1}(a+b x)^3 \log \left (1+e^{2 \text {sech}^{-1}(a+b x)}\right )+3 \text {sech}^{-1}(a+b x)^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+3 \text {sech}^{-1}(a+b x)^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\frac {3}{2} \text {sech}^{-1}(a+b x)^2 \operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(a+b x)}\right )-6 \text {sech}^{-1}(a+b x) \operatorname {PolyLog}\left (3,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )-6 \text {sech}^{-1}(a+b x) \operatorname {PolyLog}\left (3,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )+\frac {3}{2} \text {sech}^{-1}(a+b x) \operatorname {PolyLog}\left (3,-e^{2 \text {sech}^{-1}(a+b x)}\right )+6 \operatorname {PolyLog}\left (4,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+6 \operatorname {PolyLog}\left (4,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\frac {3}{4} \operatorname {PolyLog}\left (4,-e^{2 \text {sech}^{-1}(a+b x)}\right ) \] Output:

arcsech(b*x+a)^3*ln(1-a*(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2) 
)/(1-(-a^2+1)^(1/2)))+arcsech(b*x+a)^3*ln(1-a*(1/(b*x+a)+(1/(b*x+a)-1)^(1/ 
2)*(1/(b*x+a)+1)^(1/2))/(1+(-a^2+1)^(1/2)))-arcsech(b*x+a)^3*ln(1+(1/(b*x+ 
a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2))^2)+3*arcsech(b*x+a)^2*polylog( 
2,a*(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2))/(1-(-a^2+1)^(1/2)) 
)+3*arcsech(b*x+a)^2*polylog(2,a*(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a) 
+1)^(1/2))/(1+(-a^2+1)^(1/2)))-3/2*arcsech(b*x+a)^2*polylog(2,-(1/(b*x+a)+ 
(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2))^2)-6*arcsech(b*x+a)*polylog(3,a*( 
1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2))/(1-(-a^2+1)^(1/2)))-6*a 
rcsech(b*x+a)*polylog(3,a*(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/ 
2))/(1+(-a^2+1)^(1/2)))+3/2*arcsech(b*x+a)*polylog(3,-(1/(b*x+a)+(1/(b*x+a 
)-1)^(1/2)*(1/(b*x+a)+1)^(1/2))^2)+6*polylog(4,a*(1/(b*x+a)+(1/(b*x+a)-1)^ 
(1/2)*(1/(b*x+a)+1)^(1/2))/(1-(-a^2+1)^(1/2)))+6*polylog(4,a*(1/(b*x+a)+(1 
/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2))/(1+(-a^2+1)^(1/2)))-3/4*polylog(4,- 
(1/(b*x+a)+(1/(b*x+a)-1)^(1/2)*(1/(b*x+a)+1)^(1/2))^2)
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.02 \[ \int \frac {\text {sech}^{-1}(a+b x)^3}{x} \, dx=-\frac {1}{2} \text {sech}^{-1}(a+b x)^4-\text {sech}^{-1}(a+b x)^3 \log \left (1+e^{-2 \text {sech}^{-1}(a+b x)}\right )+\text {sech}^{-1}(a+b x)^3 \log \left (1+\frac {a e^{\text {sech}^{-1}(a+b x)}}{-1+\sqrt {1-a^2}}\right )+\text {sech}^{-1}(a+b x)^3 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )+\frac {3}{2} \text {sech}^{-1}(a+b x)^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(a+b x)}\right )+3 \text {sech}^{-1}(a+b x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{\text {sech}^{-1}(a+b x)}}{-1+\sqrt {1-a^2}}\right )+3 \text {sech}^{-1}(a+b x)^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )+\frac {3}{2} \text {sech}^{-1}(a+b x) \operatorname {PolyLog}\left (3,-e^{-2 \text {sech}^{-1}(a+b x)}\right )-6 \text {sech}^{-1}(a+b x) \operatorname {PolyLog}\left (3,-\frac {a e^{\text {sech}^{-1}(a+b x)}}{-1+\sqrt {1-a^2}}\right )-6 \text {sech}^{-1}(a+b x) \operatorname {PolyLog}\left (3,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )+\frac {3}{4} \operatorname {PolyLog}\left (4,-e^{-2 \text {sech}^{-1}(a+b x)}\right )+6 \operatorname {PolyLog}\left (4,-\frac {a e^{\text {sech}^{-1}(a+b x)}}{-1+\sqrt {1-a^2}}\right )+6 \operatorname {PolyLog}\left (4,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right ) \] Input:

Integrate[ArcSech[a + b*x]^3/x,x]
 

Output:

-1/2*ArcSech[a + b*x]^4 - ArcSech[a + b*x]^3*Log[1 + E^(-2*ArcSech[a + b*x 
])] + ArcSech[a + b*x]^3*Log[1 + (a*E^ArcSech[a + b*x])/(-1 + Sqrt[1 - a^2 
])] + ArcSech[a + b*x]^3*Log[1 - (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2] 
)] + (3*ArcSech[a + b*x]^2*PolyLog[2, -E^(-2*ArcSech[a + b*x])])/2 + 3*Arc 
Sech[a + b*x]^2*PolyLog[2, -((a*E^ArcSech[a + b*x])/(-1 + Sqrt[1 - a^2]))] 
 + 3*ArcSech[a + b*x]^2*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^ 
2])] + (3*ArcSech[a + b*x]*PolyLog[3, -E^(-2*ArcSech[a + b*x])])/2 - 6*Arc 
Sech[a + b*x]*PolyLog[3, -((a*E^ArcSech[a + b*x])/(-1 + Sqrt[1 - a^2]))] - 
 6*ArcSech[a + b*x]*PolyLog[3, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])] 
 + (3*PolyLog[4, -E^(-2*ArcSech[a + b*x])])/4 + 6*PolyLog[4, -((a*E^ArcSec 
h[a + b*x])/(-1 + Sqrt[1 - a^2]))] + 6*PolyLog[4, (a*E^ArcSech[a + b*x])/( 
1 + Sqrt[1 - a^2])]
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.83 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.20, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.333, Rules used = {6875, 25, 6129, 6104, 25, 3042, 26, 4201, 2620, 3011, 6096, 2620, 3011, 7163, 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.

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

\(\Big \downarrow \) 6875

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 6129

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

\(\Big \downarrow \) 6104

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

\(\displaystyle -a \int \frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)^3}{(a+b x) \left (1-\frac {a}{a+b x}\right )}d\text {sech}^{-1}(a+b x)-\int -i \text {sech}^{-1}(a+b x)^3 \tan \left (i \text {sech}^{-1}(a+b x)\right )d\text {sech}^{-1}(a+b x)\)

\(\Big \downarrow \) 26

\(\displaystyle -a \int \frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)^3}{(a+b x) \left (1-\frac {a}{a+b x}\right )}d\text {sech}^{-1}(a+b x)+i \int \text {sech}^{-1}(a+b x)^3 \tan \left (i \text {sech}^{-1}(a+b x)\right )d\text {sech}^{-1}(a+b x)\)

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 6096

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

Int[ArcSech[a + b*x]^3/x,x]
 

Output:

-(a*(ArcSech[a + b*x]^4/(4*a) - (ArcSech[a + b*x]^3*Log[1 - (a*E^ArcSech[a 
 + b*x])/(1 - Sqrt[1 - a^2])])/a - (ArcSech[a + b*x]^3*Log[1 - (a*E^ArcSec 
h[a + b*x])/(1 + Sqrt[1 - a^2])])/a + (3*(-(ArcSech[a + b*x]^2*PolyLog[2, 
(a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])]) + 2*(ArcSech[a + b*x]*PolyLog 
[3, (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])] - PolyLog[4, (a*E^ArcSech[ 
a + b*x])/(1 - Sqrt[1 - a^2])])))/a + (3*(-(ArcSech[a + b*x]^2*PolyLog[2, 
(a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])]) + 2*(ArcSech[a + b*x]*PolyLog 
[3, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])] - PolyLog[4, (a*E^ArcSech[ 
a + b*x])/(1 + Sqrt[1 - a^2])])))/a)) + I*((-1/4*I)*ArcSech[a + b*x]^4 + ( 
2*I)*((ArcSech[a + b*x]^3*Log[1 + E^(2*ArcSech[a + b*x])])/2 - (3*(-1/2*(A 
rcSech[a + b*x]^2*PolyLog[2, -E^(2*ArcSech[a + b*x])]) + (ArcSech[a + b*x] 
*PolyLog[3, -E^(2*ArcSech[a + b*x])])/2 - PolyLog[4, -E^(2*ArcSech[a + b*x 
])]/4))/2))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2620

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

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 4201

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

rule 6096

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)])/(Cosh[(c_.) + (d_ 
.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), 
 x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 - b^2, 2] + b*E^(c + d*x))) 
, x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 - b^2, 2] + b*E^(c + d*x))) 
, x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 0]
 

rule 6104

Int[(((e_.) + (f_.)*(x_))^(m_.)*Tanh[(c_.) + (d_.)*(x_)]^(n_.))/(Cosh[(c_.) 
 + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[1/a   Int[(e + f*x)^m*Tanh[ 
c + d*x]^n, x], x] - Simp[b/a   Int[(e + f*x)^m*Sinh[c + d*x]*(Tanh[c + d*x 
]^(n - 1)/(a + b*Cosh[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && 
 IGtQ[m, 0] && IGtQ[n, 0]
 

rule 6129

Int[(((e_.) + (f_.)*(x_))^(m_.)*(F_)[(c_.) + (d_.)*(x_)]^(n_.)*(G_)[(c_.) + 
 (d_.)*(x_)]^(p_.))/((a_) + (b_.)*Sech[(c_.) + (d_.)*(x_)]), x_Symbol] :> I 
nt[(e + f*x)^m*Cosh[c + d*x]*F[c + d*x]^n*(G[c + d*x]^p/(b + a*Cosh[c + d*x 
])), x] /; FreeQ[{a, b, c, d, e, f}, x] && HyperbolicQ[F] && HyperbolicQ[G] 
 && IntegersQ[m, n, p]
 

rule 6875

Int[((a_.) + ArcSech[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( 
m_.), x_Symbol] :> Simp[-(d^(m + 1))^(-1)   Subst[Int[(a + b*x)^p*Sech[x]*T 
anh[x]*(d*e - c*f + f*Sech[x])^m, x], x, ArcSech[c + d*x]], x] /; FreeQ[{a, 
 b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]
 

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]
 

rule 7163

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

Maple [F]

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

Input:

int(arcsech(b*x+a)^3/x,x)
 

Output:

int(arcsech(b*x+a)^3/x,x)
 

Fricas [F]

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

integrate(arcsech(b*x+a)^3/x,x, algorithm="fricas")
 

Output:

integral(arcsech(b*x + a)^3/x, x)
 

Sympy [F]

\[ \int \frac {\text {sech}^{-1}(a+b x)^3}{x} \, dx=\int \frac {\operatorname {asech}^{3}{\left (a + b x \right )}}{x}\, dx \] Input:

integrate(asech(b*x+a)**3/x,x)
                                                                                    
                                                                                    
 

Output:

Integral(asech(a + b*x)**3/x, x)
 

Maxima [F]

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

integrate(arcsech(b*x+a)^3/x,x, algorithm="maxima")
 

Output:

integrate(arcsech(b*x + a)^3/x, x)
 

Giac [F]

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

integrate(arcsech(b*x+a)^3/x,x, algorithm="giac")
 

Output:

integrate(arcsech(b*x + a)^3/x, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\text {sech}^{-1}(a+b x)^3}{x} \, dx=\int \frac {\mathit {asech} \left (b x +a \right )^{3}}{x}d x \] Input:

int(asech(b*x+a)^3/x,x)
 

Output:

int(asech(a + b*x)**3/x,x)