\(\int \frac {\text {arctanh}(a x)^3}{x^3 (1-a^2 x^2)^{3/2}} \, dx\) [410]

Optimal result

Integrand size = 24, antiderivative size = 360 \[ \int \frac {\text {arctanh}(a x)^3}{x^3 \left (1-a^2 x^2\right )^{3/2}} \, dx=-\frac {6 a^3 x}{\sqrt {1-a^2 x^2}}+\frac {6 a^2 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {3 a^3 x \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}-\frac {3 a \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 x}+\frac {a^2 \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{2 x^2}-3 a^2 \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^3-6 a^2 \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-\frac {9}{2} a^2 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )+\frac {9}{2} a^2 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )+3 a^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-3 a^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+9 a^2 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arctanh}(a x)}\right )-9 a^2 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arctanh}(a x)}\right )-9 a^2 \operatorname {PolyLog}\left (4,-e^{\text {arctanh}(a x)}\right )+9 a^2 \operatorname {PolyLog}\left (4,e^{\text {arctanh}(a x)}\right ) \] Output:

-6*a^3*x/(-a^2*x^2+1)^(1/2)+6*a^2*arctanh(a*x)/(-a^2*x^2+1)^(1/2)-3*a^3*x* 
arctanh(a*x)^2/(-a^2*x^2+1)^(1/2)-3/2*a*(-a^2*x^2+1)^(1/2)*arctanh(a*x)^2/ 
x+a^2*arctanh(a*x)^3/(-a^2*x^2+1)^(1/2)-1/2*(-a^2*x^2+1)^(1/2)*arctanh(a*x 
)^3/x^2-3*a^2*arctanh((a*x+1)/(-a^2*x^2+1)^(1/2))*arctanh(a*x)^3-6*a^2*arc 
tanh(a*x)*arctanh((-a*x+1)^(1/2)/(a*x+1)^(1/2))-9/2*a^2*arctanh(a*x)^2*pol 
ylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+9/2*a^2*arctanh(a*x)^2*polylog(2,(a*x+ 
1)/(-a^2*x^2+1)^(1/2))+3*a^2*polylog(2,-(-a*x+1)^(1/2)/(a*x+1)^(1/2))-3*a^ 
2*polylog(2,(-a*x+1)^(1/2)/(a*x+1)^(1/2))+9*a^2*arctanh(a*x)*polylog(3,-(a 
*x+1)/(-a^2*x^2+1)^(1/2))-9*a^2*arctanh(a*x)*polylog(3,(a*x+1)/(-a^2*x^2+1 
)^(1/2))-9*a^2*polylog(4,-(a*x+1)/(-a^2*x^2+1)^(1/2))+9*a^2*polylog(4,(a*x 
+1)/(-a^2*x^2+1)^(1/2))
 

Mathematica [A] (verified)

Time = 6.25 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.05 \[ \int \frac {\text {arctanh}(a x)^3}{x^3 \left (1-a^2 x^2\right )^{3/2}} \, dx=\frac {1}{16} a^2 \left (3 \pi ^4-\frac {96 a x}{\sqrt {1-a^2 x^2}}+\frac {96 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {48 a x \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}+\frac {16 \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-6 \text {arctanh}(a x)^4-\frac {6 a x \text {arctanh}(a x)^2 \text {csch}^2\left (\frac {1}{2} \text {arctanh}(a x)\right )}{\sqrt {1-a^2 x^2}}-2 \text {arctanh}(a x)^3 \text {csch}^2\left (\frac {1}{2} \text {arctanh}(a x)\right )+48 \text {arctanh}(a x) \log \left (1-e^{-\text {arctanh}(a x)}\right )-48 \text {arctanh}(a x) \log \left (1+e^{-\text {arctanh}(a x)}\right )-24 \text {arctanh}(a x)^3 \log \left (1+e^{-\text {arctanh}(a x)}\right )+24 \text {arctanh}(a x)^3 \log \left (1-e^{\text {arctanh}(a x)}\right )+24 \left (2+3 \text {arctanh}(a x)^2\right ) \operatorname {PolyLog}\left (2,-e^{-\text {arctanh}(a x)}\right )-48 \operatorname {PolyLog}\left (2,e^{-\text {arctanh}(a x)}\right )+72 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )+144 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,-e^{-\text {arctanh}(a x)}\right )-144 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arctanh}(a x)}\right )+144 \operatorname {PolyLog}\left (4,-e^{-\text {arctanh}(a x)}\right )+144 \operatorname {PolyLog}\left (4,e^{\text {arctanh}(a x)}\right )-2 \text {arctanh}(a x)^3 \text {sech}^2\left (\frac {1}{2} \text {arctanh}(a x)\right )+12 \text {arctanh}(a x)^2 \tanh \left (\frac {1}{2} \text {arctanh}(a x)\right )\right ) \] Input:

Integrate[ArcTanh[a*x]^3/(x^3*(1 - a^2*x^2)^(3/2)),x]
 

Output:

(a^2*(3*Pi^4 - (96*a*x)/Sqrt[1 - a^2*x^2] + (96*ArcTanh[a*x])/Sqrt[1 - a^2 
*x^2] - (48*a*x*ArcTanh[a*x]^2)/Sqrt[1 - a^2*x^2] + (16*ArcTanh[a*x]^3)/Sq 
rt[1 - a^2*x^2] - 6*ArcTanh[a*x]^4 - (6*a*x*ArcTanh[a*x]^2*Csch[ArcTanh[a* 
x]/2]^2)/Sqrt[1 - a^2*x^2] - 2*ArcTanh[a*x]^3*Csch[ArcTanh[a*x]/2]^2 + 48* 
ArcTanh[a*x]*Log[1 - E^(-ArcTanh[a*x])] - 48*ArcTanh[a*x]*Log[1 + E^(-ArcT 
anh[a*x])] - 24*ArcTanh[a*x]^3*Log[1 + E^(-ArcTanh[a*x])] + 24*ArcTanh[a*x 
]^3*Log[1 - E^ArcTanh[a*x]] + 24*(2 + 3*ArcTanh[a*x]^2)*PolyLog[2, -E^(-Ar 
cTanh[a*x])] - 48*PolyLog[2, E^(-ArcTanh[a*x])] + 72*ArcTanh[a*x]^2*PolyLo 
g[2, E^ArcTanh[a*x]] + 144*ArcTanh[a*x]*PolyLog[3, -E^(-ArcTanh[a*x])] - 1 
44*ArcTanh[a*x]*PolyLog[3, E^ArcTanh[a*x]] + 144*PolyLog[4, -E^(-ArcTanh[a 
*x])] + 144*PolyLog[4, E^ArcTanh[a*x]] - 2*ArcTanh[a*x]^3*Sech[ArcTanh[a*x 
]/2]^2 + 12*ArcTanh[a*x]^2*Tanh[ArcTanh[a*x]/2]))/16
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 4.16 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.36, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {6592, 6588, 6570, 6580, 6582, 3042, 26, 4670, 3011, 6592, 6556, 6524, 208, 6582, 3042, 26, 4670, 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.

\(\displaystyle \int \frac {\text {arctanh}(a x)^3}{x^3 \left (1-a^2 x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 6592

\(\displaystyle a^2 \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^{3/2}}dx+\int \frac {\text {arctanh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}}dx\)

\(\Big \downarrow \) 6588

\(\displaystyle \frac {3}{2} a \int \frac {\text {arctanh}(a x)^2}{x^2 \sqrt {1-a^2 x^2}}dx+a^2 \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^{3/2}}dx+\frac {1}{2} a^2 \int \frac {\text {arctanh}(a x)^3}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{2 x^2}\)

\(\Big \downarrow \) 6570

\(\displaystyle \frac {3}{2} a \left (2 a \int \frac {\text {arctanh}(a x)}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x}\right )+a^2 \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^{3/2}}dx+\frac {1}{2} a^2 \int \frac {\text {arctanh}(a x)^3}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{2 x^2}\)

\(\Big \downarrow \) 6580

\(\displaystyle a^2 \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^{3/2}}dx+\frac {1}{2} a^2 \int \frac {\text {arctanh}(a x)^3}{x \sqrt {1-a^2 x^2}}dx+\frac {3}{2} a \left (2 a \left (-2 \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{2 x^2}\)

\(\Big \downarrow \) 6582

\(\displaystyle a^2 \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^{3/2}}dx+\frac {1}{2} a^2 \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{a x}d\text {arctanh}(a x)+\frac {3}{2} a \left (2 a \left (-2 \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{2 x^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle a^2 \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^{3/2}}dx+\frac {1}{2} a^2 \int i \text {arctanh}(a x)^3 \csc (i \text {arctanh}(a x))d\text {arctanh}(a x)+\frac {3}{2} a \left (2 a \left (-2 \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{2 x^2}\)

\(\Big \downarrow \) 26

\(\displaystyle a^2 \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^{3/2}}dx+\frac {1}{2} i a^2 \int \text {arctanh}(a x)^3 \csc (i \text {arctanh}(a x))d\text {arctanh}(a x)+\frac {3}{2} a \left (2 a \left (-2 \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{2 x^2}\)

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 3011

\(\displaystyle \frac {1}{2} i a^2 \left (-3 i \left (2 \int \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )\right )+3 i \left (2 \int \text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )\right )+2 i \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^3\right )+a^2 \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^{3/2}}dx+\frac {3}{2} a \left (2 a \left (-2 \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{2 x^2}\)

\(\Big \downarrow \) 6592

\(\displaystyle \frac {1}{2} i a^2 \left (-3 i \left (2 \int \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )\right )+3 i \left (2 \int \text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )\right )+2 i \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^3\right )+a^2 \left (a^2 \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}}dx+\int \frac {\text {arctanh}(a x)^3}{x \sqrt {1-a^2 x^2}}dx\right )+\frac {3}{2} a \left (2 a \left (-2 \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{2 x^2}\)

\(\Big \downarrow \) 6556

\(\displaystyle \frac {1}{2} i a^2 \left (-3 i \left (2 \int \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )\right )+3 i \left (2 \int \text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )\right )+2 i \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^3\right )+a^2 \left (a^2 \left (\frac {\text {arctanh}(a x)^3}{a^2 \sqrt {1-a^2 x^2}}-\frac {3 \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}}dx}{a}\right )+\int \frac {\text {arctanh}(a x)^3}{x \sqrt {1-a^2 x^2}}dx\right )+\frac {3}{2} a \left (2 a \left (-2 \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{2 x^2}\)

\(\Big \downarrow \) 6524

\(\displaystyle \frac {1}{2} i a^2 \left (-3 i \left (2 \int \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )\right )+3 i \left (2 \int \text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )\right )+2 i \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^3\right )+a^2 \left (a^2 \left (\frac {\text {arctanh}(a x)^3}{a^2 \sqrt {1-a^2 x^2}}-\frac {3 \left (2 \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}-\frac {2 \text {arctanh}(a x)}{a \sqrt {1-a^2 x^2}}\right )}{a}\right )+\int \frac {\text {arctanh}(a x)^3}{x \sqrt {1-a^2 x^2}}dx\right )+\frac {3}{2} a \left (2 a \left (-2 \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x}\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{2 x^2}\)

\(\Big \downarrow \) 208

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

\(\Big \downarrow \) 6582

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

Int[ArcTanh[a*x]^3/(x^3*(1 - a^2*x^2)^(3/2)),x]
 

Output:

-1/2*(Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3)/x^2 + (3*a*(-((Sqrt[1 - a^2*x^2]*A 
rcTanh[a*x]^2)/x) + 2*a*(-2*ArcTanh[a*x]*ArcTanh[Sqrt[1 - a*x]/Sqrt[1 + a* 
x]] + PolyLog[2, -(Sqrt[1 - a*x]/Sqrt[1 + a*x])] - PolyLog[2, Sqrt[1 - a*x 
]/Sqrt[1 + a*x]])))/2 + a^2*(a^2*(ArcTanh[a*x]^3/(a^2*Sqrt[1 - a^2*x^2]) - 
 (3*((2*x)/Sqrt[1 - a^2*x^2] - (2*ArcTanh[a*x])/(a*Sqrt[1 - a^2*x^2]) + (x 
*ArcTanh[a*x]^2)/Sqrt[1 - a^2*x^2]))/a) + I*((2*I)*ArcTanh[E^ArcTanh[a*x]] 
*ArcTanh[a*x]^3 - (3*I)*(-(ArcTanh[a*x]^2*PolyLog[2, -E^ArcTanh[a*x]]) + 2 
*(ArcTanh[a*x]*PolyLog[3, -E^ArcTanh[a*x]] - PolyLog[4, -E^ArcTanh[a*x]])) 
 + (3*I)*(-(ArcTanh[a*x]^2*PolyLog[2, E^ArcTanh[a*x]]) + 2*(ArcTanh[a*x]*P 
olyLog[3, E^ArcTanh[a*x]] - PolyLog[4, E^ArcTanh[a*x]])))) + (I/2)*a^2*((2 
*I)*ArcTanh[E^ArcTanh[a*x]]*ArcTanh[a*x]^3 - (3*I)*(-(ArcTanh[a*x]^2*PolyL 
og[2, -E^ArcTanh[a*x]]) + 2*(ArcTanh[a*x]*PolyLog[3, -E^ArcTanh[a*x]] - Po 
lyLog[4, -E^ArcTanh[a*x]])) + (3*I)*(-(ArcTanh[a*x]^2*PolyLog[2, E^ArcTanh 
[a*x]]) + 2*(ArcTanh[a*x]*PolyLog[3, E^ArcTanh[a*x]] - PolyLog[4, E^ArcTan 
h[a*x]])))
 

Defintions of rubi rules used

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 208

Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), 
x] /; FreeQ[{a, b}, x]
 

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 4670

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

rule 6524

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/((d_) + (e_.)*(x_)^2)^(3/2), x 
_Symbol] :> Simp[(-b)*p*((a + b*ArcTanh[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2]) 
), x] + (Simp[x*((a + b*ArcTanh[c*x])^p/(d*Sqrt[d + e*x^2])), x] + Simp[b^2 
*p*(p - 1)   Int[(a + b*ArcTanh[c*x])^(p - 2)/(d + e*x^2)^(3/2), x], x]) /; 
 FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 1]
 

rule 6556

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q 
_.), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^p/(2*e*(q 
+ 1))), x] + Simp[b*(p/(2*c*(q + 1)))   Int[(d + e*x^2)^q*(a + b*ArcTanh[c* 
x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[c^2*d + e, 0] && 
 GtQ[p, 0] && NeQ[q, -1]
 

rule 6570

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e 
_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a 
+ b*ArcTanh[c*x])^p/(d*(m + 1))), x] - Simp[b*c*(p/(m + 1))   Int[(f*x)^(m 
+ 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, 
d, e, f, m, q}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] 
 && NeQ[m, -1]
 

rule 6580

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x 
_Symbol] :> Simp[(-2/Sqrt[d])*(a + b*ArcTanh[c*x])*ArcTanh[Sqrt[1 - c*x]/Sq 
rt[1 + c*x]], x] + (Simp[(b/Sqrt[d])*PolyLog[2, -Sqrt[1 - c*x]/Sqrt[1 + c*x 
]], x] - Simp[(b/Sqrt[d])*PolyLog[2, Sqrt[1 - c*x]/Sqrt[1 + c*x]], x]) /; F 
reeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]
 

rule 6582

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2 
]), x_Symbol] :> Simp[1/Sqrt[d]   Subst[Int[(a + b*x)^p*Csch[x], x], x, Arc 
Tanh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 
 0] && GtQ[d, 0]
 

rule 6588

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) 
 + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*A 
rcTanh[c*x])^p/(d*f*(m + 1))), x] + (-Simp[b*c*(p/(f*(m + 1)))   Int[(f*x)^ 
(m + 1)*((a + b*ArcTanh[c*x])^(p - 1)/Sqrt[d + e*x^2]), x], x] + Simp[c^2*( 
(m + 2)/(f^2*(m + 1)))   Int[(f*x)^(m + 2)*((a + b*ArcTanh[c*x])^p/Sqrt[d + 
 e*x^2]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && G 
tQ[p, 0] && LtQ[m, -1] && NeQ[m, -2]
 

rule 6592

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^ 
2)^(q_), x_Symbol] :> Simp[1/d   Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh 
[c*x])^p, x], x] - Simp[e/d   Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTanh[c* 
x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && Integers 
Q[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -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]
 

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 [A] (verified)

Time = 0.65 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.34

method	result	size
default	\(-\frac {a^{2} \left (\operatorname {arctanh}\left (a x \right )^{3}-3 \operatorname {arctanh}\left (a x \right )^{2}+6 \,\operatorname {arctanh}\left (a x \right )-6\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 \left (a x -1\right )}+\frac {\left (\operatorname {arctanh}\left (a x \right )^{3}+3 \operatorname {arctanh}\left (a x \right )^{2}+6 \,\operatorname {arctanh}\left (a x \right )+6\right ) a^{2} \sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 a x +2}-\frac {\left (3 a x +\operatorname {arctanh}\left (a x \right )\right ) \operatorname {arctanh}\left (a x \right )^{2} \sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 x^{2}}+\frac {3 a^{2} \operatorname {arctanh}\left (a x \right )^{3} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\frac {9 a^{2} \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-9 a^{2} \operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+9 a^{2} \operatorname {polylog}\left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {3 a^{2} \operatorname {arctanh}\left (a x \right )^{3} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-\frac {9 a^{2} \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+9 a^{2} \operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-9 a^{2} \operatorname {polylog}\left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 a^{2} \operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 a^{2} \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-3 a^{2} \operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-3 a^{2} \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\)	\(482\)

Input:

int(arctanh(a*x)^3/x^3/(-a^2*x^2+1)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/2*a^2*(arctanh(a*x)^3-3*arctanh(a*x)^2+6*arctanh(a*x)-6)*(-(a*x-1)*(a*x 
+1))^(1/2)/(a*x-1)+1/2*(arctanh(a*x)^3+3*arctanh(a*x)^2+6*arctanh(a*x)+6)* 
a^2*(-(a*x-1)*(a*x+1))^(1/2)/(a*x+1)-1/2*(3*a*x+arctanh(a*x))*arctanh(a*x) 
^2*(-(a*x-1)*(a*x+1))^(1/2)/x^2+3/2*a^2*arctanh(a*x)^3*ln(1-(a*x+1)/(-a^2* 
x^2+1)^(1/2))+9/2*a^2*arctanh(a*x)^2*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2)) 
-9*a^2*arctanh(a*x)*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))+9*a^2*polylog(4, 
(a*x+1)/(-a^2*x^2+1)^(1/2))-3/2*a^2*arctanh(a*x)^3*ln(1+(a*x+1)/(-a^2*x^2+ 
1)^(1/2))-9/2*a^2*arctanh(a*x)^2*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+9* 
a^2*arctanh(a*x)*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))-9*a^2*polylog(4,-( 
a*x+1)/(-a^2*x^2+1)^(1/2))+3*a^2*arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1 
/2))+3*a^2*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-3*a^2*arctanh(a*x)*ln(1+( 
a*x+1)/(-a^2*x^2+1)^(1/2))-3*a^2*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))
 

Fricas [F]

\[ \int \frac {\text {arctanh}(a x)^3}{x^3 \left (1-a^2 x^2\right )^{3/2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{3}} \,d x } \] Input:

integrate(arctanh(a*x)^3/x^3/(-a^2*x^2+1)^(3/2),x, algorithm="fricas")
 

Output:

integral(sqrt(-a^2*x^2 + 1)*arctanh(a*x)^3/(a^4*x^7 - 2*a^2*x^5 + x^3), x)
 

Sympy [F]

\[ \int \frac {\text {arctanh}(a x)^3}{x^3 \left (1-a^2 x^2\right )^{3/2}} \, dx=\int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{x^{3} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(atanh(a*x)**3/x**3/(-a**2*x**2+1)**(3/2),x)
 

Output:

Integral(atanh(a*x)**3/(x**3*(-(a*x - 1)*(a*x + 1))**(3/2)), x)
 

Maxima [F]

\[ \int \frac {\text {arctanh}(a x)^3}{x^3 \left (1-a^2 x^2\right )^{3/2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{3}} \,d x } \] Input:

integrate(arctanh(a*x)^3/x^3/(-a^2*x^2+1)^(3/2),x, algorithm="maxima")
 

Output:

integrate(arctanh(a*x)^3/((-a^2*x^2 + 1)^(3/2)*x^3), x)
                                                                                    
                                                                                    
 

Giac [F]

\[ \int \frac {\text {arctanh}(a x)^3}{x^3 \left (1-a^2 x^2\right )^{3/2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{3}} \,d x } \] Input:

integrate(arctanh(a*x)^3/x^3/(-a^2*x^2+1)^(3/2),x, algorithm="giac")
 

Output:

integrate(arctanh(a*x)^3/((-a^2*x^2 + 1)^(3/2)*x^3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arctanh}(a x)^3}{x^3 \left (1-a^2 x^2\right )^{3/2}} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{x^3\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \] Input:

int(atanh(a*x)^3/(x^3*(1 - a^2*x^2)^(3/2)),x)
 

Output:

int(atanh(a*x)^3/(x^3*(1 - a^2*x^2)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {\text {arctanh}(a x)^3}{x^3 \left (1-a^2 x^2\right )^{3/2}} \, dx=-\left (\int \frac {\mathit {atanh} \left (a x \right )^{3}}{\sqrt {-a^{2} x^{2}+1}\, a^{2} x^{5}-\sqrt {-a^{2} x^{2}+1}\, x^{3}}d x \right ) \] Input:

int(atanh(a*x)^3/x^3/(-a^2*x^2+1)^(3/2),x)
 

Output:

 - int(atanh(a*x)**3/(sqrt( - a**2*x**2 + 1)*a**2*x**5 - sqrt( - a**2*x**2 
 + 1)*x**3),x)