\(\int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x^3} \, dx\) [445]

Optimal result

Integrand size = 24, antiderivative size = 151 \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x^3} \, dx=-\frac {a \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 x^2}+a^2 \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2-a^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )+a^2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )-a^2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )-a^2 \operatorname {PolyLog}\left (3,-e^{\text {arctanh}(a x)}\right )+a^2 \operatorname {PolyLog}\left (3,e^{\text {arctanh}(a x)}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 1.00 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.25 \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x^3} \, dx=\frac {1}{8} a^2 \left (-4 \text {arctanh}(a x) \coth \left (\frac {1}{2} \text {arctanh}(a x)\right )-\text {arctanh}(a x)^2 \text {csch}^2\left (\frac {1}{2} \text {arctanh}(a x)\right )-4 \text {arctanh}(a x)^2 \log \left (1-e^{-\text {arctanh}(a x)}\right )+4 \text {arctanh}(a x)^2 \log \left (1+e^{-\text {arctanh}(a x)}\right )+8 \log \left (\tanh \left (\frac {1}{2} \text {arctanh}(a x)\right )\right )-8 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{-\text {arctanh}(a x)}\right )+8 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{-\text {arctanh}(a x)}\right )-8 \operatorname {PolyLog}\left (3,-e^{-\text {arctanh}(a x)}\right )+8 \operatorname {PolyLog}\left (3,e^{-\text {arctanh}(a x)}\right )-\text {arctanh}(a x)^2 \text {sech}^2\left (\frac {1}{2} \text {arctanh}(a x)\right )+4 \text {arctanh}(a x) \tanh \left (\frac {1}{2} \text {arctanh}(a x)\right )\right ) \] Input:

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

Output:

(a^2*(-4*ArcTanh[a*x]*Coth[ArcTanh[a*x]/2] - ArcTanh[a*x]^2*Csch[ArcTanh[a 
*x]/2]^2 - 4*ArcTanh[a*x]^2*Log[1 - E^(-ArcTanh[a*x])] + 4*ArcTanh[a*x]^2* 
Log[1 + E^(-ArcTanh[a*x])] + 8*Log[Tanh[ArcTanh[a*x]/2]] - 8*ArcTanh[a*x]* 
PolyLog[2, -E^(-ArcTanh[a*x])] + 8*ArcTanh[a*x]*PolyLog[2, E^(-ArcTanh[a*x 
])] - 8*PolyLog[3, -E^(-ArcTanh[a*x])] + 8*PolyLog[3, E^(-ArcTanh[a*x])] - 
 ArcTanh[a*x]^2*Sech[ArcTanh[a*x]/2]^2 + 4*ArcTanh[a*x]*Tanh[ArcTanh[a*x]/ 
2]))/8
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 2.19 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.04, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.792, Rules used = {6576, 6582, 3042, 26, 4670, 3011, 2720, 6588, 6570, 243, 73, 221, 6582, 3042, 26, 4670, 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.

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

\(\Big \downarrow \) 6576

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

\(\Big \downarrow \) 6582

\(\displaystyle \int \frac {\text {arctanh}(a x)^2}{x^3 \sqrt {1-a^2 x^2}}dx-a^2 \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{a x}d\text {arctanh}(a x)\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\text {arctanh}(a x)^2}{x^3 \sqrt {1-a^2 x^2}}dx-a^2 \int i \text {arctanh}(a x)^2 \csc (i \text {arctanh}(a x))d\text {arctanh}(a x)\)

\(\Big \downarrow \) 26

\(\displaystyle \int \frac {\text {arctanh}(a x)^2}{x^3 \sqrt {1-a^2 x^2}}dx-i a^2 \int \text {arctanh}(a x)^2 \csc (i \text {arctanh}(a x))d\text {arctanh}(a x)\)

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 6588

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

\(\Big \downarrow \) 6570

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 6582

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

-1/2*(Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/x^2 + a*(-((Sqrt[1 - a^2*x^2]*ArcT 
anh[a*x])/x) - a*ArcTanh[Sqrt[1 - a^2*x^2]]) - (I/2)*a^2*((2*I)*ArcTanh[E^ 
ArcTanh[a*x]]*ArcTanh[a*x]^2 - (2*I)*(-(ArcTanh[a*x]*PolyLog[2, -E^ArcTanh 
[a*x]]) + PolyLog[3, -E^ArcTanh[a*x]]) + (2*I)*(-(ArcTanh[a*x]*PolyLog[2, 
E^ArcTanh[a*x]]) + PolyLog[3, E^ArcTanh[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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

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 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 6576

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

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 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]
 

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.52

method	result	size
default	\(-\frac {\left (2 a x +\operatorname {arctanh}\left (a x \right )\right ) \operatorname {arctanh}\left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-a^{2} \operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+a^{2} \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {a^{2} \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+a^{2} \operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-a^{2} \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-2 a^{2} \operatorname {arctanh}\left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\)	\(229\)

Input:

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

Output:

-1/2*(2*a*x+arctanh(a*x))*arctanh(a*x)*(-a^2*x^2+1)^(1/2)/x^2-1/2*a^2*arct 
anh(a*x)^2*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))-a^2*arctanh(a*x)*polylog(2,(a* 
x+1)/(-a^2*x^2+1)^(1/2))+a^2*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))+1/2*a^2 
*arctanh(a*x)^2*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+a^2*arctanh(a*x)*polylog( 
2,-(a*x+1)/(-a^2*x^2+1)^(1/2))-a^2*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))- 
2*a^2*arctanh((a*x+1)/(-a^2*x^2+1)^(1/2))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x^3} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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