Integrand size = 22, antiderivative size = 277 \[ \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^3} \, dx=-\frac {3 a x}{128 \left (1-a^2 x^2\right )^2}-\frac {141 a x}{256 \left (1-a^2 x^2\right )}-\frac {141}{256} \text {arctanh}(a x)+\frac {3 \text {arctanh}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac {33 \text {arctanh}(a x)}{32 \left (1-a^2 x^2\right )}-\frac {3 a x \text {arctanh}(a x)^2}{16 \left (1-a^2 x^2\right )^2}-\frac {33 a x \text {arctanh}(a x)^2}{32 \left (1-a^2 x^2\right )}-\frac {11}{32} \text {arctanh}(a x)^3+\frac {\text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac {\text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )-\frac {3}{2} \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-\frac {3}{2} \text {arctanh}(a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+a x}\right )-\frac {3}{4} \operatorname {PolyLog}\left (4,-1+\frac {2}{1+a x}\right ) \]
-3/128*a*x/(-a^2*x^2+1)^2-141/256*a*x/(-a^2*x^2+1)-141/256*arctanh(a*x)+3/ 32*arctanh(a*x)/(-a^2*x^2+1)^2+33/32*arctanh(a*x)/(-a^2*x^2+1)-3/16*a*x*ar ctanh(a*x)^2/(-a^2*x^2+1)^2-33/32*a*x*arctanh(a*x)^2/(-a^2*x^2+1)-11/32*ar ctanh(a*x)^3+1/4*arctanh(a*x)^3/(-a^2*x^2+1)^2+1/2*arctanh(a*x)^3/(-a^2*x^ 2+1)+1/4*arctanh(a*x)^4+arctanh(a*x)^3*ln(2-2/(a*x+1))-3/2*arctanh(a*x)^2* polylog(2,-1+2/(a*x+1))-3/2*arctanh(a*x)*polylog(3,-1+2/(a*x+1))-3/4*polyl og(4,-1+2/(a*x+1))
Time = 0.41 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.68 \[ \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^3} \, dx=\frac {16 \pi ^4-256 \text {arctanh}(a x)^4+576 \text {arctanh}(a x) \cosh (2 \text {arctanh}(a x))+384 \text {arctanh}(a x)^3 \cosh (2 \text {arctanh}(a x))+12 \text {arctanh}(a x) \cosh (4 \text {arctanh}(a x))+32 \text {arctanh}(a x)^3 \cosh (4 \text {arctanh}(a x))+1024 \text {arctanh}(a x)^3 \log \left (1-e^{2 \text {arctanh}(a x)}\right )+1536 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(a x)}\right )-1536 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(a x)}\right )+768 \operatorname {PolyLog}\left (4,e^{2 \text {arctanh}(a x)}\right )-288 \sinh (2 \text {arctanh}(a x))-576 \text {arctanh}(a x)^2 \sinh (2 \text {arctanh}(a x))-3 \sinh (4 \text {arctanh}(a x))-24 \text {arctanh}(a x)^2 \sinh (4 \text {arctanh}(a x))}{1024} \]
(16*Pi^4 - 256*ArcTanh[a*x]^4 + 576*ArcTanh[a*x]*Cosh[2*ArcTanh[a*x]] + 38 4*ArcTanh[a*x]^3*Cosh[2*ArcTanh[a*x]] + 12*ArcTanh[a*x]*Cosh[4*ArcTanh[a*x ]] + 32*ArcTanh[a*x]^3*Cosh[4*ArcTanh[a*x]] + 1024*ArcTanh[a*x]^3*Log[1 - E^(2*ArcTanh[a*x])] + 1536*ArcTanh[a*x]^2*PolyLog[2, E^(2*ArcTanh[a*x])] - 1536*ArcTanh[a*x]*PolyLog[3, E^(2*ArcTanh[a*x])] + 768*PolyLog[4, E^(2*Ar cTanh[a*x])] - 288*Sinh[2*ArcTanh[a*x]] - 576*ArcTanh[a*x]^2*Sinh[2*ArcTan h[a*x]] - 3*Sinh[4*ArcTanh[a*x]] - 24*ArcTanh[a*x]^2*Sinh[4*ArcTanh[a*x]]) /1024
Time = 3.55 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.75, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.955, Rules used = {6592, 6556, 6526, 215, 215, 219, 6518, 6556, 215, 219, 6592, 6550, 6494, 6556, 6518, 6556, 215, 219, 6618, 6622, 7164}
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.
| \(\displaystyle \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 6592 |
| \(\displaystyle a^2 \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3}dx+\int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 6556 |
| \(\displaystyle a^2 \left (\frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3}dx}{4 a}\right )+\int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 6526 |
| \(\displaystyle a^2 \left (\frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {3}{4} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {1}{8} \int \frac {1}{\left (1-a^2 x^2\right )^3}dx+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}\right )}{4 a}\right )+\int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle a^2 \left (\frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {3}{4} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {1}{8} \left (\frac {3}{4} \int \frac {1}{\left (1-a^2 x^2\right )^2}dx+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}\right )}{4 a}\right )+\int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle a^2 \left (\frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {3}{4} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {1}{8} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{1-a^2 x^2}dx+\frac {x}{2 \left (1-a^2 x^2\right )}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}\right )}{4 a}\right )+\int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle a^2 \left (\frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {3}{4} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )\right )}{4 a}\right )+\int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 6518 |
| \(\displaystyle a^2 \left (\frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {3}{4} \left (-a \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )\right )}{4 a}\right )+\int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 6556 |
| \(\displaystyle a^2 \left (\frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {3}{4} \left (-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2}dx}{2 a}\right )+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )\right )}{4 a}\right )+\int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle a^2 \left (\frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {3}{4} \left (-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {1}{2} \int \frac {1}{1-a^2 x^2}dx+\frac {x}{2 \left (1-a^2 x^2\right )}}{2 a}\right )+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )\right )}{4 a}\right )+\int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^2}dx+a^2 \left (\frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )\right )}{4 a}\right )\) |
| \(\Big \downarrow \) 6592 |
| \(\displaystyle a^2 \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2}dx+\int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )}dx+a^2 \left (\frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )\right )}{4 a}\right )\) |
| \(\Big \downarrow \) 6550 |
| \(\displaystyle a^2 \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2}dx+\int \frac {\text {arctanh}(a x)^3}{x (a x+1)}dx+a^2 \left (\frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )\right )}{4 a}\right )+\frac {1}{4} \text {arctanh}(a x)^4\) |
| \(\Big \downarrow \) 6494 |
| \(\displaystyle a^2 \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2}dx-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+a^2 \left (\frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )\right )}{4 a}\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\) |
| \(\Big \downarrow \) 6556 |
| \(\displaystyle a^2 \left (\frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx}{2 a}\right )-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+a^2 \left (\frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )\right )}{4 a}\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\) |
| \(\Big \downarrow \) 6518 |
| \(\displaystyle a^2 \left (\frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \left (-a \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )}{2 a}\right )-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+a^2 \left (\frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )\right )}{4 a}\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\) |
| \(\Big \downarrow \) 6556 |
| \(\displaystyle a^2 \left (\frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \left (-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2}dx}{2 a}\right )+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )}{2 a}\right )-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+a^2 \left (\frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )\right )}{4 a}\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle a^2 \left (\frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \left (-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {1}{2} \int \frac {1}{1-a^2 x^2}dx+\frac {x}{2 \left (1-a^2 x^2\right )}}{2 a}\right )+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )}{2 a}\right )-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+a^2 \left (\frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )\right )}{4 a}\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+a^2 \left (\frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )}{2 a}\right )+a^2 \left (\frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )\right )}{4 a}\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\) |
| \(\Big \downarrow \) 6618 |
| \(\displaystyle -3 a \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}-\int \frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{1-a^2 x^2}dx\right )+a^2 \left (\frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )}{2 a}\right )+a^2 \left (\frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )\right )}{4 a}\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\) |
| \(\Big \downarrow \) 6622 |
| \(\displaystyle -3 a \left (-\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{1-a^2 x^2}dx+\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}+\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{2 a}\right )+a^2 \left (\frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )}{2 a}\right )+a^2 \left (\frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )\right )}{4 a}\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle a^2 \left (\frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )}{2 a}\right )+a^2 \left (\frac {\text {arctanh}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )\right )}{4 a}\right )-3 a \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}+\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{2 a}+\frac {\operatorname {PolyLog}\left (4,\frac {2}{a x+1}-1\right )}{4 a}\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\) |
ArcTanh[a*x]^4/4 + a^2*(ArcTanh[a*x]^3/(2*a^2*(1 - a^2*x^2)) - (3*((x*ArcT anh[a*x]^2)/(2*(1 - a^2*x^2)) + ArcTanh[a*x]^3/(6*a) - a*(ArcTanh[a*x]/(2* a^2*(1 - a^2*x^2)) - (x/(2*(1 - a^2*x^2)) + ArcTanh[a*x]/(2*a))/(2*a))))/( 2*a)) + a^2*(ArcTanh[a*x]^3/(4*a^2*(1 - a^2*x^2)^2) - (3*(-1/8*ArcTanh[a*x ]/(a*(1 - a^2*x^2)^2) + (x*ArcTanh[a*x]^2)/(4*(1 - a^2*x^2)^2) + (x/(4*(1 - a^2*x^2)^2) + (3*(x/(2*(1 - a^2*x^2)) + ArcTanh[a*x]/(2*a)))/4)/8 + (3*( (x*ArcTanh[a*x]^2)/(2*(1 - a^2*x^2)) + ArcTanh[a*x]^3/(6*a) - a*(ArcTanh[a *x]/(2*a^2*(1 - a^2*x^2)) - (x/(2*(1 - a^2*x^2)) + ArcTanh[a*x]/(2*a))/(2* a))))/4))/(4*a)) + ArcTanh[a*x]^3*Log[2 - 2/(1 + a*x)] - 3*a*((ArcTanh[a*x ]^2*PolyLog[2, -1 + 2/(1 + a*x)])/(2*a) + (ArcTanh[a*x]*PolyLog[3, -1 + 2/ (1 + a*x)])/(2*a) + PolyLog[4, -1 + 2/(1 + a*x)]/(4*a))
3.4.18.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x _Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Simp[b*c*(p/d) Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2 - 2/(1 + e*(x/d))] /(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c ^2*d^2 - e^2, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2)^2, x_Sy mbol] :> Simp[x*((a + b*ArcTanh[c*x])^p/(2*d*(d + e*x^2))), x] + (Simp[(a + b*ArcTanh[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)), x] - Simp[b*c*(p/2) Int[x*( (a + b*ArcTanh[c*x])^(p - 1)/(d + e*x^2)^2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_ Symbol] :> Simp[(-b)*p*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^(p - 1)/(4 *c*d*(q + 1)^2)), x] + (-Simp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^p /(2*d*(q + 1))), x] + Simp[(2*q + 3)/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1 )*(a + b*ArcTanh[c*x])^p, x], x] + Simp[b^2*p*((p - 1)/(4*(q + 1)^2)) Int [(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*d*(p + 1)), x] + Simp[1/ d Int[(a + b*ArcTanh[c*x])^p/(x*(1 + c*x)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
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]
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]
Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^ 2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x ] - Simp[b*(p/2) Int[(a + b*ArcTanh[c*x])^(p - 1)*(PolyLog[2, 1 - u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 + c*x))^2, 0]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_ .)*(x_)^2), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(PolyLog[k + 1, u]/ (2*c*d)), x] + Simp[b*(p/2) Int[(a + b*ArcTanh[c*x])^(p - 1)*(PolyLog[k + 1, u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] & & EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 + c*x))^2, 0]
Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /; !FalseQ[w]] /; FreeQ[n, x]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.84 (sec) , antiderivative size = 1446, normalized size of antiderivative = 5.22
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1446\) |
| default | \(\text {Expression too large to display}\) | \(1446\) |
| parts | \(\text {Expression too large to display}\) | \(1875\) |
-9/32*arctanh(a*x)^2*(a*x-1)/(a*x+1)-9/32*arctanh(a*x)*(a*x-1)/(a*x+1)-9/3 2*(a*x+1)*arctanh(a*x)/(a*x-1)+9/32*(a*x+1)*arctanh(a*x)^2/(a*x-1)+arctanh (a*x)^3*ln(a*x)-1/2*arctanh(a*x)^3*ln(a*x-1)-1/2*arctanh(a*x)^3*ln(a*x+1)+ arctanh(a*x)^3*ln((a*x+1)/(-a^2*x^2+1)^(1/2))+3*arctanh(a*x)^2*polylog(2,- (a*x+1)/(-a^2*x^2+1)^(1/2))+3*arctanh(a*x)^2*polylog(2,(a*x+1)/(-a^2*x^2+1 )^(1/2))-6*arctanh(a*x)*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))-6*arctanh(a *x)*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))+6*polylog(4,-(a*x+1)/(-a^2*x^2+1 )^(1/2))+6*polylog(4,(a*x+1)/(-a^2*x^2+1)^(1/2))-1/4*arctanh(a*x)^4+1/32*( 16*I*Pi*csgn(I*(-(a*x+1)^2/(a^2*x^2-1)-1)/(1-(a*x+1)^2/(a^2*x^2-1)))^3+16* I*Pi*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2-8* I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1)))^2*csgn(I*(a*x +1)^2/(a^2*x^2-1))+16*I*Pi*csgn(I/(1-(a*x+1)^2/(a^2*x^2-1)))^3+8*I*Pi*csgn (I*(a*x+1)^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1)))^2*csgn(I/(1-(a*x+1)^2/ (a^2*x^2-1)))+8*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3-16*I*Pi*csgn(I/(1-(a* x+1)^2/(a^2*x^2-1)))^2+8*I*Pi*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*csgn(I* (a*x+1)^2/(a^2*x^2-1))+16*I*Pi-16*I*Pi*csgn(I*(-(a*x+1)^2/(a^2*x^2-1)-1)/( 1-(a*x+1)^2/(a^2*x^2-1)))^2*csgn(I/(1-(a*x+1)^2/(a^2*x^2-1)))+8*I*Pi*csgn( I*(a*x+1)^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1)))^3+16*I*Pi*csgn(I*(-(a*x +1)^2/(a^2*x^2-1)-1))*csgn(I*(-(a*x+1)^2/(a^2*x^2-1)-1)/(1-(a*x+1)^2/(a^2* x^2-1)))*csgn(I/(1-(a*x+1)^2/(a^2*x^2-1)))-16*I*Pi*csgn(I*(-(a*x+1)^2/(...
\[ \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^3} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )}^{3} x} \,d x } \]
\[ \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^3} \, dx=- \int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{a^{6} x^{7} - 3 a^{4} x^{5} + 3 a^{2} x^{3} - x}\, dx \]
\[ \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^3} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )}^{3} x} \,d x } \]
1/64*((a^4*x^4 - 2*a^2*x^2 + 1)*log(-a*x + 1)^4 + 2*(2*a^2*x^2 + 2*(a^4*x^ 4 - 2*a^2*x^2 + 1)*log(a*x + 1) - 3)*log(-a*x + 1)^3)/(a^4*x^4 - 2*a^2*x^2 + 1) - 1/8*integrate(1/4*(4*log(a*x + 1)^3 - 12*log(a*x + 1)^2*log(-a*x + 1) + 3*(2*a^4*x^4 + 2*a^3*x^3 - 3*a^2*x^2 - 3*a*x + 2*(a^6*x^6 + a^5*x^5 - 2*a^4*x^4 - 2*a^3*x^3 + a^2*x^2 + a*x + 2)*log(a*x + 1))*log(-a*x + 1)^2 )/(a^6*x^7 - 3*a^4*x^5 + 3*a^2*x^3 - x), x)
\[ \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^3} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )}^{3} x} \,d x } \]
Timed out. \[ \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^3} \, dx=-\int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{x\,{\left (a^2\,x^2-1\right )}^3} \,d x \]