Integrand size = 22, antiderivative size = 302 \[ \int \frac {\text {arctanh}(a x)^3}{x^3 \left (1-a^2 x^2\right )^2} \, dx=-\frac {3 a^3 x}{8 \left (1-a^2 x^2\right )}-\frac {3}{8} a^2 \text {arctanh}(a x)+\frac {3 a^2 \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )}+\frac {3}{2} a^2 \text {arctanh}(a x)^2-\frac {3 a \text {arctanh}(a x)^2}{2 x}-\frac {3 a^3 x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )}+\frac {1}{4} a^2 \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^3}{2 x^2}+\frac {a^2 \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {1}{2} a^2 \text {arctanh}(a x)^4+3 a^2 \text {arctanh}(a x) \log \left (2-\frac {2}{1+a x}\right )+2 a^2 \text {arctanh}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )-\frac {3}{2} a^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-3 a^2 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-3 a^2 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+a x}\right )-\frac {3}{2} a^2 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+a x}\right ) \] Output:
-3*a^3*x/(-8*a^2*x^2+8)-3/8*a^2*arctanh(a*x)+3*a^2*arctanh(a*x)/(-4*a^2*x^ 2+4)+3/2*a^2*arctanh(a*x)^2-3/2*a*arctanh(a*x)^2/x-3*a^3*x*arctanh(a*x)^2/ (-4*a^2*x^2+4)+1/4*a^2*arctanh(a*x)^3-1/2*arctanh(a*x)^3/x^2+a^2*arctanh(a *x)^3/(-2*a^2*x^2+2)+1/2*a^2*arctanh(a*x)^4+3*a^2*arctanh(a*x)*ln(2-2/(a*x +1))+2*a^2*arctanh(a*x)^3*ln(2-2/(a*x+1))-3/2*a^2*polylog(2,-1+2/(a*x+1))- 3*a^2*arctanh(a*x)^2*polylog(2,-1+2/(a*x+1))-3*a^2*arctanh(a*x)*polylog(3, -1+2/(a*x+1))-3/2*a^2*polylog(4,-1+2/(a*x+1))
Time = 0.47 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.71 \[ \int \frac {\text {arctanh}(a x)^3}{x^3 \left (1-a^2 x^2\right )^2} \, dx=\frac {1}{32} a^2 \left (\pi ^4+48 \text {arctanh}(a x)^2-\frac {48 \text {arctanh}(a x)^2}{a x}-\frac {16 \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3}{a^2 x^2}-16 \text {arctanh}(a x)^4+12 \text {arctanh}(a x) \cosh (2 \text {arctanh}(a x))+8 \text {arctanh}(a x)^3 \cosh (2 \text {arctanh}(a x))+96 \text {arctanh}(a x) \log \left (1-e^{-2 \text {arctanh}(a x)}\right )+64 \text {arctanh}(a x)^3 \log \left (1-e^{2 \text {arctanh}(a x)}\right )-48 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(a x)}\right )+96 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(a x)}\right )-96 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(a x)}\right )+48 \operatorname {PolyLog}\left (4,e^{2 \text {arctanh}(a x)}\right )-6 \sinh (2 \text {arctanh}(a x))-12 \text {arctanh}(a x)^2 \sinh (2 \text {arctanh}(a x))\right ) \] Input:
Integrate[ArcTanh[a*x]^3/(x^3*(1 - a^2*x^2)^2),x]
Output:
(a^2*(Pi^4 + 48*ArcTanh[a*x]^2 - (48*ArcTanh[a*x]^2)/(a*x) - (16*(1 - a^2* x^2)*ArcTanh[a*x]^3)/(a^2*x^2) - 16*ArcTanh[a*x]^4 + 12*ArcTanh[a*x]*Cosh[ 2*ArcTanh[a*x]] + 8*ArcTanh[a*x]^3*Cosh[2*ArcTanh[a*x]] + 96*ArcTanh[a*x]* Log[1 - E^(-2*ArcTanh[a*x])] + 64*ArcTanh[a*x]^3*Log[1 - E^(2*ArcTanh[a*x] )] - 48*PolyLog[2, E^(-2*ArcTanh[a*x])] + 96*ArcTanh[a*x]^2*PolyLog[2, E^( 2*ArcTanh[a*x])] - 96*ArcTanh[a*x]*PolyLog[3, E^(2*ArcTanh[a*x])] + 48*Pol yLog[4, E^(2*ArcTanh[a*x])] - 6*Sinh[2*ArcTanh[a*x]] - 12*ArcTanh[a*x]^2*S inh[2*ArcTanh[a*x]]))/32
Time = 4.82 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.47, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.909, Rules used = {6592, 6544, 6452, 6544, 6452, 6510, 6550, 6494, 2897, 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.
| \(\displaystyle \int \frac {\text {arctanh}(a x)^3}{x^3 \left (1-a^2 x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 6592 |
| \(\displaystyle a^2 \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^2}dx+\int \frac {\text {arctanh}(a x)^3}{x^3 \left (1-a^2 x^2\right )}dx\) |
| \(\Big \downarrow \) 6544 |
| \(\displaystyle a^2 \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^2}dx+a^2 \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )}dx+\int \frac {\text {arctanh}(a x)^3}{x^3}dx\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {3}{2} a \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )}dx+a^2 \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^2}dx+a^2 \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)^3}{2 x^2}\) |
| \(\Big \downarrow \) 6544 |
| \(\displaystyle \frac {3}{2} a \left (a^2 \int \frac {\text {arctanh}(a x)^2}{1-a^2 x^2}dx+\int \frac {\text {arctanh}(a x)^2}{x^2}dx\right )+a^2 \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^2}dx+a^2 \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)^3}{2 x^2}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {3}{2} a \left (a^2 \int \frac {\text {arctanh}(a x)^2}{1-a^2 x^2}dx+2 a \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)^2}{x}\right )+a^2 \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^2}dx+a^2 \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)^3}{2 x^2}\) |
| \(\Big \downarrow \) 6510 |
| \(\displaystyle \frac {3}{2} a \left (2 a \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )}dx+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\right )+a^2 \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^2}dx+a^2 \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)^3}{2 x^2}\) |
| \(\Big \downarrow \) 6550 |
| \(\displaystyle a^2 \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^2}dx+a^2 \left (\int \frac {\text {arctanh}(a x)^3}{x (a x+1)}dx+\frac {1}{4} \text {arctanh}(a x)^4\right )+\frac {3}{2} a \left (2 a \left (\int \frac {\text {arctanh}(a x)}{x (a x+1)}dx+\frac {1}{2} \text {arctanh}(a x)^2\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\right )-\frac {\text {arctanh}(a x)^3}{2 x^2}\) |
| \(\Big \downarrow \) 6494 |
| \(\displaystyle a^2 \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^2}dx+\frac {3}{2} a \left (2 a \left (-a \int \frac {\log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\right )+a^2 \left (-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )-\frac {\text {arctanh}(a x)^3}{2 x^2}\) |
| \(\Big \downarrow \) 2897 |
| \(\displaystyle a^2 \int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )^2}dx+a^2 \left (-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )+\frac {3}{2} a \left (2 a \left (\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\right )-\frac {\text {arctanh}(a x)^3}{2 x^2}\) |
| \(\Big \downarrow \) 6592 |
| \(\displaystyle a^2 \left (a^2 \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2}dx+\int \frac {\text {arctanh}(a x)^3}{x \left (1-a^2 x^2\right )}dx\right )+a^2 \left (-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )+\frac {3}{2} a \left (2 a \left (\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\right )-\frac {\text {arctanh}(a x)^3}{2 x^2}\) |
| \(\Big \downarrow \) 6550 |
| \(\displaystyle a^2 \left (a^2 \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2}dx+\int \frac {\text {arctanh}(a x)^3}{x (a x+1)}dx+\frac {1}{4} \text {arctanh}(a x)^4\right )+a^2 \left (-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )+\frac {3}{2} a \left (2 a \left (\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\right )-\frac {\text {arctanh}(a x)^3}{2 x^2}\) |
| \(\Big \downarrow \) 6494 |
| \(\displaystyle a^2 \left (-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )+a^2 \left (a^2 \int \frac {x \text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2}dx-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )+\frac {3}{2} a \left (2 a \left (\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\right )-\frac {\text {arctanh}(a x)^3}{2 x^2}\) |
| \(\Big \downarrow \) 6556 |
| \(\displaystyle a^2 \left (-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )+a^2 \left (a^2 \left (\frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx}{2 a}\right )-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )+\frac {3}{2} a \left (2 a \left (\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\right )-\frac {\text {arctanh}(a x)^3}{2 x^2}\) |
| \(\Big \downarrow \) 6518 |
| \(\displaystyle a^2 \left (-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )+a^2 \left (a^2 \left (\frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \left (-a \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )}{2 a}\right )-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )+\frac {3}{2} a \left (2 a \left (\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\right )-\frac {\text {arctanh}(a x)^3}{2 x^2}\) |
| \(\Big \downarrow \) 6556 |
| \(\displaystyle a^2 \left (-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )+a^2 \left (a^2 \left (\frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \left (-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2}dx}{2 a}\right )+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )}{2 a}\right )-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )+\frac {3}{2} a \left (2 a \left (\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\right )-\frac {\text {arctanh}(a x)^3}{2 x^2}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle a^2 \left (-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )+a^2 \left (a^2 \left (\frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \left (-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {1}{2} \int \frac {1}{1-a^2 x^2}dx+\frac {x}{2 \left (1-a^2 x^2\right )}}{2 a}\right )+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )}{2 a}\right )-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )+\frac {3}{2} a \left (2 a \left (\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\right )-\frac {\text {arctanh}(a x)^3}{2 x^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle a^2 \left (-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )+a^2 \left (-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+a^2 \left (\frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )}{2 a}\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )+\frac {3}{2} a \left (2 a \left (\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\right )-\frac {\text {arctanh}(a x)^3}{2 x^2}\) |
| \(\Big \downarrow \) 6618 |
| \(\displaystyle a^2 \left (-3 a \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}-\int \frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{1-a^2 x^2}dx\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )+a^2 \left (-3 a \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}-\int \frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{1-a^2 x^2}dx\right )+a^2 \left (\frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )}{2 a}\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )+\frac {3}{2} a \left (2 a \left (\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\right )-\frac {\text {arctanh}(a x)^3}{2 x^2}\) |
| \(\Big \downarrow \) 6622 |
| \(\displaystyle a^2 \left (-3 a \left (-\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{1-a^2 x^2}dx+\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}+\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{2 a}\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )+a^2 \left (-3 a \left (-\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{1-a^2 x^2}dx+\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}+\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{2 a}\right )+a^2 \left (\frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )}{2 a}\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )+\frac {3}{2} a \left (2 a \left (\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\right )-\frac {\text {arctanh}(a x)^3}{2 x^2}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle a^2 \left (a^2 \left (\frac {\text {arctanh}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {3 \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )}{2 a}\right )-3 a \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}+\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{2 a}+\frac {\operatorname {PolyLog}\left (4,\frac {2}{a x+1}-1\right )}{4 a}\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )+a^2 \left (-3 a \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}+\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{2 a}+\frac {\operatorname {PolyLog}\left (4,\frac {2}{a x+1}-1\right )}{4 a}\right )+\frac {1}{4} \text {arctanh}(a x)^4+\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )\right )+\frac {3}{2} a \left (2 a \left (\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\right )-\frac {\text {arctanh}(a x)^3}{2 x^2}\) |
Input:
Int[ArcTanh[a*x]^3/(x^3*(1 - a^2*x^2)^2),x]
Output:
-1/2*ArcTanh[a*x]^3/x^2 + (3*a*(-(ArcTanh[a*x]^2/x) + (a*ArcTanh[a*x]^3)/3 + 2*a*(ArcTanh[a*x]^2/2 + ArcTanh[a*x]*Log[2 - 2/(1 + a*x)] - PolyLog[2, -1 + 2/(1 + a*x)]/2)))/2 + a^2*(ArcTanh[a*x]^4/4 + 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))) + a^2*(ArcTanh[a*x]^4/4 + a^2*(ArcTanh[a*x]^3/(2*a^2*(1 - a ^2*x^2)) - (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))))/(2*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)))
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[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, x][[2]], Expon[Pq, x]]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : > Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x ], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 ] && IntegerQ[m])) && NeQ[m, -1]
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), x_Symb ol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b , c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]
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_.)*((f_.)*(x_))^(m_))/((d_) + ( e_.)*(x_)^2), x_Symbol] :> Simp[1/d Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x ], x] - Simp[e/(d*f^2) Int[(f*x)^(m + 2)*((a + b*ArcTanh[c*x])^p/(d + e*x ^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
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]
Time = 90.92 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.48
| method | result | size |
| derivativedivides | \(a^{2} \left (-\frac {\operatorname {arctanh}\left (a x \right )^{4}}{2}-\frac {\left (a x +1\right ) \left (4 \operatorname {arctanh}\left (a x \right )^{3}-6 \operatorname {arctanh}\left (a x \right )^{2}+6 \,\operatorname {arctanh}\left (a x \right )-3\right )}{32 \left (a x -1\right )}-\frac {\left (4 \operatorname {arctanh}\left (a x \right )^{3}+6 \operatorname {arctanh}\left (a x \right )^{2}+6 \,\operatorname {arctanh}\left (a x \right )+3\right ) \left (a x -1\right )}{32 \left (a x +1\right )}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \left (a x \,\operatorname {arctanh}\left (a x \right )+\operatorname {arctanh}\left (a x \right )+3 a x \right ) \left (a x -1\right )}{2 a^{2} x^{2}}-3 \operatorname {arctanh}\left (a x \right )^{2}+3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 \operatorname {arctanh}\left (a x \right )^{3} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-12 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+12 \operatorname {polylog}\left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 \operatorname {arctanh}\left (a x \right )^{3} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-12 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+12 \operatorname {polylog}\left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\) | \(447\) |
| default | \(a^{2} \left (-\frac {\operatorname {arctanh}\left (a x \right )^{4}}{2}-\frac {\left (a x +1\right ) \left (4 \operatorname {arctanh}\left (a x \right )^{3}-6 \operatorname {arctanh}\left (a x \right )^{2}+6 \,\operatorname {arctanh}\left (a x \right )-3\right )}{32 \left (a x -1\right )}-\frac {\left (4 \operatorname {arctanh}\left (a x \right )^{3}+6 \operatorname {arctanh}\left (a x \right )^{2}+6 \,\operatorname {arctanh}\left (a x \right )+3\right ) \left (a x -1\right )}{32 \left (a x +1\right )}+\frac {\operatorname {arctanh}\left (a x \right )^{2} \left (a x \,\operatorname {arctanh}\left (a x \right )+\operatorname {arctanh}\left (a x \right )+3 a x \right ) \left (a x -1\right )}{2 a^{2} x^{2}}-3 \operatorname {arctanh}\left (a x \right )^{2}+3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 \operatorname {arctanh}\left (a x \right )^{3} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-12 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+12 \operatorname {polylog}\left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 \operatorname {arctanh}\left (a x \right )^{3} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-12 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+12 \operatorname {polylog}\left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\) | \(447\) |
Input:
int(arctanh(a*x)^3/x^3/(-a^2*x^2+1)^2,x,method=_RETURNVERBOSE)
Output:
a^2*(-1/2*arctanh(a*x)^4-1/32*(a*x+1)*(4*arctanh(a*x)^3-6*arctanh(a*x)^2+6 *arctanh(a*x)-3)/(a*x-1)-1/32*(4*arctanh(a*x)^3+6*arctanh(a*x)^2+6*arctanh (a*x)+3)*(a*x-1)/(a*x+1)+1/2*arctanh(a*x)^2*(a*x*arctanh(a*x)+arctanh(a*x) +3*a*x)*(a*x-1)/a^2/x^2-3*arctanh(a*x)^2+3*arctanh(a*x)*ln(1-(a*x+1)/(-a^2 *x^2+1)^(1/2))+3*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+3*arctanh(a*x)*ln(1 +(a*x+1)/(-a^2*x^2+1)^(1/2))+3*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+2*ar ctanh(a*x)^3*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+6*arctanh(a*x)^2*polylog(2,( a*x+1)/(-a^2*x^2+1)^(1/2))-12*arctanh(a*x)*polylog(3,(a*x+1)/(-a^2*x^2+1)^ (1/2))+12*polylog(4,(a*x+1)/(-a^2*x^2+1)^(1/2))+2*arctanh(a*x)^3*ln(1+(a*x +1)/(-a^2*x^2+1)^(1/2))+6*arctanh(a*x)^2*polylog(2,-(a*x+1)/(-a^2*x^2+1)^( 1/2))-12*arctanh(a*x)*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))+12*polylog(4, -(a*x+1)/(-a^2*x^2+1)^(1/2)))
\[ \int \frac {\text {arctanh}(a x)^3}{x^3 \left (1-a^2 x^2\right )^2} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )}^{2} x^{3}} \,d x } \] Input:
integrate(arctanh(a*x)^3/x^3/(-a^2*x^2+1)^2,x, algorithm="fricas")
Output:
integral(arctanh(a*x)^3/(a^4*x^7 - 2*a^2*x^5 + x^3), x)
\[ \int \frac {\text {arctanh}(a x)^3}{x^3 \left (1-a^2 x^2\right )^2} \, dx=\int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{x^{3} \left (a x - 1\right )^{2} \left (a x + 1\right )^{2}}\, dx \] Input:
integrate(atanh(a*x)**3/x**3/(-a**2*x**2+1)**2,x)
Output:
Integral(atanh(a*x)**3/(x**3*(a*x - 1)**2*(a*x + 1)**2), x)
\[ \int \frac {\text {arctanh}(a x)^3}{x^3 \left (1-a^2 x^2\right )^2} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )}^{2} x^{3}} \,d x } \] Input:
integrate(arctanh(a*x)^3/x^3/(-a^2*x^2+1)^2,x, algorithm="maxima")
Output:
1/32*((a^4*x^4 - a^2*x^2)*log(-a*x + 1)^4 + 2*(2*a^2*x^2 + 2*(a^4*x^4 - a^ 2*x^2)*log(a*x + 1) - 1)*log(-a*x + 1)^3)/(a^2*x^4 - x^2) - 1/8*integrate( -1/2*(2*log(a*x + 1)^3 - 6*log(a*x + 1)^2*log(-a*x + 1) - 3*(2*a^4*x^4 + 2 *a^3*x^3 - a^2*x^2 - a*x + 2*(a^6*x^6 + a^5*x^5 - a^4*x^4 - a^3*x^3 - 1)*l og(a*x + 1))*log(-a*x + 1)^2)/(a^4*x^7 - 2*a^2*x^5 + x^3), x)
\[ \int \frac {\text {arctanh}(a x)^3}{x^3 \left (1-a^2 x^2\right )^2} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )}^{2} x^{3}} \,d x } \] Input:
integrate(arctanh(a*x)^3/x^3/(-a^2*x^2+1)^2,x, algorithm="giac")
Output:
integrate(arctanh(a*x)^3/((a^2*x^2 - 1)^2*x^3), x)
Timed out. \[ \int \frac {\text {arctanh}(a x)^3}{x^3 \left (1-a^2 x^2\right )^2} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{x^3\,{\left (a^2\,x^2-1\right )}^2} \,d x \] Input:
int(atanh(a*x)^3/(x^3*(a^2*x^2 - 1)^2),x)
Output:
int(atanh(a*x)^3/(x^3*(a^2*x^2 - 1)^2), x)
\[ \int \frac {\text {arctanh}(a x)^3}{x^3 \left (1-a^2 x^2\right )^2} \, dx=\int \frac {\mathit {atanh} \left (a x \right )^{3}}{a^{4} x^{7}-2 a^{2} x^{5}+x^{3}}d x \] Input:
int(atanh(a*x)^3/x^3/(-a^2*x^2+1)^2,x)
Output:
int(atanh(a*x)**3/(a**4*x**7 - 2*a**2*x**5 + x**3),x)