Integrand size = 22, antiderivative size = 209 \[ \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )^3} \, dx=\frac {a^2 x}{32 \left (1-a^2 x^2\right )^2}+\frac {31 a^2 x}{64 \left (1-a^2 x^2\right )}+\frac {31}{64} a \text {arctanh}(a x)-\frac {a \text {arctanh}(a x)}{8 \left (1-a^2 x^2\right )^2}-\frac {7 a \text {arctanh}(a x)}{8 \left (1-a^2 x^2\right )}+a \text {arctanh}(a x)^2-\frac {\text {arctanh}(a x)^2}{x}+\frac {a^2 x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}+\frac {7 a^2 x \text {arctanh}(a x)^2}{8 \left (1-a^2 x^2\right )}+\frac {5}{8} a \text {arctanh}(a x)^3+2 a \text {arctanh}(a x) \log \left (2-\frac {2}{1+a x}\right )-a \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \]
1/32*a^2*x/(-a^2*x^2+1)^2+31/64*a^2*x/(-a^2*x^2+1)+31/64*a*arctanh(a*x)-1/ 8*a*arctanh(a*x)/(-a^2*x^2+1)^2-7/8*a*arctanh(a*x)/(-a^2*x^2+1)+a*arctanh( a*x)^2-arctanh(a*x)^2/x+1/4*a^2*x*arctanh(a*x)^2/(-a^2*x^2+1)^2+7/8*a^2*x* arctanh(a*x)^2/(-a^2*x^2+1)+5/8*a*arctanh(a*x)^3+2*a*arctanh(a*x)*ln(2-2/( a*x+1))-a*polylog(2,-1+2/(a*x+1))
Time = 0.47 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.61 \[ \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )^3} \, dx=-a \left (-\frac {5}{8} \text {arctanh}(a x)^3+\frac {1}{64} \text {arctanh}(a x) \left (32 \cosh (2 \text {arctanh}(a x))+\cosh (4 \text {arctanh}(a x))-128 \log \left (1-e^{-2 \text {arctanh}(a x)}\right )\right )+\operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(a x)}\right )-\frac {1}{4} \sinh (2 \text {arctanh}(a x))+\text {arctanh}(a x)^2 \left (-1+\frac {1}{a x}+\frac {a x}{-1+a^2 x^2}-\frac {1}{32} \sinh (4 \text {arctanh}(a x))\right )-\frac {1}{256} \sinh (4 \text {arctanh}(a x))\right ) \]
-(a*((-5*ArcTanh[a*x]^3)/8 + (ArcTanh[a*x]*(32*Cosh[2*ArcTanh[a*x]] + Cosh [4*ArcTanh[a*x]] - 128*Log[1 - E^(-2*ArcTanh[a*x])]))/64 + PolyLog[2, E^(- 2*ArcTanh[a*x])] - Sinh[2*ArcTanh[a*x]]/4 + ArcTanh[a*x]^2*(-1 + 1/(a*x) + (a*x)/(-1 + a^2*x^2) - Sinh[4*ArcTanh[a*x]]/32) - Sinh[4*ArcTanh[a*x]]/25 6))
Time = 2.98 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.84, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.909, Rules used = {6592, 6526, 215, 215, 219, 6518, 6556, 215, 219, 6592, 6518, 6544, 6452, 6510, 6550, 6494, 2897, 6556, 215, 219}
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.
| \(\displaystyle \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 6592 |
| \(\displaystyle a^2 \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^3}dx+\int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 6526 |
| \(\displaystyle a^2 \left (\frac {3}{4} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {1}{8} \int \frac {1}{\left (1-a^2 x^2\right )^3}dx+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}\right )+\int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle a^2 \left (\frac {3}{4} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {1}{8} \left (\frac {3}{4} \int \frac {1}{\left (1-a^2 x^2\right )^2}dx+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}\right )+\int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle a^2 \left (\frac {3}{4} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {1}{8} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{1-a^2 x^2}dx+\frac {x}{2 \left (1-a^2 x^2\right )}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}\right )+\int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle a^2 \left (\frac {3}{4} \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )\right )+\int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 6518 |
| \(\displaystyle a^2 \left (\frac {3}{4} \left (-a \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )\right )+\int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 6556 |
| \(\displaystyle a^2 \left (\frac {3}{4} \left (-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2}dx}{2 a}\right )+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )\right )+\int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle a^2 \left (\frac {3}{4} \left (-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {1}{2} \int \frac {1}{1-a^2 x^2}dx+\frac {x}{2 \left (1-a^2 x^2\right )}}{2 a}\right )+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )+\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )\right )+\int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )^2}dx+a^2 \left (\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )\right )\) |
| \(\Big \downarrow \) 6592 |
| \(\displaystyle a^2 \int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )}dx+a^2 \left (\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )\right )\) |
| \(\Big \downarrow \) 6518 |
| \(\displaystyle a^2 \left (-a \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )+\int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )}dx+a^2 \left (\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )\right )\) |
| \(\Big \downarrow \) 6544 |
| \(\displaystyle a^2 \left (-a \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )+a^2 \int \frac {\text {arctanh}(a x)^2}{1-a^2 x^2}dx+\int \frac {\text {arctanh}(a x)^2}{x^2}dx+a^2 \left (\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle a^2 \left (-a \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )+a^2 \int \frac {\text {arctanh}(a x)^2}{1-a^2 x^2}dx+2 a \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )}dx+a^2 \left (\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )\right )-\frac {\text {arctanh}(a x)^2}{x}\) |
| \(\Big \downarrow \) 6510 |
| \(\displaystyle a^2 \left (-a \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )+2 a \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )}dx+a^2 \left (\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\) |
| \(\Big \downarrow \) 6550 |
| \(\displaystyle a^2 \left (-a \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )+2 a \left (\int \frac {\text {arctanh}(a x)}{x (a x+1)}dx+\frac {1}{2} \text {arctanh}(a x)^2\right )+a^2 \left (\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\) |
| \(\Big \downarrow \) 6494 |
| \(\displaystyle a^2 \left (-a \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )+2 a \left (-a \int \frac {\log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )\right )+a^2 \left (\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\) |
| \(\Big \downarrow \) 2897 |
| \(\displaystyle a^2 \left (-a \int \frac {x \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )+a^2 \left (\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )\right )+2 a \left (\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\) |
| \(\Big \downarrow \) 6556 |
| \(\displaystyle a^2 \left (-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2}dx}{2 a}\right )+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )+a^2 \left (\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )\right )+2 a \left (\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle a^2 \left (-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {1}{2} \int \frac {1}{1-a^2 x^2}dx+\frac {x}{2 \left (1-a^2 x^2\right )}}{2 a}\right )+\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^3}{6 a}\right )+a^2 \left (\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )\right )+2 a \left (\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle a^2 \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )+a^2 \left (\frac {x \text {arctanh}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {\text {arctanh}(a x)}{8 a \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (\frac {3}{4} \left (\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}\right )+\frac {x}{4 \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^2}{2 \left (1-a^2 x^2\right )}-a \left (\frac {\text {arctanh}(a x)}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)}{2 a}}{2 a}\right )+\frac {\text {arctanh}(a x)^3}{6 a}\right )\right )+2 a \left (\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\) |
-(ArcTanh[a*x]^2/x) + (a*ArcTanh[a*x]^3)/3 + a^2*((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))) + a^2*(-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*(ArcTa nh[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) + 2*a*(ArcTanh[a*x]^2/2 + ArcTanh[a*x]*Log[2 - 2/(1 + a*x)] - PolyLog[2, -1 + 2/(1 + a*x)]/2)
3.4.13.3.1 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[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_)*((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_.)*((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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.90 (sec) , antiderivative size = 3143, normalized size of antiderivative = 15.04
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(3143\) |
| default | \(\text {Expression too large to display}\) | \(3143\) |
| parts | \(\text {Expression too large to display}\) | \(3154\) |
a*(1/4*arctanh(a*x)*(a*x-1)/(a*x+1)+1/4*(a*x+1)*arctanh(a*x)/(a*x-1)+15/16 *I*Pi*arctanh(a*x)^2-arctanh(a*x)^2/a/x+15/32*I*Pi*csgn(I*(a*x+1)^2/(a^2*x ^2-1)/(1-(a*x+1)^2/(a^2*x^2-1)))*csgn(I/(1-(a*x+1)^2/(a^2*x^2-1)))*csgn(I* (a*x+1)^2/(a^2*x^2-1))*(arctanh(a*x)*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+dilo g(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-dilog((a*x+1)/(-a^2*x^2+1)^(1/2)))-15/16*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*(ar ctanh(a*x)*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+dilog(1+(a*x+1)/(-a^2*x^2+1)^( 1/2))-dilog((a*x+1)/(-a^2*x^2+1)^(1/2)))+15/32*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))*(arctanh (a*x)*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+dilog(1+(a*x+1)/(-a^2*x^2+1)^(1/2)) -dilog((a*x+1)/(-a^2*x^2+1)^(1/2)))+1/16*arctanh(a*x)^2/(a*x-1)^2-1/16*arc tanh(a*x)^2/(a*x+1)^2+polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+polylog(2,(a* x+1)/(-a^2*x^2+1)^(1/2))+5/8*arctanh(a*x)^3-arctanh(a*x)^2-1/128*arctanh(a *x)*(a*x+1)^2/(a*x-1)^2-1/128*(a*x-1)^2*arctanh(a*x)/(a*x+1)^2+15/32*I*Pi* csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1)))*csgn(I/(1-(a*x+1)^ 2/(a^2*x^2-1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*(arctanh(a*x)^2-arctanh(a*x) *ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^( 1/2))-polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))-polylog(2,(a*x+1)/(-a^2*x^2+1 )^(1/2)))+1/512*(a*x+1)^2/(a*x-1)^2+1/8/(a*x+1)*(a*x-1)-15/32*I*Pi*csgn(I* (a*x+1)^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^2*x^2-1)))^3*(arctanh(a*x)^2-arct...
\[ \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )^3} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{2}}{{\left (a^{2} x^{2} - 1\right )}^{3} x^{2}} \,d x } \]
\[ \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )^3} \, dx=- \int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{a^{6} x^{8} - 3 a^{4} x^{6} + 3 a^{2} x^{4} - x^{2}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 534 vs. \(2 (186) = 372\).
Time = 0.20 (sec) , antiderivative size = 534, normalized size of antiderivative = 2.56 \[ \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )^3} \, dx=-\frac {1}{128} \, a^{2} {\left (\frac {2 \, {\left (31 \, a^{3} x^{3} - 5 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{3} + 5 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{3} - {\left (16 \, a^{4} x^{4} - 32 \, a^{2} x^{2} - 15 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right ) + 16\right )} \log \left (a x + 1\right )^{2} + 16 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} - 33 \, a x - {\left (15 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} - 32 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + 1\right )\right )}}{a^{5} x^{4} - 2 \, a^{3} x^{2} + a} - \frac {128 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a} + \frac {128 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )}}{a} - \frac {128 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )}}{a} - \frac {31 \, \log \left (a x + 1\right )}{a} + \frac {31 \, \log \left (a x - 1\right )}{a}\right )} + \frac {1}{32} \, a {\left (\frac {28 \, a^{2} x^{2} - 15 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 30 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 15 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} - 32}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1} - 32 \, \log \left (a x + 1\right ) - 32 \, \log \left (a x - 1\right ) + 64 \, \log \left (x\right )\right )} \operatorname {artanh}\left (a x\right ) + \frac {1}{16} \, {\left (15 \, a \log \left (a x + 1\right ) - 15 \, a \log \left (a x - 1\right ) - \frac {2 \, {\left (15 \, a^{4} x^{4} - 25 \, a^{2} x^{2} + 8\right )}}{a^{4} x^{5} - 2 \, a^{2} x^{3} + x}\right )} \operatorname {artanh}\left (a x\right )^{2} \]
-1/128*a^2*(2*(31*a^3*x^3 - 5*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^3 + 5 *(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^3 - (16*a^4*x^4 - 32*a^2*x^2 - 15* (a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1) + 16)*log(a*x + 1)^2 + 16*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^2 - 33*a*x - (15*(a^4*x^4 - 2*a^2*x^2 + 1)*l og(a*x - 1)^2 - 32*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1))*log(a*x + 1))/( a^5*x^4 - 2*a^3*x^2 + a) - 128*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1 /2*a*x + 1/2))/a + 128*(log(a*x + 1)*log(x) + dilog(-a*x))/a - 128*(log(-a *x + 1)*log(x) + dilog(a*x))/a - 31*log(a*x + 1)/a + 31*log(a*x - 1)/a) + 1/32*a*((28*a^2*x^2 - 15*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^2 + 30*(a^ 4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)*log(a*x - 1) - 15*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^2 - 32)/(a^4*x^4 - 2*a^2*x^2 + 1) - 32*log(a*x + 1) - 3 2*log(a*x - 1) + 64*log(x))*arctanh(a*x) + 1/16*(15*a*log(a*x + 1) - 15*a* log(a*x - 1) - 2*(15*a^4*x^4 - 25*a^2*x^2 + 8)/(a^4*x^5 - 2*a^2*x^3 + x))* arctanh(a*x)^2
\[ \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )^3} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{2}}{{\left (a^{2} x^{2} - 1\right )}^{3} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )^3} \, dx=-\int \frac {{\mathrm {atanh}\left (a\,x\right )}^2}{x^2\,{\left (a^2\,x^2-1\right )}^3} \,d x \]