Integrand size = 22, antiderivative size = 281 \[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^3} \, dx=-\frac {3 a}{128 \left (1-a^2 x^2\right )^2}-\frac {93 a}{128 \left (1-a^2 x^2\right )}+\frac {3 a^2 x \text {arctanh}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac {93 a^2 x \text {arctanh}(a x)}{64 \left (1-a^2 x^2\right )}+\frac {93}{128} a \text {arctanh}(a x)^2-\frac {3 a \text {arctanh}(a x)^2}{16 \left (1-a^2 x^2\right )^2}-\frac {21 a \text {arctanh}(a x)^2}{16 \left (1-a^2 x^2\right )}+a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^3}{x}+\frac {a^2 x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac {7 a^2 x \text {arctanh}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac {15}{32} a \text {arctanh}(a x)^4+3 a \text {arctanh}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-3 a \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-\frac {3}{2} a \operatorname {PolyLog}\left (3,-1+\frac {2}{1+a x}\right ) \] Output:
-3/128*a/(-a^2*x^2+1)^2-93*a/(-128*a^2*x^2+128)+3/32*a^2*x*arctanh(a*x)/(- a^2*x^2+1)^2+93*a^2*x*arctanh(a*x)/(-64*a^2*x^2+64)+93/128*a*arctanh(a*x)^ 2-3/16*a*arctanh(a*x)^2/(-a^2*x^2+1)^2-21*a*arctanh(a*x)^2/(-16*a^2*x^2+16 )+a*arctanh(a*x)^3-arctanh(a*x)^3/x+1/4*a^2*x*arctanh(a*x)^3/(-a^2*x^2+1)^ 2+7*a^2*x*arctanh(a*x)^3/(-8*a^2*x^2+8)+15/32*a*arctanh(a*x)^4+3*a*arctanh (a*x)^2*ln(2-2/(a*x+1))-3*a*arctanh(a*x)*polylog(2,-1+2/(a*x+1))-3/2*a*pol ylog(3,-1+2/(a*x+1))
Result contains complex when optimal does not.
Time = 0.53 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.78 \[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^3} \, dx=-a \left (-\frac {i \pi ^3}{8}+\text {arctanh}(a x)^3+\frac {\text {arctanh}(a x)^3}{a x}-\frac {a x \text {arctanh}(a x)^3}{1-a^2 x^2}-\frac {15}{32} \text {arctanh}(a x)^4+\frac {3}{8} \cosh (2 \text {arctanh}(a x))+\frac {3}{4} \text {arctanh}(a x)^2 \cosh (2 \text {arctanh}(a x))+\frac {3 \cosh (4 \text {arctanh}(a x))}{1024}+\frac {3}{128} \text {arctanh}(a x)^2 \cosh (4 \text {arctanh}(a x))-3 \text {arctanh}(a x)^2 \log \left (1-e^{2 \text {arctanh}(a x)}\right )-3 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(a x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(a x)}\right )-\frac {3}{4} \text {arctanh}(a x) \sinh (2 \text {arctanh}(a x))-\frac {3}{256} \text {arctanh}(a x) \sinh (4 \text {arctanh}(a x))-\frac {1}{32} \text {arctanh}(a x)^3 \sinh (4 \text {arctanh}(a x))\right ) \] Input:
Integrate[ArcTanh[a*x]^3/(x^2*(1 - a^2*x^2)^3),x]
Output:
-(a*((-1/8*I)*Pi^3 + ArcTanh[a*x]^3 + ArcTanh[a*x]^3/(a*x) - (a*x*ArcTanh[ a*x]^3)/(1 - a^2*x^2) - (15*ArcTanh[a*x]^4)/32 + (3*Cosh[2*ArcTanh[a*x]])/ 8 + (3*ArcTanh[a*x]^2*Cosh[2*ArcTanh[a*x]])/4 + (3*Cosh[4*ArcTanh[a*x]])/1 024 + (3*ArcTanh[a*x]^2*Cosh[4*ArcTanh[a*x]])/128 - 3*ArcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x])] - 3*ArcTanh[a*x]*PolyLog[2, E^(2*ArcTanh[a*x])] + ( 3*PolyLog[3, E^(2*ArcTanh[a*x])])/2 - (3*ArcTanh[a*x]*Sinh[2*ArcTanh[a*x]] )/4 - (3*ArcTanh[a*x]*Sinh[4*ArcTanh[a*x]])/256 - (ArcTanh[a*x]^3*Sinh[4*A rcTanh[a*x]])/32))
Time = 5.28 (sec) , antiderivative size = 522, normalized size of antiderivative = 1.86, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.955, Rules used = {6592, 6526, 6518, 6522, 6518, 241, 6556, 6518, 241, 6592, 6518, 6544, 6452, 6510, 6550, 6494, 6556, 6518, 241, 6618, 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.
| \(\displaystyle \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 6592 |
| \(\displaystyle a^2 \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3}dx+\int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 6526 |
| \(\displaystyle a^2 \left (\frac {3}{8} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3}dx+\frac {3}{4} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}\right )+\int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 6518 |
| \(\displaystyle a^2 \left (\frac {3}{8} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3}dx+\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}\right )+\int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 6522 |
| \(\displaystyle a^2 \left (\frac {3}{8} \left (\frac {3}{4} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}\right )+\int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 6518 |
| \(\displaystyle a^2 \left (\frac {3}{8} \left (\frac {3}{4} \left (-\frac {1}{2} a \int \frac {x}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )+\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}\right )+\int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle a^2 \left (\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {3}{8} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )\right )+\int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 6556 |
| \(\displaystyle a^2 \left (\frac {3}{4} \left (-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx}{a}\right )+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {3}{8} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )\right )+\int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 6518 |
| \(\displaystyle a^2 \left (\frac {3}{4} \left (-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {-\frac {1}{2} a \int \frac {x}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {3}{8} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )\right )+\int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^2}dx+a^2 \left (\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {3}{8} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {\text {arctanh}(a x)^4}{8 a}\right )\right )\) |
| \(\Big \downarrow \) 6592 |
| \(\displaystyle a^2 \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2}dx+\int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )}dx+a^2 \left (\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {3}{8} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {\text {arctanh}(a x)^4}{8 a}\right )\right )\) |
| \(\Big \downarrow \) 6518 |
| \(\displaystyle a^2 \left (-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+\int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )}dx+a^2 \left (\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {3}{8} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {\text {arctanh}(a x)^4}{8 a}\right )\right )\) |
| \(\Big \downarrow \) 6544 |
| \(\displaystyle a^2 \left (-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+a^2 \int \frac {\text {arctanh}(a x)^3}{1-a^2 x^2}dx+\int \frac {\text {arctanh}(a x)^3}{x^2}dx+a^2 \left (\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {3}{8} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {\text {arctanh}(a x)^4}{8 a}\right )\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle a^2 \left (-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+3 a \int \frac {\text {arctanh}(a x)^2}{x \left (1-a^2 x^2\right )}dx+a^2 \int \frac {\text {arctanh}(a x)^3}{1-a^2 x^2}dx+a^2 \left (\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {3}{8} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {\text {arctanh}(a x)^4}{8 a}\right )\right )-\frac {\text {arctanh}(a x)^3}{x}\) |
| \(\Big \downarrow \) 6510 |
| \(\displaystyle a^2 \left (-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+3 a \int \frac {\text {arctanh}(a x)^2}{x \left (1-a^2 x^2\right )}dx+a^2 \left (\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {3}{8} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {\text {arctanh}(a x)^4}{8 a}\right )\right )+\frac {1}{4} a \text {arctanh}(a x)^4-\frac {\text {arctanh}(a x)^3}{x}\) |
| \(\Big \downarrow \) 6550 |
| \(\displaystyle a^2 \left (-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+3 a \left (\int \frac {\text {arctanh}(a x)^2}{x (a x+1)}dx+\frac {1}{3} \text {arctanh}(a x)^3\right )+a^2 \left (\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {3}{8} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {\text {arctanh}(a x)^4}{8 a}\right )\right )+\frac {1}{4} a \text {arctanh}(a x)^4-\frac {\text {arctanh}(a x)^3}{x}\) |
| \(\Big \downarrow \) 6494 |
| \(\displaystyle a^2 \left (-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+3 a \left (-2 a \int \frac {\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{3} \text {arctanh}(a x)^3+\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )\right )+a^2 \left (\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {3}{8} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {\text {arctanh}(a x)^4}{8 a}\right )\right )+\frac {1}{4} a \text {arctanh}(a x)^4-\frac {\text {arctanh}(a x)^3}{x}\) |
| \(\Big \downarrow \) 6556 |
| \(\displaystyle a^2 \left (-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx}{a}\right )+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+3 a \left (-2 a \int \frac {\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{3} \text {arctanh}(a x)^3+\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )\right )+a^2 \left (\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {3}{8} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {\text {arctanh}(a x)^4}{8 a}\right )\right )+\frac {1}{4} a \text {arctanh}(a x)^4-\frac {\text {arctanh}(a x)^3}{x}\) |
| \(\Big \downarrow \) 6518 |
| \(\displaystyle a^2 \left (-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {-\frac {1}{2} a \int \frac {x}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+3 a \left (-2 a \int \frac {\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{3} \text {arctanh}(a x)^3+\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )\right )+a^2 \left (\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {3}{8} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {\text {arctanh}(a x)^4}{8 a}\right )\right )+\frac {1}{4} a \text {arctanh}(a x)^4-\frac {\text {arctanh}(a x)^3}{x}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle 3 a \left (-2 a \int \frac {\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{3} \text {arctanh}(a x)^3+\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )\right )+a^2 \left (\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {\text {arctanh}(a x)^4}{8 a}\right )+a^2 \left (\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {3}{8} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {\text {arctanh}(a x)^4}{8 a}\right )\right )+\frac {1}{4} a \text {arctanh}(a x)^4-\frac {\text {arctanh}(a x)^3}{x}\) |
| \(\Big \downarrow \) 6618 |
| \(\displaystyle 3 a \left (-2 a \left (\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}-\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{1-a^2 x^2}dx\right )+\frac {1}{3} \text {arctanh}(a x)^3+\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )\right )+a^2 \left (\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {\text {arctanh}(a x)^4}{8 a}\right )+a^2 \left (\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {3}{8} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {\text {arctanh}(a x)^4}{8 a}\right )\right )+\frac {1}{4} a \text {arctanh}(a x)^4-\frac {\text {arctanh}(a x)^3}{x}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle a^2 \left (\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {\text {arctanh}(a x)^4}{8 a}\right )+a^2 \left (\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {3}{8} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {\text {arctanh}(a x)^4}{8 a}\right )\right )+3 a \left (-2 a \left (\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}+\frac {\operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{4 a}\right )+\frac {1}{3} \text {arctanh}(a x)^3+\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )\right )+\frac {1}{4} a \text {arctanh}(a x)^4-\frac {\text {arctanh}(a x)^3}{x}\) |
Input:
Int[ArcTanh[a*x]^3/(x^2*(1 - a^2*x^2)^3),x]
Output:
-(ArcTanh[a*x]^3/x) + (a*ArcTanh[a*x]^4)/4 + a^2*((x*ArcTanh[a*x]^3)/(2*(1 - a^2*x^2)) + ArcTanh[a*x]^4/(8*a) - (3*a*(ArcTanh[a*x]^2/(2*a^2*(1 - a^2 *x^2)) - (-1/4*1/(a*(1 - a^2*x^2)) + (x*ArcTanh[a*x])/(2*(1 - a^2*x^2)) + ArcTanh[a*x]^2/(4*a))/a))/2) + a^2*((-3*ArcTanh[a*x]^2)/(16*a*(1 - a^2*x^2 )^2) + (x*ArcTanh[a*x]^3)/(4*(1 - a^2*x^2)^2) + (3*(-1/16*1/(a*(1 - a^2*x^ 2)^2) + (x*ArcTanh[a*x])/(4*(1 - a^2*x^2)^2) + (3*(-1/4*1/(a*(1 - a^2*x^2) ) + (x*ArcTanh[a*x])/(2*(1 - a^2*x^2)) + ArcTanh[a*x]^2/(4*a)))/4))/8 + (3 *((x*ArcTanh[a*x]^3)/(2*(1 - a^2*x^2)) + ArcTanh[a*x]^4/(8*a) - (3*a*(ArcT anh[a*x]^2/(2*a^2*(1 - a^2*x^2)) - (-1/4*1/(a*(1 - a^2*x^2)) + (x*ArcTanh[ a*x])/(2*(1 - a^2*x^2)) + ArcTanh[a*x]^2/(4*a))/a))/2))/4) + 3*a*(ArcTanh[ a*x]^3/3 + ArcTanh[a*x]^2*Log[2 - 2/(1 + a*x)] - 2*a*((ArcTanh[a*x]*PolyLo g[2, -1 + 2/(1 + a*x)])/(2*a) + PolyLog[3, -1 + 2/(1 + a*x)]/(4*a)))
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
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_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbo l] :> Simp[(-b)*((d + e*x^2)^(q + 1)/(4*c*d*(q + 1)^2)), x] + (-Simp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])/(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]), x], x]) /; Fre eQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && NeQ[q, -3/2]
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]
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[(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 = 7.11 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.25
| method | result | size |
| derivativedivides | \(a \left (\frac {15 \operatorname {arctanh}\left (a x \right )^{4}}{32}+\frac {\left (32 \operatorname {arctanh}\left (a x \right )^{3}-24 \operatorname {arctanh}\left (a x \right )^{2}+12 \,\operatorname {arctanh}\left (a x \right )-3\right ) \left (a x +1\right )^{2}}{2048 \left (a x -1\right )^{2}}-\frac {\left (a x +1\right ) \left (4 \operatorname {arctanh}\left (a x \right )^{3}-6 \operatorname {arctanh}\left (a x \right )^{2}+6 \,\operatorname {arctanh}\left (a x \right )-3\right )}{16 \left (a x -1\right )}+\frac {\left (4 \operatorname {arctanh}\left (a x \right )^{3}+6 \operatorname {arctanh}\left (a x \right )^{2}+6 \,\operatorname {arctanh}\left (a x \right )+3\right ) \left (a x -1\right )}{16 a x +16}-\frac {\left (32 \operatorname {arctanh}\left (a x \right )^{3}+24 \operatorname {arctanh}\left (a x \right )^{2}+12 \,\operatorname {arctanh}\left (a x \right )+3\right ) \left (a x -1\right )^{2}}{2048 \left (a x +1\right )^{2}}+\frac {\operatorname {arctanh}\left (a x \right )^{3} \left (a x -1\right )}{a x}-2 \operatorname {arctanh}\left (a x \right )^{3}+3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\) | \(351\) |
| default | \(a \left (\frac {15 \operatorname {arctanh}\left (a x \right )^{4}}{32}+\frac {\left (32 \operatorname {arctanh}\left (a x \right )^{3}-24 \operatorname {arctanh}\left (a x \right )^{2}+12 \,\operatorname {arctanh}\left (a x \right )-3\right ) \left (a x +1\right )^{2}}{2048 \left (a x -1\right )^{2}}-\frac {\left (a x +1\right ) \left (4 \operatorname {arctanh}\left (a x \right )^{3}-6 \operatorname {arctanh}\left (a x \right )^{2}+6 \,\operatorname {arctanh}\left (a x \right )-3\right )}{16 \left (a x -1\right )}+\frac {\left (4 \operatorname {arctanh}\left (a x \right )^{3}+6 \operatorname {arctanh}\left (a x \right )^{2}+6 \,\operatorname {arctanh}\left (a x \right )+3\right ) \left (a x -1\right )}{16 a x +16}-\frac {\left (32 \operatorname {arctanh}\left (a x \right )^{3}+24 \operatorname {arctanh}\left (a x \right )^{2}+12 \,\operatorname {arctanh}\left (a x \right )+3\right ) \left (a x -1\right )^{2}}{2048 \left (a x +1\right )^{2}}+\frac {\operatorname {arctanh}\left (a x \right )^{3} \left (a x -1\right )}{a x}-2 \operatorname {arctanh}\left (a x \right )^{3}+3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\) | \(351\) |
Input:
int(arctanh(a*x)^3/x^2/(-a^2*x^2+1)^3,x,method=_RETURNVERBOSE)
Output:
a*(15/32*arctanh(a*x)^4+1/2048*(32*arctanh(a*x)^3-24*arctanh(a*x)^2+12*arc tanh(a*x)-3)*(a*x+1)^2/(a*x-1)^2-1/16*(a*x+1)*(4*arctanh(a*x)^3-6*arctanh( a*x)^2+6*arctanh(a*x)-3)/(a*x-1)+1/16*(4*arctanh(a*x)^3+6*arctanh(a*x)^2+6 *arctanh(a*x)+3)*(a*x-1)/(a*x+1)-1/2048*(32*arctanh(a*x)^3+24*arctanh(a*x) ^2+12*arctanh(a*x)+3)*(a*x-1)^2/(a*x+1)^2+arctanh(a*x)^3/a/x*(a*x-1)-2*arc tanh(a*x)^3+3*arctanh(a*x)^2*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+6*arctanh(a* x)*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-6*polylog(3,(a*x+1)/(-a^2*x^2+1)^ (1/2))+3*arctanh(a*x)^2*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+6*arctanh(a*x)*po lylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))-6*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/ 2)))
\[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^3} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )}^{3} x^{2}} \,d x } \] Input:
integrate(arctanh(a*x)^3/x^2/(-a^2*x^2+1)^3,x, algorithm="fricas")
Output:
integral(-arctanh(a*x)^3/(a^6*x^8 - 3*a^4*x^6 + 3*a^2*x^4 - x^2), x)
\[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^3} \, dx=- \int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{a^{6} x^{8} - 3 a^{4} x^{6} + 3 a^{2} x^{4} - x^{2}}\, dx \] Input:
integrate(atanh(a*x)**3/x**2/(-a**2*x**2+1)**3,x)
Output:
-Integral(atanh(a*x)**3/(a**6*x**8 - 3*a**4*x**6 + 3*a**2*x**4 - x**2), x)
Exception generated. \[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(arctanh(a*x)^3/x^2/(-a^2*x^2+1)^3,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
\[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^3} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )}^{3} x^{2}} \,d x } \] Input:
integrate(arctanh(a*x)^3/x^2/(-a^2*x^2+1)^3,x, algorithm="giac")
Output:
integrate(-arctanh(a*x)^3/((a^2*x^2 - 1)^3*x^2), x)
Timed out. \[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^3} \, dx=-\int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{x^2\,{\left (a^2\,x^2-1\right )}^3} \,d x \] Input:
int(-atanh(a*x)^3/(x^2*(a^2*x^2 - 1)^3),x)
Output:
-int(atanh(a*x)^3/(x^2*(a^2*x^2 - 1)^3), x)
\[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^3} \, dx=\frac {120 \mathit {atanh} \left (a x \right )^{4} a^{5} x^{5}-240 \mathit {atanh} \left (a x \right )^{4} a^{3} x^{3}+120 \mathit {atanh} \left (a x \right )^{4} a x -480 \mathit {atanh} \left (a x \right )^{3} a^{4} x^{4}+800 \mathit {atanh} \left (a x \right )^{3} a^{2} x^{2}-256 \mathit {atanh} \left (a x \right )^{3}+450 \mathit {atanh} \left (a x \right )^{2} a^{5} x^{5}-180 \mathit {atanh} \left (a x \right )^{2} a^{3} x^{3}-510 \mathit {atanh} \left (a x \right )^{2} a x -900 \mathit {atanh} \left (a x \right ) a^{4} x^{4}+1020 \mathit {atanh} \left (a x \right ) a^{2} x^{2}-768 \left (\int \frac {\mathit {atanh} \left (a x \right )^{2}}{a^{6} x^{7}-3 a^{4} x^{5}+3 a^{2} x^{3}-x}d x \right ) a^{5} x^{5}+1536 \left (\int \frac {\mathit {atanh} \left (a x \right )^{2}}{a^{6} x^{7}-3 a^{4} x^{5}+3 a^{2} x^{3}-x}d x \right ) a^{3} x^{3}-768 \left (\int \frac {\mathit {atanh} \left (a x \right )^{2}}{a^{6} x^{7}-3 a^{4} x^{5}+3 a^{2} x^{3}-x}d x \right ) a x +225 a^{5} x^{5}-255 a x}{256 x \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right )} \] Input:
int(atanh(a*x)^3/x^2/(-a^2*x^2+1)^3,x)
Output:
(120*atanh(a*x)**4*a**5*x**5 - 240*atanh(a*x)**4*a**3*x**3 + 120*atanh(a*x )**4*a*x - 480*atanh(a*x)**3*a**4*x**4 + 800*atanh(a*x)**3*a**2*x**2 - 256 *atanh(a*x)**3 + 450*atanh(a*x)**2*a**5*x**5 - 180*atanh(a*x)**2*a**3*x**3 - 510*atanh(a*x)**2*a*x - 900*atanh(a*x)*a**4*x**4 + 1020*atanh(a*x)*a**2 *x**2 - 768*int(atanh(a*x)**2/(a**6*x**7 - 3*a**4*x**5 + 3*a**2*x**3 - x), x)*a**5*x**5 + 1536*int(atanh(a*x)**2/(a**6*x**7 - 3*a**4*x**5 + 3*a**2*x* *3 - x),x)*a**3*x**3 - 768*int(atanh(a*x)**2/(a**6*x**7 - 3*a**4*x**5 + 3* a**2*x**3 - x),x)*a*x + 225*a**5*x**5 - 255*a*x)/(256*x*(a**4*x**4 - 2*a** 2*x**2 + 1))