Integrand size = 19, antiderivative size = 338 \[ \int \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3 \, dx=-\frac {13 \left (1-a^2 x^2\right )}{210 a}-\frac {\left (1-a^2 x^2\right )^2}{140 a}-\frac {14}{15} x \text {arctanh}(a x)-\frac {13}{105} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)-\frac {1}{35} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)+\frac {12 \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{35 a}+\frac {9 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{70 a}+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}{14 a}+\frac {16 \text {arctanh}(a x)^3}{35 a}+\frac {16}{35} x \text {arctanh}(a x)^3+\frac {8}{35} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3+\frac {6}{35} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3-\frac {48 \text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{35 a}-\frac {7 \log \left (1-a^2 x^2\right )}{15 a}-\frac {48 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{35 a}+\frac {24 \operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{35 a} \]
-13/210*(-a^2*x^2+1)/a-1/140*(-a^2*x^2+1)^2/a-14/15*x*arctanh(a*x)-13/105* x*(-a^2*x^2+1)*arctanh(a*x)-1/35*x*(-a^2*x^2+1)^2*arctanh(a*x)+12/35*(-a^2 *x^2+1)*arctanh(a*x)^2/a+9/70*(-a^2*x^2+1)^2*arctanh(a*x)^2/a+1/14*(-a^2*x ^2+1)^3*arctanh(a*x)^2/a+16/35*arctanh(a*x)^3/a+16/35*x*arctanh(a*x)^3+8/3 5*x*(-a^2*x^2+1)*arctanh(a*x)^3+6/35*x*(-a^2*x^2+1)^2*arctanh(a*x)^3+1/7*x *(-a^2*x^2+1)^3*arctanh(a*x)^3-48/35*arctanh(a*x)^2*ln(2/(-a*x+1))/a-7/15* ln(-a^2*x^2+1)/a-48/35*arctanh(a*x)*polylog(2,1-2/(-a*x+1))/a+24/35*polylo g(3,1-2/(-a*x+1))/a
Time = 0.84 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.68 \[ \int \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3 \, dx=-\frac {29-32 a^2 x^2+3 a^4 x^4+456 a x \text {arctanh}(a x)-76 a^3 x^3 \text {arctanh}(a x)+12 a^5 x^5 \text {arctanh}(a x)-228 \text {arctanh}(a x)^2+342 a^2 x^2 \text {arctanh}(a x)^2-144 a^4 x^4 \text {arctanh}(a x)^2+30 a^6 x^6 \text {arctanh}(a x)^2+192 \text {arctanh}(a x)^3-420 a x \text {arctanh}(a x)^3+420 a^3 x^3 \text {arctanh}(a x)^3-252 a^5 x^5 \text {arctanh}(a x)^3+60 a^7 x^7 \text {arctanh}(a x)^3+576 \text {arctanh}(a x)^2 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )+196 \log \left (1-a^2 x^2\right )-576 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )-288 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(a x)}\right )}{420 a} \]
-1/420*(29 - 32*a^2*x^2 + 3*a^4*x^4 + 456*a*x*ArcTanh[a*x] - 76*a^3*x^3*Ar cTanh[a*x] + 12*a^5*x^5*ArcTanh[a*x] - 228*ArcTanh[a*x]^2 + 342*a^2*x^2*Ar cTanh[a*x]^2 - 144*a^4*x^4*ArcTanh[a*x]^2 + 30*a^6*x^6*ArcTanh[a*x]^2 + 19 2*ArcTanh[a*x]^3 - 420*a*x*ArcTanh[a*x]^3 + 420*a^3*x^3*ArcTanh[a*x]^3 - 2 52*a^5*x^5*ArcTanh[a*x]^3 + 60*a^7*x^7*ArcTanh[a*x]^3 + 576*ArcTanh[a*x]^2 *Log[1 + E^(-2*ArcTanh[a*x])] + 196*Log[1 - a^2*x^2] - 576*ArcTanh[a*x]*Po lyLog[2, -E^(-2*ArcTanh[a*x])] - 288*PolyLog[3, -E^(-2*ArcTanh[a*x])])/a
Time = 2.60 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.38, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.842, Rules used = {6506, 6504, 6504, 6436, 240, 6506, 6504, 6436, 240, 6506, 6436, 240, 6546, 6470, 6620, 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.
| \(\displaystyle \int \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3 \, dx\) |
| \(\Big \downarrow \) 6506 |
| \(\displaystyle -\frac {1}{7} \int \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)dx+\frac {6}{7} \int \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3dx+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}{14 a}\) |
| \(\Big \downarrow \) 6504 |
| \(\displaystyle \frac {1}{7} \left (-\frac {4}{5} \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)dx-\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)-\frac {\left (1-a^2 x^2\right )^2}{20 a}\right )+\frac {6}{7} \int \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3dx+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}{14 a}\) |
| \(\Big \downarrow \) 6504 |
| \(\displaystyle \frac {1}{7} \left (-\frac {4}{5} \left (\frac {2}{3} \int \text {arctanh}(a x)dx+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {1-a^2 x^2}{6 a}\right )-\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)-\frac {\left (1-a^2 x^2\right )^2}{20 a}\right )+\frac {6}{7} \int \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3dx+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}{14 a}\) |
| \(\Big \downarrow \) 6436 |
| \(\displaystyle \frac {1}{7} \left (-\frac {4}{5} \left (\frac {2}{3} \left (x \text {arctanh}(a x)-a \int \frac {x}{1-a^2 x^2}dx\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {1-a^2 x^2}{6 a}\right )-\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)-\frac {\left (1-a^2 x^2\right )^2}{20 a}\right )+\frac {6}{7} \int \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3dx+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}{14 a}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {6}{7} \int \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3dx+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}{14 a}+\frac {1}{7} \left (-\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)-\frac {4}{5} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}\right )-\frac {\left (1-a^2 x^2\right )^2}{20 a}\right )\) |
| \(\Big \downarrow \) 6506 |
| \(\displaystyle \frac {6}{7} \left (-\frac {3}{10} \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)dx+\frac {4}{5} \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3dx+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3+\frac {3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{20 a}\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}{14 a}+\frac {1}{7} \left (-\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)-\frac {4}{5} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}\right )-\frac {\left (1-a^2 x^2\right )^2}{20 a}\right )\) |
| \(\Big \downarrow \) 6504 |
| \(\displaystyle \frac {6}{7} \left (-\frac {3}{10} \left (\frac {2}{3} \int \text {arctanh}(a x)dx+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {1-a^2 x^2}{6 a}\right )+\frac {4}{5} \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3dx+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3+\frac {3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{20 a}\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}{14 a}+\frac {1}{7} \left (-\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)-\frac {4}{5} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}\right )-\frac {\left (1-a^2 x^2\right )^2}{20 a}\right )\) |
| \(\Big \downarrow \) 6436 |
| \(\displaystyle \frac {6}{7} \left (-\frac {3}{10} \left (\frac {2}{3} \left (x \text {arctanh}(a x)-a \int \frac {x}{1-a^2 x^2}dx\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {1-a^2 x^2}{6 a}\right )+\frac {4}{5} \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3dx+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3+\frac {3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{20 a}\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}{14 a}+\frac {1}{7} \left (-\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)-\frac {4}{5} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}\right )-\frac {\left (1-a^2 x^2\right )^2}{20 a}\right )\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {6}{7} \left (\frac {4}{5} \int \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3dx+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3+\frac {3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{20 a}-\frac {3}{10} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}\right )\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}{14 a}+\frac {1}{7} \left (-\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)-\frac {4}{5} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}\right )-\frac {\left (1-a^2 x^2\right )^2}{20 a}\right )\) |
| \(\Big \downarrow \) 6506 |
| \(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (-\int \text {arctanh}(a x)dx+\frac {2}{3} \int \text {arctanh}(a x)^3dx+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{2 a}\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3+\frac {3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{20 a}-\frac {3}{10} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}\right )\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}{14 a}+\frac {1}{7} \left (-\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)-\frac {4}{5} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}\right )-\frac {\left (1-a^2 x^2\right )^2}{20 a}\right )\) |
| \(\Big \downarrow \) 6436 |
| \(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \left (x \text {arctanh}(a x)^3-3 a \int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2}dx\right )+a \int \frac {x}{1-a^2 x^2}dx+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{2 a}-x \text {arctanh}(a x)\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3+\frac {3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{20 a}-\frac {3}{10} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}\right )\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}{14 a}+\frac {1}{7} \left (-\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)-\frac {4}{5} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}\right )-\frac {\left (1-a^2 x^2\right )^2}{20 a}\right )\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \left (x \text {arctanh}(a x)^3-3 a \int \frac {x \text {arctanh}(a x)^2}{1-a^2 x^2}dx\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{2 a}-\frac {\log \left (1-a^2 x^2\right )}{2 a}-x \text {arctanh}(a x)\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3+\frac {3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{20 a}-\frac {3}{10} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}\right )\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}{14 a}+\frac {1}{7} \left (-\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)-\frac {4}{5} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}\right )-\frac {\left (1-a^2 x^2\right )^2}{20 a}\right )\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \left (x \text {arctanh}(a x)^3-3 a \left (\frac {\int \frac {\text {arctanh}(a x)^2}{1-a x}dx}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{2 a}-\frac {\log \left (1-a^2 x^2\right )}{2 a}-x \text {arctanh}(a x)\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3+\frac {3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{20 a}-\frac {3}{10} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}\right )\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}{14 a}+\frac {1}{7} \left (-\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)-\frac {4}{5} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}\right )-\frac {\left (1-a^2 x^2\right )^2}{20 a}\right )\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \left (x \text {arctanh}(a x)^3-3 a \left (\frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \int \frac {\text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{2 a}-\frac {\log \left (1-a^2 x^2\right )}{2 a}-x \text {arctanh}(a x)\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3+\frac {3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{20 a}-\frac {3}{10} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}\right )\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}{14 a}+\frac {1}{7} \left (-\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)-\frac {4}{5} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}\right )-\frac {\left (1-a^2 x^2\right )^2}{20 a}\right )\) |
| \(\Big \downarrow \) 6620 |
| \(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \left (x \text {arctanh}(a x)^3-3 a \left (\frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \left (\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}\right )}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{2 a}-\frac {\log \left (1-a^2 x^2\right )}{2 a}-x \text {arctanh}(a x)\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3+\frac {3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{20 a}-\frac {3}{10} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}\right )\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}{14 a}+\frac {1}{7} \left (-\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)-\frac {4}{5} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}\right )-\frac {\left (1-a^2 x^2\right )^2}{20 a}\right )\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \left (x \text {arctanh}(a x)^3-3 a \left (\frac {\frac {\text {arctanh}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-2 \left (\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{4 a}-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}\right )}{a}-\frac {\text {arctanh}(a x)^3}{3 a^2}\right )\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{2 a}-\frac {\log \left (1-a^2 x^2\right )}{2 a}-x \text {arctanh}(a x)\right )+\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3+\frac {3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}{20 a}-\frac {3}{10} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}\right )\right )+\frac {1}{7} x \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3+\frac {\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}{14 a}+\frac {1}{7} \left (-\frac {1}{5} x \left (1-a^2 x^2\right )^2 \text {arctanh}(a x)-\frac {4}{5} \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \text {arctanh}(a x)+\frac {2}{3} \left (\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \text {arctanh}(a x)\right )+\frac {1-a^2 x^2}{6 a}\right )-\frac {\left (1-a^2 x^2\right )^2}{20 a}\right )\) |
((1 - a^2*x^2)^3*ArcTanh[a*x]^2)/(14*a) + (x*(1 - a^2*x^2)^3*ArcTanh[a*x]^ 3)/7 + (-1/20*(1 - a^2*x^2)^2/a - (x*(1 - a^2*x^2)^2*ArcTanh[a*x])/5 - (4* ((1 - a^2*x^2)/(6*a) + (x*(1 - a^2*x^2)*ArcTanh[a*x])/3 + (2*(x*ArcTanh[a* x] + Log[1 - a^2*x^2]/(2*a)))/3))/5)/7 + (6*((3*(1 - a^2*x^2)^2*ArcTanh[a* x]^2)/(20*a) + (x*(1 - a^2*x^2)^2*ArcTanh[a*x]^3)/5 - (3*((1 - a^2*x^2)/(6 *a) + (x*(1 - a^2*x^2)*ArcTanh[a*x])/3 + (2*(x*ArcTanh[a*x] + Log[1 - a^2* x^2]/(2*a)))/3))/10 + (4*(-(x*ArcTanh[a*x]) + ((1 - a^2*x^2)*ArcTanh[a*x]^ 2)/(2*a) + (x*(1 - a^2*x^2)*ArcTanh[a*x]^3)/3 - Log[1 - a^2*x^2]/(2*a) + ( 2*(x*ArcTanh[a*x]^3 - 3*a*(-1/3*ArcTanh[a*x]^3/a^2 + ((ArcTanh[a*x]^2*Log[ 2/(1 - a*x)])/a - 2*(-1/2*(ArcTanh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)])/a + P olyLog[3, 1 - 2/(1 - a*x)]/(4*a)))/a)))/3))/5))/7
3.3.26.3.1 Defintions of rubi rules used
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTanh[c*x^n])^p, x] - Simp[b*c*n*p Int[x^n*((a + b*ArcTanh[c*x^n]) ^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 1])
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol ] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c *(p/e) Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[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_.))*((d_) + (e_.)*(x_)^2)^(q_.), x_Symb ol] :> Simp[b*((d + e*x^2)^q/(2*c*q*(2*q + 1))), x] + (Simp[x*(d + e*x^2)^q *((a + b*ArcTanh[c*x])/(2*q + 1)), x] + Simp[2*d*(q/(2*q + 1)) Int[(d + e *x^2)^(q - 1)*(a + b*ArcTanh[c*x]), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_.), x _Symbol] :> Simp[b*p*(d + e*x^2)^q*((a + b*ArcTanh[c*x])^(p - 1)/(2*c*q*(2* q + 1))), x] + (Simp[x*(d + e*x^2)^q*((a + b*ArcTanh[c*x])^p/(2*q + 1)), x] + Simp[2*d*(q/(2*q + 1)) Int[(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x], x] - Simp[b^2*d*p*((p - 1)/(2*q*(2*q + 1))) Int[(d + e*x^2)^(q - 1)* (a + b*ArcTanh[c*x])^(p - 2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c ^2*d + e, 0] && GtQ[q, 0] && GtQ[p, 1]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*e*(p + 1)), x] + Simp[1/ (c*d) Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.40 (sec) , antiderivative size = 978, normalized size of antiderivative = 2.89
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(978\) |
| default | \(\text {Expression too large to display}\) | \(978\) |
| parts | \(\text {Expression too large to display}\) | \(983\) |
1/a*(-24/35*I*Pi*arctanh(a*x)^2-1/14*arctanh(a*x)^2*a^6*x^6-arctanh(a*x)^3 *a^3*x^3+arctanh(a*x)^3*a*x+12/35*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)^2-12/35*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+24/35*I*Pi*csgn(I/(1-(a*x+1)^2/(a^2*x^ 2-1)))^2*arctanh(a*x)^2-24/35*I*Pi*csgn(I/(1-(a*x+1)^2/(a^2*x^2-1)))^3*arc tanh(a*x)^2-12/35*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*arctanh(a*x)^2-11/1 05*(a^2*x^2-4*a*x+7)*(a*x+1)*arctanh(a*x)+9/35*(a*x-3)*(a*x+1)*arctanh(a*x )-13/105+16/35*arctanh(a*x)^3+19/35*arctanh(a*x)^2+13/105*a*x+12/35*a^4*x^ 4*arctanh(a*x)^2-57/70*a^2*x^2*arctanh(a*x)^2-48/35*arctanh(a*x)^2*ln(2)-1 2/35*I*Pi*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1 ))*arctanh(a*x)^2-12/35*I*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1-(a*x+1)^2/(a^ 2*x^2-1)))^2*csgn(I/(1-(a*x+1)^2/(a^2*x^2-1)))*arctanh(a*x)^2-24/35*I*Pi*c sgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2*arctanh( a*x)^2+12/35*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)^2-1/140*(a*x-1)^4-1/35*(a*x-1 )^3-1/35*(a^4*x^4-6*a^3*x^3+16*a^2*x^2-26*a*x+31)*(a*x+1)*arctanh(a*x)-1/7 *(a^3*x^3-5*a^2*x^2+11*a*x-15)*(a*x+1)*arctanh(a*x)+1/30*(a*x-1)^2-48/35*a rctanh(a*x)*polylog(2,-(a*x+1)^2/(-a^2*x^2+1))+24/35*arctanh(a*x)^2*ln(a*x -1)+24/35*arctanh(a*x)^2*ln(a*x+1)-48/35*arctanh(a*x)^2*ln((a*x+1)/(-a^...
\[ \int \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3 \, dx=\int { -{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{3} \,d x } \]
\[ \int \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3 \, dx=- \int 3 a^{2} x^{2} \operatorname {atanh}^{3}{\left (a x \right )}\, dx - \int \left (- 3 a^{4} x^{4} \operatorname {atanh}^{3}{\left (a x \right )}\right )\, dx - \int a^{6} x^{6} \operatorname {atanh}^{3}{\left (a x \right )}\, dx - \int \left (- \operatorname {atanh}^{3}{\left (a x \right )}\right )\, dx \]
-Integral(3*a**2*x**2*atanh(a*x)**3, x) - Integral(-3*a**4*x**4*atanh(a*x) **3, x) - Integral(a**6*x**6*atanh(a*x)**3, x) - Integral(-atanh(a*x)**3, x)
\[ \int \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3 \, dx=\int { -{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{3} \,d x } \]
1/19600*(150*a^7*x^7 - 175*a^6*x^6 - 672*a^5*x^5 + 840*a^4*x^4 + 1330*a^3* x^3 - 1995*a^2*x^2 - 3360*a*x - 210*(5*a^7*x^7 - 21*a^5*x^5 + 35*a^3*x^3 - 35*a*x - 16)*log(a*x + 1))*log(-a*x + 1)^2/a - 1/8*(log(-a*x + 1)^3 - 3*l og(-a*x + 1)^2 + 6*log(-a*x + 1) - 6)*(a*x - 1)/a + 1/691488000*(36000*(34 3*log(-a*x + 1)^3 - 147*log(-a*x + 1)^2 + 42*log(-a*x + 1) - 6)*(a*x - 1)^ 7 + 2401000*(36*log(-a*x + 1)^3 - 18*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 1 )*(a*x - 1)^6 + 2074464*(125*log(-a*x + 1)^3 - 75*log(-a*x + 1)^2 + 30*log (-a*x + 1) - 6)*(a*x - 1)^5 + 13505625*(32*log(-a*x + 1)^3 - 24*log(-a*x + 1)^2 + 12*log(-a*x + 1) - 3)*(a*x - 1)^4 + 48020000*(9*log(-a*x + 1)^3 - 9*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 2)*(a*x - 1)^3 + 64827000*(4*log(-a* x + 1)^3 - 6*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 3)*(a*x - 1)^2 + 86436000 *(log(-a*x + 1)^3 - 3*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 6)*(a*x - 1))/a - 1/480000*(288*(125*log(-a*x + 1)^3 - 75*log(-a*x + 1)^2 + 30*log(-a*x + 1) - 6)*(a*x - 1)^5 + 5625*(32*log(-a*x + 1)^3 - 24*log(-a*x + 1)^2 + 12*l og(-a*x + 1) - 3)*(a*x - 1)^4 + 40000*(9*log(-a*x + 1)^3 - 9*log(-a*x + 1) ^2 + 6*log(-a*x + 1) - 2)*(a*x - 1)^3 + 90000*(4*log(-a*x + 1)^3 - 6*log(- a*x + 1)^2 + 6*log(-a*x + 1) - 3)*(a*x - 1)^2 + 180000*(log(-a*x + 1)^3 - 3*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 6)*(a*x - 1))/a + 1/288*(4*(9*log(-a *x + 1)^3 - 9*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 2)*(a*x - 1)^3 + 27*(4*l og(-a*x + 1)^3 - 6*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 3)*(a*x - 1)^2 +...
\[ \int \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3 \, dx=\int { -{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{3} \,d x } \]
Timed out. \[ \int \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^3 \, dx=-\int {\mathrm {atanh}\left (a\,x\right )}^3\,{\left (a^2\,x^2-1\right )}^3 \,d x \]