Integrand size = 19, antiderivative size = 291 \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^4} \, dx=-\frac {1}{216 a \left (1-a^2 x^2\right )^3}-\frac {65}{2304 a \left (1-a^2 x^2\right )^2}-\frac {245}{768 a \left (1-a^2 x^2\right )}+\frac {x \text {arctanh}(a x)}{36 \left (1-a^2 x^2\right )^3}+\frac {65 x \text {arctanh}(a x)}{576 \left (1-a^2 x^2\right )^2}+\frac {245 x \text {arctanh}(a x)}{384 \left (1-a^2 x^2\right )}+\frac {245 \text {arctanh}(a x)^2}{768 a}-\frac {\text {arctanh}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}-\frac {5 \text {arctanh}(a x)^2}{32 a \left (1-a^2 x^2\right )^2}-\frac {15 \text {arctanh}(a x)^2}{32 a \left (1-a^2 x^2\right )}+\frac {x \text {arctanh}(a x)^3}{6 \left (1-a^2 x^2\right )^3}+\frac {5 x \text {arctanh}(a x)^3}{24 \left (1-a^2 x^2\right )^2}+\frac {5 x \text {arctanh}(a x)^3}{16 \left (1-a^2 x^2\right )}+\frac {5 \text {arctanh}(a x)^4}{64 a} \]
-1/216/a/(-a^2*x^2+1)^3-65/2304/a/(-a^2*x^2+1)^2-245/768/a/(-a^2*x^2+1)+1/ 36*x*arctanh(a*x)/(-a^2*x^2+1)^3+65/576*x*arctanh(a*x)/(-a^2*x^2+1)^2+245/ 384*x*arctanh(a*x)/(-a^2*x^2+1)+245/768*arctanh(a*x)^2/a-1/12*arctanh(a*x) ^2/a/(-a^2*x^2+1)^3-5/32*arctanh(a*x)^2/a/(-a^2*x^2+1)^2-15/32*arctanh(a*x )^2/a/(-a^2*x^2+1)+1/6*x*arctanh(a*x)^3/(-a^2*x^2+1)^3+5/24*x*arctanh(a*x) ^3/(-a^2*x^2+1)^2+5/16*x*arctanh(a*x)^3/(-a^2*x^2+1)+5/64*arctanh(a*x)^4/a
Time = 0.09 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.49 \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^4} \, dx=\frac {2432-4605 a^2 x^2+2205 a^4 x^4-6 a x \left (897-1600 a^2 x^2+735 a^4 x^4\right ) \text {arctanh}(a x)+9 \left (299-105 a^2 x^2-375 a^4 x^4+245 a^6 x^6\right ) \text {arctanh}(a x)^2-144 a x \left (33-40 a^2 x^2+15 a^4 x^4\right ) \text {arctanh}(a x)^3+540 \left (-1+a^2 x^2\right )^3 \text {arctanh}(a x)^4}{6912 a \left (-1+a^2 x^2\right )^3} \]
(2432 - 4605*a^2*x^2 + 2205*a^4*x^4 - 6*a*x*(897 - 1600*a^2*x^2 + 735*a^4* x^4)*ArcTanh[a*x] + 9*(299 - 105*a^2*x^2 - 375*a^4*x^4 + 245*a^6*x^6)*ArcT anh[a*x]^2 - 144*a*x*(33 - 40*a^2*x^2 + 15*a^4*x^4)*ArcTanh[a*x]^3 + 540*( -1 + a^2*x^2)^3*ArcTanh[a*x]^4)/(6912*a*(-1 + a^2*x^2)^3)
Time = 2.25 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.66, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {6526, 6522, 6522, 6518, 241, 6526, 6518, 6522, 6518, 241, 6556, 6518, 241}
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.
| \(\displaystyle \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^4} \, dx\) |
| \(\Big \downarrow \) 6526 |
| \(\displaystyle \frac {1}{6} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^4}dx+\frac {5}{6} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3}dx+\frac {x \text {arctanh}(a x)^3}{6 \left (1-a^2 x^2\right )^3}-\frac {\text {arctanh}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}\) |
| \(\Big \downarrow \) 6522 |
| \(\displaystyle \frac {1}{6} \left (\frac {5}{6} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3}dx+\frac {x \text {arctanh}(a x)}{6 \left (1-a^2 x^2\right )^3}-\frac {1}{36 a \left (1-a^2 x^2\right )^3}\right )+\frac {5}{6} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3}dx+\frac {x \text {arctanh}(a x)^3}{6 \left (1-a^2 x^2\right )^3}-\frac {\text {arctanh}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}\) |
| \(\Big \downarrow \) 6522 |
| \(\displaystyle \frac {1}{6} \left (\frac {5}{6} \left (\frac {3}{4} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {x \text {arctanh}(a x)}{6 \left (1-a^2 x^2\right )^3}-\frac {1}{36 a \left (1-a^2 x^2\right )^3}\right )+\frac {5}{6} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3}dx+\frac {x \text {arctanh}(a x)^3}{6 \left (1-a^2 x^2\right )^3}-\frac {\text {arctanh}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}\) |
| \(\Big \downarrow \) 6518 |
| \(\displaystyle \frac {1}{6} \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {1}{2} a \int \frac {x}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )+\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {x \text {arctanh}(a x)}{6 \left (1-a^2 x^2\right )^3}-\frac {1}{36 a \left (1-a^2 x^2\right )^3}\right )+\frac {5}{6} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3}dx+\frac {x \text {arctanh}(a x)^3}{6 \left (1-a^2 x^2\right )^3}-\frac {\text {arctanh}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {5}{6} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^3}dx+\frac {x \text {arctanh}(a x)^3}{6 \left (1-a^2 x^2\right )^3}-\frac {\text {arctanh}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}+\frac {1}{6} \left (\frac {x \text {arctanh}(a x)}{6 \left (1-a^2 x^2\right )^3}+\frac {5}{6} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )-\frac {1}{36 a \left (1-a^2 x^2\right )^3}\right )\) |
| \(\Big \downarrow \) 6526 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{8} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3}dx+\frac {3}{4} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {x \text {arctanh}(a x)^3}{6 \left (1-a^2 x^2\right )^3}-\frac {\text {arctanh}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}+\frac {1}{6} \left (\frac {x \text {arctanh}(a x)}{6 \left (1-a^2 x^2\right )^3}+\frac {5}{6} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )-\frac {1}{36 a \left (1-a^2 x^2\right )^3}\right )\) |
| \(\Big \downarrow \) 6518 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{8} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3}dx+\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {x \text {arctanh}(a x)^3}{6 \left (1-a^2 x^2\right )^3}-\frac {\text {arctanh}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}+\frac {1}{6} \left (\frac {x \text {arctanh}(a x)}{6 \left (1-a^2 x^2\right )^3}+\frac {5}{6} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )-\frac {1}{36 a \left (1-a^2 x^2\right )^3}\right )\) |
| \(\Big \downarrow \) 6522 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{8} \left (\frac {3}{4} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {x \text {arctanh}(a x)^3}{6 \left (1-a^2 x^2\right )^3}-\frac {\text {arctanh}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}+\frac {1}{6} \left (\frac {x \text {arctanh}(a x)}{6 \left (1-a^2 x^2\right )^3}+\frac {5}{6} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )-\frac {1}{36 a \left (1-a^2 x^2\right )^3}\right )\) |
| \(\Big \downarrow \) 6518 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{8} \left (\frac {3}{4} \left (-\frac {1}{2} a \int \frac {x}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )+\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {x \text {arctanh}(a x)^3}{6 \left (1-a^2 x^2\right )^3}-\frac {\text {arctanh}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}+\frac {1}{6} \left (\frac {x \text {arctanh}(a x)}{6 \left (1-a^2 x^2\right )^3}+\frac {5}{6} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )-\frac {1}{36 a \left (1-a^2 x^2\right )^3}\right )\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{4} \left (-\frac {3}{2} a \int \frac {x \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {3}{8} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )\right )+\frac {x \text {arctanh}(a x)^3}{6 \left (1-a^2 x^2\right )^3}-\frac {\text {arctanh}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}+\frac {1}{6} \left (\frac {x \text {arctanh}(a x)}{6 \left (1-a^2 x^2\right )^3}+\frac {5}{6} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )-\frac {1}{36 a \left (1-a^2 x^2\right )^3}\right )\) |
| \(\Big \downarrow \) 6556 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{4} \left (-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^2}dx}{a}\right )+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {3}{8} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )\right )+\frac {x \text {arctanh}(a x)^3}{6 \left (1-a^2 x^2\right )^3}-\frac {\text {arctanh}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}+\frac {1}{6} \left (\frac {x \text {arctanh}(a x)}{6 \left (1-a^2 x^2\right )^3}+\frac {5}{6} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )-\frac {1}{36 a \left (1-a^2 x^2\right )^3}\right )\) |
| \(\Big \downarrow \) 6518 |
| \(\displaystyle \frac {5}{6} \left (\frac {3}{4} \left (-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {-\frac {1}{2} a \int \frac {x}{\left (1-a^2 x^2\right )^2}dx+\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^4}{8 a}\right )+\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {3}{8} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )\right )+\frac {x \text {arctanh}(a x)^3}{6 \left (1-a^2 x^2\right )^3}-\frac {\text {arctanh}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}+\frac {1}{6} \left (\frac {x \text {arctanh}(a x)}{6 \left (1-a^2 x^2\right )^3}+\frac {5}{6} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )-\frac {1}{36 a \left (1-a^2 x^2\right )^3}\right )\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {x \text {arctanh}(a x)^3}{6 \left (1-a^2 x^2\right )^3}-\frac {\text {arctanh}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}+\frac {1}{6} \left (\frac {x \text {arctanh}(a x)}{6 \left (1-a^2 x^2\right )^3}+\frac {5}{6} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )-\frac {1}{36 a \left (1-a^2 x^2\right )^3}\right )+\frac {5}{6} \left (\frac {x \text {arctanh}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {3 \text {arctanh}(a x)^2}{16 a \left (1-a^2 x^2\right )^2}+\frac {3}{8} \left (\frac {x \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}\right )-\frac {1}{16 a \left (1-a^2 x^2\right )^2}\right )+\frac {3}{4} \left (\frac {x \text {arctanh}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3}{2} a \left (\frac {\text {arctanh}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\frac {x \text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4 a \left (1-a^2 x^2\right )}+\frac {\text {arctanh}(a x)^2}{4 a}}{a}\right )+\frac {\text {arctanh}(a x)^4}{8 a}\right )\right )\) |
-1/12*ArcTanh[a*x]^2/(a*(1 - a^2*x^2)^3) + (x*ArcTanh[a*x]^3)/(6*(1 - a^2* x^2)^3) + (-1/36*1/(a*(1 - a^2*x^2)^3) + (x*ArcTanh[a*x])/(6*(1 - a^2*x^2) ^3) + (5*(-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)) + A rcTanh[a*x]^2/(4*a)))/4))/6)/6 + (5*((-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*(Arc Tanh[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))/6
3.4.47.3.1 Defintions of rubi rules used
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_)]*(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_.)*(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]
Time = 0.70 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.73
| method | result | size |
| parallelrisch | \(-\frac {-2205 \operatorname {arctanh}\left (a x \right )^{2} a^{6} x^{6}-5760 \operatorname {arctanh}\left (a x \right )^{3} a^{3} x^{3}+4752 \operatorname {arctanh}\left (a x \right )^{3} a x +4410 \,\operatorname {arctanh}\left (a x \right ) a^{5} x^{5}-2432 a^{6} x^{6}-2691 a^{2} x^{2}+540 \operatorname {arctanh}\left (a x \right )^{4}-2691 \operatorname {arctanh}\left (a x \right )^{2}+5382 a x \,\operatorname {arctanh}\left (a x \right )-9600 a^{3} x^{3} \operatorname {arctanh}\left (a x \right )+3375 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )^{2}+945 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}+1620 \operatorname {arctanh}\left (a x \right )^{4} a^{4} x^{4}-1620 \operatorname {arctanh}\left (a x \right )^{4} a^{2} x^{2}-540 a^{6} \operatorname {arctanh}\left (a x \right )^{4} x^{6}+5091 a^{4} x^{4}+2160 \operatorname {arctanh}\left (a x \right )^{3} a^{5} x^{5}}{6912 \left (a^{2} x^{2}-1\right )^{3} a}\) | \(212\) |
| risch | \(\frac {5 \ln \left (a x +1\right )^{4}}{1024 a}-\frac {\left (15 a^{6} x^{6} \ln \left (-a x +1\right )+30 a^{5} x^{5}-45 a^{4} x^{4} \ln \left (-a x +1\right )-80 a^{3} x^{3}+45 x^{2} \ln \left (-a x +1\right ) a^{2}+66 a x -15 \ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )^{3}}{768 \left (a^{2} x^{2}-1\right )^{3} a}+\frac {\left (90 a^{6} x^{6} \ln \left (-a x +1\right )^{2}+245 a^{6} x^{6}+360 a^{5} x^{5} \ln \left (-a x +1\right )-270 a^{4} x^{4} \ln \left (-a x +1\right )^{2}-375 a^{4} x^{4}-960 a^{3} x^{3} \ln \left (-a x +1\right )+270 a^{2} x^{2} \ln \left (-a x +1\right )^{2}-105 a^{2} x^{2}+792 a x \ln \left (-a x +1\right )-90 \ln \left (-a x +1\right )^{2}+299\right ) \ln \left (a x +1\right )^{2}}{3072 \left (a^{2} x^{2}-1\right )^{2} \left (a x -1\right ) a \left (a x +1\right )}-\frac {\left (90 a^{6} x^{6} \ln \left (-a x +1\right )^{3}+735 a^{6} x^{6} \ln \left (-a x +1\right )+540 a^{5} x^{5} \ln \left (-a x +1\right )^{2}-270 a^{4} x^{4} \ln \left (-a x +1\right )^{3}+1470 a^{5} x^{5}-1125 a^{4} x^{4} \ln \left (-a x +1\right )-1440 a^{3} x^{3} \ln \left (-a x +1\right )^{2}+270 a^{2} x^{2} \ln \left (-a x +1\right )^{3}-3200 a^{3} x^{3}-315 x^{2} \ln \left (-a x +1\right ) a^{2}+1188 a \ln \left (-a x +1\right )^{2} x -90 \ln \left (-a x +1\right )^{3}+1794 a x +897 \ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )}{4608 a \left (a x -1\right )^{2} \left (a x +1\right )^{2} \left (a^{2} x^{2}-1\right )}+\frac {135 a^{6} x^{6} \ln \left (-a x +1\right )^{4}+2205 a^{6} x^{6} \ln \left (-a x +1\right )^{2}+1080 a^{5} x^{5} \ln \left (-a x +1\right )^{3}-405 a^{4} x^{4} \ln \left (-a x +1\right )^{4}+8820 a^{5} x^{5} \ln \left (-a x +1\right )-3375 a^{4} x^{4} \ln \left (-a x +1\right )^{2}-2880 a^{3} x^{3} \ln \left (-a x +1\right )^{3}+8820 a^{4} x^{4}+405 a^{2} x^{2} \ln \left (-a x +1\right )^{4}-19200 a^{3} x^{3} \ln \left (-a x +1\right )-945 a^{2} x^{2} \ln \left (-a x +1\right )^{2}+2376 a x \ln \left (-a x +1\right )^{3}-18420 a^{2} x^{2}-135 \ln \left (-a x +1\right )^{4}+10764 a x \ln \left (-a x +1\right )+2691 \ln \left (-a x +1\right )^{2}+9728}{27648 a \left (a x -1\right )^{2} \left (a x +1\right )^{2} \left (a^{2} x^{2}-1\right )}\) | \(761\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1059\) |
| default | \(\text {Expression too large to display}\) | \(1059\) |
| parts | \(\text {Expression too large to display}\) | \(1173\) |
-1/6912*(-2205*arctanh(a*x)^2*a^6*x^6-5760*arctanh(a*x)^3*a^3*x^3+4752*arc tanh(a*x)^3*a*x+4410*arctanh(a*x)*a^5*x^5-2432*a^6*x^6-2691*a^2*x^2+540*ar ctanh(a*x)^4-2691*arctanh(a*x)^2+5382*a*x*arctanh(a*x)-9600*a^3*x^3*arctan h(a*x)+3375*a^4*x^4*arctanh(a*x)^2+945*a^2*x^2*arctanh(a*x)^2+1620*arctanh (a*x)^4*a^4*x^4-1620*arctanh(a*x)^4*a^2*x^2-540*a^6*arctanh(a*x)^4*x^6+509 1*a^4*x^4+2160*arctanh(a*x)^3*a^5*x^5)/(a^2*x^2-1)^3/a
Time = 0.26 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.74 \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^4} \, dx=\frac {8820 \, a^{4} x^{4} + 135 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{4} - 18420 \, a^{2} x^{2} - 72 \, {\left (15 \, a^{5} x^{5} - 40 \, a^{3} x^{3} + 33 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 9 \, {\left (245 \, a^{6} x^{6} - 375 \, a^{4} x^{4} - 105 \, a^{2} x^{2} + 299\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 12 \, {\left (735 \, a^{5} x^{5} - 1600 \, a^{3} x^{3} + 897 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) + 9728}{27648 \, {\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )}} \]
1/27648*(8820*a^4*x^4 + 135*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(-(a* x + 1)/(a*x - 1))^4 - 18420*a^2*x^2 - 72*(15*a^5*x^5 - 40*a^3*x^3 + 33*a*x )*log(-(a*x + 1)/(a*x - 1))^3 + 9*(245*a^6*x^6 - 375*a^4*x^4 - 105*a^2*x^2 + 299)*log(-(a*x + 1)/(a*x - 1))^2 - 12*(735*a^5*x^5 - 1600*a^3*x^3 + 897 *a*x)*log(-(a*x + 1)/(a*x - 1)) + 9728)/(a^7*x^6 - 3*a^5*x^4 + 3*a^3*x^2 - a)
\[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^4} \, dx=\int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{\left (a x - 1\right )^{4} \left (a x + 1\right )^{4}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 871 vs. \(2 (251) = 502\).
Time = 0.21 (sec) , antiderivative size = 871, normalized size of antiderivative = 2.99 \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^4} \, dx=-\frac {1}{96} \, {\left (\frac {2 \, {\left (15 \, a^{4} x^{5} - 40 \, a^{2} x^{3} + 33 \, x\right )}}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1} - \frac {15 \, \log \left (a x + 1\right )}{a} + \frac {15 \, \log \left (a x - 1\right )}{a}\right )} \operatorname {artanh}\left (a x\right )^{3} + \frac {{\left (180 \, a^{4} x^{4} - 420 \, a^{2} x^{2} - 45 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} + 90 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 45 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} + 272\right )} a \operatorname {artanh}\left (a x\right )^{2}}{384 \, {\left (a^{8} x^{6} - 3 \, a^{6} x^{4} + 3 \, a^{4} x^{2} - a^{2}\right )}} + \frac {1}{27648} \, {\left (\frac {{\left (8820 \, a^{4} x^{4} - 135 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{4} + 540 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{3} \log \left (a x - 1\right ) - 135 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{4} - 18420 \, a^{2} x^{2} - 45 \, {\left (49 \, a^{6} x^{6} - 147 \, a^{4} x^{4} + 147 \, a^{2} x^{2} + 18 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 49\right )} \log \left (a x + 1\right )^{2} - 2205 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} + 90 \, {\left (6 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{3} + 49 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + 1\right ) + 9728\right )} a^{2}}{a^{10} x^{6} - 3 \, a^{8} x^{4} + 3 \, a^{6} x^{2} - a^{4}} - \frac {12 \, {\left (1470 \, a^{5} x^{5} - 3200 \, a^{3} x^{3} - 90 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{3} + 270 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} \log \left (a x - 1\right ) + 90 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{3} + 1794 \, a x - 15 \, {\left (49 \, a^{6} x^{6} - 147 \, a^{4} x^{4} + 147 \, a^{2} x^{2} + 18 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 49\right )} \log \left (a x + 1\right ) + 735 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )\right )} a \operatorname {artanh}\left (a x\right )}{a^{9} x^{6} - 3 \, a^{7} x^{4} + 3 \, a^{5} x^{2} - a^{3}}\right )} a \]
-1/96*(2*(15*a^4*x^5 - 40*a^2*x^3 + 33*x)/(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1) - 15*log(a*x + 1)/a + 15*log(a*x - 1)/a)*arctanh(a*x)^3 + 1/384*(180 *a^4*x^4 - 420*a^2*x^2 - 45*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x + 1)^2 + 90*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x + 1)*log(a*x - 1 ) - 45*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x - 1)^2 + 272)*a*arcta nh(a*x)^2/(a^8*x^6 - 3*a^6*x^4 + 3*a^4*x^2 - a^2) + 1/27648*((8820*a^4*x^4 - 135*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x + 1)^4 + 540*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x + 1)^3*log(a*x - 1) - 135*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x - 1)^4 - 18420*a^2*x^2 - 45*(49*a^6*x^ 6 - 147*a^4*x^4 + 147*a^2*x^2 + 18*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*l og(a*x - 1)^2 - 49)*log(a*x + 1)^2 - 2205*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x - 1)^2 + 90*(6*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x - 1)^3 + 49*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x - 1))*log(a*x + 1) + 9728)*a^2/(a^10*x^6 - 3*a^8*x^4 + 3*a^6*x^2 - a^4) - 12*(1470*a^5*x^ 5 - 3200*a^3*x^3 - 90*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x + 1)^3 + 270*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x + 1)^2*log(a*x - 1) + 90*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x - 1)^3 + 1794*a*x - 15*( 49*a^6*x^6 - 147*a^4*x^4 + 147*a^2*x^2 + 18*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x ^2 - 1)*log(a*x - 1)^2 - 49)*log(a*x + 1) + 735*(a^6*x^6 - 3*a^4*x^4 + 3*a ^2*x^2 - 1)*log(a*x - 1))*a*arctanh(a*x)/(a^9*x^6 - 3*a^7*x^4 + 3*a^5*x...
\[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^4} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )}^{4}} \,d x } \]
Time = 6.15 (sec) , antiderivative size = 1041, normalized size of antiderivative = 3.58 \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^4} \, dx=\frac {\frac {1216}{3\,a}-\frac {1535\,a\,x^2}{2}+\frac {735\,a^3\,x^4}{2}}{1152\,a^6\,x^6-3456\,a^4\,x^4+3456\,a^2\,x^2-1152}-{\ln \left (1-a\,x\right )}^3\,\left (\frac {5\,\ln \left (a\,x+1\right )}{256\,a}-\frac {\frac {5\,a^4\,x^5}{16}-\frac {5\,a^2\,x^3}{6}+\frac {11\,x}{16}}{8\,a^6\,x^6-24\,a^4\,x^4+24\,a^2\,x^2-8}\right )+\frac {5\,{\ln \left (a\,x+1\right )}^4}{1024\,a}+\frac {5\,{\ln \left (1-a\,x\right )}^4}{1024\,a}+{\ln \left (1-a\,x\right )}^2\,\left (\frac {15\,{\ln \left (a\,x+1\right )}^2}{512\,a}+\frac {245}{3072\,a}+\frac {\frac {37\,x}{2}-35\,a\,x^2+\frac {68}{3\,a}-\frac {82\,a^2\,x^3}{3}+15\,a^3\,x^4+\frac {23\,a^4\,x^5}{2}}{256\,a^6\,x^6-768\,a^4\,x^4+768\,a^2\,x^2-256}-\frac {\frac {37\,x}{2}+35\,a\,x^2-\frac {68}{3\,a}-\frac {82\,a^2\,x^3}{3}-15\,a^3\,x^4+\frac {23\,a^4\,x^5}{2}}{256\,a^6\,x^6-768\,a^4\,x^4+768\,a^2\,x^2-256}-\frac {\ln \left (a\,x+1\right )\,\left (30\,a^4\,x^5-80\,a^2\,x^3+66\,x\right )}{256\,a^6\,x^6-768\,a^4\,x^4+768\,a^2\,x^2-256}\right )+{\ln \left (a\,x+1\right )}^2\,\left (\frac {\frac {17}{96\,a^2}-\frac {35\,x^2}{128}+\frac {15\,a^2\,x^4}{128}}{3\,a\,x^2-\frac {1}{a}-3\,a^3\,x^4+a^5\,x^6}+\frac {245}{3072\,a}\right )+\ln \left (1-a\,x\right )\,\left (\frac {36\,x+22\,a\,x^2-\frac {23}{2\,a}-67\,a^2\,x^3-\frac {21\,a^3\,x^4}{2}+31\,a^4\,x^5}{768\,a^6\,x^6-2304\,a^4\,x^4+2304\,a^2\,x^2-768}-\frac {5\,{\ln \left (a\,x+1\right )}^3}{256\,a}-\ln \left (a\,x+1\right )\,\left (\frac {\frac {37\,x}{2}-35\,a\,x^2+\frac {68}{3\,a}-\frac {82\,a^2\,x^3}{3}+15\,a^3\,x^4+\frac {23\,a^4\,x^5}{2}}{128\,a^6\,x^6-384\,a^4\,x^4+384\,a^2\,x^2-128}-\frac {\frac {37\,x}{2}+35\,a\,x^2-\frac {68}{3\,a}-\frac {82\,a^2\,x^3}{3}-15\,a^3\,x^4+\frac {23\,a^4\,x^5}{2}}{128\,a^6\,x^6-384\,a^4\,x^4+384\,a^2\,x^2-128}+\frac {245\,\left (a^6\,x^6-3\,a^4\,x^4+3\,a^2\,x^2-1\right )}{12\,a\,\left (128\,a^6\,x^6-384\,a^4\,x^4+384\,a^2\,x^2-128\right )}\right )+\frac {\frac {227\,x}{2}+173\,a\,x^2-\frac {593}{6\,a}-\frac {599\,a^2\,x^3}{3}-\frac {159\,a^3\,x^4}{2}+\frac {183\,a^4\,x^5}{2}}{768\,a^6\,x^6-2304\,a^4\,x^4+2304\,a^2\,x^2-768}+\frac {\frac {299\,x}{2}-195\,a\,x^2+\frac {331}{3\,a}-\frac {800\,a^2\,x^3}{3}+90\,a^3\,x^4+\frac {245\,a^4\,x^5}{2}}{768\,a^6\,x^6-2304\,a^4\,x^4+2304\,a^2\,x^2-768}+\frac {{\ln \left (a\,x+1\right )}^2\,\left (30\,a^4\,x^5-80\,a^2\,x^3+66\,x\right )}{256\,a^6\,x^6-768\,a^4\,x^4+768\,a^2\,x^2-256}\right )-\frac {\ln \left (a\,x+1\right )\,\left (\frac {299\,x}{768\,a}-\frac {25\,a\,x^3}{36}+\frac {245\,a^3\,x^5}{768}\right )}{3\,a\,x^2-\frac {1}{a}-3\,a^3\,x^4+a^5\,x^6}-\frac {{\ln \left (a\,x+1\right )}^3\,\left (\frac {11\,x}{128\,a}-\frac {5\,a\,x^3}{48}+\frac {5\,a^3\,x^5}{128}\right )}{3\,a\,x^2-\frac {1}{a}-3\,a^3\,x^4+a^5\,x^6} \]
(1216/(3*a) - (1535*a*x^2)/2 + (735*a^3*x^4)/2)/(3456*a^2*x^2 - 3456*a^4*x ^4 + 1152*a^6*x^6 - 1152) - log(1 - a*x)^3*((5*log(a*x + 1))/(256*a) - ((1 1*x)/16 - (5*a^2*x^3)/6 + (5*a^4*x^5)/16)/(24*a^2*x^2 - 24*a^4*x^4 + 8*a^6 *x^6 - 8)) + (5*log(a*x + 1)^4)/(1024*a) + (5*log(1 - a*x)^4)/(1024*a) + l og(1 - a*x)^2*((15*log(a*x + 1)^2)/(512*a) + 245/(3072*a) + ((37*x)/2 - 35 *a*x^2 + 68/(3*a) - (82*a^2*x^3)/3 + 15*a^3*x^4 + (23*a^4*x^5)/2)/(768*a^2 *x^2 - 768*a^4*x^4 + 256*a^6*x^6 - 256) - ((37*x)/2 + 35*a*x^2 - 68/(3*a) - (82*a^2*x^3)/3 - 15*a^3*x^4 + (23*a^4*x^5)/2)/(768*a^2*x^2 - 768*a^4*x^4 + 256*a^6*x^6 - 256) - (log(a*x + 1)*(66*x - 80*a^2*x^3 + 30*a^4*x^5))/(7 68*a^2*x^2 - 768*a^4*x^4 + 256*a^6*x^6 - 256)) + log(a*x + 1)^2*((17/(96*a ^2) - (35*x^2)/128 + (15*a^2*x^4)/128)/(3*a*x^2 - 1/a - 3*a^3*x^4 + a^5*x^ 6) + 245/(3072*a)) + log(1 - a*x)*((36*x + 22*a*x^2 - 23/(2*a) - 67*a^2*x^ 3 - (21*a^3*x^4)/2 + 31*a^4*x^5)/(2304*a^2*x^2 - 2304*a^4*x^4 + 768*a^6*x^ 6 - 768) - (5*log(a*x + 1)^3)/(256*a) - log(a*x + 1)*(((37*x)/2 - 35*a*x^2 + 68/(3*a) - (82*a^2*x^3)/3 + 15*a^3*x^4 + (23*a^4*x^5)/2)/(384*a^2*x^2 - 384*a^4*x^4 + 128*a^6*x^6 - 128) - ((37*x)/2 + 35*a*x^2 - 68/(3*a) - (82* a^2*x^3)/3 - 15*a^3*x^4 + (23*a^4*x^5)/2)/(384*a^2*x^2 - 384*a^4*x^4 + 128 *a^6*x^6 - 128) + (245*(3*a^2*x^2 - 3*a^4*x^4 + a^6*x^6 - 1))/(12*a*(384*a ^2*x^2 - 384*a^4*x^4 + 128*a^6*x^6 - 128))) + ((227*x)/2 + 173*a*x^2 - 593 /(6*a) - (599*a^2*x^3)/3 - (159*a^3*x^4)/2 + (183*a^4*x^5)/2)/(2304*a^2...