Integrand size = 22, antiderivative size = 334 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^4 (d+c d x)} \, dx=-\frac {b^2 c^2}{3 d x}+\frac {b^2 c^3 \text {arctanh}(c x)}{3 d}-\frac {b c (a+b \text {arctanh}(c x))}{3 d x^2}+\frac {b c^2 (a+b \text {arctanh}(c x))}{d x}+\frac {5 c^3 (a+b \text {arctanh}(c x))^2}{6 d}-\frac {(a+b \text {arctanh}(c x))^2}{3 d x^3}+\frac {c (a+b \text {arctanh}(c x))^2}{2 d x^2}-\frac {c^2 (a+b \text {arctanh}(c x))^2}{d x}-\frac {b^2 c^3 \log (x)}{d}+\frac {b^2 c^3 \log \left (1-c^2 x^2\right )}{2 d}+\frac {8 b c^3 (a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{1+c x}\right )}{3 d}-\frac {c^3 (a+b \text {arctanh}(c x))^2 \log \left (2-\frac {2}{1+c x}\right )}{d}-\frac {4 b^2 c^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )}{3 d}+\frac {b c^3 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )}{d}+\frac {b^2 c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+c x}\right )}{2 d} \] Output:
-1/3*b^2*c^2/d/x+1/3*b^2*c^3*arctanh(c*x)/d-1/3*b*c*(a+b*arctanh(c*x))/d/x ^2+b*c^2*(a+b*arctanh(c*x))/d/x+5/6*c^3*(a+b*arctanh(c*x))^2/d-1/3*(a+b*ar ctanh(c*x))^2/d/x^3+1/2*c*(a+b*arctanh(c*x))^2/d/x^2-c^2*(a+b*arctanh(c*x) )^2/d/x-b^2*c^3*ln(x)/d+1/2*b^2*c^3*ln(-c^2*x^2+1)/d+8/3*b*c^3*(a+b*arctan h(c*x))*ln(2-2/(c*x+1))/d-c^3*(a+b*arctanh(c*x))^2*ln(2-2/(c*x+1))/d-4/3*b ^2*c^3*polylog(2,-1+2/(c*x+1))/d+b*c^3*(a+b*arctanh(c*x))*polylog(2,-1+2/( c*x+1))/d+1/2*b^2*c^3*polylog(3,-1+2/(c*x+1))/d
Result contains complex when optimal does not.
Time = 1.11 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.16 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^4 (d+c d x)} \, dx=\frac {-\frac {8 a^2}{x^3}+\frac {12 a^2 c}{x^2}-\frac {24 a^2 c^2}{x}-24 a^2 c^3 \log (x)+24 a^2 c^3 \log (1+c x)-\frac {8 a b \left (\text {arctanh}(c x) \left (2-3 c x+6 c^2 x^2+3 c^3 x^3+6 c^3 x^3 \log \left (1-e^{-2 \text {arctanh}(c x)}\right )\right )-c x \left (-1+3 c x+c^2 x^2+8 c^2 x^2 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )\right )-3 c^3 x^3 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )\right )}{x^3}+b^2 c^3 \left (-i \pi ^3-\frac {8}{c x}+8 \text {arctanh}(c x)-\frac {8 \text {arctanh}(c x)}{c^2 x^2}+\frac {24 \text {arctanh}(c x)}{c x}+20 \text {arctanh}(c x)^2-\frac {8 \text {arctanh}(c x)^2}{c^3 x^3}+\frac {12 \text {arctanh}(c x)^2}{c^2 x^2}-\frac {24 \text {arctanh}(c x)^2}{c x}+16 \text {arctanh}(c x)^3+64 \text {arctanh}(c x) \log \left (1-e^{-2 \text {arctanh}(c x)}\right )-24 \text {arctanh}(c x)^2 \log \left (1-e^{2 \text {arctanh}(c x)}\right )-24 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )-32 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )-24 \text {arctanh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )+12 \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )\right )}{24 d} \] Input:
Integrate[(a + b*ArcTanh[c*x])^2/(x^4*(d + c*d*x)),x]
Output:
((-8*a^2)/x^3 + (12*a^2*c)/x^2 - (24*a^2*c^2)/x - 24*a^2*c^3*Log[x] + 24*a ^2*c^3*Log[1 + c*x] - (8*a*b*(ArcTanh[c*x]*(2 - 3*c*x + 6*c^2*x^2 + 3*c^3* x^3 + 6*c^3*x^3*Log[1 - E^(-2*ArcTanh[c*x])]) - c*x*(-1 + 3*c*x + c^2*x^2 + 8*c^2*x^2*Log[(c*x)/Sqrt[1 - c^2*x^2]]) - 3*c^3*x^3*PolyLog[2, E^(-2*Arc Tanh[c*x])]))/x^3 + b^2*c^3*((-I)*Pi^3 - 8/(c*x) + 8*ArcTanh[c*x] - (8*Arc Tanh[c*x])/(c^2*x^2) + (24*ArcTanh[c*x])/(c*x) + 20*ArcTanh[c*x]^2 - (8*Ar cTanh[c*x]^2)/(c^3*x^3) + (12*ArcTanh[c*x]^2)/(c^2*x^2) - (24*ArcTanh[c*x] ^2)/(c*x) + 16*ArcTanh[c*x]^3 + 64*ArcTanh[c*x]*Log[1 - E^(-2*ArcTanh[c*x] )] - 24*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] - 24*Log[(c*x)/Sqrt[1 - c^2*x^2]] - 32*PolyLog[2, E^(-2*ArcTanh[c*x])] - 24*ArcTanh[c*x]*PolyLog[ 2, E^(2*ArcTanh[c*x])] + 12*PolyLog[3, E^(2*ArcTanh[c*x])]))/(24*d)
Time = 4.57 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.10, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6496, 27, 6452, 6496, 6452, 6496, 6452, 6494, 6544, 6452, 243, 47, 14, 16, 264, 219, 6510, 6550, 6494, 2897, 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.
| \(\displaystyle \int \frac {(a+b \text {arctanh}(c x))^2}{x^4 (c d x+d)} \, dx\) |
| \(\Big \downarrow \) 6496 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arctanh}(c x))^2}{x^4}dx}{d}-c \int \frac {(a+b \text {arctanh}(c x))^2}{d x^3 (c x+1)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arctanh}(c x))^2}{x^4}dx}{d}-\frac {c \int \frac {(a+b \text {arctanh}(c x))^2}{x^3 (c x+1)}dx}{d}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {\frac {2}{3} b c \int \frac {a+b \text {arctanh}(c x)}{x^3 \left (1-c^2 x^2\right )}dx-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}}{d}-\frac {c \int \frac {(a+b \text {arctanh}(c x))^2}{x^3 (c x+1)}dx}{d}\) |
| \(\Big \downarrow \) 6496 |
| \(\displaystyle \frac {\frac {2}{3} b c \int \frac {a+b \text {arctanh}(c x)}{x^3 \left (1-c^2 x^2\right )}dx-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}}{d}-\frac {c \left (\int \frac {(a+b \text {arctanh}(c x))^2}{x^3}dx-c \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (c x+1)}dx\right )}{d}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {\frac {2}{3} b c \int \frac {a+b \text {arctanh}(c x)}{x^3 \left (1-c^2 x^2\right )}dx-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}}{d}-\frac {c \left (b c \int \frac {a+b \text {arctanh}(c x)}{x^2 \left (1-c^2 x^2\right )}dx-c \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (c x+1)}dx-\frac {(a+b \text {arctanh}(c x))^2}{2 x^2}\right )}{d}\) |
| \(\Big \downarrow \) 6496 |
| \(\displaystyle \frac {\frac {2}{3} b c \int \frac {a+b \text {arctanh}(c x)}{x^3 \left (1-c^2 x^2\right )}dx-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}}{d}-\frac {c \left (b c \int \frac {a+b \text {arctanh}(c x)}{x^2 \left (1-c^2 x^2\right )}dx-c \left (\int \frac {(a+b \text {arctanh}(c x))^2}{x^2}dx-c \int \frac {(a+b \text {arctanh}(c x))^2}{x (c x+1)}dx\right )-\frac {(a+b \text {arctanh}(c x))^2}{2 x^2}\right )}{d}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {\frac {2}{3} b c \int \frac {a+b \text {arctanh}(c x)}{x^3 \left (1-c^2 x^2\right )}dx-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}}{d}-\frac {c \left (b c \int \frac {a+b \text {arctanh}(c x)}{x^2 \left (1-c^2 x^2\right )}dx-c \left (2 b c \int \frac {a+b \text {arctanh}(c x)}{x \left (1-c^2 x^2\right )}dx-c \int \frac {(a+b \text {arctanh}(c x))^2}{x (c x+1)}dx-\frac {(a+b \text {arctanh}(c x))^2}{x}\right )-\frac {(a+b \text {arctanh}(c x))^2}{2 x^2}\right )}{d}\) |
| \(\Big \downarrow \) 6494 |
| \(\displaystyle \frac {\frac {2}{3} b c \int \frac {a+b \text {arctanh}(c x)}{x^3 \left (1-c^2 x^2\right )}dx-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}}{d}-\frac {c \left (b c \int \frac {a+b \text {arctanh}(c x)}{x^2 \left (1-c^2 x^2\right )}dx-c \left (2 b c \int \frac {a+b \text {arctanh}(c x)}{x \left (1-c^2 x^2\right )}dx-c \left (\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2-2 b c \int \frac {(a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{c x+1}\right )}{1-c^2 x^2}dx\right )-\frac {(a+b \text {arctanh}(c x))^2}{x}\right )-\frac {(a+b \text {arctanh}(c x))^2}{2 x^2}\right )}{d}\) |
| \(\Big \downarrow \) 6544 |
| \(\displaystyle \frac {\frac {2}{3} b c \left (c^2 \int \frac {a+b \text {arctanh}(c x)}{x \left (1-c^2 x^2\right )}dx+\int \frac {a+b \text {arctanh}(c x)}{x^3}dx\right )-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}}{d}-\frac {c \left (b c \left (c^2 \int \frac {a+b \text {arctanh}(c x)}{1-c^2 x^2}dx+\int \frac {a+b \text {arctanh}(c x)}{x^2}dx\right )-c \left (2 b c \int \frac {a+b \text {arctanh}(c x)}{x \left (1-c^2 x^2\right )}dx-c \left (\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2-2 b c \int \frac {(a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{c x+1}\right )}{1-c^2 x^2}dx\right )-\frac {(a+b \text {arctanh}(c x))^2}{x}\right )-\frac {(a+b \text {arctanh}(c x))^2}{2 x^2}\right )}{d}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {\frac {2}{3} b c \left (c^2 \int \frac {a+b \text {arctanh}(c x)}{x \left (1-c^2 x^2\right )}dx+\frac {1}{2} b c \int \frac {1}{x^2 \left (1-c^2 x^2\right )}dx-\frac {a+b \text {arctanh}(c x)}{2 x^2}\right )-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}}{d}-\frac {c \left (b c \left (c^2 \int \frac {a+b \text {arctanh}(c x)}{1-c^2 x^2}dx+b c \int \frac {1}{x \left (1-c^2 x^2\right )}dx-\frac {a+b \text {arctanh}(c x)}{x}\right )-c \left (2 b c \int \frac {a+b \text {arctanh}(c x)}{x \left (1-c^2 x^2\right )}dx-c \left (\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2-2 b c \int \frac {(a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{c x+1}\right )}{1-c^2 x^2}dx\right )-\frac {(a+b \text {arctanh}(c x))^2}{x}\right )-\frac {(a+b \text {arctanh}(c x))^2}{2 x^2}\right )}{d}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {2}{3} b c \left (c^2 \int \frac {a+b \text {arctanh}(c x)}{x \left (1-c^2 x^2\right )}dx+\frac {1}{2} b c \int \frac {1}{x^2 \left (1-c^2 x^2\right )}dx-\frac {a+b \text {arctanh}(c x)}{2 x^2}\right )-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}}{d}-\frac {c \left (b c \left (c^2 \int \frac {a+b \text {arctanh}(c x)}{1-c^2 x^2}dx+\frac {1}{2} b c \int \frac {1}{x^2 \left (1-c^2 x^2\right )}dx^2-\frac {a+b \text {arctanh}(c x)}{x}\right )-c \left (2 b c \int \frac {a+b \text {arctanh}(c x)}{x \left (1-c^2 x^2\right )}dx-c \left (\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2-2 b c \int \frac {(a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{c x+1}\right )}{1-c^2 x^2}dx\right )-\frac {(a+b \text {arctanh}(c x))^2}{x}\right )-\frac {(a+b \text {arctanh}(c x))^2}{2 x^2}\right )}{d}\) |
| \(\Big \downarrow \) 47 |
| \(\displaystyle \frac {\frac {2}{3} b c \left (c^2 \int \frac {a+b \text {arctanh}(c x)}{x \left (1-c^2 x^2\right )}dx+\frac {1}{2} b c \int \frac {1}{x^2 \left (1-c^2 x^2\right )}dx-\frac {a+b \text {arctanh}(c x)}{2 x^2}\right )-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}}{d}-\frac {c \left (b c \left (c^2 \int \frac {a+b \text {arctanh}(c x)}{1-c^2 x^2}dx+\frac {1}{2} b c \left (c^2 \int \frac {1}{1-c^2 x^2}dx^2+\int \frac {1}{x^2}dx^2\right )-\frac {a+b \text {arctanh}(c x)}{x}\right )-c \left (2 b c \int \frac {a+b \text {arctanh}(c x)}{x \left (1-c^2 x^2\right )}dx-c \left (\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2-2 b c \int \frac {(a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{c x+1}\right )}{1-c^2 x^2}dx\right )-\frac {(a+b \text {arctanh}(c x))^2}{x}\right )-\frac {(a+b \text {arctanh}(c x))^2}{2 x^2}\right )}{d}\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle \frac {\frac {2}{3} b c \left (c^2 \int \frac {a+b \text {arctanh}(c x)}{x \left (1-c^2 x^2\right )}dx+\frac {1}{2} b c \int \frac {1}{x^2 \left (1-c^2 x^2\right )}dx-\frac {a+b \text {arctanh}(c x)}{2 x^2}\right )-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}}{d}-\frac {c \left (b c \left (c^2 \int \frac {a+b \text {arctanh}(c x)}{1-c^2 x^2}dx+\frac {1}{2} b c \left (c^2 \int \frac {1}{1-c^2 x^2}dx^2+\log \left (x^2\right )\right )-\frac {a+b \text {arctanh}(c x)}{x}\right )-c \left (2 b c \int \frac {a+b \text {arctanh}(c x)}{x \left (1-c^2 x^2\right )}dx-c \left (\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2-2 b c \int \frac {(a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{c x+1}\right )}{1-c^2 x^2}dx\right )-\frac {(a+b \text {arctanh}(c x))^2}{x}\right )-\frac {(a+b \text {arctanh}(c x))^2}{2 x^2}\right )}{d}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {2}{3} b c \left (c^2 \int \frac {a+b \text {arctanh}(c x)}{x \left (1-c^2 x^2\right )}dx+\frac {1}{2} b c \int \frac {1}{x^2 \left (1-c^2 x^2\right )}dx-\frac {a+b \text {arctanh}(c x)}{2 x^2}\right )-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}}{d}-\frac {c \left (b c \left (c^2 \int \frac {a+b \text {arctanh}(c x)}{1-c^2 x^2}dx-\frac {a+b \text {arctanh}(c x)}{x}+\frac {1}{2} b c \left (\log \left (x^2\right )-\log \left (1-c^2 x^2\right )\right )\right )-c \left (2 b c \int \frac {a+b \text {arctanh}(c x)}{x \left (1-c^2 x^2\right )}dx-c \left (\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2-2 b c \int \frac {(a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{c x+1}\right )}{1-c^2 x^2}dx\right )-\frac {(a+b \text {arctanh}(c x))^2}{x}\right )-\frac {(a+b \text {arctanh}(c x))^2}{2 x^2}\right )}{d}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {\frac {2}{3} b c \left (c^2 \int \frac {a+b \text {arctanh}(c x)}{x \left (1-c^2 x^2\right )}dx+\frac {1}{2} b c \left (c^2 \int \frac {1}{1-c^2 x^2}dx-\frac {1}{x}\right )-\frac {a+b \text {arctanh}(c x)}{2 x^2}\right )-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}}{d}-\frac {c \left (b c \left (c^2 \int \frac {a+b \text {arctanh}(c x)}{1-c^2 x^2}dx-\frac {a+b \text {arctanh}(c x)}{x}+\frac {1}{2} b c \left (\log \left (x^2\right )-\log \left (1-c^2 x^2\right )\right )\right )-c \left (2 b c \int \frac {a+b \text {arctanh}(c x)}{x \left (1-c^2 x^2\right )}dx-c \left (\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2-2 b c \int \frac {(a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{c x+1}\right )}{1-c^2 x^2}dx\right )-\frac {(a+b \text {arctanh}(c x))^2}{x}\right )-\frac {(a+b \text {arctanh}(c x))^2}{2 x^2}\right )}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {2}{3} b c \left (c^2 \int \frac {a+b \text {arctanh}(c x)}{x \left (1-c^2 x^2\right )}dx-\frac {a+b \text {arctanh}(c x)}{2 x^2}+\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )\right )-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}}{d}-\frac {c \left (b c \left (c^2 \int \frac {a+b \text {arctanh}(c x)}{1-c^2 x^2}dx-\frac {a+b \text {arctanh}(c x)}{x}+\frac {1}{2} b c \left (\log \left (x^2\right )-\log \left (1-c^2 x^2\right )\right )\right )-c \left (2 b c \int \frac {a+b \text {arctanh}(c x)}{x \left (1-c^2 x^2\right )}dx-c \left (\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2-2 b c \int \frac {(a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{c x+1}\right )}{1-c^2 x^2}dx\right )-\frac {(a+b \text {arctanh}(c x))^2}{x}\right )-\frac {(a+b \text {arctanh}(c x))^2}{2 x^2}\right )}{d}\) |
| \(\Big \downarrow \) 6510 |
| \(\displaystyle \frac {\frac {2}{3} b c \left (c^2 \int \frac {a+b \text {arctanh}(c x)}{x \left (1-c^2 x^2\right )}dx-\frac {a+b \text {arctanh}(c x)}{2 x^2}+\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )\right )-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}}{d}-\frac {c \left (-c \left (2 b c \int \frac {a+b \text {arctanh}(c x)}{x \left (1-c^2 x^2\right )}dx-c \left (\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2-2 b c \int \frac {(a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{c x+1}\right )}{1-c^2 x^2}dx\right )-\frac {(a+b \text {arctanh}(c x))^2}{x}\right )+b c \left (\frac {c (a+b \text {arctanh}(c x))^2}{2 b}-\frac {a+b \text {arctanh}(c x)}{x}+\frac {1}{2} b c \left (\log \left (x^2\right )-\log \left (1-c^2 x^2\right )\right )\right )-\frac {(a+b \text {arctanh}(c x))^2}{2 x^2}\right )}{d}\) |
| \(\Big \downarrow \) 6550 |
| \(\displaystyle \frac {\frac {2}{3} b c \left (c^2 \left (\int \frac {a+b \text {arctanh}(c x)}{x (c x+1)}dx+\frac {(a+b \text {arctanh}(c x))^2}{2 b}\right )-\frac {a+b \text {arctanh}(c x)}{2 x^2}+\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )\right )-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}}{d}-\frac {c \left (-c \left (-c \left (\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2-2 b c \int \frac {(a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{c x+1}\right )}{1-c^2 x^2}dx\right )+2 b c \left (\int \frac {a+b \text {arctanh}(c x)}{x (c x+1)}dx+\frac {(a+b \text {arctanh}(c x))^2}{2 b}\right )-\frac {(a+b \text {arctanh}(c x))^2}{x}\right )+b c \left (\frac {c (a+b \text {arctanh}(c x))^2}{2 b}-\frac {a+b \text {arctanh}(c x)}{x}+\frac {1}{2} b c \left (\log \left (x^2\right )-\log \left (1-c^2 x^2\right )\right )\right )-\frac {(a+b \text {arctanh}(c x))^2}{2 x^2}\right )}{d}\) |
| \(\Big \downarrow \) 6494 |
| \(\displaystyle \frac {\frac {2}{3} b c \left (c^2 \left (-b c \int \frac {\log \left (2-\frac {2}{c x+1}\right )}{1-c^2 x^2}dx+\frac {(a+b \text {arctanh}(c x))^2}{2 b}+\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))\right )-\frac {a+b \text {arctanh}(c x)}{2 x^2}+\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )\right )-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}}{d}-\frac {c \left (-c \left (2 b c \left (-b c \int \frac {\log \left (2-\frac {2}{c x+1}\right )}{1-c^2 x^2}dx+\frac {(a+b \text {arctanh}(c x))^2}{2 b}+\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))\right )-c \left (\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2-2 b c \int \frac {(a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{c x+1}\right )}{1-c^2 x^2}dx\right )-\frac {(a+b \text {arctanh}(c x))^2}{x}\right )+b c \left (\frac {c (a+b \text {arctanh}(c x))^2}{2 b}-\frac {a+b \text {arctanh}(c x)}{x}+\frac {1}{2} b c \left (\log \left (x^2\right )-\log \left (1-c^2 x^2\right )\right )\right )-\frac {(a+b \text {arctanh}(c x))^2}{2 x^2}\right )}{d}\) |
| \(\Big \downarrow \) 2897 |
| \(\displaystyle \frac {\frac {2}{3} b c \left (c^2 \left (\frac {(a+b \text {arctanh}(c x))^2}{2 b}+\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right )\right )-\frac {a+b \text {arctanh}(c x)}{2 x^2}+\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )\right )-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}}{d}-\frac {c \left (-c \left (-c \left (\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2-2 b c \int \frac {(a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{c x+1}\right )}{1-c^2 x^2}dx\right )+2 b c \left (\frac {(a+b \text {arctanh}(c x))^2}{2 b}+\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right )\right )-\frac {(a+b \text {arctanh}(c x))^2}{x}\right )+b c \left (\frac {c (a+b \text {arctanh}(c x))^2}{2 b}-\frac {a+b \text {arctanh}(c x)}{x}+\frac {1}{2} b c \left (\log \left (x^2\right )-\log \left (1-c^2 x^2\right )\right )\right )-\frac {(a+b \text {arctanh}(c x))^2}{2 x^2}\right )}{d}\) |
| \(\Big \downarrow \) 6618 |
| \(\displaystyle \frac {\frac {2}{3} b c \left (c^2 \left (\frac {(a+b \text {arctanh}(c x))^2}{2 b}+\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right )\right )-\frac {a+b \text {arctanh}(c x)}{2 x^2}+\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )\right )-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}}{d}-\frac {c \left (-c \left (-c \left (\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2-2 b c \left (\frac {\operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right ) (a+b \text {arctanh}(c x))}{2 c}-\frac {1}{2} b \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right )}{1-c^2 x^2}dx\right )\right )+2 b c \left (\frac {(a+b \text {arctanh}(c x))^2}{2 b}+\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right )\right )-\frac {(a+b \text {arctanh}(c x))^2}{x}\right )+b c \left (\frac {c (a+b \text {arctanh}(c x))^2}{2 b}-\frac {a+b \text {arctanh}(c x)}{x}+\frac {1}{2} b c \left (\log \left (x^2\right )-\log \left (1-c^2 x^2\right )\right )\right )-\frac {(a+b \text {arctanh}(c x))^2}{2 x^2}\right )}{d}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {\frac {2}{3} b c \left (c^2 \left (\frac {(a+b \text {arctanh}(c x))^2}{2 b}+\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right )\right )-\frac {a+b \text {arctanh}(c x)}{2 x^2}+\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )\right )-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}}{d}-\frac {c \left (b c \left (\frac {c (a+b \text {arctanh}(c x))^2}{2 b}-\frac {a+b \text {arctanh}(c x)}{x}+\frac {1}{2} b c \left (\log \left (x^2\right )-\log \left (1-c^2 x^2\right )\right )\right )-c \left (2 b c \left (\frac {(a+b \text {arctanh}(c x))^2}{2 b}+\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right )\right )-c \left (\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2-2 b c \left (\frac {\operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right ) (a+b \text {arctanh}(c x))}{2 c}+\frac {b \operatorname {PolyLog}\left (3,\frac {2}{c x+1}-1\right )}{4 c}\right )\right )-\frac {(a+b \text {arctanh}(c x))^2}{x}\right )-\frac {(a+b \text {arctanh}(c x))^2}{2 x^2}\right )}{d}\) |
Input:
Int[(a + b*ArcTanh[c*x])^2/(x^4*(d + c*d*x)),x]
Output:
(-1/3*(a + b*ArcTanh[c*x])^2/x^3 + (2*b*c*(-1/2*(a + b*ArcTanh[c*x])/x^2 + (b*c*(-x^(-1) + c*ArcTanh[c*x]))/2 + c^2*((a + b*ArcTanh[c*x])^2/(2*b) + (a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)] - (b*PolyLog[2, -1 + 2/(1 + c*x) ])/2)))/3)/d - (c*(-1/2*(a + b*ArcTanh[c*x])^2/x^2 + b*c*(-((a + b*ArcTanh [c*x])/x) + (c*(a + b*ArcTanh[c*x])^2)/(2*b) + (b*c*(Log[x^2] - Log[1 - c^ 2*x^2]))/2) - c*(-((a + b*ArcTanh[c*x])^2/x) + 2*b*c*((a + b*ArcTanh[c*x]) ^2/(2*b) + (a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)] - (b*PolyLog[2, -1 + 2/(1 + c*x)])/2) - c*((a + b*ArcTanh[c*x])^2*Log[2 - 2/(1 + c*x)] - 2*b*c* (((a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 + c*x)])/(2*c) + (b*PolyLog[3, -1 + 2/(1 + c*x)])/(4*c))))))/d
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[1/(c + d*x), x ], x] /; FreeQ[{a, b, c, d}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, x][[2]], Expon[Pq, x]]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : > Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x ], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 ] && IntegerQ[m])) && NeQ[m, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x _Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Simp[b*c*(p/d) Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2 - 2/(1 + e*(x/d))] /(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c ^2*d^2 - e^2, 0]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( e_.)*(x_)), x_Symbol] :> Simp[1/d Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x], x] - Simp[e/(d*f) Int[(f*x)^(m + 1)*((a + b*ArcTanh[c*x])^p/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0] && LtQ[m, -1]
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_.)*((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[(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 = 7.85 (sec) , antiderivative size = 1718, normalized size of antiderivative = 5.14
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1718\) |
| default | \(\text {Expression too large to display}\) | \(1718\) |
| parts | \(\text {Expression too large to display}\) | \(1720\) |
Input:
int((a+b*arctanh(c*x))^2/x^4/(c*d*x+d),x,method=_RETURNVERBOSE)
Output:
c^3*(a^2/d*(-1/3/c^3/x^3-1/c/x+1/2/c^2/x^2-ln(c*x)+ln(c*x+1))+b^2/d*(-arct anh(c*x)^2*ln(1-(c*x+1)/(-c^2*x^2+1)^(1/2))-2*arctanh(c*x)*polylog(2,(c*x+ 1)/(-c^2*x^2+1)^(1/2))-arctanh(c*x)^2*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))-2*a rctanh(c*x)*polylog(2,-(c*x+1)/(-c^2*x^2+1)^(1/2))+arctanh(c*x)^2*ln((c*x+ 1)^2/(-c^2*x^2+1)-1)+1/2*I*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1) ^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2*x^2-1)))*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)) )*arctanh(c*x)^2-arctanh(c*x)^2*ln(c*x)+arctanh(c*x)^2*ln(c*x+1)+8/3*arcta nh(c*x)*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))-ln((c*x+1)/(-c^2*x^2+1)^(1/2)-1)- 1/3*arctanh(c*x)^2/c^3/x^3+1/2*arctanh(c*x)^2/c^2/x^2+2/3*arctanh(c*x)^3-1 /2*I*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3*arctanh(c*x)^2-1/2*I*Pi*csgn(I*(c* x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^3*arctanh(c*x)^2-8/3*dilog(( c*x+1)/(-c^2*x^2+1)^(1/2))+8/3*dilog(1+(c*x+1)/(-c^2*x^2+1)^(1/2))-11/6*ar ctanh(c*x)^2-ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+2*polylog(3,-(c*x+1)/(-c^2*x ^2+1)^(1/2))+2*polylog(3,(c*x+1)/(-c^2*x^2+1)^(1/2))-arctanh(c*x)^2/c/x+1/ 6*((-c^2*x^2+1)^(1/2)*c*x+2*c^2*x^2-(-c^2*x^2+1)^(1/2)+c*x-1)*arctanh(c*x) /c^2/x^2+1/6*(-(-c^2*x^2+1)^(1/2)*c*x+2*c^2*x^2+(-c^2*x^2+1)^(1/2)+c*x-1)* arctanh(c*x)/c^2/x^2-1/2*I*Pi*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1)/(1-(c*x+1) ^2/(c^2*x^2-1)))^3*arctanh(c*x)^2-2*arctanh(c*x)^2*ln((c*x+1)/(-c^2*x^2+1) ^(1/2))-ln(2)*arctanh(c*x)^2+1/3*(c*x-(-c^2*x^2+1)^(1/2)+1)/c/x*arctanh(c* x)+1/3*arctanh(c*x)*(c*x+(-c^2*x^2+1)^(1/2)+1)/c/x+1/2*I*Pi*csgn(I*(c*x...
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^4 (d+c d x)} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )} x^{4}} \,d x } \] Input:
integrate((a+b*arctanh(c*x))^2/x^4/(c*d*x+d),x, algorithm="fricas")
Output:
integral((b^2*arctanh(c*x)^2 + 2*a*b*arctanh(c*x) + a^2)/(c*d*x^5 + d*x^4) , x)
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^4 (d+c d x)} \, dx=\frac {\int \frac {a^{2}}{c x^{5} + x^{4}}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{c x^{5} + x^{4}}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{c x^{5} + x^{4}}\, dx}{d} \] Input:
integrate((a+b*atanh(c*x))**2/x**4/(c*d*x+d),x)
Output:
(Integral(a**2/(c*x**5 + x**4), x) + Integral(b**2*atanh(c*x)**2/(c*x**5 + x**4), x) + Integral(2*a*b*atanh(c*x)/(c*x**5 + x**4), x))/d
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^4 (d+c d x)} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )} x^{4}} \,d x } \] Input:
integrate((a+b*arctanh(c*x))^2/x^4/(c*d*x+d),x, algorithm="maxima")
Output:
1/6*(6*c^3*log(c*x + 1)/d - 6*c^3*log(x)/d - (6*c^2*x^2 - 3*c*x + 2)/(d*x^ 3))*a^2 + 1/24*(6*b^2*c^3*x^3*log(c*x + 1) - 6*b^2*c^2*x^2 + 3*b^2*c*x - 2 *b^2)*log(-c*x + 1)^2/(d*x^3) - integrate(-1/12*(3*(b^2*c*x - b^2)*log(c*x + 1)^2 + 12*(a*b*c*x - a*b)*log(c*x + 1) + (6*b^2*c^4*x^4 + 3*b^2*c^3*x^3 - b^2*c^2*x^2 + 12*a*b - 2*(6*a*b*c - b^2*c)*x - 6*(b^2*c^5*x^5 + b^2*c^4 *x^4 + b^2*c*x - b^2)*log(c*x + 1))*log(-c*x + 1))/(c^2*d*x^6 - d*x^4), x)
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^4 (d+c d x)} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )} x^{4}} \,d x } \] Input:
integrate((a+b*arctanh(c*x))^2/x^4/(c*d*x+d),x, algorithm="giac")
Output:
integrate((b*arctanh(c*x) + a)^2/((c*d*x + d)*x^4), x)
Timed out. \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^4 (d+c d x)} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{x^4\,\left (d+c\,d\,x\right )} \,d x \] Input:
int((a + b*atanh(c*x))^2/(x^4*(d + c*d*x)),x)
Output:
int((a + b*atanh(c*x))^2/(x^4*(d + c*d*x)), x)
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^4 (d+c d x)} \, dx=\frac {6 \left (\int \frac {\mathit {atanh} \left (c x \right )^{2}}{c^{2} x^{3}-x}d x \right ) b^{2} c^{3} x^{3}-6 \,\mathrm {log}\left (x \right ) a^{2} c^{3} x^{3}+2 \mathit {atanh} \left (c x \right )^{3} b^{2} c^{3} x^{3}+6 \,\mathrm {log}\left (c x +1\right ) a^{2} c^{3} x^{3}-4 \mathit {atanh} \left (c x \right ) a b +6 b^{2} c^{2} x^{2}+6 \mathit {atanh} \left (c x \right ) b^{2} c x -2 a b c x +3 \mathit {atanh} \left (c x \right )^{2} b^{2} c x +6 \mathit {atanh} \left (c x \right ) b^{2} c^{2} x^{2}+6 \,\mathrm {log}\left (c^{2} x -c \right ) b^{2} c^{3} x^{3}-6 \,\mathrm {log}\left (x \right ) b^{2} c^{3} x^{3}-16 \left (\int \frac {\mathit {atanh} \left (c x \right )}{c^{2} x^{5}-x^{3}}d x \right ) b^{2} c \,x^{3}-2 a^{2}-3 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{3} x^{3}-16 \,\mathrm {log}\left (c^{2} x -c \right ) a b \,c^{3} x^{3}+16 \,\mathrm {log}\left (x \right ) a b \,c^{3} x^{3}+12 \left (\int \frac {\mathit {atanh} \left (c x \right )}{c^{2} x^{5}-x^{3}}d x \right ) a b c \,x^{3}-6 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{2} x^{2}-16 \mathit {atanh} \left (c x \right ) a b \,c^{3} x^{3}+3 a^{2} c x -2 \mathit {atanh} \left (c x \right )^{2} b^{2}-12 \mathit {atanh} \left (c x \right ) a b \,c^{2} x^{2}-6 a^{2} c^{2} x^{2}+6 \mathit {atanh} \left (c x \right )^{2} a b \,c^{3} x^{3}}{6 d \,x^{3}} \] Input:
int((a+b*atanh(c*x))^2/x^4/(c*d*x+d),x)
Output:
(2*atanh(c*x)**3*b**2*c**3*x**3 + 6*atanh(c*x)**2*a*b*c**3*x**3 - 3*atanh( c*x)**2*b**2*c**3*x**3 - 6*atanh(c*x)**2*b**2*c**2*x**2 + 3*atanh(c*x)**2* b**2*c*x - 2*atanh(c*x)**2*b**2 - 16*atanh(c*x)*a*b*c**3*x**3 - 12*atanh(c *x)*a*b*c**2*x**2 - 4*atanh(c*x)*a*b + 6*atanh(c*x)*b**2*c**2*x**2 + 6*ata nh(c*x)*b**2*c*x + 12*int(atanh(c*x)/(c**2*x**5 - x**3),x)*a*b*c*x**3 - 16 *int(atanh(c*x)/(c**2*x**5 - x**3),x)*b**2*c*x**3 + 6*int(atanh(c*x)**2/(c **2*x**3 - x),x)*b**2*c**3*x**3 - 16*log(c**2*x - c)*a*b*c**3*x**3 + 6*log (c**2*x - c)*b**2*c**3*x**3 + 6*log(c*x + 1)*a**2*c**3*x**3 - 6*log(x)*a** 2*c**3*x**3 + 16*log(x)*a*b*c**3*x**3 - 6*log(x)*b**2*c**3*x**3 - 6*a**2*c **2*x**2 + 3*a**2*c*x - 2*a**2 - 2*a*b*c*x + 6*b**2*c**2*x**2)/(6*d*x**3)