Integrand size = 14, antiderivative size = 262 \[ \int x^4 (a+b \text {arctanh}(c x))^3 \, dx=\frac {9 a b^2 x}{10 c^4}+\frac {b^3 x^2}{20 c^3}+\frac {9 b^3 x \text {arctanh}(c x)}{10 c^4}+\frac {b^2 x^3 (a+b \text {arctanh}(c x))}{10 c^2}-\frac {9 b (a+b \text {arctanh}(c x))^2}{20 c^5}+\frac {3 b x^2 (a+b \text {arctanh}(c x))^2}{10 c^3}+\frac {3 b x^4 (a+b \text {arctanh}(c x))^2}{20 c}+\frac {(a+b \text {arctanh}(c x))^3}{5 c^5}+\frac {1}{5} x^5 (a+b \text {arctanh}(c x))^3-\frac {3 b (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1-c x}\right )}{5 c^5}+\frac {b^3 \log \left (1-c^2 x^2\right )}{2 c^5}-\frac {3 b^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{5 c^5}+\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{10 c^5} \] Output:
9/10*a*b^2*x/c^4+1/20*b^3*x^2/c^3+9/10*b^3*x*arctanh(c*x)/c^4+1/10*b^2*x^3 *(a+b*arctanh(c*x))/c^2-9/20*b*(a+b*arctanh(c*x))^2/c^5+3/10*b*x^2*(a+b*ar ctanh(c*x))^2/c^3+3/20*b*x^4*(a+b*arctanh(c*x))^2/c+1/5*(a+b*arctanh(c*x)) ^3/c^5+1/5*x^5*(a+b*arctanh(c*x))^3-3/5*b*(a+b*arctanh(c*x))^2*ln(2/(-c*x+ 1))/c^5+1/2*b^3*ln(-c^2*x^2+1)/c^5-3/5*b^2*(a+b*arctanh(c*x))*polylog(2,1- 2/(-c*x+1))/c^5+3/10*b^3*polylog(3,1-2/(-c*x+1))/c^5
Time = 0.51 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.46 \[ \int x^4 (a+b \text {arctanh}(c x))^3 \, dx=\frac {-b^3+18 a b^2 c x+6 a^2 b c^2 x^2+b^3 c^2 x^2+2 a b^2 c^3 x^3+3 a^2 b c^4 x^4+4 a^3 c^5 x^5-18 a b^2 \text {arctanh}(c x)+18 b^3 c x \text {arctanh}(c x)+12 a b^2 c^2 x^2 \text {arctanh}(c x)+2 b^3 c^3 x^3 \text {arctanh}(c x)+6 a b^2 c^4 x^4 \text {arctanh}(c x)+12 a^2 b c^5 x^5 \text {arctanh}(c x)-12 a b^2 \text {arctanh}(c x)^2-9 b^3 \text {arctanh}(c x)^2+6 b^3 c^2 x^2 \text {arctanh}(c x)^2+3 b^3 c^4 x^4 \text {arctanh}(c x)^2+12 a b^2 c^5 x^5 \text {arctanh}(c x)^2-4 b^3 \text {arctanh}(c x)^3+4 b^3 c^5 x^5 \text {arctanh}(c x)^3-24 a b^2 \text {arctanh}(c x) \log \left (1+e^{-2 \text {arctanh}(c x)}\right )-12 b^3 \text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+6 a^2 b \log \left (1-c^2 x^2\right )+10 b^3 \log \left (1-c^2 x^2\right )+12 b^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+6 b^3 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )}{20 c^5} \] Input:
Integrate[x^4*(a + b*ArcTanh[c*x])^3,x]
Output:
(-b^3 + 18*a*b^2*c*x + 6*a^2*b*c^2*x^2 + b^3*c^2*x^2 + 2*a*b^2*c^3*x^3 + 3 *a^2*b*c^4*x^4 + 4*a^3*c^5*x^5 - 18*a*b^2*ArcTanh[c*x] + 18*b^3*c*x*ArcTan h[c*x] + 12*a*b^2*c^2*x^2*ArcTanh[c*x] + 2*b^3*c^3*x^3*ArcTanh[c*x] + 6*a* b^2*c^4*x^4*ArcTanh[c*x] + 12*a^2*b*c^5*x^5*ArcTanh[c*x] - 12*a*b^2*ArcTan h[c*x]^2 - 9*b^3*ArcTanh[c*x]^2 + 6*b^3*c^2*x^2*ArcTanh[c*x]^2 + 3*b^3*c^4 *x^4*ArcTanh[c*x]^2 + 12*a*b^2*c^5*x^5*ArcTanh[c*x]^2 - 4*b^3*ArcTanh[c*x] ^3 + 4*b^3*c^5*x^5*ArcTanh[c*x]^3 - 24*a*b^2*ArcTanh[c*x]*Log[1 + E^(-2*Ar cTanh[c*x])] - 12*b^3*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + 6*a^2* b*Log[1 - c^2*x^2] + 10*b^3*Log[1 - c^2*x^2] + 12*b^2*(a + b*ArcTanh[c*x]) *PolyLog[2, -E^(-2*ArcTanh[c*x])] + 6*b^3*PolyLog[3, -E^(-2*ArcTanh[c*x])] )/(20*c^5)
Time = 2.90 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.39, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.071, Rules used = {6452, 6542, 6452, 6542, 6452, 243, 49, 2009, 6542, 2009, 6510, 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.
| \(\displaystyle \int x^4 (a+b \text {arctanh}(c x))^3 \, dx\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^3-\frac {3}{5} b c \int \frac {x^5 (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^3-\frac {3}{5} b c \left (\frac {\int \frac {x^3 (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {\int x^3 (a+b \text {arctanh}(c x))^2dx}{c^2}\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^3-\frac {3}{5} b c \left (\frac {\int \frac {x^3 (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))^2-\frac {1}{2} b c \int \frac {x^4 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}\right )\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^3-\frac {3}{5} b c \left (\frac {\frac {\int \frac {x (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {\int x (a+b \text {arctanh}(c x))^2dx}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))^2-\frac {1}{2} b c \left (\frac {\int \frac {x^2 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\int x^2 (a+b \text {arctanh}(c x))dx}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^3-\frac {3}{5} b c \left (\frac {\frac {\int \frac {x (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))^2-b c \int \frac {x^2 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))^2-\frac {1}{2} b c \left (\frac {\int \frac {x^2 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))-\frac {1}{3} b c \int \frac {x^3}{1-c^2 x^2}dx}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^3-\frac {3}{5} b c \left (\frac {\frac {\int \frac {x (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))^2-b c \int \frac {x^2 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))^2-\frac {1}{2} b c \left (\frac {\int \frac {x^2 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))-\frac {1}{6} b c \int \frac {x^2}{1-c^2 x^2}dx^2}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^3-\frac {3}{5} b c \left (\frac {\frac {\int \frac {x (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))^2-b c \int \frac {x^2 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))^2-\frac {1}{2} b c \left (\frac {\int \frac {x^2 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))-\frac {1}{6} b c \int \left (-\frac {1}{c^2}-\frac {1}{c^2 \left (c^2 x^2-1\right )}\right )dx^2}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^3-\frac {3}{5} b c \left (\frac {\frac {\int \frac {x (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))^2-b c \int \frac {x^2 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))^2-\frac {1}{2} b c \left (\frac {\int \frac {x^2 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))-\frac {1}{6} b c \left (-\frac {x^2}{c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^3-\frac {3}{5} b c \left (\frac {\frac {\int \frac {x (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))^2-b c \left (\frac {\int \frac {a+b \text {arctanh}(c x)}{1-c^2 x^2}dx}{c^2}-\frac {\int (a+b \text {arctanh}(c x))dx}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))^2-\frac {1}{2} b c \left (\frac {\frac {\int \frac {a+b \text {arctanh}(c x)}{1-c^2 x^2}dx}{c^2}-\frac {\int (a+b \text {arctanh}(c x))dx}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))-\frac {1}{6} b c \left (-\frac {x^2}{c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^3-\frac {3}{5} b c \left (\frac {\frac {\int \frac {x (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))^2-b c \left (\frac {\int \frac {a+b \text {arctanh}(c x)}{1-c^2 x^2}dx}{c^2}-\frac {a x+b x \text {arctanh}(c x)+\frac {b \log \left (1-c^2 x^2\right )}{2 c}}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))^2-\frac {1}{2} b c \left (\frac {\frac {\int \frac {a+b \text {arctanh}(c x)}{1-c^2 x^2}dx}{c^2}-\frac {a x+b x \text {arctanh}(c x)+\frac {b \log \left (1-c^2 x^2\right )}{2 c}}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))-\frac {1}{6} b c \left (-\frac {x^2}{c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6510 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^3-\frac {3}{5} b c \left (\frac {\frac {\int \frac {x (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))^2-b c \left (\frac {(a+b \text {arctanh}(c x))^2}{2 b c^3}-\frac {a x+b x \text {arctanh}(c x)+\frac {b \log \left (1-c^2 x^2\right )}{2 c}}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))^2-\frac {1}{2} b c \left (\frac {\frac {(a+b \text {arctanh}(c x))^2}{2 b c^3}-\frac {a x+b x \text {arctanh}(c x)+\frac {b \log \left (1-c^2 x^2\right )}{2 c}}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))-\frac {1}{6} b c \left (-\frac {x^2}{c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^3-\frac {3}{5} b c \left (\frac {\frac {\frac {\int \frac {(a+b \text {arctanh}(c x))^2}{1-c x}dx}{c}-\frac {(a+b \text {arctanh}(c x))^3}{3 b c^2}}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))^2-b c \left (\frac {(a+b \text {arctanh}(c x))^2}{2 b c^3}-\frac {a x+b x \text {arctanh}(c x)+\frac {b \log \left (1-c^2 x^2\right )}{2 c}}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))^2-\frac {1}{2} b c \left (\frac {\frac {(a+b \text {arctanh}(c x))^2}{2 b c^3}-\frac {a x+b x \text {arctanh}(c x)+\frac {b \log \left (1-c^2 x^2\right )}{2 c}}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))-\frac {1}{6} b c \left (-\frac {x^2}{c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^3-\frac {3}{5} b c \left (\frac {\frac {\frac {\frac {\log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2}{c}-2 b \int \frac {(a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2}dx}{c}-\frac {(a+b \text {arctanh}(c x))^3}{3 b c^2}}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))^2-b c \left (\frac {(a+b \text {arctanh}(c x))^2}{2 b c^3}-\frac {a x+b x \text {arctanh}(c x)+\frac {b \log \left (1-c^2 x^2\right )}{2 c}}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))^2-\frac {1}{2} b c \left (\frac {\frac {(a+b \text {arctanh}(c x))^2}{2 b c^3}-\frac {a x+b x \text {arctanh}(c x)+\frac {b \log \left (1-c^2 x^2\right )}{2 c}}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))-\frac {1}{6} b c \left (-\frac {x^2}{c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6620 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^3-\frac {3}{5} b c \left (\frac {\frac {\frac {\frac {\log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2}{c}-2 b \left (\frac {1}{2} b \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{1-c^2 x^2}dx-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{2 c}\right )}{c}-\frac {(a+b \text {arctanh}(c x))^3}{3 b c^2}}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))^2-b c \left (\frac {(a+b \text {arctanh}(c x))^2}{2 b c^3}-\frac {a x+b x \text {arctanh}(c x)+\frac {b \log \left (1-c^2 x^2\right )}{2 c}}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))^2-\frac {1}{2} b c \left (\frac {\frac {(a+b \text {arctanh}(c x))^2}{2 b c^3}-\frac {a x+b x \text {arctanh}(c x)+\frac {b \log \left (1-c^2 x^2\right )}{2 c}}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))-\frac {1}{6} b c \left (-\frac {x^2}{c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x))^3-\frac {3}{5} b c \left (\frac {\frac {\frac {\frac {\log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2}{c}-2 b \left (\frac {b \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{4 c}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{2 c}\right )}{c}-\frac {(a+b \text {arctanh}(c x))^3}{3 b c^2}}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))^2-b c \left (\frac {(a+b \text {arctanh}(c x))^2}{2 b c^3}-\frac {a x+b x \text {arctanh}(c x)+\frac {b \log \left (1-c^2 x^2\right )}{2 c}}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))^2-\frac {1}{2} b c \left (\frac {\frac {(a+b \text {arctanh}(c x))^2}{2 b c^3}-\frac {a x+b x \text {arctanh}(c x)+\frac {b \log \left (1-c^2 x^2\right )}{2 c}}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))-\frac {1}{6} b c \left (-\frac {x^2}{c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{c^2}\right )}{c^2}\right )\) |
Input:
Int[x^4*(a + b*ArcTanh[c*x])^3,x]
Output:
(x^5*(a + b*ArcTanh[c*x])^3)/5 - (3*b*c*(-(((x^4*(a + b*ArcTanh[c*x])^2)/4 - (b*c*(-(((x^3*(a + b*ArcTanh[c*x]))/3 - (b*c*(-(x^2/c^2) - Log[1 - c^2* x^2]/c^4))/6)/c^2) + ((a + b*ArcTanh[c*x])^2/(2*b*c^3) - (a*x + b*x*ArcTan h[c*x] + (b*Log[1 - c^2*x^2])/(2*c))/c^2)/c^2))/2)/c^2) + (-(((x^2*(a + b* ArcTanh[c*x])^2)/2 - b*c*((a + b*ArcTanh[c*x])^2/(2*b*c^3) - (a*x + b*x*Ar cTanh[c*x] + (b*Log[1 - c^2*x^2])/(2*c))/c^2))/c^2) + (-1/3*(a + b*ArcTanh [c*x])^3/(b*c^2) + (((a + b*ArcTanh[c*x])^2*Log[2/(1 - c*x)])/c - 2*b*(-1/ 2*((a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/c + (b*PolyLog[3, 1 - 2/(1 - c*x)])/(4*c)))/c)/c^2)/c^2))/5
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 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[((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_.)/((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_.))^(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[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcTanh[c* x])^p, x], x] - Simp[d*(f^2/e) 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] && GtQ[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*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 = 12.01 (sec) , antiderivative size = 1070, normalized size of antiderivative = 4.08
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1070\) |
| default | \(\text {Expression too large to display}\) | \(1070\) |
| parts | \(\text {Expression too large to display}\) | \(1072\) |
Input:
int(x^4*(a+b*arctanh(c*x))^3,x,method=_RETURNVERBOSE)
Output:
1/c^5*(1/5*a^3*c^5*x^5+b^3*(3/20*c^4*x^4*arctanh(c*x)^2-1/10-9/20*arctanh( c*x)^2+1/10*(c^2*x^2-4*c*x+7)*(c*x+1)*arctanh(c*x)+3/20*I*Pi*csgn(I/(1-(c* x+1)^2/(c^2*x^2-1)))*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)))*arctanh(c*x)^2+1/10*c*x+1/20*(c*x-1)^2-ln (1+(c*x+1)^2/(-c^2*x^2+1))+3/10*polylog(3,-(c*x+1)^2/(-c^2*x^2+1))-3/5*arc tanh(c*x)*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))+3/10*c^2*x^2*arctanh(c*x)^2-3 /20*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+3/10*I*Pi*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))^2*arctanh(c*x)^2-3/10* I*Pi*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))^3*arctanh(c*x)^2-3/10*I*Pi*csgn(I*( c*x+1)/(-c^2*x^2+1)^(1/2))*csgn(I*(c*x+1)^2/(c^2*x^2-1))^2*arctanh(c*x)^2- 3/20*I*Pi*csgn(I/(1-(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)))^2*arctanh(c*x)^2+3/20*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)))^2*arctanh (c*x)^2-3/20*I*Pi*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))^2*csgn(I*(c*x+1)^2/(c ^2*x^2-1))*arctanh(c*x)^2+3/10*arctanh(c*x)^2*ln(c*x-1)+3/10*arctanh(c*x)^ 2*ln(c*x+1)-3/5*arctanh(c*x)^2*ln((c*x+1)/(-c^2*x^2+1)^(1/2))-3/5*ln(2)*ar ctanh(c*x)^2+6/5*(c*x+1)*arctanh(c*x)+1/5*c^5*x^5*arctanh(c*x)^3-3/10*I*Pi *arctanh(c*x)^2+1/5*arctanh(c*x)^3-3/20*I*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1)) ^3*arctanh(c*x)^2+3/10*(c*x-3)*(c*x+1)*arctanh(c*x))+3*a*b^2*(1/5*c^5*x^5* arctanh(c*x)^2+1/10*c^4*x^4*arctanh(c*x)+1/5*c^2*x^2*arctanh(c*x)+1/5*a...
\[ \int x^4 (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} x^{4} \,d x } \] Input:
integrate(x^4*(a+b*arctanh(c*x))^3,x, algorithm="fricas")
Output:
integral(b^3*x^4*arctanh(c*x)^3 + 3*a*b^2*x^4*arctanh(c*x)^2 + 3*a^2*b*x^4 *arctanh(c*x) + a^3*x^4, x)
\[ \int x^4 (a+b \text {arctanh}(c x))^3 \, dx=\int x^{4} \left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3}\, dx \] Input:
integrate(x**4*(a+b*atanh(c*x))**3,x)
Output:
Integral(x**4*(a + b*atanh(c*x))**3, x)
\[ \int x^4 (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} x^{4} \,d x } \] Input:
integrate(x^4*(a+b*arctanh(c*x))^3,x, algorithm="maxima")
Output:
1/5*a^3*x^5 + 3/20*(4*x^5*arctanh(c*x) + c*((c^2*x^4 + 2*x^2)/c^4 + 2*log( c^2*x^2 - 1)/c^6))*a^2*b - 1/80*(2*(b^3*c^5*x^5 - b^3)*log(-c*x + 1)^3 - 3 *(4*a*b^2*c^5*x^5 + b^3*c^4*x^4 + 2*b^3*c^2*x^2 + 2*(b^3*c^5*x^5 + b^3)*lo g(c*x + 1))*log(-c*x + 1)^2)/c^5 - integrate(-1/40*(5*(b^3*c^5*x^5 - b^3*c ^4*x^4)*log(c*x + 1)^3 + 30*(a*b^2*c^5*x^5 - a*b^2*c^4*x^4)*log(c*x + 1)^2 - 3*(4*a*b^2*c^5*x^5 + b^3*c^4*x^4 + 2*b^3*c^2*x^2 + 5*(b^3*c^5*x^5 - b^3 *c^4*x^4)*log(c*x + 1)^2 - 2*(10*a*b^2*c^4*x^4 - (10*a*b^2*c^5 + b^3*c^5)* x^5 - b^3)*log(c*x + 1))*log(-c*x + 1))/(c^5*x - c^4), x)
\[ \int x^4 (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} x^{4} \,d x } \] Input:
integrate(x^4*(a+b*arctanh(c*x))^3,x, algorithm="giac")
Output:
integrate((b*arctanh(c*x) + a)^3*x^4, x)
Timed out. \[ \int x^4 (a+b \text {arctanh}(c x))^3 \, dx=\int x^4\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3 \,d x \] Input:
int(x^4*(a + b*atanh(c*x))^3,x)
Output:
int(x^4*(a + b*atanh(c*x))^3, x)
\[ \int x^4 (a+b \text {arctanh}(c x))^3 \, dx=\frac {12 \left (\int \mathit {atanh} \left (c x \right )^{2}d x \right ) a \,b^{2} c +20 \mathit {atanh} \left (c x \right ) b^{3}+b^{3} c^{2} x^{2}+4 \mathit {atanh} \left (c x \right )^{3} b^{3} c^{5} x^{5}-4 \mathit {atanh} \left (c x \right )^{3} b^{3} c x +3 \mathit {atanh} \left (c x \right )^{2} b^{3} c^{4} x^{4}+6 \mathit {atanh} \left (c x \right )^{2} b^{3} c^{2} x^{2}+2 \mathit {atanh} \left (c x \right ) b^{3} c^{3} x^{3}+18 \mathit {atanh} \left (c x \right ) b^{3} c x +3 a^{2} b \,c^{4} x^{4}+6 a^{2} b \,c^{2} x^{2}+2 a \,b^{2} c^{3} x^{3}+18 a \,b^{2} c x -9 \mathit {atanh} \left (c x \right )^{2} b^{3}+20 \,\mathrm {log}\left (c^{2} x -c \right ) b^{3}+4 \left (\int \mathit {atanh} \left (c x \right )^{3}d x \right ) b^{3} c +12 \mathit {atanh} \left (c x \right ) a^{2} b -18 \mathit {atanh} \left (c x \right ) a \,b^{2}+12 \mathit {atanh} \left (c x \right )^{2} a \,b^{2} c^{5} x^{5}-12 \mathit {atanh} \left (c x \right )^{2} a \,b^{2} c x +12 \mathit {atanh} \left (c x \right ) a^{2} b \,c^{5} x^{5}+6 \mathit {atanh} \left (c x \right ) a \,b^{2} c^{4} x^{4}+12 \mathit {atanh} \left (c x \right ) a \,b^{2} c^{2} x^{2}+12 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2} b +4 a^{3} c^{5} x^{5}}{20 c^{5}} \] Input:
int(x^4*(a+b*atanh(c*x))^3,x)
Output:
(4*atanh(c*x)**3*b**3*c**5*x**5 - 4*atanh(c*x)**3*b**3*c*x + 12*atanh(c*x) **2*a*b**2*c**5*x**5 - 12*atanh(c*x)**2*a*b**2*c*x + 3*atanh(c*x)**2*b**3* c**4*x**4 + 6*atanh(c*x)**2*b**3*c**2*x**2 - 9*atanh(c*x)**2*b**3 + 12*ata nh(c*x)*a**2*b*c**5*x**5 + 12*atanh(c*x)*a**2*b + 6*atanh(c*x)*a*b**2*c**4 *x**4 + 12*atanh(c*x)*a*b**2*c**2*x**2 - 18*atanh(c*x)*a*b**2 + 2*atanh(c* x)*b**3*c**3*x**3 + 18*atanh(c*x)*b**3*c*x + 20*atanh(c*x)*b**3 + 4*int(at anh(c*x)**3,x)*b**3*c + 12*int(atanh(c*x)**2,x)*a*b**2*c + 12*log(c**2*x - c)*a**2*b + 20*log(c**2*x - c)*b**3 + 4*a**3*c**5*x**5 + 3*a**2*b*c**4*x* *4 + 6*a**2*b*c**2*x**2 + 2*a*b**2*c**3*x**3 + 18*a*b**2*c*x + b**3*c**2*x **2)/(20*c**5)