Integrand size = 14, antiderivative size = 247 \[ \int x^5 (a+b \text {arctanh}(c x))^3 \, dx=\frac {19 b^3 x}{60 c^5}+\frac {b^3 x^3}{60 c^3}-\frac {19 b^3 \text {arctanh}(c x)}{60 c^6}+\frac {4 b^2 x^2 (a+b \text {arctanh}(c x))}{15 c^4}+\frac {b^2 x^4 (a+b \text {arctanh}(c x))}{20 c^2}+\frac {23 b (a+b \text {arctanh}(c x))^2}{30 c^6}+\frac {b x (a+b \text {arctanh}(c x))^2}{2 c^5}+\frac {b x^3 (a+b \text {arctanh}(c x))^2}{6 c^3}+\frac {b x^5 (a+b \text {arctanh}(c x))^2}{10 c}-\frac {(a+b \text {arctanh}(c x))^3}{6 c^6}+\frac {1}{6} x^6 (a+b \text {arctanh}(c x))^3-\frac {23 b^2 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{15 c^6}-\frac {23 b^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{30 c^6} \] Output:
19/60*b^3*x/c^5+1/60*b^3*x^3/c^3-19/60*b^3*arctanh(c*x)/c^6+4/15*b^2*x^2*( a+b*arctanh(c*x))/c^4+1/20*b^2*x^4*(a+b*arctanh(c*x))/c^2+23/30*b*(a+b*arc tanh(c*x))^2/c^6+1/2*b*x*(a+b*arctanh(c*x))^2/c^5+1/6*b*x^3*(a+b*arctanh(c *x))^2/c^3+1/10*b*x^5*(a+b*arctanh(c*x))^2/c-1/6*(a+b*arctanh(c*x))^3/c^6+ 1/6*x^6*(a+b*arctanh(c*x))^3-23/15*b^2*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/c ^6-23/30*b^3*polylog(2,1-2/(-c*x+1))/c^6
Time = 0.48 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.23 \[ \int x^5 (a+b \text {arctanh}(c x))^3 \, dx=\frac {-19 a b^2+30 a^2 b c x+19 b^3 c x+16 a b^2 c^2 x^2+10 a^2 b c^3 x^3+b^3 c^3 x^3+3 a b^2 c^4 x^4+6 a^2 b c^5 x^5+10 a^3 c^6 x^6+2 b^2 \left (b \left (-23+15 c x+5 c^3 x^3+3 c^5 x^5\right )+15 a \left (-1+c^6 x^6\right )\right ) \text {arctanh}(c x)^2+10 b^3 \left (-1+c^6 x^6\right ) \text {arctanh}(c x)^3+b \text {arctanh}(c x) \left (30 a^2 c^6 x^6+4 a b c x \left (15+5 c^2 x^2+3 c^4 x^4\right )+b^2 \left (-19+16 c^2 x^2+3 c^4 x^4\right )-92 b^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+15 a^2 b \log (1-c x)-15 a^2 b \log (1+c x)+46 a b^2 \log \left (1-c^2 x^2\right )+46 b^3 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )}{60 c^6} \] Input:
Integrate[x^5*(a + b*ArcTanh[c*x])^3,x]
Output:
(-19*a*b^2 + 30*a^2*b*c*x + 19*b^3*c*x + 16*a*b^2*c^2*x^2 + 10*a^2*b*c^3*x ^3 + b^3*c^3*x^3 + 3*a*b^2*c^4*x^4 + 6*a^2*b*c^5*x^5 + 10*a^3*c^6*x^6 + 2* b^2*(b*(-23 + 15*c*x + 5*c^3*x^3 + 3*c^5*x^5) + 15*a*(-1 + c^6*x^6))*ArcTa nh[c*x]^2 + 10*b^3*(-1 + c^6*x^6)*ArcTanh[c*x]^3 + b*ArcTanh[c*x]*(30*a^2* c^6*x^6 + 4*a*b*c*x*(15 + 5*c^2*x^2 + 3*c^4*x^4) + b^2*(-19 + 16*c^2*x^2 + 3*c^4*x^4) - 92*b^2*Log[1 + E^(-2*ArcTanh[c*x])]) + 15*a^2*b*Log[1 - c*x] - 15*a^2*b*Log[1 + c*x] + 46*a*b^2*Log[1 - c^2*x^2] + 46*b^3*PolyLog[2, - E^(-2*ArcTanh[c*x])])/(60*c^6)
Leaf count is larger than twice the leaf count of optimal. \(499\) vs. \(2(247)=494\).
Time = 3.53 (sec) , antiderivative size = 499, normalized size of antiderivative = 2.02, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.214, Rules used = {6452, 6542, 6452, 6542, 6452, 254, 2009, 6542, 6436, 6452, 262, 219, 6510, 6546, 6470, 2849, 2752}
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.
| \(\displaystyle \int x^5 (a+b \text {arctanh}(c x))^3 \, dx\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^3-\frac {1}{2} b c \int \frac {x^6 (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^3-\frac {1}{2} b c \left (\frac {\int \frac {x^4 (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {\int x^4 (a+b \text {arctanh}(c x))^2dx}{c^2}\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^3-\frac {1}{2} b c \left (\frac {\int \frac {x^4 (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2}{5} b c \int \frac {x^5 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}\right )\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^3-\frac {1}{2} b c \left (\frac {\frac {\int \frac {x^2 (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {\int x^2 (a+b \text {arctanh}(c x))^2dx}{c^2}}{c^2}-\frac {\frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2}{5} b c \left (\frac {\int \frac {x^3 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\int x^3 (a+b \text {arctanh}(c x))dx}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^3-\frac {1}{2} b c \left (\frac {\frac {\int \frac {x^2 (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}}{c^2}-\frac {\frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2}{5} b c \left (\frac {\int \frac {x^3 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))-\frac {1}{4} b c \int \frac {x^4}{1-c^2 x^2}dx}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 254 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^3-\frac {1}{2} b c \left (\frac {\frac {\int \frac {x^2 (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}}{c^2}-\frac {\frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2}{5} b c \left (\frac {\int \frac {x^3 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))-\frac {1}{4} b c \int \left (-\frac {x^2}{c^2}+\frac {1}{c^4 \left (1-c^2 x^2\right )}-\frac {1}{c^4}\right )dx}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^3-\frac {1}{2} b c \left (\frac {\frac {\int \frac {x^2 (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))^2-\frac {2}{3} b c \int \frac {x^3 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}}{c^2}-\frac {\frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2}{5} b c \left (\frac {\int \frac {x^3 (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))-\frac {1}{4} b c \left (\frac {\text {arctanh}(c x)}{c^5}-\frac {x}{c^4}-\frac {x^3}{3 c^2}\right )}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^3-\frac {1}{2} b c \left (\frac {\frac {\frac {\int \frac {(a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {\int (a+b \text {arctanh}(c x))^2dx}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))^2-\frac {2}{3} b c \left (\frac {\int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\int x (a+b \text {arctanh}(c x))dx}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2}{5} b c \left (\frac {\frac {\int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\int x (a+b \text {arctanh}(c x))dx}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))-\frac {1}{4} b c \left (\frac {\text {arctanh}(c x)}{c^5}-\frac {x}{c^4}-\frac {x^3}{3 c^2}\right )}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6436 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^3-\frac {1}{2} b c \left (\frac {\frac {\frac {\int \frac {(a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {x (a+b \text {arctanh}(c x))^2-2 b c \int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))^2-\frac {2}{3} b c \left (\frac {\int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\int x (a+b \text {arctanh}(c x))dx}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2}{5} b c \left (\frac {\frac {\int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\int x (a+b \text {arctanh}(c x))dx}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))-\frac {1}{4} b c \left (\frac {\text {arctanh}(c x)}{c^5}-\frac {x}{c^4}-\frac {x^3}{3 c^2}\right )}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^3-\frac {1}{2} b c \left (\frac {\frac {\frac {\int \frac {(a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {x (a+b \text {arctanh}(c x))^2-2 b c \int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))^2-\frac {2}{3} b c \left (\frac {\int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \int \frac {x^2}{1-c^2 x^2}dx}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2}{5} b c \left (\frac {\frac {\int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \int \frac {x^2}{1-c^2 x^2}dx}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))-\frac {1}{4} b c \left (\frac {\text {arctanh}(c x)}{c^5}-\frac {x}{c^4}-\frac {x^3}{3 c^2}\right )}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^3-\frac {1}{2} b c \left (\frac {\frac {\frac {\int \frac {(a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {x (a+b \text {arctanh}(c x))^2-2 b c \int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))^2-\frac {2}{3} b c \left (\frac {\int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\int \frac {1}{1-c^2 x^2}dx}{c^2}-\frac {x}{c^2}\right )}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2}{5} b c \left (\frac {\frac {\int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\int \frac {1}{1-c^2 x^2}dx}{c^2}-\frac {x}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))-\frac {1}{4} b c \left (\frac {\text {arctanh}(c x)}{c^5}-\frac {x}{c^4}-\frac {x^3}{3 c^2}\right )}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^3-\frac {1}{2} b c \left (\frac {\frac {\frac {\int \frac {(a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {x (a+b \text {arctanh}(c x))^2-2 b c \int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))^2-\frac {2}{3} b c \left (\frac {\int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2}{5} b c \left (\frac {\frac {\int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))-\frac {1}{4} b c \left (\frac {\text {arctanh}(c x)}{c^5}-\frac {x}{c^4}-\frac {x^3}{3 c^2}\right )}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6510 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^3-\frac {1}{2} b c \left (\frac {\frac {\frac {(a+b \text {arctanh}(c x))^3}{3 b c^3}-\frac {x (a+b \text {arctanh}(c x))^2-2 b c \int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))^2-\frac {2}{3} b c \left (\frac {\int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2}{5} b c \left (\frac {\frac {\int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))-\frac {1}{4} b c \left (\frac {\text {arctanh}(c x)}{c^5}-\frac {x}{c^4}-\frac {x^3}{3 c^2}\right )}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^3-\frac {1}{2} b c \left (\frac {\frac {\frac {(a+b \text {arctanh}(c x))^3}{3 b c^3}-\frac {x (a+b \text {arctanh}(c x))^2-2 b c \left (\frac {\int \frac {a+b \text {arctanh}(c x)}{1-c x}dx}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))^2-\frac {2}{3} b c \left (\frac {\frac {\int \frac {a+b \text {arctanh}(c x)}{1-c x}dx}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2}{5} b c \left (\frac {\frac {\frac {\int \frac {a+b \text {arctanh}(c x)}{1-c x}dx}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))-\frac {1}{4} b c \left (\frac {\text {arctanh}(c x)}{c^5}-\frac {x}{c^4}-\frac {x^3}{3 c^2}\right )}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^3-\frac {1}{2} b c \left (\frac {\frac {\frac {(a+b \text {arctanh}(c x))^3}{3 b c^3}-\frac {x (a+b \text {arctanh}(c x))^2-2 b c \left (\frac {\frac {\log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c}-b \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2}dx}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))^2-\frac {2}{3} b c \left (\frac {\frac {\frac {\log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c}-b \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2}dx}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2}{5} b c \left (\frac {\frac {\frac {\frac {\log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c}-b \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2}dx}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))-\frac {1}{4} b c \left (\frac {\text {arctanh}(c x)}{c^5}-\frac {x}{c^4}-\frac {x^3}{3 c^2}\right )}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^3-\frac {1}{2} b c \left (\frac {\frac {\frac {(a+b \text {arctanh}(c x))^3}{3 b c^3}-\frac {x (a+b \text {arctanh}(c x))^2-2 b c \left (\frac {\frac {b \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-\frac {2}{1-c x}}d\frac {1}{1-c x}}{c}+\frac {\log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c}}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))^2-\frac {2}{3} b c \left (\frac {\frac {\frac {b \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-\frac {2}{1-c x}}d\frac {1}{1-c x}}{c}+\frac {\log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c}}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2}{5} b c \left (\frac {\frac {\frac {\frac {b \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-\frac {2}{1-c x}}d\frac {1}{1-c x}}{c}+\frac {\log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c}}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))-\frac {1}{4} b c \left (\frac {\text {arctanh}(c x)}{c^5}-\frac {x}{c^4}-\frac {x^3}{3 c^2}\right )}{c^2}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))^3-\frac {1}{2} b c \left (\frac {\frac {\frac {(a+b \text {arctanh}(c x))^3}{3 b c^3}-\frac {x (a+b \text {arctanh}(c x))^2-2 b c \left (\frac {\frac {\log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{2 c}}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{3} x^3 (a+b \text {arctanh}(c x))^2-\frac {2}{3} b c \left (\frac {\frac {\frac {\log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{2 c}}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{5} x^5 (a+b \text {arctanh}(c x))^2-\frac {2}{5} b c \left (\frac {\frac {\frac {\frac {\log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{2 c}}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x^2 (a+b \text {arctanh}(c x))-\frac {1}{2} b c \left (\frac {\text {arctanh}(c x)}{c^3}-\frac {x}{c^2}\right )}{c^2}}{c^2}-\frac {\frac {1}{4} x^4 (a+b \text {arctanh}(c x))-\frac {1}{4} b c \left (\frac {\text {arctanh}(c x)}{c^5}-\frac {x}{c^4}-\frac {x^3}{3 c^2}\right )}{c^2}\right )}{c^2}\right )\) |
Input:
Int[x^5*(a + b*ArcTanh[c*x])^3,x]
Output:
(x^6*(a + b*ArcTanh[c*x])^3)/6 - (b*c*(-(((x^5*(a + b*ArcTanh[c*x])^2)/5 - (2*b*c*(-(((x^4*(a + b*ArcTanh[c*x]))/4 - (b*c*(-(x/c^4) - x^3/(3*c^2) + ArcTanh[c*x]/c^5))/4)/c^2) + (-(((x^2*(a + b*ArcTanh[c*x]))/2 - (b*c*(-(x/ c^2) + ArcTanh[c*x]/c^3))/2)/c^2) + (-1/2*(a + b*ArcTanh[c*x])^2/(b*c^2) + (((a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/c + (b*PolyLog[2, 1 - 2/(1 - c*x )])/(2*c))/c)/c^2)/c^2))/5)/c^2) + (-(((x^3*(a + b*ArcTanh[c*x])^2)/3 - (2 *b*c*(-(((x^2*(a + b*ArcTanh[c*x]))/2 - (b*c*(-(x/c^2) + ArcTanh[c*x]/c^3) )/2)/c^2) + (-1/2*(a + b*ArcTanh[c*x])^2/(b*c^2) + (((a + b*ArcTanh[c*x])* Log[2/(1 - c*x)])/c + (b*PolyLog[2, 1 - 2/(1 - c*x)])/(2*c))/c)/c^2))/3)/c ^2) + ((a + b*ArcTanh[c*x])^3/(3*b*c^3) - (x*(a + b*ArcTanh[c*x])^2 - 2*b* c*(-1/2*(a + b*ArcTanh[c*x])^2/(b*c^2) + (((a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/c + (b*PolyLog[2, 1 - 2/(1 - c*x)])/(2*c))/c))/c^2)/c^2)/c^2))/2
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), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp [-e/g Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ {c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
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_)^(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]
Leaf count of result is larger than twice the leaf count of optimal. \(802\) vs. \(2(221)=442\).
Time = 9.29 (sec) , antiderivative size = 803, normalized size of antiderivative = 3.25
| method | result | size |
| risch | \(-\frac {19 b^{3} \ln \left (-c x -1\right )}{120 c^{6}}+\frac {a^{3} x^{6}}{6}+\left (\frac {b^{3} \left (c^{6} x^{6}-1\right ) \ln \left (-c x +1\right )^{2}}{16 c^{6}}-\frac {b^{2} x \left (15 a \,c^{5} x^{5}+3 b \,c^{4} x^{4}+5 b \,c^{2} x^{2}+15 b \right ) \ln \left (-c x +1\right )}{60 c^{5}}-\frac {b \left (-30 a^{2} c^{6} x^{6}-12 a b \,c^{5} x^{5}-3 b^{2} c^{4} x^{4}-20 a b \,c^{3} x^{3}-16 b^{2} c^{2} x^{2}-60 a b c x -30 b \ln \left (-c x +1\right ) a -46 b^{2} \ln \left (-c x +1\right )\right )}{120 c^{6}}\right ) \ln \left (c x +1\right )-\frac {a \,b^{2} \ln \left (-c x +1\right ) x^{5}}{10 c}-\frac {a^{3}}{6 c^{6}}+\frac {19 b^{3} x}{60 c^{5}}+\frac {b^{3} x^{3}}{60 c^{3}}+\frac {a \,b^{2} x^{4}}{20 c^{2}}+\frac {4 a \,b^{2} x^{2}}{15 c^{4}}+\frac {a^{2} b \,x^{5}}{10 c}+\frac {a^{2} b \,x^{3}}{6 c^{3}}+\frac {a^{2} b x}{2 c^{5}}-\frac {a \,b^{2} \ln \left (-c x +1\right )^{2}}{8 c^{6}}-\frac {19 a \,b^{2}}{60 c^{6}}-\frac {23 a^{2} b}{30 c^{6}}-\frac {b^{3}}{3 c^{6}}-\frac {b^{3} \ln \left (-c x +1\right )^{3} x^{6}}{48}+\frac {b^{3} \ln \left (-c x +1\right )^{3}}{48 c^{6}}-\frac {23 b^{3} \ln \left (-c x +1\right )^{2}}{120 c^{6}}+\frac {19 b^{3} \ln \left (-c x +1\right )}{120 c^{6}}-\frac {a \,b^{2} \ln \left (-c x +1\right ) x^{3}}{6 c^{3}}-\frac {a \,b^{2} \ln \left (-c x +1\right ) x}{2 c^{5}}+\frac {23 a \,b^{2} \ln \left (-c x +1\right )}{30 c^{6}}+\frac {a^{2} b \ln \left (-c x +1\right )}{4 c^{6}}+\frac {b^{3} \ln \left (-c x +1\right )^{2} x^{5}}{40 c}+\frac {b^{3} \ln \left (-c x +1\right )^{2} x^{3}}{24 c^{3}}+\frac {b^{3} \ln \left (-c x +1\right )^{2} x}{8 c^{5}}-\frac {b^{3} \ln \left (-c x +1\right ) x^{4}}{40 c^{2}}-\frac {2 b^{3} \ln \left (-c x +1\right ) x^{2}}{15 c^{4}}+\frac {a \,b^{2} \ln \left (-c x +1\right )^{2} x^{6}}{8}-\frac {a^{2} b \ln \left (-c x +1\right ) x^{6}}{4}+\frac {23 b^{3} \operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right )}{30 c^{6}}+\frac {23 b^{2} \ln \left (-c x -1\right ) a}{30 c^{6}}-\frac {b \ln \left (-c x -1\right ) a^{2}}{4 c^{6}}-\frac {23 b^{3} \ln \left (-c x +1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{30 c^{6}}+\frac {23 b^{3} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{30 c^{6}}+\frac {b^{3} \left (c^{6} x^{6}-1\right ) \ln \left (c x +1\right )^{3}}{48 c^{6}}+\frac {b^{2} \left (-15 b \,x^{6} \ln \left (-c x +1\right ) c^{6}+30 a \,c^{6} x^{6}+6 b \,c^{5} x^{5}+10 b \,c^{3} x^{3}+30 b c x +15 b \ln \left (-c x +1\right )-30 a +46 b \right ) \ln \left (c x +1\right )^{2}}{240 c^{6}}\) | \(803\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1167\) |
| default | \(\text {Expression too large to display}\) | \(1167\) |
| parts | \(\text {Expression too large to display}\) | \(1230\) |
Input:
int(x^5*(a+b*arctanh(c*x))^3,x,method=_RETURNVERBOSE)
Output:
-19/120/c^6*b^3*ln(-c*x-1)+1/6*a^3*x^6+(1/16*b^3*(c^6*x^6-1)/c^6*ln(-c*x+1 )^2-1/60*b^2*x*(15*a*c^5*x^5+3*b*c^4*x^4+5*b*c^2*x^2+15*b)/c^5*ln(-c*x+1)- 1/120*b*(-30*a^2*c^6*x^6-12*a*b*c^5*x^5-3*b^2*c^4*x^4-20*a*b*c^3*x^3-16*b^ 2*c^2*x^2-60*a*b*c*x-30*b*ln(-c*x+1)*a-46*b^2*ln(-c*x+1))/c^6)*ln(c*x+1)-1 /10/c*a*b^2*ln(-c*x+1)*x^5-1/6/c^6*a^3+19/60*b^3*x/c^5+1/60*b^3*x^3/c^3+1/ 20/c^2*a*b^2*x^4+4/15/c^4*a*b^2*x^2+1/10/c*a^2*b*x^5+1/6/c^3*a^2*b*x^3+1/2 /c^5*a^2*b*x-1/8/c^6*a*b^2*ln(-c*x+1)^2-19/60/c^6*a*b^2-23/30/c^6*a^2*b-1/ 3*b^3/c^6-1/48*b^3*ln(-c*x+1)^3*x^6+1/48/c^6*b^3*ln(-c*x+1)^3-23/120/c^6*b ^3*ln(-c*x+1)^2+19/120/c^6*b^3*ln(-c*x+1)-1/6/c^3*a*b^2*ln(-c*x+1)*x^3-1/2 /c^5*a*b^2*ln(-c*x+1)*x+23/30/c^6*a*b^2*ln(-c*x+1)+1/4/c^6*a^2*b*ln(-c*x+1 )+1/40/c*b^3*ln(-c*x+1)^2*x^5+1/24/c^3*b^3*ln(-c*x+1)^2*x^3+1/8/c^5*b^3*ln (-c*x+1)^2*x-1/40/c^2*b^3*ln(-c*x+1)*x^4-2/15/c^4*b^3*ln(-c*x+1)*x^2+1/8*a *b^2*ln(-c*x+1)^2*x^6-1/4*a^2*b*ln(-c*x+1)*x^6+23/30*b^3/c^6*dilog(-1/2*c* x+1/2)+23/30*b^2/c^6*ln(-c*x-1)*a-1/4*b/c^6*ln(-c*x-1)*a^2-23/30*b^3/c^6*l n(-c*x+1)*ln(1/2*c*x+1/2)+23/30*b^3/c^6*ln(-1/2*c*x+1/2)*ln(1/2*c*x+1/2)+1 /48*b^3*(c^6*x^6-1)/c^6*ln(c*x+1)^3+1/240*b^2*(-15*b*x^6*ln(-c*x+1)*c^6+30 *a*c^6*x^6+6*b*c^5*x^5+10*b*c^3*x^3+30*b*c*x+15*b*ln(-c*x+1)-30*a+46*b)/c^ 6*ln(c*x+1)^2
\[ \int x^5 (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} x^{5} \,d x } \] Input:
integrate(x^5*(a+b*arctanh(c*x))^3,x, algorithm="fricas")
Output:
integral(b^3*x^5*arctanh(c*x)^3 + 3*a*b^2*x^5*arctanh(c*x)^2 + 3*a^2*b*x^5 *arctanh(c*x) + a^3*x^5, x)
\[ \int x^5 (a+b \text {arctanh}(c x))^3 \, dx=\int x^{5} \left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3}\, dx \] Input:
integrate(x**5*(a+b*atanh(c*x))**3,x)
Output:
Integral(x**5*(a + b*atanh(c*x))**3, x)
\[ \int x^5 (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} x^{5} \,d x } \] Input:
integrate(x^5*(a+b*arctanh(c*x))^3,x, algorithm="maxima")
Output:
1/2*a*b^2*x^6*arctanh(c*x)^2 + 1/6*a^3*x^6 + 1/60*(30*x^6*arctanh(c*x) + c *(2*(3*c^4*x^5 + 5*c^2*x^3 + 15*x)/c^6 - 15*log(c*x + 1)/c^7 + 15*log(c*x - 1)/c^7))*a^2*b + 1/120*(4*c*(2*(3*c^4*x^5 + 5*c^2*x^3 + 15*x)/c^6 - 15*l og(c*x + 1)/c^7 + 15*log(c*x - 1)/c^7)*arctanh(c*x) + (6*c^4*x^4 + 32*c^2* x^2 - 2*(15*log(c*x - 1) - 46)*log(c*x + 1) + 15*log(c*x + 1)^2 + 15*log(c *x - 1)^2 + 92*log(c*x - 1))/c^6)*a*b^2 - 1/1728000*(500*c^7*((2*c^4*x^6 + 3*c^2*x^4 + 6*x^2)/c^11 + 6*log(c^2*x^2 - 1)/c^13) + 728*c^6*(2*(3*c^4*x^ 5 + 5*c^2*x^3 + 15*x)/c^11 - 15*log(c*x + 1)/c^12 + 15*log(c*x - 1)/c^12) + 1485*c^5*((c^2*x^4 + 2*x^2)/c^9 + 2*log(c^2*x^2 - 1)/c^11) - 622080000*c ^5*integrate(1/3600*x^5*log(c*x + 1)/(c^7*x^2 - c^5), x) + 9750*c^4*(2*(c^ 2*x^3 + 3*x)/c^9 - 3*log(c*x + 1)/c^10 + 3*log(c*x - 1)/c^10) - 2700*c^3*( x^2/c^7 + log(c^2*x^2 - 1)/c^9) - 1036800000*c^3*integrate(1/3600*x^3*log( c*x + 1)/(c^7*x^2 - c^5), x) + 227700*c^2*(2*x/c^7 - log(c*x + 1)/c^8 + lo g(c*x - 1)/c^8) - 5495040000*c*integrate(1/3600*x*log(c*x + 1)/(c^7*x^2 - c^5), x) + (1000*(36*log(-c*x + 1)^3 - 18*log(-c*x + 1)^2 + 6*log(-c*x + 1 ) - 1)*(c*x - 1)^6 + 1728*(125*log(-c*x + 1)^3 - 75*log(-c*x + 1)^2 + 30*l og(-c*x + 1) - 6)*(c*x - 1)^5 + 16875*(32*log(-c*x + 1)^3 - 24*log(-c*x + 1)^2 + 12*log(-c*x + 1) - 3)*(c*x - 1)^4 + 80000*(9*log(-c*x + 1)^3 - 9*lo g(-c*x + 1)^2 + 6*log(-c*x + 1) - 2)*(c*x - 1)^3 + 135000*(4*log(-c*x + 1) ^3 - 6*log(-c*x + 1)^2 + 6*log(-c*x + 1) - 3)*(c*x - 1)^2 + 216000*(log...
\[ \int x^5 (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} x^{5} \,d x } \] Input:
integrate(x^5*(a+b*arctanh(c*x))^3,x, algorithm="giac")
Output:
integrate((b*arctanh(c*x) + a)^3*x^5, x)
Timed out. \[ \int x^5 (a+b \text {arctanh}(c x))^3 \, dx=\int x^5\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3 \,d x \] Input:
int(x^5*(a + b*atanh(c*x))^3,x)
Output:
int(x^5*(a + b*atanh(c*x))^3, x)
\[ \int x^5 (a+b \text {arctanh}(c x))^3 \, dx=\frac {-10 \mathit {atanh} \left (c x \right )^{3} b^{3}-19 \mathit {atanh} \left (c x \right ) b^{3}+30 \mathit {atanh} \left (c x \right )^{2} a \,b^{2} c^{6} x^{6}+30 \mathit {atanh} \left (c x \right ) a^{2} b \,c^{6} x^{6}+12 \mathit {atanh} \left (c x \right ) a \,b^{2} c^{5} x^{5}+20 \mathit {atanh} \left (c x \right ) a \,b^{2} c^{3} x^{3}+60 \mathit {atanh} \left (c x \right ) a \,b^{2} c x +b^{3} c^{3} x^{3}+10 \mathit {atanh} \left (c x \right )^{3} b^{3} c^{6} x^{6}+6 \mathit {atanh} \left (c x \right )^{2} b^{3} c^{5} x^{5}+10 \mathit {atanh} \left (c x \right )^{2} b^{3} c^{3} x^{3}+30 \mathit {atanh} \left (c x \right )^{2} b^{3} c x +3 \mathit {atanh} \left (c x \right ) b^{3} c^{4} x^{4}+16 \mathit {atanh} \left (c x \right ) b^{3} c^{2} x^{2}+6 a^{2} b \,c^{5} x^{5}+10 a^{2} b \,c^{3} x^{3}+30 a^{2} b c x +3 a \,b^{2} c^{4} x^{4}+16 a \,b^{2} c^{2} x^{2}+92 \left (\int \frac {\mathit {atanh} \left (c x \right ) x}{c^{2} x^{2}-1}d x \right ) b^{3} c^{2}-30 \mathit {atanh} \left (c x \right )^{2} a \,b^{2}-30 \mathit {atanh} \left (c x \right ) a^{2} b +92 \mathit {atanh} \left (c x \right ) a \,b^{2}+92 \,\mathrm {log}\left (c^{2} x -c \right ) a \,b^{2}+10 a^{3} c^{6} x^{6}+19 b^{3} c x}{60 c^{6}} \] Input:
int(x^5*(a+b*atanh(c*x))^3,x)
Output:
(10*atanh(c*x)**3*b**3*c**6*x**6 - 10*atanh(c*x)**3*b**3 + 30*atanh(c*x)** 2*a*b**2*c**6*x**6 - 30*atanh(c*x)**2*a*b**2 + 6*atanh(c*x)**2*b**3*c**5*x **5 + 10*atanh(c*x)**2*b**3*c**3*x**3 + 30*atanh(c*x)**2*b**3*c*x + 30*ata nh(c*x)*a**2*b*c**6*x**6 - 30*atanh(c*x)*a**2*b + 12*atanh(c*x)*a*b**2*c** 5*x**5 + 20*atanh(c*x)*a*b**2*c**3*x**3 + 60*atanh(c*x)*a*b**2*c*x + 92*at anh(c*x)*a*b**2 + 3*atanh(c*x)*b**3*c**4*x**4 + 16*atanh(c*x)*b**3*c**2*x* *2 - 19*atanh(c*x)*b**3 + 92*int((atanh(c*x)*x)/(c**2*x**2 - 1),x)*b**3*c* *2 + 92*log(c**2*x - c)*a*b**2 + 10*a**3*c**6*x**6 + 6*a**2*b*c**5*x**5 + 10*a**2*b*c**3*x**3 + 30*a**2*b*c*x + 3*a*b**2*c**4*x**4 + 16*a*b**2*c**2* x**2 + b**3*c**3*x**3 + 19*b**3*c*x)/(60*c**6)