Integrand size = 10, antiderivative size = 186 \[ \int x^5 \coth ^{-1}(a x)^3 \, dx=\frac {19 x}{60 a^5}+\frac {x^3}{60 a^3}+\frac {4 x^2 \coth ^{-1}(a x)}{15 a^4}+\frac {x^4 \coth ^{-1}(a x)}{20 a^2}+\frac {23 \coth ^{-1}(a x)^2}{30 a^6}+\frac {x \coth ^{-1}(a x)^2}{2 a^5}+\frac {x^3 \coth ^{-1}(a x)^2}{6 a^3}+\frac {x^5 \coth ^{-1}(a x)^2}{10 a}-\frac {\coth ^{-1}(a x)^3}{6 a^6}+\frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {19 \text {arctanh}(a x)}{60 a^6}-\frac {23 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{15 a^6}-\frac {23 \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{30 a^6} \]
19/60*x/a^5+1/60*x^3/a^3+4/15*x^2*arccoth(a*x)/a^4+1/20*x^4*arccoth(a*x)/a ^2+23/30*arccoth(a*x)^2/a^6+1/2*x*arccoth(a*x)^2/a^5+1/6*x^3*arccoth(a*x)^ 2/a^3+1/10*x^5*arccoth(a*x)^2/a-1/6*arccoth(a*x)^3/a^6+1/6*x^6*arccoth(a*x )^3-19/60*arctanh(a*x)/a^6-23/15*arccoth(a*x)*ln(2/(-a*x+1))/a^6-23/30*pol ylog(2,1-2/(-a*x+1))/a^6
Time = 0.39 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.63 \[ \int x^5 \coth ^{-1}(a x)^3 \, dx=\frac {a x \left (19+a^2 x^2\right )+2 \left (-23+15 a x+5 a^3 x^3+3 a^5 x^5\right ) \coth ^{-1}(a x)^2+10 \left (-1+a^6 x^6\right ) \coth ^{-1}(a x)^3+\coth ^{-1}(a x) \left (-19+16 a^2 x^2+3 a^4 x^4-92 \log \left (1-e^{-2 \coth ^{-1}(a x)}\right )\right )+46 \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(a x)}\right )}{60 a^6} \]
(a*x*(19 + a^2*x^2) + 2*(-23 + 15*a*x + 5*a^3*x^3 + 3*a^5*x^5)*ArcCoth[a*x ]^2 + 10*(-1 + a^6*x^6)*ArcCoth[a*x]^3 + ArcCoth[a*x]*(-19 + 16*a^2*x^2 + 3*a^4*x^4 - 92*Log[1 - E^(-2*ArcCoth[a*x])]) + 46*PolyLog[2, E^(-2*ArcCoth [a*x])])/(60*a^6)
Leaf count is larger than twice the leaf count of optimal. \(421\) vs. \(2(186)=372\).
Time = 2.90 (sec) , antiderivative size = 421, normalized size of antiderivative = 2.26, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.700, Rules used = {6453, 6543, 6453, 6543, 6453, 254, 2009, 6543, 6437, 6453, 262, 219, 6511, 6547, 6471, 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.
| \(\displaystyle \int x^5 \coth ^{-1}(a x)^3 \, dx\) |
| \(\Big \downarrow \) 6453 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {1}{2} a \int \frac {x^6 \coth ^{-1}(a x)^2}{1-a^2 x^2}dx\) |
| \(\Big \downarrow \) 6543 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {1}{2} a \left (\frac {\int \frac {x^4 \coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\int x^4 \coth ^{-1}(a x)^2dx}{a^2}\right )\) |
| \(\Big \downarrow \) 6453 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {1}{2} a \left (\frac {\int \frac {x^4 \coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2}{5} a \int \frac {x^5 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}\right )\) |
| \(\Big \downarrow \) 6543 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {1}{2} a \left (\frac {\frac {\int \frac {x^2 \coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\int x^2 \coth ^{-1}(a x)^2dx}{a^2}}{a^2}-\frac {\frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2}{5} a \left (\frac {\int \frac {x^3 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\int x^3 \coth ^{-1}(a x)dx}{a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 6453 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {1}{2} a \left (\frac {\frac {\int \frac {x^2 \coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)^2-\frac {2}{3} a \int \frac {x^3 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}}{a^2}-\frac {\frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2}{5} a \left (\frac {\int \frac {x^3 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {1}{4} a \int \frac {x^4}{1-a^2 x^2}dx}{a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 254 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {1}{2} a \left (\frac {\frac {\int \frac {x^2 \coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)^2-\frac {2}{3} a \int \frac {x^3 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}}{a^2}-\frac {\frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2}{5} a \left (\frac {\int \frac {x^3 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {1}{4} a \int \left (-\frac {x^2}{a^2}+\frac {1}{a^4 \left (1-a^2 x^2\right )}-\frac {1}{a^4}\right )dx}{a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {1}{2} a \left (\frac {\frac {\int \frac {x^2 \coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)^2-\frac {2}{3} a \int \frac {x^3 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}}{a^2}-\frac {\frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2}{5} a \left (\frac {\int \frac {x^3 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 6543 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {1}{2} a \left (\frac {\frac {\frac {\int \frac {\coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {\int \coth ^{-1}(a x)^2dx}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)^2-\frac {2}{3} a \left (\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\int x \coth ^{-1}(a x)dx}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2}{5} a \left (\frac {\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\int x \coth ^{-1}(a x)dx}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 6437 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {1}{2} a \left (\frac {\frac {\frac {\int \frac {\coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {x \coth ^{-1}(a x)^2-2 a \int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)^2-\frac {2}{3} a \left (\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\int x \coth ^{-1}(a x)dx}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2}{5} a \left (\frac {\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\int x \coth ^{-1}(a x)dx}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 6453 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {1}{2} a \left (\frac {\frac {\frac {\int \frac {\coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {x \coth ^{-1}(a x)^2-2 a \int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)^2-\frac {2}{3} a \left (\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \int \frac {x^2}{1-a^2 x^2}dx}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2}{5} a \left (\frac {\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \int \frac {x^2}{1-a^2 x^2}dx}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {1}{2} a \left (\frac {\frac {\frac {\int \frac {\coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {x \coth ^{-1}(a x)^2-2 a \int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)^2-\frac {2}{3} a \left (\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \left (\frac {\int \frac {1}{1-a^2 x^2}dx}{a^2}-\frac {x}{a^2}\right )}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2}{5} a \left (\frac {\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \left (\frac {\int \frac {1}{1-a^2 x^2}dx}{a^2}-\frac {x}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {1}{2} a \left (\frac {\frac {\frac {\int \frac {\coth ^{-1}(a x)^2}{1-a^2 x^2}dx}{a^2}-\frac {x \coth ^{-1}(a x)^2-2 a \int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)^2-\frac {2}{3} a \left (\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2}{5} a \left (\frac {\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 6511 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {1}{2} a \left (\frac {\frac {\frac {\coth ^{-1}(a x)^3}{3 a^3}-\frac {x \coth ^{-1}(a x)^2-2 a \int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)^2-\frac {2}{3} a \left (\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2}{5} a \left (\frac {\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 6547 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {1}{2} a \left (\frac {\frac {\frac {\coth ^{-1}(a x)^3}{3 a^3}-\frac {x \coth ^{-1}(a x)^2-2 a \left (\frac {\int \frac {\coth ^{-1}(a x)}{1-a x}dx}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)^2-\frac {2}{3} a \left (\frac {\frac {\int \frac {\coth ^{-1}(a x)}{1-a x}dx}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2}{5} a \left (\frac {\frac {\frac {\int \frac {\coth ^{-1}(a x)}{1-a x}dx}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 6471 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {1}{2} a \left (\frac {\frac {\frac {\coth ^{-1}(a x)^3}{3 a^3}-\frac {x \coth ^{-1}(a x)^2-2 a \left (\frac {\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a}-\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)^2-\frac {2}{3} a \left (\frac {\frac {\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a}-\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2}{5} a \left (\frac {\frac {\frac {\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a}-\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {1}{2} a \left (\frac {\frac {\frac {\coth ^{-1}(a x)^3}{3 a^3}-\frac {x \coth ^{-1}(a x)^2-2 a \left (\frac {\frac {\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-\frac {2}{1-a x}}d\frac {1}{1-a x}}{a}+\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a}}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)^2-\frac {2}{3} a \left (\frac {\frac {\frac {\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-\frac {2}{1-a x}}d\frac {1}{1-a x}}{a}+\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a}}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2}{5} a \left (\frac {\frac {\frac {\frac {\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-\frac {2}{1-a x}}d\frac {1}{1-a x}}{a}+\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a}}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {1}{2} a \left (\frac {\frac {\frac {\coth ^{-1}(a x)^3}{3 a^3}-\frac {x \coth ^{-1}(a x)^2-2 a \left (\frac {\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}+\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a}}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{3} x^3 \coth ^{-1}(a x)^2-\frac {2}{3} a \left (\frac {\frac {\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}+\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a}}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2}{5} a \left (\frac {\frac {\frac {\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}+\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a}}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )}{a^2}\right )\) |
(x^6*ArcCoth[a*x]^3)/6 - (a*(-(((x^5*ArcCoth[a*x]^2)/5 - (2*a*(-(((x^4*Arc Coth[a*x])/4 - (a*(-(x/a^4) - x^3/(3*a^2) + ArcTanh[a*x]/a^5))/4)/a^2) + ( -(((x^2*ArcCoth[a*x])/2 - (a*(-(x/a^2) + ArcTanh[a*x]/a^3))/2)/a^2) + (-1/ 2*ArcCoth[a*x]^2/a^2 + ((ArcCoth[a*x]*Log[2/(1 - a*x)])/a + PolyLog[2, 1 - 2/(1 - a*x)]/(2*a))/a)/a^2)/a^2))/5)/a^2) + (-(((x^3*ArcCoth[a*x]^2)/3 - (2*a*(-(((x^2*ArcCoth[a*x])/2 - (a*(-(x/a^2) + ArcTanh[a*x]/a^3))/2)/a^2) + (-1/2*ArcCoth[a*x]^2/a^2 + ((ArcCoth[a*x]*Log[2/(1 - a*x)])/a + PolyLog[ 2, 1 - 2/(1 - a*x)]/(2*a))/a)/a^2))/3)/a^2) + (ArcCoth[a*x]^3/(3*a^3) - (x *ArcCoth[a*x]^2 - 2*a*(-1/2*ArcCoth[a*x]^2/a^2 + ((ArcCoth[a*x]*Log[2/(1 - a*x)])/a + PolyLog[2, 1 - 2/(1 - a*x)]/(2*a))/a))/a^2)/a^2)/a^2))/2
3.1.23.3.1 Defintions of rubi rules used
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_.) + ArcCoth[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCoth[c*x^n])^p, x] - Simp[b*c*n*p Int[x^n*((a + b*ArcCoth[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_.) + ArcCoth[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : > Simp[x^(m + 1)*((a + b*ArcCoth[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcCoth[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_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol ] :> Simp[(-(a + b*ArcCoth[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c *(p/e) Int[(a + b*ArcCoth[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_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[(a + b*ArcCoth[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_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( e_.)*(x_)^2), x_Symbol] :> Simp[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcCoth[c* x])^p, x], x] - Simp[d*(f^2/e) Int[(f*x)^(m - 2)*((a + b*ArcCoth[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_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcCoth[c*x])^(p + 1)/(b*e*(p + 1)), x] + Simp[1/ (c*d) Int[(a + b*ArcCoth[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 5.82 (sec) , antiderivative size = 2133, normalized size of antiderivative = 11.47
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(2133\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(2135\) |
| default | \(\text {Expression too large to display}\) | \(2135\) |
1/6*x^6*arccoth(a*x)^3+1/2/a^6*(arccoth(a*x)^2*a*x+3/20*(2*((a*x-1)/(a*x+1 ))^(1/2)*a^2*x^2+2*((a*x-1)/(a*x+1))^(1/2)*a*x-2*a^2*x^2+1)*arccoth(a*x)*( a*x+1)*(a*x-1)-3/20*(2*((a*x-1)/(a*x+1))^(1/2)*a^2*x^2+2*((a*x-1)/(a*x+1)) ^(1/2)*a*x+2*a^2*x^2-1)*arccoth(a*x)*(a*x+1)*(a*x-1)+1/10*(2*((a*x-1)/(a*x +1))^(1/2)*a^2*x^2+2*a^2*x^2-((a*x-1)/(a*x+1))^(1/2)-2*a*x)*(a*x-1)*arccot h(a*x)*(a*x+1)+1/3*a^3*x^3*arccoth(a*x)^2+1/5*a^5*x^5*arccoth(a*x)^2-1/4*I *Pi*csgn(I/((a*x+1)/(a*x-1)-1))*csgn(I*(a*x+1)/(a*x-1))*csgn(I/(a*x-1)*(a* x+1)/((a*x+1)/(a*x-1)-1))*arccoth(a*x)^2-46/15*arccoth(a*x)*ln(1+1/((a*x-1 )/(a*x+1))^(1/2))-1/3*arccoth(a*x)^3-41/120*(((a*x-1)/(a*x+1))^(1/2)*a*x+( (a*x-1)/(a*x+1))^(1/2)-a*x-1)*arccoth(a*x)-1/40*(a*x-1)/(((a*x-1)/(a*x+1)) ^(1/2)*a*x+((a*x-1)/(a*x+1))^(1/2)-a*x)-41/60/(((a*x-1)/(a*x+1))^(1/2)-1)* ((a*x-1)/(a*x+1))^(1/2)+41/120*(((a*x-1)/(a*x+1))^(1/2)*a*x+((a*x-1)/(a*x+ 1))^(1/2)+a*x+1)*arccoth(a*x)-41/60/(((a*x-1)/(a*x+1))^(1/2)+1)*((a*x-1)/( a*x+1))^(1/2)+1/40*(a*x-1)/(((a*x-1)/(a*x+1))^(1/2)*a*x+((a*x-1)/(a*x+1))^ (1/2)+a*x)+23/15*arccoth(a*x)^2-1/4*I*Pi*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/( a*x-1)-1))^3*arccoth(a*x)^2+1/4*I*Pi*csgn(I/((a*x+1)/(a*x-1)-1))*csgn(I/(a *x-1)*(a*x+1)/((a*x+1)/(a*x-1)-1))^2*arccoth(a*x)^2+1/2*I*Pi*csgn(I/((a*x- 1)/(a*x+1))^(1/2))*csgn(I*(a*x+1)/(a*x-1))^2*arccoth(a*x)^2-1/4*I*Pi*csgn( I/((a*x-1)/(a*x+1))^(1/2))^2*csgn(I*(a*x+1)/(a*x-1))*arccoth(a*x)^2+1/4*I* Pi*csgn(I*(a*x+1)/(a*x-1))*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/(a*x-1)-1))^...
\[ \int x^5 \coth ^{-1}(a x)^3 \, dx=\int { x^{5} \operatorname {arcoth}\left (a x\right )^{3} \,d x } \]
\[ \int x^5 \coth ^{-1}(a x)^3 \, dx=\int x^{5} \operatorname {acoth}^{3}{\left (a x \right )}\, dx \]
Time = 0.24 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.55 \[ \int x^5 \coth ^{-1}(a x)^3 \, dx=\frac {1}{6} \, x^{6} \operatorname {arcoth}\left (a x\right )^{3} + \frac {1}{60} \, a {\left (\frac {2 \, {\left (3 \, a^{4} x^{5} + 5 \, a^{2} x^{3} + 15 \, x\right )}}{a^{6}} - \frac {15 \, \log \left (a x + 1\right )}{a^{7}} + \frac {15 \, \log \left (a x - 1\right )}{a^{7}}\right )} \operatorname {arcoth}\left (a x\right )^{2} + \frac {1}{240} \, a {\left (\frac {\frac {4 \, a^{3} x^{3} + {\left (15 \, \log \left (a x - 1\right ) - 46\right )} \log \left (a x + 1\right )^{2} - 5 \, \log \left (a x + 1\right )^{3} + 5 \, \log \left (a x - 1\right )^{3} + 76 \, a x - {\left (15 \, \log \left (a x - 1\right )^{2} - 92 \, \log \left (a x - 1\right )\right )} \log \left (a x + 1\right ) + 46 \, \log \left (a x - 1\right )^{2} + 38 \, \log \left (a x - 1\right )}{a} - \frac {184 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a} - \frac {38 \, \log \left (a x + 1\right )}{a}}{a^{6}} + \frac {2 \, {\left (6 \, a^{4} x^{4} + 32 \, a^{2} x^{2} - 2 \, {\left (15 \, \log \left (a x - 1\right ) - 46\right )} \log \left (a x + 1\right ) + 15 \, \log \left (a x + 1\right )^{2} + 15 \, \log \left (a x - 1\right )^{2} + 92 \, \log \left (a x - 1\right )\right )} \operatorname {arcoth}\left (a x\right )}{a^{7}}\right )} \]
1/6*x^6*arccoth(a*x)^3 + 1/60*a*(2*(3*a^4*x^5 + 5*a^2*x^3 + 15*x)/a^6 - 15 *log(a*x + 1)/a^7 + 15*log(a*x - 1)/a^7)*arccoth(a*x)^2 + 1/240*a*(((4*a^3 *x^3 + (15*log(a*x - 1) - 46)*log(a*x + 1)^2 - 5*log(a*x + 1)^3 + 5*log(a* x - 1)^3 + 76*a*x - (15*log(a*x - 1)^2 - 92*log(a*x - 1))*log(a*x + 1) + 4 6*log(a*x - 1)^2 + 38*log(a*x - 1))/a - 184*(log(a*x - 1)*log(1/2*a*x + 1/ 2) + dilog(-1/2*a*x + 1/2))/a - 38*log(a*x + 1)/a)/a^6 + 2*(6*a^4*x^4 + 32 *a^2*x^2 - 2*(15*log(a*x - 1) - 46)*log(a*x + 1) + 15*log(a*x + 1)^2 + 15* log(a*x - 1)^2 + 92*log(a*x - 1))*arccoth(a*x)/a^7)
\[ \int x^5 \coth ^{-1}(a x)^3 \, dx=\int { x^{5} \operatorname {arcoth}\left (a x\right )^{3} \,d x } \]
Timed out. \[ \int x^5 \coth ^{-1}(a x)^3 \, dx=\int x^5\,{\mathrm {acoth}\left (a\,x\right )}^3 \,d x \]