Integrand size = 10, antiderivative size = 194 \[ \int x^5 \cot ^{-1}(a x)^3 \, dx=-\frac {19 x}{60 a^5}+\frac {x^3}{60 a^3}-\frac {4 x^2 \cot ^{-1}(a x)}{15 a^4}+\frac {x^4 \cot ^{-1}(a x)}{20 a^2}+\frac {23 i \cot ^{-1}(a x)^2}{30 a^6}+\frac {x \cot ^{-1}(a x)^2}{2 a^5}-\frac {x^3 \cot ^{-1}(a x)^2}{6 a^3}+\frac {x^5 \cot ^{-1}(a x)^2}{10 a}+\frac {\cot ^{-1}(a x)^3}{6 a^6}+\frac {1}{6} x^6 \cot ^{-1}(a x)^3+\frac {19 \arctan (a x)}{60 a^6}-\frac {23 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^6}+\frac {23 i \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{30 a^6} \]
-19/60*x/a^5+1/60*x^3/a^3-4/15*x^2*arccot(a*x)/a^4+1/20*x^4*arccot(a*x)/a^ 2+23/30*I*arccot(a*x)^2/a^6+1/2*x*arccot(a*x)^2/a^5-1/6*x^3*arccot(a*x)^2/ a^3+1/10*x^5*arccot(a*x)^2/a+1/6*arccot(a*x)^3/a^6+1/6*x^6*arccot(a*x)^3+1 9/60*arctan(a*x)/a^6-23/15*arccot(a*x)*ln(2/(1+I*a*x))/a^6+23/30*I*polylog (2,1-2/(1+I*a*x))/a^6
Time = 0.52 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.64 \[ \int x^5 \cot ^{-1}(a x)^3 \, dx=\frac {a x \left (-19+a^2 x^2\right )+2 \left (23 i+15 a x-5 a^3 x^3+3 a^5 x^5\right ) \cot ^{-1}(a x)^2+10 \left (1+a^6 x^6\right ) \cot ^{-1}(a x)^3+\cot ^{-1}(a x) \left (-19-16 a^2 x^2+3 a^4 x^4-92 \log \left (1-e^{2 i \cot ^{-1}(a x)}\right )\right )+46 i \operatorname {PolyLog}\left (2,e^{2 i \cot ^{-1}(a x)}\right )}{60 a^6} \]
(a*x*(-19 + a^2*x^2) + 2*(23*I + 15*a*x - 5*a^3*x^3 + 3*a^5*x^5)*ArcCot[a* x]^2 + 10*(1 + a^6*x^6)*ArcCot[a*x]^3 + ArcCot[a*x]*(-19 - 16*a^2*x^2 + 3* a^4*x^4 - 92*Log[1 - E^((2*I)*ArcCot[a*x])]) + (46*I)*PolyLog[2, E^((2*I)* ArcCot[a*x])])/(60*a^6)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(447\) vs. \(2(194)=388\).
Time = 2.66 (sec) , antiderivative size = 447, normalized size of antiderivative = 2.30, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.700, Rules used = {5362, 5452, 5362, 5452, 5362, 254, 2009, 5452, 5346, 5362, 262, 216, 5420, 5456, 5380, 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.
| \(\displaystyle \int x^5 \cot ^{-1}(a x)^3 \, dx\) |
| \(\Big \downarrow \) 5362 |
| \(\displaystyle \frac {1}{2} a \int \frac {x^6 \cot ^{-1}(a x)^2}{a^2 x^2+1}dx+\frac {1}{6} x^6 \cot ^{-1}(a x)^3\) |
| \(\Big \downarrow \) 5452 |
| \(\displaystyle \frac {1}{2} a \left (\frac {\int x^4 \cot ^{-1}(a x)^2dx}{a^2}-\frac {\int \frac {x^4 \cot ^{-1}(a x)^2}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{6} x^6 \cot ^{-1}(a x)^3\) |
| \(\Big \downarrow \) 5362 |
| \(\displaystyle \frac {1}{2} a \left (\frac {\frac {2}{5} a \int \frac {x^5 \cot ^{-1}(a x)}{a^2 x^2+1}dx+\frac {1}{5} x^5 \cot ^{-1}(a x)^2}{a^2}-\frac {\int \frac {x^4 \cot ^{-1}(a x)^2}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{6} x^6 \cot ^{-1}(a x)^3\) |
| \(\Big \downarrow \) 5452 |
| \(\displaystyle \frac {1}{2} a \left (\frac {\frac {2}{5} a \left (\frac {\int x^3 \cot ^{-1}(a x)dx}{a^2}-\frac {\int \frac {x^3 \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{5} x^5 \cot ^{-1}(a x)^2}{a^2}-\frac {\frac {\int x^2 \cot ^{-1}(a x)^2dx}{a^2}-\frac {\int \frac {x^2 \cot ^{-1}(a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )+\frac {1}{6} x^6 \cot ^{-1}(a x)^3\) |
| \(\Big \downarrow \) 5362 |
| \(\displaystyle \frac {1}{2} a \left (\frac {\frac {2}{5} a \left (\frac {\frac {1}{4} a \int \frac {x^4}{a^2 x^2+1}dx+\frac {1}{4} x^4 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x^3 \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{5} x^5 \cot ^{-1}(a x)^2}{a^2}-\frac {\frac {\frac {2}{3} a \int \frac {x^3 \cot ^{-1}(a x)}{a^2 x^2+1}dx+\frac {1}{3} x^3 \cot ^{-1}(a x)^2}{a^2}-\frac {\int \frac {x^2 \cot ^{-1}(a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )+\frac {1}{6} x^6 \cot ^{-1}(a x)^3\) |
| \(\Big \downarrow \) 254 |
| \(\displaystyle \frac {1}{2} a \left (\frac {\frac {2}{5} a \left (\frac {\frac {1}{4} a \int \left (\frac {x^2}{a^2}+\frac {1}{a^4 \left (a^2 x^2+1\right )}-\frac {1}{a^4}\right )dx+\frac {1}{4} x^4 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x^3 \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{5} x^5 \cot ^{-1}(a x)^2}{a^2}-\frac {\frac {\frac {2}{3} a \int \frac {x^3 \cot ^{-1}(a x)}{a^2 x^2+1}dx+\frac {1}{3} x^3 \cot ^{-1}(a x)^2}{a^2}-\frac {\int \frac {x^2 \cot ^{-1}(a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )+\frac {1}{6} x^6 \cot ^{-1}(a x)^3\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} a \left (\frac {\frac {2}{5} a \left (\frac {\frac {1}{4} a \left (\frac {\arctan (a x)}{a^5}-\frac {x}{a^4}+\frac {x^3}{3 a^2}\right )+\frac {1}{4} x^4 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x^3 \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{5} x^5 \cot ^{-1}(a x)^2}{a^2}-\frac {\frac {\frac {2}{3} a \int \frac {x^3 \cot ^{-1}(a x)}{a^2 x^2+1}dx+\frac {1}{3} x^3 \cot ^{-1}(a x)^2}{a^2}-\frac {\int \frac {x^2 \cot ^{-1}(a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )+\frac {1}{6} x^6 \cot ^{-1}(a x)^3\) |
| \(\Big \downarrow \) 5452 |
| \(\displaystyle \frac {1}{2} a \left (\frac {\frac {2}{5} a \left (\frac {\frac {1}{4} a \left (\frac {\arctan (a x)}{a^5}-\frac {x}{a^4}+\frac {x^3}{3 a^2}\right )+\frac {1}{4} x^4 \cot ^{-1}(a x)}{a^2}-\frac {\frac {\int x \cot ^{-1}(a x)dx}{a^2}-\frac {\int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}}{a^2}\right )+\frac {1}{5} x^5 \cot ^{-1}(a x)^2}{a^2}-\frac {\frac {\frac {2}{3} a \left (\frac {\int x \cot ^{-1}(a x)dx}{a^2}-\frac {\int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)^2}{a^2}-\frac {\frac {\int \cot ^{-1}(a x)^2dx}{a^2}-\frac {\int \frac {\cot ^{-1}(a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}}{a^2}\right )+\frac {1}{6} x^6 \cot ^{-1}(a x)^3\) |
| \(\Big \downarrow \) 5346 |
| \(\displaystyle \frac {1}{2} a \left (\frac {\frac {2}{5} a \left (\frac {\frac {1}{4} a \left (\frac {\arctan (a x)}{a^5}-\frac {x}{a^4}+\frac {x^3}{3 a^2}\right )+\frac {1}{4} x^4 \cot ^{-1}(a x)}{a^2}-\frac {\frac {\int x \cot ^{-1}(a x)dx}{a^2}-\frac {\int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}}{a^2}\right )+\frac {1}{5} x^5 \cot ^{-1}(a x)^2}{a^2}-\frac {\frac {\frac {2}{3} a \left (\frac {\int x \cot ^{-1}(a x)dx}{a^2}-\frac {\int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)^2}{a^2}-\frac {\frac {2 a \int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx+x \cot ^{-1}(a x)^2}{a^2}-\frac {\int \frac {\cot ^{-1}(a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}}{a^2}\right )+\frac {1}{6} x^6 \cot ^{-1}(a x)^3\) |
| \(\Big \downarrow \) 5362 |
| \(\displaystyle \frac {1}{2} a \left (\frac {\frac {2}{5} a \left (\frac {\frac {1}{4} a \left (\frac {\arctan (a x)}{a^5}-\frac {x}{a^4}+\frac {x^3}{3 a^2}\right )+\frac {1}{4} x^4 \cot ^{-1}(a x)}{a^2}-\frac {\frac {\frac {1}{2} a \int \frac {x^2}{a^2 x^2+1}dx+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}}{a^2}\right )+\frac {1}{5} x^5 \cot ^{-1}(a x)^2}{a^2}-\frac {\frac {\frac {2}{3} a \left (\frac {\frac {1}{2} a \int \frac {x^2}{a^2 x^2+1}dx+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)^2}{a^2}-\frac {\frac {2 a \int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx+x \cot ^{-1}(a x)^2}{a^2}-\frac {\int \frac {\cot ^{-1}(a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}}{a^2}\right )+\frac {1}{6} x^6 \cot ^{-1}(a x)^3\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {1}{2} a \left (\frac {\frac {2}{5} a \left (\frac {\frac {1}{4} a \left (\frac {\arctan (a x)}{a^5}-\frac {x}{a^4}+\frac {x^3}{3 a^2}\right )+\frac {1}{4} x^4 \cot ^{-1}(a x)}{a^2}-\frac {\frac {\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\int \frac {1}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}}{a^2}\right )+\frac {1}{5} x^5 \cot ^{-1}(a x)^2}{a^2}-\frac {\frac {\frac {2}{3} a \left (\frac {\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\int \frac {1}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)^2}{a^2}-\frac {\frac {2 a \int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx+x \cot ^{-1}(a x)^2}{a^2}-\frac {\int \frac {\cot ^{-1}(a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}}{a^2}\right )+\frac {1}{6} x^6 \cot ^{-1}(a x)^3\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{2} a \left (\frac {\frac {2}{5} a \left (\frac {\frac {1}{4} a \left (\frac {\arctan (a x)}{a^5}-\frac {x}{a^4}+\frac {x^3}{3 a^2}\right )+\frac {1}{4} x^4 \cot ^{-1}(a x)}{a^2}-\frac {\frac {\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}}{a^2}\right )+\frac {1}{5} x^5 \cot ^{-1}(a x)^2}{a^2}-\frac {\frac {\frac {2}{3} a \left (\frac {\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)^2}{a^2}-\frac {\frac {2 a \int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx+x \cot ^{-1}(a x)^2}{a^2}-\frac {\int \frac {\cot ^{-1}(a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}}{a^2}\right )+\frac {1}{6} x^6 \cot ^{-1}(a x)^3\) |
| \(\Big \downarrow \) 5420 |
| \(\displaystyle \frac {1}{2} a \left (\frac {\frac {2}{5} a \left (\frac {\frac {1}{4} a \left (\frac {\arctan (a x)}{a^5}-\frac {x}{a^4}+\frac {x^3}{3 a^2}\right )+\frac {1}{4} x^4 \cot ^{-1}(a x)}{a^2}-\frac {\frac {\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}}{a^2}\right )+\frac {1}{5} x^5 \cot ^{-1}(a x)^2}{a^2}-\frac {\frac {\frac {2}{3} a \left (\frac {\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)^2}{a^2}-\frac {\frac {2 a \int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx+x \cot ^{-1}(a x)^2}{a^2}+\frac {\cot ^{-1}(a x)^3}{3 a^3}}{a^2}}{a^2}\right )+\frac {1}{6} x^6 \cot ^{-1}(a x)^3\) |
| \(\Big \downarrow \) 5456 |
| \(\displaystyle \frac {1}{6} x^6 \cot ^{-1}(a x)^3+\frac {1}{2} a \left (\frac {\frac {1}{5} x^5 \cot ^{-1}(a x)^2+\frac {2}{5} a \left (\frac {\frac {1}{4} a \left (\frac {\arctan (a x)}{a^5}-\frac {x}{a^4}+\frac {x^3}{3 a^2}\right )+\frac {1}{4} x^4 \cot ^{-1}(a x)}{a^2}-\frac {\frac {\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\frac {i \cot ^{-1}(a x)^2}{2 a^2}-\frac {\int \frac {\cot ^{-1}(a x)}{i-a x}dx}{a}}{a^2}}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{3} x^3 \cot ^{-1}(a x)^2+\frac {2}{3} a \left (\frac {\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\frac {i \cot ^{-1}(a x)^2}{2 a^2}-\frac {\int \frac {\cot ^{-1}(a x)}{i-a x}dx}{a}}{a^2}\right )}{a^2}-\frac {\frac {\cot ^{-1}(a x)^3}{3 a^3}+\frac {x \cot ^{-1}(a x)^2+2 a \left (\frac {i \cot ^{-1}(a x)^2}{2 a^2}-\frac {\int \frac {\cot ^{-1}(a x)}{i-a x}dx}{a}\right )}{a^2}}{a^2}}{a^2}\right )\) |
| \(\Big \downarrow \) 5380 |
| \(\displaystyle \frac {1}{6} x^6 \cot ^{-1}(a x)^3+\frac {1}{2} a \left (\frac {\frac {1}{5} x^5 \cot ^{-1}(a x)^2+\frac {2}{5} a \left (\frac {\frac {1}{4} a \left (\frac {\arctan (a x)}{a^5}-\frac {x}{a^4}+\frac {x^3}{3 a^2}\right )+\frac {1}{4} x^4 \cot ^{-1}(a x)}{a^2}-\frac {\frac {\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\frac {i \cot ^{-1}(a x)^2}{2 a^2}-\frac {\int \frac {\log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx+\frac {\log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a}}{a}}{a^2}}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{3} x^3 \cot ^{-1}(a x)^2+\frac {2}{3} a \left (\frac {\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\frac {i \cot ^{-1}(a x)^2}{2 a^2}-\frac {\int \frac {\log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx+\frac {\log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a}}{a}}{a^2}\right )}{a^2}-\frac {\frac {\cot ^{-1}(a x)^3}{3 a^3}+\frac {x \cot ^{-1}(a x)^2+2 a \left (\frac {i \cot ^{-1}(a x)^2}{2 a^2}-\frac {\int \frac {\log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx+\frac {\log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a}}{a}\right )}{a^2}}{a^2}}{a^2}\right )\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {1}{6} x^6 \cot ^{-1}(a x)^3+\frac {1}{2} a \left (\frac {\frac {1}{5} x^5 \cot ^{-1}(a x)^2+\frac {2}{5} a \left (\frac {\frac {1}{4} a \left (\frac {\arctan (a x)}{a^5}-\frac {x}{a^4}+\frac {x^3}{3 a^2}\right )+\frac {1}{4} x^4 \cot ^{-1}(a x)}{a^2}-\frac {\frac {\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\frac {i \cot ^{-1}(a x)^2}{2 a^2}-\frac {\frac {\log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a}-\frac {i \int \frac {\log \left (\frac {2}{i a x+1}\right )}{1-\frac {2}{i a x+1}}d\frac {1}{i a x+1}}{a}}{a}}{a^2}}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{3} x^3 \cot ^{-1}(a x)^2+\frac {2}{3} a \left (\frac {\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\frac {i \cot ^{-1}(a x)^2}{2 a^2}-\frac {\frac {\log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a}-\frac {i \int \frac {\log \left (\frac {2}{i a x+1}\right )}{1-\frac {2}{i a x+1}}d\frac {1}{i a x+1}}{a}}{a}}{a^2}\right )}{a^2}-\frac {\frac {\cot ^{-1}(a x)^3}{3 a^3}+\frac {x \cot ^{-1}(a x)^2+2 a \left (\frac {i \cot ^{-1}(a x)^2}{2 a^2}-\frac {\frac {\log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a}-\frac {i \int \frac {\log \left (\frac {2}{i a x+1}\right )}{1-\frac {2}{i a x+1}}d\frac {1}{i a x+1}}{a}}{a}\right )}{a^2}}{a^2}}{a^2}\right )\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {1}{6} x^6 \cot ^{-1}(a x)^3+\frac {1}{2} a \left (\frac {\frac {1}{5} x^5 \cot ^{-1}(a x)^2+\frac {2}{5} a \left (\frac {\frac {1}{4} a \left (\frac {\arctan (a x)}{a^5}-\frac {x}{a^4}+\frac {x^3}{3 a^2}\right )+\frac {1}{4} x^4 \cot ^{-1}(a x)}{a^2}-\frac {\frac {\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\frac {i \cot ^{-1}(a x)^2}{2 a^2}-\frac {\frac {\log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}}{a}}{a^2}}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{3} x^3 \cot ^{-1}(a x)^2+\frac {2}{3} a \left (\frac {\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\frac {i \cot ^{-1}(a x)^2}{2 a^2}-\frac {\frac {\log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}}{a}}{a^2}\right )}{a^2}-\frac {\frac {\cot ^{-1}(a x)^3}{3 a^3}+\frac {x \cot ^{-1}(a x)^2+2 a \left (\frac {i \cot ^{-1}(a x)^2}{2 a^2}-\frac {\frac {\log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}}{a}\right )}{a^2}}{a^2}}{a^2}\right )\) |
(x^6*ArcCot[a*x]^3)/6 + (a*(((x^5*ArcCot[a*x]^2)/5 + (2*a*(((x^4*ArcCot[a* x])/4 + (a*(-(x/a^4) + x^3/(3*a^2) + ArcTan[a*x]/a^5))/4)/a^2 - (((x^2*Arc Cot[a*x])/2 + (a*(x/a^2 - ArcTan[a*x]/a^3))/2)/a^2 - (((I/2)*ArcCot[a*x]^2 )/a^2 - ((ArcCot[a*x]*Log[2/(1 + I*a*x)])/a - ((I/2)*PolyLog[2, 1 - 2/(1 + I*a*x)])/a)/a)/a^2)/a^2))/5)/a^2 - (((x^3*ArcCot[a*x]^2)/3 + (2*a*(((x^2* ArcCot[a*x])/2 + (a*(x/a^2 - ArcTan[a*x]/a^3))/2)/a^2 - (((I/2)*ArcCot[a*x ]^2)/a^2 - ((ArcCot[a*x]*Log[2/(1 + I*a*x)])/a - ((I/2)*PolyLog[2, 1 - 2/( 1 + I*a*x)])/a)/a)/a^2))/3)/a^2 - (ArcCot[a*x]^3/(3*a^3) + (x*ArcCot[a*x]^ 2 + 2*a*(((I/2)*ArcCot[a*x]^2)/a^2 - ((ArcCot[a*x]*Log[2/(1 + I*a*x)])/a - ((I/2)*PolyLog[2, 1 - 2/(1 + I*a*x)])/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]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[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_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCot[c*x^n])^p, x] + Simp[b*c*n*p Int[x^n*((a + b*ArcCot[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_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCot[c*x^n])^p/(m + 1)), x] + Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcCot[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_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCot[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] - Simp[b*c*( p/e) Int[(a + b*ArcCot[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_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbo l] :> Simp[-(a + b*ArcCot[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]
Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e _.)*(x_)^2), x_Symbol] :> Simp[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcCot[c*x] )^p, x], x] - Simp[d*(f^2/e) Int[(f*x)^(m - 2)*((a + b*ArcCot[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_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[I*((a + b*ArcCot[c*x])^(p + 1)/(b*e*(p + 1))), x] - Simp[ 1/(c*d) Int[(a + b*ArcCot[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1103 vs. \(2 (164 ) = 328\).
Time = 29.34 (sec) , antiderivative size = 1104, normalized size of antiderivative = 5.69
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1104\) |
| parts | \(\text {Expression too large to display}\) | \(2453\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(2455\) |
| default | \(\text {Expression too large to display}\) | \(2455\) |
23/120*I/a^6*Pi^2+8929/57600*I/a^6*ln(a^2*x^2+1)-1/4*I/a^5*Pi*ln(1-I*a*x)* x-61/1920*I/a^2*ln(1-I*a*x)*x^4+151/960*I/a^4*ln(1-I*a*x)*x^2+37/480*I/a^6 *Pi*arctan(a*x)+1/40/a^2*Pi*x^4+1/40/a*Pi^2*x^5+1/12*I/a^3*Pi*ln(1-I*a*x)* x^3+23/30*I/a^6*dilog(1/2-1/2*I*a*x)-1291/3600*I/a^6*ln(1-I*a*x)-1/8/a^6*P i^2*arctan(a*x)+331/960/a^6*Pi*ln(a^2*x^2+1)-1/240*(-15*I*x^6*ln(1-I*a*x)* a^6+15*Pi*a^6*x^6+6*a^5*x^5-10*a^3*x^3-15*I*ln(1-I*a*x)+30*a*x+15*Pi-46*I) /a^6*ln(1+I*a*x)^2+1/48/a^3*ln(1-I*a*x)^2*x^3+7/1440/a^3*ln(1-I*a*x)*x^3-1 /16/a^5*ln(1-I*a*x)^2*x-1/1200/a*ln(1-I*a*x)*x^5-37/480/a^5*ln(1-I*a*x)*x- 1/80/a*ln(1-I*a*x)^2*x^5-1/3*I/a^6+23/30*I/a^6*ln(1/2-1/2*I*a*x)*ln(1/2+1/ 2*I*a*x)-23/30*I/a^6*ln(1-I*a*x)*ln(1/2+1/2*I*a*x)+1/48*x^6*Pi^3+1/48/a^6* Pi^3-19/120/a^6*Pi+(-1/16*I*(a^6*x^6+1)/a^6*ln(1-I*a*x)^2+1/120*x*(15*Pi*a ^5*x^5+6*a^4*x^4-10*a^2*x^2+30)/a^5*ln(1-I*a*x)-1/240*(-15*I*Pi^2*a^6*x^6- 12*I*Pi*a^5*x^5-6*I*a^4*x^4+20*I*Pi*a^3*x^3+32*I*a^2*x^2-60*I*Pi*a*x-30*ln (1-I*a*x)*Pi-92*I*ln(1-I*a*x))/a^6)*ln(1+I*a*x)-1/16*I*Pi^2*ln(1-I*a*x)*x^ 6-1/20*I/a*Pi*ln(1-I*a*x)*x^5-1/32*I/a^4*ln(1-I*a*x)^2*x^2+1/64*I/a^2*ln(1 -I*a*x)^2*x^4-1/48*I*(a^6*x^6+1)/a^6*ln(1+I*a*x)^3-1/24/a^3*Pi^2*x^3+1/8/a ^5*Pi^2*x-1/16/a^6*Pi*ln(1-I*a*x)^2+37/480/a^6*Pi*ln(1-I*a*x)-1/16*Pi*ln(1 -I*a*x)^2*x^6-2/15/a^4*Pi*x^2+1/48*I/a^6*ln(1-I*a*x)^3-49/320*I/a^6*ln(1-I *a*x)^2+1/48*I*ln(1-I*a*x)^3*x^6-1/96*I*ln(1-I*a*x)^2*x^6+1/288*I*ln(1-I*a *x)*x^6-1/50*I/a^6*(1-I*a*x)^5*ln(1-I*a*x)+1/8*I/a^6*(1-I*a*x)^3*ln(1-I...
\[ \int x^5 \cot ^{-1}(a x)^3 \, dx=\int { x^{5} \operatorname {arccot}\left (a x\right )^{3} \,d x } \]
\[ \int x^5 \cot ^{-1}(a x)^3 \, dx=\int x^{5} \operatorname {acot}^{3}{\left (a x \right )}\, dx \]
\[ \int x^5 \cot ^{-1}(a x)^3 \, dx=\int { x^{5} \operatorname {arccot}\left (a x\right )^{3} \,d x } \]
1/480*(40*a^6*x^6*arctan2(1, a*x)^3 + 12*a^5*x^5*arctan2(1, a*x)^2 - 20*a^ 3*x^3*arctan2(1, a*x)^2 + 20*(5760*a^7*integrate(1/480*x^7*arctan(1/(a*x)) ^3/(a^7*x^2 + a^5), x) + 1440*a^6*integrate(1/480*x^6*arctan(1/(a*x))^2/(a ^7*x^2 + a^5), x) + 360*a^6*integrate(1/480*x^6*log(a^2*x^2 + 1)^2/(a^7*x^ 2 + a^5), x) + 288*a^6*integrate(1/480*x^6*log(a^2*x^2 + 1)/(a^7*x^2 + a^5 ), x) + 5760*a^5*integrate(1/480*x^5*arctan(1/(a*x))^3/(a^7*x^2 + a^5), x) + 576*a^5*integrate(1/480*x^5*arctan(1/(a*x))/(a^7*x^2 + a^5), x) - 480*a ^4*integrate(1/480*x^4*log(a^2*x^2 + 1)/(a^7*x^2 + a^5), x) - 960*a^3*inte grate(1/480*x^3*arctan(1/(a*x))/(a^7*x^2 + a^5), x) + 1440*a^2*integrate(1 /480*x^2*log(a^2*x^2 + 1)/(a^7*x^2 + a^5), x) + 2880*a*integrate(1/480*x*a rctan(1/(a*x))/(a^7*x^2 + a^5), x) + arctan(a*x)^3/a^6 + 3*arctan(a*x)^2*a rctan(1/(a*x))/a^6 + 3*arctan(a*x)*arctan(1/(a*x))^2/a^6 + 360*integrate(1 /480*log(a^2*x^2 + 1)^2/(a^7*x^2 + a^5), x))*a^6 + 60*a*x*arctan2(1, a*x)^ 2 + 40*arctan2(1, a*x)^3 - (3*a^5*x^5 - 5*a^3*x^3 + 15*a*x)*log(a^2*x^2 + 1)^2)/a^6
\[ \int x^5 \cot ^{-1}(a x)^3 \, dx=\int { x^{5} \operatorname {arccot}\left (a x\right )^{3} \,d x } \]
Timed out. \[ \int x^5 \cot ^{-1}(a x)^3 \, dx=\int x^5\,{\mathrm {acot}\left (a\,x\right )}^3 \,d x \]