Integrand size = 20, antiderivative size = 219 \[ \int x^3 \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=\frac {c x}{15 a^3}-\frac {c x^3}{60 a}-\frac {c \arctan (a x)}{15 a^4}-\frac {c x^2 \arctan (a x)}{60 a^2}+\frac {1}{20} c x^4 \arctan (a x)+\frac {7 i c \arctan (a x)^2}{30 a^4}+\frac {c x \arctan (a x)^2}{4 a^3}-\frac {c x^3 \arctan (a x)^2}{12 a}-\frac {1}{10} a c x^5 \arctan (a x)^2-\frac {c \arctan (a x)^3}{12 a^4}+\frac {1}{4} c x^4 \arctan (a x)^3+\frac {1}{6} a^2 c x^6 \arctan (a x)^3+\frac {7 c \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^4}+\frac {7 i c \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{30 a^4} \] Output:
1/15*c*x/a^3-1/60*c*x^3/a-1/15*c*arctan(a*x)/a^4-1/60*c*x^2*arctan(a*x)/a^ 2+1/20*c*x^4*arctan(a*x)+7/30*I*c*arctan(a*x)^2/a^4+1/4*c*x*arctan(a*x)^2/ a^3-1/12*c*x^3*arctan(a*x)^2/a-1/10*a*c*x^5*arctan(a*x)^2-1/12*c*arctan(a* x)^3/a^4+1/4*c*x^4*arctan(a*x)^3+1/6*a^2*c*x^6*arctan(a*x)^3+7/15*c*arctan (a*x)*ln(2/(1+I*a*x))/a^4+7/30*I*c*polylog(2,1-2/(1+I*a*x))/a^4
Time = 0.48 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.62 \[ \int x^3 \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=\frac {c \left (4 a x-a^3 x^3-\left (14 i-15 a x+5 a^3 x^3+6 a^5 x^5\right ) \arctan (a x)^2+5 \left (-1+3 a^4 x^4+2 a^6 x^6\right ) \arctan (a x)^3+\arctan (a x) \left (-4-a^2 x^2+3 a^4 x^4+28 \log \left (1+e^{2 i \arctan (a x)}\right )\right )-14 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )\right )}{60 a^4} \] Input:
Integrate[x^3*(c + a^2*c*x^2)*ArcTan[a*x]^3,x]
Output:
(c*(4*a*x - a^3*x^3 - (14*I - 15*a*x + 5*a^3*x^3 + 6*a^5*x^5)*ArcTan[a*x]^ 2 + 5*(-1 + 3*a^4*x^4 + 2*a^6*x^6)*ArcTan[a*x]^3 + ArcTan[a*x]*(-4 - a^2*x ^2 + 3*a^4*x^4 + 28*Log[1 + E^((2*I)*ArcTan[a*x])]) - (14*I)*PolyLog[2, -E ^((2*I)*ArcTan[a*x])]))/(60*a^4)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(711\) vs. \(2(219)=438\).
Time = 4.75 (sec) , antiderivative size = 711, normalized size of antiderivative = 3.25, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {5485, 5361, 5451, 5361, 5451, 5345, 5361, 254, 262, 216, 2009, 5419, 5451, 5345, 5361, 262, 216, 5419, 5455, 5379, 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.
| \(\displaystyle \int x^3 \arctan (a x)^3 \left (a^2 c x^2+c\right ) \, dx\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle a^2 c \int x^5 \arctan (a x)^3dx+c \int x^3 \arctan (a x)^3dx\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle a^2 c \left (\frac {1}{6} x^6 \arctan (a x)^3-\frac {1}{2} a \int \frac {x^6 \arctan (a x)^2}{a^2 x^2+1}dx\right )+c \left (\frac {1}{4} x^4 \arctan (a x)^3-\frac {3}{4} a \int \frac {x^4 \arctan (a x)^2}{a^2 x^2+1}dx\right )\) |
| \(\Big \downarrow \) 5451 |
| \(\displaystyle c \left (\frac {1}{4} x^4 \arctan (a x)^3-\frac {3}{4} a \left (\frac {\int x^2 \arctan (a x)^2dx}{a^2}-\frac {\int \frac {x^2 \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}\right )\right )+a^2 c \left (\frac {1}{6} x^6 \arctan (a x)^3-\frac {1}{2} a \left (\frac {\int x^4 \arctan (a x)^2dx}{a^2}-\frac {\int \frac {x^4 \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle c \left (\frac {1}{4} x^4 \arctan (a x)^3-\frac {3}{4} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \int \frac {x^3 \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {x^2 \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}\right )\right )+a^2 c \left (\frac {1}{6} x^6 \arctan (a x)^3-\frac {1}{2} a \left (\frac {\frac {1}{5} x^5 \arctan (a x)^2-\frac {2}{5} a \int \frac {x^5 \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {x^4 \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 5451 |
| \(\displaystyle c \left (\frac {1}{4} x^4 \arctan (a x)^3-\frac {3}{4} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\int x \arctan (a x)dx}{a^2}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {\int \arctan (a x)^2dx}{a^2}-\frac {\int \frac {\arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )\right )+a^2 c \left (\frac {1}{6} x^6 \arctan (a x)^3-\frac {1}{2} a \left (\frac {\frac {1}{5} x^5 \arctan (a x)^2-\frac {2}{5} a \left (\frac {\int x^3 \arctan (a x)dx}{a^2}-\frac {\int \frac {x^3 \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {\int x^2 \arctan (a x)^2dx}{a^2}-\frac {\int \frac {x^2 \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 5345 |
| \(\displaystyle c \left (\frac {1}{4} x^4 \arctan (a x)^3-\frac {3}{4} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\int x \arctan (a x)dx}{a^2}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {x \arctan (a x)^2-2 a \int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {\arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )\right )+a^2 c \left (\frac {1}{6} x^6 \arctan (a x)^3-\frac {1}{2} a \left (\frac {\frac {1}{5} x^5 \arctan (a x)^2-\frac {2}{5} a \left (\frac {\int x^3 \arctan (a x)dx}{a^2}-\frac {\int \frac {x^3 \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {\int x^2 \arctan (a x)^2dx}{a^2}-\frac {\int \frac {x^2 \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle c \left (\frac {1}{4} x^4 \arctan (a x)^3-\frac {3}{4} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \int \frac {x^2}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {x \arctan (a x)^2-2 a \int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {\arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )\right )+a^2 c \left (\frac {1}{6} x^6 \arctan (a x)^3-\frac {1}{2} a \left (\frac {\frac {1}{5} x^5 \arctan (a x)^2-\frac {2}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)-\frac {1}{4} a \int \frac {x^4}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {x^3 \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \int \frac {x^3 \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {x^2 \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 254 |
| \(\displaystyle c \left (\frac {1}{4} x^4 \arctan (a x)^3-\frac {3}{4} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \int \frac {x^2}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {x \arctan (a x)^2-2 a \int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {\arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )\right )+a^2 c \left (\frac {1}{6} x^6 \arctan (a x)^3-\frac {1}{2} a \left (\frac {\frac {1}{5} x^5 \arctan (a x)^2-\frac {2}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)-\frac {1}{4} a \int \left (\frac {x^2}{a^2}+\frac {1}{a^4 \left (a^2 x^2+1\right )}-\frac {1}{a^4}\right )dx}{a^2}-\frac {\int \frac {x^3 \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \int \frac {x^3 \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {x^2 \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle c \left (\frac {1}{4} x^4 \arctan (a x)^3-\frac {3}{4} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\int \frac {1}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {x \arctan (a x)^2-2 a \int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {\arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )\right )+a^2 c \left (\frac {1}{6} x^6 \arctan (a x)^3-\frac {1}{2} a \left (\frac {\frac {1}{5} x^5 \arctan (a x)^2-\frac {2}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)-\frac {1}{4} a \int \left (\frac {x^2}{a^2}+\frac {1}{a^4 \left (a^2 x^2+1\right )}-\frac {1}{a^4}\right )dx}{a^2}-\frac {\int \frac {x^3 \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \int \frac {x^3 \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {x^2 \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle a^2 c \left (\frac {1}{6} x^6 \arctan (a x)^3-\frac {1}{2} a \left (\frac {\frac {1}{5} x^5 \arctan (a x)^2-\frac {2}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)-\frac {1}{4} a \int \left (\frac {x^2}{a^2}+\frac {1}{a^4 \left (a^2 x^2+1\right )}-\frac {1}{a^4}\right )dx}{a^2}-\frac {\int \frac {x^3 \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \int \frac {x^3 \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {x^2 \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )\right )+c \left (\frac {1}{4} x^4 \arctan (a x)^3-\frac {3}{4} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {x \arctan (a x)^2-2 a \int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {\arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle c \left (\frac {1}{4} x^4 \arctan (a x)^3-\frac {3}{4} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {x \arctan (a x)^2-2 a \int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {\arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )\right )+a^2 c \left (\frac {1}{6} x^6 \arctan (a x)^3-\frac {1}{2} a \left (\frac {\frac {1}{5} x^5 \arctan (a x)^2-\frac {2}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)-\frac {1}{4} a \left (\frac {\arctan (a x)}{a^5}-\frac {x}{a^4}+\frac {x^3}{3 a^2}\right )}{a^2}-\frac {\int \frac {x^3 \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \int \frac {x^3 \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {x^2 \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle c \left (\frac {1}{4} x^4 \arctan (a x)^3-\frac {3}{4} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {x \arctan (a x)^2-2 a \int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\arctan (a x)^3}{3 a^3}}{a^2}\right )\right )+a^2 c \left (\frac {1}{6} x^6 \arctan (a x)^3-\frac {1}{2} a \left (\frac {\frac {1}{5} x^5 \arctan (a x)^2-\frac {2}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)-\frac {1}{4} a \left (\frac {\arctan (a x)}{a^5}-\frac {x}{a^4}+\frac {x^3}{3 a^2}\right )}{a^2}-\frac {\int \frac {x^3 \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \int \frac {x^3 \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {x^2 \arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 5451 |
| \(\displaystyle c \left (\frac {1}{4} x^4 \arctan (a x)^3-\frac {3}{4} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {x \arctan (a x)^2-2 a \int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\arctan (a x)^3}{3 a^3}}{a^2}\right )\right )+a^2 c \left (\frac {1}{6} x^6 \arctan (a x)^3-\frac {1}{2} a \left (\frac {\frac {1}{5} x^5 \arctan (a x)^2-\frac {2}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)-\frac {1}{4} a \left (\frac {\arctan (a x)}{a^5}-\frac {x}{a^4}+\frac {x^3}{3 a^2}\right )}{a^2}-\frac {\frac {\int x \arctan (a x)dx}{a^2}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\int x \arctan (a x)dx}{a^2}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {\int \arctan (a x)^2dx}{a^2}-\frac {\int \frac {\arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 5345 |
| \(\displaystyle c \left (\frac {1}{4} x^4 \arctan (a x)^3-\frac {3}{4} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {x \arctan (a x)^2-2 a \int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\arctan (a x)^3}{3 a^3}}{a^2}\right )\right )+a^2 c \left (\frac {1}{6} x^6 \arctan (a x)^3-\frac {1}{2} a \left (\frac {\frac {1}{5} x^5 \arctan (a x)^2-\frac {2}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)-\frac {1}{4} a \left (\frac {\arctan (a x)}{a^5}-\frac {x}{a^4}+\frac {x^3}{3 a^2}\right )}{a^2}-\frac {\frac {\int x \arctan (a x)dx}{a^2}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\int x \arctan (a x)dx}{a^2}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {x \arctan (a x)^2-2 a \int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {\arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle c \left (\frac {1}{4} x^4 \arctan (a x)^3-\frac {3}{4} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {x \arctan (a x)^2-2 a \int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\arctan (a x)^3}{3 a^3}}{a^2}\right )\right )+a^2 c \left (\frac {1}{6} x^6 \arctan (a x)^3-\frac {1}{2} a \left (\frac {\frac {1}{5} x^5 \arctan (a x)^2-\frac {2}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)-\frac {1}{4} a \left (\frac {\arctan (a x)}{a^5}-\frac {x}{a^4}+\frac {x^3}{3 a^2}\right )}{a^2}-\frac {\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \int \frac {x^2}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \int \frac {x^2}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {x \arctan (a x)^2-2 a \int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {\arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle c \left (\frac {1}{4} x^4 \arctan (a x)^3-\frac {3}{4} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {x \arctan (a x)^2-2 a \int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\arctan (a x)^3}{3 a^3}}{a^2}\right )\right )+a^2 c \left (\frac {1}{6} x^6 \arctan (a x)^3-\frac {1}{2} a \left (\frac {\frac {1}{5} x^5 \arctan (a x)^2-\frac {2}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)-\frac {1}{4} a \left (\frac {\arctan (a x)}{a^5}-\frac {x}{a^4}+\frac {x^3}{3 a^2}\right )}{a^2}-\frac {\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\int \frac {1}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\int \frac {1}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {x \arctan (a x)^2-2 a \int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {\arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle c \left (\frac {1}{4} x^4 \arctan (a x)^3-\frac {3}{4} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {x \arctan (a x)^2-2 a \int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\arctan (a x)^3}{3 a^3}}{a^2}\right )\right )+a^2 c \left (\frac {1}{6} x^6 \arctan (a x)^3-\frac {1}{2} a \left (\frac {\frac {1}{5} x^5 \arctan (a x)^2-\frac {2}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)-\frac {1}{4} a \left (\frac {\arctan (a x)}{a^5}-\frac {x}{a^4}+\frac {x^3}{3 a^2}\right )}{a^2}-\frac {\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {x \arctan (a x)^2-2 a \int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\int \frac {\arctan (a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle c \left (\frac {1}{4} x^4 \arctan (a x)^3-\frac {3}{4} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {x \arctan (a x)^2-2 a \int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\arctan (a x)^3}{3 a^3}}{a^2}\right )\right )+a^2 c \left (\frac {1}{6} x^6 \arctan (a x)^3-\frac {1}{2} a \left (\frac {\frac {1}{5} x^5 \arctan (a x)^2-\frac {2}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)-\frac {1}{4} a \left (\frac {\arctan (a x)}{a^5}-\frac {x}{a^4}+\frac {x^3}{3 a^2}\right )}{a^2}-\frac {\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}\right )}{a^2}-\frac {\frac {x \arctan (a x)^2-2 a \int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2}-\frac {\arctan (a x)^3}{3 a^3}}{a^2}}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 5455 |
| \(\displaystyle c \left (\frac {1}{4} x^4 \arctan (a x)^3-\frac {3}{4} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2}-\frac {-\frac {\int \frac {\arctan (a x)}{i-a x}dx}{a}-\frac {i \arctan (a x)^2}{2 a^2}}{a^2}\right )}{a^2}-\frac {-\frac {\arctan (a x)^3}{3 a^3}+\frac {x \arctan (a x)^2-2 a \left (-\frac {\int \frac {\arctan (a x)}{i-a x}dx}{a}-\frac {i \arctan (a x)^2}{2 a^2}\right )}{a^2}}{a^2}\right )\right )+a^2 c \left (\frac {1}{6} x^6 \arctan (a x)^3-\frac {1}{2} a \left (\frac {\frac {1}{5} x^5 \arctan (a x)^2-\frac {2}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)-\frac {1}{4} a \left (\frac {\arctan (a x)}{a^5}-\frac {x}{a^4}+\frac {x^3}{3 a^2}\right )}{a^2}-\frac {\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2}-\frac {-\frac {\int \frac {\arctan (a x)}{i-a x}dx}{a}-\frac {i \arctan (a x)^2}{2 a^2}}{a^2}}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2}-\frac {-\frac {\int \frac {\arctan (a x)}{i-a x}dx}{a}-\frac {i \arctan (a x)^2}{2 a^2}}{a^2}\right )}{a^2}-\frac {-\frac {\arctan (a x)^3}{3 a^3}+\frac {x \arctan (a x)^2-2 a \left (-\frac {\int \frac {\arctan (a x)}{i-a x}dx}{a}-\frac {i \arctan (a x)^2}{2 a^2}\right )}{a^2}}{a^2}}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 5379 |
| \(\displaystyle c \left (\frac {1}{4} x^4 \arctan (a x)^3-\frac {3}{4} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2}-\frac {-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}-\int \frac {\log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx}{a}-\frac {i \arctan (a x)^2}{2 a^2}}{a^2}\right )}{a^2}-\frac {-\frac {\arctan (a x)^3}{3 a^3}+\frac {x \arctan (a x)^2-2 a \left (-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}-\int \frac {\log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx}{a}-\frac {i \arctan (a x)^2}{2 a^2}\right )}{a^2}}{a^2}\right )\right )+a^2 c \left (\frac {1}{6} x^6 \arctan (a x)^3-\frac {1}{2} a \left (\frac {\frac {1}{5} x^5 \arctan (a x)^2-\frac {2}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)-\frac {1}{4} a \left (\frac {\arctan (a x)}{a^5}-\frac {x}{a^4}+\frac {x^3}{3 a^2}\right )}{a^2}-\frac {\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2}-\frac {-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}-\int \frac {\log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx}{a}-\frac {i \arctan (a x)^2}{2 a^2}}{a^2}}{a^2}\right )}{a^2}-\frac {\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2}-\frac {-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}-\int \frac {\log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx}{a}-\frac {i \arctan (a x)^2}{2 a^2}}{a^2}\right )}{a^2}-\frac {-\frac {\arctan (a x)^3}{3 a^3}+\frac {x \arctan (a x)^2-2 a \left (-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}-\int \frac {\log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx}{a}-\frac {i \arctan (a x)^2}{2 a^2}\right )}{a^2}}{a^2}}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle c \left (\frac {1}{6} x^6 \arctan (a x)^3-\frac {1}{2} a \left (\frac {\frac {1}{5} x^5 \arctan (a x)^2-\frac {2}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)-\frac {1}{4} a \left (\frac {x^3}{3 a^2}-\frac {x}{a^4}+\frac {\arctan (a x)}{a^5}\right )}{a^2}-\frac {\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2}-\frac {-\frac {i \arctan (a x)^2}{2 a^2}-\frac {\frac {\arctan (a x) \log \left (\frac {2}{i a x+1}\right )}{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 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2}-\frac {-\frac {i \arctan (a x)^2}{2 a^2}-\frac {\frac {\arctan (a x) \log \left (\frac {2}{i a x+1}\right )}{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 {x \arctan (a x)^2-2 a \left (-\frac {i \arctan (a x)^2}{2 a^2}-\frac {\frac {\arctan (a x) \log \left (\frac {2}{i a x+1}\right )}{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}-\frac {\arctan (a x)^3}{3 a^3}}{a^2}}{a^2}\right )\right ) a^2+c \left (\frac {1}{4} x^4 \arctan (a x)^3-\frac {3}{4} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2}-\frac {-\frac {i \arctan (a x)^2}{2 a^2}-\frac {\frac {\arctan (a x) \log \left (\frac {2}{i a x+1}\right )}{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 {x \arctan (a x)^2-2 a \left (-\frac {i \arctan (a x)^2}{2 a^2}-\frac {\frac {\arctan (a x) \log \left (\frac {2}{i a x+1}\right )}{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}-\frac {\arctan (a x)^3}{3 a^3}}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle c \left (\frac {1}{4} x^4 \arctan (a x)^3-\frac {3}{4} a \left (\frac {\frac {1}{3} x^3 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2}-\frac {-\frac {i \arctan (a x)^2}{2 a^2}-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{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 {\arctan (a x)^3}{3 a^3}+\frac {x \arctan (a x)^2-2 a \left (-\frac {i \arctan (a x)^2}{2 a^2}-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}}{a}\right )}{a^2}}{a^2}\right )\right )+a^2 c \left (\frac {1}{6} x^6 \arctan (a x)^3-\frac {1}{2} a \left (\frac {\frac {1}{5} x^5 \arctan (a x)^2-\frac {2}{5} a \left (\frac {\frac {1}{4} x^4 \arctan (a x)-\frac {1}{4} a \left (\frac {\arctan (a x)}{a^5}-\frac {x}{a^4}+\frac {x^3}{3 a^2}\right )}{a^2}-\frac {\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2}-\frac {-\frac {i \arctan (a x)^2}{2 a^2}-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{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 \arctan (a x)^2-\frac {2}{3} a \left (\frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2}-\frac {-\frac {i \arctan (a x)^2}{2 a^2}-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{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 {\arctan (a x)^3}{3 a^3}+\frac {x \arctan (a x)^2-2 a \left (-\frac {i \arctan (a x)^2}{2 a^2}-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{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 )\right )\) |
Input:
Int[x^3*(c + a^2*c*x^2)*ArcTan[a*x]^3,x]
Output:
c*((x^4*ArcTan[a*x]^3)/4 - (3*a*(((x^3*ArcTan[a*x]^2)/3 - (2*a*(((x^2*ArcT an[a*x])/2 - (a*(x/a^2 - ArcTan[a*x]/a^3))/2)/a^2 - (((-1/2*I)*ArcTan[a*x] ^2)/a^2 - ((ArcTan[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 - (-1/3*ArcTan[a*x]^3/a^3 + (x*ArcTan[a*x]^ 2 - 2*a*(((-1/2*I)*ArcTan[a*x]^2)/a^2 - ((ArcTan[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))/4) + a^2*c*((x ^6*ArcTan[a*x]^3)/6 - (a*(((x^5*ArcTan[a*x]^2)/5 - (2*a*(((x^4*ArcTan[a*x] )/4 - (a*(-(x/a^4) + x^3/(3*a^2) + ArcTan[a*x]/a^5))/4)/a^2 - (((x^2*ArcTa n[a*x])/2 - (a*(x/a^2 - ArcTan[a*x]/a^3))/2)/a^2 - (((-1/2*I)*ArcTan[a*x]^ 2)/a^2 - ((ArcTan[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*ArcTan[a*x]^2)/3 - (2*a*(((x^2 *ArcTan[a*x])/2 - (a*(x/a^2 - ArcTan[a*x]/a^3))/2)/a^2 - (((-1/2*I)*ArcTan [a*x]^2)/a^2 - ((ArcTan[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 - (-1/3*ArcTan[a*x]^3/a^3 + (x*ArcTan[ a*x]^2 - 2*a*(((-1/2*I)*ArcTan[a*x]^2)/a^2 - ((ArcTan[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)
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_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Simp[b*c*n*p Int[x^n*((a + b*ArcTan[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_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTan[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_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c*( p/e) Int[(a + b*ArcTan[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_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbo l] :> Simp[(a + b*ArcTan[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_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e _.)*(x_)^2), x_Symbol] :> Simp[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x] )^p, x], x] - Simp[d*(f^2/e) Int[(f*x)^(m - 2)*((a + b*ArcTan[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_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*((a + b*ArcTan[c*x])^(p + 1)/(b*e*(p + 1))), x] - Si mp[1/(c*d) Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(q_.), x_Symbol] :> Simp[d Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Simp[c^2*(d/f^2) Int[(f*x)^(m + 2)*(d + e*x^2 )^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || (EqQ[p, 1] && IntegerQ[q]))
Time = 3.29 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.25
| method | result | size |
| derivativedivides | \(\frac {\frac {c \arctan \left (a x \right )^{3} a^{6} x^{6}}{6}+\frac {c \arctan \left (a x \right )^{3} a^{4} x^{4}}{4}-\frac {c \arctan \left (a x \right )^{3}}{12}-\frac {c \left (\frac {2 a^{5} \arctan \left (a x \right )^{2} x^{5}}{5}+\frac {a^{3} \arctan \left (a x \right )^{2} x^{3}}{3}-a \arctan \left (a x \right )^{2} x -\frac {x^{4} \arctan \left (a x \right ) a^{4}}{5}+\frac {x^{2} a^{2} \arctan \left (a x \right )}{15}+\frac {14 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{15}+\frac {a^{3} x^{3}}{15}-\frac {4 a x}{15}+\frac {4 \arctan \left (a x \right )}{15}+\frac {7 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{15}-\frac {7 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{15}\right )}{4}}{a^{4}}\) | \(273\) |
| default | \(\frac {\frac {c \arctan \left (a x \right )^{3} a^{6} x^{6}}{6}+\frac {c \arctan \left (a x \right )^{3} a^{4} x^{4}}{4}-\frac {c \arctan \left (a x \right )^{3}}{12}-\frac {c \left (\frac {2 a^{5} \arctan \left (a x \right )^{2} x^{5}}{5}+\frac {a^{3} \arctan \left (a x \right )^{2} x^{3}}{3}-a \arctan \left (a x \right )^{2} x -\frac {x^{4} \arctan \left (a x \right ) a^{4}}{5}+\frac {x^{2} a^{2} \arctan \left (a x \right )}{15}+\frac {14 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{15}+\frac {a^{3} x^{3}}{15}-\frac {4 a x}{15}+\frac {4 \arctan \left (a x \right )}{15}+\frac {7 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{15}-\frac {7 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{15}\right )}{4}}{a^{4}}\) | \(273\) |
| parts | \(\frac {a^{2} c \,x^{6} \arctan \left (a x \right )^{3}}{6}+\frac {c \,x^{4} \arctan \left (a x \right )^{3}}{4}-\frac {c \left (\frac {2 a \arctan \left (a x \right )^{2} x^{5}}{5}+\frac {\arctan \left (a x \right )^{2} x^{3}}{3 a}-\frac {\arctan \left (a x \right )^{2} x}{a^{3}}+\frac {\arctan \left (a x \right )^{3}}{a^{4}}-\frac {2 \left (\frac {3 x^{4} \arctan \left (a x \right ) a^{4}}{2}-\frac {x^{2} a^{2} \arctan \left (a x \right )}{2}-7 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a^{3} x^{3}}{2}+2 a x -2 \arctan \left (a x \right )-\frac {7 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{2}+\frac {7 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{2}+5 \arctan \left (a x \right )^{3}\right )}{15 a^{4}}\right )}{4}\) | \(281\) |
Input:
int(x^3*(a^2*c*x^2+c)*arctan(a*x)^3,x,method=_RETURNVERBOSE)
Output:
1/a^4*(1/6*c*arctan(a*x)^3*a^6*x^6+1/4*c*arctan(a*x)^3*a^4*x^4-1/12*c*arct an(a*x)^3-1/4*c*(2/5*a^5*arctan(a*x)^2*x^5+1/3*a^3*arctan(a*x)^2*x^3-a*arc tan(a*x)^2*x-1/5*x^4*arctan(a*x)*a^4+1/15*x^2*a^2*arctan(a*x)+14/15*arctan (a*x)*ln(a^2*x^2+1)+1/15*a^3*x^3-4/15*a*x+4/15*arctan(a*x)+7/15*I*(ln(a*x- I)*ln(a^2*x^2+1)-1/2*ln(a*x-I)^2-dilog(-1/2*I*(a*x+I))-ln(a*x-I)*ln(-1/2*I *(a*x+I)))-7/15*I*(ln(a*x+I)*ln(a^2*x^2+1)-1/2*ln(a*x+I)^2-dilog(1/2*I*(a* x-I))-ln(a*x+I)*ln(1/2*I*(a*x-I)))))
\[ \int x^3 \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )} x^{3} \arctan \left (a x\right )^{3} \,d x } \] Input:
integrate(x^3*(a^2*c*x^2+c)*arctan(a*x)^3,x, algorithm="fricas")
Output:
integral((a^2*c*x^5 + c*x^3)*arctan(a*x)^3, x)
\[ \int x^3 \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=c \left (\int x^{3} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{2} x^{5} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \] Input:
integrate(x**3*(a**2*c*x**2+c)*atan(a*x)**3,x)
Output:
c*(Integral(x**3*atan(a*x)**3, x) + Integral(a**2*x**5*atan(a*x)**3, x))
\[ \int x^3 \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )} x^{3} \arctan \left (a x\right )^{3} \,d x } \] Input:
integrate(x^3*(a^2*c*x^2+c)*arctan(a*x)^3,x, algorithm="maxima")
Output:
1/960*(20*(23040*a^7*c*integrate(1/960*x^7*arctan(a*x)^3/(a^5*x^2 + a^3), x) - 5760*a^6*c*integrate(1/960*x^6*arctan(a*x)^2/(a^5*x^2 + a^3), x) - 14 40*a^6*c*integrate(1/960*x^6*log(a^2*x^2 + 1)^2/(a^5*x^2 + a^3), x) - 1152 *a^6*c*integrate(1/960*x^6*log(a^2*x^2 + 1)/(a^5*x^2 + a^3), x) + 46080*a^ 5*c*integrate(1/960*x^5*arctan(a*x)^3/(a^5*x^2 + a^3), x) + 2304*a^5*c*int egrate(1/960*x^5*arctan(a*x)/(a^5*x^2 + a^3), x) - 8640*a^4*c*integrate(1/ 960*x^4*arctan(a*x)^2/(a^5*x^2 + a^3), x) - 2160*a^4*c*integrate(1/960*x^4 *log(a^2*x^2 + 1)^2/(a^5*x^2 + a^3), x) - 960*a^4*c*integrate(1/960*x^4*lo g(a^2*x^2 + 1)/(a^5*x^2 + a^3), x) + 23040*a^3*c*integrate(1/960*x^3*arcta n(a*x)^3/(a^5*x^2 + a^3), x) + 1920*a^3*c*integrate(1/960*x^3*arctan(a*x)/ (a^5*x^2 + a^3), x) + 2880*a^2*c*integrate(1/960*x^2*log(a^2*x^2 + 1)/(a^5 *x^2 + a^3), x) - 5760*a*c*integrate(1/960*x*arctan(a*x)/(a^5*x^2 + a^3), x) + 720*c*integrate(1/960*log(a^2*x^2 + 1)^2/(a^5*x^2 + a^3), x) + c*arct an(a*x)^3/a^4)*a^4 + 40*(2*a^6*c*x^6 + 3*a^4*c*x^4 - c)*arctan(a*x)^3 - 4* (6*a^5*c*x^5 + 5*a^3*c*x^3 - 15*a*c*x)*arctan(a*x)^2 + (6*a^5*c*x^5 + 5*a^ 3*c*x^3 - 15*a*c*x)*log(a^2*x^2 + 1)^2)/a^4
\[ \int x^3 \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )} x^{3} \arctan \left (a x\right )^{3} \,d x } \] Input:
integrate(x^3*(a^2*c*x^2+c)*arctan(a*x)^3,x, algorithm="giac")
Output:
integrate((a^2*c*x^2 + c)*x^3*arctan(a*x)^3, x)
Timed out. \[ \int x^3 \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=\int x^3\,{\mathrm {atan}\left (a\,x\right )}^3\,\left (c\,a^2\,x^2+c\right ) \,d x \] Input:
int(x^3*atan(a*x)^3*(c + a^2*c*x^2),x)
Output:
int(x^3*atan(a*x)^3*(c + a^2*c*x^2), x)
\[ \int x^3 \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=\frac {c \left (10 \mathit {atan} \left (a x \right )^{3} a^{6} x^{6}+15 \mathit {atan} \left (a x \right )^{3} a^{4} x^{4}-5 \mathit {atan} \left (a x \right )^{3}-6 \mathit {atan} \left (a x \right )^{2} a^{5} x^{5}-5 \mathit {atan} \left (a x \right )^{2} a^{3} x^{3}+15 \mathit {atan} \left (a x \right )^{2} a x +3 \mathit {atan} \left (a x \right ) a^{4} x^{4}-\mathit {atan} \left (a x \right ) a^{2} x^{2}-4 \mathit {atan} \left (a x \right )-28 \left (\int \frac {\mathit {atan} \left (a x \right ) x}{a^{2} x^{2}+1}d x \right ) a^{2}-a^{3} x^{3}+4 a x \right )}{60 a^{4}} \] Input:
int(x^3*(a^2*c*x^2+c)*atan(a*x)^3,x)
Output:
(c*(10*atan(a*x)**3*a**6*x**6 + 15*atan(a*x)**3*a**4*x**4 - 5*atan(a*x)**3 - 6*atan(a*x)**2*a**5*x**5 - 5*atan(a*x)**2*a**3*x**3 + 15*atan(a*x)**2*a *x + 3*atan(a*x)*a**4*x**4 - atan(a*x)*a**2*x**2 - 4*atan(a*x) - 28*int((a tan(a*x)*x)/(a**2*x**2 + 1),x)*a**2 - a**3*x**3 + 4*a*x))/(60*a**4)