Integrand size = 19, antiderivative size = 388 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=-\frac {13 c^3 \left (1+a^2 x^2\right )}{210 a}-\frac {c^3 \left (1+a^2 x^2\right )^2}{140 a}+\frac {14}{15} c^3 x \arctan (a x)+\frac {13}{105} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)+\frac {1}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)-\frac {12 c^3 \left (1+a^2 x^2\right ) \arctan (a x)^2}{35 a}-\frac {9 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{70 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)^2}{14 a}+\frac {16 i c^3 \arctan (a x)^3}{35 a}+\frac {16}{35} c^3 x \arctan (a x)^3+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^3+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^3+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^3+\frac {48 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{35 a}-\frac {7 c^3 \log \left (1+a^2 x^2\right )}{15 a}+\frac {48 i c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{35 a}+\frac {24 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{35 a} \]
-13/210*c^3*(a^2*x^2+1)/a-1/140*c^3*(a^2*x^2+1)^2/a+14/15*c^3*x*arctan(a*x )+13/105*c^3*x*(a^2*x^2+1)*arctan(a*x)+1/35*c^3*x*(a^2*x^2+1)^2*arctan(a*x )-12/35*c^3*(a^2*x^2+1)*arctan(a*x)^2/a-9/70*c^3*(a^2*x^2+1)^2*arctan(a*x) ^2/a-1/14*c^3*(a^2*x^2+1)^3*arctan(a*x)^2/a+48/35*I*c^3*arctan(a*x)*polylo g(2,1-2/(1+I*a*x))/a+16/35*c^3*x*arctan(a*x)^3+8/35*c^3*x*(a^2*x^2+1)*arct an(a*x)^3+6/35*c^3*x*(a^2*x^2+1)^2*arctan(a*x)^3+1/7*c^3*x*(a^2*x^2+1)^3*a rctan(a*x)^3+48/35*c^3*arctan(a*x)^2*ln(2/(1+I*a*x))/a-7/15*c^3*ln(a^2*x^2 +1)/a+16/35*I*c^3*arctan(a*x)^3/a+24/35*c^3*polylog(3,1-2/(1+I*a*x))/a
Time = 1.00 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.63 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\frac {c^3 \left (-29-32 a^2 x^2-3 a^4 x^4+456 a x \arctan (a x)+76 a^3 x^3 \arctan (a x)+12 a^5 x^5 \arctan (a x)-228 \arctan (a x)^2-342 a^2 x^2 \arctan (a x)^2-144 a^4 x^4 \arctan (a x)^2-30 a^6 x^6 \arctan (a x)^2-192 i \arctan (a x)^3+420 a x \arctan (a x)^3+420 a^3 x^3 \arctan (a x)^3+252 a^5 x^5 \arctan (a x)^3+60 a^7 x^7 \arctan (a x)^3+576 \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )-196 \log \left (1+a^2 x^2\right )-576 i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+288 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )\right )}{420 a} \]
(c^3*(-29 - 32*a^2*x^2 - 3*a^4*x^4 + 456*a*x*ArcTan[a*x] + 76*a^3*x^3*ArcT an[a*x] + 12*a^5*x^5*ArcTan[a*x] - 228*ArcTan[a*x]^2 - 342*a^2*x^2*ArcTan[ a*x]^2 - 144*a^4*x^4*ArcTan[a*x]^2 - 30*a^6*x^6*ArcTan[a*x]^2 - (192*I)*Ar cTan[a*x]^3 + 420*a*x*ArcTan[a*x]^3 + 420*a^3*x^3*ArcTan[a*x]^3 + 252*a^5* x^5*ArcTan[a*x]^3 + 60*a^7*x^7*ArcTan[a*x]^3 + 576*ArcTan[a*x]^2*Log[1 + E ^((2*I)*ArcTan[a*x])] - 196*Log[1 + a^2*x^2] - (576*I)*ArcTan[a*x]*PolyLog [2, -E^((2*I)*ArcTan[a*x])] + 288*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(42 0*a)
Time = 2.22 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.22, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.895, Rules used = {5415, 27, 5413, 5413, 5345, 240, 5415, 5413, 5345, 240, 5415, 5345, 240, 5455, 5379, 5529, 7164}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \arctan (a x)^3 \left (a^2 c x^2+c\right )^3 \, dx\) |
| \(\Big \downarrow \) 5415 |
| \(\displaystyle \frac {1}{7} c \int c^2 \left (a^2 x^2+1\right )^2 \arctan (a x)dx+\frac {6}{7} c \int c^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^3dx+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^3-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)^2}{14 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} c^3 \int \left (a^2 x^2+1\right )^2 \arctan (a x)dx+\frac {6}{7} c^3 \int \left (a^2 x^2+1\right )^2 \arctan (a x)^3dx+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^3-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)^2}{14 a}\) |
| \(\Big \downarrow \) 5413 |
| \(\displaystyle \frac {1}{7} c^3 \left (\frac {4}{5} \int \left (a^2 x^2+1\right ) \arctan (a x)dx+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)-\frac {\left (a^2 x^2+1\right )^2}{20 a}\right )+\frac {6}{7} c^3 \int \left (a^2 x^2+1\right )^2 \arctan (a x)^3dx+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^3-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)^2}{14 a}\) |
| \(\Big \downarrow \) 5413 |
| \(\displaystyle \frac {1}{7} c^3 \left (\frac {4}{5} \left (\frac {2}{3} \int \arctan (a x)dx+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)-\frac {a^2 x^2+1}{6 a}\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)-\frac {\left (a^2 x^2+1\right )^2}{20 a}\right )+\frac {6}{7} c^3 \int \left (a^2 x^2+1\right )^2 \arctan (a x)^3dx+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^3-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)^2}{14 a}\) |
| \(\Big \downarrow \) 5345 |
| \(\displaystyle \frac {1}{7} c^3 \left (\frac {4}{5} \left (\frac {2}{3} \left (x \arctan (a x)-a \int \frac {x}{a^2 x^2+1}dx\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)-\frac {a^2 x^2+1}{6 a}\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)-\frac {\left (a^2 x^2+1\right )^2}{20 a}\right )+\frac {6}{7} c^3 \int \left (a^2 x^2+1\right )^2 \arctan (a x)^3dx+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^3-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)^2}{14 a}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {6}{7} c^3 \int \left (a^2 x^2+1\right )^2 \arctan (a x)^3dx+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^3-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)^2}{14 a}+\frac {1}{7} c^3 \left (\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)+\frac {4}{5} \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )-\frac {\left (a^2 x^2+1\right )^2}{20 a}\right )\) |
| \(\Big \downarrow \) 5415 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {3}{10} \int \left (a^2 x^2+1\right ) \arctan (a x)dx+\frac {4}{5} \int \left (a^2 x^2+1\right ) \arctan (a x)^3dx+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^3-\frac {3 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{20 a}\right )+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^3-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)^2}{14 a}+\frac {1}{7} c^3 \left (\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)+\frac {4}{5} \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )-\frac {\left (a^2 x^2+1\right )^2}{20 a}\right )\) |
| \(\Big \downarrow \) 5413 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {3}{10} \left (\frac {2}{3} \int \arctan (a x)dx+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)-\frac {a^2 x^2+1}{6 a}\right )+\frac {4}{5} \int \left (a^2 x^2+1\right ) \arctan (a x)^3dx+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^3-\frac {3 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{20 a}\right )+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^3-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)^2}{14 a}+\frac {1}{7} c^3 \left (\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)+\frac {4}{5} \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )-\frac {\left (a^2 x^2+1\right )^2}{20 a}\right )\) |
| \(\Big \downarrow \) 5345 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {3}{10} \left (\frac {2}{3} \left (x \arctan (a x)-a \int \frac {x}{a^2 x^2+1}dx\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)-\frac {a^2 x^2+1}{6 a}\right )+\frac {4}{5} \int \left (a^2 x^2+1\right ) \arctan (a x)^3dx+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^3-\frac {3 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{20 a}\right )+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^3-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)^2}{14 a}+\frac {1}{7} c^3 \left (\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)+\frac {4}{5} \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )-\frac {\left (a^2 x^2+1\right )^2}{20 a}\right )\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \int \left (a^2 x^2+1\right ) \arctan (a x)^3dx+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^3-\frac {3 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{20 a}+\frac {3}{10} \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )\right )+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^3-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)^2}{14 a}+\frac {1}{7} c^3 \left (\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)+\frac {4}{5} \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )-\frac {\left (a^2 x^2+1\right )^2}{20 a}\right )\) |
| \(\Big \downarrow \) 5415 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \left (\int \arctan (a x)dx+\frac {2}{3} \int \arctan (a x)^3dx+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)^3-\frac {\left (a^2 x^2+1\right ) \arctan (a x)^2}{2 a}\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^3-\frac {3 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{20 a}+\frac {3}{10} \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )\right )+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^3-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)^2}{14 a}+\frac {1}{7} c^3 \left (\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)+\frac {4}{5} \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )-\frac {\left (a^2 x^2+1\right )^2}{20 a}\right )\) |
| \(\Big \downarrow \) 5345 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \left (\frac {2}{3} \left (x \arctan (a x)^3-3 a \int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx\right )-a \int \frac {x}{a^2 x^2+1}dx+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)^3-\frac {\left (a^2 x^2+1\right ) \arctan (a x)^2}{2 a}+x \arctan (a x)\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^3-\frac {3 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{20 a}+\frac {3}{10} \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )\right )+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^3-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)^2}{14 a}+\frac {1}{7} c^3 \left (\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)+\frac {4}{5} \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )-\frac {\left (a^2 x^2+1\right )^2}{20 a}\right )\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \left (\frac {2}{3} \left (x \arctan (a x)^3-3 a \int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)^3-\frac {\left (a^2 x^2+1\right ) \arctan (a x)^2}{2 a}-\frac {\log \left (a^2 x^2+1\right )}{2 a}+x \arctan (a x)\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^3-\frac {3 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{20 a}+\frac {3}{10} \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )\right )+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^3-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)^2}{14 a}+\frac {1}{7} c^3 \left (\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)+\frac {4}{5} \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )-\frac {\left (a^2 x^2+1\right )^2}{20 a}\right )\) |
| \(\Big \downarrow \) 5455 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \left (\frac {2}{3} \left (x \arctan (a x)^3-3 a \left (-\frac {\int \frac {\arctan (a x)^2}{i-a x}dx}{a}-\frac {i \arctan (a x)^3}{3 a^2}\right )\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)^3-\frac {\left (a^2 x^2+1\right ) \arctan (a x)^2}{2 a}-\frac {\log \left (a^2 x^2+1\right )}{2 a}+x \arctan (a x)\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^3-\frac {3 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{20 a}+\frac {3}{10} \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )\right )+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^3-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)^2}{14 a}+\frac {1}{7} c^3 \left (\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)+\frac {4}{5} \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )-\frac {\left (a^2 x^2+1\right )^2}{20 a}\right )\) |
| \(\Big \downarrow \) 5379 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \left (\frac {2}{3} \left (x \arctan (a x)^3-3 a \left (-\frac {\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-2 \int \frac {\arctan (a x) \log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx}{a}-\frac {i \arctan (a x)^3}{3 a^2}\right )\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)^3-\frac {\left (a^2 x^2+1\right ) \arctan (a x)^2}{2 a}-\frac {\log \left (a^2 x^2+1\right )}{2 a}+x \arctan (a x)\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^3-\frac {3 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{20 a}+\frac {3}{10} \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )\right )+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^3-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)^2}{14 a}+\frac {1}{7} c^3 \left (\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)+\frac {4}{5} \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )-\frac {\left (a^2 x^2+1\right )^2}{20 a}\right )\) |
| \(\Big \downarrow \) 5529 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \left (\frac {2}{3} \left (x \arctan (a x)^3-3 a \left (-\frac {\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-2 \left (\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}\right )}{a}-\frac {i \arctan (a x)^3}{3 a^2}\right )\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)^3-\frac {\left (a^2 x^2+1\right ) \arctan (a x)^2}{2 a}-\frac {\log \left (a^2 x^2+1\right )}{2 a}+x \arctan (a x)\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^3-\frac {3 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{20 a}+\frac {3}{10} \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )\right )+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^3-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)^2}{14 a}+\frac {1}{7} c^3 \left (\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)+\frac {4}{5} \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )-\frac {\left (a^2 x^2+1\right )^2}{20 a}\right )\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \left (\frac {2}{3} \left (x \arctan (a x)^3-3 a \left (-\frac {i \arctan (a x)^3}{3 a^2}-\frac {\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-2 \left (-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}-\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{4 a}\right )}{a}\right )\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)^3-\frac {\left (a^2 x^2+1\right ) \arctan (a x)^2}{2 a}-\frac {\log \left (a^2 x^2+1\right )}{2 a}+x \arctan (a x)\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^3-\frac {3 \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{20 a}+\frac {3}{10} \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )\right )+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^3-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)^2}{14 a}+\frac {1}{7} c^3 \left (\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)+\frac {4}{5} \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )-\frac {\left (a^2 x^2+1\right )^2}{20 a}\right )\) |
-1/14*(c^3*(1 + a^2*x^2)^3*ArcTan[a*x]^2)/a + (c^3*x*(1 + a^2*x^2)^3*ArcTa n[a*x]^3)/7 + (c^3*(-1/20*(1 + a^2*x^2)^2/a + (x*(1 + a^2*x^2)^2*ArcTan[a* x])/5 + (4*(-1/6*(1 + a^2*x^2)/a + (x*(1 + a^2*x^2)*ArcTan[a*x])/3 + (2*(x *ArcTan[a*x] - Log[1 + a^2*x^2]/(2*a)))/3))/5))/7 + (6*c^3*((-3*(1 + a^2*x ^2)^2*ArcTan[a*x]^2)/(20*a) + (x*(1 + a^2*x^2)^2*ArcTan[a*x]^3)/5 + (3*(-1 /6*(1 + a^2*x^2)/a + (x*(1 + a^2*x^2)*ArcTan[a*x])/3 + (2*(x*ArcTan[a*x] - Log[1 + a^2*x^2]/(2*a)))/3))/10 + (4*(x*ArcTan[a*x] - ((1 + a^2*x^2)*ArcT an[a*x]^2)/(2*a) + (x*(1 + a^2*x^2)*ArcTan[a*x]^3)/3 - Log[1 + a^2*x^2]/(2 *a) + (2*(x*ArcTan[a*x]^3 - 3*a*(((-1/3*I)*ArcTan[a*x]^3)/a^2 - ((ArcTan[a *x]^2*Log[2/(1 + I*a*x)])/a - 2*(((-1/2*I)*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a - PolyLog[3, 1 - 2/(1 + I*a*x)]/(4*a)))/a)))/3))/5))/7
3.4.82.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
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_)]*(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_.))*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbo l] :> Simp[(-b)*((d + e*x^2)^q/(2*c*q*(2*q + 1))), x] + (Simp[x*(d + e*x^2) ^q*((a + b*ArcTan[c*x])/(2*q + 1)), x] + Simp[2*d*(q/(2*q + 1)) Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x]), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_.), x_ Symbol] :> Simp[(-b)*p*(d + e*x^2)^q*((a + b*ArcTan[c*x])^(p - 1)/(2*c*q*(2 *q + 1))), x] + (Simp[x*(d + e*x^2)^q*((a + b*ArcTan[c*x])^p/(2*q + 1)), x] + Simp[2*d*(q/(2*q + 1)) Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Simp[b^2*d*p*((p - 1)/(2*q*(2*q + 1))) Int[(d + e*x^2)^(q - 1)*( a + b*ArcTan[c*x])^(p - 2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && GtQ[p, 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[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2 ), x_Symbol] :> Simp[(-I)*(a + b*ArcTan[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)) , x] + Simp[b*p*(I/2) Int[(a + b*ArcTan[c*x])^(p - 1)*(PolyLog[2, 1 - u]/ (d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c ^2*d] && EqQ[(1 - u)^2 - (1 - 2*(I/(I - c*x)))^2, 0]
Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /; !FalseQ[w]] /; FreeQ[n, x]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 86.79 (sec) , antiderivative size = 1267, normalized size of antiderivative = 3.27
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1267\) |
| default | \(\text {Expression too large to display}\) | \(1267\) |
| parts | \(\text {Expression too large to display}\) | \(1268\) |
1/a*(1/7*c^3*arctan(a*x)^3*a^7*x^7+3/5*c^3*arctan(a*x)^3*a^5*x^5+c^3*arcta n(a*x)^3*a^3*x^3+c^3*arctan(a*x)^3*a*x-3/35*c^3*(4*I*Pi*csgn(I/((1+I*a*x)^ 2/(a^2*x^2+1)+1)^2)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*csgn(I*(1+I*a*x)^2/(a^ 2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*arctan(a*x)^2+19/3*arctan(a*x)^2+1 9/2*x^2*arctan(a*x)^2*a^2-16*arctan(a*x)^2*ln(2)+1/12*(I+a*x)^4+5/6*a^6*x^ 6*arctan(a*x)^2-11/3*arctan(a*x)*(a*x-I)^2*(I+a*x)+5/3*arctan(a*x)*(a*x-I) ^4*(I+a*x)+10/3*arctan(a*x)*(a*x-I)^2*(I+a*x)^3+11/3*arctan(a*x)*(a*x-I)*( I+a*x)^2-10/3*arctan(a*x)*(a*x-I)^3*(I+a*x)^2-5/3*arctan(a*x)*(a*x-I)*(I+a *x)^4+4*a^4*arctan(a*x)^2*x^4+11/9*arctan(a*x)*(a*x-I)^3-8*arctan(a*x)*(a* x-I)-1/3*arctan(a*x)*(a*x-I)^5-8*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))-98/9* ln((1+I*a*x)^2/(a^2*x^2+1)+1)+8*arctan(a*x)^2*ln(a^2*x^2+1)+7/18*(I+a*x)^2 -5/3*I*arctan(a*x)*(a*x-I)^4-4*I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^ 3*arctan(a*x)^2-10*I*arctan(a*x)*(a*x-I)^2*(I+a*x)^2+20/3*I*arctan(a*x)*(a *x-I)^3*(I+a*x)+20/3*I*arctan(a*x)*(a*x-I)*(I+a*x)^3+6*I*arctan(a*x)*(a*x- I)*(I+a*x)+4*I*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+ 1)^2)^3*arctan(a*x)^2+4*I*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3*arctan(a*x) ^2-8*I*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*csgn(I*(1+I*a*x)^2/(a^2*x^2+ 1))^2*arctan(a*x)^2+8*I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(I*((1+ I*a*x)^2/(a^2*x^2+1)+1)^2)^2*arctan(a*x)^2-4*I*Pi*csgn(I/((1+I*a*x)^2/(a^2 *x^2+1)+1)^2)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1...
\[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{3} \,d x } \]
\[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=c^{3} \left (\int 3 a^{2} x^{2} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 3 a^{4} x^{4} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{6} x^{6} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]
c**3*(Integral(3*a**2*x**2*atan(a*x)**3, x) + Integral(3*a**4*x**4*atan(a* x)**3, x) + Integral(a**6*x**6*atan(a*x)**3, x) + Integral(atan(a*x)**3, x ))
\[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{3} \,d x } \]
980*a^8*c^3*integrate(1/1120*x^8*arctan(a*x)^3/(a^2*x^2 + 1), x) + 105*a^8 *c^3*integrate(1/1120*x^8*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 60*a^8*c^3*integrate(1/1120*x^8*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) - 60*a^7*c^3*integrate(1/1120*x^7*arctan(a*x)^2/(a^2*x^2 + 1), x) + 15*a^7*c^3*integrate(1/1120*x^7*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 3 920*a^6*c^3*integrate(1/1120*x^6*arctan(a*x)^3/(a^2*x^2 + 1), x) + 420*a^6 *c^3*integrate(1/1120*x^6*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 252*a^6*c^3*integrate(1/1120*x^6*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) - 252*a^5*c^3*integrate(1/1120*x^5*arctan(a*x)^2/(a^2*x^2 + 1), x ) + 63*a^5*c^3*integrate(1/1120*x^5*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 5880*a^4*c^3*integrate(1/1120*x^4*arctan(a*x)^3/(a^2*x^2 + 1), x) + 630*a ^4*c^3*integrate(1/1120*x^4*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 420*a^4*c^3*integrate(1/1120*x^4*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^ 2 + 1), x) - 420*a^3*c^3*integrate(1/1120*x^3*arctan(a*x)^2/(a^2*x^2 + 1), x) + 105*a^3*c^3*integrate(1/1120*x^3*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x ) + 7/32*c^3*arctan(a*x)^4/a + 3920*a^2*c^3*integrate(1/1120*x^2*arctan(a* x)^3/(a^2*x^2 + 1), x) + 420*a^2*c^3*integrate(1/1120*x^2*arctan(a*x)*log( a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 420*a^2*c^3*integrate(1/1120*x^2*arctan (a*x)*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) - 420*a*c^3*integrate(1/1120*x*ar ctan(a*x)^2/(a^2*x^2 + 1), x) + 105*a*c^3*integrate(1/1120*x*log(a^2*x^...
\[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{3} \,d x } \]
Timed out. \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\int {\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^3 \,d x \]