Integrand size = 8, antiderivative size = 153 \[ \int x^3 \tan ^6(x) \, dx=\frac {19 x^2}{20}-\frac {23 i x^3}{15}-\frac {x^4}{4}+\frac {23}{5} x^2 \log \left (1+e^{2 i x}\right )-2 \log (\cos (x))-\frac {23}{5} i x \operatorname {PolyLog}\left (2,-e^{2 i x}\right )+\frac {23}{10} \operatorname {PolyLog}\left (3,-e^{2 i x}\right )-\frac {19}{10} x \tan (x)+x^3 \tan (x)-\frac {\tan ^2(x)}{20}+\frac {4}{5} x^2 \tan ^2(x)+\frac {1}{10} x \tan ^3(x)-\frac {1}{3} x^3 \tan ^3(x)-\frac {3}{20} x^2 \tan ^4(x)+\frac {1}{5} x^3 \tan ^5(x) \]
19/20*x^2-23/15*I*x^3-1/4*x^4+23/5*x^2*ln(1+exp(2*I*x))-2*ln(cos(x))-23/5* I*x*polylog(2,-exp(2*I*x))+23/10*polylog(3,-exp(2*I*x))-19/10*x*tan(x)+x^3 *tan(x)-1/20*tan(x)^2+4/5*x^2*tan(x)^2+1/10*x*tan(x)^3-1/3*x^3*tan(x)^3-3/ 20*x^2*tan(x)^4+1/5*x^3*tan(x)^5
Time = 0.23 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.87 \[ \int x^3 \tan ^6(x) \, dx=\frac {1}{60} \left (-92 i x^3-15 x^4+276 x^2 \log \left (1+e^{2 i x}\right )-120 \log (\cos (x))-276 i x \operatorname {PolyLog}\left (2,-e^{2 i x}\right )+138 \operatorname {PolyLog}\left (3,-e^{2 i x}\right )-3 \sec ^2(x)+66 x^2 \sec ^2(x)-9 x^2 \sec ^4(x)-120 x \tan (x)+92 x^3 \tan (x)+6 x \sec ^2(x) \tan (x)-44 x^3 \sec ^2(x) \tan (x)+12 x^3 \sec ^4(x) \tan (x)\right ) \]
((-92*I)*x^3 - 15*x^4 + 276*x^2*Log[1 + E^((2*I)*x)] - 120*Log[Cos[x]] - ( 276*I)*x*PolyLog[2, -E^((2*I)*x)] + 138*PolyLog[3, -E^((2*I)*x)] - 3*Sec[x ]^2 + 66*x^2*Sec[x]^2 - 9*x^2*Sec[x]^4 - 120*x*Tan[x] + 92*x^3*Tan[x] + 6* x*Sec[x]^2*Tan[x] - 44*x^3*Sec[x]^2*Tan[x] + 12*x^3*Sec[x]^4*Tan[x])/60
Time = 1.57 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.92, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 2.500, Rules used = {3042, 4203, 3042, 4203, 3042, 4203, 15, 3042, 3954, 3042, 3956, 4202, 2620, 3011, 2720, 4203, 15, 3042, 3956, 7143}
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.
| \(\displaystyle \int x^3 \tan ^6(x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int x^3 \tan (x)^6dx\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle -\int x^3 \tan ^4(x)dx-\frac {3}{5} \int x^2 \tan ^5(x)dx+\frac {1}{5} x^3 \tan ^5(x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int x^3 \tan (x)^4dx-\frac {3}{5} \int x^2 \tan (x)^5dx+\frac {1}{5} x^3 \tan ^5(x)\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle \int x^3 \tan ^2(x)dx+\int x^2 \tan ^3(x)dx-\frac {3}{5} \left (-\int x^2 \tan ^3(x)dx-\frac {1}{2} \int x \tan ^4(x)dx+\frac {1}{4} x^2 \tan ^4(x)\right )+\frac {1}{5} x^3 \tan ^5(x)-\frac {1}{3} x^3 \tan ^3(x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int x^3 \tan (x)^2dx-\frac {3}{5} \left (-\int x^2 \tan (x)^3dx-\frac {1}{2} \int x \tan (x)^4dx+\frac {1}{4} x^2 \tan ^4(x)\right )+\int x^2 \tan (x)^3dx+\frac {1}{5} x^3 \tan ^5(x)-\frac {1}{3} x^3 \tan ^3(x)\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle -\int x^3dx-\frac {3}{5} \left (\int x^2 \tan (x)dx+\int x \tan ^2(x)dx+\frac {1}{2} \left (\frac {1}{3} \int \tan ^3(x)dx+\int x \tan ^2(x)dx-\frac {1}{3} x \tan ^3(x)\right )+\frac {1}{4} x^2 \tan ^4(x)-\frac {1}{2} x^2 \tan ^2(x)\right )-4 \int x^2 \tan (x)dx-\int x \tan ^2(x)dx+\frac {1}{5} x^3 \tan ^5(x)-\frac {1}{3} x^3 \tan ^3(x)+x^3 \tan (x)+\frac {1}{2} x^2 \tan ^2(x)\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {3}{5} \left (\int x^2 \tan (x)dx+\int x \tan ^2(x)dx+\frac {1}{2} \left (\frac {1}{3} \int \tan ^3(x)dx+\int x \tan ^2(x)dx-\frac {1}{3} x \tan ^3(x)\right )+\frac {1}{4} x^2 \tan ^4(x)-\frac {1}{2} x^2 \tan ^2(x)\right )-4 \int x^2 \tan (x)dx-\int x \tan ^2(x)dx-\frac {x^4}{4}+\frac {1}{5} x^3 \tan ^5(x)-\frac {1}{3} x^3 \tan ^3(x)+x^3 \tan (x)+\frac {1}{2} x^2 \tan ^2(x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3}{5} \left (\int x^2 \tan (x)dx+\frac {1}{2} \left (\int x \tan (x)^2dx+\frac {1}{3} \int \tan (x)^3dx-\frac {1}{3} x \tan ^3(x)\right )+\int x \tan (x)^2dx+\frac {1}{4} x^2 \tan ^4(x)-\frac {1}{2} x^2 \tan ^2(x)\right )-4 \int x^2 \tan (x)dx-\int x \tan (x)^2dx-\frac {x^4}{4}+\frac {1}{5} x^3 \tan ^5(x)-\frac {1}{3} x^3 \tan ^3(x)+x^3 \tan (x)+\frac {1}{2} x^2 \tan ^2(x)\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle -\frac {3}{5} \left (\int x^2 \tan (x)dx+\frac {1}{2} \left (\frac {1}{3} \left (\frac {\tan ^2(x)}{2}-\int \tan (x)dx\right )+\int x \tan (x)^2dx-\frac {1}{3} x \tan ^3(x)\right )+\int x \tan (x)^2dx+\frac {1}{4} x^2 \tan ^4(x)-\frac {1}{2} x^2 \tan ^2(x)\right )-4 \int x^2 \tan (x)dx-\int x \tan (x)^2dx-\frac {x^4}{4}+\frac {1}{5} x^3 \tan ^5(x)-\frac {1}{3} x^3 \tan ^3(x)+x^3 \tan (x)+\frac {1}{2} x^2 \tan ^2(x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3}{5} \left (\int x^2 \tan (x)dx+\frac {1}{2} \left (\frac {1}{3} \left (\frac {\tan ^2(x)}{2}-\int \tan (x)dx\right )+\int x \tan (x)^2dx-\frac {1}{3} x \tan ^3(x)\right )+\int x \tan (x)^2dx+\frac {1}{4} x^2 \tan ^4(x)-\frac {1}{2} x^2 \tan ^2(x)\right )-4 \int x^2 \tan (x)dx-\int x \tan (x)^2dx-\frac {x^4}{4}+\frac {1}{5} x^3 \tan ^5(x)-\frac {1}{3} x^3 \tan ^3(x)+x^3 \tan (x)+\frac {1}{2} x^2 \tan ^2(x)\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle -4 \int x^2 \tan (x)dx-\frac {3}{5} \left (\int x^2 \tan (x)dx+\int x \tan (x)^2dx+\frac {1}{2} \left (\int x \tan (x)^2dx-\frac {1}{3} x \tan ^3(x)+\frac {1}{3} \left (\frac {\tan ^2(x)}{2}+\log (\cos (x))\right )\right )+\frac {1}{4} x^2 \tan ^4(x)-\frac {1}{2} x^2 \tan ^2(x)\right )-\int x \tan (x)^2dx-\frac {x^4}{4}+\frac {1}{5} x^3 \tan ^5(x)-\frac {1}{3} x^3 \tan ^3(x)+x^3 \tan (x)+\frac {1}{2} x^2 \tan ^2(x)\) |
| \(\Big \downarrow \) 4202 |
| \(\displaystyle -4 \left (\frac {i x^3}{3}-2 i \int \frac {e^{2 i x} x^2}{1+e^{2 i x}}dx\right )-\frac {3}{5} \left (-2 i \int \frac {e^{2 i x} x^2}{1+e^{2 i x}}dx+\int x \tan (x)^2dx+\frac {1}{2} \left (\int x \tan (x)^2dx-\frac {1}{3} x \tan ^3(x)+\frac {1}{3} \left (\frac {\tan ^2(x)}{2}+\log (\cos (x))\right )\right )+\frac {i x^3}{3}+\frac {1}{4} x^2 \tan ^4(x)-\frac {1}{2} x^2 \tan ^2(x)\right )-\int x \tan (x)^2dx-\frac {x^4}{4}+\frac {1}{5} x^3 \tan ^5(x)-\frac {1}{3} x^3 \tan ^3(x)+x^3 \tan (x)+\frac {1}{2} x^2 \tan ^2(x)\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -4 \left (\frac {i x^3}{3}-2 i \left (i \int x \log \left (1+e^{2 i x}\right )dx-\frac {1}{2} i x^2 \log \left (1+e^{2 i x}\right )\right )\right )-\frac {3}{5} \left (-2 i \left (i \int x \log \left (1+e^{2 i x}\right )dx-\frac {1}{2} i x^2 \log \left (1+e^{2 i x}\right )\right )+\int x \tan (x)^2dx+\frac {1}{2} \left (\int x \tan (x)^2dx-\frac {1}{3} x \tan ^3(x)+\frac {1}{3} \left (\frac {\tan ^2(x)}{2}+\log (\cos (x))\right )\right )+\frac {i x^3}{3}+\frac {1}{4} x^2 \tan ^4(x)-\frac {1}{2} x^2 \tan ^2(x)\right )-\int x \tan (x)^2dx-\frac {x^4}{4}+\frac {1}{5} x^3 \tan ^5(x)-\frac {1}{3} x^3 \tan ^3(x)+x^3 \tan (x)+\frac {1}{2} x^2 \tan ^2(x)\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -4 \left (\frac {i x^3}{3}-2 i \left (i \left (\frac {1}{2} i x \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{2} i \int \operatorname {PolyLog}\left (2,-e^{2 i x}\right )dx\right )-\frac {1}{2} i x^2 \log \left (1+e^{2 i x}\right )\right )\right )-\frac {3}{5} \left (-2 i \left (i \left (\frac {1}{2} i x \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{2} i \int \operatorname {PolyLog}\left (2,-e^{2 i x}\right )dx\right )-\frac {1}{2} i x^2 \log \left (1+e^{2 i x}\right )\right )+\int x \tan (x)^2dx+\frac {1}{2} \left (\int x \tan (x)^2dx-\frac {1}{3} x \tan ^3(x)+\frac {1}{3} \left (\frac {\tan ^2(x)}{2}+\log (\cos (x))\right )\right )+\frac {i x^3}{3}+\frac {1}{4} x^2 \tan ^4(x)-\frac {1}{2} x^2 \tan ^2(x)\right )-\int x \tan (x)^2dx-\frac {x^4}{4}+\frac {1}{5} x^3 \tan ^5(x)-\frac {1}{3} x^3 \tan ^3(x)+x^3 \tan (x)+\frac {1}{2} x^2 \tan ^2(x)\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -4 \left (\frac {i x^3}{3}-2 i \left (i \left (\frac {1}{2} i x \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{4} \int e^{-2 i x} \operatorname {PolyLog}\left (2,-e^{2 i x}\right )de^{2 i x}\right )-\frac {1}{2} i x^2 \log \left (1+e^{2 i x}\right )\right )\right )-\frac {3}{5} \left (-2 i \left (i \left (\frac {1}{2} i x \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{4} \int e^{-2 i x} \operatorname {PolyLog}\left (2,-e^{2 i x}\right )de^{2 i x}\right )-\frac {1}{2} i x^2 \log \left (1+e^{2 i x}\right )\right )+\int x \tan (x)^2dx+\frac {1}{2} \left (\int x \tan (x)^2dx-\frac {1}{3} x \tan ^3(x)+\frac {1}{3} \left (\frac {\tan ^2(x)}{2}+\log (\cos (x))\right )\right )+\frac {i x^3}{3}+\frac {1}{4} x^2 \tan ^4(x)-\frac {1}{2} x^2 \tan ^2(x)\right )-\int x \tan (x)^2dx-\frac {x^4}{4}+\frac {1}{5} x^3 \tan ^5(x)-\frac {1}{3} x^3 \tan ^3(x)+x^3 \tan (x)+\frac {1}{2} x^2 \tan ^2(x)\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle -4 \left (\frac {i x^3}{3}-2 i \left (i \left (\frac {1}{2} i x \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{4} \int e^{-2 i x} \operatorname {PolyLog}\left (2,-e^{2 i x}\right )de^{2 i x}\right )-\frac {1}{2} i x^2 \log \left (1+e^{2 i x}\right )\right )\right )-\frac {3}{5} \left (-2 i \left (i \left (\frac {1}{2} i x \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{4} \int e^{-2 i x} \operatorname {PolyLog}\left (2,-e^{2 i x}\right )de^{2 i x}\right )-\frac {1}{2} i x^2 \log \left (1+e^{2 i x}\right )\right )-\int xdx-\int \tan (x)dx+\frac {1}{2} \left (-\int xdx-\int \tan (x)dx-\frac {1}{3} x \tan ^3(x)+x \tan (x)+\frac {1}{3} \left (\frac {\tan ^2(x)}{2}+\log (\cos (x))\right )\right )+\frac {i x^3}{3}+\frac {1}{4} x^2 \tan ^4(x)-\frac {1}{2} x^2 \tan ^2(x)+x \tan (x)\right )+\int xdx+\int \tan (x)dx-\frac {x^4}{4}+\frac {1}{5} x^3 \tan ^5(x)-\frac {1}{3} x^3 \tan ^3(x)+x^3 \tan (x)+\frac {1}{2} x^2 \tan ^2(x)-x \tan (x)\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -4 \left (\frac {i x^3}{3}-2 i \left (i \left (\frac {1}{2} i x \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{4} \int e^{-2 i x} \operatorname {PolyLog}\left (2,-e^{2 i x}\right )de^{2 i x}\right )-\frac {1}{2} i x^2 \log \left (1+e^{2 i x}\right )\right )\right )-\frac {3}{5} \left (-2 i \left (i \left (\frac {1}{2} i x \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{4} \int e^{-2 i x} \operatorname {PolyLog}\left (2,-e^{2 i x}\right )de^{2 i x}\right )-\frac {1}{2} i x^2 \log \left (1+e^{2 i x}\right )\right )+\frac {1}{2} \left (-\int \tan (x)dx-\frac {x^2}{2}-\frac {1}{3} x \tan ^3(x)+x \tan (x)+\frac {1}{3} \left (\frac {\tan ^2(x)}{2}+\log (\cos (x))\right )\right )-\int \tan (x)dx+\frac {i x^3}{3}-\frac {x^2}{2}+\frac {1}{4} x^2 \tan ^4(x)-\frac {1}{2} x^2 \tan ^2(x)+x \tan (x)\right )+\int \tan (x)dx-\frac {x^4}{4}+\frac {1}{5} x^3 \tan ^5(x)-\frac {1}{3} x^3 \tan ^3(x)+x^3 \tan (x)+\frac {x^2}{2}+\frac {1}{2} x^2 \tan ^2(x)-x \tan (x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -4 \left (\frac {i x^3}{3}-2 i \left (i \left (\frac {1}{2} i x \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{4} \int e^{-2 i x} \operatorname {PolyLog}\left (2,-e^{2 i x}\right )de^{2 i x}\right )-\frac {1}{2} i x^2 \log \left (1+e^{2 i x}\right )\right )\right )-\frac {3}{5} \left (-2 i \left (i \left (\frac {1}{2} i x \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{4} \int e^{-2 i x} \operatorname {PolyLog}\left (2,-e^{2 i x}\right )de^{2 i x}\right )-\frac {1}{2} i x^2 \log \left (1+e^{2 i x}\right )\right )+\frac {1}{2} \left (-\int \tan (x)dx-\frac {x^2}{2}-\frac {1}{3} x \tan ^3(x)+x \tan (x)+\frac {1}{3} \left (\frac {\tan ^2(x)}{2}+\log (\cos (x))\right )\right )-\int \tan (x)dx+\frac {i x^3}{3}-\frac {x^2}{2}+\frac {1}{4} x^2 \tan ^4(x)-\frac {1}{2} x^2 \tan ^2(x)+x \tan (x)\right )+\int \tan (x)dx-\frac {x^4}{4}+\frac {1}{5} x^3 \tan ^5(x)-\frac {1}{3} x^3 \tan ^3(x)+x^3 \tan (x)+\frac {x^2}{2}+\frac {1}{2} x^2 \tan ^2(x)-x \tan (x)\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle -4 \left (\frac {i x^3}{3}-2 i \left (i \left (\frac {1}{2} i x \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{4} \int e^{-2 i x} \operatorname {PolyLog}\left (2,-e^{2 i x}\right )de^{2 i x}\right )-\frac {1}{2} i x^2 \log \left (1+e^{2 i x}\right )\right )\right )-\frac {3}{5} \left (-2 i \left (i \left (\frac {1}{2} i x \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{4} \int e^{-2 i x} \operatorname {PolyLog}\left (2,-e^{2 i x}\right )de^{2 i x}\right )-\frac {1}{2} i x^2 \log \left (1+e^{2 i x}\right )\right )+\frac {i x^3}{3}-\frac {x^2}{2}+\frac {1}{4} x^2 \tan ^4(x)-\frac {1}{2} x^2 \tan ^2(x)+\frac {1}{2} \left (-\frac {x^2}{2}-\frac {1}{3} x \tan ^3(x)+x \tan (x)+\log (\cos (x))+\frac {1}{3} \left (\frac {\tan ^2(x)}{2}+\log (\cos (x))\right )\right )+x \tan (x)+\log (\cos (x))\right )-\frac {x^4}{4}+\frac {1}{5} x^3 \tan ^5(x)-\frac {1}{3} x^3 \tan ^3(x)+x^3 \tan (x)+\frac {x^2}{2}+\frac {1}{2} x^2 \tan ^2(x)-x \tan (x)-\log (\cos (x))\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -4 \left (\frac {i x^3}{3}-2 i \left (i \left (\frac {1}{2} i x \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{4} \operatorname {PolyLog}\left (3,-e^{2 i x}\right )\right )-\frac {1}{2} i x^2 \log \left (1+e^{2 i x}\right )\right )\right )-\frac {3}{5} \left (-2 i \left (i \left (\frac {1}{2} i x \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{4} \operatorname {PolyLog}\left (3,-e^{2 i x}\right )\right )-\frac {1}{2} i x^2 \log \left (1+e^{2 i x}\right )\right )+\frac {i x^3}{3}-\frac {x^2}{2}+\frac {1}{4} x^2 \tan ^4(x)-\frac {1}{2} x^2 \tan ^2(x)+\frac {1}{2} \left (-\frac {x^2}{2}-\frac {1}{3} x \tan ^3(x)+x \tan (x)+\log (\cos (x))+\frac {1}{3} \left (\frac {\tan ^2(x)}{2}+\log (\cos (x))\right )\right )+x \tan (x)+\log (\cos (x))\right )-\frac {x^4}{4}+\frac {1}{5} x^3 \tan ^5(x)-\frac {1}{3} x^3 \tan ^3(x)+x^3 \tan (x)+\frac {x^2}{2}+\frac {1}{2} x^2 \tan ^2(x)-x \tan (x)-\log (\cos (x))\) |
x^2/2 - x^4/4 - Log[Cos[x]] - 4*((I/3)*x^3 - (2*I)*((-1/2*I)*x^2*Log[1 + E ^((2*I)*x)] + I*((I/2)*x*PolyLog[2, -E^((2*I)*x)] - PolyLog[3, -E^((2*I)*x )]/4))) - x*Tan[x] + x^3*Tan[x] + (x^2*Tan[x]^2)/2 - (x^3*Tan[x]^3)/3 + (x ^3*Tan[x]^5)/5 - (3*(-1/2*x^2 + (I/3)*x^3 + Log[Cos[x]] - (2*I)*((-1/2*I)* x^2*Log[1 + E^((2*I)*x)] + I*((I/2)*x*PolyLog[2, -E^((2*I)*x)] - PolyLog[3 , -E^((2*I)*x)]/4)) + x*Tan[x] - (x^2*Tan[x]^2)/2 + (x^2*Tan[x]^4)/4 + (-1 /2*x^2 + Log[Cos[x]] + x*Tan[x] - (x*Tan[x]^3)/3 + (Log[Cos[x]] + Tan[x]^2 /2)/3)/2))/5
3.2.29.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I *((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I Int[(c + d*x)^m*(E^(2*I*( e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt Q[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symb ol] :> Simp[b*(c + d*x)^m*((b*Tan[e + f*x])^(n - 1)/(f*(n - 1))), x] + (-Si mp[b*d*(m/(f*(n - 1))) Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1), x] , x] - Simp[b^2 Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; Free Q[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 0]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Time = 0.17 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.55
| method | result | size |
| risch | \(-\frac {x^{4}}{4}+\frac {i \left (9 i {\mathrm e}^{6 i x}+90 x^{3} {\mathrm e}^{8 i x}-162 i x^{2} {\mathrm e}^{4 i x}-66 i x^{2} {\mathrm e}^{2 i x}+180 x^{3} {\mathrm e}^{6 i x}-66 x \,{\mathrm e}^{8 i x}+3 i {\mathrm e}^{8 i x}+9 i {\mathrm e}^{4 i x}+280 \,{\mathrm e}^{4 i x} x^{3}-246 x \,{\mathrm e}^{6 i x}-162 i x^{2} {\mathrm e}^{6 i x}+3 i {\mathrm e}^{2 i x}+140 \,{\mathrm e}^{2 i x} x^{3}-354 x \,{\mathrm e}^{4 i x}-66 i x^{2} {\mathrm e}^{8 i x}+46 x^{3}-234 x \,{\mathrm e}^{2 i x}-60 x \right )}{15 \left ({\mathrm e}^{2 i x}+1\right )^{5}}-2 \ln \left ({\mathrm e}^{2 i x}+1\right )+4 \ln \left ({\mathrm e}^{i x}\right )-\frac {46 i x^{3}}{15}+\frac {23 x^{2} \ln \left ({\mathrm e}^{2 i x}+1\right )}{5}-\frac {23 i x \,\operatorname {Li}_{2}\left (-{\mathrm e}^{2 i x}\right )}{5}+\frac {23 \,\operatorname {Li}_{3}\left (-{\mathrm e}^{2 i x}\right )}{10}\) | \(237\) |
-1/4*x^4+1/15*I*(9*I*exp(6*I*x)+90*x^3*exp(8*I*x)-162*I*x^2*exp(4*I*x)-66* I*x^2*exp(2*I*x)+180*x^3*exp(6*I*x)-66*x*exp(8*I*x)+3*I*exp(8*I*x)+9*I*exp (4*I*x)+280*exp(4*I*x)*x^3-246*x*exp(6*I*x)-162*I*x^2*exp(6*I*x)+3*I*exp(2 *I*x)+140*exp(2*I*x)*x^3-354*x*exp(4*I*x)-66*I*x^2*exp(8*I*x)+46*x^3-234*x *exp(2*I*x)-60*x)/(exp(2*I*x)+1)^5-2*ln(exp(2*I*x)+1)+4*ln(exp(I*x))-46/15 *I*x^3+23/5*x^2*ln(exp(2*I*x)+1)-23/5*I*x*polylog(2,-exp(2*I*x))+23/10*pol ylog(3,-exp(2*I*x))
Time = 0.25 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.39 \[ \int x^3 \tan ^6(x) \, dx=\frac {1}{5} \, x^{3} \tan \left (x\right )^{5} - \frac {3}{20} \, x^{2} \tan \left (x\right )^{4} - \frac {1}{4} \, x^{4} - \frac {1}{30} \, {\left (10 \, x^{3} - 3 \, x\right )} \tan \left (x\right )^{3} + \frac {1}{20} \, {\left (16 \, x^{2} - 1\right )} \tan \left (x\right )^{2} + \frac {19}{20} \, x^{2} + \frac {23}{10} i \, x {\rm Li}_2\left (\frac {2 \, {\left (i \, \tan \left (x\right ) - 1\right )}}{\tan \left (x\right )^{2} + 1} + 1\right ) - \frac {23}{10} i \, x {\rm Li}_2\left (\frac {2 \, {\left (-i \, \tan \left (x\right ) - 1\right )}}{\tan \left (x\right )^{2} + 1} + 1\right ) + \frac {1}{10} \, {\left (23 \, x^{2} - 10\right )} \log \left (-\frac {2 \, {\left (i \, \tan \left (x\right ) - 1\right )}}{\tan \left (x\right )^{2} + 1}\right ) + \frac {1}{10} \, {\left (23 \, x^{2} - 10\right )} \log \left (-\frac {2 \, {\left (-i \, \tan \left (x\right ) - 1\right )}}{\tan \left (x\right )^{2} + 1}\right ) + \frac {1}{10} \, {\left (10 \, x^{3} - 19 \, x\right )} \tan \left (x\right ) + \frac {23}{20} \, {\rm polylog}\left (3, \frac {\tan \left (x\right )^{2} + 2 i \, \tan \left (x\right ) - 1}{\tan \left (x\right )^{2} + 1}\right ) + \frac {23}{20} \, {\rm polylog}\left (3, \frac {\tan \left (x\right )^{2} - 2 i \, \tan \left (x\right ) - 1}{\tan \left (x\right )^{2} + 1}\right ) \]
1/5*x^3*tan(x)^5 - 3/20*x^2*tan(x)^4 - 1/4*x^4 - 1/30*(10*x^3 - 3*x)*tan(x )^3 + 1/20*(16*x^2 - 1)*tan(x)^2 + 19/20*x^2 + 23/10*I*x*dilog(2*(I*tan(x) - 1)/(tan(x)^2 + 1) + 1) - 23/10*I*x*dilog(2*(-I*tan(x) - 1)/(tan(x)^2 + 1) + 1) + 1/10*(23*x^2 - 10)*log(-2*(I*tan(x) - 1)/(tan(x)^2 + 1)) + 1/10* (23*x^2 - 10)*log(-2*(-I*tan(x) - 1)/(tan(x)^2 + 1)) + 1/10*(10*x^3 - 19*x )*tan(x) + 23/20*polylog(3, (tan(x)^2 + 2*I*tan(x) - 1)/(tan(x)^2 + 1)) + 23/20*polylog(3, (tan(x)^2 - 2*I*tan(x) - 1)/(tan(x)^2 + 1))
\[ \int x^3 \tan ^6(x) \, dx=\int x^{3} \tan ^{6}{\left (x \right )}\, dx \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 777 vs. \(2 (113) = 226\).
Time = 0.57 (sec) , antiderivative size = 777, normalized size of antiderivative = 5.08 \[ \int x^3 \tan ^6(x) \, dx=\text {Too large to display} \]
(15*I*x^4 + 12*(23*x^2 + (23*x^2 - 10)*cos(10*x) + 5*(23*x^2 - 10)*cos(8*x ) + 10*(23*x^2 - 10)*cos(6*x) + 10*(23*x^2 - 10)*cos(4*x) + 5*(23*x^2 - 10 )*cos(2*x) - (-23*I*x^2 + 10*I)*sin(10*x) - 5*(-23*I*x^2 + 10*I)*sin(8*x) - 10*(-23*I*x^2 + 10*I)*sin(6*x) - 10*(-23*I*x^2 + 10*I)*sin(4*x) - 5*(-23 *I*x^2 + 10*I)*sin(2*x) - 10)*arctan2(sin(2*x), cos(2*x) + 1) + (15*I*x^4 - 184*x^3 + 240*x)*cos(10*x) + (75*I*x^4 - 560*x^3 - 264*I*x^2 + 936*x + 1 2*I)*cos(8*x) - 2*(-75*I*x^4 + 560*x^3 + 324*I*x^2 - 708*x - 18*I)*cos(6*x ) - 6*(-25*I*x^4 + 120*x^3 + 108*I*x^2 - 164*x - 6*I)*cos(4*x) - 3*(-25*I* x^4 + 120*x^3 + 88*I*x^2 - 88*x - 4*I)*cos(2*x) - 276*(x*cos(10*x) + 5*x*c os(8*x) + 10*x*cos(6*x) + 10*x*cos(4*x) + 5*x*cos(2*x) + I*x*sin(10*x) + 5 *I*x*sin(8*x) + 10*I*x*sin(6*x) + 10*I*x*sin(4*x) + 5*I*x*sin(2*x) + x)*di log(-e^(2*I*x)) - 6*(23*I*x^2 + (23*I*x^2 - 10*I)*cos(10*x) + 5*(23*I*x^2 - 10*I)*cos(8*x) + 10*(23*I*x^2 - 10*I)*cos(6*x) + 10*(23*I*x^2 - 10*I)*co s(4*x) + 5*(23*I*x^2 - 10*I)*cos(2*x) - (23*x^2 - 10)*sin(10*x) - 5*(23*x^ 2 - 10)*sin(8*x) - 10*(23*x^2 - 10)*sin(6*x) - 10*(23*x^2 - 10)*sin(4*x) - 5*(23*x^2 - 10)*sin(2*x) - 10*I)*log(cos(2*x)^2 + sin(2*x)^2 + 2*cos(2*x) + 1) - 138*(I*cos(10*x) + 5*I*cos(8*x) + 10*I*cos(6*x) + 10*I*cos(4*x) + 5*I*cos(2*x) - sin(10*x) - 5*sin(8*x) - 10*sin(6*x) - 10*sin(4*x) - 5*sin( 2*x) + I)*polylog(3, -e^(2*I*x)) - (15*x^4 + 184*I*x^3 - 240*I*x)*sin(10*x ) - (75*x^4 + 560*I*x^3 - 264*x^2 - 936*I*x + 12)*sin(8*x) - 2*(75*x^4 ...
\[ \int x^3 \tan ^6(x) \, dx=\int { x^{3} \tan \left (x\right )^{6} \,d x } \]
Timed out. \[ \int x^3 \tan ^6(x) \, dx=\int x^3\,{\mathrm {tan}\left (x\right )}^6 \,d x \]