Integrand size = 21, antiderivative size = 361 \[ \int \cot ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx=-\frac {\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{f}+\frac {\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{f}-\frac {83 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )}{64 f}-\frac {\log \left (1+\sqrt {2}+\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f}+\frac {\log \left (1+\sqrt {2}+\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f}+\frac {83 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {15 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{32 f}-\frac {3 \cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f} \]
-83/64*arctanh((1+tan(f*x+e))^(1/2))/f-1/2*ln(1+2^(1/2)-(2+2*2^(1/2))^(1/2 )*(1+tan(f*x+e))^(1/2)+tan(f*x+e))/f/(1+2^(1/2))^(1/2)+1/2*ln(1+2^(1/2)+(2 +2*2^(1/2))^(1/2)*(1+tan(f*x+e))^(1/2)+tan(f*x+e))/f/(1+2^(1/2))^(1/2)-arc tan(((2+2*2^(1/2))^(1/2)-2*(1+tan(f*x+e))^(1/2))/(-2+2*2^(1/2))^(1/2))*(1+ 2^(1/2))^(1/2)/f+arctan(((2+2*2^(1/2))^(1/2)+2*(1+tan(f*x+e))^(1/2))/(-2+2 *2^(1/2))^(1/2))*(1+2^(1/2))^(1/2)/f+83/64*cot(f*x+e)*(1+tan(f*x+e))^(1/2) /f+15/32*cot(f*x+e)^2*(1+tan(f*x+e))^(1/2)/f-3/8*cot(f*x+e)^3*(1+tan(f*x+e ))^(1/2)/f-1/4*cot(f*x+e)^4*(1+tan(f*x+e))^(1/2)/f
Result contains complex when optimal does not.
Time = 1.63 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.47 \[ \int \cot ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\frac {-83 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )+64 (1-i)^{3/2} \text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1-i}}\right )+64 (1+i)^{3/2} \text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1+i}}\right )+83 \cot (e+f x) \sqrt {1+\tan (e+f x)}+30 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}-24 \cot ^3(e+f x) \sqrt {1+\tan (e+f x)}-16 \cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{64 f} \]
(-83*ArcTanh[Sqrt[1 + Tan[e + f*x]]] + 64*(1 - I)^(3/2)*ArcTanh[Sqrt[1 + T an[e + f*x]]/Sqrt[1 - I]] + 64*(1 + I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x] ]/Sqrt[1 + I]] + 83*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]] + 30*Cot[e + f*x]^ 2*Sqrt[1 + Tan[e + f*x]] - 24*Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]] - 16*C ot[e + f*x]^4*Sqrt[1 + Tan[e + f*x]])/(64*f)
Time = 1.64 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.11, number of steps used = 28, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.286, Rules used = {3042, 4050, 27, 3042, 4133, 27, 3042, 4132, 27, 3042, 4133, 27, 3042, 4136, 27, 3042, 3966, 484, 1407, 1142, 25, 1083, 217, 1103, 4117, 73, 220}
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.
\(\displaystyle \int (\tan (e+f x)+1)^{3/2} \cot ^5(e+f x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(\tan (e+f x)+1)^{3/2}}{\tan (e+f x)^5}dx\) |
\(\Big \downarrow \) 4050 |
\(\displaystyle -\frac {1}{4} \int -\frac {\cot ^4(e+f x) \left (9-7 \tan ^2(e+f x)\right )}{2 \sqrt {\tan (e+f x)+1}}dx-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{8} \int \frac {\cot ^4(e+f x) \left (9-7 \tan ^2(e+f x)\right )}{\sqrt {\tan (e+f x)+1}}dx-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{8} \int \frac {9-7 \tan (e+f x)^2}{\tan (e+f x)^4 \sqrt {\tan (e+f x)+1}}dx-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}\) |
\(\Big \downarrow \) 4133 |
\(\displaystyle \frac {1}{8} \left (-\frac {1}{3} \int \frac {3 \cot ^3(e+f x) \left (15 \tan ^2(e+f x)+32 \tan (e+f x)+15\right )}{2 \sqrt {\tan (e+f x)+1}}dx-\frac {3 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{8} \left (-\frac {1}{2} \int \frac {\cot ^3(e+f x) \left (15 \tan ^2(e+f x)+32 \tan (e+f x)+15\right )}{\sqrt {\tan (e+f x)+1}}dx-\frac {3 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{8} \left (-\frac {1}{2} \int \frac {15 \tan (e+f x)^2+32 \tan (e+f x)+15}{\tan (e+f x)^3 \sqrt {\tan (e+f x)+1}}dx-\frac {3 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}\) |
\(\Big \downarrow \) 4132 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{2} \int -\frac {\cot ^2(e+f x) \left (83-45 \tan ^2(e+f x)\right )}{2 \sqrt {\tan (e+f x)+1}}dx+\frac {15 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {15 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}-\frac {1}{4} \int \frac {\cot ^2(e+f x) \left (83-45 \tan ^2(e+f x)\right )}{\sqrt {\tan (e+f x)+1}}dx\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {15 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}-\frac {1}{4} \int \frac {83-45 \tan (e+f x)^2}{\tan (e+f x)^2 \sqrt {\tan (e+f x)+1}}dx\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}\) |
\(\Big \downarrow \) 4133 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\int \frac {\cot (e+f x) \left (83 \tan ^2(e+f x)+256 \tan (e+f x)+83\right )}{2 \sqrt {\tan (e+f x)+1}}dx+\frac {83 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )+\frac {15 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \int \frac {\cot (e+f x) \left (83 \tan ^2(e+f x)+256 \tan (e+f x)+83\right )}{\sqrt {\tan (e+f x)+1}}dx+\frac {83 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )+\frac {15 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \int \frac {83 \tan (e+f x)^2+256 \tan (e+f x)+83}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx+\frac {83 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )+\frac {15 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}\) |
\(\Big \downarrow \) 4136 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (\int \frac {256}{\sqrt {\tan (e+f x)+1}}dx+83 \int \frac {\cot (e+f x) \left (\tan ^2(e+f x)+1\right )}{\sqrt {\tan (e+f x)+1}}dx\right )+\frac {83 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )+\frac {15 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (256 \int \frac {1}{\sqrt {\tan (e+f x)+1}}dx+83 \int \frac {\cot (e+f x) \left (\tan ^2(e+f x)+1\right )}{\sqrt {\tan (e+f x)+1}}dx\right )+\frac {83 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )+\frac {15 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (256 \int \frac {1}{\sqrt {\tan (e+f x)+1}}dx+83 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx\right )+\frac {83 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )+\frac {15 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}\) |
\(\Big \downarrow \) 3966 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {256 \int \frac {1}{\sqrt {\tan (e+f x)+1} \left (\tan ^2(e+f x)+1\right )}d\tan (e+f x)}{f}+83 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx\right )+\frac {83 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )+\frac {15 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}\) |
\(\Big \downarrow \) 484 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (83 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx+\frac {512 \int \frac {1}{(\tan (e+f x)+1)^2-2 (\tan (e+f x)+1)+2}d\sqrt {\tan (e+f x)+1}}{f}\right )+\frac {83 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )+\frac {15 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}\) |
\(\Big \downarrow \) 1407 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (83 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx+\frac {512 \left (\frac {\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-\sqrt {\tan (e+f x)+1}}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}+\frac {\int \frac {\sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )+\frac {83 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )+\frac {15 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (83 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx+\frac {512 \left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \int \frac {1}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}-\frac {1}{2} \int -\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \int \frac {1}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}+\frac {1}{2} \int \frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )+\frac {83 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )+\frac {15 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (83 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx+\frac {512 \left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \int \frac {1}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}+\frac {1}{2} \int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \int \frac {1}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}+\frac {1}{2} \int \frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )+\frac {83 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )+\frac {15 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (83 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx+\frac {512 \left (\frac {\frac {1}{2} \int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )} \int \frac {1}{-\tan (e+f x)+2 \left (1-\sqrt {2}\right )-1}d\left (2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\frac {1}{2} \int \frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )} \int \frac {1}{-\tan (e+f x)+2 \left (1-\sqrt {2}\right )-1}d\left (2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}\right )}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )+\frac {83 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )+\frac {15 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {512 \left (\frac {\frac {1}{2} \int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}+\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\frac {1}{2} \int \frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}+\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{4 \sqrt {1+\sqrt {2}}}\right )}{f}+83 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx\right )+\frac {83 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )+\frac {15 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (83 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx+\frac {512 \left (\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{2} \log \left (\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \log \left (\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )+\frac {83 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )+\frac {15 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}\) |
\(\Big \downarrow \) 4117 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {83 \int \frac {\cot (e+f x)}{\sqrt {\tan (e+f x)+1}}d\tan (e+f x)}{f}+\frac {512 \left (\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{2} \log \left (\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \log \left (\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )+\frac {83 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )+\frac {15 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {166 \int \cot (e+f x)d\sqrt {\tan (e+f x)+1}}{f}+\frac {512 \left (\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{2} \log \left (\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \log \left (\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )+\frac {83 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )+\frac {15 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}\) |
\(\Big \downarrow \) 220 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {512 \left (\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{2} \log \left (\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \log \left (\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}\right )}{f}-\frac {166 \text {arctanh}\left (\sqrt {\tan (e+f x)+1}\right )}{f}\right )+\frac {83 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )+\frac {15 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}\) |
-1/4*(Cot[e + f*x]^4*Sqrt[1 + Tan[e + f*x]])/f + ((-3*Cot[e + f*x]^3*Sqrt[ 1 + Tan[e + f*x]])/f + ((15*Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]])/(2*f) + (((-166*ArcTanh[Sqrt[1 + Tan[e + f*x]]])/f + (512*((Sqrt[(1 + Sqrt[2])/(- 1 + Sqrt[2])]*ArcTan[(-Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Tan[e + f*x]])/S qrt[2*(-1 + Sqrt[2])]] - Log[1 + Sqrt[2] + Tan[e + f*x] - Sqrt[2*(1 + Sqrt [2])]*Sqrt[1 + Tan[e + f*x]]]/2)/(4*Sqrt[1 + Sqrt[2]]) + (Sqrt[(1 + Sqrt[2 ])/(-1 + Sqrt[2])]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Tan[e + f*x] ])/Sqrt[2*(-1 + Sqrt[2])]] + Log[1 + Sqrt[2] + Tan[e + f*x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/2)/(4*Sqrt[1 + Sqrt[2]])))/f)/2 + (83*Co t[e + f*x]*Sqrt[1 + Tan[e + f*x]])/f)/4)/2)/8
3.4.94.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^2)), x_Symbol] :> Simp[2* d Subst[Int[1/(b*c^2 + a*d^2 - 2*b*c*x^2 + b*x^4), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/ c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) Int[(r - x)/(q - r* x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(r + x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d Su bst[Int[(a + x)^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c , d, n}, x] && NeQ[a^2 + b^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n - 1)/(f*(m + 1)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^ (n - 2)*Simp[a*c^2*(m + 1) + a*d^2*(n - 1) + b*c*d*(m - n + 2) - (b*c^2 - 2 *a*c*d - b*d^2)*(m + 1)*Tan[e + f*x] - d*(b*c - a*d)*(m + n)*Tan[e + f*x]^2 , x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^ 2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && LtQ[1, n, 2] && IntegerQ[ 2*m]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A/f Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* (m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d )*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ [b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 + a^2*C)*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d) *(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Sim p[A*(a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2)) - a*C*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - b*C)*Tan[e + f*x] - d*(A*b^2 + a^2*C)*( m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m , -1] && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^ n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Simp[ (A*b^2 - a*b*B + a^2*C)/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^n*((1 + Tan[ e + f*x]^2)/(a + b*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] & & !GtQ[n, 0] && !LeQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(4798\) vs. \(2(285)=570\).
Time = 88.37 (sec) , antiderivative size = 4799, normalized size of antiderivative = 13.29
1/128/f*(1+tan(f*x+e))^(1/2)/(cot(f*x+e)^2+cot(f*x+e))^(1/2)/((cos(f*x+e)+ sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(128*cot(f*x+e)^2*csc(f*x+e )*(-2+2*2^(1/2))^(1/2)*(1+2^(1/2))^(1/2)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+ e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*co s(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctan(1/4*((4+3*2^(1/2))*(cos(f*x+e)+si n(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/ 2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1)*(3*2^(1/2)-4))^(1/2)/(2*cos(f *x+e)^2-1)*(4*sin(f*x+e)*cos(f*x+e)-1-tan(f*x+e))*(-2+2*2^(1/2))^(1/2)*(2* 2^(1/2)+3)*(3*2^(1/2)-4))-332*cot(f*x+e)*csc(f*x+e)^2*((cos(f*x+e)+sin(f*x +e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(1+2^(1/2))^(1/2)*(cot(f*x+e)^2+co t(f*x+e))^(1/2)*2^(1/2)-64*cot(f*x+e)^2*csc(f*x+e)*(-2+2*2^(1/2))^(1/2)*(1 +2^(1/2))^(1/2)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)* sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1 ))^(1/2)*arctan(1/4*((4+3*2^(1/2))*(cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2 ^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e )+2*sin(f*x+e)^2+1)*(3*2^(1/2)-4))^(1/2)/(2*cos(f*x+e)^2-1)*(4*sin(f*x+e)* cos(f*x+e)-1-tan(f*x+e))*(-2+2*2^(1/2))^(1/2)*(2*2^(1/2)+3)*(3*2^(1/2)-4)) *2^(1/2)-96*cot(f*x+e)^2*csc(f*x+e)^2*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/ (cos(f*x+e)+1)^2)^(1/2)*(1+2^(1/2))^(1/2)*(cot(f*x+e)^2+cot(f*x+e))^(1/2)+ 498*cot(f*x+e)*csc(f*x+e)^2*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*...
Time = 0.27 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.23 \[ \int \cot ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\frac {64 \, \sqrt {2} f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (\sqrt {2} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} + f\right )} \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{4} - 64 \, \sqrt {2} f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (-\sqrt {2} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} + f\right )} \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{4} - 64 \, \sqrt {2} f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (\sqrt {2} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} - f\right )} \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{4} + 64 \, \sqrt {2} f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (-\sqrt {2} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} - f\right )} \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{4} - 83 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} + 1\right ) \tan \left (f x + e\right )^{4} + 83 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} - 1\right ) \tan \left (f x + e\right )^{4} + 2 \, {\left (83 \, \tan \left (f x + e\right )^{3} + 30 \, \tan \left (f x + e\right )^{2} - 24 \, \tan \left (f x + e\right ) - 16\right )} \sqrt {\tan \left (f x + e\right ) + 1}}{128 \, f \tan \left (f x + e\right )^{4}} \]
1/128*(64*sqrt(2)*f*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f^2)*log(sqrt(2)*(f^3*sqr t(-1/f^4) + f)*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f^2) + 2*sqrt(tan(f*x + e) + 1 ))*tan(f*x + e)^4 - 64*sqrt(2)*f*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f^2)*log(-sq rt(2)*(f^3*sqrt(-1/f^4) + f)*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f^2) + 2*sqrt(ta n(f*x + e) + 1))*tan(f*x + e)^4 - 64*sqrt(2)*f*sqrt((f^2*sqrt(-1/f^4) - 1) /f^2)*log(sqrt(2)*(f^3*sqrt(-1/f^4) - f)*sqrt((f^2*sqrt(-1/f^4) - 1)/f^2) + 2*sqrt(tan(f*x + e) + 1))*tan(f*x + e)^4 + 64*sqrt(2)*f*sqrt((f^2*sqrt(- 1/f^4) - 1)/f^2)*log(-sqrt(2)*(f^3*sqrt(-1/f^4) - f)*sqrt((f^2*sqrt(-1/f^4 ) - 1)/f^2) + 2*sqrt(tan(f*x + e) + 1))*tan(f*x + e)^4 - 83*log(sqrt(tan(f *x + e) + 1) + 1)*tan(f*x + e)^4 + 83*log(sqrt(tan(f*x + e) + 1) - 1)*tan( f*x + e)^4 + 2*(83*tan(f*x + e)^3 + 30*tan(f*x + e)^2 - 24*tan(f*x + e) - 16)*sqrt(tan(f*x + e) + 1))/(f*tan(f*x + e)^4)
\[ \int \cot ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\int \left (\tan {\left (e + f x \right )} + 1\right )^{\frac {3}{2}} \cot ^{5}{\left (e + f x \right )}\, dx \]
Timed out. \[ \int \cot ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\text {Timed out} \]
Timed out. \[ \int \cot ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\text {Timed out} \]
Time = 4.96 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.53 \[ \int \cot ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\frac {\mathrm {atan}\left (\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,1{}\mathrm {i}\right )\,83{}\mathrm {i}}{64\,f}-\frac {\frac {45\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}}{64}-\frac {165\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{3/2}}{64}+\frac {219\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{5/2}}{64}-\frac {83\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{7/2}}{64}}{f-4\,f\,\left (\mathrm {tan}\left (e+f\,x\right )+1\right )+6\,f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^2-4\,f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^3+f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^4}+\mathrm {atan}\left (f\,\sqrt {\frac {-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\right )\,\sqrt {\frac {-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (f\,\sqrt {\frac {-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\right )\,\sqrt {\frac {-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i} \]
(atan((tan(e + f*x) + 1)^(1/2)*1i)*83i)/(64*f) - ((45*(tan(e + f*x) + 1)^( 1/2))/64 - (165*(tan(e + f*x) + 1)^(3/2))/64 + (219*(tan(e + f*x) + 1)^(5/ 2))/64 - (83*(tan(e + f*x) + 1)^(7/2))/64)/(f - 4*f*(tan(e + f*x) + 1) + 6 *f*(tan(e + f*x) + 1)^2 - 4*f*(tan(e + f*x) + 1)^3 + f*(tan(e + f*x) + 1)^ 4) + atan(f*((- 1/2 - 1i/2)/f^2)^(1/2)*(tan(e + f*x) + 1)^(1/2))*((- 1/2 - 1i/2)/f^2)^(1/2)*2i - atan(f*((- 1/2 + 1i/2)/f^2)^(1/2)*(tan(e + f*x) + 1 )^(1/2))*((- 1/2 + 1i/2)/f^2)^(1/2)*2i