\(\int \tan ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx\) [389]

Optimal result

Integrand size = 21, antiderivative size = 310 \[ \int \tan ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\frac {\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{f}-\frac {\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{f}-\frac {\text {arctanh}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}{1+\sqrt {2}+\tan (e+f x)}\right )}{\sqrt {1+\sqrt {2}} f}+\frac {2 \sqrt {1+\tan (e+f x)}}{f}+\frac {2 (1+\tan (e+f x))^{3/2}}{3 f}+\frac {20 (1+\tan (e+f x))^{5/2}}{231 f}-\frac {50 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{231 f}-\frac {4 \tan ^2(e+f x) (1+\tan (e+f x))^{5/2}}{33 f}+\frac {2 \tan ^3(e+f x) (1+\tan (e+f x))^{5/2}}{11 f} \] Output:

(1+2^(1/2))^(1/2)*arctan(((2+2*2^(1/2))^(1/2)-2*(1+tan(f*x+e))^(1/2))/(-2+ 
2*2^(1/2))^(1/2))/f-(1+2^(1/2))^(1/2)*arctan(((2+2*2^(1/2))^(1/2)+2*(1+tan 
(f*x+e))^(1/2))/(-2+2*2^(1/2))^(1/2))/f-arctanh((2+2*2^(1/2))^(1/2)*(1+tan 
(f*x+e))^(1/2)/(1+2^(1/2)+tan(f*x+e)))/(1+2^(1/2))^(1/2)/f+2*(1+tan(f*x+e) 
)^(1/2)/f+2/3*(1+tan(f*x+e))^(3/2)/f+20/231*(1+tan(f*x+e))^(5/2)/f-50/231* 
tan(f*x+e)*(1+tan(f*x+e))^(5/2)/f-4/33*tan(f*x+e)^2*(1+tan(f*x+e))^(5/2)/f 
+2/11*tan(f*x+e)^3*(1+tan(f*x+e))^(5/2)/f
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.74 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.43 \[ \int \tan ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\frac {-231 (1-i)^{3/2} \text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1-i}}\right )-231 (1+i)^{3/2} \text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1+i}}\right )+2 \sqrt {1+\tan (e+f x)} \left (318+72 \tan (e+f x)-54 \tan ^2(e+f x)-32 \tan ^3(e+f x)+28 \tan ^4(e+f x)+21 \tan ^5(e+f x)\right )}{231 f} \] Input:

Integrate[Tan[e + f*x]^5*(1 + Tan[e + f*x])^(3/2),x]
 

Output:

(-231*(1 - I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] - 231*(1 + 
 I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]] + 2*Sqrt[1 + Tan[e + 
 f*x]]*(318 + 72*Tan[e + f*x] - 54*Tan[e + f*x]^2 - 32*Tan[e + f*x]^3 + 28 
*Tan[e + f*x]^4 + 21*Tan[e + f*x]^5))/(231*f)
 

Rubi [A] (verified)

Time = 1.41 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.32, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.238, Rules used = {3042, 4049, 27, 3042, 4130, 27, 3042, 4131, 27, 3042, 4113, 27, 3042, 4011, 3042, 4011, 27, 3042, 3966, 484, 1407, 1142, 25, 1083, 217, 1103}

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.

\(\displaystyle \int \tan ^5(e+f x) (\tan (e+f x)+1)^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \tan (e+f x)^5 (\tan (e+f x)+1)^{3/2}dx\)

\(\Big \downarrow \) 4049

\(\displaystyle \frac {2}{11} \int -\frac {1}{2} \tan ^2(e+f x) (\tan (e+f x)+1)^{3/2} \left (6 \tan ^2(e+f x)+11 \tan (e+f x)+6\right )dx+\frac {2 (\tan (e+f x)+1)^{5/2} \tan ^3(e+f x)}{11 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \tan ^3(e+f x) (\tan (e+f x)+1)^{5/2}}{11 f}-\frac {1}{11} \int \tan ^2(e+f x) (\tan (e+f x)+1)^{3/2} \left (6 \tan ^2(e+f x)+11 \tan (e+f x)+6\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \tan ^3(e+f x) (\tan (e+f x)+1)^{5/2}}{11 f}-\frac {1}{11} \int \tan (e+f x)^2 (\tan (e+f x)+1)^{3/2} \left (6 \tan (e+f x)^2+11 \tan (e+f x)+6\right )dx\)

\(\Big \downarrow \) 4130

\(\displaystyle \frac {1}{11} \left (-\frac {2}{9} \int -\frac {3}{2} \tan (e+f x) (\tan (e+f x)+1)^{3/2} \left (8-25 \tan ^2(e+f x)\right )dx-\frac {4 \tan ^2(e+f x) (\tan (e+f x)+1)^{5/2}}{3 f}\right )+\frac {2 (\tan (e+f x)+1)^{5/2} \tan ^3(e+f x)}{11 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{11} \left (\frac {1}{3} \int \tan (e+f x) (\tan (e+f x)+1)^{3/2} \left (8-25 \tan ^2(e+f x)\right )dx-\frac {4 \tan ^2(e+f x) (\tan (e+f x)+1)^{5/2}}{3 f}\right )+\frac {2 (\tan (e+f x)+1)^{5/2} \tan ^3(e+f x)}{11 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{11} \left (\frac {1}{3} \int \tan (e+f x) (\tan (e+f x)+1)^{3/2} \left (8-25 \tan (e+f x)^2\right )dx-\frac {4 \tan ^2(e+f x) (\tan (e+f x)+1)^{5/2}}{3 f}\right )+\frac {2 (\tan (e+f x)+1)^{5/2} \tan ^3(e+f x)}{11 f}\)

\(\Big \downarrow \) 4131

\(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {2}{7} \int \frac {1}{2} (\tan (e+f x)+1)^{3/2} \left (50 \tan ^2(e+f x)+231 \tan (e+f x)+50\right )dx-\frac {50 \tan (e+f x) (\tan (e+f x)+1)^{5/2}}{7 f}\right )-\frac {4 \tan ^2(e+f x) (\tan (e+f x)+1)^{5/2}}{3 f}\right )+\frac {2 (\tan (e+f x)+1)^{5/2} \tan ^3(e+f x)}{11 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {1}{7} \int (\tan (e+f x)+1)^{3/2} \left (50 \tan ^2(e+f x)+231 \tan (e+f x)+50\right )dx-\frac {50 \tan (e+f x) (\tan (e+f x)+1)^{5/2}}{7 f}\right )-\frac {4 \tan ^2(e+f x) (\tan (e+f x)+1)^{5/2}}{3 f}\right )+\frac {2 (\tan (e+f x)+1)^{5/2} \tan ^3(e+f x)}{11 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {1}{7} \int (\tan (e+f x)+1)^{3/2} \left (50 \tan (e+f x)^2+231 \tan (e+f x)+50\right )dx-\frac {50 \tan (e+f x) (\tan (e+f x)+1)^{5/2}}{7 f}\right )-\frac {4 \tan ^2(e+f x) (\tan (e+f x)+1)^{5/2}}{3 f}\right )+\frac {2 (\tan (e+f x)+1)^{5/2} \tan ^3(e+f x)}{11 f}\)

\(\Big \downarrow \) 4113

\(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {1}{7} \left (\int 231 \tan (e+f x) (\tan (e+f x)+1)^{3/2}dx+\frac {20 (\tan (e+f x)+1)^{5/2}}{f}\right )-\frac {50 \tan (e+f x) (\tan (e+f x)+1)^{5/2}}{7 f}\right )-\frac {4 \tan ^2(e+f x) (\tan (e+f x)+1)^{5/2}}{3 f}\right )+\frac {2 (\tan (e+f x)+1)^{5/2} \tan ^3(e+f x)}{11 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {1}{7} \left (231 \int \tan (e+f x) (\tan (e+f x)+1)^{3/2}dx+\frac {20 (\tan (e+f x)+1)^{5/2}}{f}\right )-\frac {50 \tan (e+f x) (\tan (e+f x)+1)^{5/2}}{7 f}\right )-\frac {4 \tan ^2(e+f x) (\tan (e+f x)+1)^{5/2}}{3 f}\right )+\frac {2 (\tan (e+f x)+1)^{5/2} \tan ^3(e+f x)}{11 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {1}{7} \left (231 \int \tan (e+f x) (\tan (e+f x)+1)^{3/2}dx+\frac {20 (\tan (e+f x)+1)^{5/2}}{f}\right )-\frac {50 \tan (e+f x) (\tan (e+f x)+1)^{5/2}}{7 f}\right )-\frac {4 \tan ^2(e+f x) (\tan (e+f x)+1)^{5/2}}{3 f}\right )+\frac {2 (\tan (e+f x)+1)^{5/2} \tan ^3(e+f x)}{11 f}\)

\(\Big \downarrow \) 4011

\(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {1}{7} \left (231 \left (\int (\tan (e+f x)-1) \sqrt {\tan (e+f x)+1}dx+\frac {2 (\tan (e+f x)+1)^{3/2}}{3 f}\right )+\frac {20 (\tan (e+f x)+1)^{5/2}}{f}\right )-\frac {50 \tan (e+f x) (\tan (e+f x)+1)^{5/2}}{7 f}\right )-\frac {4 \tan ^2(e+f x) (\tan (e+f x)+1)^{5/2}}{3 f}\right )+\frac {2 (\tan (e+f x)+1)^{5/2} \tan ^3(e+f x)}{11 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {1}{7} \left (231 \left (\int (\tan (e+f x)-1) \sqrt {\tan (e+f x)+1}dx+\frac {2 (\tan (e+f x)+1)^{3/2}}{3 f}\right )+\frac {20 (\tan (e+f x)+1)^{5/2}}{f}\right )-\frac {50 \tan (e+f x) (\tan (e+f x)+1)^{5/2}}{7 f}\right )-\frac {4 \tan ^2(e+f x) (\tan (e+f x)+1)^{5/2}}{3 f}\right )+\frac {2 (\tan (e+f x)+1)^{5/2} \tan ^3(e+f x)}{11 f}\)

\(\Big \downarrow \) 4011

\(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {1}{7} \left (231 \left (\int -\frac {2}{\sqrt {\tan (e+f x)+1}}dx+\frac {2 (\tan (e+f x)+1)^{3/2}}{3 f}+\frac {2 \sqrt {\tan (e+f x)+1}}{f}\right )+\frac {20 (\tan (e+f x)+1)^{5/2}}{f}\right )-\frac {50 \tan (e+f x) (\tan (e+f x)+1)^{5/2}}{7 f}\right )-\frac {4 \tan ^2(e+f x) (\tan (e+f x)+1)^{5/2}}{3 f}\right )+\frac {2 (\tan (e+f x)+1)^{5/2} \tan ^3(e+f x)}{11 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {1}{7} \left (231 \left (-2 \int \frac {1}{\sqrt {\tan (e+f x)+1}}dx+\frac {2 (\tan (e+f x)+1)^{3/2}}{3 f}+\frac {2 \sqrt {\tan (e+f x)+1}}{f}\right )+\frac {20 (\tan (e+f x)+1)^{5/2}}{f}\right )-\frac {50 \tan (e+f x) (\tan (e+f x)+1)^{5/2}}{7 f}\right )-\frac {4 \tan ^2(e+f x) (\tan (e+f x)+1)^{5/2}}{3 f}\right )+\frac {2 (\tan (e+f x)+1)^{5/2} \tan ^3(e+f x)}{11 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {1}{7} \left (231 \left (-2 \int \frac {1}{\sqrt {\tan (e+f x)+1}}dx+\frac {2 (\tan (e+f x)+1)^{3/2}}{3 f}+\frac {2 \sqrt {\tan (e+f x)+1}}{f}\right )+\frac {20 (\tan (e+f x)+1)^{5/2}}{f}\right )-\frac {50 \tan (e+f x) (\tan (e+f x)+1)^{5/2}}{7 f}\right )-\frac {4 \tan ^2(e+f x) (\tan (e+f x)+1)^{5/2}}{3 f}\right )+\frac {2 (\tan (e+f x)+1)^{5/2} \tan ^3(e+f x)}{11 f}\)

\(\Big \downarrow \) 3966

\(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {1}{7} \left (231 \left (-\frac {2 \int \frac {1}{\sqrt {\tan (e+f x)+1} \left (\tan ^2(e+f x)+1\right )}d\tan (e+f x)}{f}+\frac {2 (\tan (e+f x)+1)^{3/2}}{3 f}+\frac {2 \sqrt {\tan (e+f x)+1}}{f}\right )+\frac {20 (\tan (e+f x)+1)^{5/2}}{f}\right )-\frac {50 \tan (e+f x) (\tan (e+f x)+1)^{5/2}}{7 f}\right )-\frac {4 \tan ^2(e+f x) (\tan (e+f x)+1)^{5/2}}{3 f}\right )+\frac {2 (\tan (e+f x)+1)^{5/2} \tan ^3(e+f x)}{11 f}\)

\(\Big \downarrow \) 484

\(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {1}{7} \left (231 \left (-\frac {4 \int \frac {1}{(\tan (e+f x)+1)^2-2 (\tan (e+f x)+1)+2}d\sqrt {\tan (e+f x)+1}}{f}+\frac {2 (\tan (e+f x)+1)^{3/2}}{3 f}+\frac {2 \sqrt {\tan (e+f x)+1}}{f}\right )+\frac {20 (\tan (e+f x)+1)^{5/2}}{f}\right )-\frac {50 \tan (e+f x) (\tan (e+f x)+1)^{5/2}}{7 f}\right )-\frac {4 \tan ^2(e+f x) (\tan (e+f x)+1)^{5/2}}{3 f}\right )+\frac {2 (\tan (e+f x)+1)^{5/2} \tan ^3(e+f x)}{11 f}\)

\(\Big \downarrow \) 1407

\(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {1}{7} \left (231 \left (-\frac {4 \left (\frac {\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-\sqrt {\tan (e+f x)+1}}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}+\frac {\int \frac {\sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}\right )}{f}+\frac {2 (\tan (e+f x)+1)^{3/2}}{3 f}+\frac {2 \sqrt {\tan (e+f x)+1}}{f}\right )+\frac {20 (\tan (e+f x)+1)^{5/2}}{f}\right )-\frac {50 \tan (e+f x) (\tan (e+f x)+1)^{5/2}}{7 f}\right )-\frac {4 \tan ^2(e+f x) (\tan (e+f x)+1)^{5/2}}{3 f}\right )+\frac {2 (\tan (e+f x)+1)^{5/2} \tan ^3(e+f x)}{11 f}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {1}{7} \left (231 \left (-\frac {4 \left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \int \frac {1}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}-\frac {1}{2} \int -\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \int \frac {1}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}+\frac {1}{2} \int \frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}\right )}{f}+\frac {2 (\tan (e+f x)+1)^{3/2}}{3 f}+\frac {2 \sqrt {\tan (e+f x)+1}}{f}\right )+\frac {20 (\tan (e+f x)+1)^{5/2}}{f}\right )-\frac {50 \tan (e+f x) (\tan (e+f x)+1)^{5/2}}{7 f}\right )-\frac {4 \tan ^2(e+f x) (\tan (e+f x)+1)^{5/2}}{3 f}\right )+\frac {2 (\tan (e+f x)+1)^{5/2} \tan ^3(e+f x)}{11 f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {1}{7} \left (231 \left (-\frac {4 \left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \int \frac {1}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}+\frac {1}{2} \int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \int \frac {1}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}+\frac {1}{2} \int \frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}\right )}{f}+\frac {2 (\tan (e+f x)+1)^{3/2}}{3 f}+\frac {2 \sqrt {\tan (e+f x)+1}}{f}\right )+\frac {20 (\tan (e+f x)+1)^{5/2}}{f}\right )-\frac {50 \tan (e+f x) (\tan (e+f x)+1)^{5/2}}{7 f}\right )-\frac {4 \tan ^2(e+f x) (\tan (e+f x)+1)^{5/2}}{3 f}\right )+\frac {2 (\tan (e+f x)+1)^{5/2} \tan ^3(e+f x)}{11 f}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {1}{7} \left (231 \left (-\frac {4 \left (\frac {\frac {1}{2} \int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )} \int \frac {1}{-\tan (e+f x)+2 \left (1-\sqrt {2}\right )-1}d\left (2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\frac {1}{2} \int \frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )} \int \frac {1}{-\tan (e+f x)+2 \left (1-\sqrt {2}\right )-1}d\left (2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}\right )}{4 \sqrt {1+\sqrt {2}}}\right )}{f}+\frac {2 (\tan (e+f x)+1)^{3/2}}{3 f}+\frac {2 \sqrt {\tan (e+f x)+1}}{f}\right )+\frac {20 (\tan (e+f x)+1)^{5/2}}{f}\right )-\frac {50 \tan (e+f x) (\tan (e+f x)+1)^{5/2}}{7 f}\right )-\frac {4 \tan ^2(e+f x) (\tan (e+f x)+1)^{5/2}}{3 f}\right )+\frac {2 (\tan (e+f x)+1)^{5/2} \tan ^3(e+f x)}{11 f}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {1}{7} \left (231 \left (-\frac {4 \left (\frac {\frac {1}{2} \int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}+\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\frac {1}{2} \int \frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}+\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{4 \sqrt {1+\sqrt {2}}}\right )}{f}+\frac {2 (\tan (e+f x)+1)^{3/2}}{3 f}+\frac {2 \sqrt {\tan (e+f x)+1}}{f}\right )+\frac {20 (\tan (e+f x)+1)^{5/2}}{f}\right )-\frac {50 \tan (e+f x) (\tan (e+f x)+1)^{5/2}}{7 f}\right )-\frac {4 \tan ^2(e+f x) (\tan (e+f x)+1)^{5/2}}{3 f}\right )+\frac {2 (\tan (e+f x)+1)^{5/2} \tan ^3(e+f x)}{11 f}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {1}{7} \left (231 \left (-\frac {4 \left (\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{2} \log \left (\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \log \left (\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}\right )}{f}+\frac {2 (\tan (e+f x)+1)^{3/2}}{3 f}+\frac {2 \sqrt {\tan (e+f x)+1}}{f}\right )+\frac {20 (\tan (e+f x)+1)^{5/2}}{f}\right )-\frac {50 \tan (e+f x) (\tan (e+f x)+1)^{5/2}}{7 f}\right )-\frac {4 \tan ^2(e+f x) (\tan (e+f x)+1)^{5/2}}{3 f}\right )+\frac {2 (\tan (e+f x)+1)^{5/2} \tan ^3(e+f x)}{11 f}\)

Input:

Int[Tan[e + f*x]^5*(1 + Tan[e + f*x])^(3/2),x]
 

Output:

(2*Tan[e + f*x]^3*(1 + Tan[e + f*x])^(5/2))/(11*f) + ((-4*Tan[e + f*x]^2*( 
1 + Tan[e + f*x])^(5/2))/(3*f) + ((-50*Tan[e + f*x]*(1 + Tan[e + f*x])^(5/ 
2))/(7*f) + ((20*(1 + Tan[e + f*x])^(5/2))/f + 231*((-4*((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]]) + (Sqrt[(1 + S 
qrt[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*S 
qrt[1 + Tan[e + f*x]])/f + (2*(1 + Tan[e + f*x])^(3/2))/(3*f)))/7)/3)/11
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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])
 

rule 484

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]
 

rule 1083

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]
 

rule 1103

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]
 

rule 1142

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]
 

rule 1407

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3966

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]
 

rule 4011

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)]), x_Symbol] :> Simp[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Int 
[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x], x] 
, x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 
 0] && GtQ[m, 0]
 

rule 4049

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m - 2)*((c 
+ d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Simp[1/(d*(m + n - 1)) 
 Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n 
- 1) - b^2*(b*c*(m - 2) + a*d*(1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[ 
e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2, x], x], 
x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2 
, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || I 
ntegerQ[m]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])) 
)
 

rule 4113

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) 
+ (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + 
 b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Si 
mp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && 
NeQ[A*b^2 - a*b*B + a^2*C, 0] &&  !LeQ[m, -1]
 

rule 4130

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[C*(a + b*Tan[e + f*x])^m*((c + d*Tan[ 
e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1))   Int[(a 
+ b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C 
*(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f*x] - (C* 
m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[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] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[ 
c, 0] && NeQ[a, 0])))
 

rule 4131

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[C*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n + 
1))), x] + Simp[1/(d*(m + n + 1))   Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d 
*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C*(b*c*m + a*d*(n + 1)) + d*(A*b 
- b*C)*(m + n + 1)*Tan[e + f*x] - C*m*(b*c - a*d)*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] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ 
[m] || (EqQ[c, 0] && NeQ[a, 0])))
 

Maple [A] (verified)

Time = 1.05 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.20

method	result	size
derivativedivides	\(\frac {2 \left (\tan \left (f x +e \right )+1\right )^{\frac {11}{2}}}{11 f}-\frac {2 \left (\tan \left (f x +e \right )+1\right )^{\frac {9}{2}}}{3 f}+\frac {4 \left (\tan \left (f x +e \right )+1\right )^{\frac {7}{2}}}{7 f}+\frac {2 \left (\tan \left (f x +e \right )+1\right )^{\frac {3}{2}}}{3 f}+\frac {2 \sqrt {\tan \left (f x +e \right )+1}}{f}+\frac {\sqrt {2 \sqrt {2}+2}\, \sqrt {2}\, \ln \left (\tan \left (f x +e \right )+1+\sqrt {\tan \left (f x +e \right )+1}\, \sqrt {2 \sqrt {2}+2}+\sqrt {2}\right )}{4 f}-\frac {\sqrt {2 \sqrt {2}+2}\, \ln \left (\tan \left (f x +e \right )+1+\sqrt {\tan \left (f x +e \right )+1}\, \sqrt {2 \sqrt {2}+2}+\sqrt {2}\right )}{2 f}-\frac {\arctan \left (\frac {\sqrt {2 \sqrt {2}+2}+2 \sqrt {\tan \left (f x +e \right )+1}}{\sqrt {-2+2 \sqrt {2}}}\right ) \sqrt {2}}{f \sqrt {-2+2 \sqrt {2}}}-\frac {\sqrt {2 \sqrt {2}+2}\, \sqrt {2}\, \ln \left (\tan \left (f x +e \right )+1-\sqrt {\tan \left (f x +e \right )+1}\, \sqrt {2 \sqrt {2}+2}+\sqrt {2}\right )}{4 f}+\frac {\sqrt {2 \sqrt {2}+2}\, \ln \left (\tan \left (f x +e \right )+1-\sqrt {\tan \left (f x +e \right )+1}\, \sqrt {2 \sqrt {2}+2}+\sqrt {2}\right )}{2 f}-\frac {\arctan \left (\frac {2 \sqrt {\tan \left (f x +e \right )+1}-\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right ) \sqrt {2}}{f \sqrt {-2+2 \sqrt {2}}}\)	\(371\)
default	\(\frac {2 \left (\tan \left (f x +e \right )+1\right )^{\frac {11}{2}}}{11 f}-\frac {2 \left (\tan \left (f x +e \right )+1\right )^{\frac {9}{2}}}{3 f}+\frac {4 \left (\tan \left (f x +e \right )+1\right )^{\frac {7}{2}}}{7 f}+\frac {2 \left (\tan \left (f x +e \right )+1\right )^{\frac {3}{2}}}{3 f}+\frac {2 \sqrt {\tan \left (f x +e \right )+1}}{f}+\frac {\sqrt {2 \sqrt {2}+2}\, \sqrt {2}\, \ln \left (\tan \left (f x +e \right )+1+\sqrt {\tan \left (f x +e \right )+1}\, \sqrt {2 \sqrt {2}+2}+\sqrt {2}\right )}{4 f}-\frac {\sqrt {2 \sqrt {2}+2}\, \ln \left (\tan \left (f x +e \right )+1+\sqrt {\tan \left (f x +e \right )+1}\, \sqrt {2 \sqrt {2}+2}+\sqrt {2}\right )}{2 f}-\frac {\arctan \left (\frac {\sqrt {2 \sqrt {2}+2}+2 \sqrt {\tan \left (f x +e \right )+1}}{\sqrt {-2+2 \sqrt {2}}}\right ) \sqrt {2}}{f \sqrt {-2+2 \sqrt {2}}}-\frac {\sqrt {2 \sqrt {2}+2}\, \sqrt {2}\, \ln \left (\tan \left (f x +e \right )+1-\sqrt {\tan \left (f x +e \right )+1}\, \sqrt {2 \sqrt {2}+2}+\sqrt {2}\right )}{4 f}+\frac {\sqrt {2 \sqrt {2}+2}\, \ln \left (\tan \left (f x +e \right )+1-\sqrt {\tan \left (f x +e \right )+1}\, \sqrt {2 \sqrt {2}+2}+\sqrt {2}\right )}{2 f}-\frac {\arctan \left (\frac {2 \sqrt {\tan \left (f x +e \right )+1}-\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right ) \sqrt {2}}{f \sqrt {-2+2 \sqrt {2}}}\)	\(371\)

Input:

int(tan(f*x+e)^5*(tan(f*x+e)+1)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

2/11/f*(tan(f*x+e)+1)^(11/2)-2/3/f*(tan(f*x+e)+1)^(9/2)+4/7/f*(tan(f*x+e)+ 
1)^(7/2)+2/3*(tan(f*x+e)+1)^(3/2)/f+2*(tan(f*x+e)+1)^(1/2)/f+1/4/f*(2*2^(1 
/2)+2)^(1/2)*2^(1/2)*ln(tan(f*x+e)+1+(tan(f*x+e)+1)^(1/2)*(2*2^(1/2)+2)^(1 
/2)+2^(1/2))-1/2/f*(2*2^(1/2)+2)^(1/2)*ln(tan(f*x+e)+1+(tan(f*x+e)+1)^(1/2 
)*(2*2^(1/2)+2)^(1/2)+2^(1/2))-1/f/(-2+2*2^(1/2))^(1/2)*arctan(((2*2^(1/2) 
+2)^(1/2)+2*(tan(f*x+e)+1)^(1/2))/(-2+2*2^(1/2))^(1/2))*2^(1/2)-1/4/f*(2*2 
^(1/2)+2)^(1/2)*2^(1/2)*ln(tan(f*x+e)+1-(tan(f*x+e)+1)^(1/2)*(2*2^(1/2)+2) 
^(1/2)+2^(1/2))+1/2/f*(2*2^(1/2)+2)^(1/2)*ln(tan(f*x+e)+1-(tan(f*x+e)+1)^( 
1/2)*(2*2^(1/2)+2)^(1/2)+2^(1/2))-1/f/(-2+2*2^(1/2))^(1/2)*arctan((2*(tan( 
f*x+e)+1)^(1/2)-(2*2^(1/2)+2)^(1/2))/(-2+2*2^(1/2))^(1/2))*2^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.22 \[ \int \tan ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx=-\frac {231 \, \sqrt {2} f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (\sqrt {2} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} + f\right )} \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right ) - 231 \, \sqrt {2} f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (-\sqrt {2} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} + f\right )} \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right ) - 231 \, \sqrt {2} f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (\sqrt {2} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} - f\right )} \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right ) + 231 \, \sqrt {2} f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (-\sqrt {2} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} - f\right )} \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right ) - 4 \, {\left (21 \, \tan \left (f x + e\right )^{5} + 28 \, \tan \left (f x + e\right )^{4} - 32 \, \tan \left (f x + e\right )^{3} - 54 \, \tan \left (f x + e\right )^{2} + 72 \, \tan \left (f x + e\right ) + 318\right )} \sqrt {\tan \left (f x + e\right ) + 1}}{462 \, f} \] Input:

integrate(tan(f*x+e)^5*(1+tan(f*x+e))^(3/2),x, algorithm="fricas")
 

Output:

-1/462*(231*sqrt(2)*f*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f^2)*log(sqrt(2)*(f^3*s 
qrt(-1/f^4) + f)*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f^2) + 2*sqrt(tan(f*x + e) + 
 1)) - 231*sqrt(2)*f*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f^2)*log(-sqrt(2)*(f^3*s 
qrt(-1/f^4) + f)*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f^2) + 2*sqrt(tan(f*x + e) + 
 1)) - 231*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) 
) + 231*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)) 
- 4*(21*tan(f*x + e)^5 + 28*tan(f*x + e)^4 - 32*tan(f*x + e)^3 - 54*tan(f* 
x + e)^2 + 72*tan(f*x + e) + 318)*sqrt(tan(f*x + e) + 1))/f
 

Sympy [F]

\[ \int \tan ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\int \left (\tan {\left (e + f x \right )} + 1\right )^{\frac {3}{2}} \tan ^{5}{\left (e + f x \right )}\, dx \] Input:

integrate(tan(f*x+e)**5*(1+tan(f*x+e))**(3/2),x)
 

Output:

Integral((tan(e + f*x) + 1)**(3/2)*tan(e + f*x)**5, x)
 

Maxima [F(-1)]

Timed out. \[ \int \tan ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\text {Timed out} \] Input:

integrate(tan(f*x+e)^5*(1+tan(f*x+e))^(3/2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F(-2)]

Exception generated. \[ \int \tan ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(tan(f*x+e)^5*(1+tan(f*x+e))^(3/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{%%%{1,[34]%%%}+%%%{14,[32]%%%}+%%%{91,[30]%%%}+%%%{364,[28 
]%%%}+%%%
 

Mupad [B] (verification not implemented)

Time = 4.30 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.46 \[ \int \tan ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\frac {2\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}}{f}+\frac {2\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{3/2}}{3\,f}+\frac {4\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{7/2}}{7\,f}-\frac {2\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{9/2}}{3\,f}+\frac {2\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{11/2}}{11\,f}-\mathrm {atan}\left (f\,\sqrt {\frac {-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\right )\,\sqrt {\frac {-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (f\,\sqrt {\frac {-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\right )\,\sqrt {\frac {-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i} \] Input:

int(tan(e + f*x)^5*(tan(e + f*x) + 1)^(3/2),x)
 

Output:

(2*(tan(e + f*x) + 1)^(1/2))/f + (2*(tan(e + f*x) + 1)^(3/2))/(3*f) + (4*( 
tan(e + f*x) + 1)^(7/2))/(7*f) - (2*(tan(e + f*x) + 1)^(9/2))/(3*f) + (2*( 
tan(e + f*x) + 1)^(11/2))/(11*f) - 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
 

Reduce [F]

\[ \int \tan ^5(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\int \sqrt {\tan \left (f x +e \right )+1}\, \tan \left (f x +e \right )^{6}d x +\int \sqrt {\tan \left (f x +e \right )+1}\, \tan \left (f x +e \right )^{5}d x \] Input:

int(tan(f*x+e)^5*(1+tan(f*x+e))^(3/2),x)
 

Output:

int(sqrt(tan(e + f*x) + 1)*tan(e + f*x)**6,x) + int(sqrt(tan(e + f*x) + 1) 
*tan(e + f*x)**5,x)