\(\int \frac {\cot ^4(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx\) [410]

Optimal result

Integrand size = 21, antiderivative size = 282 \[ \int \frac {\cot ^4(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=-\frac {\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{2 f}+\frac {\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{2 f}-\frac {3 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )}{8 f}+\frac {\text {arctanh}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}{1+\sqrt {2}+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f}+\frac {3 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}+\frac {5 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f} \] Output:

-1/2*(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))^(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-3/8*arctanh((1+tan(f*x+e) 
)^(1/2))/f+1/2*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+3/8*cot(f*x+e)*(1+tan(f*x+e))^(1/2)/f+5/ 
12*cot(f*x+e)^2*(1+tan(f*x+e))^(1/2)/f-1/3*cot(f*x+e)^3*(1+tan(f*x+e))^(1/ 
2)/f
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.41 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.52 \[ \int \frac {\cot ^4(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=\frac {-9 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )+12 (1-i)^{3/2} \text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1-i}}\right )+12 (1+i)^{3/2} \text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1+i}}\right )+9 \cot (e+f x) \sqrt {1+\tan (e+f x)}+10 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}-8 \cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{24 f} \] Input:

Integrate[Cot[e + f*x]^4/Sqrt[1 + Tan[e + f*x]],x]
 

Output:

(-9*ArcTanh[Sqrt[1 + Tan[e + f*x]]] + 12*(1 - I)^(3/2)*ArcTanh[Sqrt[1 + Ta 
n[e + f*x]]/Sqrt[1 - I]] + 12*(1 + I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]] 
/Sqrt[1 + I]] + 9*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]] + 10*Cot[e + f*x]^2* 
Sqrt[1 + Tan[e + f*x]] - 8*Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]])/(24*f)
 

Rubi [A] (verified)

Time = 1.36 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.31, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.143, Rules used = {3042, 4052, 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.

\(\displaystyle \int \frac {\cot ^4(e+f x)}{\sqrt {\tan (e+f x)+1}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\tan (e+f x)^4 \sqrt {\tan (e+f x)+1}}dx\)

\(\Big \downarrow \) 4052

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{6} \int \frac {5 \tan (e+f x)^2+6 \tan (e+f x)+5}{\tan (e+f x)^3 \sqrt {\tan (e+f x)+1}}dx-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (\frac {5 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}-\frac {3}{4} \int \frac {\cot ^2(e+f x) \left (3-5 \tan ^2(e+f x)\right )}{\sqrt {\tan (e+f x)+1}}dx\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (\frac {5 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}-\frac {3}{4} \int \frac {3-5 \tan (e+f x)^2}{\tan (e+f x)^2 \sqrt {\tan (e+f x)+1}}dx\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 4133

\(\displaystyle \frac {1}{6} \left (\frac {5 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}-\frac {3}{4} \left (-\int \frac {\cot (e+f x) \left (3 \tan ^2(e+f x)+16 \tan (e+f x)+3\right )}{2 \sqrt {\tan (e+f x)+1}}dx-\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (\frac {5 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}-\frac {3}{4} \left (-\frac {1}{2} \int \frac {\cot (e+f x) \left (3 \tan ^2(e+f x)+16 \tan (e+f x)+3\right )}{\sqrt {\tan (e+f x)+1}}dx-\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (\frac {5 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}-\frac {3}{4} \left (-\frac {1}{2} \int \frac {3 \tan (e+f x)^2+16 \tan (e+f x)+3}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 4136

\(\displaystyle \frac {1}{6} \left (\frac {5 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}-\frac {3}{4} \left (\frac {1}{2} \left (-\int \frac {16}{\sqrt {\tan (e+f x)+1}}dx-3 \int \frac {\cot (e+f x) \left (\tan ^2(e+f x)+1\right )}{\sqrt {\tan (e+f x)+1}}dx\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (\frac {5 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}-\frac {3}{4} \left (\frac {1}{2} \left (-16 \int \frac {1}{\sqrt {\tan (e+f x)+1}}dx-3 \int \frac {\cot (e+f x) \left (\tan ^2(e+f x)+1\right )}{\sqrt {\tan (e+f x)+1}}dx\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (\frac {5 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}-\frac {3}{4} \left (\frac {1}{2} \left (-16 \int \frac {1}{\sqrt {\tan (e+f x)+1}}dx-3 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 3966

\(\displaystyle \frac {1}{6} \left (\frac {5 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}-\frac {3}{4} \left (\frac {1}{2} \left (-\frac {16 \int \frac {1}{\sqrt {\tan (e+f x)+1} \left (\tan ^2(e+f x)+1\right )}d\tan (e+f x)}{f}-3 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 484

\(\displaystyle \frac {1}{6} \left (\frac {5 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}-\frac {3}{4} \left (\frac {1}{2} \left (-3 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {32 \int \frac {1}{(\tan (e+f x)+1)^2-2 (\tan (e+f x)+1)+2}d\sqrt {\tan (e+f x)+1}}{f}\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 1407

\(\displaystyle \frac {1}{6} \left (\frac {5 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}-\frac {3}{4} \left (\frac {1}{2} \left (-3 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {32 \left (\frac {\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-\sqrt {\tan (e+f x)+1}}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}+\frac {\int \frac {\sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {1}{6} \left (\frac {5 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}-\frac {3}{4} \left (\frac {1}{2} \left (-3 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {32 \left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \int \frac {1}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}-\frac {1}{2} \int -\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \int \frac {1}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}+\frac {1}{2} \int \frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{6} \left (\frac {5 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}-\frac {3}{4} \left (\frac {1}{2} \left (-3 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {32 \left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \int \frac {1}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}+\frac {1}{2} \int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \int \frac {1}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}+\frac {1}{2} \int \frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {1}{6} \left (\frac {5 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}-\frac {3}{4} \left (\frac {1}{2} \left (-3 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {32 \left (\frac {\frac {1}{2} \int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )} \int \frac {1}{-\tan (e+f x)+2 \left (1-\sqrt {2}\right )-1}d\left (2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\frac {1}{2} \int \frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )} \int \frac {1}{-\tan (e+f x)+2 \left (1-\sqrt {2}\right )-1}d\left (2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}\right )}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{6} \left (\frac {5 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}-\frac {3}{4} \left (\frac {1}{2} \left (-\frac {32 \left (\frac {\frac {1}{2} \int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}+\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\frac {1}{2} \int \frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}+\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{4 \sqrt {1+\sqrt {2}}}\right )}{f}-3 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {1}{6} \left (\frac {5 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}-\frac {3}{4} \left (\frac {1}{2} \left (-3 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {32 \left (\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{2} \log \left (\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \log \left (\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 4117

\(\displaystyle \frac {1}{6} \left (\frac {5 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}-\frac {3}{4} \left (\frac {1}{2} \left (-\frac {3 \int \frac {\cot (e+f x)}{\sqrt {\tan (e+f x)+1}}d\tan (e+f x)}{f}-\frac {32 \left (\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{2} \log \left (\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \log \left (\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{6} \left (\frac {5 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}-\frac {3}{4} \left (\frac {1}{2} \left (-\frac {6 \int \cot (e+f x)d\sqrt {\tan (e+f x)+1}}{f}-\frac {32 \left (\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{2} \log \left (\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \log \left (\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 220

\(\displaystyle \frac {1}{6} \left (\frac {5 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}-\frac {3}{4} \left (\frac {1}{2} \left (\frac {6 \text {arctanh}\left (\sqrt {\tan (e+f x)+1}\right )}{f}-\frac {32 \left (\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{2} \log \left (\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \log \left (\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )-\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

Input:

Int[Cot[e + f*x]^4/Sqrt[1 + Tan[e + f*x]],x]
 

Output:

-1/3*(Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]])/f + ((5*Cot[e + f*x]^2*Sqrt[1 
 + Tan[e + f*x]])/(2*f) - (3*(((6*ArcTanh[Sqrt[1 + Tan[e + f*x]]])/f - (32 
*((Sqrt[(1 + Sqrt[2])/(-1 + Sqrt[2])]*ArcTan[(-Sqrt[2*(1 + Sqrt[2])] + 2*S 
qrt[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 + 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 + S 
qrt[2]])))/f)/2 - (3*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]])/f))/4)/6
 

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 73

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]
 

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 220

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])
 

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 4052

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 + 1)*((c 
+ d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d))), x] + Simp[1 
/((m + 1)*(a^2 + b^2)*(b*c - a*d))   Int[(a + b*Tan[e + f*x])^(m + 1)*(c + 
d*Tan[e + f*x])^n*Simp[a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - 
 a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*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] && LtQ[m, -1] && (LtQ[n, 0] || Integ 
erQ[m]) &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
 

rule 4117

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]
 

rule 4132

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])))
 

rule 4133

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])))
 

rule 4136

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]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(927\) vs. \(2(220)=440\).

Time = 182.58 (sec) , antiderivative size = 928, normalized size of antiderivative = 3.29

method	result	size
default	\(\text {Expression too large to display}\)	\(928\)

Input:

int(cot(f*x+e)^4/(tan(f*x+e)+1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/48/f*cot(f*x+e)*csc(f*x+e)^2*(12*(-2+2*2^(1/2))^(1/2)*(1+2^(1/2))^(1/2)* 
((3*(1-cos(f*x+e))*sin(f*x+e)+cos(f*x+e)*(cos(f*x+e)-1))*2^(1/2)+2*(cos(f* 
x+e)-1)*(2*sin(f*x+e)-cos(f*x+e)))*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2* 
2^(1/2)*sin(f*x+e)*cos(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)*(3*2^(1/2)-4)/(2*2^(1/2)*sin(f*x+e)*cos(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)/(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))+9*sin(f*x+e)*(1+2^(1/2))^(1/2)*(2*2^(1/2)*(cos(f*x+e)- 
1)-3*cos(f*x+e)+3)*ln(2*cot(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(1+ 
cos(f*x+e))^2)^(1/2)+2*csc(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(1+c 
os(f*x+e))^2)^(1/2)-2*cot(f*x+e)-1)*(cot(f*x+e)+cot(f*x+e)^2)^(1/2)+24*((5 
*(1-cos(f*x+e))*sin(f*x+e)+2*cos(f*x+e)*(cos(f*x+e)-1))*2^(1/2)+7*(cos(f*x 
+e)-1)*sin(f*x+e)-3*cos(f*x+e)*(cos(f*x+e)-1))*((cos(f*x+e)+sin(f*x+e))*co 
s(f*x+e)/(2*2^(1/2)*sin(f*x+e)*cos(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)*arctanh(((cos(f*x+e)+sin(f*x+e))*c 
os(f*x+e)/(2*2^(1/2)*sin(f*x+e)*cos(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)*2^(1/2)/(1+2^(1/2))^(1/2))+2*(1+2 
^(1/2))^(1/2)*(2*(-9+17*cos(f*x+e)^2-10*sin(f*x+e)*cos(f*x+e))*2^(1/2)+27- 
51*cos(f*x+e)^2+30*sin(f*x+e)*cos(f*x+e))*((cos(f*x+e)+sin(f*x+e))*cos(...
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.51 \[ \int \frac {\cot ^4(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=\frac {24 \, \sqrt {\frac {1}{2}} f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (\sqrt {\frac {1}{2}} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} + f\right )} \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{3} - 24 \, \sqrt {\frac {1}{2}} f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (-\sqrt {\frac {1}{2}} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} + f\right )} \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{3} - 24 \, \sqrt {\frac {1}{2}} f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (\sqrt {\frac {1}{2}} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} - f\right )} \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{3} + 24 \, \sqrt {\frac {1}{2}} f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (-\sqrt {\frac {1}{2}} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} - f\right )} \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{3} - 9 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} + 1\right ) \tan \left (f x + e\right )^{3} + 9 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} - 1\right ) \tan \left (f x + e\right )^{3} + 2 \, {\left (9 \, \tan \left (f x + e\right )^{2} + 10 \, \tan \left (f x + e\right ) - 8\right )} \sqrt {\tan \left (f x + e\right ) + 1}}{48 \, f \tan \left (f x + e\right )^{3}} \] Input:

integrate(cot(f*x+e)^4/(1+tan(f*x+e))^(1/2),x, algorithm="fricas")
 

Output:

1/48*(24*sqrt(1/2)*f*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f^2)*log(sqrt(1/2)*(f^3* 
sqrt(-1/f^4) + f)*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f^2) + sqrt(tan(f*x + e) + 
1))*tan(f*x + e)^3 - 24*sqrt(1/2)*f*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f^2)*log( 
-sqrt(1/2)*(f^3*sqrt(-1/f^4) + f)*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f^2) + sqrt 
(tan(f*x + e) + 1))*tan(f*x + e)^3 - 24*sqrt(1/2)*f*sqrt((f^2*sqrt(-1/f^4) 
 - 1)/f^2)*log(sqrt(1/2)*(f^3*sqrt(-1/f^4) - f)*sqrt((f^2*sqrt(-1/f^4) - 1 
)/f^2) + sqrt(tan(f*x + e) + 1))*tan(f*x + e)^3 + 24*sqrt(1/2)*f*sqrt((f^2 
*sqrt(-1/f^4) - 1)/f^2)*log(-sqrt(1/2)*(f^3*sqrt(-1/f^4) - f)*sqrt((f^2*sq 
rt(-1/f^4) - 1)/f^2) + sqrt(tan(f*x + e) + 1))*tan(f*x + e)^3 - 9*log(sqrt 
(tan(f*x + e) + 1) + 1)*tan(f*x + e)^3 + 9*log(sqrt(tan(f*x + e) + 1) - 1) 
*tan(f*x + e)^3 + 2*(9*tan(f*x + e)^2 + 10*tan(f*x + e) - 8)*sqrt(tan(f*x 
+ e) + 1))/(f*tan(f*x + e)^3)
 

Sympy [F]

\[ \int \frac {\cot ^4(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=\int \frac {\cot ^{4}{\left (e + f x \right )}}{\sqrt {\tan {\left (e + f x \right )} + 1}}\, dx \] Input:

integrate(cot(f*x+e)**4/(1+tan(f*x+e))**(1/2),x)
 

Output:

Integral(cot(e + f*x)**4/sqrt(tan(e + f*x) + 1), x)
 

Maxima [F]

\[ \int \frac {\cot ^4(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=\int { \frac {\cot \left (f x + e\right )^{4}}{\sqrt {\tan \left (f x + e\right ) + 1}} \,d x } \] Input:

integrate(cot(f*x+e)^4/(1+tan(f*x+e))^(1/2),x, algorithm="maxima")
 

Output:

integrate(cot(f*x + e)^4/sqrt(tan(f*x + e) + 1), x)
 

Giac [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.29 \[ \int \frac {\cot ^4(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=-\frac {3 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} + 1\right )}{16 \, f} + \frac {3 \, \log \left ({\left | \sqrt {\tan \left (f x + e\right ) + 1} - 1 \right |}\right )}{16 \, f} + \frac {{\left (f^{2} \sqrt {2 \, \sqrt {2} - 2} + f \sqrt {2 \, \sqrt {2} + 2} {\left | f \right |}\right )} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right )}{4 \, f^{3}} + \frac {{\left (f^{2} \sqrt {2 \, \sqrt {2} - 2} + f \sqrt {2 \, \sqrt {2} + 2} {\left | f \right |}\right )} \arctan \left (-\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} - 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right )}{4 \, f^{3}} + \frac {{\left (f^{2} \sqrt {2 \, \sqrt {2} + 2} - f \sqrt {2 \, \sqrt {2} - 2} {\left | f \right |}\right )} \log \left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} \sqrt {\tan \left (f x + e\right ) + 1} + \sqrt {2} + \tan \left (f x + e\right ) + 1\right )}{8 \, f^{3}} - \frac {{\left (f^{2} \sqrt {2 \, \sqrt {2} + 2} - f \sqrt {2 \, \sqrt {2} - 2} {\left | f \right |}\right )} \log \left (-2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} \sqrt {\tan \left (f x + e\right ) + 1} + \sqrt {2} + \tan \left (f x + e\right ) + 1\right )}{8 \, f^{3}} + \frac {9 \, {\left (\tan \left (f x + e\right ) + 1\right )}^{\frac {5}{2}} - 8 \, {\left (\tan \left (f x + e\right ) + 1\right )}^{\frac {3}{2}} - 9 \, \sqrt {\tan \left (f x + e\right ) + 1}}{24 \, f \tan \left (f x + e\right )^{3}} \] Input:

integrate(cot(f*x+e)^4/(1+tan(f*x+e))^(1/2),x, algorithm="giac")
 

Output:

-3/16*log(sqrt(tan(f*x + e) + 1) + 1)/f + 3/16*log(abs(sqrt(tan(f*x + e) + 
 1) - 1))/f + 1/4*(f^2*sqrt(2*sqrt(2) - 2) + f*sqrt(2*sqrt(2) + 2)*abs(f)) 
*arctan(1/2*2^(3/4)*(2^(1/4)*sqrt(sqrt(2) + 2) + 2*sqrt(tan(f*x + e) + 1)) 
/sqrt(-sqrt(2) + 2))/f^3 + 1/4*(f^2*sqrt(2*sqrt(2) - 2) + f*sqrt(2*sqrt(2) 
 + 2)*abs(f))*arctan(-1/2*2^(3/4)*(2^(1/4)*sqrt(sqrt(2) + 2) - 2*sqrt(tan( 
f*x + e) + 1))/sqrt(-sqrt(2) + 2))/f^3 + 1/8*(f^2*sqrt(2*sqrt(2) + 2) - f* 
sqrt(2*sqrt(2) - 2)*abs(f))*log(2^(1/4)*sqrt(sqrt(2) + 2)*sqrt(tan(f*x + e 
) + 1) + sqrt(2) + tan(f*x + e) + 1)/f^3 - 1/8*(f^2*sqrt(2*sqrt(2) + 2) - 
f*sqrt(2*sqrt(2) - 2)*abs(f))*log(-2^(1/4)*sqrt(sqrt(2) + 2)*sqrt(tan(f*x 
+ e) + 1) + sqrt(2) + tan(f*x + e) + 1)/f^3 + 1/24*(9*(tan(f*x + e) + 1)^( 
5/2) - 8*(tan(f*x + e) + 1)^(3/2) - 9*sqrt(tan(f*x + e) + 1))/(f*tan(f*x + 
 e)^3)
 

Mupad [B] (verification not implemented)

Time = 1.11 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.60 \[ \int \frac {\cot ^4(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=\frac {\mathrm {atan}\left (\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{8\,f}+\frac {\frac {3\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}}{8}+\frac {{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{3/2}}{3}-\frac {3\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{5/2}}{8}}{f-3\,f\,\left (\mathrm {tan}\left (e+f\,x\right )+1\right )+3\,f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^2-f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^3}+\mathrm {atan}\left (2\,f\,\sqrt {\frac {-\frac {1}{8}-\frac {1}{8}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\right )\,\sqrt {\frac {-\frac {1}{8}-\frac {1}{8}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (2\,f\,\sqrt {\frac {-\frac {1}{8}+\frac {1}{8}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\right )\,\sqrt {\frac {-\frac {1}{8}+\frac {1}{8}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i} \] Input:

int(cot(e + f*x)^4/(tan(e + f*x) + 1)^(1/2),x)
 

Output:

(atan((tan(e + f*x) + 1)^(1/2)*1i)*3i)/(8*f) + ((3*(tan(e + f*x) + 1)^(1/2 
))/8 + (tan(e + f*x) + 1)^(3/2)/3 - (3*(tan(e + f*x) + 1)^(5/2))/8)/(f - 3 
*f*(tan(e + f*x) + 1) + 3*f*(tan(e + f*x) + 1)^2 - f*(tan(e + f*x) + 1)^3) 
 + atan(2*f*((- 1/8 - 1i/8)/f^2)^(1/2)*(tan(e + f*x) + 1)^(1/2))*((- 1/8 - 
 1i/8)/f^2)^(1/2)*2i - atan(2*f*((- 1/8 + 1i/8)/f^2)^(1/2)*(tan(e + f*x) + 
 1)^(1/2))*((- 1/8 + 1i/8)/f^2)^(1/2)*2i
 

Reduce [F]

\[ \int \frac {\cot ^4(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=\int \frac {\sqrt {\tan \left (f x +e \right )+1}\, \cot \left (f x +e \right )^{4}}{\tan \left (f x +e \right )+1}d x \] Input:

int(cot(f*x+e)^4/(1+tan(f*x+e))^(1/2),x)
 

Output:

int((sqrt(tan(e + f*x) + 1)*cot(e + f*x)**4)/(tan(e + f*x) + 1),x)