\(\int \cot ^4(e+f x) \sqrt {1+\tan (e+f x)} \, dx\) [388]

Optimal result

Integrand size = 21, antiderivative size = 287 \[ \int \cot ^4(e+f x) \sqrt {1+\tan (e+f x)} \, dx=-\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{f}+\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{f}+\frac {7 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )}{8 f}-\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}{1+\sqrt {2}+\tan (e+f x)}\right )}{f}+\frac {9 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {\cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f} \] Output:

-1/2*(2+2*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*(2+2*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+7/8*arctanh((1+tan(f* 
x+e))^(1/2))/f-1/2*(-2+2*2^(1/2))^(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)))/f+9/8*cot(f*x+e)*(1+tan(f*x+e))^(1/ 
2)/f-1/12*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] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 20.07 (sec) , antiderivative size = 4078, normalized size of antiderivative = 14.21 \[ \int \cot ^4(e+f x) \sqrt {1+\tan (e+f x)} \, dx=\text {Result too large to show} \] Input:

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

Output:

((1/12 + (35*Cot[e + f*x])/24 - Csc[e + f*x]^2/12 - (Cot[e + f*x]*Csc[e + 
f*x]^2)/3)*Sqrt[1 + Tan[e + f*x]])/f + ((9*EllipticF[ArcSin[(2^(1/4)*Sqrt[ 
(1 + Tan[(e + f*x)/2])/(-1 + Tan[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]], -3 - 
2*Sqrt[2]] + 7*EllipticPi[-1 - Sqrt[2], ArcSin[(2^(1/4)*Sqrt[(1 + Tan[(e + 
 f*x)/2])/(-1 + Tan[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]], -3 - 2*Sqrt[2]] - 
(16 + 16*I)*EllipticPi[(-I)*(1 + Sqrt[2]), ArcSin[(2^(1/4)*Sqrt[(1 + Tan[( 
e + f*x)/2])/(-1 + Tan[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]], -3 - 2*Sqrt[2]] 
 - (16 - 16*I)*EllipticPi[I*(1 + Sqrt[2]), ArcSin[(2^(1/4)*Sqrt[(1 + Tan[( 
e + f*x)/2])/(-1 + Tan[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]], -3 - 2*Sqrt[2]] 
 + 7*EllipticPi[1 + Sqrt[2], ArcSin[(2^(1/4)*Sqrt[(1 + Tan[(e + f*x)/2])/( 
-1 + Tan[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]], -3 - 2*Sqrt[2]])*((Csc[e + f* 
x]*Sqrt[Sec[e + f*x]])/(16*Sqrt[Cos[e + f*x] + Sin[e + f*x]]) - (Cos[2*(e 
+ f*x)]*Csc[e + f*x]*Sqrt[Sec[e + f*x]])/(2*Sqrt[Cos[e + f*x] + Sin[e + f* 
x]]) + (Csc[e + f*x]*Sqrt[Sec[e + f*x]]*Sin[2*(e + f*x)])/(2*Sqrt[Cos[e + 
f*x] + Sin[e + f*x]]))*Sqrt[-((1 + Tan[(e + f*x)/2])/((-2 + Sqrt[2])*(-1 + 
 Tan[(e + f*x)/2])))]*Sqrt[1 + Tan[e + f*x]])/(4*2^(1/4)*f*Sqrt[(Cos[e + f 
*x] + Sin[e + f*x])/(-1 + Sin[e + f*x])]*(((9*EllipticF[ArcSin[(2^(1/4)*Sq 
rt[(1 + Tan[(e + f*x)/2])/(-1 + Tan[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]], -3 
 - 2*Sqrt[2]] + 7*EllipticPi[-1 - Sqrt[2], ArcSin[(2^(1/4)*Sqrt[(1 + Tan[( 
e + f*x)/2])/(-1 + Tan[(e + f*x)/2])])/Sqrt[2 + Sqrt[2]]], -3 - 2*Sqrt[...
 

Rubi [A] (verified)

Time = 1.45 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.25, number of steps used = 26, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.190, Rules used = {3042, 4051, 27, 3042, 4132, 27, 3042, 4132, 27, 3042, 4136, 27, 3042, 3966, 483, 1447, 1475, 1083, 217, 1478, 25, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4051

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

\(\displaystyle \frac {1}{6} \left (-\frac {1}{2} \int \frac {3 \cot ^2(e+f x) \left (\tan ^2(e+f x)+8 \tan (e+f x)+9\right )}{2 \sqrt {\tan (e+f x)+1}}dx-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \int \frac {\cot ^2(e+f x) \left (\tan ^2(e+f x)+8 \tan (e+f x)+9\right )}{\sqrt {\tan (e+f x)+1}}dx-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \int \frac {\tan (e+f x)^2+8 \tan (e+f x)+9}{\tan (e+f x)^2 \sqrt {\tan (e+f x)+1}}dx-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 4132

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (-\int -\frac {\cot (e+f x) \left (-9 \tan ^2(e+f x)-16 \tan (e+f x)+7\right )}{2 \sqrt {\tan (e+f x)+1}}dx-\frac {9 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \int \frac {\cot (e+f x) \left (-9 \tan ^2(e+f x)-16 \tan (e+f x)+7\right )}{\sqrt {\tan (e+f x)+1}}dx-\frac {9 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \int \frac {-9 \tan (e+f x)^2-16 \tan (e+f x)+7}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {9 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 4136

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \left (\int -16 \sqrt {\tan (e+f x)+1}dx+7 \int \frac {\cot (e+f x) \left (\tan ^2(e+f x)+1\right )}{\sqrt {\tan (e+f x)+1}}dx\right )-\frac {9 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \left (7 \int \frac {\cot (e+f x) \left (\tan ^2(e+f x)+1\right )}{\sqrt {\tan (e+f x)+1}}dx-16 \int \sqrt {\tan (e+f x)+1}dx\right )-\frac {9 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \left (7 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-16 \int \sqrt {\tan (e+f x)+1}dx\right )-\frac {9 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 3966

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \left (7 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {16 \int \frac {\sqrt {\tan (e+f x)+1}}{\tan ^2(e+f x)+1}d\tan (e+f x)}{f}\right )-\frac {9 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 483

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \left (7 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {32 \int \frac {\tan (e+f x)+1}{(\tan (e+f x)+1)^2-2 (\tan (e+f x)+1)+2}d\sqrt {\tan (e+f x)+1}}{f}\right )-\frac {9 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 1447

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \left (7 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {32 \left (\frac {1}{2} \int \frac {\tan (e+f x)+\sqrt {2}+1}{(\tan (e+f x)+1)^2-2 (\tan (e+f x)+1)+2}d\sqrt {\tan (e+f x)+1}-\frac {1}{2} \int \frac {-\tan (e+f x)+\sqrt {2}-1}{(\tan (e+f x)+1)^2-2 (\tan (e+f x)+1)+2}d\sqrt {\tan (e+f x)+1}\right )}{f}\right )-\frac {9 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 1475

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \left (7 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {32 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}+\frac {1}{2} \int \frac {1}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}\right )-\frac {1}{2} \int \frac {-\tan (e+f x)+\sqrt {2}-1}{(\tan (e+f x)+1)^2-2 (\tan (e+f x)+1)+2}d\sqrt {\tan (e+f x)+1}\right )}{f}\right )-\frac {9 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \left (7 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {32 \left (\frac {1}{2} \left (-\int \frac {1}{-\tan (e+f x)+2 \left (1-\sqrt {2}\right )-1}d\left (2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}\right )-\int \frac {1}{-\tan (e+f x)+2 \left (1-\sqrt {2}\right )-1}d\left (2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}\right )\right )-\frac {1}{2} \int \frac {-\tan (e+f x)+\sqrt {2}-1}{(\tan (e+f x)+1)^2-2 (\tan (e+f x)+1)+2}d\sqrt {\tan (e+f x)+1}\right )}{f}\right )-\frac {9 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \left (7 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {32 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}+\frac {\arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{2} \int \frac {-\tan (e+f x)+\sqrt {2}-1}{(\tan (e+f x)+1)^2-2 (\tan (e+f x)+1)+2}d\sqrt {\tan (e+f x)+1}\right )}{f}\right )-\frac {9 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 1478

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \left (7 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {32 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\int -\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}+\frac {\arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )\right )}{f}\right )-\frac {9 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \left (7 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {32 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\int \frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}+\frac {\arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )\right )}{f}\right )-\frac {9 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \left (7 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {32 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}+\frac {\arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\log \left (\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}\right )\right )}{f}\right )-\frac {9 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 4117

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \left (\frac {7 \int \frac {\cot (e+f x)}{\sqrt {\tan (e+f x)+1}}d\tan (e+f x)}{f}-\frac {32 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}+\frac {\arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\log \left (\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}\right )\right )}{f}\right )-\frac {9 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \left (\frac {14 \int \cot (e+f x)d\sqrt {\tan (e+f x)+1}}{f}-\frac {32 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}+\frac {\arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\log \left (\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}\right )\right )}{f}\right )-\frac {9 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 220

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \left (-\frac {32 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}+\frac {\arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\log \left (\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}\right )\right )}{f}-\frac {14 \text {arctanh}\left (\sqrt {\tan (e+f x)+1}\right )}{f}\right )-\frac {9 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

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 + (-1/2*(Cot[e + f*x]^2*Sqr 
t[1 + Tan[e + f*x]])/f - (3*(((-14*ArcTanh[Sqrt[1 + Tan[e + f*x]]])/f - (3 
2*((ArcTan[(-Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 
+ Sqrt[2])]]/Sqrt[2*(-1 + Sqrt[2])] + ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sq 
rt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]]/Sqrt[2*(-1 + Sqrt[2])])/2 + 
(Log[1 + Sqrt[2] + Tan[e + f*x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f 
*x]]]/(2*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*Sqrt[2*(1 + Sqrt[2])]))/2))/f)/2 
 - (9*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 483

Int[Sqrt[(c_) + (d_.)*(x_)]/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[2*d 
Subst[Int[x^2/(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 1447

Int[(x_)^2/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
a/c, 2]}, Simp[1/2   Int[(q + x^2)/(a + b*x^2 + c*x^4), x], x] - Simp[1/2 
 Int[(q - x^2)/(a + b*x^2 + c*x^4), x], x]] /; FreeQ[{a, b, c}, x] && LtQ[b 
^2 - 4*a*c, 0] && PosQ[a*c]
 

rule 1475

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[2*(d/e) - b/c, 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^ 
2, x], x], x] + Simp[e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; F 
reeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && 
 (GtQ[2*(d/e) - b/c, 0] || ( !LtQ[2*(d/e) - b/c, 0] && EqQ[d - e*Rt[a/c, 2] 
, 0]))
 

rule 1478

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[-2*(d/e) - b/c, 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e 
 + q*x - x^2, x], x], x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - 
x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ 
[c*d^2 - a*e^2, 0] &&  !GtQ[b^2 - 4*a*c, 0]
 

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 4051

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

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 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. \(930\) vs. \(2(226)=452\).

Time = 123.91 (sec) , antiderivative size = 931, normalized size of antiderivative = 3.24

method	result	size
default	\(\text {Expression too large to display}\)	\(931\)

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*(24*(-2+2*2^(1/2))^(1/2)*(1+2^(1/2))^(1/2)* 
((cos(f*x+e)-1)*(2*sin(f*x+e)-cos(f*x+e))*2^(1/2)+3*(1-cos(f*x+e))*sin(f*x 
+e)+cos(f*x+e)*(cos(f*x+e)-1))*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*sin(f*x+e)^2*2^(1/2)-2*2^(1/2)*sin(f*x 
+e)*cos(f*x+e)-2*sin(f*x+e)^2+2*sin(f*x+e)*cos(f*x+e)-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))*(-(cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*sin(f*x+e) 
^2*2^(1/2)-2*2^(1/2)*sin(f*x+e)*cos(f*x+e)-2*sin(f*x+e)^2+2*sin(f*x+e)*cos 
(f*x+e)-1))^(1/2)+21*sin(f*x+e)*(1+2^(1/2))^(1/2)*(2*2^(1/2)*(1-cos(f*x+e) 
)-3+3*cos(f*x+e))*ln(2*cot(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*csc(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(1+co 
s(f*x+e))^2)^(1/2)-2*cot(f*x+e)-1)*(cot(f*x+e)+cot(f*x+e)^2)^(1/2)+24*((7* 
(1-cos(f*x+e))*sin(f*x+e)+3*cos(f*x+e)*(cos(f*x+e)-1))*2^(1/2)+10*(cos(f*x 
+e)-1)*sin(f*x+e)-4*cos(f*x+e)*(cos(f*x+e)-1))*(-(cos(f*x+e)+sin(f*x+e))*c 
os(f*x+e)/(2*sin(f*x+e)^2*2^(1/2)-2*2^(1/2)*sin(f*x+e)*cos(f*x+e)-2*sin(f* 
x+e)^2+2*sin(f*x+e)*cos(f*x+e)-1))^(1/2)*arctanh((-(cos(f*x+e)+sin(f*x+e)) 
*cos(f*x+e)/(2*sin(f*x+e)^2*2^(1/2)-2*2^(1/2)*sin(f*x+e)*cos(f*x+e)-2*sin( 
f*x+e)^2+2*sin(f*x+e)*cos(f*x+e)-1))^(1/2)*2^(1/2)/(1+2^(1/2))^(1/2))+2*(1 
+2^(1/2))^(1/2)*(2*(-27+35*cos(f*x+e)^2+2*sin(f*x+e)*cos(f*x+e))*2^(1/2)+8 
1-105*cos(f*x+e)^2-6*sin(f*x+e)*cos(f*x+e))*((cos(f*x+e)+sin(f*x+e))*co...
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.34 \[ \int \cot ^4(e+f x) \sqrt {1+\tan (e+f x)} \, dx=-\frac {24 \, f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (f^{3} \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \sqrt {-\frac {1}{f^{4}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{3} - 24 \, f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (-f^{3} \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \sqrt {-\frac {1}{f^{4}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{3} - 24 \, f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (f^{3} \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \sqrt {-\frac {1}{f^{4}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{3} + 24 \, f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (-f^{3} \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \sqrt {-\frac {1}{f^{4}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{3} - 21 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} + 1\right ) \tan \left (f x + e\right )^{3} + 21 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} - 1\right ) \tan \left (f x + e\right )^{3} - 2 \, {\left (27 \, \tan \left (f x + e\right )^{2} - 2 \, \tan \left (f x + e\right ) - 8\right )} \sqrt {\tan \left (f x + e\right ) + 1}}{48 \, f \tan \left (f x + e\right )^{3}} \] Input:

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

Output:

-1/48*(24*f*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f^2)*log(f^3*sqrt(-(f^2*sqrt(-1/f 
^4) + 1)/f^2)*sqrt(-1/f^4) + sqrt(tan(f*x + e) + 1))*tan(f*x + e)^3 - 24*f 
*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f^2)*log(-f^3*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f 
^2)*sqrt(-1/f^4) + sqrt(tan(f*x + e) + 1))*tan(f*x + e)^3 - 24*f*sqrt((f^2 
*sqrt(-1/f^4) - 1)/f^2)*log(f^3*sqrt((f^2*sqrt(-1/f^4) - 1)/f^2)*sqrt(-1/f 
^4) + sqrt(tan(f*x + e) + 1))*tan(f*x + e)^3 + 24*f*sqrt((f^2*sqrt(-1/f^4) 
 - 1)/f^2)*log(-f^3*sqrt((f^2*sqrt(-1/f^4) - 1)/f^2)*sqrt(-1/f^4) + sqrt(t 
an(f*x + e) + 1))*tan(f*x + e)^3 - 21*log(sqrt(tan(f*x + e) + 1) + 1)*tan( 
f*x + e)^3 + 21*log(sqrt(tan(f*x + e) + 1) - 1)*tan(f*x + e)^3 - 2*(27*tan 
(f*x + e)^2 - 2*tan(f*x + e) - 8)*sqrt(tan(f*x + e) + 1))/(f*tan(f*x + e)^ 
3)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.55 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.21 \[ \int \cot ^4(e+f x) \sqrt {1+\tan (e+f x)} \, dx=\frac {7 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} + 1\right )}{16 \, f} - \frac {7 \, \log \left ({\left | \sqrt {\tan \left (f x + e\right ) + 1} - 1 \right |}\right )}{16 \, f} + \frac {{\left (f^{2} \sqrt {\sqrt {2} + 1} + f \sqrt {\sqrt {2} - 1} {\left | f \right |}\right )} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right )}{2 \, f^{3}} + \frac {{\left (f^{2} \sqrt {\sqrt {2} + 1} + f \sqrt {\sqrt {2} - 1} {\left | f \right |}\right )} \arctan \left (-\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} - 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right )}{2 \, f^{3}} + \frac {{\left (f^{2} \sqrt {\sqrt {2} - 1} - f \sqrt {\sqrt {2} + 1} {\left | f \right |}\right )} \log \left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} \sqrt {\tan \left (f x + e\right ) + 1} + \sqrt {2} + \tan \left (f x + e\right ) + 1\right )}{4 \, f^{3}} - \frac {{\left (f^{2} \sqrt {\sqrt {2} - 1} - f \sqrt {\sqrt {2} + 1} {\left | f \right |}\right )} \log \left (-2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} \sqrt {\tan \left (f x + e\right ) + 1} + \sqrt {2} + \tan \left (f x + e\right ) + 1\right )}{4 \, f^{3}} + \frac {27 \, {\left (\tan \left (f x + e\right ) + 1\right )}^{\frac {5}{2}} - 56 \, {\left (\tan \left (f x + e\right ) + 1\right )}^{\frac {3}{2}} + 21 \, \sqrt {\tan \left (f x + e\right ) + 1}}{24 \, f \tan \left (f x + e\right )^{3}} \] Input:

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

Output:

7/16*log(sqrt(tan(f*x + e) + 1) + 1)/f - 7/16*log(abs(sqrt(tan(f*x + e) + 
1) - 1))/f + 1/2*(f^2*sqrt(sqrt(2) + 1) + f*sqrt(sqrt(2) - 1)*abs(f))*arct 
an(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 + 1/2*(f^2*sqrt(sqrt(2) + 1) + f*sqrt(sqrt(2) - 1)*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(sqrt(2) - 1) - f*sqrt(sqrt(2 
) + 1)*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 - 1/4*(f^2*sqrt(sqrt(2) - 1) - f*sqrt(sqrt(2) 
+ 1)*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*(27*(tan(f*x + e) + 1)^(5/2) - 56*(tan(f 
*x + e) + 1)^(3/2) + 21*sqrt(tan(f*x + e) + 1))/(f*tan(f*x + e)^3)
 

Mupad [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.61 \[ \int \cot ^4(e+f x) \sqrt {1+\tan (e+f x)} \, dx=-\frac {\mathrm {atan}\left (\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,1{}\mathrm {i}\right )\,7{}\mathrm {i}}{8\,f}-\frac {\frac {7\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}}{8}-\frac {7\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{3/2}}{3}+\frac {9\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{5/2}}{8}}{f-3\,f\,\left (\mathrm {tan}\left (e+f\,x\right )+1\right )+3\,f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^2-f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^3}-\mathrm {atan}\left (f\,\sqrt {\frac {-\frac {1}{4}-\frac {1}{4}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,\left (1-\mathrm {i}\right )\right )\,\sqrt {\frac {-\frac {1}{4}-\frac {1}{4}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (f\,\sqrt {\frac {-\frac {1}{4}+\frac {1}{4}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,\left (1+1{}\mathrm {i}\right )\right )\,\sqrt {\frac {-\frac {1}{4}+\frac {1}{4}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i} \] Input:

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

Output:

atan(f*((- 1/4 + 1i/4)/f^2)^(1/2)*(tan(e + f*x) + 1)^(1/2)*(1 + 1i))*((- 1 
/4 + 1i/4)/f^2)^(1/2)*2i - ((7*(tan(e + f*x) + 1)^(1/2))/8 - (7*(tan(e + f 
*x) + 1)^(3/2))/3 + (9*(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(f*((- 1/ 
4 - 1i/4)/f^2)^(1/2)*(tan(e + f*x) + 1)^(1/2)*(1 - 1i))*((- 1/4 - 1i/4)/f^ 
2)^(1/2)*2i - (atan((tan(e + f*x) + 1)^(1/2)*1i)*7i)/(8*f)
 

Reduce [F]

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