\(\int \cot (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx\) [686]

Optimal result

Integrand size = 21, antiderivative size = 402 \[ \int \cot (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=-\frac {1}{4} i \sqrt [3]{c-i d} x+\frac {1}{4} i \sqrt [3]{c+i d} x-\frac {\sqrt {3} \sqrt [3]{c} \arctan \left (\frac {\sqrt [3]{c}+2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt {3} \sqrt [3]{c}}\right )}{f}+\frac {\sqrt {3} \sqrt [3]{c-i d} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt {3}}\right )}{2 f}+\frac {\sqrt {3} \sqrt [3]{c+i d} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt {3}}\right )}{2 f}-\frac {\sqrt [3]{c-i d} \log (\cos (e+f x))}{4 f}-\frac {\sqrt [3]{c+i d} \log (\cos (e+f x))}{4 f}-\frac {\sqrt [3]{c} \log (\tan (e+f x))}{2 f}+\frac {3 \sqrt [3]{c} \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 f}-\frac {3 \sqrt [3]{c-i d} \log \left (\sqrt [3]{c-i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f}-\frac {3 \sqrt [3]{c+i d} \log \left (\sqrt [3]{c+i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 f} \] Output:

-1/4*I*(c-I*d)^(1/3)*x+1/4*I*(c+I*d)^(1/3)*x-3^(1/2)*c^(1/3)*arctan(1/3*(c 
^(1/3)+2*(c+d*tan(f*x+e))^(1/3))*3^(1/2)/c^(1/3))/f+1/2*3^(1/2)*(c-I*d)^(1 
/3)*arctan(1/3*(1+2*(c+d*tan(f*x+e))^(1/3)/(c-I*d)^(1/3))*3^(1/2))/f+1/2*3 
^(1/2)*(c+I*d)^(1/3)*arctan(1/3*(1+2*(c+d*tan(f*x+e))^(1/3)/(c+I*d)^(1/3)) 
*3^(1/2))/f-1/4*(c-I*d)^(1/3)*ln(cos(f*x+e))/f-1/4*(c+I*d)^(1/3)*ln(cos(f* 
x+e))/f-1/2*c^(1/3)*ln(tan(f*x+e))/f+3/2*c^(1/3)*ln(c^(1/3)-(c+d*tan(f*x+e 
))^(1/3))/f-3/4*(c-I*d)^(1/3)*ln((c-I*d)^(1/3)-(c+d*tan(f*x+e))^(1/3))/f-3 
/4*(c+I*d)^(1/3)*ln((c+I*d)^(1/3)-(c+d*tan(f*x+e))^(1/3))/f
 

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 744, normalized size of antiderivative = 1.85 \[ \int \cot (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\frac {-4 \sqrt {3} \sqrt [3]{c} \arctan \left (\frac {\sqrt [3]{c}+2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt {3} \sqrt [3]{c}}\right )+2 \sqrt {3} \sqrt [3]{c-i d} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt {3}}\right )+\frac {2 \sqrt {3} c \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt {3}}\right )}{(c+i d)^{2/3}}+\frac {2 i \sqrt {3} d \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt {3}}\right )}{(c+i d)^{2/3}}+4 \sqrt [3]{c} \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )-\frac {2 c \log \left (\sqrt [3]{c-i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{(c-i d)^{2/3}}+\frac {2 i d \log \left (\sqrt [3]{c-i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{(c-i d)^{2/3}}-\frac {2 c \log \left (\sqrt [3]{c+i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{(c+i d)^{2/3}}-\frac {2 i d \log \left (\sqrt [3]{c+i d}-\sqrt [3]{c+d \tan (e+f x)}\right )}{(c+i d)^{2/3}}-2 \sqrt [3]{c} \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right )+\frac {c \log \left ((c-i d)^{2/3}+\sqrt [3]{c-i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right )}{(c-i d)^{2/3}}-\frac {i d \log \left ((c-i d)^{2/3}+\sqrt [3]{c-i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right )}{(c-i d)^{2/3}}+\frac {c \log \left ((c+i d)^{2/3}+\sqrt [3]{c+i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right )}{(c+i d)^{2/3}}+\frac {i d \log \left ((c+i d)^{2/3}+\sqrt [3]{c+i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right )}{(c+i d)^{2/3}}}{4 f} \] Input:

Integrate[Cot[e + f*x]*(c + d*Tan[e + f*x])^(1/3),x]
 

Output:

(-4*Sqrt[3]*c^(1/3)*ArcTan[(c^(1/3) + 2*(c + d*Tan[e + f*x])^(1/3))/(Sqrt[ 
3]*c^(1/3))] + 2*Sqrt[3]*(c - I*d)^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f*x 
])^(1/3))/(c - I*d)^(1/3))/Sqrt[3]] + (2*Sqrt[3]*c*ArcTan[(1 + (2*(c + d*T 
an[e + f*x])^(1/3))/(c + I*d)^(1/3))/Sqrt[3]])/(c + I*d)^(2/3) + ((2*I)*Sq 
rt[3]*d*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c + I*d)^(1/3))/Sqrt[3 
]])/(c + I*d)^(2/3) + 4*c^(1/3)*Log[c^(1/3) - (c + d*Tan[e + f*x])^(1/3)] 
- (2*c*Log[(c - I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(c - I*d)^(2/3) 
+ ((2*I)*d*Log[(c - I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(c - I*d)^(2 
/3) - (2*c*Log[(c + I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(c + I*d)^(2 
/3) - ((2*I)*d*Log[(c + I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(c + I*d 
)^(2/3) - 2*c^(1/3)*Log[c^(2/3) + c^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c 
+ d*Tan[e + f*x])^(2/3)] + (c*Log[(c - I*d)^(2/3) + (c - I*d)^(1/3)*(c + d 
*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x])^(2/3)])/(c - I*d)^(2/3) - (I*d 
*Log[(c - I*d)^(2/3) + (c - I*d)^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c + d 
*Tan[e + f*x])^(2/3)])/(c - I*d)^(2/3) + (c*Log[(c + I*d)^(2/3) + (c + I*d 
)^(1/3)*(c + d*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x])^(2/3)])/(c + I*d 
)^(2/3) + (I*d*Log[(c + I*d)^(2/3) + (c + I*d)^(1/3)*(c + d*Tan[e + f*x])^ 
(1/3) + (c + d*Tan[e + f*x])^(2/3)])/(c + I*d)^(2/3))/(4*f)
 

Rubi [A] (warning: unable to verify)

Time = 1.33 (sec) , antiderivative size = 368, normalized size of antiderivative = 0.92, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.952, Rules used = {3042, 4057, 25, 3042, 4011, 3042, 4022, 3042, 4020, 25, 69, 16, 1082, 217, 4117, 60, 69, 16, 1082, 217}

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.

\(\displaystyle \int \cot (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sqrt [3]{c+d \tan (e+f x)}}{\tan (e+f x)}dx\)

\(\Big \downarrow \) 4057

\(\displaystyle \int -\tan (e+f x) \sqrt [3]{c+d \tan (e+f x)}dx+\int \cot (e+f x) \sqrt [3]{c+d \tan (e+f x)} \left (\tan ^2(e+f x)+1\right )dx\)

\(\Big \downarrow \) 25

\(\displaystyle \int \cot (e+f x) \sqrt [3]{c+d \tan (e+f x)} \left (\tan ^2(e+f x)+1\right )dx-\int \tan (e+f x) \sqrt [3]{c+d \tan (e+f x)}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sqrt [3]{c+d \tan (e+f x)} \left (\tan (e+f x)^2+1\right )}{\tan (e+f x)}dx-\int \tan (e+f x) \sqrt [3]{c+d \tan (e+f x)}dx\)

\(\Big \downarrow \) 4011

\(\displaystyle \int \frac {\sqrt [3]{c+d \tan (e+f x)} \left (\tan (e+f x)^2+1\right )}{\tan (e+f x)}dx-\int \frac {c \tan (e+f x)-d}{(c+d \tan (e+f x))^{2/3}}dx-\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\int \frac {c \tan (e+f x)-d}{(c+d \tan (e+f x))^{2/3}}dx+\int \frac {\sqrt [3]{c+d \tan (e+f x)} \left (\tan (e+f x)^2+1\right )}{\tan (e+f x)}dx-\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}\)

\(\Big \downarrow \) 4022

\(\displaystyle \int \frac {\sqrt [3]{c+d \tan (e+f x)} \left (\tan (e+f x)^2+1\right )}{\tan (e+f x)}dx-\frac {1}{2} (-d+i c) \int \frac {1-i \tan (e+f x)}{(c+d \tan (e+f x))^{2/3}}dx+\frac {1}{2} (d+i c) \int \frac {i \tan (e+f x)+1}{(c+d \tan (e+f x))^{2/3}}dx-\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{2} (-d+i c) \int \frac {1-i \tan (e+f x)}{(c+d \tan (e+f x))^{2/3}}dx+\frac {1}{2} (d+i c) \int \frac {i \tan (e+f x)+1}{(c+d \tan (e+f x))^{2/3}}dx+\int \frac {\sqrt [3]{c+d \tan (e+f x)} \left (\tan (e+f x)^2+1\right )}{\tan (e+f x)}dx-\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}\)

\(\Big \downarrow \) 4020

\(\displaystyle \int \frac {\sqrt [3]{c+d \tan (e+f x)} \left (\tan (e+f x)^2+1\right )}{\tan (e+f x)}dx+\frac {i (d+i c) \int -\frac {1}{(1-i \tan (e+f x)) (c+d \tan (e+f x))^{2/3}}d(i \tan (e+f x))}{2 f}+\frac {i (-d+i c) \int -\frac {1}{(i \tan (e+f x)+1) (c+d \tan (e+f x))^{2/3}}d(-i \tan (e+f x))}{2 f}-\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle \int \frac {\sqrt [3]{c+d \tan (e+f x)} \left (\tan (e+f x)^2+1\right )}{\tan (e+f x)}dx-\frac {i (d+i c) \int \frac {1}{(1-i \tan (e+f x)) (c+d \tan (e+f x))^{2/3}}d(i \tan (e+f x))}{2 f}-\frac {i (-d+i c) \int \frac {1}{(i \tan (e+f x)+1) (c+d \tan (e+f x))^{2/3}}d(-i \tan (e+f x))}{2 f}-\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}\)

\(\Big \downarrow \) 69

\(\displaystyle \int \frac {\sqrt [3]{c+d \tan (e+f x)} \left (\tan (e+f x)^2+1\right )}{\tan (e+f x)}dx+\frac {i (d+i c) \left (-\frac {3 \int \frac {1}{-\tan ^2(e+f x)+(c-i d)^{2/3}+\sqrt [3]{c-i d} \sqrt [3]{c+d \tan (e+f x)}}d\sqrt [3]{c+d \tan (e+f x)}}{2 \sqrt [3]{c-i d}}-\frac {3 \int \frac {1}{\sqrt [3]{c-i d}-i \tan (e+f x)}d\sqrt [3]{c+d \tan (e+f x)}}{2 (c-i d)^{2/3}}-\frac {\log (1-i \tan (e+f x))}{2 (c-i d)^{2/3}}\right )}{2 f}+\frac {i (-d+i c) \left (-\frac {3 \int \frac {1}{-\tan ^2(e+f x)+(c+i d)^{2/3}+\sqrt [3]{c+i d} \sqrt [3]{c+d \tan (e+f x)}}d\sqrt [3]{c+d \tan (e+f x)}}{2 \sqrt [3]{c+i d}}-\frac {3 \int \frac {1}{i \tan (e+f x)+\sqrt [3]{c+i d}}d\sqrt [3]{c+d \tan (e+f x)}}{2 (c+i d)^{2/3}}-\frac {\log (1+i \tan (e+f x))}{2 (c+i d)^{2/3}}\right )}{2 f}-\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}\)

\(\Big \downarrow \) 16

\(\displaystyle \int \frac {\sqrt [3]{c+d \tan (e+f x)} \left (\tan (e+f x)^2+1\right )}{\tan (e+f x)}dx+\frac {i (d+i c) \left (-\frac {3 \int \frac {1}{-\tan ^2(e+f x)+(c-i d)^{2/3}+\sqrt [3]{c-i d} \sqrt [3]{c+d \tan (e+f x)}}d\sqrt [3]{c+d \tan (e+f x)}}{2 \sqrt [3]{c-i d}}-\frac {\log (1-i \tan (e+f x))}{2 (c-i d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{c-i d}-i \tan (e+f x)\right )}{2 (c-i d)^{2/3}}\right )}{2 f}+\frac {i (-d+i c) \left (-\frac {3 \int \frac {1}{-\tan ^2(e+f x)+(c+i d)^{2/3}+\sqrt [3]{c+i d} \sqrt [3]{c+d \tan (e+f x)}}d\sqrt [3]{c+d \tan (e+f x)}}{2 \sqrt [3]{c+i d}}-\frac {\log (1+i \tan (e+f x))}{2 (c+i d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{c+i d}+i \tan (e+f x)\right )}{2 (c+i d)^{2/3}}\right )}{2 f}-\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}\)

\(\Big \downarrow \) 1082

\(\displaystyle \int \frac {\sqrt [3]{c+d \tan (e+f x)} \left (\tan (e+f x)^2+1\right )}{\tan (e+f x)}dx+\frac {i (d+i c) \left (\frac {3 \int \frac {1}{\tan ^2(e+f x)-3}d\left (\frac {2 i \tan (e+f x)}{\sqrt [3]{c-i d}}+1\right )}{(c-i d)^{2/3}}-\frac {\log (1-i \tan (e+f x))}{2 (c-i d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{c-i d}-i \tan (e+f x)\right )}{2 (c-i d)^{2/3}}\right )}{2 f}+\frac {i (-d+i c) \left (\frac {3 \int \frac {1}{\tan ^2(e+f x)-3}d\left (1-\frac {2 i \tan (e+f x)}{\sqrt [3]{c+i d}}\right )}{(c+i d)^{2/3}}-\frac {\log (1+i \tan (e+f x))}{2 (c+i d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{c+i d}+i \tan (e+f x)\right )}{2 (c+i d)^{2/3}}\right )}{2 f}-\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}\)

\(\Big \downarrow \) 217

\(\displaystyle \int \frac {\sqrt [3]{c+d \tan (e+f x)} \left (\tan (e+f x)^2+1\right )}{\tan (e+f x)}dx+\frac {i (d+i c) \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {\tan (e+f x)}{\sqrt {3}}\right )}{(c-i d)^{2/3}}-\frac {\log (1-i \tan (e+f x))}{2 (c-i d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{c-i d}-i \tan (e+f x)\right )}{2 (c-i d)^{2/3}}\right )}{2 f}+\frac {i (-d+i c) \left (\frac {i \sqrt {3} \text {arctanh}\left (\frac {\tan (e+f x)}{\sqrt {3}}\right )}{(c+i d)^{2/3}}-\frac {\log (1+i \tan (e+f x))}{2 (c+i d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{c+i d}+i \tan (e+f x)\right )}{2 (c+i d)^{2/3}}\right )}{2 f}-\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}\)

\(\Big \downarrow \) 4117

\(\displaystyle \frac {\int \cot (e+f x) \sqrt [3]{c+d \tan (e+f x)}d\tan (e+f x)}{f}+\frac {i (d+i c) \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {\tan (e+f x)}{\sqrt {3}}\right )}{(c-i d)^{2/3}}-\frac {\log (1-i \tan (e+f x))}{2 (c-i d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{c-i d}-i \tan (e+f x)\right )}{2 (c-i d)^{2/3}}\right )}{2 f}+\frac {i (-d+i c) \left (\frac {i \sqrt {3} \text {arctanh}\left (\frac {\tan (e+f x)}{\sqrt {3}}\right )}{(c+i d)^{2/3}}-\frac {\log (1+i \tan (e+f x))}{2 (c+i d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{c+i d}+i \tan (e+f x)\right )}{2 (c+i d)^{2/3}}\right )}{2 f}-\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {c \int \frac {\cot (e+f x)}{(c+d \tan (e+f x))^{2/3}}d\tan (e+f x)+3 \sqrt [3]{c+d \tan (e+f x)}}{f}+\frac {i (d+i c) \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {\tan (e+f x)}{\sqrt {3}}\right )}{(c-i d)^{2/3}}-\frac {\log (1-i \tan (e+f x))}{2 (c-i d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{c-i d}-i \tan (e+f x)\right )}{2 (c-i d)^{2/3}}\right )}{2 f}+\frac {i (-d+i c) \left (\frac {i \sqrt {3} \text {arctanh}\left (\frac {\tan (e+f x)}{\sqrt {3}}\right )}{(c+i d)^{2/3}}-\frac {\log (1+i \tan (e+f x))}{2 (c+i d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{c+i d}+i \tan (e+f x)\right )}{2 (c+i d)^{2/3}}\right )}{2 f}-\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}\)

\(\Big \downarrow \) 69

\(\displaystyle \frac {c \left (-\frac {3 \int \frac {1}{\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}}d\sqrt [3]{c+d \tan (e+f x)}}{2 c^{2/3}}-\frac {3 \int \frac {1}{c^{2/3}+\sqrt [3]{c+d \tan (e+f x)} \sqrt [3]{c}+(c+d \tan (e+f x))^{2/3}}d\sqrt [3]{c+d \tan (e+f x)}}{2 \sqrt [3]{c}}-\frac {\log (\tan (e+f x))}{2 c^{2/3}}\right )+3 \sqrt [3]{c+d \tan (e+f x)}}{f}+\frac {i (d+i c) \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {\tan (e+f x)}{\sqrt {3}}\right )}{(c-i d)^{2/3}}-\frac {\log (1-i \tan (e+f x))}{2 (c-i d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{c-i d}-i \tan (e+f x)\right )}{2 (c-i d)^{2/3}}\right )}{2 f}+\frac {i (-d+i c) \left (\frac {i \sqrt {3} \text {arctanh}\left (\frac {\tan (e+f x)}{\sqrt {3}}\right )}{(c+i d)^{2/3}}-\frac {\log (1+i \tan (e+f x))}{2 (c+i d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{c+i d}+i \tan (e+f x)\right )}{2 (c+i d)^{2/3}}\right )}{2 f}-\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {c \left (-\frac {3 \int \frac {1}{c^{2/3}+\sqrt [3]{c+d \tan (e+f x)} \sqrt [3]{c}+(c+d \tan (e+f x))^{2/3}}d\sqrt [3]{c+d \tan (e+f x)}}{2 \sqrt [3]{c}}+\frac {3 \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 c^{2/3}}-\frac {\log (\tan (e+f x))}{2 c^{2/3}}\right )+3 \sqrt [3]{c+d \tan (e+f x)}}{f}+\frac {i (d+i c) \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {\tan (e+f x)}{\sqrt {3}}\right )}{(c-i d)^{2/3}}-\frac {\log (1-i \tan (e+f x))}{2 (c-i d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{c-i d}-i \tan (e+f x)\right )}{2 (c-i d)^{2/3}}\right )}{2 f}+\frac {i (-d+i c) \left (\frac {i \sqrt {3} \text {arctanh}\left (\frac {\tan (e+f x)}{\sqrt {3}}\right )}{(c+i d)^{2/3}}-\frac {\log (1+i \tan (e+f x))}{2 (c+i d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{c+i d}+i \tan (e+f x)\right )}{2 (c+i d)^{2/3}}\right )}{2 f}-\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {c \left (\frac {3 \int \frac {1}{-(c+d \tan (e+f x))^{2/3}-3}d\left (\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c}}+1\right )}{c^{2/3}}+\frac {3 \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 c^{2/3}}-\frac {\log (\tan (e+f x))}{2 c^{2/3}}\right )+3 \sqrt [3]{c+d \tan (e+f x)}}{f}+\frac {i (d+i c) \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {\tan (e+f x)}{\sqrt {3}}\right )}{(c-i d)^{2/3}}-\frac {\log (1-i \tan (e+f x))}{2 (c-i d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{c-i d}-i \tan (e+f x)\right )}{2 (c-i d)^{2/3}}\right )}{2 f}+\frac {i (-d+i c) \left (\frac {i \sqrt {3} \text {arctanh}\left (\frac {\tan (e+f x)}{\sqrt {3}}\right )}{(c+i d)^{2/3}}-\frac {\log (1+i \tan (e+f x))}{2 (c+i d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{c+i d}+i \tan (e+f x)\right )}{2 (c+i d)^{2/3}}\right )}{2 f}-\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {c \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c}}+1}{\sqrt {3}}\right )}{c^{2/3}}+\frac {3 \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 c^{2/3}}-\frac {\log (\tan (e+f x))}{2 c^{2/3}}\right )+3 \sqrt [3]{c+d \tan (e+f x)}}{f}+\frac {i (d+i c) \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {\tan (e+f x)}{\sqrt {3}}\right )}{(c-i d)^{2/3}}-\frac {\log (1-i \tan (e+f x))}{2 (c-i d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{c-i d}-i \tan (e+f x)\right )}{2 (c-i d)^{2/3}}\right )}{2 f}+\frac {i (-d+i c) \left (\frac {i \sqrt {3} \text {arctanh}\left (\frac {\tan (e+f x)}{\sqrt {3}}\right )}{(c+i d)^{2/3}}-\frac {\log (1+i \tan (e+f x))}{2 (c+i d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{c+i d}+i \tan (e+f x)\right )}{2 (c+i d)^{2/3}}\right )}{2 f}-\frac {3 \sqrt [3]{c+d \tan (e+f x)}}{f}\)

Input:

Int[Cot[e + f*x]*(c + d*Tan[e + f*x])^(1/3),x]
 

Output:

((I/2)*(I*c + d)*(((-I)*Sqrt[3]*ArcTanh[Tan[e + f*x]/Sqrt[3]])/(c - I*d)^( 
2/3) - Log[1 - I*Tan[e + f*x]]/(2*(c - I*d)^(2/3)) + (3*Log[(c - I*d)^(1/3 
) - I*Tan[e + f*x]])/(2*(c - I*d)^(2/3))))/f + ((I/2)*(I*c - d)*((I*Sqrt[3 
]*ArcTanh[Tan[e + f*x]/Sqrt[3]])/(c + I*d)^(2/3) - Log[1 + I*Tan[e + f*x]] 
/(2*(c + I*d)^(2/3)) + (3*Log[(c + I*d)^(1/3) + I*Tan[e + f*x]])/(2*(c + I 
*d)^(2/3))))/f - (3*(c + d*Tan[e + f*x])^(1/3))/f + (c*(-((Sqrt[3]*ArcTan[ 
(1 + (2*(c + d*Tan[e + f*x])^(1/3))/c^(1/3))/Sqrt[3]])/c^(2/3)) - Log[Tan[ 
e + f*x]]/(2*c^(2/3)) + (3*Log[c^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(2*c 
^(2/3))) + 3*(c + d*Tan[e + f*x])^(1/3))/f
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 25

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

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 69

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ 
{q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2), 
 x] + (-Simp[3/(2*b*q)   Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1 
/3)], x] - Simp[3/(2*b*q^2)   Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], 
 x])] /; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
 

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 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 3042

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

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 4020

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

rule 4022

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

rule 4057

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

Maple [F]

\[\int \cot \left (f x +e \right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {1}{3}}d x\]

Input:

int(cot(f*x+e)*(c+d*tan(f*x+e))^(1/3),x)
 

Output:

int(cot(f*x+e)*(c+d*tan(f*x+e))^(1/3),x)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 562, normalized size of antiderivative = 1.40 \[ \int \cot (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx =\text {Too large to display} \] Input:

integrate(cot(f*x+e)*(c+d*tan(f*x+e))^(1/3),x, algorithm="fricas")
 

Output:

-1/4*((sqrt(-3)*f + f)*(-(f^3*sqrt(-d^2/f^6) + c)/f^3)^(1/3)*log(-1/2*(sqr 
t(-3)*f + f)*(-(f^3*sqrt(-d^2/f^6) + c)/f^3)^(1/3) + (d*tan(f*x + e) + c)^ 
(1/3)) - (sqrt(-3)*f - f)*(-(f^3*sqrt(-d^2/f^6) + c)/f^3)^(1/3)*log(1/2*(s 
qrt(-3)*f - f)*(-(f^3*sqrt(-d^2/f^6) + c)/f^3)^(1/3) + (d*tan(f*x + e) + c 
)^(1/3)) - 2*f*(-(f^3*sqrt(-d^2/f^6) + c)/f^3)^(1/3)*log(f*(-(f^3*sqrt(-d^ 
2/f^6) + c)/f^3)^(1/3) + (d*tan(f*x + e) + c)^(1/3)) + (sqrt(-3)*f + f)*(( 
f^3*sqrt(-d^2/f^6) - c)/f^3)^(1/3)*log(-1/2*(sqrt(-3)*f + f)*((f^3*sqrt(-d 
^2/f^6) - c)/f^3)^(1/3) + (d*tan(f*x + e) + c)^(1/3)) - (sqrt(-3)*f - f)*( 
(f^3*sqrt(-d^2/f^6) - c)/f^3)^(1/3)*log(1/2*(sqrt(-3)*f - f)*((f^3*sqrt(-d 
^2/f^6) - c)/f^3)^(1/3) + (d*tan(f*x + e) + c)^(1/3)) - 2*f*((f^3*sqrt(-d^ 
2/f^6) - c)/f^3)^(1/3)*log(f*((f^3*sqrt(-d^2/f^6) - c)/f^3)^(1/3) + (d*tan 
(f*x + e) + c)^(1/3)) + 4*sqrt(3)*c^(1/3)*arctan(1/3*(2*sqrt(3)*(d*tan(f*x 
 + e) + c)^(1/3)*c^(2/3) + sqrt(3)*c)/c) + 2*c^(1/3)*log((d*tan(f*x + e) + 
 c)^(2/3) + (d*tan(f*x + e) + c)^(1/3)*c^(1/3) + c^(2/3)) - 4*c^(1/3)*log( 
(d*tan(f*x + e) + c)^(1/3) - c^(1/3)))/f
 

Sympy [F]

\[ \int \cot (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\int \sqrt [3]{c + d \tan {\left (e + f x \right )}} \cot {\left (e + f x \right )}\, dx \] Input:

integrate(cot(f*x+e)*(c+d*tan(f*x+e))**(1/3),x)
 

Output:

Integral((c + d*tan(e + f*x))**(1/3)*cot(e + f*x), x)
 

Maxima [F]

\[ \int \cot (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\int { {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}} \cot \left (f x + e\right ) \,d x } \] Input:

integrate(cot(f*x+e)*(c+d*tan(f*x+e))^(1/3),x, algorithm="maxima")
 

Output:

integrate((d*tan(f*x + e) + c)^(1/3)*cot(f*x + e), x)
 

Giac [F]

\[ \int \cot (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\int { {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}} \cot \left (f x + e\right ) \,d x } \] Input:

integrate(cot(f*x+e)*(c+d*tan(f*x+e))^(1/3),x, algorithm="giac")
 

Output:

undef
 

Mupad [B] (verification not implemented)

Time = 13.03 (sec) , antiderivative size = 2133, normalized size of antiderivative = 5.31 \[ \int \cot (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\text {Too large to display} \] Input:

int(cot(e + f*x)*(c + d*tan(e + f*x))^(1/3),x)
 

Output:

log((c + d*tan(e + f*x))^(1/3) - f*(c/f^3)^(1/3))*(c/f^3)^(1/3) + log(((-( 
c - d*1i)/f^3)^(1/3)*(((-(c - d*1i)/f^3)^(2/3)*(((((104976*c*d^14*(-(c - d 
*1i)/f^3)^(1/3)*(3*c^4 + 2*d^4 + 5*c^2*d^2) - (104976*c*d^14*(c + d*tan(e 
+ f*x))^(1/3)*(3*c^4 + 4*d^4 + 7*c^2*d^2))/f)*(-(c - d*1i)/f^3)^(2/3))/4 - 
 (78732*c^2*d^14*(c^4 - d^4))/f^3)*(-(c - d*1i)/f^3)^(1/3))/2 - (39366*c^2 
*d^14*(c + d*tan(e + f*x))^(1/3)*(5*c^4 + 3*d^4 + 8*c^2*d^2))/f^4))/4 + (6 
561*c*d^14*(d^6 - 3*c^6 + 7*c^2*d^4 + 3*c^4*d^2))/f^6))/2 - (6561*c*d^14*( 
c + d*tan(e + f*x))^(1/3)*(3*c^6 + d^6 + c^2*d^4 + 3*c^4*d^2))/f^7)*(-(c - 
 d*1i)/(8*f^3))^(1/3) + log(((-(c + d*1i)/f^3)^(1/3)*(((-(c + d*1i)/f^3)^( 
2/3)*(((((104976*c*d^14*(-(c + d*1i)/f^3)^(1/3)*(3*c^4 + 2*d^4 + 5*c^2*d^2 
) - (104976*c*d^14*(c + d*tan(e + f*x))^(1/3)*(3*c^4 + 4*d^4 + 7*c^2*d^2)) 
/f)*(-(c + d*1i)/f^3)^(2/3))/4 - (78732*c^2*d^14*(c^4 - d^4))/f^3)*(-(c + 
d*1i)/f^3)^(1/3))/2 - (39366*c^2*d^14*(c + d*tan(e + f*x))^(1/3)*(5*c^4 + 
3*d^4 + 8*c^2*d^2))/f^4))/4 + (6561*c*d^14*(d^6 - 3*c^6 + 7*c^2*d^4 + 3*c^ 
4*d^2))/f^6))/2 - (6561*c*d^14*(c + d*tan(e + f*x))^(1/3)*(3*c^6 + d^6 + c 
^2*d^4 + 3*c^4*d^2))/f^7)*(-(c + d*1i)/(8*f^3))^(1/3) + (log(2*(c + d*tan( 
e + f*x))^(1/3) + f*(c/f^3)^(1/3) - 3^(1/2)*f*(c/f^3)^(1/3)*1i)*(3^(1/2)*1 
i - 1)*(c/f^3)^(1/3))/2 - (log(2*(c + d*tan(e + f*x))^(1/3) + f*(c/f^3)^(1 
/3) + 3^(1/2)*f*(c/f^3)^(1/3)*1i)*(3^(1/2)*1i + 1)*(c/f^3)^(1/3))/2 + log( 
(((3^(1/2)*1i)/2 - 1/2)*((((((3^(1/2)*1i)/2 - 1/2)*((((3^(1/2)*1i)/2 + ...
 

Reduce [F]

\[ \int \cot (e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\int \left (d \tan \left (f x +e \right )+c \right )^{\frac {1}{3}} \cot \left (f x +e \right )d x \] Input:

int(cot(f*x+e)*(c+d*tan(f*x+e))^(1/3),x)
 

Output:

int((tan(e + f*x)*d + c)**(1/3)*cot(e + f*x),x)