\(\int \cot ^2(e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx\) [687]

Optimal result

Integrand size = 23, antiderivative size = 546 \[ \int \cot ^2(e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\frac {1}{4} \sqrt [3]{c-\sqrt {-d^2}} x+\frac {1}{4} \sqrt [3]{c+\sqrt {-d^2}} x-\frac {d \arctan \left (\frac {\sqrt [3]{c}+2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{2/3} f}-\frac {\sqrt {3} d \sqrt [3]{c-\sqrt {-d^2}} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-\sqrt {-d^2}}}}{\sqrt {3}}\right )}{2 \sqrt {-d^2} f}+\frac {\sqrt {3} d \sqrt [3]{c+\sqrt {-d^2}} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+\sqrt {-d^2}}}}{\sqrt {3}}\right )}{2 \sqrt {-d^2} f}+\frac {d \sqrt [3]{c-\sqrt {-d^2}} \log (\cos (e+f x))}{4 \sqrt {-d^2} f}-\frac {d \sqrt [3]{c+\sqrt {-d^2}} \log (\cos (e+f x))}{4 \sqrt {-d^2} f}-\frac {d \log (\tan (e+f x))}{6 c^{2/3} f}+\frac {d \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 c^{2/3} f}+\frac {3 d \sqrt [3]{c-\sqrt {-d^2}} \log \left (\sqrt [3]{c-\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt {-d^2} f}-\frac {3 d \sqrt [3]{c+\sqrt {-d^2}} \log \left (\sqrt [3]{c+\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt {-d^2} f}-\frac {\cot (e+f x) \sqrt [3]{c+d \tan (e+f x)}}{f} \] Output:

1/4*(c-(-d^2)^(1/2))^(1/3)*x+1/4*(c+(-d^2)^(1/2))^(1/3)*x-1/3*d*arctan(1/3 
*(c^(1/3)+2*(c+d*tan(f*x+e))^(1/3))*3^(1/2)/c^(1/3))*3^(1/2)/c^(2/3)/f-1/2 
*3^(1/2)*d*(c-(-d^2)^(1/2))^(1/3)*arctan(1/3*(1+2*(c+d*tan(f*x+e))^(1/3)/( 
c-(-d^2)^(1/2))^(1/3))*3^(1/2))/(-d^2)^(1/2)/f+1/2*3^(1/2)*d*(c+(-d^2)^(1/ 
2))^(1/3)*arctan(1/3*(1+2*(c+d*tan(f*x+e))^(1/3)/(c+(-d^2)^(1/2))^(1/3))*3 
^(1/2))/(-d^2)^(1/2)/f+1/4*d*(c-(-d^2)^(1/2))^(1/3)*ln(cos(f*x+e))/(-d^2)^ 
(1/2)/f-1/4*d*(c+(-d^2)^(1/2))^(1/3)*ln(cos(f*x+e))/(-d^2)^(1/2)/f-1/6*d*l 
n(tan(f*x+e))/c^(2/3)/f+1/2*d*ln(c^(1/3)-(c+d*tan(f*x+e))^(1/3))/c^(2/3)/f 
+3/4*d*(c-(-d^2)^(1/2))^(1/3)*ln((c-(-d^2)^(1/2))^(1/3)-(c+d*tan(f*x+e))^( 
1/3))/(-d^2)^(1/2)/f-3/4*d*(c+(-d^2)^(1/2))^(1/3)*ln((c+(-d^2)^(1/2))^(1/3 
)-(c+d*tan(f*x+e))^(1/3))/(-d^2)^(1/2)/f-cot(f*x+e)*(c+d*tan(f*x+e))^(1/3) 
/f
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 25.15 (sec) , antiderivative size = 941, normalized size of antiderivative = 1.72 \[ \int \cot ^2(e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=-\frac {\cot (e+f x) \sqrt [3]{c+d \tan (e+f x)}}{f}+\frac {\cos (e+f x) \left (-4 \sqrt {3} (c-i d)^{2/3} (c+i d)^{2/3} d \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c}}}{\sqrt {3}}\right )+6 \sqrt {3} c^{2/3} (c+i d)^{2/3} (i c+d) \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt {3}}\right )-6 i \sqrt {3} c^{5/3} (c-i d)^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt {3}}\right )+6 \sqrt {3} c^{2/3} (c-i d)^{2/3} d \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt {3}}\right )+4 (c-i d)^{2/3} (c+i d)^{2/3} d \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )-6 i c^{5/3} (c+i d)^{2/3} \log \left (\sqrt [3]{c-i d}-\sqrt [3]{c+d \tan (e+f x)}\right )-6 c^{2/3} (c+i d)^{2/3} d \log \left (\sqrt [3]{c-i d}-\sqrt [3]{c+d \tan (e+f x)}\right )+6 i c^{5/3} (c-i d)^{2/3} \log \left (\sqrt [3]{c+i d}-\sqrt [3]{c+d \tan (e+f x)}\right )-6 c^{2/3} (c-i d)^{2/3} d \log \left (\sqrt [3]{c+i d}-\sqrt [3]{c+d \tan (e+f x)}\right )-2 (c-i d)^{2/3} (c+i d)^{2/3} d \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right )+3 i c^{5/3} (c+i d)^{2/3} \log \left ((c-i d)^{2/3}+\sqrt [3]{c-i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right )+3 c^{2/3} (c+i d)^{2/3} d \log \left ((c-i d)^{2/3}+\sqrt [3]{c-i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right )-3 i c^{5/3} (c-i d)^{2/3} \log \left ((c+i d)^{2/3}+\sqrt [3]{c+i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right )+3 c^{2/3} (c-i d)^{2/3} d \log \left ((c+i d)^{2/3}+\sqrt [3]{c+i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right )\right ) (c+d \tan (e+f x))}{12 c^{2/3} (c-i d)^{2/3} (c+i d)^{2/3} f (c \cos (e+f x)+d \sin (e+f x))} \] Input:

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

Output:

-((Cot[e + f*x]*(c + d*Tan[e + f*x])^(1/3))/f) + (Cos[e + f*x]*(-4*Sqrt[3] 
*(c - I*d)^(2/3)*(c + I*d)^(2/3)*d*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/ 
3))/c^(1/3))/Sqrt[3]] + 6*Sqrt[3]*c^(2/3)*(c + I*d)^(2/3)*(I*c + d)*ArcTan 
[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c - I*d)^(1/3))/Sqrt[3]] - (6*I)*Sqr 
t[3]*c^(5/3)*(c - I*d)^(2/3)*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c 
 + I*d)^(1/3))/Sqrt[3]] + 6*Sqrt[3]*c^(2/3)*(c - I*d)^(2/3)*d*ArcTan[(1 + 
(2*(c + d*Tan[e + f*x])^(1/3))/(c + I*d)^(1/3))/Sqrt[3]] + 4*(c - I*d)^(2/ 
3)*(c + I*d)^(2/3)*d*Log[c^(1/3) - (c + d*Tan[e + f*x])^(1/3)] - (6*I)*c^( 
5/3)*(c + I*d)^(2/3)*Log[(c - I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)] - 6 
*c^(2/3)*(c + I*d)^(2/3)*d*Log[(c - I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3 
)] + (6*I)*c^(5/3)*(c - I*d)^(2/3)*Log[(c + I*d)^(1/3) - (c + d*Tan[e + f* 
x])^(1/3)] - 6*c^(2/3)*(c - I*d)^(2/3)*d*Log[(c + I*d)^(1/3) - (c + d*Tan[ 
e + f*x])^(1/3)] - 2*(c - I*d)^(2/3)*(c + I*d)^(2/3)*d*Log[c^(2/3) + c^(1/ 
3)*(c + d*Tan[e + f*x])^(1/3) + (c + d*Tan[e + f*x])^(2/3)] + (3*I)*c^(5/3 
)*(c + I*d)^(2/3)*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)] + 3*c^(2/3)*(c + I*d)^(2/3)*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)] - (3*I)*c^(5/3)*(c - I*d)^(2/3)*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)] + 3*c 
^(2/3)*(c - I*d)^(2/3)*d*Log[(c + I*d)^(2/3) + (c + I*d)^(1/3)*(c + d*T...
 

Rubi [A] (verified)

Time = 1.33 (sec) , antiderivative size = 507, normalized size of antiderivative = 0.93, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {3042, 4051, 27, 3042, 4136, 27, 3042, 3966, 485, 2009, 4117, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4051

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4136

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3966

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

\(\Big \downarrow \) 485

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} \left (d \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) (c+d \tan (e+f x))^{2/3}}dx-\frac {3 d \left (\frac {\sqrt {3} \sqrt [3]{c-\sqrt {-d^2}} \arctan \left (\frac {\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-\sqrt {-d^2}}}+1}{\sqrt {3}}\right )}{2 \sqrt {-d^2}}-\frac {\sqrt {3} \sqrt [3]{c+\sqrt {-d^2}} \arctan \left (\frac {\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+\sqrt {-d^2}}}+1}{\sqrt {3}}\right )}{2 \sqrt {-d^2}}-\frac {\sqrt [3]{c+\sqrt {-d^2}} \log \left (\sqrt {-d^2}-d \tan (e+f x)\right )}{4 \sqrt {-d^2}}+\frac {\sqrt [3]{c-\sqrt {-d^2}} \log \left (\sqrt {-d^2}+d \tan (e+f x)\right )}{4 \sqrt {-d^2}}-\frac {3 \sqrt [3]{c-\sqrt {-d^2}} \log \left (\sqrt [3]{c-\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt {-d^2}}+\frac {3 \sqrt [3]{c+\sqrt {-d^2}} \log \left (\sqrt [3]{c+\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt {-d^2}}\right )}{f}\right )-\frac {\cot (e+f x) \sqrt [3]{c+d \tan (e+f x)}}{f}\)

\(\Big \downarrow \) 4117

\(\displaystyle \frac {1}{3} \left (\frac {d \int \frac {\cot (e+f x)}{(c+d \tan (e+f x))^{2/3}}d\tan (e+f x)}{f}-\frac {3 d \left (\frac {\sqrt {3} \sqrt [3]{c-\sqrt {-d^2}} \arctan \left (\frac {\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-\sqrt {-d^2}}}+1}{\sqrt {3}}\right )}{2 \sqrt {-d^2}}-\frac {\sqrt {3} \sqrt [3]{c+\sqrt {-d^2}} \arctan \left (\frac {\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+\sqrt {-d^2}}}+1}{\sqrt {3}}\right )}{2 \sqrt {-d^2}}-\frac {\sqrt [3]{c+\sqrt {-d^2}} \log \left (\sqrt {-d^2}-d \tan (e+f x)\right )}{4 \sqrt {-d^2}}+\frac {\sqrt [3]{c-\sqrt {-d^2}} \log \left (\sqrt {-d^2}+d \tan (e+f x)\right )}{4 \sqrt {-d^2}}-\frac {3 \sqrt [3]{c-\sqrt {-d^2}} \log \left (\sqrt [3]{c-\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt {-d^2}}+\frac {3 \sqrt [3]{c+\sqrt {-d^2}} \log \left (\sqrt [3]{c+\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt {-d^2}}\right )}{f}\right )-\frac {\cot (e+f x) \sqrt [3]{c+d \tan (e+f x)}}{f}\)

\(\Big \downarrow \) 69

\(\displaystyle \frac {1}{3} \left (\frac {d \left (-\frac {3 \int \frac {1}{\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}}d\sqrt [3]{c+d \tan (e+f x)}}{2 c^{2/3}}-\frac {3 \int \frac {1}{c^{2/3}+\sqrt [3]{c+d \tan (e+f x)} \sqrt [3]{c}+(c+d \tan (e+f x))^{2/3}}d\sqrt [3]{c+d \tan (e+f x)}}{2 \sqrt [3]{c}}-\frac {\log (\tan (e+f x))}{2 c^{2/3}}\right )}{f}-\frac {3 d \left (\frac {\sqrt {3} \sqrt [3]{c-\sqrt {-d^2}} \arctan \left (\frac {\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-\sqrt {-d^2}}}+1}{\sqrt {3}}\right )}{2 \sqrt {-d^2}}-\frac {\sqrt {3} \sqrt [3]{c+\sqrt {-d^2}} \arctan \left (\frac {\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+\sqrt {-d^2}}}+1}{\sqrt {3}}\right )}{2 \sqrt {-d^2}}-\frac {\sqrt [3]{c+\sqrt {-d^2}} \log \left (\sqrt {-d^2}-d \tan (e+f x)\right )}{4 \sqrt {-d^2}}+\frac {\sqrt [3]{c-\sqrt {-d^2}} \log \left (\sqrt {-d^2}+d \tan (e+f x)\right )}{4 \sqrt {-d^2}}-\frac {3 \sqrt [3]{c-\sqrt {-d^2}} \log \left (\sqrt [3]{c-\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt {-d^2}}+\frac {3 \sqrt [3]{c+\sqrt {-d^2}} \log \left (\sqrt [3]{c+\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt {-d^2}}\right )}{f}\right )-\frac {\cot (e+f x) \sqrt [3]{c+d \tan (e+f x)}}{f}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {1}{3} \left (\frac {d \left (-\frac {3 \int \frac {1}{c^{2/3}+\sqrt [3]{c+d \tan (e+f x)} \sqrt [3]{c}+(c+d \tan (e+f x))^{2/3}}d\sqrt [3]{c+d \tan (e+f x)}}{2 \sqrt [3]{c}}+\frac {3 \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 c^{2/3}}-\frac {\log (\tan (e+f x))}{2 c^{2/3}}\right )}{f}-\frac {3 d \left (\frac {\sqrt {3} \sqrt [3]{c-\sqrt {-d^2}} \arctan \left (\frac {\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-\sqrt {-d^2}}}+1}{\sqrt {3}}\right )}{2 \sqrt {-d^2}}-\frac {\sqrt {3} \sqrt [3]{c+\sqrt {-d^2}} \arctan \left (\frac {\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+\sqrt {-d^2}}}+1}{\sqrt {3}}\right )}{2 \sqrt {-d^2}}-\frac {\sqrt [3]{c+\sqrt {-d^2}} \log \left (\sqrt {-d^2}-d \tan (e+f x)\right )}{4 \sqrt {-d^2}}+\frac {\sqrt [3]{c-\sqrt {-d^2}} \log \left (\sqrt {-d^2}+d \tan (e+f x)\right )}{4 \sqrt {-d^2}}-\frac {3 \sqrt [3]{c-\sqrt {-d^2}} \log \left (\sqrt [3]{c-\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt {-d^2}}+\frac {3 \sqrt [3]{c+\sqrt {-d^2}} \log \left (\sqrt [3]{c+\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt {-d^2}}\right )}{f}\right )-\frac {\cot (e+f x) \sqrt [3]{c+d \tan (e+f x)}}{f}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {1}{3} \left (\frac {d \left (\frac {3 \int \frac {1}{-(c+d \tan (e+f x))^{2/3}-3}d\left (\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c}}+1\right )}{c^{2/3}}+\frac {3 \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 c^{2/3}}-\frac {\log (\tan (e+f x))}{2 c^{2/3}}\right )}{f}-\frac {3 d \left (\frac {\sqrt {3} \sqrt [3]{c-\sqrt {-d^2}} \arctan \left (\frac {\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-\sqrt {-d^2}}}+1}{\sqrt {3}}\right )}{2 \sqrt {-d^2}}-\frac {\sqrt {3} \sqrt [3]{c+\sqrt {-d^2}} \arctan \left (\frac {\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+\sqrt {-d^2}}}+1}{\sqrt {3}}\right )}{2 \sqrt {-d^2}}-\frac {\sqrt [3]{c+\sqrt {-d^2}} \log \left (\sqrt {-d^2}-d \tan (e+f x)\right )}{4 \sqrt {-d^2}}+\frac {\sqrt [3]{c-\sqrt {-d^2}} \log \left (\sqrt {-d^2}+d \tan (e+f x)\right )}{4 \sqrt {-d^2}}-\frac {3 \sqrt [3]{c-\sqrt {-d^2}} \log \left (\sqrt [3]{c-\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt {-d^2}}+\frac {3 \sqrt [3]{c+\sqrt {-d^2}} \log \left (\sqrt [3]{c+\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt {-d^2}}\right )}{f}\right )-\frac {\cot (e+f x) \sqrt [3]{c+d \tan (e+f x)}}{f}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{3} \left (\frac {d \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c}}+1}{\sqrt {3}}\right )}{c^{2/3}}+\frac {3 \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 c^{2/3}}-\frac {\log (\tan (e+f x))}{2 c^{2/3}}\right )}{f}-\frac {3 d \left (\frac {\sqrt {3} \sqrt [3]{c-\sqrt {-d^2}} \arctan \left (\frac {\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-\sqrt {-d^2}}}+1}{\sqrt {3}}\right )}{2 \sqrt {-d^2}}-\frac {\sqrt {3} \sqrt [3]{c+\sqrt {-d^2}} \arctan \left (\frac {\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+\sqrt {-d^2}}}+1}{\sqrt {3}}\right )}{2 \sqrt {-d^2}}-\frac {\sqrt [3]{c+\sqrt {-d^2}} \log \left (\sqrt {-d^2}-d \tan (e+f x)\right )}{4 \sqrt {-d^2}}+\frac {\sqrt [3]{c-\sqrt {-d^2}} \log \left (\sqrt {-d^2}+d \tan (e+f x)\right )}{4 \sqrt {-d^2}}-\frac {3 \sqrt [3]{c-\sqrt {-d^2}} \log \left (\sqrt [3]{c-\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt {-d^2}}+\frac {3 \sqrt [3]{c+\sqrt {-d^2}} \log \left (\sqrt [3]{c+\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt {-d^2}}\right )}{f}\right )-\frac {\cot (e+f x) \sqrt [3]{c+d \tan (e+f x)}}{f}\)

Input:

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

Output:

((d*(-((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))))/f - (3*d*((Sqrt[3]*(c - Sqrt[-d^2])^(1/3) 
*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c - Sqrt[-d^2])^(1/3))/Sqrt[3 
]])/(2*Sqrt[-d^2]) - (Sqrt[3]*(c + Sqrt[-d^2])^(1/3)*ArcTan[(1 + (2*(c + d 
*Tan[e + f*x])^(1/3))/(c + Sqrt[-d^2])^(1/3))/Sqrt[3]])/(2*Sqrt[-d^2]) - ( 
(c + Sqrt[-d^2])^(1/3)*Log[Sqrt[-d^2] - d*Tan[e + f*x]])/(4*Sqrt[-d^2]) + 
((c - Sqrt[-d^2])^(1/3)*Log[Sqrt[-d^2] + d*Tan[e + f*x]])/(4*Sqrt[-d^2]) - 
 (3*(c - Sqrt[-d^2])^(1/3)*Log[(c - Sqrt[-d^2])^(1/3) - (c + d*Tan[e + f*x 
])^(1/3)])/(4*Sqrt[-d^2]) + (3*(c + Sqrt[-d^2])^(1/3)*Log[(c + Sqrt[-d^2]) 
^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(4*Sqrt[-d^2])))/f)/3 - (Cot[e + f*x 
]*(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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 485

Int[((c_) + (d_.)*(x_))^(n_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[Expand 
Integrand[(c + d*x)^n, 1/(a + b*x^2), x], x] /; FreeQ[{a, b, c, d, n}, x] & 
&  !IntegerQ[2*n]
 

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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 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 [F]

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

Input:

int(cot(f*x+e)^2*(c+d*tan(f*x+e))^(1/3),x)
 

Output:

int(cot(f*x+e)^2*(c+d*tan(f*x+e))^(1/3),x)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 801, normalized size of antiderivative = 1.47 \[ \int \cot ^2(e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\text {Too large to display} \] Input:

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

Output:

1/12*(6*c^2*f*((f^3*sqrt(-c^2/f^6) + d)/f^3)^(1/3)*log(-f^4*((f^3*sqrt(-c^ 
2/f^6) + d)/f^3)^(1/3)*sqrt(-c^2/f^6) + (d*tan(f*x + e) + c)^(1/3)*c)*tan( 
f*x + e) + 6*c^2*f*(-(f^3*sqrt(-c^2/f^6) - d)/f^3)^(1/3)*log(f^4*(-(f^3*sq 
rt(-c^2/f^6) - d)/f^3)^(1/3)*sqrt(-c^2/f^6) + (d*tan(f*x + e) + c)^(1/3)*c 
)*tan(f*x + e) - 12*sqrt(1/3)*(c^2)^(1/6)*c*d*arctan(sqrt(1/3)*(c^2)^(1/6) 
*((c^2)^(1/3)*c + 2*(c^2)^(2/3)*(d*tan(f*x + e) + c)^(1/3))/c^2)*tan(f*x + 
 e) - 2*(c^2)^(2/3)*d*log((d*tan(f*x + e) + c)^(2/3)*c + (c^2)^(1/3)*c + ( 
c^2)^(2/3)*(d*tan(f*x + e) + c)^(1/3))*tan(f*x + e) + 4*(c^2)^(2/3)*d*log( 
(d*tan(f*x + e) + c)^(1/3)*c - (c^2)^(2/3))*tan(f*x + e) - 3*(sqrt(-3)*c^2 
*f + c^2*f)*((f^3*sqrt(-c^2/f^6) + d)/f^3)^(1/3)*log(1/2*(sqrt(-3)*f^4 + f 
^4)*((f^3*sqrt(-c^2/f^6) + d)/f^3)^(1/3)*sqrt(-c^2/f^6) + (d*tan(f*x + e) 
+ c)^(1/3)*c)*tan(f*x + e) + 3*(sqrt(-3)*c^2*f - c^2*f)*((f^3*sqrt(-c^2/f^ 
6) + d)/f^3)^(1/3)*log(-1/2*(sqrt(-3)*f^4 - f^4)*((f^3*sqrt(-c^2/f^6) + d) 
/f^3)^(1/3)*sqrt(-c^2/f^6) + (d*tan(f*x + e) + c)^(1/3)*c)*tan(f*x + e) - 
3*(sqrt(-3)*c^2*f + c^2*f)*(-(f^3*sqrt(-c^2/f^6) - d)/f^3)^(1/3)*log(-1/2* 
(sqrt(-3)*f^4 + f^4)*(-(f^3*sqrt(-c^2/f^6) - d)/f^3)^(1/3)*sqrt(-c^2/f^6) 
+ (d*tan(f*x + e) + c)^(1/3)*c)*tan(f*x + e) + 3*(sqrt(-3)*c^2*f - c^2*f)* 
(-(f^3*sqrt(-c^2/f^6) - d)/f^3)^(1/3)*log(1/2*(sqrt(-3)*f^4 - f^4)*(-(f^3* 
sqrt(-c^2/f^6) - d)/f^3)^(1/3)*sqrt(-c^2/f^6) + (d*tan(f*x + e) + c)^(1/3) 
*c)*tan(f*x + e) - 12*(d*tan(f*x + e) + c)^(1/3)*c^2)/(c^2*f*tan(f*x + ...
 

Sympy [F]

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

integrate(cot(f*x+e)**2*(c+d*tan(f*x+e))**(1/3),x)
 

Output:

Integral((c + d*tan(e + f*x))**(1/3)*cot(e + f*x)**2, x)
 

Maxima [F]

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

integrate(cot(f*x+e)^2*(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)^2, x)
 

Giac [F]

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

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

Output:

undef
 

Mupad [B] (verification not implemented)

Time = 18.99 (sec) , antiderivative size = 3239, normalized size of antiderivative = 5.93 \[ \int \cot ^2(e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\text {Too large to display} \] Input:

int(cot(e + f*x)^2*(c + d*tan(e + f*x))^(1/3),x)
 

Output:

log((((((243*(c + d*tan(e + f*x))^(1/3)*(576*d^19*f^6 + 1584*c^2*d^17*f^6 
+ 1008*c^4*d^15*f^6))/f^7 - (243*(1728*c*d^18*f^6 + 4320*c^3*d^16*f^6 + 25 
92*c^5*d^14*f^6)*(d^3/(27*c^2*f^3))^(1/3))/f^6)*(d^3/(27*c^2*f^3))^(2/3) + 
 (243*(464*c*d^19*f^3 + 1112*c^3*d^17*f^3 + 648*c^5*d^15*f^3))/f^6)*(d^3/( 
27*c^2*f^3))^(1/3) - (243*(c + d*tan(e + f*x))^(1/3)*(160*d^20*f^3 + 358*c 
^2*d^18*f^3 + 144*c^4*d^16*f^3 - 54*c^6*d^14*f^3))/f^7)*(d^3/(27*c^2*f^3)) 
^(2/3) - (243*(35*c*d^20 + 89*c^3*d^18 + 81*c^5*d^16 + 27*c^7*d^14))/f^6)* 
(d^3/(27*c^2*f^3))^(1/3) + (243*(c + d*tan(e + f*x))^(1/3)*(11*d^21 + 27*c 
^2*d^19 + 25*c^4*d^17 + 9*c^6*d^15))/f^7)*(d^3/(27*c^2*f^3))^(1/3) + log(( 
243*d^15*(c + d*tan(e + f*x))^(1/3)*(9*c^6 + 11*d^6 + 27*c^2*d^4 + 25*c^4* 
d^2))/f^7 - (((c*1i + d)/f^3)^(1/3)*((((((c*1i + d)/f^3)^(1/3)*(((104976*c 
*d^14*((c*1i + d)/f^3)^(1/3)*(3*c^4 + 2*d^4 + 5*c^2*d^2) - (34992*d^15*(c 
+ d*tan(e + f*x))^(1/3)*(7*c^4 + 4*d^4 + 11*c^2*d^2))/f)*((c*1i + d)/f^3)^ 
(2/3))/4 - (1944*c*d^15*(81*c^4 + 58*d^4 + 139*c^2*d^2))/f^3))/2 + (486*d^ 
14*(c + d*tan(e + f*x))^(1/3)*(80*d^6 - 27*c^6 + 179*c^2*d^4 + 72*c^4*d^2) 
)/f^4)*((c*1i + d)/f^3)^(2/3))/4 + (243*c*d^14*(27*c^6 + 35*d^6 + 89*c^2*d 
^4 + 81*c^4*d^2))/f^6))/2)*((c*1i + d)/(8*f^3))^(1/3) + log(((-(c*1i - d)/ 
f^3)^(1/3)*(((-(c*1i - d)/f^3)^(2/3)*(((-(c*1i - d)/f^3)^(1/3)*(((-(c*1i - 
 d)/f^3)^(2/3)*(104976*c*d^14*(-(c*1i - d)/f^3)^(1/3)*(3*c^4 + 2*d^4 + 5*c 
^2*d^2) - (34992*d^15*(c + d*tan(e + f*x))^(1/3)*(7*c^4 + 4*d^4 + 11*c^...
 

Reduce [F]

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

int(cot(f*x+e)^2*(c+d*tan(f*x+e))^(1/3),x)
 

Output:

int((tan(e + f*x)*d + c)**(1/3)*cot(e + f*x)**2,x)