Integrand size = 23, antiderivative size = 546 \[ \int \cot ^2(e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\frac {1}{4} \sqrt [3]{c-\sqrt {-d^2}} x+\frac {1}{4} \sqrt [3]{c+\sqrt {-d^2}} x-\frac {d \arctan \left (\frac {\sqrt [3]{c}+2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{2/3} f}-\frac {\sqrt {3} d \sqrt [3]{c-\sqrt {-d^2}} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-\sqrt {-d^2}}}}{\sqrt {3}}\right )}{2 \sqrt {-d^2} f}+\frac {\sqrt {3} d \sqrt [3]{c+\sqrt {-d^2}} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+\sqrt {-d^2}}}}{\sqrt {3}}\right )}{2 \sqrt {-d^2} f}+\frac {d \sqrt [3]{c-\sqrt {-d^2}} \log (\cos (e+f x))}{4 \sqrt {-d^2} f}-\frac {d \sqrt [3]{c+\sqrt {-d^2}} \log (\cos (e+f x))}{4 \sqrt {-d^2} f}-\frac {d \log (\tan (e+f x))}{6 c^{2/3} f}+\frac {d \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 c^{2/3} f}+\frac {3 d \sqrt [3]{c-\sqrt {-d^2}} \log \left (\sqrt [3]{c-\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt {-d^2} f}-\frac {3 d \sqrt [3]{c+\sqrt {-d^2}} \log \left (\sqrt [3]{c+\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt {-d^2} f}-\frac {\cot (e+f x) \sqrt [3]{c+d \tan (e+f x)}}{f} \]
-1/6*d*ln(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-1/3*d*arctan(1/3*(c^(1/3)+2*(c+d*tan(f*x+e))^(1/3))/c^(1/3)*3^(1/ 2))/c^(2/3)/f*3^(1/2)+1/4*x*(c-(-d^2)^(1/2))^(1/3)+1/4*d*ln(cos(f*x+e))*(c -(-d^2)^(1/2))^(1/3)/f/(-d^2)^(1/2)+3/4*d*ln((c-(-d^2)^(1/2))^(1/3)-(c+d*t an(f*x+e))^(1/3))*(c-(-d^2)^(1/2))^(1/3)/f/(-d^2)^(1/2)-1/2*d*arctan(1/3*( 1+2*(c+d*tan(f*x+e))^(1/3)/(c-(-d^2)^(1/2))^(1/3))*3^(1/2))*3^(1/2)*(c-(-d ^2)^(1/2))^(1/3)/f/(-d^2)^(1/2)+1/4*x*(c+(-d^2)^(1/2))^(1/3)-1/4*d*ln(cos( f*x+e))*(c+(-d^2)^(1/2))^(1/3)/f/(-d^2)^(1/2)-3/4*d*ln((c+(-d^2)^(1/2))^(1 /3)-(c+d*tan(f*x+e))^(1/3))*(c+(-d^2)^(1/2))^(1/3)/f/(-d^2)^(1/2)+1/2*d*ar ctan(1/3*(1+2*(c+d*tan(f*x+e))^(1/3)/(c+(-d^2)^(1/2))^(1/3))*3^(1/2))*3^(1 /2)*(c+(-d^2)^(1/2))^(1/3)/f/(-d^2)^(1/2)-cot(f*x+e)*(c+d*tan(f*x+e))^(1/3 )/f
Result contains complex when optimal does not.
Time = 3.53 (sec) , antiderivative size = 460, normalized size of antiderivative = 0.84 \[ \int \cot ^2(e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\frac {-\frac {1}{2} \sqrt [3]{c} d \left (2 \sqrt {3} \arctan \left (\frac {\sqrt [3]{c}+2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt {3} \sqrt [3]{c}}\right )-2 \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )+\log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right )\right )+\frac {3}{4} i c \sqrt [3]{c-i d} \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-i d}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{c-i d}-\sqrt [3]{c+d \tan (e+f x)}\right )+\log \left ((c-i d)^{2/3}+\sqrt [3]{c-i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right )\right )-\frac {3}{4} i c \sqrt [3]{c+i d} \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+i d}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{c+i d}-\sqrt [3]{c+d \tan (e+f x)}\right )+\log \left ((c+i d)^{2/3}+\sqrt [3]{c+i d} \sqrt [3]{c+d \tan (e+f x)}+(c+d \tan (e+f x))^{2/3}\right )\right )+3 d \sqrt [3]{c+d \tan (e+f x)}-3 \cot (e+f x) (c+d \tan (e+f x))^{4/3}}{3 c f} \]
(-1/2*(c^(1/3)*d*(2*Sqrt[3]*ArcTan[(c^(1/3) + 2*(c + d*Tan[e + f*x])^(1/3) )/(Sqrt[3]*c^(1/3))] - 2*Log[c^(1/3) - (c + d*Tan[e + f*x])^(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)]) ) + ((3*I)/4)*c*(c - I*d)^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(c + d*Tan[e + f *x])^(1/3))/(c - I*d)^(1/3))/Sqrt[3]] - 2*Log[(c - I*d)^(1/3) - (c + d*Tan [e + f*x])^(1/3)] + 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*I)/4)*c*(c + I*d)^(1/3)*(2 *Sqrt[3]*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c + I*d)^(1/3))/Sqrt[ 3]] - 2*Log[(c + I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)] + 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*d*(c + d*Tan[e + f*x])^(1/3) - 3*Cot[e + f*x]*(c + d*Tan[e + f *x])^(4/3))/(3*c*f)
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 |
| \(\displaystyle \frac {1}{3} \left (\frac {d \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c}}+1}{\sqrt {3}}\right )}{c^{2/3}}+\frac {3 \log \left (\sqrt [3]{c}-\sqrt [3]{c+d \tan (e+f x)}\right )}{2 c^{2/3}}-\frac {\log (\tan (e+f x))}{2 c^{2/3}}\right )}{f}-\frac {3 d \left (\frac {\sqrt {3} \sqrt [3]{c-\sqrt {-d^2}} \arctan \left (\frac {\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c-\sqrt {-d^2}}}+1}{\sqrt {3}}\right )}{2 \sqrt {-d^2}}-\frac {\sqrt {3} \sqrt [3]{c+\sqrt {-d^2}} \arctan \left (\frac {\frac {2 \sqrt [3]{c+d \tan (e+f x)}}{\sqrt [3]{c+\sqrt {-d^2}}}+1}{\sqrt {3}}\right )}{2 \sqrt {-d^2}}-\frac {\sqrt [3]{c+\sqrt {-d^2}} \log \left (\sqrt {-d^2}-d \tan (e+f x)\right )}{4 \sqrt {-d^2}}+\frac {\sqrt [3]{c-\sqrt {-d^2}} \log \left (\sqrt {-d^2}+d \tan (e+f x)\right )}{4 \sqrt {-d^2}}-\frac {3 \sqrt [3]{c-\sqrt {-d^2}} \log \left (\sqrt [3]{c-\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt {-d^2}}+\frac {3 \sqrt [3]{c+\sqrt {-d^2}} \log \left (\sqrt [3]{c+\sqrt {-d^2}}-\sqrt [3]{c+d \tan (e+f x)}\right )}{4 \sqrt {-d^2}}\right )}{f}\right )-\frac {\cot (e+f x) \sqrt [3]{c+d \tan (e+f x)}}{f}\) |
((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
3.7.87.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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])
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]
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]
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]
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]
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]
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]
\[\int \left (\cot ^{2}\left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {1}{3}}d x\]
Time = 0.30 (sec) , antiderivative size = 805, normalized size of antiderivative = 1.47 \[ \int \cot ^2(e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\text {Too large to display} \]
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) - 4*sqrt(3)*(c^2)^(1/6)*c*d*arctan(1/3*(c^2)^(1/6)*(sqrt(3) *(c^2)^(1/3)*c + 2*sqrt(3)*(c^2)^(2/3)*(d*tan(f*x + e) + c)^(1/3))/c^2)*ta n(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)*lo g(-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...
\[ \int \cot ^2(e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\int \sqrt [3]{c + d \tan {\left (e + f x \right )}} \cot ^{2}{\left (e + f x \right )}\, dx \]
\[ \int \cot ^2(e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\int { {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}} \cot \left (f x + e\right )^{2} \,d x } \]
\[ \int \cot ^2(e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\int { {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {1}{3}} \cot \left (f x + e\right )^{2} \,d x } \]
Time = 25.33 (sec) , antiderivative size = 3239, normalized size of antiderivative = 5.93 \[ \int \cot ^2(e+f x) \sqrt [3]{c+d \tan (e+f x)} \, dx=\text {Too large to display} \]
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^...