Integrand size = 26, antiderivative size = 321 \[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+i a \tan (c+d x))} \, dx=\frac {5 i \arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}-\frac {5 i \arctan \left (\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{12 a d}+\frac {2 \arctan \left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{\sqrt {3} a d}-\frac {5 i \arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{6 a d}+\frac {2 \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{3 a d}+\frac {5 i \log \left (1-\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{8 \sqrt {3} a d}-\frac {5 i \log \left (1+\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{8 \sqrt {3} a d}-\frac {\log \left (1-\tan ^{\frac {2}{3}}(c+d x)+\tan ^{\frac {4}{3}}(c+d x)\right )}{3 a d}-\frac {2}{a d \tan ^{\frac {2}{3}}(c+d x)}+\frac {1}{2 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))} \]
-5/12*I*arctan(-3^(1/2)+2*tan(d*x+c)^(1/3))/a/d-5/12*I*arctan(3^(1/2)+2*ta n(d*x+c)^(1/3))/a/d-5/6*I*arctan(tan(d*x+c)^(1/3))/a/d+2/3*ln(1+tan(d*x+c) ^(2/3))/a/d-1/3*ln(1-tan(d*x+c)^(2/3)+tan(d*x+c)^(4/3))/a/d+2/3*arctan(1/3 *(1-2*tan(d*x+c)^(2/3))*3^(1/2))/a/d*3^(1/2)+5/24*I*ln(1-3^(1/2)*tan(d*x+c )^(1/3)+tan(d*x+c)^(2/3))/a/d*3^(1/2)-5/24*I*ln(1+3^(1/2)*tan(d*x+c)^(1/3) +tan(d*x+c)^(2/3))/a/d*3^(1/2)-2/a/d/tan(d*x+c)^(2/3)+1/2/d/tan(d*x+c)^(2/ 3)/(a+I*a*tan(d*x+c))
Time = 6.35 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.45 \[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+i a \tan (c+d x))} \, dx=\frac {1}{2 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))}-\frac {\frac {5 i a \sqrt [3]{\tan (c+d x)} \left (\frac {i \log \left (1-i \sqrt [6]{\tan ^2(c+d x)}\right )}{6 \sqrt [6]{\tan ^2(c+d x)}}-\frac {i \log \left (1+i \sqrt [6]{\tan ^2(c+d x)}\right )}{6 \sqrt [6]{\tan ^2(c+d x)}}-\frac {\sqrt [6]{-1} \log \left (1-e^{-\frac {i \pi }{6}} \sqrt [6]{\tan ^2(c+d x)}\right )}{6 \sqrt [6]{\tan ^2(c+d x)}}+\frac {(-1)^{5/6} \log \left (1-e^{\frac {i \pi }{6}} \sqrt [6]{\tan ^2(c+d x)}\right )}{6 \sqrt [6]{\tan ^2(c+d x)}}-\frac {(-1)^{5/6} \log \left (1-e^{-\frac {5 i \pi }{6}} \sqrt [6]{\tan ^2(c+d x)}\right )}{6 \sqrt [6]{\tan ^2(c+d x)}}+\frac {\sqrt [6]{-1} \log \left (1-e^{\frac {5 i \pi }{6}} \sqrt [6]{\tan ^2(c+d x)}\right )}{6 \sqrt [6]{\tan ^2(c+d x)}}\right )}{d}+\frac {4 a \left (1-\frac {1}{3} \log \left (1+\sqrt [3]{\tan ^2(c+d x)}\right ) \sqrt [3]{\tan ^2(c+d x)}-\frac {1}{3} (-1)^{2/3} \log \left (1-e^{-\frac {i \pi }{3}} \sqrt [3]{\tan ^2(c+d x)}\right ) \sqrt [3]{\tan ^2(c+d x)}+\frac {1}{3} \sqrt [3]{-1} \log \left (1-e^{\frac {i \pi }{3}} \sqrt [3]{\tan ^2(c+d x)}\right ) \sqrt [3]{\tan ^2(c+d x)}\right )}{d \tan ^{\frac {2}{3}}(c+d x)}}{2 a^2} \]
1/(2*d*Tan[c + d*x]^(2/3)*(a + I*a*Tan[c + d*x])) - (((5*I)*a*Tan[c + d*x] ^(1/3)*(((I/6)*Log[1 - I*(Tan[c + d*x]^2)^(1/6)])/(Tan[c + d*x]^2)^(1/6) - ((I/6)*Log[1 + I*(Tan[c + d*x]^2)^(1/6)])/(Tan[c + d*x]^2)^(1/6) - ((-1)^ (1/6)*Log[1 - (Tan[c + d*x]^2)^(1/6)/E^((I/6)*Pi)])/(6*(Tan[c + d*x]^2)^(1 /6)) + ((-1)^(5/6)*Log[1 - E^((I/6)*Pi)*(Tan[c + d*x]^2)^(1/6)])/(6*(Tan[c + d*x]^2)^(1/6)) - ((-1)^(5/6)*Log[1 - (Tan[c + d*x]^2)^(1/6)/E^(((5*I)/6 )*Pi)])/(6*(Tan[c + d*x]^2)^(1/6)) + ((-1)^(1/6)*Log[1 - E^(((5*I)/6)*Pi)* (Tan[c + d*x]^2)^(1/6)])/(6*(Tan[c + d*x]^2)^(1/6))))/d + (4*a*(1 - (Log[1 + (Tan[c + d*x]^2)^(1/3)]*(Tan[c + d*x]^2)^(1/3))/3 - ((-1)^(2/3)*Log[1 - (Tan[c + d*x]^2)^(1/3)/E^((I/3)*Pi)]*(Tan[c + d*x]^2)^(1/3))/3 + ((-1)^(1 /3)*Log[1 - E^((I/3)*Pi)*(Tan[c + d*x]^2)^(1/3)]*(Tan[c + d*x]^2)^(1/3))/3 ))/(d*Tan[c + d*x]^(2/3)))/(2*a^2)
Time = 0.86 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.81, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {3042, 4035, 27, 3042, 4012, 25, 3042, 4021, 3042, 3957, 266, 753, 27, 216, 807, 821, 16, 1142, 25, 1083, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+i a \tan (c+d x))} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan (c+d x)^{5/3} (a+i a \tan (c+d x))}dx\) |
\(\Big \downarrow \) 4035 |
\(\displaystyle \frac {1}{2 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))}-\frac {\int -\frac {8 a-5 i a \tan (c+d x)}{3 \tan ^{\frac {5}{3}}(c+d x)}dx}{2 a^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {8 a-5 i a \tan (c+d x)}{\tan ^{\frac {5}{3}}(c+d x)}dx}{6 a^2}+\frac {1}{2 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {8 a-5 i a \tan (c+d x)}{\tan (c+d x)^{5/3}}dx}{6 a^2}+\frac {1}{2 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle \frac {-\frac {12 a}{d \tan ^{\frac {2}{3}}(c+d x)}+\int -\frac {8 \tan (c+d x) a+5 i a}{\tan ^{\frac {2}{3}}(c+d x)}dx}{6 a^2}+\frac {1}{2 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {-\frac {12 a}{d \tan ^{\frac {2}{3}}(c+d x)}-\int \frac {8 \tan (c+d x) a+5 i a}{\tan ^{\frac {2}{3}}(c+d x)}dx}{6 a^2}+\frac {1}{2 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {12 a}{d \tan ^{\frac {2}{3}}(c+d x)}-\int \frac {8 \tan (c+d x) a+5 i a}{\tan (c+d x)^{2/3}}dx}{6 a^2}+\frac {1}{2 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
\(\Big \downarrow \) 4021 |
\(\displaystyle \frac {-5 i a \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)}dx-8 a \int \sqrt [3]{\tan (c+d x)}dx-\frac {12 a}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-5 i a \int \frac {1}{\tan (c+d x)^{2/3}}dx-8 a \int \sqrt [3]{\tan (c+d x)}dx-\frac {12 a}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
\(\Big \downarrow \) 3957 |
\(\displaystyle \frac {-\frac {8 a \int \frac {\sqrt [3]{\tan (c+d x)}}{\tan ^2(c+d x)+1}d\tan (c+d x)}{d}-\frac {5 i a \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x) \left (\tan ^2(c+d x)+1\right )}d\tan (c+d x)}{d}-\frac {12 a}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {-\frac {15 i a \int \frac {1}{\tan ^2(c+d x)+1}d\sqrt [3]{\tan (c+d x)}}{d}-\frac {24 a \int \frac {\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt [3]{\tan (c+d x)}}{d}-\frac {12 a}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
\(\Big \downarrow \) 753 |
\(\displaystyle \frac {-\frac {15 i a \left (\frac {1}{3} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{3} \int \frac {2-\sqrt {3} \sqrt [3]{\tan (c+d x)}}{2 \left (\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )}d\sqrt [3]{\tan (c+d x)}+\frac {1}{3} \int \frac {\sqrt {3} \sqrt [3]{\tan (c+d x)}+2}{2 \left (\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )}d\sqrt [3]{\tan (c+d x)}\right )}{d}-\frac {24 a \int \frac {\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt [3]{\tan (c+d x)}}{d}-\frac {12 a}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {-\frac {15 i a \left (\frac {1}{3} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{6} \int \frac {2-\sqrt {3} \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{6} \int \frac {\sqrt {3} \sqrt [3]{\tan (c+d x)}+2}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )}{d}-\frac {24 a \int \frac {\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt [3]{\tan (c+d x)}}{d}-\frac {12 a}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {-\frac {15 i a \left (\frac {1}{6} \int \frac {2-\sqrt {3} \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{6} \int \frac {\sqrt {3} \sqrt [3]{\tan (c+d x)}+2}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}-\frac {24 a \int \frac {\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt [3]{\tan (c+d x)}}{d}-\frac {12 a}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {-\frac {15 i a \left (\frac {1}{6} \int \frac {2-\sqrt {3} \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{6} \int \frac {\sqrt {3} \sqrt [3]{\tan (c+d x)}+2}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}-\frac {12 a \int \frac {\tan ^{\frac {2}{3}}(c+d x)}{\tan (c+d x)+1}d\tan ^{\frac {2}{3}}(c+d x)}{d}-\frac {12 a}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
\(\Big \downarrow \) 821 |
\(\displaystyle \frac {-\frac {15 i a \left (\frac {1}{6} \int \frac {2-\sqrt {3} \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{6} \int \frac {\sqrt {3} \sqrt [3]{\tan (c+d x)}+2}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}-\frac {12 a \left (\frac {1}{3} \int \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )d\tan ^{\frac {2}{3}}(c+d x)-\frac {1}{3} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)+1}d\tan ^{\frac {2}{3}}(c+d x)\right )}{d}-\frac {12 a}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {-\frac {15 i a \left (\frac {1}{6} \int \frac {2-\sqrt {3} \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{6} \int \frac {\sqrt {3} \sqrt [3]{\tan (c+d x)}+2}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}-\frac {12 a \left (\frac {1}{3} \int \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )d\tan ^{\frac {2}{3}}(c+d x)-\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{d}-\frac {12 a}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {-\frac {15 i a \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}-\frac {1}{2} \sqrt {3} \int -\frac {\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}-\frac {12 a \left (\frac {1}{3} \left (\frac {3}{2} \int 1d\tan ^{\frac {2}{3}}(c+d x)+\frac {1}{2} \int \left (2 \tan ^{\frac {2}{3}}(c+d x)-1\right )d\tan ^{\frac {2}{3}}(c+d x)\right )-\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{d}-\frac {12 a}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {-\frac {15 i a \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}-\frac {12 a \left (\frac {1}{3} \left (\frac {3}{2} \int 1d\tan ^{\frac {2}{3}}(c+d x)-\frac {1}{2} \int \left (1-2 \tan ^{\frac {2}{3}}(c+d x)\right )d\tan ^{\frac {2}{3}}(c+d x)\right )-\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{d}-\frac {12 a}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {-\frac {15 i a \left (\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}-\int \frac {1}{-\tan ^{\frac {2}{3}}(c+d x)-1}d\left (2 \sqrt [3]{\tan (c+d x)}-\sqrt {3}\right )\right )+\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}-\int \frac {1}{-\tan ^{\frac {2}{3}}(c+d x)-1}d\left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )\right )+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}-\frac {12 a \left (\frac {1}{3} \left (-3 \int \frac {1}{-2 \tan ^{\frac {2}{3}}(c+d x)-2}d\left (2 \tan ^{\frac {2}{3}}(c+d x)-1\right )-\frac {1}{2} \int \left (1-2 \tan ^{\frac {2}{3}}(c+d x)\right )d\tan ^{\frac {2}{3}}(c+d x)\right )-\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{d}-\frac {12 a}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {-\frac {15 i a \left (\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}-\arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )\right )+\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\arctan \left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )\right )+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}-\frac {12 a \left (\frac {1}{3} \left (\sqrt {3} \arctan \left (\frac {2 \tan ^{\frac {2}{3}}(c+d x)-1}{\sqrt {3}}\right )-\frac {1}{2} \int \left (1-2 \tan ^{\frac {2}{3}}(c+d x)\right )d\tan ^{\frac {2}{3}}(c+d x)\right )-\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{d}-\frac {12 a}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {-\frac {12 a \left (\frac {\arctan \left (\frac {2 \tan ^{\frac {2}{3}}(c+d x)-1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{d}-\frac {15 i a \left (\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{6} \left (-\arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )-\frac {1}{2} \sqrt {3} \log \left (\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )\right )+\frac {1}{6} \left (\arctan \left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )+\frac {1}{2} \sqrt {3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )\right )\right )}{d}-\frac {12 a}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{2 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))}\) |
((-12*a*(ArcTan[(-1 + 2*Tan[c + d*x]^(2/3))/Sqrt[3]]/Sqrt[3] - Log[1 + Tan [c + d*x]^(2/3)]/3))/d - ((15*I)*a*(ArcTan[Tan[c + d*x]^(1/3)]/3 + (-ArcTa n[Sqrt[3] - 2*Tan[c + d*x]^(1/3)] - (Sqrt[3]*Log[1 - Sqrt[3]*Tan[c + d*x]^ (1/3) + Tan[c + d*x]^(2/3)])/2)/6 + (ArcTan[Sqrt[3] + 2*Tan[c + d*x]^(1/3) ] + (Sqrt[3]*Log[1 + Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/2)/ 6))/d - (12*a)/(d*Tan[c + d*x]^(2/3)))/(6*a^2) + 1/(2*d*Tan[c + d*x]^(2/3) *(a + I*a*Tan[c + d*x]))
3.3.38.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/ b, n]], s = Denominator[Rt[a/b, n]], k, u, v}, Simp[u = Int[(r - s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[ (r + s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2* x^2), x]; 2*(r^2/(a*n)) Int[1/(r^2 + s^2*x^2), x] + 2*(r/(a*n)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && PosQ[a /b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2 *x^2), x], x] /; FreeQ[{a, b}, x]
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]
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]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d Subst[Int [x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
Int[((b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Tan[e + f*x])^m, x], x] + Simp[d/b Int [(b*Tan[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x] && NeQ[c^ 2 + d^2, 0] && !IntegerQ[2*m]
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-a)*((c + d*Tan[e + f*x])^(n + 1)/(2*f*(b* c - a*d)*(a + b*Tan[e + f*x]))), x] + Simp[1/(2*a*(b*c - a*d)) Int[(c + d *Tan[e + f*x])^n*Simp[b*c + a*d*(n - 1) - b*d*n*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !GtQ[n, 0]
Time = 0.53 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.62
method | result | size |
derivativedivides | \(\frac {\frac {2 i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )-2}{-12 i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+12 \left (\tan ^{\frac {2}{3}}\left (d x +c \right )\right )-12}-\frac {13 \ln \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{24}-\frac {13 i \sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (-i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{12}+\frac {\ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )-i\right )}{4}-\frac {3}{2 \tan \left (d x +c \right )^{\frac {2}{3}}}-\frac {i}{6 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}+\frac {13 \ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}{12}-\frac {\ln \left (i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{8}+\frac {i \sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{4}}{d a}\) | \(199\) |
default | \(\frac {\frac {2 i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )-2}{-12 i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+12 \left (\tan ^{\frac {2}{3}}\left (d x +c \right )\right )-12}-\frac {13 \ln \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{24}-\frac {13 i \sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (-i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{12}+\frac {\ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )-i\right )}{4}-\frac {3}{2 \tan \left (d x +c \right )^{\frac {2}{3}}}-\frac {i}{6 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}+\frac {13 \ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}{12}-\frac {\ln \left (i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{8}+\frac {i \sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{4}}{d a}\) | \(199\) |
1/d/a*(1/12*(2*I*tan(d*x+c)^(1/3)-2)/(-I*tan(d*x+c)^(1/3)+tan(d*x+c)^(2/3) -1)-13/24*ln(-I*tan(d*x+c)^(1/3)+tan(d*x+c)^(2/3)-1)-13/12*I*3^(1/2)*arcta nh(1/3*(-I+2*tan(d*x+c)^(1/3))*3^(1/2))+1/4*ln(tan(d*x+c)^(1/3)-I)-3/2/tan (d*x+c)^(2/3)-1/6*I/(tan(d*x+c)^(1/3)+I)+13/12*ln(tan(d*x+c)^(1/3)+I)-1/8* ln(I*tan(d*x+c)^(1/3)+tan(d*x+c)^(2/3)-1)+1/4*I*3^(1/2)*arctanh(1/3*(I+2*t an(d*x+c)^(1/3))*3^(1/2)))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 642 vs. \(2 (254) = 508\).
Time = 0.27 (sec) , antiderivative size = 642, normalized size of antiderivative = 2.00 \[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+i a \tan (c+d x))} \, dx=-\frac {3 \, {\left (\sqrt {3} {\left (-i \, a d e^{\left (4 i \, d x + 4 i \, c\right )} + i \, a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \sqrt {\frac {1}{a^{2} d^{2}}} + e^{\left (4 i \, d x + 4 i \, c\right )} - e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \log \left (\frac {1}{2} \, \sqrt {3} a d \sqrt {\frac {1}{a^{2} d^{2}}} + \left (\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} + \frac {1}{2} i\right ) + 3 \, {\left (\sqrt {3} {\left (i \, a d e^{\left (4 i \, d x + 4 i \, c\right )} - i \, a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \sqrt {\frac {1}{a^{2} d^{2}}} + e^{\left (4 i \, d x + 4 i \, c\right )} - e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \log \left (-\frac {1}{2} \, \sqrt {3} a d \sqrt {\frac {1}{a^{2} d^{2}}} + \left (\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} + \frac {1}{2} i\right ) + 13 \, {\left (3 \, \sqrt {\frac {1}{3}} {\left (i \, a d e^{\left (4 i \, d x + 4 i \, c\right )} - i \, a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \sqrt {\frac {1}{a^{2} d^{2}}} + e^{\left (4 i \, d x + 4 i \, c\right )} - e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \log \left (\frac {3}{2} \, \sqrt {\frac {1}{3}} a d \sqrt {\frac {1}{a^{2} d^{2}}} + \left (\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} - \frac {1}{2} i\right ) + 13 \, {\left (3 \, \sqrt {\frac {1}{3}} {\left (-i \, a d e^{\left (4 i \, d x + 4 i \, c\right )} + i \, a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \sqrt {\frac {1}{a^{2} d^{2}}} + e^{\left (4 i \, d x + 4 i \, c\right )} - e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \log \left (-\frac {3}{2} \, \sqrt {\frac {1}{3}} a d \sqrt {\frac {1}{a^{2} d^{2}}} + \left (\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} - \frac {1}{2} i\right ) - 26 \, {\left (e^{\left (4 i \, d x + 4 i \, c\right )} - e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \log \left (\left (\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} + i\right ) - 6 \, {\left (e^{\left (4 i \, d x + 4 i \, c\right )} - e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \log \left (\left (\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} - i\right ) + 6 \, \left (\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} {\left (7 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 6 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - i\right )}}{24 \, {\left (a d e^{\left (4 i \, d x + 4 i \, c\right )} - a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )}} \]
-1/24*(3*(sqrt(3)*(-I*a*d*e^(4*I*d*x + 4*I*c) + I*a*d*e^(2*I*d*x + 2*I*c)) *sqrt(1/(a^2*d^2)) + e^(4*I*d*x + 4*I*c) - e^(2*I*d*x + 2*I*c))*log(1/2*sq rt(3)*a*d*sqrt(1/(a^2*d^2)) + ((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1))^(1/3) + 1/2*I) + 3*(sqrt(3)*(I*a*d*e^(4*I*d*x + 4*I*c) - I*a* d*e^(2*I*d*x + 2*I*c))*sqrt(1/(a^2*d^2)) + e^(4*I*d*x + 4*I*c) - e^(2*I*d* x + 2*I*c))*log(-1/2*sqrt(3)*a*d*sqrt(1/(a^2*d^2)) + ((-I*e^(2*I*d*x + 2*I *c) + I)/(e^(2*I*d*x + 2*I*c) + 1))^(1/3) + 1/2*I) + 13*(3*sqrt(1/3)*(I*a* d*e^(4*I*d*x + 4*I*c) - I*a*d*e^(2*I*d*x + 2*I*c))*sqrt(1/(a^2*d^2)) + e^( 4*I*d*x + 4*I*c) - e^(2*I*d*x + 2*I*c))*log(3/2*sqrt(1/3)*a*d*sqrt(1/(a^2* d^2)) + ((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1))^(1/3) - 1 /2*I) + 13*(3*sqrt(1/3)*(-I*a*d*e^(4*I*d*x + 4*I*c) + I*a*d*e^(2*I*d*x + 2 *I*c))*sqrt(1/(a^2*d^2)) + e^(4*I*d*x + 4*I*c) - e^(2*I*d*x + 2*I*c))*log( -3/2*sqrt(1/3)*a*d*sqrt(1/(a^2*d^2)) + ((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2 *I*d*x + 2*I*c) + 1))^(1/3) - 1/2*I) - 26*(e^(4*I*d*x + 4*I*c) - e^(2*I*d* x + 2*I*c))*log(((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1))^( 1/3) + I) - 6*(e^(4*I*d*x + 4*I*c) - e^(2*I*d*x + 2*I*c))*log(((-I*e^(2*I* d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1))^(1/3) - I) + 6*((-I*e^(2*I*d* x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*(7*I*e^(4*I*d*x + 4*I*c) + 6*I*e^(2*I*d*x + 2*I*c) - I))/(a*d*e^(4*I*d*x + 4*I*c) - a*d*e^(2*I*d*x + 2*I*c))
\[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+i a \tan (c+d x))} \, dx=- \frac {i \int \frac {1}{\tan ^{\frac {8}{3}}{\left (c + d x \right )} - i \tan ^{\frac {5}{3}}{\left (c + d x \right )}}\, dx}{a} \]
Exception generated. \[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+i a \tan (c+d x))} \, dx=\text {Exception raised: RuntimeError} \]
Time = 1.27 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+i a \tan (c+d x))} \, dx=\frac {13 i \, \sqrt {3} \log \left (-\frac {\sqrt {3} - 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} + i}{\sqrt {3} + 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} - i}\right )}{24 \, a d} - \frac {i \, \sqrt {3} \log \left (-\frac {\sqrt {3} - 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} - i}{\sqrt {3} + 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} + i}\right )}{8 \, a d} - \frac {\log \left (\tan \left (d x + c\right )^{\frac {2}{3}} + i \, \tan \left (d x + c\right )^{\frac {1}{3}} - 1\right )}{8 \, a d} - \frac {13 \, \log \left (\tan \left (d x + c\right )^{\frac {2}{3}} - i \, \tan \left (d x + c\right )^{\frac {1}{3}} - 1\right )}{24 \, a d} + \frac {13 \, \log \left (\tan \left (d x + c\right )^{\frac {1}{3}} + i\right )}{12 \, a d} + \frac {\log \left (\tan \left (d x + c\right )^{\frac {1}{3}} - i\right )}{4 \, a d} - \frac {3}{2 \, a d \tan \left (d x + c\right )^{\frac {2}{3}}} - \frac {\tan \left (d x + c\right )^{\frac {1}{3}}}{2 \, a d {\left (\tan \left (d x + c\right ) - i\right )}} \]
13/24*I*sqrt(3)*log(-(sqrt(3) - 2*tan(d*x + c)^(1/3) + I)/(sqrt(3) + 2*tan (d*x + c)^(1/3) - I))/(a*d) - 1/8*I*sqrt(3)*log(-(sqrt(3) - 2*tan(d*x + c) ^(1/3) - I)/(sqrt(3) + 2*tan(d*x + c)^(1/3) + I))/(a*d) - 1/8*log(tan(d*x + c)^(2/3) + I*tan(d*x + c)^(1/3) - 1)/(a*d) - 13/24*log(tan(d*x + c)^(2/3 ) - I*tan(d*x + c)^(1/3) - 1)/(a*d) + 13/12*log(tan(d*x + c)^(1/3) + I)/(a *d) + 1/4*log(tan(d*x + c)^(1/3) - I)/(a*d) - 3/2/(a*d*tan(d*x + c)^(2/3)) - 1/2*tan(d*x + c)^(1/3)/(a*d*(tan(d*x + c) - I))
Time = 6.01 (sec) , antiderivative size = 630, normalized size of antiderivative = 1.96 \[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+i a \tan (c+d x))} \, dx=-\frac {\frac {3}{2\,a\,d}+\frac {\mathrm {tan}\left (c+d\,x\right )\,2{}\mathrm {i}}{a\,d}}{{\mathrm {tan}\left (c+d\,x\right )}^{2/3}+{\mathrm {tan}\left (c+d\,x\right )}^{5/3}\,1{}\mathrm {i}}+\ln \left (\left (331776\,a^5\,d^5\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,{\left (\frac {1}{64\,a^3\,d^3}\right )}^{2/3}+a^3\,d^3\,312480{}\mathrm {i}\right )\,{\left (\frac {1}{64\,a^3\,d^3}\right )}^{1/3}-83304\,a^2\,d^2\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\right )\,{\left (\frac {1}{64\,a^3\,d^3}\right )}^{1/3}+\frac {13\,\ln \left (\frac {13\,\left (389376\,a^5\,d^5\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,{\left (\frac {1}{a^3\,d^3}\right )}^{2/3}+a^3\,d^3\,312480{}\mathrm {i}\right )\,{\left (\frac {1}{a^3\,d^3}\right )}^{1/3}}{12}-83304\,a^2\,d^2\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\right )\,{\left (\frac {1}{a^3\,d^3}\right )}^{1/3}}{12}+\frac {13\,\ln \left (\frac {13\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (a^3\,d^3\,312480{}\mathrm {i}+97344\,a^5\,d^5\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,{\left (\frac {1}{a^3\,d^3}\right )}^{2/3}\right )\,{\left (\frac {1}{a^3\,d^3}\right )}^{1/3}}{24}-83304\,a^2\,d^2\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {1}{a^3\,d^3}\right )}^{1/3}}{24}-\frac {13\,\ln \left (\frac {13\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (a^3\,d^3\,312480{}\mathrm {i}+97344\,a^5\,d^5\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,{\left (\frac {1}{a^3\,d^3}\right )}^{2/3}\right )\,{\left (\frac {1}{a^3\,d^3}\right )}^{1/3}}{24}+83304\,a^2\,d^2\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {1}{a^3\,d^3}\right )}^{1/3}}{24}+\ln \left (\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (a^3\,d^3\,312480{}\mathrm {i}+331776\,a^5\,d^5\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\,{\left (\frac {1}{64\,a^3\,d^3}\right )}^{2/3}\right )\,{\left (\frac {1}{64\,a^3\,d^3}\right )}^{1/3}-83304\,a^2\,d^2\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {1}{64\,a^3\,d^3}\right )}^{1/3}-\ln \left (\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (a^3\,d^3\,312480{}\mathrm {i}+331776\,a^5\,d^5\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\,{\left (\frac {1}{64\,a^3\,d^3}\right )}^{2/3}\right )\,{\left (\frac {1}{64\,a^3\,d^3}\right )}^{1/3}+83304\,a^2\,d^2\,{\mathrm {tan}\left (c+d\,x\right )}^{1/3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {1}{64\,a^3\,d^3}\right )}^{1/3} \]
log((a^3*d^3*312480i + 331776*a^5*d^5*tan(c + d*x)^(1/3)*(1/(64*a^3*d^3))^ (2/3))*(1/(64*a^3*d^3))^(1/3) - 83304*a^2*d^2*tan(c + d*x)^(1/3))*(1/(64*a ^3*d^3))^(1/3) - (3/(2*a*d) + (tan(c + d*x)*2i)/(a*d))/(tan(c + d*x)^(2/3) + tan(c + d*x)^(5/3)*1i) + (13*log((13*(a^3*d^3*312480i + 389376*a^5*d^5* tan(c + d*x)^(1/3)*(1/(a^3*d^3))^(2/3))*(1/(a^3*d^3))^(1/3))/12 - 83304*a^ 2*d^2*tan(c + d*x)^(1/3))*(1/(a^3*d^3))^(1/3))/12 + (13*log((13*(3^(1/2)*1 i - 1)*(a^3*d^3*312480i + 97344*a^5*d^5*tan(c + d*x)^(1/3)*(3^(1/2)*1i - 1 )^2*(1/(a^3*d^3))^(2/3))*(1/(a^3*d^3))^(1/3))/24 - 83304*a^2*d^2*tan(c + d *x)^(1/3))*(3^(1/2)*1i - 1)*(1/(a^3*d^3))^(1/3))/24 - (13*log((13*(3^(1/2) *1i + 1)*(a^3*d^3*312480i + 97344*a^5*d^5*tan(c + d*x)^(1/3)*(3^(1/2)*1i + 1)^2*(1/(a^3*d^3))^(2/3))*(1/(a^3*d^3))^(1/3))/24 + 83304*a^2*d^2*tan(c + d*x)^(1/3))*(3^(1/2)*1i + 1)*(1/(a^3*d^3))^(1/3))/24 + log(((3^(1/2)*1i)/ 2 - 1/2)*(a^3*d^3*312480i + 331776*a^5*d^5*tan(c + d*x)^(1/3)*((3^(1/2)*1i )/2 - 1/2)^2*(1/(64*a^3*d^3))^(2/3))*(1/(64*a^3*d^3))^(1/3) - 83304*a^2*d^ 2*tan(c + d*x)^(1/3))*((3^(1/2)*1i)/2 - 1/2)*(1/(64*a^3*d^3))^(1/3) - log( ((3^(1/2)*1i)/2 + 1/2)*(a^3*d^3*312480i + 331776*a^5*d^5*tan(c + d*x)^(1/3 )*((3^(1/2)*1i)/2 + 1/2)^2*(1/(64*a^3*d^3))^(2/3))*(1/(64*a^3*d^3))^(1/3) + 83304*a^2*d^2*tan(c + d*x)^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(1/(64*a^3*d^3) )^(1/3)