Integrand size = 26, antiderivative size = 359 \[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\frac {55 i \arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}-\frac {55 i \arctan \left (\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{72 a^2 d}+\frac {8 \arctan \left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{3 \sqrt {3} a^2 d}-\frac {55 i \arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{36 a^2 d}+\frac {8 \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{9 a^2 d}+\frac {55 i \log \left (1-\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{48 \sqrt {3} a^2 d}-\frac {55 i \log \left (1+\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{48 \sqrt {3} a^2 d}-\frac {4 \log \left (1-\tan ^{\frac {2}{3}}(c+d x)+\tan ^{\frac {4}{3}}(c+d x)\right )}{9 a^2 d}-\frac {8}{3 a^2 d \tan ^{\frac {2}{3}}(c+d x)}+\frac {11}{12 a^2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2} \]
-55/72*I*arctan(-3^(1/2)+2*tan(d*x+c)^(1/3))/a^2/d-55/72*I*arctan(3^(1/2)+ 2*tan(d*x+c)^(1/3))/a^2/d-55/36*I*arctan(tan(d*x+c)^(1/3))/a^2/d+8/9*ln(1+ tan(d*x+c)^(2/3))/a^2/d-4/9*ln(1-tan(d*x+c)^(2/3)+tan(d*x+c)^(4/3))/a^2/d+ 8/9*arctan(1/3*(1-2*tan(d*x+c)^(2/3))*3^(1/2))/a^2/d*3^(1/2)+55/144*I*ln(1 -3^(1/2)*tan(d*x+c)^(1/3)+tan(d*x+c)^(2/3))/a^2/d*3^(1/2)-55/144*I*ln(1+3^ (1/2)*tan(d*x+c)^(1/3)+tan(d*x+c)^(2/3))/a^2/d*3^(1/2)-8/3/a^2/d/tan(d*x+c )^(2/3)+11/12/a^2/d/(1+I*tan(d*x+c))/tan(d*x+c)^(2/3)+1/4/d/tan(d*x+c)^(2/ 3)/(a+I*a*tan(d*x+c))^2
Time = 4.84 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\frac {18 a^2 \sqrt [6]{\tan ^2(c+d x)}+(a+i a \tan (c+d x)) \left (66 a \sqrt [6]{\tan ^2(c+d x)}-(a+i a \tan (c+d x)) \left (-55 \left (\log \left (1-i \sqrt [6]{\tan ^2(c+d x)}\right )-\log \left (1+i \sqrt [6]{\tan ^2(c+d x)}\right )+\sqrt [3]{-1} \left (\log \left (1-\sqrt [6]{-1} \sqrt [6]{\tan ^2(c+d x)}\right )-\log \left (1+\sqrt [6]{-1} \sqrt [6]{\tan ^2(c+d x)}\right )+\sqrt [3]{-1} \left (-\log \left (1-(-1)^{5/6} \sqrt [6]{\tan ^2(c+d x)}\right )+\log \left (1+(-1)^{5/6} \sqrt [6]{\tan ^2(c+d x)}\right )\right )\right )\right ) \tan (c+d x)-64 \sqrt [6]{\tan ^2(c+d x)} \left (-3+\log \left (1+\sqrt [3]{\tan ^2(c+d x)}\right ) \sqrt [3]{\tan ^2(c+d x)}-\sqrt [3]{-1} \log \left (1-\sqrt [3]{-1} \sqrt [3]{\tan ^2(c+d x)}\right ) \sqrt [3]{\tan ^2(c+d x)}+(-1)^{2/3} \log \left (1+(-1)^{2/3} \sqrt [3]{\tan ^2(c+d x)}\right ) \sqrt [3]{\tan ^2(c+d x)}\right )\right )\right )}{72 a^2 d \tan ^{\frac {2}{3}}(c+d x) \sqrt [6]{\tan ^2(c+d x)} (a+i a \tan (c+d x))^2} \]
(18*a^2*(Tan[c + d*x]^2)^(1/6) + (a + I*a*Tan[c + d*x])*(66*a*(Tan[c + d*x ]^2)^(1/6) - (a + I*a*Tan[c + d*x])*(-55*(Log[1 - I*(Tan[c + d*x]^2)^(1/6) ] - Log[1 + I*(Tan[c + d*x]^2)^(1/6)] + (-1)^(1/3)*(Log[1 - (-1)^(1/6)*(Ta n[c + d*x]^2)^(1/6)] - Log[1 + (-1)^(1/6)*(Tan[c + d*x]^2)^(1/6)] + (-1)^( 1/3)*(-Log[1 - (-1)^(5/6)*(Tan[c + d*x]^2)^(1/6)] + Log[1 + (-1)^(5/6)*(Ta n[c + d*x]^2)^(1/6)])))*Tan[c + d*x] - 64*(Tan[c + d*x]^2)^(1/6)*(-3 + Log [1 + (Tan[c + d*x]^2)^(1/3)]*(Tan[c + d*x]^2)^(1/3) - (-1)^(1/3)*Log[1 - ( -1)^(1/3)*(Tan[c + d*x]^2)^(1/3)]*(Tan[c + d*x]^2)^(1/3) + (-1)^(2/3)*Log[ 1 + (-1)^(2/3)*(Tan[c + d*x]^2)^(1/3)]*(Tan[c + d*x]^2)^(1/3)))))/(72*a^2* d*Tan[c + d*x]^(2/3)*(Tan[c + d*x]^2)^(1/6)*(a + I*a*Tan[c + d*x])^2)
Time = 1.14 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.85, number of steps used = 26, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.962, Rules used = {3042, 4042, 27, 3042, 4079, 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.
\(\displaystyle \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+i a \tan (c+d x))^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan (c+d x)^{5/3} (a+i a \tan (c+d x))^2}dx\) |
\(\Big \downarrow \) 4042 |
\(\displaystyle \frac {\int \frac {2 (7 a-4 i a \tan (c+d x))}{3 \tan ^{\frac {5}{3}}(c+d x) (i \tan (c+d x) a+a)}dx}{4 a^2}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {7 a-4 i a \tan (c+d x)}{\tan ^{\frac {5}{3}}(c+d x) (i \tan (c+d x) a+a)}dx}{6 a^2}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {7 a-4 i a \tan (c+d x)}{\tan (c+d x)^{5/3} (i \tan (c+d x) a+a)}dx}{6 a^2}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 4079 |
\(\displaystyle \frac {\frac {\int \frac {64 a^2-55 i a^2 \tan (c+d x)}{3 \tan ^{\frac {5}{3}}(c+d x)}dx}{2 a^2}+\frac {11}{2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\int \frac {64 a^2-55 i a^2 \tan (c+d x)}{\tan ^{\frac {5}{3}}(c+d x)}dx}{6 a^2}+\frac {11}{2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\int \frac {64 a^2-55 i a^2 \tan (c+d x)}{\tan (c+d x)^{5/3}}dx}{6 a^2}+\frac {11}{2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle \frac {\frac {-\frac {96 a^2}{d \tan ^{\frac {2}{3}}(c+d x)}+\int -\frac {64 \tan (c+d x) a^2+55 i a^2}{\tan ^{\frac {2}{3}}(c+d x)}dx}{6 a^2}+\frac {11}{2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {-\frac {96 a^2}{d \tan ^{\frac {2}{3}}(c+d x)}-\int \frac {64 \tan (c+d x) a^2+55 i a^2}{\tan ^{\frac {2}{3}}(c+d x)}dx}{6 a^2}+\frac {11}{2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {-\frac {96 a^2}{d \tan ^{\frac {2}{3}}(c+d x)}-\int \frac {64 \tan (c+d x) a^2+55 i a^2}{\tan (c+d x)^{2/3}}dx}{6 a^2}+\frac {11}{2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 4021 |
\(\displaystyle \frac {\frac {-55 i a^2 \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)}dx-64 a^2 \int \sqrt [3]{\tan (c+d x)}dx-\frac {96 a^2}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {11}{2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {-55 i a^2 \int \frac {1}{\tan (c+d x)^{2/3}}dx-64 a^2 \int \sqrt [3]{\tan (c+d x)}dx-\frac {96 a^2}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {11}{2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 3957 |
\(\displaystyle \frac {\frac {-\frac {64 a^2 \int \frac {\sqrt [3]{\tan (c+d x)}}{\tan ^2(c+d x)+1}d\tan (c+d x)}{d}-\frac {55 i a^2 \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x) \left (\tan ^2(c+d x)+1\right )}d\tan (c+d x)}{d}-\frac {96 a^2}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {11}{2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {\frac {-\frac {165 i a^2 \int \frac {1}{\tan ^2(c+d x)+1}d\sqrt [3]{\tan (c+d x)}}{d}-\frac {192 a^2 \int \frac {\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt [3]{\tan (c+d x)}}{d}-\frac {96 a^2}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {11}{2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 753 |
\(\displaystyle \frac {\frac {-\frac {165 i a^2 \left (\frac {1}{3} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{3} \int \frac {2-\sqrt {3} \sqrt [3]{\tan (c+d x)}}{2 \left (\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )}d\sqrt [3]{\tan (c+d x)}+\frac {1}{3} \int \frac {\sqrt {3} \sqrt [3]{\tan (c+d x)}+2}{2 \left (\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )}d\sqrt [3]{\tan (c+d x)}\right )}{d}-\frac {192 a^2 \int \frac {\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt [3]{\tan (c+d x)}}{d}-\frac {96 a^2}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {11}{2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {-\frac {165 i a^2 \left (\frac {1}{3} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{6} \int \frac {2-\sqrt {3} \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{6} \int \frac {\sqrt {3} \sqrt [3]{\tan (c+d x)}+2}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )}{d}-\frac {192 a^2 \int \frac {\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt [3]{\tan (c+d x)}}{d}-\frac {96 a^2}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {11}{2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\frac {-\frac {165 i a^2 \left (\frac {1}{6} \int \frac {2-\sqrt {3} \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{6} \int \frac {\sqrt {3} \sqrt [3]{\tan (c+d x)}+2}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}-\frac {192 a^2 \int \frac {\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt [3]{\tan (c+d x)}}{d}-\frac {96 a^2}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {11}{2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {\frac {-\frac {165 i a^2 \left (\frac {1}{6} \int \frac {2-\sqrt {3} \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{6} \int \frac {\sqrt {3} \sqrt [3]{\tan (c+d x)}+2}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}-\frac {96 a^2 \int \frac {\tan ^{\frac {2}{3}}(c+d x)}{\tan (c+d x)+1}d\tan ^{\frac {2}{3}}(c+d x)}{d}-\frac {96 a^2}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {11}{2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 821 |
\(\displaystyle \frac {\frac {-\frac {165 i a^2 \left (\frac {1}{6} \int \frac {2-\sqrt {3} \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{6} \int \frac {\sqrt {3} \sqrt [3]{\tan (c+d x)}+2}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}-\frac {96 a^2 \left (\frac {1}{3} \int \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )d\tan ^{\frac {2}{3}}(c+d x)-\frac {1}{3} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)+1}d\tan ^{\frac {2}{3}}(c+d x)\right )}{d}-\frac {96 a^2}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {11}{2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {\frac {-\frac {165 i a^2 \left (\frac {1}{6} \int \frac {2-\sqrt {3} \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{6} \int \frac {\sqrt {3} \sqrt [3]{\tan (c+d x)}+2}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}-\frac {96 a^2 \left (\frac {1}{3} \int \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )d\tan ^{\frac {2}{3}}(c+d x)-\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{d}-\frac {96 a^2}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {11}{2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {\frac {-\frac {165 i a^2 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}-\frac {1}{2} \sqrt {3} \int -\frac {\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}-\frac {96 a^2 \left (\frac {1}{3} \left (\frac {3}{2} \int 1d\tan ^{\frac {2}{3}}(c+d x)+\frac {1}{2} \int \left (2 \tan ^{\frac {2}{3}}(c+d x)-1\right )d\tan ^{\frac {2}{3}}(c+d x)\right )-\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{d}-\frac {96 a^2}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {11}{2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {-\frac {165 i a^2 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}-\frac {96 a^2 \left (\frac {1}{3} \left (\frac {3}{2} \int 1d\tan ^{\frac {2}{3}}(c+d x)-\frac {1}{2} \int \left (1-2 \tan ^{\frac {2}{3}}(c+d x)\right )d\tan ^{\frac {2}{3}}(c+d x)\right )-\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{d}-\frac {96 a^2}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {11}{2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {\frac {-\frac {165 i a^2 \left (\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}-\int \frac {1}{-\tan ^{\frac {2}{3}}(c+d x)-1}d\left (2 \sqrt [3]{\tan (c+d x)}-\sqrt {3}\right )\right )+\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}-\int \frac {1}{-\tan ^{\frac {2}{3}}(c+d x)-1}d\left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )\right )+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}-\frac {96 a^2 \left (\frac {1}{3} \left (-3 \int \frac {1}{-2 \tan ^{\frac {2}{3}}(c+d x)-2}d\left (2 \tan ^{\frac {2}{3}}(c+d x)-1\right )-\frac {1}{2} \int \left (1-2 \tan ^{\frac {2}{3}}(c+d x)\right )d\tan ^{\frac {2}{3}}(c+d x)\right )-\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{d}-\frac {96 a^2}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {11}{2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\frac {-\frac {165 i a^2 \left (\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}-\arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )\right )+\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\arctan \left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )\right )+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}-\frac {96 a^2 \left (\frac {1}{3} \left (\sqrt {3} \arctan \left (\frac {2 \tan ^{\frac {2}{3}}(c+d x)-1}{\sqrt {3}}\right )-\frac {1}{2} \int \left (1-2 \tan ^{\frac {2}{3}}(c+d x)\right )d\tan ^{\frac {2}{3}}(c+d x)\right )-\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{d}-\frac {96 a^2}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {11}{2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {\frac {-\frac {96 a^2 \left (\frac {\arctan \left (\frac {2 \tan ^{\frac {2}{3}}(c+d x)-1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{d}-\frac {165 i a^2 \left (\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{6} \left (-\arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )-\frac {1}{2} \sqrt {3} \log \left (\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )\right )+\frac {1}{6} \left (\arctan \left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )+\frac {1}{2} \sqrt {3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )\right )\right )}{d}-\frac {96 a^2}{d \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {11}{2 d (1+i \tan (c+d x)) \tan ^{\frac {2}{3}}(c+d x)}}{6 a^2}+\frac {1}{4 d \tan ^{\frac {2}{3}}(c+d x) (a+i a \tan (c+d x))^2}\) |
(((-96*a^2*(ArcTan[(-1 + 2*Tan[c + d*x]^(2/3))/Sqrt[3]]/Sqrt[3] - Log[1 + Tan[c + d*x]^(2/3)]/3))/d - ((165*I)*a^2*(ArcTan[Tan[c + d*x]^(1/3)]/3 + ( -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 + (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 - (96*a^2)/(d*Tan[c + d*x]^(2/3)))/(6*a^2) + 11/(2*d*(1 + I*Ta n[c + d*x])*Tan[c + d*x]^(2/3)))/(6*a^2) + 1/(4*d*Tan[c + d*x]^(2/3)*(a + I*a*Tan[c + d*x])^2)
3.3.46.3.1 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[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*f*m*(b*c - a*d))), x] + Simp[1/(2*a*m*(b*c - a*d)) In t[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b*c*m - a*d*(2*m + n + 1) + b*d*(m + n + 1)*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] && LtQ[m, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(a*A + b*B)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*f*m*( b*c - a*d))), x] + Simp[1/(2*a*m*(b*c - a*d)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x], x] /; Free Q[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] && !GtQ[n, 0]
Time = 0.58 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.65
method | result | size |
derivativedivides | \(\frac {\frac {\ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )-i\right )}{8}+\frac {30 i \tan \left (d x +c \right )+4 \left (\tan ^{\frac {2}{3}}\left (d x +c \right )\right )-4 i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+32}{72 {\left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}^{2}}-\frac {119 \ln \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{144}-\frac {119 i \sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (-i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{72}-\frac {\ln \left (i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{16}+\frac {i \sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{8}-\frac {5 i}{12 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}+\frac {1}{36 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )^{2}}+\frac {119 \ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}{72}-\frac {3}{2 \tan \left (d x +c \right )^{\frac {2}{3}}}}{d \,a^{2}}\) | \(233\) |
default | \(\frac {\frac {\ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )-i\right )}{8}+\frac {30 i \tan \left (d x +c \right )+4 \left (\tan ^{\frac {2}{3}}\left (d x +c \right )\right )-4 i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+32}{72 {\left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}^{2}}-\frac {119 \ln \left (-i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{144}-\frac {119 i \sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (-i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{72}-\frac {\ln \left (i \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )-1\right )}{16}+\frac {i \sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (i+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right ) \sqrt {3}}{3}\right )}{8}-\frac {5 i}{12 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}+\frac {1}{36 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )^{2}}+\frac {119 \ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+i\right )}{72}-\frac {3}{2 \tan \left (d x +c \right )^{\frac {2}{3}}}}{d \,a^{2}}\) | \(233\) |
1/d/a^2*(1/8*ln(tan(d*x+c)^(1/3)-I)+1/72*(30*I*tan(d*x+c)+4*tan(d*x+c)^(2/ 3)-4*I*tan(d*x+c)^(1/3)+32)/(-I*tan(d*x+c)^(1/3)+tan(d*x+c)^(2/3)-1)^2-119 /144*ln(-I*tan(d*x+c)^(1/3)+tan(d*x+c)^(2/3)-1)-119/72*I*3^(1/2)*arctanh(1 /3*(-I+2*tan(d*x+c)^(1/3))*3^(1/2))-1/16*ln(I*tan(d*x+c)^(1/3)+tan(d*x+c)^ (2/3)-1)+1/8*I*3^(1/2)*arctanh(1/3*(I+2*tan(d*x+c)^(1/3))*3^(1/2))-5/12*I/ (tan(d*x+c)^(1/3)+I)+1/36/(tan(d*x+c)^(1/3)+I)^2+119/72*ln(tan(d*x+c)^(1/3 )+I)-3/2/tan(d*x+c)^(2/3))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 681 vs. \(2 (282) = 564\).
Time = 0.27 (sec) , antiderivative size = 681, normalized size of antiderivative = 1.90 \[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=-\frac {9 \, {\left (\sqrt {3} {\left (-i \, a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + i \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt {\frac {1}{a^{4} d^{2}}} + e^{\left (6 i \, d x + 6 i \, c\right )} - e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \log \left (\frac {1}{2} \, \sqrt {3} a^{2} d \sqrt {\frac {1}{a^{4} 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 ) + 9 \, {\left (\sqrt {3} {\left (i \, a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} - i \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt {\frac {1}{a^{4} d^{2}}} + e^{\left (6 i \, d x + 6 i \, c\right )} - e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \log \left (-\frac {1}{2} \, \sqrt {3} a^{2} d \sqrt {\frac {1}{a^{4} 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 ) + 119 \, {\left (3 \, \sqrt {\frac {1}{3}} {\left (i \, a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} - i \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt {\frac {1}{a^{4} d^{2}}} + e^{\left (6 i \, d x + 6 i \, c\right )} - e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \log \left (\frac {3}{2} \, \sqrt {\frac {1}{3}} a^{2} d \sqrt {\frac {1}{a^{4} 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 ) + 119 \, {\left (3 \, \sqrt {\frac {1}{3}} {\left (-i \, a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + i \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt {\frac {1}{a^{4} d^{2}}} + e^{\left (6 i \, d x + 6 i \, c\right )} - e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \log \left (-\frac {3}{2} \, \sqrt {\frac {1}{3}} a^{2} d \sqrt {\frac {1}{a^{4} 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 ) - 238 \, {\left (e^{\left (6 i \, d x + 6 i \, c\right )} - e^{\left (4 i \, d x + 4 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 ) - 18 \, {\left (e^{\left (6 i \, d x + 6 i \, c\right )} - e^{\left (4 i \, d x + 4 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 ) + 3 \, \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 (103 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 75 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 31 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 3 i\right )}}{144 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} - a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )}} \]
-1/144*(9*(sqrt(3)*(-I*a^2*d*e^(6*I*d*x + 6*I*c) + I*a^2*d*e^(4*I*d*x + 4* I*c))*sqrt(1/(a^4*d^2)) + e^(6*I*d*x + 6*I*c) - e^(4*I*d*x + 4*I*c))*log(1 /2*sqrt(3)*a^2*d*sqrt(1/(a^4*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) + 9*(sqrt(3)*(I*a^2*d*e^(6*I*d*x + 6*I* c) - I*a^2*d*e^(4*I*d*x + 4*I*c))*sqrt(1/(a^4*d^2)) + e^(6*I*d*x + 6*I*c) - e^(4*I*d*x + 4*I*c))*log(-1/2*sqrt(3)*a^2*d*sqrt(1/(a^4*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) + 119*(3*s qrt(1/3)*(I*a^2*d*e^(6*I*d*x + 6*I*c) - I*a^2*d*e^(4*I*d*x + 4*I*c))*sqrt( 1/(a^4*d^2)) + e^(6*I*d*x + 6*I*c) - e^(4*I*d*x + 4*I*c))*log(3/2*sqrt(1/3 )*a^2*d*sqrt(1/(a^4*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) + 119*(3*sqrt(1/3)*(-I*a^2*d*e^(6*I*d*x + 6*I*c) + I*a^2*d*e^(4*I*d*x + 4*I*c))*sqrt(1/(a^4*d^2)) + e^(6*I*d*x + 6*I*c) - e^(4*I*d*x + 4*I*c))*log(-3/2*sqrt(1/3)*a^2*d*sqrt(1/(a^4*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) - 238*(e^( 6*I*d*x + 6*I*c) - e^(4*I*d*x + 4*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) - 18*(e^(6*I*d*x + 6*I*c) - e^(4*I*d *x + 4*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) + 3*((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1))^(1 /3)*(103*I*e^(6*I*d*x + 6*I*c) + 75*I*e^(4*I*d*x + 4*I*c) - 31*I*e^(2*I*d* x + 2*I*c) - 3*I))/(a^2*d*e^(6*I*d*x + 6*I*c) - a^2*d*e^(4*I*d*x + 4*I*...
Timed out. \[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \]
Time = 2.00 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.66 \[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\frac {119 i \, \sqrt {3} \log \left (-\frac {\sqrt {3} - 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} + i}{\sqrt {3} + 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} - i}\right )}{144 \, a^{2} d} - \frac {i \, \sqrt {3} \log \left (-\frac {\sqrt {3} - 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} - i}{\sqrt {3} + 2 \, \tan \left (d x + c\right )^{\frac {1}{3}} + i}\right )}{16 \, a^{2} d} - \frac {\log \left (\tan \left (d x + c\right )^{\frac {2}{3}} + i \, \tan \left (d x + c\right )^{\frac {1}{3}} - 1\right )}{16 \, a^{2} d} - \frac {119 \, \log \left (\tan \left (d x + c\right )^{\frac {2}{3}} - i \, \tan \left (d x + c\right )^{\frac {1}{3}} - 1\right )}{144 \, a^{2} d} + \frac {119 \, \log \left (\tan \left (d x + c\right )^{\frac {1}{3}} + i\right )}{72 \, a^{2} d} + \frac {\log \left (\tan \left (d x + c\right )^{\frac {1}{3}} - i\right )}{8 \, a^{2} d} - \frac {32 \, \tan \left (d x + c\right )^{2} - 53 i \, \tan \left (d x + c\right ) - 18}{12 \, {\left (\tan \left (d x + c\right )^{\frac {4}{3}} - i \, \tan \left (d x + c\right )^{\frac {1}{3}}\right )}^{2} a^{2} d} \]
119/144*I*sqrt(3)*log(-(sqrt(3) - 2*tan(d*x + c)^(1/3) + I)/(sqrt(3) + 2*t an(d*x + c)^(1/3) - I))/(a^2*d) - 1/16*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^2*d) - 1/16*log( tan(d*x + c)^(2/3) + I*tan(d*x + c)^(1/3) - 1)/(a^2*d) - 119/144*log(tan(d *x + c)^(2/3) - I*tan(d*x + c)^(1/3) - 1)/(a^2*d) + 119/72*log(tan(d*x + c )^(1/3) + I)/(a^2*d) + 1/8*log(tan(d*x + c)^(1/3) - I)/(a^2*d) - 1/12*(32* tan(d*x + c)^2 - 53*I*tan(d*x + c) - 18)/((tan(d*x + c)^(4/3) - I*tan(d*x + c)^(1/3))^2*a^2*d)
Time = 6.10 (sec) , antiderivative size = 660, normalized size of antiderivative = 1.84 \[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+i a \tan (c+d x))^2} \, dx=\text {Too large to display} \]
log(((a^6*d^3*215607040i)/3 + 28311552*a^10*d^5*tan(c + d*x)^(1/3)*(1/(512 *a^6*d^3))^(2/3))*(1/(512*a^6*d^3))^(1/3) - (27116768*a^4*d^2*tan(c + d*x) ^(1/3))/3)*(1/(512*a^6*d^3))^(1/3) + (119*log((119*((a^6*d^3*215607040i)/3 + (232013824*a^10*d^5*tan(c + d*x)^(1/3)*(1/(a^6*d^3))^(2/3))/3)*(1/(a^6* d^3))^(1/3))/72 - (27116768*a^4*d^2*tan(c + d*x)^(1/3))/3)*(1/(a^6*d^3))^( 1/3))/72 - ((53*tan(c + d*x))/(12*a^2*d) - 3i/(2*a^2*d) + (tan(c + d*x)^2* 8i)/(3*a^2*d))/(2*tan(c + d*x)^(5/3) - tan(c + d*x)^(2/3)*1i + tan(c + d*x )^(8/3)*1i) + log(((3^(1/2)*1i)/2 - 1/2)*((a^6*d^3*215607040i)/3 + 2831155 2*a^10*d^5*tan(c + d*x)^(1/3)*((3^(1/2)*1i)/2 - 1/2)^2*(1/(512*a^6*d^3))^( 2/3))*(1/(512*a^6*d^3))^(1/3) - (27116768*a^4*d^2*tan(c + d*x)^(1/3))/3)*( (3^(1/2)*1i)/2 - 1/2)*(1/(512*a^6*d^3))^(1/3) - log(((3^(1/2)*1i)/2 + 1/2) *((a^6*d^3*215607040i)/3 + 28311552*a^10*d^5*tan(c + d*x)^(1/3)*((3^(1/2)* 1i)/2 + 1/2)^2*(1/(512*a^6*d^3))^(2/3))*(1/(512*a^6*d^3))^(1/3) + (2711676 8*a^4*d^2*tan(c + d*x)^(1/3))/3)*((3^(1/2)*1i)/2 + 1/2)*(1/(512*a^6*d^3))^ (1/3) + (119*log((119*(3^(1/2)*1i - 1)*((a^6*d^3*215607040i)/3 + (58003456 *a^10*d^5*tan(c + d*x)^(1/3)*(3^(1/2)*1i - 1)^2*(1/(a^6*d^3))^(2/3))/3)*(1 /(a^6*d^3))^(1/3))/144 - (27116768*a^4*d^2*tan(c + d*x)^(1/3))/3)*(3^(1/2) *1i - 1)*(1/(a^6*d^3))^(1/3))/144 - (119*log((119*(3^(1/2)*1i + 1)*((a^6*d ^3*215607040i)/3 + (58003456*a^10*d^5*tan(c + d*x)^(1/3)*(3^(1/2)*1i + 1)^ 2*(1/(a^6*d^3))^(2/3))/3)*(1/(a^6*d^3))^(1/3))/144 + (27116768*a^4*d^2*...