Integrand size = 24, antiderivative size = 313 \[ \int \frac {\cot (c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=-\frac {i x}{8 \sqrt [3]{2} a^{4/3}}+\frac {\sqrt {3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{4/3} d}-\frac {\sqrt {3} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{4 \sqrt [3]{2} a^{4/3} d}-\frac {\log (\cos (c+d x))}{8 \sqrt [3]{2} a^{4/3} d}-\frac {\log (\tan (c+d x))}{2 a^{4/3} d}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 a^{4/3} d}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{8 \sqrt [3]{2} a^{4/3} d}+\frac {3}{8 d (a+i a \tan (c+d x))^{4/3}}+\frac {9}{4 a d \sqrt [3]{a+i a \tan (c+d x)}} \] Output:
-1/16*I*x*2^(2/3)/a^(4/3)+3^(1/2)*arctan(1/3*(a^(1/3)+2*(a+I*a*tan(d*x+c)) ^(1/3))*3^(1/2)/a^(1/3))/a^(4/3)/d-1/8*3^(1/2)*arctan(1/3*(a^(1/3)+2^(2/3) *(a+I*a*tan(d*x+c))^(1/3))*3^(1/2)/a^(1/3))*2^(2/3)/a^(4/3)/d-1/16*ln(cos( d*x+c))*2^(2/3)/a^(4/3)/d-1/2*ln(tan(d*x+c))/a^(4/3)/d+3/2*ln(a^(1/3)-(a+I *a*tan(d*x+c))^(1/3))/a^(4/3)/d-3/16*ln(2^(1/3)*a^(1/3)-(a+I*a*tan(d*x+c)) ^(1/3))*2^(2/3)/a^(4/3)/d+3/8/d/(a+I*a*tan(d*x+c))^(4/3)+9/4/a/d/(a+I*a*ta n(d*x+c))^(1/3)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.22 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.60 \[ \int \frac {\cot (c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\frac {\frac {8 \sqrt {3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{4/3}}-\frac {4 \log (\tan (c+d x))}{a^{4/3}}+\frac {12 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{a^{4/3}}+\frac {6}{(a+i a \tan (c+d x))^{4/3}}-\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},1,-\frac {1}{3},\frac {1}{2} (1+i \tan (c+d x))\right )}{(a+i a \tan (c+d x))^{4/3}}+\frac {24}{a \sqrt [3]{a+i a \tan (c+d x)}}}{8 d} \] Input:
Integrate[Cot[c + d*x]/(a + I*a*Tan[c + d*x])^(4/3),x]
Output:
((8*Sqrt[3]*ArcTan[(a^(1/3) + 2*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^( 1/3))])/a^(4/3) - (4*Log[Tan[c + d*x]])/a^(4/3) + (12*Log[a^(1/3) - (a + I *a*Tan[c + d*x])^(1/3)])/a^(4/3) + 6/(a + I*a*Tan[c + d*x])^(4/3) - (3*Hyp ergeometric2F1[-4/3, 1, -1/3, (1 + I*Tan[c + d*x])/2])/(a + I*a*Tan[c + d* x])^(4/3) + 24/(a*(a + I*a*Tan[c + d*x])^(1/3)))/(8*d)
Time = 1.75 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.11, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.958, Rules used = {3042, 4045, 3042, 3960, 3042, 3960, 3042, 3962, 67, 16, 1082, 217, 4079, 27, 3042, 4079, 27, 3042, 4082, 67, 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.
| \(\displaystyle \int \frac {\cot (c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (c+d x) (a+i a \tan (c+d x))^{4/3}}dx\) |
| \(\Big \downarrow \) 4045 |
| \(\displaystyle i \int \frac {1}{(i \tan (c+d x) a+a)^{4/3}}dx-\frac {i \int \frac {\cot (c+d x) (\tan (c+d x) a+i a)}{(i \tan (c+d x) a+a)^{4/3}}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \int \frac {1}{(i \tan (c+d x) a+a)^{4/3}}dx-\frac {i \int \frac {\tan (c+d x) a+i a}{\tan (c+d x) (i \tan (c+d x) a+a)^{4/3}}dx}{a}\) |
| \(\Big \downarrow \) 3960 |
| \(\displaystyle i \left (\frac {\int \frac {1}{\sqrt [3]{i \tan (c+d x) a+a}}dx}{2 a}+\frac {3 i}{8 d (a+i a \tan (c+d x))^{4/3}}\right )-\frac {i \int \frac {\tan (c+d x) a+i a}{\tan (c+d x) (i \tan (c+d x) a+a)^{4/3}}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {\int \frac {1}{\sqrt [3]{i \tan (c+d x) a+a}}dx}{2 a}+\frac {3 i}{8 d (a+i a \tan (c+d x))^{4/3}}\right )-\frac {i \int \frac {\tan (c+d x) a+i a}{\tan (c+d x) (i \tan (c+d x) a+a)^{4/3}}dx}{a}\) |
| \(\Big \downarrow \) 3960 |
| \(\displaystyle i \left (\frac {\frac {\int (i \tan (c+d x) a+a)^{2/3}dx}{2 a}+\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{2 a}+\frac {3 i}{8 d (a+i a \tan (c+d x))^{4/3}}\right )-\frac {i \int \frac {\tan (c+d x) a+i a}{\tan (c+d x) (i \tan (c+d x) a+a)^{4/3}}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {\frac {\int (i \tan (c+d x) a+a)^{2/3}dx}{2 a}+\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{2 a}+\frac {3 i}{8 d (a+i a \tan (c+d x))^{4/3}}\right )-\frac {i \int \frac {\tan (c+d x) a+i a}{\tan (c+d x) (i \tan (c+d x) a+a)^{4/3}}dx}{a}\) |
| \(\Big \downarrow \) 3962 |
| \(\displaystyle i \left (\frac {\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \int \frac {1}{(a-i a \tan (c+d x)) \sqrt [3]{i \tan (c+d x) a+a}}d(i a \tan (c+d x))}{2 d}}{2 a}+\frac {3 i}{8 d (a+i a \tan (c+d x))^{4/3}}\right )-\frac {i \int \frac {\tan (c+d x) a+i a}{\tan (c+d x) (i \tan (c+d x) a+a)^{4/3}}dx}{a}\) |
| \(\Big \downarrow \) 67 |
| \(\displaystyle i \left (\frac {\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \left (-\frac {3}{2} \int \frac {1}{-a^2 \tan ^2(c+d x)+i \sqrt [3]{2} a^{4/3} \tan (c+d x)+2^{2/3} a^{2/3}}d\sqrt [3]{i \tan (c+d x) a+a}+\frac {3 \int \frac {1}{\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)}d\sqrt [3]{i \tan (c+d x) a+a}}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{2 d}}{2 a}+\frac {3 i}{8 d (a+i a \tan (c+d x))^{4/3}}\right )-\frac {i \int \frac {\tan (c+d x) a+i a}{\tan (c+d x) (i \tan (c+d x) a+a)^{4/3}}dx}{a}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle i \left (\frac {\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \left (-\frac {3}{2} \int \frac {1}{-a^2 \tan ^2(c+d x)+i \sqrt [3]{2} a^{4/3} \tan (c+d x)+2^{2/3} a^{2/3}}d\sqrt [3]{i \tan (c+d x) a+a}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{2 d}}{2 a}+\frac {3 i}{8 d (a+i a \tan (c+d x))^{4/3}}\right )-\frac {i \int \frac {\tan (c+d x) a+i a}{\tan (c+d x) (i \tan (c+d x) a+a)^{4/3}}dx}{a}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle i \left (\frac {\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \left (\frac {3 \int \frac {1}{a^2 \tan ^2(c+d x)-3}d\left (i 2^{2/3} a^{2/3} \tan (c+d x)+1\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{2 d}}{2 a}+\frac {3 i}{8 d (a+i a \tan (c+d x))^{4/3}}\right )-\frac {i \int \frac {\tan (c+d x) a+i a}{\tan (c+d x) (i \tan (c+d x) a+a)^{4/3}}dx}{a}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle i \left (\frac {\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{2 d}}{2 a}+\frac {3 i}{8 d (a+i a \tan (c+d x))^{4/3}}\right )-\frac {i \int \frac {\tan (c+d x) a+i a}{\tan (c+d x) (i \tan (c+d x) a+a)^{4/3}}dx}{a}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle i \left (\frac {\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{2 d}}{2 a}+\frac {3 i}{8 d (a+i a \tan (c+d x))^{4/3}}\right )-\frac {i \left (\frac {3 \int \frac {8 \cot (c+d x) \left (\tan (c+d x) a^2+i a^2\right )}{3 \sqrt [3]{i \tan (c+d x) a+a}}dx}{8 a^2}+\frac {3 i a}{4 d (a+i a \tan (c+d x))^{4/3}}\right )}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle i \left (\frac {\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{2 d}}{2 a}+\frac {3 i}{8 d (a+i a \tan (c+d x))^{4/3}}\right )-\frac {i \left (\frac {\int \frac {\cot (c+d x) \left (\tan (c+d x) a^2+i a^2\right )}{\sqrt [3]{i \tan (c+d x) a+a}}dx}{a^2}+\frac {3 i a}{4 d (a+i a \tan (c+d x))^{4/3}}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{2 d}}{2 a}+\frac {3 i}{8 d (a+i a \tan (c+d x))^{4/3}}\right )-\frac {i \left (\frac {\int \frac {\tan (c+d x) a^2+i a^2}{\tan (c+d x) \sqrt [3]{i \tan (c+d x) a+a}}dx}{a^2}+\frac {3 i a}{4 d (a+i a \tan (c+d x))^{4/3}}\right )}{a}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle i \left (\frac {\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{2 d}}{2 a}+\frac {3 i}{8 d (a+i a \tan (c+d x))^{4/3}}\right )-\frac {i \left (\frac {\frac {3 \int \frac {2}{3} \cot (c+d x) (i \tan (c+d x) a+a)^{2/3} \left (\tan (c+d x) a^3+i a^3\right )dx}{2 a^2}+\frac {3 i a^2}{d \sqrt [3]{a+i a \tan (c+d x)}}}{a^2}+\frac {3 i a}{4 d (a+i a \tan (c+d x))^{4/3}}\right )}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle i \left (\frac {\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{2 d}}{2 a}+\frac {3 i}{8 d (a+i a \tan (c+d x))^{4/3}}\right )-\frac {i \left (\frac {\frac {\int \cot (c+d x) (i \tan (c+d x) a+a)^{2/3} \left (\tan (c+d x) a^3+i a^3\right )dx}{a^2}+\frac {3 i a^2}{d \sqrt [3]{a+i a \tan (c+d x)}}}{a^2}+\frac {3 i a}{4 d (a+i a \tan (c+d x))^{4/3}}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{2 d}}{2 a}+\frac {3 i}{8 d (a+i a \tan (c+d x))^{4/3}}\right )-\frac {i \left (\frac {\frac {\int \frac {(i \tan (c+d x) a+a)^{2/3} \left (\tan (c+d x) a^3+i a^3\right )}{\tan (c+d x)}dx}{a^2}+\frac {3 i a^2}{d \sqrt [3]{a+i a \tan (c+d x)}}}{a^2}+\frac {3 i a}{4 d (a+i a \tan (c+d x))^{4/3}}\right )}{a}\) |
| \(\Big \downarrow \) 4082 |
| \(\displaystyle i \left (\frac {\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{2 d}}{2 a}+\frac {3 i}{8 d (a+i a \tan (c+d x))^{4/3}}\right )-\frac {i \left (\frac {\frac {i a^2 \int \frac {\cot (c+d x)}{\sqrt [3]{i \tan (c+d x) a+a}}d\tan (c+d x)}{d}+\frac {3 i a^2}{d \sqrt [3]{a+i a \tan (c+d x)}}}{a^2}+\frac {3 i a}{4 d (a+i a \tan (c+d x))^{4/3}}\right )}{a}\) |
| \(\Big \downarrow \) 67 |
| \(\displaystyle i \left (\frac {\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{2 d}}{2 a}+\frac {3 i}{8 d (a+i a \tan (c+d x))^{4/3}}\right )-\frac {i \left (\frac {\frac {i a^2 \left (\frac {3}{2} \int \frac {1}{a^{2/3}+\sqrt [3]{i \tan (c+d x) a+a} \sqrt [3]{a}+(i \tan (c+d x) a+a)^{2/3}}d\sqrt [3]{i \tan (c+d x) a+a}-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{i \tan (c+d x) a+a}}d\sqrt [3]{i \tan (c+d x) a+a}}{2 \sqrt [3]{a}}-\frac {\log (\tan (c+d x))}{2 \sqrt [3]{a}}\right )}{d}+\frac {3 i a^2}{d \sqrt [3]{a+i a \tan (c+d x)}}}{a^2}+\frac {3 i a}{4 d (a+i a \tan (c+d x))^{4/3}}\right )}{a}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle i \left (\frac {\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{2 d}}{2 a}+\frac {3 i}{8 d (a+i a \tan (c+d x))^{4/3}}\right )-\frac {i \left (\frac {\frac {i a^2 \left (\frac {3}{2} \int \frac {1}{a^{2/3}+\sqrt [3]{i \tan (c+d x) a+a} \sqrt [3]{a}+(i \tan (c+d x) a+a)^{2/3}}d\sqrt [3]{i \tan (c+d x) a+a}-\frac {\log (\tan (c+d x))}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{a}}\right )}{d}+\frac {3 i a^2}{d \sqrt [3]{a+i a \tan (c+d x)}}}{a^2}+\frac {3 i a}{4 d (a+i a \tan (c+d x))^{4/3}}\right )}{a}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle i \left (\frac {\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{2 d}}{2 a}+\frac {3 i}{8 d (a+i a \tan (c+d x))^{4/3}}\right )-\frac {i \left (\frac {\frac {i a^2 \left (-\frac {3 \int \frac {1}{-(i \tan (c+d x) a+a)^{2/3}-3}d\left (\frac {2 \sqrt [3]{i \tan (c+d x) a+a}}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{a}}-\frac {\log (\tan (c+d x))}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{a}}\right )}{d}+\frac {3 i a^2}{d \sqrt [3]{a+i a \tan (c+d x)}}}{a^2}+\frac {3 i a}{4 d (a+i a \tan (c+d x))^{4/3}}\right )}{a}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle i \left (\frac {\frac {3 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {i \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{2 d}}{2 a}+\frac {3 i}{8 d (a+i a \tan (c+d x))^{4/3}}\right )-\frac {i \left (\frac {\frac {i a^2 \left (\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}-\frac {\log (\tan (c+d x))}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{a}}\right )}{d}+\frac {3 i a^2}{d \sqrt [3]{a+i a \tan (c+d x)}}}{a^2}+\frac {3 i a}{4 d (a+i a \tan (c+d x))^{4/3}}\right )}{a}\) |
Input:
Int[Cot[c + d*x]/(a + I*a*Tan[c + d*x])^(4/3),x]
Output:
I*(((3*I)/8)/(d*(a + I*a*Tan[c + d*x])^(4/3)) + (((-1/2*I)*(((-I)*Sqrt[3]* ArcTanh[(a*Tan[c + d*x])/Sqrt[3]])/(2^(1/3)*a^(1/3)) - (3*Log[2^(1/3)*a^(1 /3) - I*a*Tan[c + d*x]])/(2*2^(1/3)*a^(1/3)) + Log[a - I*a*Tan[c + d*x]]/( 2*2^(1/3)*a^(1/3))))/d + ((3*I)/2)/(d*(a + I*a*Tan[c + d*x])^(1/3)))/(2*a) ) - (I*((((3*I)/4)*a)/(d*(a + I*a*Tan[c + d*x])^(4/3)) + ((I*a^2*((Sqrt[3] *ArcTan[(1 + (2*(a + I*a*Tan[c + d*x])^(1/3))/a^(1/3))/Sqrt[3]])/a^(1/3) - Log[Tan[c + d*x]]/(2*a^(1/3)) + (3*Log[a^(1/3) - (a + I*a*Tan[c + d*x])^( 1/3)])/(2*a^(1/3))))/d + ((3*I)*a^2)/(d*(a + I*a*Tan[c + d*x])^(1/3)))/a^2 ))/a
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_))^(1/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x ] + (Simp[3/(2*b) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], x] - Simp[3/(2*b*q) 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[((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[a*((a + b*Tan[c + d*x])^n/(2*b*d*n)), x] + Simp[1/(2*a) Int[(a + b*Tan[c + d*x])^ (n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && LtQ[n, 0]
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[-b/d S ubst[Int[(a + x)^(n - 1)/(a - x), x], x, b*Tan[c + d*x]], x] /; FreeQ[{a, b , c, d, n}, x] && EqQ[a^2 + b^2, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)/((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a/(a*c - b*d) Int[(a + b*Tan[e + f*x])^m, x], x] - Simp[d/(a*c - b*d) Int[(a + b*Tan[e + f*x])^m*((b + a*Tan[e + f *x])/(c + d*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && Ne Q[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
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]
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[b*(B/f) Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[ a^2 + b^2, 0] && EqQ[A*b + a*B, 0]
Time = 1.16 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.91
| method | result | size |
| derivativedivides | \(-\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{8 d \,a^{\frac {4}{3}}}+\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{16 d \,a^{\frac {4}{3}}}-\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{8 d \,a^{\frac {4}{3}}}+\frac {9}{4 a d \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}+\frac {3}{8 d \left (a +i a \tan \left (d x +c \right )\right )^{\frac {4}{3}}}+\frac {\ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{d \,a^{\frac {4}{3}}}-\frac {\ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2 d \,a^{\frac {4}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{d \,a^{\frac {4}{3}}}\) | \(286\) |
| default | \(-\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{8 d \,a^{\frac {4}{3}}}+\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{16 d \,a^{\frac {4}{3}}}-\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{8 d \,a^{\frac {4}{3}}}+\frac {9}{4 a d \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}+\frac {3}{8 d \left (a +i a \tan \left (d x +c \right )\right )^{\frac {4}{3}}}+\frac {\ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{d \,a^{\frac {4}{3}}}-\frac {\ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2 d \,a^{\frac {4}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{d \,a^{\frac {4}{3}}}\) | \(286\) |
Input:
int(cot(d*x+c)/(a+I*a*tan(d*x+c))^(4/3),x,method=_RETURNVERBOSE)
Output:
-1/8/d/a^(4/3)*2^(2/3)*ln((a+I*a*tan(d*x+c))^(1/3)-2^(1/3)*a^(1/3))+1/16/d /a^(4/3)*2^(2/3)*ln((a+I*a*tan(d*x+c))^(2/3)+2^(1/3)*a^(1/3)*(a+I*a*tan(d* x+c))^(1/3)+2^(2/3)*a^(2/3))-1/8/d/a^(4/3)*3^(1/2)*2^(2/3)*arctan(1/3*3^(1 /2)*(2^(2/3)/a^(1/3)*(a+I*a*tan(d*x+c))^(1/3)+1))+9/4/a/d/(a+I*a*tan(d*x+c ))^(1/3)+3/8/d/(a+I*a*tan(d*x+c))^(4/3)+1/d/a^(4/3)*ln((a+I*a*tan(d*x+c))^ (1/3)-a^(1/3))-1/2/d/a^(4/3)*ln((a+I*a*tan(d*x+c))^(2/3)+a^(1/3)*(a+I*a*ta n(d*x+c))^(1/3)+a^(2/3))+1/d/a^(4/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/a^(1/3) *(a+I*a*tan(d*x+c))^(1/3)+1))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 633 vs. \(2 (227) = 454\).
Time = 0.09 (sec) , antiderivative size = 633, normalized size of antiderivative = 2.02 \[ \int \frac {\cot (c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx =\text {Too large to display} \] Input:
integrate(cot(d*x+c)/(a+I*a*tan(d*x+c))^(4/3),x, algorithm="fricas")
Output:
1/32*(8*(1/2)^(1/3)*a^2*d*(-1/(a^4*d^3))^(1/3)*e^(4*I*d*x + 4*I*c)*log(-2* (1/2)^(2/3)*a^3*d^2*(-1/(a^4*d^3))^(2/3) + 2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c)) + 32*a^2*d*(1/(a^4*d^3))^(1/3)*e^(4* I*d*x + 4*I*c)*log(-a^3*d^2*(1/(a^4*d^3))^(2/3) + 2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c)) - 4*(1/2)^(1/3)*(I*sqrt(3)*a^ 2*d + a^2*d)*(-1/(a^4*d^3))^(1/3)*e^(4*I*d*x + 4*I*c)*log(-(1/2)^(2/3)*(I* sqrt(3)*a^3*d^2 - a^3*d^2)*(-1/(a^4*d^3))^(2/3) + 2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c)) - 4*(1/2)^(1/3)*(-I*sqrt(3)*a ^2*d + a^2*d)*(-1/(a^4*d^3))^(1/3)*e^(4*I*d*x + 4*I*c)*log(-(1/2)^(2/3)*(- I*sqrt(3)*a^3*d^2 - a^3*d^2)*(-1/(a^4*d^3))^(2/3) + 2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c)) + 3*2^(2/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(2/3)*(13*e^(4*I*d*x + 4*I*c) + 14*e^(2*I*d*x + 2*I*c) + 1) *e^(4/3*I*d*x + 4/3*I*c) - 16*(-I*sqrt(3)*a^2*d + a^2*d)*(1/(a^4*d^3))^(1/ 3)*e^(4*I*d*x + 4*I*c)*log(1/2*(I*sqrt(3)*a^3*d^2 + a^3*d^2)*(1/(a^4*d^3)) ^(2/3) + 2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I* c)) - 16*(I*sqrt(3)*a^2*d + a^2*d)*(1/(a^4*d^3))^(1/3)*e^(4*I*d*x + 4*I*c) *log(1/2*(-I*sqrt(3)*a^3*d^2 + a^3*d^2)*(1/(a^4*d^3))^(2/3) + 2^(1/3)*(a/( e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c)))*e^(-4*I*d*x - 4* I*c)/(a^2*d)
\[ \int \frac {\cot (c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\int \frac {\cot {\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {4}{3}}}\, dx \] Input:
integrate(cot(d*x+c)/(a+I*a*tan(d*x+c))**(4/3),x)
Output:
Integral(cot(c + d*x)/(I*a*(tan(c + d*x) - I))**(4/3), x)
Time = 0.12 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.85 \[ \int \frac {\cot (c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=-\frac {\frac {2 \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right )}{a^{\frac {4}{3}}} - \frac {2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right )}{a^{\frac {4}{3}}} + \frac {2 \cdot 2^{\frac {2}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}{a^{\frac {4}{3}}} - \frac {16 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {4}{3}}} + \frac {8 \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {4}{3}}} - \frac {16 \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{a^{\frac {4}{3}}} - \frac {6 \, {\left (6 i \, a \tan \left (d x + c\right ) + 7 \, a\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}} a}}{16 \, d} \] Input:
integrate(cot(d*x+c)/(a+I*a*tan(d*x+c))^(4/3),x, algorithm="maxima")
Output:
-1/16*(2*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3)*a^(1/3) + 2*( I*a*tan(d*x + c) + a)^(1/3))/a^(1/3))/a^(4/3) - 2^(2/3)*log(2^(2/3)*a^(2/3 ) + 2^(1/3)*(I*a*tan(d*x + c) + a)^(1/3)*a^(1/3) + (I*a*tan(d*x + c) + a)^ (2/3))/a^(4/3) + 2*2^(2/3)*log(-2^(1/3)*a^(1/3) + (I*a*tan(d*x + c) + a)^( 1/3))/a^(4/3) - 16*sqrt(3)*arctan(1/3*sqrt(3)*(2*(I*a*tan(d*x + c) + a)^(1 /3) + a^(1/3))/a^(1/3))/a^(4/3) + 8*log((I*a*tan(d*x + c) + a)^(2/3) + (I* a*tan(d*x + c) + a)^(1/3)*a^(1/3) + a^(2/3))/a^(4/3) - 16*log((I*a*tan(d*x + c) + a)^(1/3) - a^(1/3))/a^(4/3) - 6*(6*I*a*tan(d*x + c) + 7*a)/((I*a*t an(d*x + c) + a)^(4/3)*a))/d
Time = 0.63 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.85 \[ \int \frac {\cot (c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\frac {i \, {\left (\frac {2 i \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right )}{a^{\frac {4}{3}}} - \frac {i \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right )}{a^{\frac {4}{3}}} + \frac {2 i \cdot 2^{\frac {2}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}{a^{\frac {4}{3}}} - \frac {16 i \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {4}{3}}} + \frac {8 i \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {4}{3}}} - \frac {16 i \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{a^{\frac {4}{3}}} + \frac {6 \, {\left (6 \, a \tan \left (d x + c\right ) - 7 i \, a\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}} a}\right )}}{16 \, d} \] Input:
integrate(cot(d*x+c)/(a+I*a*tan(d*x+c))^(4/3),x, algorithm="giac")
Output:
1/16*I*(2*I*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3)*a^(1/3) + 2*(I*a*tan(d*x + c) + a)^(1/3))/a^(1/3))/a^(4/3) - I*2^(2/3)*log(2^(2/3)*a ^(2/3) + 2^(1/3)*(I*a*tan(d*x + c) + a)^(1/3)*a^(1/3) + (I*a*tan(d*x + c) + a)^(2/3))/a^(4/3) + 2*I*2^(2/3)*log(-2^(1/3)*a^(1/3) + (I*a*tan(d*x + c) + a)^(1/3))/a^(4/3) - 16*I*sqrt(3)*arctan(1/3*sqrt(3)*(2*(I*a*tan(d*x + c ) + a)^(1/3) + a^(1/3))/a^(1/3))/a^(4/3) + 8*I*log((I*a*tan(d*x + c) + a)^ (2/3) + (I*a*tan(d*x + c) + a)^(1/3)*a^(1/3) + a^(2/3))/a^(4/3) - 16*I*log ((I*a*tan(d*x + c) + a)^(1/3) - a^(1/3))/a^(4/3) + 6*(6*a*tan(d*x + c) - 7 *I*a)/((I*a*tan(d*x + c) + a)^(4/3)*a))/d
Time = 0.30 (sec) , antiderivative size = 822, normalized size of antiderivative = 2.63 \[ \int \frac {\cot (c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\text {Too large to display} \] Input:
int(cot(c + d*x)/(a + a*tan(c + d*x)*1i)^(4/3),x)
Output:
log(((382205952*a^16*d^9*(1/(a^4*d^3))^(2/3) - 258785280*a^13*d^7*(a + a*t an(c + d*x)*1i)^(1/3))*(1/(a^4*d^3))^(1/3) - 125411328*a^12*d^6)*(1/(a^4*d ^3))^(2/3) + 1990656*a^9*d^4*(a + a*tan(c + d*x)*1i)^(1/3))*(1/(a^4*d^3))^ (1/3) + log(((382205952*a^16*d^9*(-1/(128*a^4*d^3))^(2/3) - 258785280*a^13 *d^7*(a + a*tan(c + d*x)*1i)^(1/3))*(-1/(128*a^4*d^3))^(1/3) - 125411328*a ^12*d^6)*(-1/(128*a^4*d^3))^(2/3) + 1990656*a^9*d^4*(a + a*tan(c + d*x)*1i )^(1/3))*(-1/(128*a^4*d^3))^(1/3) + (log(1990656*a^9*d^4*(a + a*tan(c + d* x)*1i)^(1/3) - ((3^(1/2)*1i - 1)^2*(125411328*a^12*d^6 + ((3^(1/2)*1i - 1) *(258785280*a^13*d^7*(a + a*tan(c + d*x)*1i)^(1/3) - 95551488*a^16*d^9*(3^ (1/2)*1i - 1)^2*(1/(a^4*d^3))^(2/3))*(1/(a^4*d^3))^(1/3))/2)*(1/(a^4*d^3)) ^(2/3))/4)*(3^(1/2)*1i - 1)*(1/(a^4*d^3))^(1/3))/2 - (log(1990656*a^9*d^4* (a + a*tan(c + d*x)*1i)^(1/3) - ((3^(1/2)*1i + 1)^2*(125411328*a^12*d^6 - ((3^(1/2)*1i + 1)*(258785280*a^13*d^7*(a + a*tan(c + d*x)*1i)^(1/3) - 9555 1488*a^16*d^9*(3^(1/2)*1i + 1)^2*(1/(a^4*d^3))^(2/3))*(1/(a^4*d^3))^(1/3)) /2)*(1/(a^4*d^3))^(2/3))/4)*(3^(1/2)*1i + 1)*(1/(a^4*d^3))^(1/3))/2 + log( 1990656*a^9*d^4*(a + a*tan(c + d*x)*1i)^(1/3) - ((3^(1/2)*1i)/2 - 1/2)^2*( 125411328*a^12*d^6 + ((3^(1/2)*1i)/2 - 1/2)*(258785280*a^13*d^7*(a + a*tan (c + d*x)*1i)^(1/3) - 382205952*a^16*d^9*((3^(1/2)*1i)/2 - 1/2)^2*(-1/(128 *a^4*d^3))^(2/3))*(-1/(128*a^4*d^3))^(1/3))*(-1/(128*a^4*d^3))^(2/3))*((3^ (1/2)*1i)/2 - 1/2)*(-1/(128*a^4*d^3))^(1/3) - log(1990656*a^9*d^4*(a + ...
\[ \int \frac {\cot (c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\frac {\int \frac {\cot \left (d x +c \right )}{\left (\tan \left (d x +c \right ) i +1\right )^{\frac {1}{3}} \tan \left (d x +c \right ) i +\left (\tan \left (d x +c \right ) i +1\right )^{\frac {1}{3}}}d x}{a^{\frac {4}{3}}} \] Input:
int(cot(d*x+c)/(a+I*a*tan(d*x+c))^(4/3),x)
Output:
int(cot(c + d*x)/((tan(c + d*x)*i + 1)**(1/3)*tan(c + d*x)*i + (tan(c + d* x)*i + 1)**(1/3)),x)/(a**(1/3)*a)