Integrand size = 13, antiderivative size = 28 \[ \int \frac {\csc ^5(x)}{i+\cot (x)} \, dx=\frac {1}{2} i \text {arctanh}(\cos (x))+\frac {1}{2} i \cot (x) \csc (x)-\frac {\csc ^3(x)}{3} \]
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Time = 0.05 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3582, 3853, 3855} \[ \int \frac {\csc ^5(x)}{i+\cot (x)} \, dx=\frac {1}{2} i \text {arctanh}(\cos (x))-\frac {1}{3} \csc ^3(x)+\frac {1}{2} i \cot (x) \csc (x) \]
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Rule 3582
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{3} \csc ^3(x)-i \int \csc ^3(x) \, dx \\ & = \frac {1}{2} i \cot (x) \csc (x)-\frac {\csc ^3(x)}{3}-\frac {1}{2} i \int \csc (x) \, dx \\ & = \frac {1}{2} i \text {arctanh}(\cos (x))+\frac {1}{2} i \cot (x) \csc (x)-\frac {\csc ^3(x)}{3} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(67\) vs. \(2(28)=56\).
Time = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.39 \[ \int \frac {\csc ^5(x)}{i+\cot (x)} \, dx=\frac {1}{24} i \csc ^3(x) \left (8 i+9 \left (\log \left (\cos \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )\right )\right ) \sin (x)+6 \sin (2 x)-3 \log \left (\cos \left (\frac {x}{2}\right )\right ) \sin (3 x)+3 \log \left (\sin \left (\frac {x}{2}\right )\right ) \sin (3 x)\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (20 ) = 40\).
Time = 0.10 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.07
method | result | size |
default | \(-\frac {\tan \left (\frac {x}{2}\right )}{8}-\frac {\tan \left (\frac {x}{2}\right )^{3}}{24}-\frac {i \tan \left (\frac {x}{2}\right )^{2}}{8}-\frac {i \ln \left (\tan \left (\frac {x}{2}\right )\right )}{2}+\frac {i}{8 \tan \left (\frac {x}{2}\right )^{2}}-\frac {1}{24 \tan \left (\frac {x}{2}\right )^{3}}-\frac {1}{8 \tan \left (\frac {x}{2}\right )}\) | \(58\) |
risch | \(-\frac {i \left (3 \,{\mathrm e}^{5 i x}-8 \,{\mathrm e}^{3 i x}-3 \,{\mathrm e}^{i x}\right )}{3 \left ({\mathrm e}^{2 i x}-1\right )^{3}}+\frac {i \ln \left ({\mathrm e}^{i x}+1\right )}{2}-\frac {i \ln \left ({\mathrm e}^{i x}-1\right )}{2}\) | \(58\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (18) = 36\).
Time = 0.28 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.54 \[ \int \frac {\csc ^5(x)}{i+\cot (x)} \, dx=-\frac {3 \, {\left (-i \, e^{\left (6 i \, x\right )} + 3 i \, e^{\left (4 i \, x\right )} - 3 i \, e^{\left (2 i \, x\right )} + i\right )} \log \left (e^{\left (i \, x\right )} + 1\right ) + 3 \, {\left (i \, e^{\left (6 i \, x\right )} - 3 i \, e^{\left (4 i \, x\right )} + 3 i \, e^{\left (2 i \, x\right )} - i\right )} \log \left (e^{\left (i \, x\right )} - 1\right ) + 6 i \, e^{\left (5 i \, x\right )} - 16 i \, e^{\left (3 i \, x\right )} - 6 i \, e^{\left (i \, x\right )}}{6 \, {\left (e^{\left (6 i \, x\right )} - 3 \, e^{\left (4 i \, x\right )} + 3 \, e^{\left (2 i \, x\right )} - 1\right )}} \]
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\[ \int \frac {\csc ^5(x)}{i+\cot (x)} \, dx=\int \frac {\csc ^{5}{\left (x \right )}}{\cot {\left (x \right )} + i}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (18) = 36\).
Time = 0.24 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.96 \[ \int \frac {\csc ^5(x)}{i+\cot (x)} \, dx=\frac {{\left (\frac {3 i \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {3 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (x\right ) + 1\right )}^{3}}{24 \, \sin \left (x\right )^{3}} - \frac {\sin \left (x\right )}{8 \, {\left (\cos \left (x\right ) + 1\right )}} - \frac {i \, \sin \left (x\right )^{2}}{8 \, {\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {\sin \left (x\right )^{3}}{24 \, {\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac {1}{2} i \, \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (18) = 36\).
Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.21 \[ \int \frac {\csc ^5(x)}{i+\cot (x)} \, dx=-\frac {1}{24} \, \tan \left (\frac {1}{2} \, x\right )^{3} - \frac {1}{8} i \, \tan \left (\frac {1}{2} \, x\right )^{2} - \frac {-22 i \, \tan \left (\frac {1}{2} \, x\right )^{3} + 3 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 3 i \, \tan \left (\frac {1}{2} \, x\right ) + 1}{24 \, \tan \left (\frac {1}{2} \, x\right )^{3}} - \frac {1}{2} i \, \log \left (\tan \left (\frac {1}{2} \, x\right )\right ) - \frac {1}{8} \, \tan \left (\frac {1}{2} \, x\right ) \]
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Time = 12.75 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.96 \[ \int \frac {\csc ^5(x)}{i+\cot (x)} \, dx=-\frac {\mathrm {tan}\left (\frac {x}{2}\right )}{8}-\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )\,1{}\mathrm {i}}{2}-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,1{}\mathrm {i}}{8}-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{24}-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2-\mathrm {tan}\left (\frac {x}{2}\right )\,1{}\mathrm {i}+\frac {1}{3}}{8\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3} \]
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