Optimal. Leaf size=28 \[ -\frac {1}{3} \csc ^3(x)+\frac {1}{2} i \tanh ^{-1}(\cos (x))+\frac {1}{2} i \cot (x) \csc (x) \]
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Rubi [A] time = 0.04, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3501, 3768, 3770} \[ -\frac {1}{3} \csc ^3(x)+\frac {1}{2} i \tanh ^{-1}(\cos (x))+\frac {1}{2} i \cot (x) \csc (x) \]
Antiderivative was successfully verified.
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Rule 3501
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\csc ^5(x)}{i+\cot (x)} \, dx &=-\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 \tanh ^{-1}(\cos (x))+\frac {1}{2} i \cot (x) \csc (x)-\frac {\csc ^3(x)}{3}\\ \end {align*}
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Mathematica [B] time = 0.10, size = 67, normalized size = 2.39 \[ \frac {1}{24} i \csc ^3(x) \left (6 \sin (2 x)+3 \sin (3 x) \log \left (\sin \left (\frac {x}{2}\right )\right )+9 \sin (x) \left (\log \left (\cos \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )\right )\right )-3 \sin (3 x) \log \left (\cos \left (\frac {x}{2}\right )\right )+8 i\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 97, normalized size = 3.46 \[ \frac {{\left (3 i \, e^{\left (6 i \, x\right )} - 9 i \, e^{\left (4 i \, x\right )} + 9 i \, e^{\left (2 i \, x\right )} - 3 i\right )} \log \left (e^{\left (i \, x\right )} + 1\right ) + {\left (-3 i \, e^{\left (6 i \, x\right )} + 9 i \, e^{\left (4 i \, x\right )} - 9 i \, e^{\left (2 i \, x\right )} + 3 i\right )} \log \left (e^{\left (i \, x\right )} - 1\right ) - 6 i \, e^{\left (5 i \, x\right )} + 16 i \, e^{\left (3 i \, x\right )} + 6 i \, e^{\left (i \, x\right )}}{6 \, {\left (e^{\left (6 i \, x\right )} - 3 \, e^{\left (4 i \, x\right )} + 3 \, e^{\left (2 i \, x\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.65, size = 62, normalized size = 2.21 \[ -\frac {1}{24} \, \tan \left (\frac {1}{2} \, x\right )^{3} - \frac {1}{8} i \, \tan \left (\frac {1}{2} \, x\right )^{2} - \frac {-22 i \, \tan \left (\frac {1}{2} \, x\right )^{3} + 3 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 3 i \, \tan \left (\frac {1}{2} \, x\right ) + 1}{24 \, \tan \left (\frac {1}{2} \, x\right )^{3}} - \frac {1}{2} i \, \log \left (\tan \left (\frac {1}{2} \, x\right )\right ) - \frac {1}{8} \, \tan \left (\frac {1}{2} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.30, size = 58, normalized size = 2.07 \[ -\frac {\tan \left (\frac {x}{2}\right )}{8}-\frac {\left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{24}-\frac {i \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{8}+\frac {i}{8 \tan \left (\frac {x}{2}\right )^{2}}-\frac {1}{24 \tan \left (\frac {x}{2}\right )^{3}}-\frac {i \ln \left (\tan \left (\frac {x}{2}\right )\right )}{2}-\frac {1}{8 \tan \left (\frac {x}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 83, normalized size = 2.96 \[ -\frac {{\left (-\frac {6 i \, \sin \relax (x)}{\cos \relax (x) + 1} + \frac {6 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 2\right )} {\left (\cos \relax (x) + 1\right )}^{3}}{48 \, \sin \relax (x)^{3}} - \frac {\sin \relax (x)}{8 \, {\left (\cos \relax (x) + 1\right )}} - \frac {i \, \sin \relax (x)^{2}}{8 \, {\left (\cos \relax (x) + 1\right )}^{2}} - \frac {\sin \relax (x)^{3}}{24 \, {\left (\cos \relax (x) + 1\right )}^{3}} - \frac {1}{2} i \, \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 55, normalized size = 1.96 \[ -\frac {\mathrm {tan}\left (\frac {x}{2}\right )}{8}-\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )\,1{}\mathrm {i}}{2}-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,1{}\mathrm {i}}{8}-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{24}-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2-\mathrm {tan}\left (\frac {x}{2}\right )\,1{}\mathrm {i}+\frac {1}{3}}{8\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{5}{\relax (x )}}{\cot {\relax (x )} + i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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