Optimal. Leaf size=37 \[ -\frac {4}{15} i \cos ^3(x)+\frac {4}{5} i \cos (x)+\frac {i \sin ^3(x)}{5 (\cot (x)+i)} \]
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Rubi [A] time = 0.04, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3502, 2633} \[ -\frac {4}{15} i \cos ^3(x)+\frac {4}{5} i \cos (x)+\frac {i \sin ^3(x)}{5 (\cot (x)+i)} \]
Antiderivative was successfully verified.
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Rule 2633
Rule 3502
Rubi steps
\begin {align*} \int \frac {\sin ^3(x)}{i+\cot (x)} \, dx &=\frac {i \sin ^3(x)}{5 (i+\cot (x))}-\frac {4}{5} i \int \sin ^3(x) \, dx\\ &=\frac {i \sin ^3(x)}{5 (i+\cot (x))}+\frac {4}{5} i \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (x)\right )\\ &=\frac {4}{5} i \cos (x)-\frac {4}{15} i \cos ^3(x)+\frac {i \sin ^3(x)}{5 (i+\cot (x))}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 46, normalized size = 1.24 \[ \frac {\csc (x) (-40 \sin (2 x)+4 \sin (4 x)+20 i \cos (2 x)-i \cos (4 x)+45 i)}{120 (\cot (x)+i)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 32, normalized size = 0.86 \[ \frac {1}{240} \, {\left (-5 i \, e^{\left (8 i \, x\right )} + 60 i \, e^{\left (6 i \, x\right )} + 90 i \, e^{\left (4 i \, x\right )} - 20 i \, e^{\left (2 i \, x\right )} + 3 i\right )} e^{\left (-5 i \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 69, normalized size = 1.86 \[ -\frac {9 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 24 i \, \tan \left (\frac {1}{2} \, x\right ) - 11}{24 \, {\left (\tan \left (\frac {1}{2} \, x\right ) + i\right )}^{3}} + \frac {45 \, \tan \left (\frac {1}{2} \, x\right )^{4} - 240 i \, \tan \left (\frac {1}{2} \, x\right )^{3} - 490 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 320 i \, \tan \left (\frac {1}{2} \, x\right ) + 73}{120 \, {\left (\tan \left (\frac {1}{2} \, x\right ) - i\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.32, size = 93, normalized size = 2.51 \[ -\frac {i}{4 \left (\tan \left (\frac {x}{2}\right )+i\right )^{2}}-\frac {1}{6 \left (\tan \left (\frac {x}{2}\right )+i\right )^{3}}-\frac {3}{8 \left (\tan \left (\frac {x}{2}\right )+i\right )}-\frac {i}{\left (\tan \left (\frac {x}{2}\right )-i\right )^{4}}-\frac {i}{2 \left (\tan \left (\frac {x}{2}\right )-i\right )^{2}}+\frac {2}{5 \left (\tan \left (\frac {x}{2}\right )-i\right )^{5}}-\frac {1}{3 \left (\tan \left (\frac {x}{2}\right )-i\right )^{3}}+\frac {3}{8 \left (\tan \left (\frac {x}{2}\right )-i\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 49, normalized size = 1.32 \[ \frac {\left (6\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3-{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,2{}\mathrm {i}+2\,\mathrm {tan}\left (\frac {x}{2}\right )-\mathrm {i}\right )\,16{}\mathrm {i}}{15\,{\left (1+\mathrm {tan}\left (\frac {x}{2}\right )\,1{}\mathrm {i}\right )}^5\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1{}\mathrm {i}\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 48, normalized size = 1.30 \[ - \frac {i e^{3 i x}}{48} + \frac {i e^{i x}}{4} + \frac {3 i e^{- i x}}{8} - \frac {i e^{- 3 i x}}{12} + \frac {i e^{- 5 i x}}{80} \]
Verification of antiderivative is not currently implemented for this CAS.
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