Optimal. Leaf size=25 \[ \frac {2}{3} i \cos (x)+\frac {i \sin (x)}{3 (\cot (x)+i)} \]
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Rubi [A] time = 0.03, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3502, 2638} \[ \frac {2}{3} i \cos (x)+\frac {i \sin (x)}{3 (\cot (x)+i)} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3502
Rubi steps
\begin {align*} \int \frac {\sin (x)}{i+\cot (x)} \, dx &=\frac {i \sin (x)}{3 (i+\cot (x))}-\frac {2}{3} i \int \sin (x) \, dx\\ &=\frac {2}{3} i \cos (x)+\frac {i \sin (x)}{3 (i+\cot (x))}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 27, normalized size = 1.08 \[ \frac {1}{6} (\sin (x)+i \cos (x)) (2 i \sin (2 x)+\cos (2 x)+3) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 20, normalized size = 0.80 \[ \frac {1}{12} \, {\left (3 i \, e^{\left (4 i \, x\right )} + 6 i \, e^{\left (2 i \, x\right )} - i\right )} e^{\left (-3 i \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.57, size = 37, normalized size = 1.48 \[ -\frac {1}{2 \, {\left (\tan \left (\frac {1}{2} \, x\right ) + i\right )}} + \frac {3 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 12 i \, \tan \left (\frac {1}{2} \, x\right ) - 5}{6 \, {\left (\tan \left (\frac {1}{2} \, x\right ) - i\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.31, size = 47, normalized size = 1.88 \[ -\frac {1}{2 \left (\tan \left (\frac {x}{2}\right )+i\right )}-\frac {i}{\left (\tan \left (\frac {x}{2}\right )-i\right )^{2}}+\frac {2}{3 \left (\tan \left (\frac {x}{2}\right )-i\right )^{3}}+\frac {1}{2 \tan \left (\frac {x}{2}\right )-2 i} \]
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 = 0.30, size = 31, normalized size = 1.24 \[ -\frac {\frac {4}{3}+\frac {\mathrm {tan}\left (\frac {x}{2}\right )\,8{}\mathrm {i}}{3}}{{\left (1+\mathrm {tan}\left (\frac {x}{2}\right )\,1{}\mathrm {i}\right )}^3\,\left (\mathrm {tan}\left (\frac {x}{2}\right )+1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 26, normalized size = 1.04 \[ \frac {i e^{i x}}{4} + \frac {i e^{- i x}}{2} - \frac {i e^{- 3 i x}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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