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 = 0.0299948, antiderivative size = 25, normalized size of antiderivative = 1., 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 3502
Rule 2638
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*}
Mathematica [A] time = 0.0431948, 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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Maple [B] time = 0.055, size = 47, normalized size = 1.9 \begin{align*}{-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} \left ( \tan \left ({\frac{x}{2}} \right ) -i \right ) ^{-1}}-{\frac{1}{2} \left ( \tan \left ({\frac{x}{2}} \right ) +i \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56445, size = 96, normalized size = 3.84 \begin{align*} \frac{1}{12} \,{\left ({\left (3 i \, e^{\left (2 i \, x\right )} + 3 i\right )} e^{\left (2 i \, x\right )} + 3 i \, e^{\left (2 i \, x\right )} - i\right )} e^{\left (-3 i \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.373806, size = 26, normalized size = 1.04 \begin{align*} \frac{i e^{i x}}{4} + \frac{i e^{- i x}}{2} - \frac{i e^{- 3 i x}}{12} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22285, size = 50, normalized size = 2. \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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