Optimal. Leaf size=19 \[ \frac{\sin ^3(x)}{3}+\frac{1}{3} i \cos ^3(x) \]
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Rubi [A] time = 0.0893613, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {3518, 3108, 3107, 2565, 30, 2564} \[ \frac{\sin ^3(x)}{3}+\frac{1}{3} i \cos ^3(x) \]
Antiderivative was successfully verified.
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Rule 3518
Rule 3108
Rule 3107
Rule 2565
Rule 30
Rule 2564
Rubi steps
\begin{align*} \int \frac{\sin (x)}{i+\tan (x)} \, dx &=\int \frac{\cos (x) \sin (x)}{i \cos (x)+\sin (x)} \, dx\\ &=-(i \int \cos (x) (\cos (x)+i \sin (x)) \sin (x) \, dx)\\ &=-\left (i \int \left (\cos ^2(x) \sin (x)+i \cos (x) \sin ^2(x)\right ) \, dx\right )\\ &=-\left (i \int \cos ^2(x) \sin (x) \, dx\right )+\int \cos (x) \sin ^2(x) \, dx\\ &=i \operatorname{Subst}\left (\int x^2 \, dx,x,\cos (x)\right )+\operatorname{Subst}\left (\int x^2 \, dx,x,\sin (x)\right )\\ &=\frac{1}{3} i \cos ^3(x)+\frac{\sin ^3(x)}{3}\\ \end{align*}
Mathematica [A] time = 0.01213, size = 33, normalized size = 1.74 \[ \frac{\sin (x)}{4}-\frac{1}{12} \sin (3 x)+\frac{1}{4} i \cos (x)+\frac{1}{12} i \cos (3 x) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.035, size = 47, normalized size = 2.5 \begin{align*}{\frac{1}{2} \left ( \tan \left ({\frac{x}{2}} \right ) -i \right ) ^{-1}}+{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}} \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 = 1.99361, size = 47, normalized size = 2.47 \begin{align*} \frac{1}{12} \,{\left (i \, e^{\left (4 i \, x\right )} + 3 i\right )} e^{\left (-i \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.17362, size = 17, normalized size = 0.89 \begin{align*} \frac{i e^{3 i x}}{12} + \frac{i e^{- i x}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.3401, size = 45, normalized size = 2.37 \begin{align*} -\frac{i}{2 \,{\left (-i \, \tan \left (\frac{1}{2} \, x\right ) - 1\right )}} - \frac{3 \, \tan \left (\frac{1}{2} \, x\right )^{2} - 1}{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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