Optimal. Leaf size=31 \[ \frac{8}{3} \csc (x) \sqrt{\sin (x) \tan (x)}-\frac{2}{3} \sin (x) \sqrt{\sin (x) \tan (x)} \]
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Rubi [A] time = 0.0534873, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4400, 2598, 2589} \[ \frac{8}{3} \csc (x) \sqrt{\sin (x) \tan (x)}-\frac{2}{3} \sin (x) \sqrt{\sin (x) \tan (x)} \]
Antiderivative was successfully verified.
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Rule 4400
Rule 2598
Rule 2589
Rubi steps
\begin{align*} \int (\sin (x) \tan (x))^{3/2} \, dx &=\frac{\sqrt{\sin (x) \tan (x)} \int \sin ^{\frac{3}{2}}(x) \tan ^{\frac{3}{2}}(x) \, dx}{\sqrt{\sin (x)} \sqrt{\tan (x)}}\\ &=-\frac{2}{3} \sin (x) \sqrt{\sin (x) \tan (x)}+\frac{\left (4 \sqrt{\sin (x) \tan (x)}\right ) \int \frac{\tan ^{\frac{3}{2}}(x)}{\sqrt{\sin (x)}} \, dx}{3 \sqrt{\sin (x)} \sqrt{\tan (x)}}\\ &=\frac{8}{3} \csc (x) \sqrt{\sin (x) \tan (x)}-\frac{2}{3} \sin (x) \sqrt{\sin (x) \tan (x)}\\ \end{align*}
Mathematica [A] time = 0.0409342, size = 23, normalized size = 0.74 \[ \frac{2}{3} \sin (x) \left (4 \csc ^2(x)-1\right ) \sqrt{\sin (x) \tan (x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.13, size = 587, normalized size = 18.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.55491, size = 77, normalized size = 2.48 \begin{align*} -\frac{8 \,{\left (\frac{\sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - 1\right )}}{3 \,{\left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right )}^{\frac{3}{2}}{\left (-\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right )}^{\frac{3}{2}}{\left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.41241, size = 76, normalized size = 2.45 \begin{align*} \frac{2 \,{\left (\cos \left (x\right )^{2} + 3\right )} \sqrt{-\frac{\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}}}{3 \, \sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\sin \left (x\right ) \tan \left (x\right )\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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