Optimal. Leaf size=32 \[ -\frac{2 b (a \sin (e+f x))^{3/2}}{3 f (b \tan (e+f x))^{3/2}} \]
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Rubi [A] time = 0.0493257, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {2589} \[ -\frac{2 b (a \sin (e+f x))^{3/2}}{3 f (b \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2589
Rubi steps
\begin{align*} \int \frac{(a \sin (e+f x))^{3/2}}{\sqrt{b \tan (e+f x)}} \, dx &=-\frac{2 b (a \sin (e+f x))^{3/2}}{3 f (b \tan (e+f x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.131211, size = 32, normalized size = 1. \[ -\frac{2 b (a \sin (e+f x))^{3/2}}{3 f (b \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.135, size = 48, normalized size = 1.5 \begin{align*} -{\frac{2\,\cos \left ( fx+e \right ) }{3\,f\sin \left ( fx+e \right ) } \left ( a\sin \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{{\frac{b\sin \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a \sin \left (f x + e\right )\right )^{\frac{3}{2}}}{\sqrt{b \tan \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.63189, size = 131, normalized size = 4.09 \begin{align*} -\frac{2 \, \sqrt{a \sin \left (f x + e\right )} a \sqrt{\frac{b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )^{2}}{3 \, b f \sin \left (f x + e\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 \frac{\left (a \sin \left (f x + e\right )\right )^{\frac{3}{2}}}{\sqrt{b \tan \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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