Optimal. Leaf size=68 \[ \frac{8 a^2 b \sqrt{b \tan (e+f x)}}{3 f \sqrt{a \sin (e+f x)}}-\frac{2 b (a \sin (e+f x))^{3/2} \sqrt{b \tan (e+f x)}}{3 f} \]
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Rubi [A] time = 0.104562, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2598, 2589} \[ \frac{8 a^2 b \sqrt{b \tan (e+f x)}}{3 f \sqrt{a \sin (e+f x)}}-\frac{2 b (a \sin (e+f x))^{3/2} \sqrt{b \tan (e+f x)}}{3 f} \]
Antiderivative was successfully verified.
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Rule 2598
Rule 2589
Rubi steps
\begin{align*} \int (a \sin (e+f x))^{3/2} (b \tan (e+f x))^{3/2} \, dx &=-\frac{2 b (a \sin (e+f x))^{3/2} \sqrt{b \tan (e+f x)}}{3 f}+\frac{1}{3} \left (4 a^2\right ) \int \frac{(b \tan (e+f x))^{3/2}}{\sqrt{a \sin (e+f x)}} \, dx\\ &=\frac{8 a^2 b \sqrt{b \tan (e+f x)}}{3 f \sqrt{a \sin (e+f x)}}-\frac{2 b (a \sin (e+f x))^{3/2} \sqrt{b \tan (e+f x)}}{3 f}\\ \end{align*}
Mathematica [A] time = 0.15863, size = 45, normalized size = 0.66 \[ \frac{a^2 b (\cos (2 (e+f x))+7) \sqrt{b \tan (e+f x)}}{3 f \sqrt{a \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.155, size = 492, normalized size = 7.2 \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \sin \left (f x + e\right )\right )^{\frac{3}{2}} \left (b \tan \left (f x + e\right )\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65223, size = 143, normalized size = 2.1 \begin{align*} \frac{2 \,{\left (a b \cos \left (f x + e\right )^{2} + 3 \, a b\right )} \sqrt{a \sin \left (f x + e\right )} \sqrt{\frac{b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}}}{3 \, 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(-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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