Optimal. Leaf size=79 \[ -\frac{2 (a \sin (e+f x))^m \, _2F_1\left (-\frac{1}{4},\frac{1}{4} (2 m-1);\frac{1}{4} (2 m+3);\sin ^2(e+f x)\right )}{b f (1-2 m) \sqrt [4]{\cos ^2(e+f x)} \sqrt{b \tan (e+f x)}} \]
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Rubi [A] time = 0.119777, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2602, 2577} \[ -\frac{2 (a \sin (e+f x))^m \, _2F_1\left (-\frac{1}{4},\frac{1}{4} (2 m-1);\frac{1}{4} (2 m+3);\sin ^2(e+f x)\right )}{b f (1-2 m) \sqrt [4]{\cos ^2(e+f x)} \sqrt{b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2602
Rule 2577
Rubi steps
\begin{align*} \int \frac{(a \sin (e+f x))^m}{(b \tan (e+f x))^{3/2}} \, dx &=\frac{\left (a \sqrt{a \sin (e+f x)}\right ) \int \cos ^{\frac{3}{2}}(e+f x) (a \sin (e+f x))^{-\frac{3}{2}+m} \, dx}{b \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}\\ &=-\frac{2 \, _2F_1\left (-\frac{1}{4},\frac{1}{4} (-1+2 m);\frac{1}{4} (3+2 m);\sin ^2(e+f x)\right ) (a \sin (e+f x))^m}{b f (1-2 m) \sqrt [4]{\cos ^2(e+f x)} \sqrt{b \tan (e+f x)}}\\ \end{align*}
Mathematica [B] time = 5.1392, size = 225, normalized size = 2.85 \[ \frac{\sec ^4(e+f x) \sec ^2(e+f x)^{\frac{m-4}{2}} (a \sin (e+f x))^m \left (\, _2F_1\left (\frac{m}{2},\frac{1}{4} (2 m-1);\frac{1}{4} (2 m+3);-\tan ^2(e+f x)\right )+\frac{\cos (2 (e+f x)) \sec ^2(e+f x) \left ((2 m-1) \tan ^2(e+f x) \, _2F_1\left (\frac{m+2}{2},\frac{1}{4} (2 m+3);\frac{1}{4} (2 m+7);-\tan ^2(e+f x)\right )-(2 m+3) \, _2F_1\left (\frac{m+2}{2},\frac{1}{4} (2 m-1);\frac{1}{4} (2 m+3);-\tan ^2(e+f x)\right )\right )}{(2 m+3) \left (\tan ^2(e+f x)-1\right )}\right )}{b f (2 m-1) \sqrt{b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.125, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a\sin \left ( fx+e \right ) \right ) ^{m} \left ( b\tan \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \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 )^{m}}{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b \tan \left (f x + e\right )} \left (a \sin \left (f x + e\right )\right )^{m}}{b^{2} \tan \left (f x + e\right )^{2}}, 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 \frac{\left (a \sin \left (f x + e\right )\right )^{m}}{\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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