Optimal. Leaf size=79 \[ \frac{2 \sqrt [4]{\cos ^2(e+f x)} \sqrt{b \tan (e+f x)} (a \sin (e+f x))^m \, _2F_1\left (\frac{1}{4},\frac{1}{4} (2 m+1);\frac{1}{4} (2 m+5);\sin ^2(e+f x)\right )}{b f (2 m+1)} \]
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Rubi [A] time = 0.102909, 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} \[ \frac{2 \sqrt [4]{\cos ^2(e+f x)} \sqrt{b \tan (e+f x)} (a \sin (e+f x))^m \, _2F_1\left (\frac{1}{4},\frac{1}{4} (2 m+1);\frac{1}{4} (2 m+5);\sin ^2(e+f x)\right )}{b f (2 m+1)} \]
Antiderivative was successfully verified.
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Rule 2602
Rule 2577
Rubi steps
\begin{align*} \int \frac{(a \sin (e+f x))^m}{\sqrt{b \tan (e+f x)}} \, dx &=\frac{\left (a \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}\right ) \int \sqrt{\cos (e+f x)} (a \sin (e+f x))^{-\frac{1}{2}+m} \, dx}{b \sqrt{a \sin (e+f x)}}\\ &=\frac{2 \sqrt [4]{\cos ^2(e+f x)} \, _2F_1\left (\frac{1}{4},\frac{1}{4} (1+2 m);\frac{1}{4} (5+2 m);\sin ^2(e+f x)\right ) (a \sin (e+f x))^m \sqrt{b \tan (e+f x)}}{b f (1+2 m)}\\ \end{align*}
Mathematica [A] time = 2.81801, size = 87, normalized size = 1.1 \[ \frac{2 \sqrt{b \tan (e+f x)} \sec ^2(e+f x)^{m/2} (a \sin (e+f x))^m \, _2F_1\left (\frac{m+2}{2},\frac{1}{4} (2 m+1);\frac{1}{4} (2 m+5);-\tan ^2(e+f x)\right )}{b f (2 m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.132, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a\sin \left ( fx+e \right ) \right ) ^{m}{\frac{1}{\sqrt{b\tan \left ( fx+e \right ) }}}}\, 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}}{\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 [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 \tan \left (f x + e\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a \sin{\left (e + f x \right )}\right )^{m}}{\sqrt{b \tan{\left (e + f x \right )}}}\, dx \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}}{\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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