Optimal. Leaf size=83 \[ -\frac{\left (\frac{7}{2}\right )^{-m-1} \cos (e+f x) (\sin (e+f x)+1)^{-m-1} (a \sin (e+f x)+a)^m \, _2F_1\left (\frac{1}{2},m+1;\frac{3}{2};\frac{a-a \sin (e+f x)}{7 (\sin (e+f x) a+a)}\right )}{f} \]
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Rubi [A] time = 0.11348, antiderivative size = 118, normalized size of antiderivative = 1.42, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {2788, 132} \[ -\frac{\sqrt{\frac{1-\sin (e+f x)}{\sin (e+f x)+1}} \cos (e+f x) (4 \sin (e+f x)+3)^{-m} (a \sin (e+f x)+a)^m \, _2F_1\left (\frac{1}{2},-m;1-m;\frac{2 (4 \sin (e+f x)+3)}{7 (\sin (e+f x)+1)}\right )}{\sqrt{7} f m (1-\sin (e+f x))} \]
Antiderivative was successfully verified.
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Rule 2788
Rule 132
Rubi steps
\begin{align*} \int (3+4 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx &=\frac{\left (a^2 \cos (e+f x)\right ) \operatorname{Subst}\left (\int \frac{(3+4 x)^{-1-m} (a+a x)^{-\frac{1}{2}+m}}{\sqrt{a-a x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}\\ &=-\frac{\cos (e+f x) \, _2F_1\left (\frac{1}{2},-m;1-m;\frac{2 (3+4 \sin (e+f x))}{7 (1+\sin (e+f x))}\right ) \sqrt{\frac{1-\sin (e+f x)}{1+\sin (e+f x)}} (3+4 \sin (e+f x))^{-m} (a+a \sin (e+f x))^m}{\sqrt{7} f m (1-\sin (e+f x))}\\ \end{align*}
Mathematica [A] time = 0.455197, size = 90, normalized size = 1.08 \[ -\frac{2\ 7^{-m-1} \cot \left (\frac{1}{4} (2 e+2 f x+\pi )\right ) \sin ^2\left (\frac{1}{4} (2 e+2 f x+\pi )\right )^{-m} (a (\sin (e+f x)+1))^m \, _2F_1\left (\frac{1}{2},m+1;\frac{3}{2};\frac{1}{7} \tan ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )\right )}{f} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.251, size = 0, normalized size = 0. \begin{align*} \int \left ( 3+4\,\sin \left ( fx+e \right ) \right ) ^{-1-m} \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m}\, 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{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (4 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1}\,{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 ({\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (4 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1}, 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 (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (4 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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