Optimal. Leaf size=122 \[ -\frac{2^{m+\frac{1}{2}} 5^{-m-\frac{1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{m-1} \left (\frac{\sin (e+f x)+1}{2 \sin (e+f x)+3}\right )^{\frac{1}{2}-m} (2 \sin (e+f x)+3)^{-m} \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1-\sin (e+f x)}{2 (2 \sin (e+f x)+3)}\right )}{f} \]
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Rubi [A] time = 0.114781, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {2788, 132} \[ -\frac{2^{m+\frac{1}{2}} 5^{-m-\frac{1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{m-1} \left (\frac{\sin (e+f x)+1}{2 \sin (e+f x)+3}\right )^{\frac{1}{2}-m} (2 \sin (e+f x)+3)^{-m} \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1-\sin (e+f x)}{2 (2 \sin (e+f x)+3)}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 2788
Rule 132
Rubi steps
\begin{align*} \int (1+\sin (e+f x))^m (3+2 \sin (e+f x))^{-1-m} \, dx &=\frac{\cos (e+f x) \operatorname{Subst}\left (\int \frac{(1+x)^{-\frac{1}{2}+m} (3+2 x)^{-1-m}}{\sqrt{1-x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt{1-\sin (e+f x)} \sqrt{1+\sin (e+f x)}}\\ &=-\frac{2^{\frac{1}{2}+m} 5^{-\frac{1}{2}-m} \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1-\sin (e+f x)}{2 (3+2 \sin (e+f x))}\right ) (1+\sin (e+f x))^{-1+m} \left (\frac{1+\sin (e+f x)}{3+2 \sin (e+f x)}\right )^{\frac{1}{2}-m} (3+2 \sin (e+f x))^{-m}}{f}\\ \end{align*}
Mathematica [A] time = 0.530042, size = 131, normalized size = 1.07 \[ \frac{2\ 5^{-m-1} \tan \left (\frac{1}{4} (2 e+2 f x-\pi )\right ) (\sin (e+f x)+1)^m (2 \sin (e+f x)+3)^{-m} \left ((2 \sin (e+f x)+3) \sec ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )\right )^m \, _2F_1\left (\frac{1}{2},m+1;\frac{3}{2};-\frac{1}{5} \cos ^2\left (\frac{1}{4} (2 e+2 f x+\pi )\right ) \sec ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )\right )}{f} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.25, size = 0, normalized size = 0. \begin{align*} \int \left ( 1+\sin \left ( fx+e \right ) \right ) ^{m} \left ( 3+2\,\sin \left ( fx+e \right ) \right ) ^{-1-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 (2 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1}{\left (\sin \left (f x + e\right ) + 1\right )}^{m}\,{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 (2 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1}{\left (\sin \left (f x + e\right ) + 1\right )}^{m}, 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 (2 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1}{\left (\sin \left (f x + e\right ) + 1\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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