Optimal. Leaf size=94 \[ -\frac{\cos (e+f x) (3-\sin (e+f x))^{-m-1} \left (\frac{3-\sin (e+f x)}{\sin (e+f x)+1}\right )^{m+1} (\sin (e+f x)+1)^m \, _2F_1\left (\frac{1}{2},m+1;\frac{3}{2};-\frac{2 (1-\sin (e+f x))}{\sin (e+f x)+1}\right )}{f} \]
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Rubi [A] time = 0.0922346, antiderivative size = 94, 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} \[ -\frac{\cos (e+f x) (3-\sin (e+f x))^{-m-1} \left (\frac{3-\sin (e+f x)}{\sin (e+f x)+1}\right )^{m+1} (\sin (e+f x)+1)^m \, _2F_1\left (\frac{1}{2},m+1;\frac{3}{2};-\frac{2 (1-\sin (e+f x))}{\sin (e+f x)+1}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 2788
Rule 132
Rubi steps
\begin{align*} \int (3-\sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx &=\frac{\cos (e+f x) \operatorname{Subst}\left (\int \frac{(3-x)^{-1-m} (1+x)^{-\frac{1}{2}+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{\cos (e+f x) \, _2F_1\left (\frac{1}{2},1+m;\frac{3}{2};-\frac{2 (1-\sin (e+f x))}{1+\sin (e+f x)}\right ) (3-\sin (e+f x))^{-1-m} \left (\frac{3-\sin (e+f x)}{1+\sin (e+f x)}\right )^{1+m} (1+\sin (e+f x))^m}{f}\\ \end{align*}
Mathematica [A] time = 1.06608, size = 182, normalized size = 1.94 \[ -\frac{2^{\frac{1}{2}-m} \cot \left (\frac{1}{4} (2 e+2 f x+\pi )\right ) (3-\sin (e+f x))^{-m} (\sin (e+f x)+1)^m \sin ^2\left (\frac{1}{4} (2 e+2 f x+\pi )\right )^{\frac{1}{2}-m} \cos ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )^{m-\frac{1}{2}} \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};-\frac{4 \sin ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )}{\sin (e+f x)-3}\right ) \left (-\frac{\cos ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )}{\sin (e+f x)-3}\right )^{\frac{1}{2}-m}}{f} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.258, size = 0, normalized size = 0. \begin{align*} \int \left ( 3-\sin \left ( fx+e \right ) \right ) ^{-1-m} \left ( 1+\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 (\sin \left (f x + e\right ) + 1\right )}^{m}{\left (-\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 (\sin \left (f x + e\right ) + 1\right )}^{m}{\left (-\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 (\sin \left (f x + e\right ) + 1\right )}^{m}{\left (-\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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