Optimal. Leaf size=43 \[ \frac{\cos (e+f x) (3-3 \sin (e+f x))^{-m-1} (\sin (e+f x)+1)^m}{f (2 m+1)} \]
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Rubi [A] time = 0.0541109, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037, Rules used = {2742} \[ \frac{\cos (e+f x) (3-3 \sin (e+f x))^{-m-1} (\sin (e+f x)+1)^m}{f (2 m+1)} \]
Antiderivative was successfully verified.
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Rule 2742
Rubi steps
\begin{align*} \int (3-3 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx &=\frac{\cos (e+f x) (3-3 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m}{f (1+2 m)}\\ \end{align*}
Mathematica [B] time = 0.522703, size = 97, normalized size = 2.26 \[ \frac{\sin \left (\frac{1}{4} (2 e+2 f x+\pi )\right ) (6-6 \sin (e+f x))^{-m} (\sin (e+f x)+1)^m \cos ^{-2 m-1}\left (\frac{1}{4} (2 e+2 f x+\pi )\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{2 m}}{3 (2 f m+f)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.262, size = 0, normalized size = 0. \begin{align*} \int \left ( 3-3\,\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 (-3 \, \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 [A] time = 1.06304, size = 105, normalized size = 2.44 \begin{align*} \frac{{\left (\sin \left (f x + e\right ) + 1\right )}^{m}{\left (-3 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1} \cos \left (f x + e\right )}{2 \, f m + f} \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*} \mathit{sage}_{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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