Optimal. Leaf size=39 \[ -\frac{\cos (e+f x) (3 \sin (e+f x)+3)^{-m-1} (a \sin (e+f x)+a)^m}{f} \]
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Rubi [A] time = 0.0170718, antiderivative size = 39, normalized size of antiderivative = 1., 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 = {23, 2648} \[ -\frac{\cos (e+f x) (3 \sin (e+f x)+3)^{-m-1} (a \sin (e+f x)+a)^m}{f} \]
Antiderivative was successfully verified.
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Rule 23
Rule 2648
Rubi steps
\begin{align*} \int (3+3 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m \, dx &=\left ((3+3 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^{1+m}\right ) \int \frac{1}{a+a \sin (e+f x)} \, dx\\ &=-\frac{\cos (e+f x) (3+3 \sin (e+f x))^{-1-m} (a+a \sin (e+f x))^m}{f}\\ \end{align*}
Mathematica [B] time = 5.03509, size = 104, normalized size = 2.67 \[ -\frac{2^{-m} 3^{-m-1} \cos \left (\frac{1}{4} (2 e+2 f x+\pi )\right ) (\sin (e+f x)+1)^{-m-1} \sin ^{-2 m-1}\left (\frac{1}{4} (2 e+2 f x+\pi )\right ) (a (\sin (e+f x)+1))^m \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^{2 (m+1)}}{f} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.254, size = 0, normalized size = 0. \begin{align*} \int \left ( 3+3\,\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 [A] time = 1.75836, size = 51, normalized size = 1.31 \begin{align*} -\frac{2 \, a^{m}}{{\left (3^{m + 1} + \frac{3^{m + 1} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.0135, size = 119, normalized size = 3.05 \begin{align*} -\frac{\left (\frac{1}{3} \, a\right )^{m}{\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )}}{3 \,{\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\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 (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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