Optimal. Leaf size=28 \[ -\frac{3^{-m-1} \cos (e+f x)}{f (\sin (e+f x)+1)} \]
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Rubi [A] time = 0.0196123, antiderivative size = 28, 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 = {22, 2648} \[ -\frac{3^{-m-1} \cos (e+f x)}{f (\sin (e+f x)+1)} \]
Antiderivative was successfully verified.
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Rule 22
Rule 2648
Rubi steps
\begin{align*} \int (1+\sin (e+f x))^m (3+3 \sin (e+f x))^{-1-m} \, dx &=3^{-m} \int \frac{1}{3+3 \sin (e+f x)} \, dx\\ &=-\frac{3^{-1-m} \cos (e+f x)}{f (1+\sin (e+f x))}\\ \end{align*}
Mathematica [A] time = 0.037168, size = 45, normalized size = 1.61 \[ \frac{2\ 3^{-m-1} \sin \left (\frac{1}{2} (e+f x)\right )}{f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.22, size = 0, normalized size = 0. \begin{align*} \int \left ( 1+\sin \left ( fx+e \right ) \right ) ^{m} \left ( 3+3\,\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 [A] time = 1.79773, size = 47, normalized size = 1.68 \begin{align*} -\frac{2}{{\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 = 1.00792, size = 132, normalized size = 4.71 \begin{align*} -\frac{3^{-m - 1}{\left (\cos \left (f x + e\right ) + 1\right )} - 3^{-m - 1} \sin \left (f x + e\right )}{f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + 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*} \int{\left (3 \, \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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