Optimal. Leaf size=28 \[ -\frac {3^{-1-m} \cos (e+f x)}{f (1+\sin (e+f x))} \]
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Rubi [A]
time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, 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, 2727}
\begin {gather*} -\frac {3^{-m-1} \cos (e+f x)}{f (\sin (e+f x)+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 22
Rule 2727
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*}
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Mathematica [A]
time = 0.04, size = 45, normalized size = 1.61 \begin {gather*} \frac {2\ 3^{-1-m} \sin \left (\frac {1}{2} (e+f x)\right )}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \left (1+\sin \left (f x +e \right )\right )^{m} \left (3+3 \sin \left (f x +e \right )\right )^{-1-m}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 37, normalized size = 1.32 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 58, normalized size = 2.07 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 3^{- m - 1} \int \frac {1}{\sin {\left (e + f x \right )} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.64, size = 45, normalized size = 1.61 \begin {gather*} \frac {3^{-m - 1} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3^{-m - 1}}{f \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.43, size = 41, normalized size = 1.46 \begin {gather*} \frac {\frac {1}{3^{m+1}}\,\left (-\cos \left (e+f\,x\right )+\sin \left (e+f\,x\right )\,1{}\mathrm {i}+1{}\mathrm {i}\right )}{f\,\left (\sin \left (e+f\,x\right )+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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