Optimal. Leaf size=46 \[ \frac {\cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1}}{f (2 m+1)} \]
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Rubi [A] time = 0.07, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2742} \[ \frac {\cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1}}{f (2 m+1)} \]
Antiderivative was successfully verified.
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Rule 2742
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1-m} \, dx &=\frac {\cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1-m}}{f (1+2 m)}\\ \end {align*}
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Mathematica [B] time = 1.53, size = 107, normalized size = 2.33 \[ \frac {2^{-m} \sin \left (\frac {1}{4} (2 e+2 f x+\pi )\right ) \cos ^{-2 m-1}\left (\frac {1}{4} (2 e+2 f x+\pi )\right ) (a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^{-m} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{2 m}}{c f (2 m+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 44, normalized size = 0.96 \[ \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-m - 1} \cos \left (f x + e\right )}{2 \, f m + f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-m - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.61, size = 0, normalized size = 0.00 \[ \int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{-1-m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (-c \sin \left (f x + e\right ) + c\right )}^{-m - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.41, size = 58, normalized size = 1.26 \[ -\frac {\cos \left (e+f\,x\right )\,{\left (a\,\left (\sin \left (e+f\,x\right )+1\right )\right )}^m}{c\,f\,\left (2\,m+1\right )\,{\left (-c\,\left (\sin \left (e+f\,x\right )-1\right )\right )}^m\,\left (\sin \left (e+f\,x\right )-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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