Optimal. Leaf size=39 \[ -\frac{\cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{-m-1}}{f} \]
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Rubi [A] time = 0.171788, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 55, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.018, Rules used = {2974} \[ -\frac{\cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{-m-1}}{f} \]
Antiderivative was successfully verified.
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Rule 2974
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-2-m} (d-(c-d) m+(c+(c-d) m) \sin (e+f x)) \, dx &=-\frac{\cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m}}{f}\\ \end{align*}
Mathematica [A] time = 0.664572, size = 39, normalized size = 1. \[ -\frac{\cos (e+f x) (a (\sin (e+f x)+1))^m (c+d \sin (e+f x))^{-m-1}}{f} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.589, size = 0, normalized size = 0. \begin{align*} \int \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( c+d\sin \left ( fx+e \right ) \right ) ^{-2-m} \left ( d- \left ( c-d \right ) m+ \left ( c+ \left ( c-d \right ) m \right ) \sin \left ( fx+e \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [A] time = 1.99149, size = 140, normalized size = 3.59 \begin{align*} -\frac{{\left (d \cos \left (f x + e\right ) \sin \left (f x + e\right ) + c \cos \left (f x + e\right )\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (d \sin \left (f x + e\right ) + c\right )}^{-m - 2}}{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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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