Optimal. Leaf size=34 \[ \frac{(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-m}}{d e m} \]
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Rubi [A] time = 0.0506929, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037, Rules used = {2671} \[ \frac{(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-m}}{d e m} \]
Antiderivative was successfully verified.
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Rule 2671
Rubi steps
\begin{align*} \int (e \cos (c+d x))^{-1-m} (a+a \sin (c+d x))^m \, dx &=\frac{(e \cos (c+d x))^{-m} (a+a \sin (c+d x))^m}{d e m}\\ \end{align*}
Mathematica [A] time = 0.0531585, size = 34, normalized size = 1. \[ \frac{(a (\sin (c+d x)+1))^m (e \cos (c+d x))^{-m}}{d e m} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.144, size = 0, normalized size = 0. \begin{align*} \int \left ( e\cos \left ( dx+c \right ) \right ) ^{-1-m} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.57164, size = 88, normalized size = 2.59 \begin{align*} \frac{a^{m} e^{-m - 1} e^{\left (m \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right ) - m \log \left (-\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )\right )}}{d m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.28788, size = 93, normalized size = 2.74 \begin{align*} \frac{\left (e \cos \left (d x + c\right )\right )^{-m - 1}{\left (a \sin \left (d x + c\right ) + a\right )}^{m} \cos \left (d x + c\right )}{d m} \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 (e \cos \left (d x + c\right )\right )^{-m - 1}{\left (a \sin \left (d x + c\right ) + a\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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