Optimal. Leaf size=44 \[ -\frac{a (a \sin (c+d x)+a)^{m-1} (e \cos (c+d x))^{2-2 m}}{d e (1-m)} \]
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Rubi [A] time = 0.0554556, antiderivative size = 44, 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 = {2673} \[ -\frac{a (a \sin (c+d x)+a)^{m-1} (e \cos (c+d x))^{2-2 m}}{d e (1-m)} \]
Antiderivative was successfully verified.
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Rule 2673
Rubi steps
\begin{align*} \int (e \cos (c+d x))^{1-2 m} (a+a \sin (c+d x))^m \, dx &=-\frac{a (e \cos (c+d x))^{2-2 m} (a+a \sin (c+d x))^{-1+m}}{d e (1-m)}\\ \end{align*}
Mathematica [A] time = 0.153695, size = 43, normalized size = 0.98 \[ -\frac{e (\sin (c+d x)-1) (a (\sin (c+d x)+1))^m (e \cos (c+d x))^{-2 m}}{d (m-1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.25, size = 0, normalized size = 0. \begin{align*} \int \left ( e\cos \left ( dx+c \right ) \right ) ^{1-2\,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 [B] time = 1.50738, size = 194, normalized size = 4.41 \begin{align*} \frac{{\left (a^{m} e - \frac{2 \, a^{m} e \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{a^{m} e \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} e^{\left (-2 \, 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 )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )\right )}}{{\left (e^{2 \, m}{\left (m - 1\right )} + \frac{e^{2 \, m}{\left (m - 1\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.29197, size = 197, normalized size = 4.48 \begin{align*} \frac{\left (e \cos \left (d x + c\right )\right )^{-2 \, m + 1}{\left (a \sin \left (d x + c\right ) + a\right )}^{m}{\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )}}{d m +{\left (d m - d\right )} \cos \left (d x + c\right ) +{\left (d m - d\right )} \sin \left (d x + c\right ) - d} \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 )^{-2 \, 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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