Optimal. Leaf size=51 \[ \frac{\tanh ^{-1}(\sin (e+f x)) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^m (g \cos (e+f x))^{-2 m}}{f g} \]
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Rubi [A] time = 0.172097, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.075, Rules used = {2847, 12, 3770} \[ \frac{\tanh ^{-1}(\sin (e+f x)) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^m (g \cos (e+f x))^{-2 m}}{f g} \]
Antiderivative was successfully verified.
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Rule 2847
Rule 12
Rule 3770
Rubi steps
\begin{align*} \int (g \cos (e+f x))^{-1-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^m \, dx &=\left ((g \cos (e+f x))^{-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^m\right ) \int \frac{\sec (e+f x)}{g} \, dx\\ &=\frac{\left ((g \cos (e+f x))^{-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^m\right ) \int \sec (e+f x) \, dx}{g}\\ &=\frac{\tanh ^{-1}(\sin (e+f x)) (g \cos (e+f x))^{-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^m}{f g}\\ \end{align*}
Mathematica [A] time = 0.95267, size = 94, normalized size = 1.84 \[ \frac{\sqrt{-\tan ^2(e+f x)} \csc (e+f x) \cos ^{2 (m+1)}(e+f x) \sin ^{-1}(\sec (e+f x)) (g \cos (e+f x))^{-2 m-1} \exp (m (\log (a (\sin (e+f x)+1))+\log (c-c \sin (e+f x))-2 \log (\cos (e+f x))))}{f} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.446, size = 0, normalized size = 0. \begin{align*} \int \left ( g\cos \left ( fx+e \right ) \right ) ^{-1-2\,m} \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( c-c\sin \left ( fx+e \right ) \right ) ^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (g \cos \left (f x + e\right )\right )^{-2 \, m - 1}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (-c \sin \left (f x + e\right ) + c\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77073, size = 113, normalized size = 2.22 \begin{align*} \frac{\left (\frac{a c}{g^{2}}\right )^{m} \log \left (\sin \left (f x + e\right ) + 1\right ) - \left (\frac{a c}{g^{2}}\right )^{m} \log \left (-\sin \left (f x + e\right ) + 1\right )}{2 \, f g} \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 (g \cos \left (f x + e\right )\right )^{-2 \, m - 1}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (-c \sin \left (f x + e\right ) + c\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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