Optimal. Leaf size=135 \[ \frac{c 2^{n+\frac{p}{2}+\frac{1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \cos (e+f x))^{p+1} (1-\sin (e+f x))^{\frac{1}{2} (-2 n-p+1)} \, _2F_1\left (\frac{1}{2} (-2 n-p+1),\frac{1}{2} (2 m+p+1);\frac{1}{2} (2 m+p+3);\frac{1}{2} (\sin (e+f x)+1)\right )}{f g (2 m+p+1)} \]
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Rubi [A] time = 0.286558, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2853, 2689, 70, 69} \[ \frac{c 2^{n+\frac{p}{2}+\frac{1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} (g \cos (e+f x))^{p+1} (1-\sin (e+f x))^{\frac{1}{2} (-2 n-p+1)} \, _2F_1\left (\frac{1}{2} (-2 n-p+1),\frac{1}{2} (2 m+p+1);\frac{1}{2} (2 m+p+3);\frac{1}{2} (\sin (e+f x)+1)\right )}{f g (2 m+p+1)} \]
Antiderivative was successfully verified.
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Rule 2853
Rule 2689
Rule 70
Rule 69
Rubi steps
\begin{align*} \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n \, 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 (g \cos (e+f x))^{2 m+p} (c-c \sin (e+f x))^{-m+n} \, dx\\ &=\frac{\left (c^2 (g \cos (e+f x))^{1+p} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{m+\frac{1}{2} (-1-2 m-p)} (c+c \sin (e+f x))^{\frac{1}{2} (-1-2 m-p)}\right ) \operatorname{Subst}\left (\int (c-c x)^{-m+n+\frac{1}{2} (-1+2 m+p)} (c+c x)^{\frac{1}{2} (-1+2 m+p)} \, dx,x,\sin (e+f x)\right )}{f g}\\ &=\frac{\left (2^{-\frac{1}{2}+n+\frac{p}{2}} c^2 (g \cos (e+f x))^{1+p} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-\frac{1}{2}+m+n+\frac{1}{2} (-1-2 m-p)+\frac{p}{2}} \left (\frac{c-c \sin (e+f x)}{c}\right )^{\frac{1}{2}-n-\frac{p}{2}} (c+c \sin (e+f x))^{\frac{1}{2} (-1-2 m-p)}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2}-\frac{x}{2}\right )^{-m+n+\frac{1}{2} (-1+2 m+p)} (c+c x)^{\frac{1}{2} (-1+2 m+p)} \, dx,x,\sin (e+f x)\right )}{f g}\\ &=\frac{2^{\frac{1}{2}+n+\frac{p}{2}} c (g \cos (e+f x))^{1+p} \, _2F_1\left (\frac{1}{2} (1-2 n-p),\frac{1}{2} (1+2 m+p);\frac{1}{2} (3+2 m+p);\frac{1}{2} (1+\sin (e+f x))\right ) (1-\sin (e+f x))^{\frac{1}{2} (1-2 n-p)} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1+n}}{f g (1+2 m+p)}\\ \end{align*}
Mathematica [A] time = 40.6333, size = 133, normalized size = 0.99 \[ \frac{2 \tan \left (\frac{1}{4} (2 e+2 f x-\pi )\right ) (a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^n (g \cos (e+f x))^p \sec ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )^{m+n+p} \, _2F_1\left (m+n+p+1,\frac{1}{2} (2 n+p+1);\frac{1}{2} (2 n+p+3);-\tan ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )\right )}{f (2 n+p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 5.265, size = 0, normalized size = 0. \begin{align*} \int \left ( g\cos \left ( fx+e \right ) \right ) ^{p} \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( c-c\sin \left ( fx+e \right ) \right ) ^{n}\, 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 )^{p}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (-c \sin \left (f x + e\right ) + c\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (g \cos \left (f x + e\right )\right )^{p}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (-c \sin \left (f x + e\right ) + c\right )}^{n}, x\right ) \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 )^{p}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (-c \sin \left (f x + e\right ) + c\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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