Optimal. Leaf size=119 \[ \frac{c 2^{n+\frac{9}{4}} (g \cos (e+f x))^{5/2} (1-\sin (e+f x))^{-n-\frac{1}{4}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} \, _2F_1\left (\frac{1}{4} (4 m+5),\frac{1}{4} (-4 n-1);\frac{1}{4} (4 m+9);\frac{1}{2} (\sin (e+f x)+1)\right )}{f g (4 m+5)} \]
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Rubi [A] time = 0.292605, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2853, 2689, 70, 69} \[ \frac{c 2^{n+\frac{9}{4}} (g \cos (e+f x))^{5/2} (1-\sin (e+f x))^{-n-\frac{1}{4}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{n-1} \, _2F_1\left (\frac{1}{4} (4 m+5),\frac{1}{4} (-4 n-1);\frac{1}{4} (4 m+9);\frac{1}{2} (\sin (e+f x)+1)\right )}{f g (4 m+5)} \]
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))^{3/2} (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))^{\frac{3}{2}+2 m} (c-c \sin (e+f x))^{-m+n} \, dx\\ &=\frac{\left (c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{\frac{1}{2} \left (-\frac{5}{2}-2 m\right )+m} (c+c \sin (e+f x))^{\frac{1}{2} \left (-\frac{5}{2}-2 m\right )}\right ) \operatorname{Subst}\left (\int (c-c x)^{-m+\frac{1}{2} \left (\frac{1}{2}+2 m\right )+n} (c+c x)^{\frac{1}{2} \left (\frac{1}{2}+2 m\right )} \, dx,x,\sin (e+f x)\right )}{f g}\\ &=\frac{\left (2^{\frac{1}{4}+n} c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{\frac{1}{4}+\frac{1}{2} \left (-\frac{5}{2}-2 m\right )+m+n} \left (\frac{c-c \sin (e+f x)}{c}\right )^{-\frac{1}{4}-n} (c+c \sin (e+f x))^{\frac{1}{2} \left (-\frac{5}{2}-2 m\right )}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2}-\frac{x}{2}\right )^{-m+\frac{1}{2} \left (\frac{1}{2}+2 m\right )+n} (c+c x)^{\frac{1}{2} \left (\frac{1}{2}+2 m\right )} \, dx,x,\sin (e+f x)\right )}{f g}\\ &=\frac{2^{\frac{9}{4}+n} c (g \cos (e+f x))^{5/2} \, _2F_1\left (\frac{1}{4} (5+4 m),\frac{1}{4} (-1-4 n);\frac{1}{4} (9+4 m);\frac{1}{2} (1+\sin (e+f x))\right ) (1-\sin (e+f x))^{-\frac{1}{4}-n} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1+n}}{f g (5+4 m)}\\ \end{align*}
Mathematica [A] time = 2.1797, size = 126, normalized size = 1.06 \[ -\frac{8 g \cos ^2\left (\frac{1}{4} (2 e+2 f x+\pi )\right ) \sqrt{g \cos (e+f x)} (a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^n \csc ^2\left (\frac{1}{4} (2 e+2 f x+\pi )\right )^{m+n+\frac{3}{2}} \, _2F_1\left (n+\frac{5}{4},m+n+\frac{5}{2};n+\frac{9}{4};-\tan ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )\right )}{f (4 n+5)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.227, size = 0, normalized size = 0. \begin{align*} \int \left ( g\cos \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}} \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 )^{\frac{3}{2}}{\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 (\sqrt{g \cos \left (f x + e\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (-c \sin \left (f x + e\right ) + c\right )}^{n} g \cos \left (f x + e\right ), 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*} \mathit{sage}_{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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