Optimal. Leaf size=106 \[ -\frac{c 2^{m+\frac{9}{4}} \cos (e+f x) (g \cos (e+f x))^{3/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (c \sin (e+f x)+c)^{m-1} \, _2F_1\left (\frac{3}{4},-m-\frac{1}{4};\frac{7}{4};\frac{1}{2} (1-\sin (e+f x))\right )}{3 f \sqrt{a-a \sin (e+f x)}} \]
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Rubi [A] time = 0.313294, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2853, 2689, 70, 69} \[ -\frac{c 2^{m+\frac{9}{4}} \cos (e+f x) (g \cos (e+f x))^{3/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (c \sin (e+f x)+c)^{m-1} \, _2F_1\left (\frac{3}{4},-m-\frac{1}{4};\frac{7}{4};\frac{1}{2} (1-\sin (e+f x))\right )}{3 f \sqrt{a-a \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2853
Rule 2689
Rule 70
Rule 69
Rubi steps
\begin{align*} \int \frac{(g \cos (e+f x))^{3/2} (c+c \sin (e+f x))^m}{\sqrt{a-a \sin (e+f x)}} \, dx &=\frac{(g \cos (e+f x)) \int \sqrt{g \cos (e+f x)} (c+c \sin (e+f x))^{\frac{1}{2}+m} \, dx}{\sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}\\ &=\frac{\left (c^2 \cos (e+f x) (g \cos (e+f x))^{3/2}\right ) \operatorname{Subst}\left (\int \frac{(c+c x)^{\frac{1}{4}+m}}{\sqrt [4]{c-c x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt{a-a \sin (e+f x)} (c-c \sin (e+f x))^{3/4} (c+c \sin (e+f x))^{5/4}}\\ &=\frac{\left (2^{\frac{1}{4}+m} c^2 \cos (e+f x) (g \cos (e+f x))^{3/2} (c+c \sin (e+f x))^{-1+m} \left (\frac{c+c \sin (e+f x)}{c}\right )^{-\frac{1}{4}-m}\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{1}{2}+\frac{x}{2}\right )^{\frac{1}{4}+m}}{\sqrt [4]{c-c x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt{a-a \sin (e+f x)} (c-c \sin (e+f x))^{3/4}}\\ &=-\frac{2^{\frac{9}{4}+m} c \cos (e+f x) (g \cos (e+f x))^{3/2} \, _2F_1\left (\frac{3}{4},-\frac{1}{4}-m;\frac{7}{4};\frac{1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{-\frac{1}{4}-m} (c+c \sin (e+f x))^{-1+m}}{3 f \sqrt{a-a \sin (e+f x)}}\\ \end{align*}
Mathematica [F] time = 180.135, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
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Maple [F] time = 0.286, size = 0, normalized size = 0. \begin{align*} \int{ \left ( c+c\sin \left ( fx+e \right ) \right ) ^{m} \left ( g\cos \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{a-a\sin \left ( fx+e \right ) }}}}\, 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 \frac{\left (g \cos \left (f x + e\right )\right )^{\frac{3}{2}}{\left (c \sin \left (f x + e\right ) + c\right )}^{m}}{\sqrt{-a \sin \left (f x + e\right ) + a}}\,{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 (-\frac{\sqrt{g \cos \left (f x + e\right )} \sqrt{-a \sin \left (f x + e\right ) + a}{\left (c \sin \left (f x + e\right ) + c\right )}^{m} g \cos \left (f x + e\right )}{a \sin \left (f x + e\right ) - a}, 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 \frac{\left (g \cos \left (f x + e\right )\right )^{\frac{3}{2}}{\left (c \sin \left (f x + e\right ) + c\right )}^{m}}{\sqrt{-a \sin \left (f x + e\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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