Optimal. Leaf size=91 \[ -\frac{a^2 c 2^{m+\frac{9}{4}} (g \cos (e+f x))^{9/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m-2} \, _2F_1\left (\frac{9}{4},-m-\frac{1}{4};\frac{13}{4};\frac{1}{2} (1-\sin (e+f x))\right )}{9 f g^3} \]
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Rubi [A] time = 0.207681, antiderivative size = 91, 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 = {2840, 2689, 70, 69} \[ -\frac{a^2 c 2^{m+\frac{9}{4}} (g \cos (e+f x))^{9/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m-2} \, _2F_1\left (\frac{9}{4},-m-\frac{1}{4};\frac{13}{4};\frac{1}{2} (1-\sin (e+f x))\right )}{9 f g^3} \]
Antiderivative was successfully verified.
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Rule 2840
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)) \, dx &=\frac{(a c) \int (g \cos (e+f x))^{7/2} (a+a \sin (e+f x))^{-1+m} \, dx}{g^2}\\ &=\frac{\left (a^3 c (g \cos (e+f x))^{9/2}\right ) \operatorname{Subst}\left (\int (a-a x)^{5/4} (a+a x)^{\frac{1}{4}+m} \, dx,x,\sin (e+f x)\right )}{f g^3 (a-a \sin (e+f x))^{9/4} (a+a \sin (e+f x))^{9/4}}\\ &=\frac{\left (2^{\frac{1}{4}+m} a^3 c (g \cos (e+f x))^{9/2} (a+a \sin (e+f x))^{-2+m} \left (\frac{a+a \sin (e+f x)}{a}\right )^{-\frac{1}{4}-m}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2}+\frac{x}{2}\right )^{\frac{1}{4}+m} (a-a x)^{5/4} \, dx,x,\sin (e+f x)\right )}{f g^3 (a-a \sin (e+f x))^{9/4}}\\ &=-\frac{2^{\frac{9}{4}+m} a^2 c (g \cos (e+f x))^{9/2} \, _2F_1\left (\frac{9}{4},-\frac{1}{4}-m;\frac{13}{4};\frac{1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{-\frac{1}{4}-m} (a+a \sin (e+f x))^{-2+m}}{9 f g^3}\\ \end{align*}
Mathematica [F] time = 168.115, size = 0, normalized size = 0. \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x)) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.283, 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 ) \, 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 (c \sin \left (f x + e\right ) - c\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{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 (c g \cos \left (f x + e\right ) \sin \left (f x + e\right ) - c g \cos \left (f x + e\right )\right )} \sqrt{g \cos \left (f x + e\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}, 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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