Optimal. Leaf size=114 \[ -\frac{a^3 c^2 2^{m+\frac{9}{4}} \sec (e+f x) \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{15/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m-3} \, _2F_1\left (\frac{15}{4},-m-\frac{1}{4};\frac{19}{4};\frac{1}{2} (1-\sin (e+f x))\right )}{15 f g^6} \]
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Rubi [A] time = 0.356974, antiderivative size = 114, 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} \[ -\frac{a^3 c^2 2^{m+\frac{9}{4}} \sec (e+f x) \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{15/2} (\sin (e+f x)+1)^{-m-\frac{1}{4}} (a \sin (e+f x)+a)^{m-3} \, _2F_1\left (\frac{15}{4},-m-\frac{1}{4};\frac{19}{4};\frac{1}{2} (1-\sin (e+f x))\right )}{15 f g^6} \]
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))^{5/2} \, dx &=\frac{\left (a^2 c^2 \sec (e+f x) \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}\right ) \int (g \cos (e+f x))^{13/2} (a+a \sin (e+f x))^{-\frac{5}{2}+m} \, dx}{g^5}\\ &=\frac{\left (a^4 c^2 (g \cos (e+f x))^{15/2} \sec (e+f x) \sqrt{c-c \sin (e+f x)}\right ) \operatorname{Subst}\left (\int (a-a x)^{11/4} (a+a x)^{\frac{1}{4}+m} \, dx,x,\sin (e+f x)\right )}{f g^6 (a-a \sin (e+f x))^{15/4} (a+a \sin (e+f x))^{13/4}}\\ &=\frac{\left (2^{\frac{1}{4}+m} a^4 c^2 (g \cos (e+f x))^{15/2} \sec (e+f x) (a+a \sin (e+f x))^{-3+m} \left (\frac{a+a \sin (e+f x)}{a}\right )^{-\frac{1}{4}-m} \sqrt{c-c \sin (e+f x)}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2}+\frac{x}{2}\right )^{\frac{1}{4}+m} (a-a x)^{11/4} \, dx,x,\sin (e+f x)\right )}{f g^6 (a-a \sin (e+f x))^{15/4}}\\ &=-\frac{2^{\frac{9}{4}+m} a^3 c^2 (g \cos (e+f x))^{15/2} \, _2F_1\left (\frac{15}{4},-\frac{1}{4}-m;\frac{19}{4};\frac{1}{2} (1-\sin (e+f x))\right ) \sec (e+f x) (1+\sin (e+f x))^{-\frac{1}{4}-m} (a+a \sin (e+f x))^{-3+m} \sqrt{c-c \sin (e+f x)}}{15 f g^6}\\ \end{align*}
Mathematica [F] time = 180.037, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
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Maple [F] time = 0.243, 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 ) ^{{\frac{5}{2}}}\, 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 )}^{\frac{5}{2}}{\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^{2} g \cos \left (f x + e\right )^{3} + 2 \, c^{2} g \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, c^{2} g \cos \left (f x + e\right )\right )} \sqrt{g \cos \left (f x + e\right )} \sqrt{-c \sin \left (f x + e\right ) + c}{\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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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