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 = 0.21, antiderivative size = 91, normalized size of antiderivative = 1.00, 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 69
Rule 70
Rule 2689
Rule 2840
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*}
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Mathematica [F] time = 169.60, size = 0, normalized size = 0.00 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x)) \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (c g \cos \left (f x + e\right ) \sin \left (f x + e\right ) - c g \cos \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (c \sin \left (f x + e\right ) - c\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.81, size = 0, normalized size = 0.00 \[ \int \left (g \cos \left (f x +e \right )\right )^{\frac {3}{2}} \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (c \sin \left (f x + e\right ) - c\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (c-c\,\sin \left (e+f\,x\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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