Optimal. Leaf size=93 \[ \frac {g^3 2^{m+\frac {9}{4}} (\sin (e+f x)+1)^{-m-\frac {1}{4}} (a \sin (e+f x)+a)^{m+1} \, _2F_1\left (-\frac {3}{4},-m-\frac {1}{4};\frac {1}{4};\frac {1}{2} (1-\sin (e+f x))\right )}{3 a c^2 f (g \cos (e+f x))^{3/2}} \]
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Rubi [A] time = 0.28, antiderivative size = 93, normalized size of antiderivative = 1.00, 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 = {2840, 2689, 70, 69} \[ \frac {g^3 2^{m+\frac {9}{4}} (\sin (e+f x)+1)^{-m-\frac {1}{4}} (a \sin (e+f x)+a)^{m+1} \, _2F_1\left (-\frac {3}{4},-m-\frac {1}{4};\frac {1}{4};\frac {1}{2} (1-\sin (e+f x))\right )}{3 a c^2 f (g \cos (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 2689
Rule 2840
Rubi steps
\begin {align*} \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^2} \, dx &=\frac {g^4 \int \frac {(a+a \sin (e+f x))^{2+m}}{(g \cos (e+f x))^{5/2}} \, dx}{a^2 c^2}\\ &=\frac {\left (g^3 (a-a \sin (e+f x))^{3/4} (a+a \sin (e+f x))^{3/4}\right ) \operatorname {Subst}\left (\int \frac {(a+a x)^{\frac {1}{4}+m}}{(a-a x)^{7/4}} \, dx,x,\sin (e+f x)\right )}{c^2 f (g \cos (e+f x))^{3/2}}\\ &=\frac {\left (2^{\frac {1}{4}+m} g^3 (a-a \sin (e+f x))^{3/4} (a+a \sin (e+f x))^{1+m} \left (\frac {a+a \sin (e+f x)}{a}\right )^{-\frac {1}{4}-m}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {1}{2}+\frac {x}{2}\right )^{\frac {1}{4}+m}}{(a-a x)^{7/4}} \, dx,x,\sin (e+f x)\right )}{c^2 f (g \cos (e+f x))^{3/2}}\\ &=\frac {2^{\frac {9}{4}+m} g^3 \, _2F_1\left (-\frac {3}{4},-\frac {1}{4}-m;\frac {1}{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))^{1+m}}{3 a c^2 f (g \cos (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 96, normalized size = 1.03 \[ -\frac {g 2^{m+\frac {9}{4}} \sqrt {g \cos (e+f x)} (\sin (e+f x)+1)^{-m-\frac {1}{4}} (a (\sin (e+f x)+1))^m \, _2F_1\left (-\frac {3}{4},-m-\frac {1}{4};\frac {1}{4};\frac {1}{2} (1-\sin (e+f x))\right )}{3 c^2 f (\sin (e+f x)-1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {g \cos \left (f x + e\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} g \cos \left (f x + e\right )}{c^{2} \cos \left (f x + e\right )^{2} + 2 \, c^{2} \sin \left (f x + e\right ) - 2 \, c^{2}}, x\right ) \]
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 \[ \int \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (c \sin \left (f x + e\right ) - c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.36, size = 0, normalized size = 0.00 \[ \int \frac {\left (g \cos \left (f x +e \right )\right )^{\frac {3}{2}} \left (a +a \sin \left (f x +e \right )\right )^{m}}{\left (c -c \sin \left (f x +e \right )\right )^{2}}\, dx \]
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 \[ \int \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (c \sin \left (f x + e\right ) - c\right )}^{2}}\,{d x} \]
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 \[ \int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]
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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