Optimal. Leaf size=123 \[ \frac{2^{-m-\frac{3}{4}} (g \cos (e+f x))^{5/2} (1-\sin (e+f x))^{m-\frac{1}{4}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \, _2F_1\left (\frac{1}{4} (4 m+5),\frac{1}{4} (4 m+11);\frac{1}{4} (4 m+9);\frac{1}{2} (\sin (e+f x)+1)\right )}{c^2 f g (4 m+5)} \]
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Rubi [A] time = 0.369605, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2853, 2689, 70, 69} \[ \frac{2^{-m-\frac{3}{4}} (g \cos (e+f x))^{5/2} (1-\sin (e+f x))^{m-\frac{1}{4}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-1} \, _2F_1\left (\frac{1}{4} (4 m+5),\frac{1}{4} (4 m+11);\frac{1}{4} (4 m+9);\frac{1}{2} (\sin (e+f x)+1)\right )}{c^2 f g (4 m+5)} \]
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))^{-3-m} \, dx &=\left ((g \cos (e+f x))^{-2 m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^m\right ) \int (g \cos (e+f x))^{\frac{3}{2}+2 m} (c-c \sin (e+f x))^{-3-2 m} \, dx\\ &=\frac{\left (c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{\frac{1}{2} \left (-\frac{5}{2}-2 m\right )+m} (c+c \sin (e+f x))^{\frac{1}{2} \left (-\frac{5}{2}-2 m\right )}\right ) \operatorname{Subst}\left (\int (c-c x)^{-3-2 m+\frac{1}{2} \left (\frac{1}{2}+2 m\right )} (c+c x)^{\frac{1}{2} \left (\frac{1}{2}+2 m\right )} \, dx,x,\sin (e+f x)\right )}{f g}\\ &=\frac{\left (2^{-\frac{11}{4}-m} (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{\frac{1}{4}+\frac{1}{2} \left (-\frac{5}{2}-2 m\right )} \left (\frac{c-c \sin (e+f x)}{c}\right )^{-\frac{1}{4}+m} (c+c \sin (e+f x))^{\frac{1}{2} \left (-\frac{5}{2}-2 m\right )}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2}-\frac{x}{2}\right )^{-3-2 m+\frac{1}{2} \left (\frac{1}{2}+2 m\right )} (c+c x)^{\frac{1}{2} \left (\frac{1}{2}+2 m\right )} \, dx,x,\sin (e+f x)\right )}{c f g}\\ &=\frac{2^{-\frac{3}{4}-m} (g \cos (e+f x))^{5/2} \, _2F_1\left (\frac{1}{4} (5+4 m),\frac{1}{4} (11+4 m);\frac{1}{4} (9+4 m);\frac{1}{2} (1+\sin (e+f x))\right ) (1-\sin (e+f x))^{-\frac{1}{4}+m} (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{-1-m}}{c^2 f g (5+4 m)}\\ \end{align*}
Mathematica [B] time = 17.7088, size = 382, normalized size = 3.11 \[ \frac{2^{-m-4} \sec ^2\left (\frac{1}{4} \left (-e-f x+\frac{\pi }{2}\right )\right ) \sec (e+f x) (g \cos (e+f x))^{3/2} \sin ^{-2 m}\left (\frac{1}{2} \left (-e-f x+\frac{\pi }{2}\right )\right ) \left (1-\tan ^2\left (\frac{1}{4} \left (-e-f x+\frac{\pi }{2}\right )\right )\right )^{-2 m-\frac{1}{2}} (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^{-m-3} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{-2 (-m-3)} \left (\left (16 m^2+8 m-3\right ) \cot ^4\left (\frac{1}{4} \left (-e-f x+\frac{\pi }{2}\right )\right ) \, _2F_1\left (-2 m-\frac{3}{2},-m-\frac{7}{4};-m-\frac{3}{4};\tan ^2\left (\frac{1}{4} \left (-e-f x+\frac{\pi }{2}\right )\right )\right )+(4 m+7) \left ((4 m+3) \, _2F_1\left (-2 m-\frac{3}{2},\frac{1}{4}-m;\frac{5}{4}-m;\tan ^2\left (\frac{1}{4} \left (-e-f x+\frac{\pi }{2}\right )\right )\right )+2 (4 m-1) \cot ^2\left (\frac{1}{4} \left (-e-f x+\frac{\pi }{2}\right )\right ) \, _2F_1\left (-2 m-\frac{3}{2},-m-\frac{3}{4};\frac{1}{4}-m;\tan ^2\left (\frac{1}{4} \left (-e-f x+\frac{\pi }{2}\right )\right )\right )\right )\right )}{f (4 m-1) (4 m+3) (4 m+7)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.385, 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 ) ^{-3-m}\, 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 (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (-c \sin \left (f x + e\right ) + c\right )}^{-m - 3}\,{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 (\sqrt{g \cos \left (f x + e\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (-c \sin \left (f x + e\right ) + c\right )}^{-m - 3} g \cos \left (f x + e\right ), 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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