Optimal. Leaf size=76 \[ \frac{\cos (e+f x) (a \sin (e+f x)+a)^{m+1} \, _2F_1\left (1,m+\frac{3}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1)\right )}{a c f (2 m+3) \sqrt{c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.361743, antiderivative size = 76, 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 = {2841, 2745, 2667, 68} \[ \frac{\cos (e+f x) (a \sin (e+f x)+a)^{m+1} \, _2F_1\left (1,m+\frac{3}{2};m+\frac{5}{2};\frac{1}{2} (\sin (e+f x)+1)\right )}{a c f (2 m+3) \sqrt{c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2841
Rule 2745
Rule 2667
Rule 68
Rubi steps
\begin{align*} \int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^{3/2}} \, dx &=\frac{\int \frac{(a+a \sin (e+f x))^{1+m}}{\sqrt{c-c \sin (e+f x)}} \, dx}{a c}\\ &=\frac{\cos (e+f x) \int \sec (e+f x) (a+a \sin (e+f x))^{\frac{3}{2}+m} \, dx}{a c \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{\cos (e+f x) \operatorname{Subst}\left (\int \frac{(a+x)^{\frac{1}{2}+m}}{a-x} \, dx,x,a \sin (e+f x)\right )}{c f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{\cos (e+f x) \, _2F_1\left (1,\frac{3}{2}+m;\frac{5}{2}+m;\frac{1}{2} (1+\sin (e+f x))\right ) (a+a \sin (e+f x))^{1+m}}{a c f (3+2 m) \sqrt{c-c \sin (e+f x)}}\\ \end{align*}
Mathematica [B] time = 6.50331, size = 218, normalized size = 2.87 \[ -\frac{2^{-2 m-\frac{5}{2}} \cos ^2\left (\frac{1}{2} \left (-e-f x+\frac{\pi }{2}\right )\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^3 (a \sin (e+f x)+a)^m \left (\sec ^4\left (\frac{1}{4} \left (-e-f x+\frac{\pi }{2}\right )\right ) \sec ^2\left (\frac{1}{4} \left (-e-f x+\frac{\pi }{2}\right )\right )^{2 m} \, _2F_1\left (2 m+2,2 m+2;2 m+3;\frac{1}{2} \left (1-\tan ^2\left (\frac{1}{4} \left (-e-f x+\frac{\pi }{2}\right )\right )\right )\right )-4^{m+1} \, _2F_1\left (1,2 m+2;2 m+3;\cos \left (\frac{1}{2} \left (-e-f x+\frac{\pi }{2}\right )\right )\right )\right )}{f (m+1) (c-c \sin (e+f x))^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.376, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \cos \left ( fx+e \right ) \right ) ^{2} \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( c-c\sin \left ( fx+e \right ) \right ) ^{-{\frac{3}{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 \frac{{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}\,{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 (-\frac{\sqrt{-c \sin \left (f x + e\right ) + c}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2}}{c^{2} \cos \left (f x + e\right )^{2} + 2 \, c^{2} \sin \left (f x + e\right ) - 2 \, c^{2}}, 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*} \int \frac{{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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