Optimal. Leaf size=123 \[ \frac{(A+C) \cos (e+f x) (a \sin (e+f x)+a)^m \, _2F_1\left (1,m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1)\right )}{f (2 m+1) \sqrt{c-c \sin (e+f x)}}-\frac{2 C \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (2 m+3) \sqrt{c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.292738, antiderivative size = 123, 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 = {3038, 2745, 2667, 68} \[ \frac{(A+C) \cos (e+f x) (a \sin (e+f x)+a)^m \, _2F_1\left (1,m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1)\right )}{f (2 m+1) \sqrt{c-c \sin (e+f x)}}-\frac{2 C \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (2 m+3) \sqrt{c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3038
Rule 2745
Rule 2667
Rule 68
Rubi steps
\begin{align*} \int \frac{(a+a \sin (e+f x))^m \left (A+C \sin ^2(e+f x)\right )}{\sqrt{c-c \sin (e+f x)}} \, dx &=-\frac{2 C \cos (e+f x) (a+a \sin (e+f x))^{1+m}}{a f (3+2 m) \sqrt{c-c \sin (e+f x)}}+(A+C) \int \frac{(a+a \sin (e+f x))^m}{\sqrt{c-c \sin (e+f x)}} \, dx\\ &=-\frac{2 C \cos (e+f x) (a+a \sin (e+f x))^{1+m}}{a f (3+2 m) \sqrt{c-c \sin (e+f x)}}+\frac{((A+C) \cos (e+f x)) \int \sec (e+f x) (a+a \sin (e+f x))^{\frac{1}{2}+m} \, dx}{\sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=-\frac{2 C \cos (e+f x) (a+a \sin (e+f x))^{1+m}}{a f (3+2 m) \sqrt{c-c \sin (e+f x)}}+\frac{(a (A+C) \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 )}{f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{(A+C) \cos (e+f x) \, _2F_1\left (1,\frac{1}{2}+m;\frac{3}{2}+m;\frac{1}{2} (1+\sin (e+f x))\right ) (a+a \sin (e+f x))^m}{f (1+2 m) \sqrt{c-c \sin (e+f x)}}-\frac{2 C \cos (e+f x) (a+a \sin (e+f x))^{1+m}}{a f (3+2 m) \sqrt{c-c \sin (e+f x)}}\\ \end{align*}
Mathematica [F] time = 180.002, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
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Maple [F] time = 0.735, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( A+C \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ){\frac{1}{\sqrt{c-c\sin \left ( fx+e \right ) }}}}\, 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 (C \sin \left (f x + e\right )^{2} + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{\sqrt{-c \sin \left (f x + e\right ) + c}}\,{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{{\left (C \cos \left (f x + e\right )^{2} - A - C\right )} \sqrt{-c \sin \left (f x + e\right ) + c}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{c \sin \left (f x + e\right ) - c}, 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 (C \sin \left (f x + e\right )^{2} + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{\sqrt{-c \sin \left (f x + e\right ) + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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