Optimal. Leaf size=118 \[ \frac{(A+B) \cos (e+f x) (c \sin (e+f x)+c)^m \, _2F_1\left (1,m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1)\right )}{f (2 m+1) \sqrt{a-a \sin (e+f x)}}-\frac{2 B \cos (e+f x) (c \sin (e+f x)+c)^m}{f (2 m+1) \sqrt{a-a \sin (e+f x)}} \]
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Rubi [A] time = 0.288117, antiderivative size = 118, normalized size of antiderivative = 1., 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 = {2973, 2745, 2667, 68} \[ \frac{(A+B) \cos (e+f x) (c \sin (e+f x)+c)^m \, _2F_1\left (1,m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1)\right )}{f (2 m+1) \sqrt{a-a \sin (e+f x)}}-\frac{2 B \cos (e+f x) (c \sin (e+f x)+c)^m}{f (2 m+1) \sqrt{a-a \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2973
Rule 2745
Rule 2667
Rule 68
Rubi steps
\begin{align*} \int \frac{(A+B \sin (e+f x)) (c+c \sin (e+f x))^m}{\sqrt{a-a \sin (e+f x)}} \, dx &=-\frac{2 B \cos (e+f x) (c+c \sin (e+f x))^m}{f (1+2 m) \sqrt{a-a \sin (e+f x)}}-(-A-B) \int \frac{(c+c \sin (e+f x))^m}{\sqrt{a-a \sin (e+f x)}} \, dx\\ &=-\frac{2 B \cos (e+f x) (c+c \sin (e+f x))^m}{f (1+2 m) \sqrt{a-a \sin (e+f x)}}-\frac{((-A-B) \cos (e+f x)) \int \sec (e+f x) (c+c \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 B \cos (e+f x) (c+c \sin (e+f x))^m}{f (1+2 m) \sqrt{a-a \sin (e+f x)}}-\frac{((-A-B) c \cos (e+f x)) \operatorname{Subst}\left (\int \frac{(c+x)^{-\frac{1}{2}+m}}{c-x} \, dx,x,c \sin (e+f x)\right )}{f \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}\\ &=-\frac{2 B \cos (e+f x) (c+c \sin (e+f x))^m}{f (1+2 m) \sqrt{a-a \sin (e+f x)}}+\frac{(A+B) \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 ) (c+c \sin (e+f x))^m}{f (1+2 m) \sqrt{a-a \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 2.35267, size = 200, normalized size = 1.69 \[ \frac{2^{-2 m-\frac{3}{2}} \sin \left (\frac{1}{4} (2 e+2 f x+\pi )\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) (c (\sin (e+f x)+1))^m \left (2^{2 m+1} (A+B) \, _2F_1\left (1,2 m+1;2 (m+1);\sin \left (\frac{1}{4} (2 e+2 f x+\pi )\right )\right )+(A+B) \sec ^2\left (\frac{1}{8} (2 e+2 f x-\pi )\right )^{2 m+1} \, _2F_1\left (2 m+1,2 m+1;2 (m+1);\frac{1}{2} \left (1-\tan ^2\left (\frac{1}{8} (2 e+2 f x-\pi )\right )\right )\right )-B 2^{2 m+3}\right )}{(2 f m+f) \sqrt{a-a \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.312, size = 0, normalized size = 0. \begin{align*} \int{ \left ( A+B\sin \left ( fx+e \right ) \right ) \left ( c+c\sin \left ( fx+e \right ) \right ) ^{m}{\frac{1}{\sqrt{a-a\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 (B \sin \left (f x + e\right ) + A\right )}{\left (c \sin \left (f x + e\right ) + c\right )}^{m}}{\sqrt{-a \sin \left (f x + e\right ) + a}}\,{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 (B \sin \left (f x + e\right ) + A\right )} \sqrt{-a \sin \left (f x + e\right ) + a}{\left (c \sin \left (f x + e\right ) + c\right )}^{m}}{a \sin \left (f x + e\right ) - a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c \left (\sin{\left (e + f x \right )} + 1\right )\right )^{m} \left (A + B \sin{\left (e + f x \right )}\right )}{\sqrt{- a \left (\sin{\left (e + f x \right )} - 1\right )}}\, dx \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 (B \sin \left (f x + e\right ) + A\right )}{\left (c \sin \left (f x + e\right ) + c\right )}^{m}}{\sqrt{-a \sin \left (f x + e\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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