Optimal. Leaf size=73 \[ \frac{2 a c^2 (5 A+B) \cos ^3(e+f x)}{15 f (c-c \sin (e+f x))^{3/2}}-\frac{2 a B c \cos ^3(e+f x)}{5 f \sqrt{c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.239825, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2967, 2856, 2673} \[ \frac{2 a c^2 (5 A+B) \cos ^3(e+f x)}{15 f (c-c \sin (e+f x))^{3/2}}-\frac{2 a B c \cos ^3(e+f x)}{5 f \sqrt{c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2967
Rule 2856
Rule 2673
Rubi steps
\begin{align*} \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) \sqrt{c-c \sin (e+f x)} \, dx &=(a c) \int \frac{\cos ^2(e+f x) (A+B \sin (e+f x))}{\sqrt{c-c \sin (e+f x)}} \, dx\\ &=-\frac{2 a B c \cos ^3(e+f x)}{5 f \sqrt{c-c \sin (e+f x)}}+\frac{1}{5} (a (5 A+B) c) \int \frac{\cos ^2(e+f x)}{\sqrt{c-c \sin (e+f x)}} \, dx\\ &=\frac{2 a (5 A+B) c^2 \cos ^3(e+f x)}{15 f (c-c \sin (e+f x))^{3/2}}-\frac{2 a B c \cos ^3(e+f x)}{5 f \sqrt{c-c \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.423134, size = 87, normalized size = 1.19 \[ \frac{2 a \sqrt{c-c \sin (e+f x)} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3 (5 A+3 B \sin (e+f x)-2 B)}{15 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.963, size = 63, normalized size = 0.9 \begin{align*} -{\frac{ \left ( -2+2\,\sin \left ( fx+e \right ) \right ) c \left ( 1+\sin \left ( fx+e \right ) \right ) ^{2}a \left ( 3\,B\sin \left ( fx+e \right ) +5\,A-2\,B \right ) }{15\,f\cos \left ( fx+e \right ) }{\frac{1}{\sqrt{c-c\sin \left ( fx+e \right ) }}}} \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 (B \sin \left (f x + e\right ) + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )} \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 [A] time = 1.66284, size = 336, normalized size = 4.6 \begin{align*} -\frac{2 \,{\left (3 \, B a \cos \left (f x + e\right )^{3} +{\left (5 \, A + 4 \, B\right )} a \cos \left (f x + e\right )^{2} -{\left (5 \, A + B\right )} a \cos \left (f x + e\right ) - 2 \,{\left (5 \, A + B\right )} a +{\left (3 \, B a \cos \left (f x + e\right )^{2} -{\left (5 \, A + B\right )} a \cos \left (f x + e\right ) - 2 \,{\left (5 \, A + B\right )} a\right )} \sin \left (f x + e\right )\right )} \sqrt{-c \sin \left (f x + e\right ) + c}}{15 \,{\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\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*} a \left (\int A \sqrt{- c \sin{\left (e + f x \right )} + c}\, dx + \int A \sqrt{- c \sin{\left (e + f x \right )} + c} \sin{\left (e + f x \right )}\, dx + \int B \sqrt{- c \sin{\left (e + f x \right )} + c} \sin{\left (e + f x \right )}\, dx + \int B \sqrt{- c \sin{\left (e + f x \right )} + c} \sin ^{2}{\left (e + f x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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