Optimal. Leaf size=69 \[ \frac{2 a c^2 \cos ^3(e+f x)}{5 f \sqrt{c-c \sin (e+f x)}}+\frac{8 a c^3 \cos ^3(e+f x)}{15 f (c-c \sin (e+f x))^{3/2}} \]
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Rubi [A] time = 0.157268, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {2736, 2674, 2673} \[ \frac{2 a c^2 \cos ^3(e+f x)}{5 f \sqrt{c-c \sin (e+f x)}}+\frac{8 a c^3 \cos ^3(e+f x)}{15 f (c-c \sin (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2736
Rule 2674
Rule 2673
Rubi steps
\begin{align*} \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx &=(a c) \int \cos ^2(e+f x) \sqrt{c-c \sin (e+f x)} \, dx\\ &=\frac{2 a c^2 \cos ^3(e+f x)}{5 f \sqrt{c-c \sin (e+f x)}}+\frac{1}{5} \left (4 a c^2\right ) \int \frac{\cos ^2(e+f x)}{\sqrt{c-c \sin (e+f x)}} \, dx\\ &=\frac{8 a c^3 \cos ^3(e+f x)}{15 f (c-c \sin (e+f x))^{3/2}}+\frac{2 a c^2 \cos ^3(e+f x)}{5 f \sqrt{c-c \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.336405, size = 82, normalized size = 1.19 \[ -\frac{2 a c (3 \sin (e+f x)-7) \sqrt{c-c \sin (e+f x)} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3}{15 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.506, size = 59, normalized size = 0.9 \begin{align*}{\frac{ \left ( -2+2\,\sin \left ( fx+e \right ) \right ){c}^{2} \left ( 1+\sin \left ( fx+e \right ) \right ) ^{2}a \left ( 3\,\sin \left ( fx+e \right ) -7 \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 (a \sin \left (f x + e\right ) + a\right )}{\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 [A] time = 1.07368, size = 284, normalized size = 4.12 \begin{align*} \frac{2 \,{\left (3 \, a c \cos \left (f x + e\right )^{3} - a c \cos \left (f x + e\right )^{2} + 4 \, a c \cos \left (f x + e\right ) + 8 \, a c +{\left (3 \, a c \cos \left (f x + e\right )^{2} + 4 \, a c \cos \left (f x + e\right ) + 8 \, a c\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 c \sqrt{- c \sin{\left (e + f x \right )} + c}\, dx + \int - c \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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