Optimal. Leaf size=43 \[ -\frac{a \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt{a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.0805129, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033, Rules used = {2738} \[ -\frac{a \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt{a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 2738
Rubi steps
\begin{align*} \int \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx &=-\frac{a \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt{a+a \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.21214, size = 60, normalized size = 1.4 \[ \frac{c \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (4 \sin (e+f x)+\cos (2 (e+f x)))}{4 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.176, size = 61, normalized size = 1.4 \begin{align*}{\frac{\sin \left ( fx+e \right ) \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{2}+\sin \left ( fx+e \right ) +1 \right ) }{2\,f \left ( \cos \left ( fx+e \right ) \right ) ^{3}} \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{3}{2}}}\sqrt{a \left ( 1+\sin \left ( fx+e \right ) \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 \sqrt{a \sin \left (f x + e\right ) + a}{\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 = 1.09567, size = 155, normalized size = 3.6 \begin{align*} \frac{{\left (c \cos \left (f x + e\right )^{2} + 2 \, c \sin \left (f x + e\right ) - c\right )} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c}}{2 \, f \cos \left (f x + e\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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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