Optimal. Leaf size=89 \[ -\frac{a^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{a \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}{3 f} \]
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Rubi [A] time = 0.178479, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2740, 2738} \[ -\frac{a^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{a \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}{3 f} \]
Antiderivative was successfully verified.
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Rule 2740
Rule 2738
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2} \, dx &=-\frac{a \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}{3 f}+\frac{1}{3} (2 a) \int \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx\\ &=-\frac{a^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{3 f \sqrt{a+a \sin (e+f x)}}-\frac{a \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}{3 f}\\ \end{align*}
Mathematica [A] time = 0.429396, size = 70, normalized size = 0.79 \[ -\frac{c (\sin (e+f x)-1) (9 \sin (e+f x)+\sin (3 (e+f x))) \sec ^3(e+f x) (a (\sin (e+f x)+1))^{3/2} \sqrt{c-c \sin (e+f x)}}{12 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.141, size = 55, normalized size = 0.6 \begin{align*}{\frac{ \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{2}+2 \right ) \sin \left ( fx+e \right ) }{3\,f \left ( \cos \left ( fx+e \right ) \right ) ^{3}} \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{3}{2}}} \left ( a \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{3}{2}}}} \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 )}^{\frac{3}{2}}{\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 = 1.44806, size = 155, normalized size = 1.74 \begin{align*} \frac{{\left (a c \cos \left (f x + e\right )^{2} + 2 \, a c\right )} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )}{3 \, 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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