Optimal. Leaf size=43 \[ \frac{c \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt{c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.0818158, 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} \[ \frac{c \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt{c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2738
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)} \, dx &=\frac{c \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 f \sqrt{c-c \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.203004, size = 60, normalized size = 1.4 \[ -\frac{a \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (\cos (2 (e+f x))-4 \sin (e+f x))}{4 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.158, size = 63, normalized size = 1.5 \begin{align*} -{\frac{\sin \left ( fx+e \right ) \left ( -1- \left ( \cos \left ( fx+e \right ) \right ) ^{2}+\sin \left ( fx+e \right ) \right ) }{2\,f \left ( \cos \left ( fx+e \right ) \right ) ^{3}}\sqrt{-c \left ( -1+\sin \left ( fx+e \right ) \right ) } \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}} \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 = 1.35472, size = 157, normalized size = 3.65 \begin{align*} -\frac{{\left (a \cos \left (f x + e\right )^{2} - 2 \, a \sin \left (f x + e\right ) - a\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}} \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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