Optimal. Leaf size=41 \[ -\frac{c \cos (e+f x)}{f (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.0854564, antiderivative size = 41, 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} \[ -\frac{c \cos (e+f x)}{f (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2738
Rubi steps
\begin{align*} \int \frac{\sqrt{c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx &=-\frac{c \cos (e+f x)}{f (a+a \sin (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)}}\\ \end{align*}
Mathematica [B] time = 0.19293, size = 85, normalized size = 2.07 \[ -\frac{\sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)}}{a^2 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.161, size = 69, normalized size = 1.7 \begin{align*} -{\frac{\sin \left ( fx+e \right ) \left ( -1+\cos \left ( fx+e \right ) -\sin \left ( fx+e \right ) \right ) }{f \left ( -1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) \right ) }\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 \frac{\sqrt{-c \sin \left (f x + e\right ) + c}}{{\left (a \sin \left (f x + e\right ) + a\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 = 1.03902, size = 146, normalized size = 3.56 \begin{align*} -\frac{\sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c}}{a^{2} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a^{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- c \left (\sin{\left (e + f x \right )} - 1\right )}}{\left (a \left (\sin{\left (e + f x \right )} + 1\right )\right )^{\frac{3}{2}}}\, dx \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 \frac{\sqrt{-c \sin \left (f x + e\right ) + c}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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