Optimal. Leaf size=45 \[ -\frac{\cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 c f \sqrt{a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.283838, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2841, 2738} \[ -\frac{\cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 c f \sqrt{a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 2841
Rule 2738
Rubi steps
\begin{align*} \int \frac{\cos ^2(e+f x) \sqrt{c-c \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx &=\frac{\int \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx}{a c}\\ &=-\frac{\cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 c f \sqrt{a+a \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.279765, size = 62, normalized size = 1.38 \[ \frac{\sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (4 \sin (e+f x)+\cos (2 (e+f x)))}{4 a f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.211, size = 90, normalized size = 2. \begin{align*}{\frac{ \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{2}+\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) -2\,\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) +1 \right ) \sin \left ( fx+e \right ) }{2\,f \left ( -1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) \right ) }\sqrt{-c \left ( -1+\sin \left ( fx+e \right ) \right ) }{\frac{1}{\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 [B] time = 1.857, size = 524, normalized size = 11.64 \begin{align*} -\frac{\frac{2 \, \sqrt{a} \sqrt{c} + \frac{\sqrt{a} \sqrt{c} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{\sqrt{a} \sqrt{c} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{\sqrt{a} \sqrt{c} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a + \frac{2 \, a \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{a \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}} - \frac{2 \, \sqrt{a} \sqrt{c} - \frac{\sqrt{a} \sqrt{c} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{3 \, \sqrt{a} \sqrt{c} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{\sqrt{a} \sqrt{c} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a + \frac{2 \, a \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{a \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}} + \frac{2 \,{\left (\frac{\sqrt{a} \sqrt{c} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac{\sqrt{a} \sqrt{c} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{\sqrt{a} \sqrt{c} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a + \frac{2 \, a \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{a \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62901, size = 153, normalized size = 3.4 \begin{align*} \frac{{\left (\cos \left (f x + e\right )^{2} + 2 \, \sin \left (f x + e\right ) - 1\right )} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c}}{2 \, a 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 )} \cos ^{2}{\left (e + f x \right )}}{\sqrt{a \left (\sin{\left (e + f x \right )} + 1\right )}}\, 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} \cos \left (f x + e\right )^{2}}{\sqrt{a \sin \left (f x + e\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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