Optimal. Leaf size=43 \[ -\frac{\cos (e+f x) \sqrt{c-c \sin (e+f x)}}{c f \sqrt{a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.288419, antiderivative size = 43, 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} \[ -\frac{\cos (e+f x) \sqrt{c-c \sin (e+f x)}}{c f \sqrt{a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 2841
Rule 2738
Rubi steps
\begin{align*} \int \frac{\cos ^2(e+f x)}{\sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}} \, dx &=\frac{\int \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)} \, dx}{a c}\\ &=-\frac{\cos (e+f x) \sqrt{c-c \sin (e+f x)}}{c f \sqrt{a+a \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.296315, size = 44, normalized size = 1.02 \[ \frac{\sin (2 (e+f x))}{2 f \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.21, size = 42, normalized size = 1. \begin{align*}{\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }{f}{\frac{1}{\sqrt{a \left ( 1+\sin \left ( fx+e \right ) \right ) }}}{\frac{1}{\sqrt{-c \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 \frac{\cos \left (f x + e\right )^{2}}{\sqrt{a \sin \left (f x + e\right ) + a} \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 = 1.66291, size = 116, normalized size = 2.7 \begin{align*} \frac{\sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )}{a c 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{\cos ^{2}{\left (e + f x \right )}}{\sqrt{a \left (\sin{\left (e + f x \right )} + 1\right )} \sqrt{- c \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{\cos \left (f x + e\right )^{2}}{\sqrt{a \sin \left (f x + e\right ) + a} \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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