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 = 0.29, antiderivative size = 43, normalized size of antiderivative = 1.00, 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 2738
Rule 2841
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*}
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Mathematica [A] time = 0.30, 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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fricas [A] time = 0.47, size = 49, normalized size = 1.14 \[ \frac {\sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )}{a c f \cos \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (f x + e\right )^{2}}{\sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 42, normalized size = 0.98 \[ \frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{f \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (f x + e\right )^{2}}{\sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.87, size = 52, normalized size = 1.21 \[ -\frac {\sin \left (2\,e+2\,f\,x\right )\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}}{2\,c\,f\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\left (\sin \left (e+f\,x\right )-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos ^{2}{\left (e + f x \right )}}{\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \sqrt {- c \left (\sin {\left (e + f x \right )} - 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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