Optimal. Leaf size=43 \[ -\frac {\cos (e+f x)}{a f (a \sin (e+f x)+a)^{3/2} \sqrt {c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.32, 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} \[ -\frac {\cos (e+f x)}{a f (a \sin (e+f x)+a)^{3/2} \sqrt {c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2738
Rule 2841
Rubi steps
\begin {align*} \int \frac {\cos ^2(e+f x)}{(a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}} \, dx &=\frac {\int \frac {\sqrt {c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx}{a c}\\ &=-\frac {\cos (e+f x)}{a f (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.52, size = 80, normalized size = 1.86 \[ -\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3}{f (a (\sin (e+f x)+1))^{5/2} \sqrt {c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 60, normalized size = 1.40 \[ -\frac {\sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{a^{3} c f \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a^{3} c f \cos \left (f x + e\right )} \]
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 \[ \int \frac {\cos \left (f x + e\right )^{2}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} \sqrt {-c \sin \left (f x + e\right ) + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 50, normalized size = 1.16 \[ \frac {\left (1+\sin \left (f x +e \right )\right ) \cos \left (f x +e \right ) \sin \left (f x +e \right )}{f \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {5}{2}} \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}} \]
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 \[ \int \frac {\cos \left (f x + e\right )^{2}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} \sqrt {-c \sin \left (f x + e\right ) + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.26, size = 55, normalized size = 1.28 \[ -\frac {2\,\cos \left (e+f\,x\right )\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}}{a^2\,c\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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