Optimal. Leaf size=43 \[ -\frac {\cos (e+f x) \sqrt {c-c \sin (e+f x)}}{c f \sqrt {a+a \sin (e+f x)}} \]
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Rubi [A]
time = 0.19, 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 = {2920, 2817}
\begin {gather*} -\frac {\cos (e+f x) \sqrt {c-c \sin (e+f x)}}{c f \sqrt {a \sin (e+f x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2817
Rule 2920
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 = 0.20, size = 44, normalized size = 1.02 \begin {gather*} \frac {\sin (2 (e+f x))}{2 f \sqrt {a (1+\sin (e+f x))} \sqrt {c-c \sin (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 42, normalized size = 0.98
method | result | size |
default | \(\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{f \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}}\) | \(42\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 53, normalized size = 1.23 \begin {gather*} \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 {gather*}
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 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.55, size = 58, normalized size = 1.35 \begin {gather*} -\frac {2 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}{\sqrt {a} \sqrt {c} f \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} \end {gather*}
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 \begin {gather*} -\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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