Optimal. Leaf size=31 \[ \frac {1}{2} \log \left (\sin (x)+\sqrt {\sin (2 x)}+\cos (x)\right )-\frac {1}{2} \sin ^{-1}(\cos (x)-\sin (x)) \]
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Rubi [A] time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4305} \[ \frac {1}{2} \log \left (\sin (x)+\sqrt {\sin (2 x)}+\cos (x)\right )-\frac {1}{2} \sin ^{-1}(\cos (x)-\sin (x)) \]
Antiderivative was successfully verified.
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Rule 4305
Rubi steps
\begin {align*} \int \frac {\cos (x)}{\sqrt {\sin (2 x)}} \, dx &=-\frac {1}{2} \sin ^{-1}(\cos (x)-\sin (x))+\frac {1}{2} \log \left (\cos (x)+\sin (x)+\sqrt {\sin (2 x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 29, normalized size = 0.94 \[ \frac {1}{2} \left (\log \left (\sin (x)+\sqrt {\sin (2 x)}+\cos (x)\right )-\sin ^{-1}(\cos (x)-\sin (x))\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 137, normalized size = 4.42 \[ \frac {1}{4} \, \arctan \left (-\frac {\sqrt {2} \sqrt {\cos \relax (x) \sin \relax (x)} {\left (\cos \relax (x) - \sin \relax (x)\right )} + \cos \relax (x) \sin \relax (x)}{\cos \relax (x)^{2} + 2 \, \cos \relax (x) \sin \relax (x) - 1}\right ) - \frac {1}{4} \, \arctan \left (-\frac {2 \, \sqrt {2} \sqrt {\cos \relax (x) \sin \relax (x)} - \cos \relax (x) - \sin \relax (x)}{\cos \relax (x) - \sin \relax (x)}\right ) - \frac {1}{8} \, \log \left (-32 \, \cos \relax (x)^{4} + 4 \, \sqrt {2} {\left (4 \, \cos \relax (x)^{3} - {\left (4 \, \cos \relax (x)^{2} + 1\right )} \sin \relax (x) - 5 \, \cos \relax (x)\right )} \sqrt {\cos \relax (x) \sin \relax (x)} + 32 \, \cos \relax (x)^{2} + 16 \, \cos \relax (x) \sin \relax (x) + 1\right ) \]
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 \[ \int \frac {\cos \relax (x)}{\sqrt {\sin \left (2 \, x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.81, size = 98, normalized size = 3.16 \[ \frac {\sqrt {-\frac {\tan \left (\frac {x}{2}\right )}{\tan ^{2}\left (\frac {x}{2}\right )-1}}\, \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right ) \sqrt {\tan \left (\frac {x}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \EllipticF \left (\sqrt {\tan \left (\frac {x}{2}\right )+1}, \frac {\sqrt {2}}{2}\right )}{\sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}} \]
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 \[ \int \frac {\cos \relax (x)}{\sqrt {\sin \left (2 \, x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\cos \relax (x)}{\sqrt {\sin \left (2\,x\right )}} \,d x \]
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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