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 = 0.0129827, antiderivative size = 31, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0248818, 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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Maple [C] time = 0.046, size = 98, normalized size = 3.2 \begin{align*}{\sqrt{-{\tan \left ({\frac{x}{2}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}-1 \right ) ^{-1}}} \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}-1 \right ) \sqrt{\tan \left ({\frac{x}{2}} \right ) +1}\sqrt{-2\,\tan \left ( x/2 \right ) +2}\sqrt{-\tan \left ({\frac{x}{2}} \right ) }{\it EllipticF} \left ( \sqrt{\tan \left ({\frac{x}{2}} \right ) +1},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{\tan \left ({\frac{x}{2}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}-1 \right ) }}}{\frac{1}{\sqrt{ \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3}-\tan \left ({\frac{x}{2}} \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 (x\right )}{\sqrt{\sin \left (2 \, x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.507912, size = 455, normalized size = 14.68 \begin{align*} \frac{1}{4} \, \arctan \left (-\frac{\sqrt{2} \sqrt{\cos \left (x\right ) \sin \left (x\right )}{\left (\cos \left (x\right ) - \sin \left (x\right )\right )} + \cos \left (x\right ) \sin \left (x\right )}{\cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) \sin \left (x\right ) - 1}\right ) - \frac{1}{4} \, \arctan \left (-\frac{2 \, \sqrt{2} \sqrt{\cos \left (x\right ) \sin \left (x\right )} - \cos \left (x\right ) - \sin \left (x\right )}{\cos \left (x\right ) - \sin \left (x\right )}\right ) - \frac{1}{8} \, \log \left (-32 \, \cos \left (x\right )^{4} + 4 \, \sqrt{2}{\left (4 \, \cos \left (x\right )^{3} -{\left (4 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right ) - 5 \, \cos \left (x\right )\right )} \sqrt{\cos \left (x\right ) \sin \left (x\right )} + 32 \, \cos \left (x\right )^{2} + 16 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (x\right )}{\sqrt{\sin \left (2 \, x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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