Optimal. Leaf size=31 \[ -\frac{1}{2} \sin ^{-1}(\cos (x)-\sin (x))-\frac{1}{2} \log \left (\sin (x)+\sqrt{\sin (2 x)}+\cos (x)\right ) \]
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Rubi [A] time = 0.015072, 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 = {4306} \[ -\frac{1}{2} \sin ^{-1}(\cos (x)-\sin (x))-\frac{1}{2} \log \left (\sin (x)+\sqrt{\sin (2 x)}+\cos (x)\right ) \]
Antiderivative was successfully verified.
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Rule 4306
Rubi steps
\begin{align*} \int \frac{\sin (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 = 0.0376623, size = 31, normalized size = 1. \[ \frac{1}{2} \left (-\sin ^{-1}(\cos (x)-\sin (x))-\log \left (\sin (x)+\sqrt{\sin (2 x)}+\cos (x)\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.041, size = 266, normalized size = 8.6 \begin{align*} -{\frac{1}{2}\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 ) \left ( 2\,\sqrt{1+\tan \left ( x/2 \right ) }\sqrt{-2\,\tan \left ( x/2 \right ) +2}\sqrt{-\tan \left ( x/2 \right ) }{\it EllipticE} \left ( \sqrt{1+\tan \left ( x/2 \right ) },1/2\,\sqrt{2} \right ) \left ( \tan \left ( x/2 \right ) \right ) ^{2}-\sqrt{1+\tan \left ({\frac{x}{2}} \right ) }\sqrt{-2\,\tan \left ( x/2 \right ) +2}\sqrt{-\tan \left ({\frac{x}{2}} \right ) }{\it EllipticF} \left ( \sqrt{1+\tan \left ({\frac{x}{2}} \right ) },{\frac{\sqrt{2}}{2}} \right ) \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+2\,\sqrt{1+\tan \left ( x/2 \right ) }\sqrt{-2\,\tan \left ( x/2 \right ) +2}\sqrt{-\tan \left ( x/2 \right ) }{\it EllipticE} \left ( \sqrt{1+\tan \left ( x/2 \right ) },1/2\,\sqrt{2} \right ) -\sqrt{1+\tan \left ({\frac{x}{2}} \right ) }\sqrt{-2\,\tan \left ( x/2 \right ) +2}\sqrt{-\tan \left ({\frac{x}{2}} \right ) }{\it EllipticF} \left ( \sqrt{1+\tan \left ({\frac{x}{2}} \right ) },{\frac{\sqrt{2}}{2}} \right ) +2\, \left ( \tan \left ( x/2 \right ) \right ) ^{4}-2\, \left ( \tan \left ( x/2 \right ) \right ) ^{2} \right ){\frac{1}{\sqrt{\tan \left ({\frac{x}{2}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}-1 \right ) }}} \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-1}{\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{\sin \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 = 2.01286, 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{\sin \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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