Optimal. Leaf size=31 \[ \frac{1}{2} \log \left (\sin (x)+\sqrt{\sin (2 x)}+\cos (x)\right )-\frac{1}{2} \sin ^{-1}(\cos (x)-\sin (x)) \]
[Out]
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Rubi [A] time = 0.0269563, 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 \[ \frac{1}{2} \log \left (\sin (x)+\sqrt{\sin (2 x)}+\cos (x)\right )-\frac{1}{2} \sin ^{-1}(\cos (x)-\sin (x)) \]
Antiderivative was successfully verified.
[In] Int[Cos[x]/Sqrt[Sin[2*x]],x]
[Out]
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Rubi in Sympy [A] time = 1.3834, size = 27, normalized size = 0.87 \[ \frac{\log{\left (\sin{\left (x \right )} + \sqrt{\sin{\left (2 x \right )}} + \cos{\left (x \right )} \right )}}{2} + \frac{\operatorname{asin}{\left (\sin{\left (x \right )} - \cos{\left (x \right )} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(cos(x)/sin(2*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0342881, 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.
[In] Integrate[Cos[x]/Sqrt[Sin[2*x]],x]
[Out]
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Maple [C] time = 0.112, size = 98, normalized size = 3.2 \[{1\sqrt{-{1\tan \left ({\frac{x}{2}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}-1 \right ) ^{-1}}} \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}-1 \right ) \sqrt{1+\tan \left ({\frac{x}{2}} \right ) }\sqrt{-2\,\tan \left ( x/2 \right ) +2}\sqrt{-\tan \left ({\frac{x}{2}} \right ) }{\it EllipticF} \left ( \sqrt{1+\tan \left ({\frac{x}{2}} \right ) },{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{\tan \left ({\frac{x}{2}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}-1 \right ) }}}{\frac{1}{\sqrt{ \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3}-\tan \left ({\frac{x}{2}} \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(cos(x)/sin(2*x)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(x)/sqrt(sin(2*x)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.253368, size = 342, normalized size = 11.03 \[ \frac{1}{8} \, \arctan \left (-\frac{\sqrt{2} \sqrt{\cos \left (x\right ) \sin \left (x\right )}{\left (\cos \left (x\right ) - \sin \left (x\right )\right )} + 2 \, \cos \left (x\right )^{2} - 1}{\sqrt{2} \sqrt{\cos \left (x\right ) \sin \left (x\right )}{\left (\cos \left (x\right ) + \sin \left (x\right )\right )} + 2 \, \cos \left (x\right ) \sin \left (x\right )}\right ) - \frac{1}{8} \, \arctan \left (-\frac{\sqrt{2} \sqrt{\cos \left (x\right ) \sin \left (x\right )}{\left (\cos \left (x\right ) - \sin \left (x\right )\right )} - 2 \, \cos \left (x\right )^{2} + 1}{\sqrt{2} \sqrt{\cos \left (x\right ) \sin \left (x\right )}{\left (\cos \left (x\right ) + \sin \left (x\right )\right )} - 2 \, \cos \left (x\right ) \sin \left (x\right )}\right ) + \frac{1}{8} \, \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{\cos \left (x\right ) \sin \left (x\right )}{\left (\cos \left (x\right ) + \sin \left (x\right )\right )} + 1\right )}}{2 \, \sqrt{\cos \left (x\right ) \sin \left (x\right )}{\left (\cos \left (x\right ) - \sin \left (x\right )\right )}}\right ) - \frac{1}{8} \, \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{\cos \left (x\right ) \sin \left (x\right )}{\left (\cos \left (x\right ) + \sin \left (x\right )\right )} - 1\right )}}{2 \, \sqrt{\cos \left (x\right ) \sin \left (x\right )}{\left (\cos \left (x\right ) - \sin \left (x\right )\right )}}\right ) + \frac{1}{8} \, \log \left (2 \, \sqrt{2} \sqrt{\cos \left (x\right ) \sin \left (x\right )}{\left (\cos \left (x\right ) + \sin \left (x\right )\right )} + 4 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) - \frac{1}{8} \, \log \left (-2 \, \sqrt{2} \sqrt{\cos \left (x\right ) \sin \left (x\right )}{\left (\cos \left (x\right ) + \sin \left (x\right )\right )} + 4 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(x)/sqrt(sin(2*x)),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(x)/sin(2*x)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\cos \left (x\right )}{\sqrt{\sin \left (2 \, x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(x)/sqrt(sin(2*x)),x, algorithm="giac")
[Out]