3.403 \(\int \frac{\cos (x)}{\sqrt{\sin (2 x)}} \, dx\)

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]

-ArcSin[Cos[x] - Sin[x]]/2 + Log[Cos[x] + Sin[x] + Sqrt[Sin[2*x]]]/2

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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]

-ArcSin[Cos[x] - Sin[x]]/2 + Log[Cos[x] + Sin[x] + Sqrt[Sin[2*x]]]/2

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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]

log(sin(x) + sqrt(sin(2*x)) + cos(x))/2 + asin(sin(x) - cos(x))/2

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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]

(-ArcSin[Cos[x] - Sin[x]] + Log[Cos[x] + Sin[x] + Sqrt[Sin[2*x]]])/2

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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]

(-tan(1/2*x)/(tan(1/2*x)^2-1))^(1/2)*(tan(1/2*x)^2-1)/(tan(1/2*x)*(tan(1/2*x)^2-
1))^(1/2)*(1+tan(1/2*x))^(1/2)*(-2*tan(1/2*x)+2)^(1/2)*(-tan(1/2*x))^(1/2)/(tan(
1/2*x)^3-tan(1/2*x))^(1/2)*EllipticF((1+tan(1/2*x))^(1/2),1/2*2^(1/2))

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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]

Timed 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]

1/8*arctan(-(sqrt(2)*sqrt(cos(x)*sin(x))*(cos(x) - sin(x)) + 2*cos(x)^2 - 1)/(sq
rt(2)*sqrt(cos(x)*sin(x))*(cos(x) + sin(x)) + 2*cos(x)*sin(x))) - 1/8*arctan(-(s
qrt(2)*sqrt(cos(x)*sin(x))*(cos(x) - sin(x)) - 2*cos(x)^2 + 1)/(sqrt(2)*sqrt(cos
(x)*sin(x))*(cos(x) + sin(x)) - 2*cos(x)*sin(x))) + 1/8*arctan(-1/2*sqrt(2)*(sqr
t(2)*sqrt(cos(x)*sin(x))*(cos(x) + sin(x)) + 1)/(sqrt(cos(x)*sin(x))*(cos(x) - s
in(x)))) - 1/8*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(cos(x)*sin(x))*(cos(x) + sin(x)
) - 1)/(sqrt(cos(x)*sin(x))*(cos(x) - sin(x)))) + 1/8*log(2*sqrt(2)*sqrt(cos(x)*
sin(x))*(cos(x) + sin(x)) + 4*cos(x)*sin(x) + 1) - 1/8*log(-2*sqrt(2)*sqrt(cos(x
)*sin(x))*(cos(x) + sin(x)) + 4*cos(x)*sin(x) + 1)

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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]

Timed 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]

integrate(cos(x)/sqrt(sin(2*x)), x)