3.392 \(\int \frac{1}{\sqrt{1+\cos (2 x)}} \, dx\)

Optimal. Leaf size=27 \[ \frac{\tanh ^{-1}\left (\frac{\sin (2 x)}{\sqrt{2} \sqrt{\cos (2 x)+1}}\right )}{\sqrt{2}} \]

[Out]

ArcTanh[Sin[2*x]/(Sqrt[2]*Sqrt[1 + Cos[2*x]])]/Sqrt[2]

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Rubi [A]  time = 0.0234976, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\tanh ^{-1}\left (\frac{\sin (2 x)}{\sqrt{2} \sqrt{\cos (2 x)+1}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[1 + Cos[2*x]],x]

[Out]

ArcTanh[Sin[2*x]/(Sqrt[2]*Sqrt[1 + Cos[2*x]])]/Sqrt[2]

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Rubi in Sympy [A]  time = 0.574884, size = 29, normalized size = 1.07 \[ \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sin{\left (2 x \right )}}{2 \sqrt{\cos{\left (2 x \right )} + 1}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1+cos(2*x))**(1/2),x)

[Out]

sqrt(2)*atanh(sqrt(2)*sin(2*x)/(2*sqrt(cos(2*x) + 1)))/2

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Mathematica [A]  time = 0.0257318, size = 47, normalized size = 1.74 \[ -\frac{\cos (x) \left (\log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )-\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )\right )}{\sqrt{\cos (2 x)+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[1 + Cos[2*x]],x]

[Out]

-((Cos[x]*(Log[Cos[x/2] - Sin[x/2]] - Log[Cos[x/2] + Sin[x/2]]))/Sqrt[1 + Cos[2*
x]])

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Maple [C]  time = 0.033, size = 9, normalized size = 0.3 \[{\frac{\sqrt{2}{\it InverseJacobiAM} \left ( x,1 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+cos(2*x))^(1/2),x)

[Out]

1/2*2^(1/2)*InverseJacobiAM(x,1)

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Maxima [A]  time = 1.73778, size = 26, normalized size = 0.96 \[ \frac{1}{2} \, \sqrt{2} \operatorname{arsinh}\left (\frac{\sin \left (2 \, x\right )}{\cos \left (2 \, x\right ) + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(cos(2*x) + 1),x, algorithm="maxima")

[Out]

1/2*sqrt(2)*arcsinh(sin(2*x)/(cos(2*x) + 1))

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Fricas [A]  time = 0.216912, size = 55, normalized size = 2.04 \[ \frac{1}{4} \, \sqrt{2} \log \left (-\frac{\sqrt{\cos \left (2 \, x\right ) + 1}{\left (\cos \left (2 \, x\right ) - 3\right )} - 2 \, \sqrt{2} \sin \left (2 \, x\right )}{{\left (\cos \left (2 \, x\right ) + 1\right )}^{\frac{3}{2}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(cos(2*x) + 1),x, algorithm="fricas")

[Out]

1/4*sqrt(2)*log(-(sqrt(cos(2*x) + 1)*(cos(2*x) - 3) - 2*sqrt(2)*sin(2*x))/(cos(2
*x) + 1)^(3/2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\cos{\left (2 x \right )} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1+cos(2*x))**(1/2),x)

[Out]

Integral(1/sqrt(cos(2*x) + 1), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\cos \left (2 \, x\right ) + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(cos(2*x) + 1),x, algorithm="giac")

[Out]

integrate(1/sqrt(cos(2*x) + 1), x)