3.393 \(\int \frac{1}{\sqrt{1-\cos (2 x)}} \, dx\)

Optimal. Leaf size=30 \[ -\frac{\tanh ^{-1}\left (\frac{\sin (2 x)}{\sqrt{2} \sqrt{1-\cos (2 x)}}\right )}{\sqrt{2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0277617, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{\tanh ^{-1}\left (\frac{\sin (2 x)}{\sqrt{2} \sqrt{1-\cos (2 x)}}\right )}{\sqrt{2}} \]

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])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 0.607716, size = 31, normalized size = 1.03 \[ - \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sin{\left (2 x \right )}}{2 \sqrt{- \cos{\left (2 x \right )} + 1}} \right )}}{2} \]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0261692, size = 33, normalized size = 1.1 \[ -\frac{\sin (x) \left (\log \left (\cos \left (\frac{x}{2}\right )\right )-\log \left (\sin \left (\frac{x}{2}\right )\right )\right )}{\sqrt{1-\cos (2 x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.055, size = 17, normalized size = 0.6 \[ -{\frac{\sin \left ( x \right ){\it Artanh} \left ( \cos \left ( x \right ) \right ) \sqrt{2}}{2}{\frac{1}{\sqrt{ \left ( \sin \left ( x \right ) \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-cos(2*x))^(1/2),x)

[Out]

-1/2*sin(x)*arctanh(cos(x))*2^(1/2)/(sin(x)^2)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.7707, size = 51, normalized size = 1.7 \[ -\frac{1}{4} \, \sqrt{2}{\left (\log \left (\frac{\sqrt{2}}{\sqrt{\cos \left (2 \, x\right ) + 1}} + 1\right ) - \log \left (\frac{\sqrt{2}}{\sqrt{\cos \left (2 \, x\right ) + 1}} - 1\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.214218, size = 72, normalized size = 2.4 \[ \frac{1}{4} \, \sqrt{2} \log \left (-\frac{{\left (\cos \left (2 \, x\right ) + 3\right )} \sqrt{-\cos \left (2 \, x\right ) + 1} - 2 \, \sqrt{2} \sin \left (2 \, x\right )}{{\left (\cos \left (2 \, x\right ) - 1\right )} \sqrt{-\cos \left (2 \, x\right ) + 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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)

_______________________________________________________________________________________

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)