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])

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Rubi [A]  time = 0.0131854, 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, Rules used = {2649, 206} \[ -\frac{\tanh ^{-1}\left (\frac{\sin (2 x)}{\sqrt{2} \sqrt{1-\cos (2 x)}}\right )}{\sqrt{2}} \]

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])

Rule 2649

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, (b*C
os[c + d*x])/Sqrt[a + b*Sin[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{1-\cos (2 x)}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\frac{\sin (2 x)}{\sqrt{1-\cos (2 x)}}\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{\sin (2 x)}{\sqrt{2} \sqrt{1-\cos (2 x)}}\right )}{\sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.0154276, 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]])

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Maple [A]  time = 0.034, size = 17, normalized size = 0.6 \begin{align*} -{\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}}}}} \end{align*}

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)

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Maxima [B]  time = 1.61261, size = 136, normalized size = 4.53 \begin{align*} -\frac{1}{4} \, \sqrt{2} \log \left (\cos \left (\frac{1}{2} \, \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right )\right )\right )^{2} + \sin \left (\frac{1}{2} \, \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right )\right )\right )^{2} + 2 \, \cos \left (\frac{1}{2} \, \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right )\right )\right ) + 1\right ) + \frac{1}{4} \, \sqrt{2} \log \left (\cos \left (\frac{1}{2} \, \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right )\right )\right )^{2} + \sin \left (\frac{1}{2} \, \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right )\right )\right )^{2} - 2 \, \cos \left (\frac{1}{2} \, \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right )\right )\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cos(2*x))^(1/2),x, algorithm="maxima")

[Out]

-1/4*sqrt(2)*log(cos(1/2*arctan2(sin(2*x), cos(2*x)))^2 + sin(1/2*arctan2(sin(2*x), cos(2*x)))^2 + 2*cos(1/2*a
rctan2(sin(2*x), cos(2*x))) + 1) + 1/4*sqrt(2)*log(cos(1/2*arctan2(sin(2*x), cos(2*x)))^2 + sin(1/2*arctan2(si
n(2*x), cos(2*x)))^2 - 2*cos(1/2*arctan2(sin(2*x), cos(2*x))) + 1)

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Fricas [B]  time = 1.81252, size = 167, normalized size = 5.57 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (-\frac{{\left (\cos \left (2 \, x\right ) + 3\right )} \sin \left (2 \, x\right ) - 2 \,{\left (\sqrt{2} \cos \left (2 \, x\right ) + \sqrt{2}\right )} \sqrt{-\cos \left (2 \, x\right ) + 1}}{{\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cos(2*x))^(1/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cos(2*x))**(1/2),x)

[Out]

Integral(1/sqrt(1 - cos(2*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-\cos \left (2 \, x\right ) + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cos(2*x))^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(-cos(2*x) + 1), x)