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

Optimal. Leaf size=19 \[ \frac{\sin (x) \log (\sin (x))}{\sqrt{2} \sqrt{\sin ^2(x)}} \]

[Out]

(Log[Sin[x]]*Sin[x])/(Sqrt[2]*Sqrt[Sin[x]^2])

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Rubi [A]  time = 0.0303875, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {4356, 12, 15, 29} \[ \frac{\sin (x) \log (\sin (x))}{\sqrt{2} \sqrt{\sin ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]/Sqrt[1 - Cos[2*x]],x]

[Out]

(Log[Sin[x]]*Sin[x])/(Sqrt[2]*Sqrt[Sin[x]^2])

Rule 4356

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[d/(b
*c), Subst[Int[SubstFor[1, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a +
b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cos] || EqQ[F, cos])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

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

Mathematica [A]  time = 0.0137804, size = 18, normalized size = 0.95 \[ \frac{\sin (x) \log (\sin (x))}{\sqrt{1-\cos (2 x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 1.99, size = 25, normalized size = 1.3 \begin{align*}{\frac{\sin \left ( x \right ) \left ( \ln \left ( -1+\cos \left ( x \right ) \right ) +\ln \left ( 1+\cos \left ( x \right ) \right ) \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{ \left ( \sin \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)/(1-cos(2*x))^(1/2),x)

[Out]

1/4*sin(x)*(ln(-1+cos(x))+ln(1+cos(x)))*2^(1/2)/(sin(x)^2)^(1/2)

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Maxima [B]  time = 1.48877, size = 55, normalized size = 2.89 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + \frac{1}{4} \, \sqrt{2} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.99703, size = 68, normalized size = 3.58 \begin{align*} \frac{\sqrt{-2 \, \cos \left (x\right )^{2} + 2} \log \left (\frac{1}{2} \, \sin \left (x\right )\right )}{2 \, \sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.0917, size = 19, normalized size = 1. \begin{align*} \frac{\sqrt{2} \log \left ({\left | \sin \left (x\right ) \right |}\right )}{2 \, \mathrm{sgn}\left (\sin \left (x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*log(abs(sin(x)))/sgn(sin(x))