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]

1/2*ln(sin(x))*sin(x)*2^(1/2)/(sin(x)^2)^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 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]

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])

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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 1.33, size = 21, normalized size = 1.11 \[ \frac {\sqrt {-2 \, \cos \relax (x)^{2} + 2} \log \left (\frac {1}{2} \, \sin \relax (x)\right )}{2 \, \sin \relax (x)} \]

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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giac [A]  time = 0.16, size = 14, normalized size = 0.74 \[ \frac {\sqrt {2} \log \left ({\left | \sin \relax (x) \right |}\right )}{2 \, \mathrm {sgn}\left (\sin \relax (x)\right )} \]

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))

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maple [A]  time = 0.45, size = 25, normalized size = 1.32 \[ \frac {\sin \relax (x ) \left (\ln \left (-1+\cos \relax (x )\right )+\ln \left (1+\cos \relax (x )\right )\right ) \sqrt {2}}{2 \sqrt {2-2 \cos \left (2 x \right )}} \]

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 = 0.42, size = 41, normalized size = 2.16 \[ \frac {1}{4} \, \sqrt {2} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) + \frac {1}{4} \, \sqrt {2} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {\cos \relax (x)}{\sqrt {1-\cos \left (2\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)/(1 - cos(2*x))^(1/2),x)

[Out]

int(cos(x)/(1 - cos(2*x))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos {\relax (x )}}{\sqrt {1 - \cos {\left (2 x \right )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(x)/sqrt(1 - cos(2*x)), x)

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