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

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

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Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2649, 206} \[ \frac {\tanh ^{-1}\left (\frac {\sin (2 x)}{\sqrt {2} \sqrt {\cos (2 x)+1}}\right )}{\sqrt {2}} \]

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]

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])

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]

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*}

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Mathematica [A]  time = 0.02, 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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {1+\cos (2 x)}} \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[1/Sqrt[1 + Cos[2*x]],x]

[Out]

Could not integrate

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fricas [B]  time = 1.15, size = 55, normalized size = 2.04 \[ \frac {1}{4} \, \sqrt {2} \log \left (-\frac {\cos \left (2 \, x\right )^{2} - 2 \, \sqrt {2} \sqrt {\cos \left (2 \, x\right ) + 1} \sin \left (2 \, x\right ) - 2 \, \cos \left (2 \, x\right ) - 3}{\cos \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1}\right ) \]

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

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giac [A]  time = 0.63, size = 41, normalized size = 1.52 \[ \frac {\sqrt {2} \log \left ({\left | \frac {1}{\sin \relax (x)} + \sin \relax (x) + 2 \right |}\right )}{8 \, \mathrm {sgn}\left (\cos \relax (x)\right )} - \frac {\sqrt {2} \log \left ({\left | \frac {1}{\sin \relax (x)} + \sin \relax (x) - 2 \right |}\right )}{8 \, \mathrm {sgn}\left (\cos \relax (x)\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*sqrt(2)*log(abs(1/sin(x) + sin(x) + 2))/sgn(cos(x)) - 1/8*sqrt(2)*log(abs(1/sin(x) + sin(x) - 2))/sgn(cos(
x))

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maple [C]  time = 0.09, size = 9, normalized size = 0.33

method	result	size
default	\(\frac {\sqrt {2}\, \mathrm {am}^{-1}\left (x | 1\right )}{2}\)	\(9\)
risch	\(-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{i x}-i\right ) \cos \relax (x )}{\sqrt {\left (1+{\mathrm e}^{2 i x}\right )^{2} {\mathrm e}^{-2 i x}}}+\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{i x}+i\right ) \cos \relax (x )}{\sqrt {\left (1+{\mathrm e}^{2 i x}\right )^{2} {\mathrm e}^{-2 i x}}}\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+cos(2*x))^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 1.14, size = 41, normalized size = 1.52 \[ \frac {1}{4} \, \sqrt {2} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \sin \relax (x) + 1\right ) - \frac {1}{4} \, \sqrt {2} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \sin \relax (x) + 1\right ) \]

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

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mupad [B]  time = 0.05, size = 13, normalized size = 0.48 \[ \frac {\sqrt {2}\,\mathrm {asinh}\left (\frac {\sin \relax (x)}{\cos \relax (x)}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(2*x) + 1)^(1/2),x)

[Out]

(2^(1/2)*asinh(sin(x)/cos(x)))/2

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

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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