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

Optimal. Leaf size=56 \[ \frac {\log \left (\sin \left (\frac {x}{2}\right )+\sqrt {3} \cos \left (\frac {x}{2}\right )\right )}{\sqrt {3}}-\frac {\log \left (\sqrt {3} \cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )}{\sqrt {3}} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2659, 206} \[ \frac {\log \left (\sin \left (\frac {x}{2}\right )+\sqrt {3} \cos \left (\frac {x}{2}\right )\right )}{\sqrt {3}}-\frac {\log \left (\sqrt {3} \cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )}{\sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + 2*Cos[x])^(-1),x]

[Out]

-(Log[Sqrt[3]*Cos[x/2] - Sin[x/2]]/Sqrt[3]) + Log[Sqrt[3]*Cos[x/2] + Sin[x/2]]/Sqrt[3]

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 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rubi steps

\begin {align*} \int \frac {1}{1+2 \cos (x)} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{3-x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=-\frac {\log \left (\sqrt {3} \cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )}{\sqrt {3}}+\frac {\log \left (\sqrt {3} \cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )}{\sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 20, normalized size = 0.36 \[ \frac {2 \tanh ^{-1}\left (\frac {\tan \left (\frac {x}{2}\right )}{\sqrt {3}}\right )}{\sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*Cos[x])^(-1),x]

[Out]

(2*ArcTanh[Tan[x/2]/Sqrt[3]])/Sqrt[3]

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fricas [A]  time = 0.41, size = 50, normalized size = 0.89 \[ \frac {1}{6} \, \sqrt {3} \log \left (-\frac {2 \, \cos \relax (x)^{2} - 2 \, {\left (\sqrt {3} \cos \relax (x) + 2 \, \sqrt {3}\right )} \sin \relax (x) - 4 \, \cos \relax (x) - 7}{4 \, \cos \relax (x)^{2} + 4 \, \cos \relax (x) + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(3)*log(-(2*cos(x)^2 - 2*(sqrt(3)*cos(x) + 2*sqrt(3))*sin(x) - 4*cos(x) - 7)/(4*cos(x)^2 + 4*cos(x) +
1))

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giac [A]  time = 0.03, size = 35, normalized size = 0.62 \[ -\frac {1}{3} \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, \tan \left (\frac {1}{2} \, x\right ) \right |}}{{\left | 2 \, \sqrt {3} + 2 \, \tan \left (\frac {1}{2} \, x\right ) \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(3)*log(abs(-2*sqrt(3) + 2*tan(1/2*x))/abs(2*sqrt(3) + 2*tan(1/2*x)))

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maple [A]  time = 0.02, size = 16, normalized size = 0.29 \[ \frac {2 \sqrt {3}\, \arctanh \left (\frac {\sqrt {3}\, \tan \left (\frac {x}{2}\right )}{3}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+2*cos(x)),x)

[Out]

2/3*3^(1/2)*arctanh(1/3*3^(1/2)*tan(1/2*x))

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maxima [A]  time = 1.35, size = 37, normalized size = 0.66 \[ -\frac {1}{3} \, \sqrt {3} \log \left (-\frac {\sqrt {3} - \frac {\sin \relax (x)}{\cos \relax (x) + 1}}{\sqrt {3} + \frac {\sin \relax (x)}{\cos \relax (x) + 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.24, size = 15, normalized size = 0.27 \[ \frac {2\,\sqrt {3}\,\mathrm {atanh}\left (\frac {\sqrt {3}\,\mathrm {tan}\left (\frac {x}{2}\right )}{3}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*cos(x) + 1),x)

[Out]

(2*3^(1/2)*atanh((3^(1/2)*tan(x/2))/3))/3

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sympy [A]  time = 0.34, size = 36, normalized size = 0.64 \[ - \frac {\sqrt {3} \log {\left (\tan {\left (\frac {x}{2} \right )} - \sqrt {3} \right )}}{3} + \frac {\sqrt {3} \log {\left (\tan {\left (\frac {x}{2} \right )} + \sqrt {3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(3)*log(tan(x/2) - sqrt(3))/3 + sqrt(3)*log(tan(x/2) + sqrt(3))/3

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