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]

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

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Rubi [A]  time = 0.0154401, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 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]

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}{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*}

Mathematica [A]  time = 0.0124048, 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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Maple [A]  time = 0.007, size = 16, normalized size = 0.3 \begin{align*}{\frac{2\,\sqrt{3}}{3}{\it Artanh} \left ({\frac{\sqrt{3}}{3}\tan \left ({\frac{x}{2}} \right ) } \right ) } \end{align*}

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 = 1.41467, size = 50, normalized size = 0.89 \begin{align*} -\frac{1}{3} \, \sqrt{3} \log \left (-\frac{\sqrt{3} - \frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}}{\sqrt{3} + \frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}}\right ) \end{align*}

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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Fricas [A]  time = 1.03689, size = 155, normalized size = 2.77 \begin{align*} \frac{1}{6} \, \sqrt{3} \log \left (-\frac{2 \, \cos \left (x\right )^{2} - 2 \,{\left (\sqrt{3} \cos \left (x\right ) + 2 \, \sqrt{3}\right )} \sin \left (x\right ) - 4 \, \cos \left (x\right ) - 7}{4 \, \cos \left (x\right )^{2} + 4 \, \cos \left (x\right ) + 1}\right ) \end{align*}

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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Sympy [A]  time = 0.352938, size = 36, normalized size = 0.64 \begin{align*} - \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} \end{align*}

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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Giac [A]  time = 1.16382, size = 47, normalized size = 0.84 \begin{align*} -\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 ) \end{align*}

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)))