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

Optimal. Leaf size=31 \[ \frac{2 x}{\sqrt{3}}-\frac{4 \tan ^{-1}\left (\frac{\sin (x)}{\cos (x)+\sqrt{3}+2}\right )}{\sqrt{3}} \]

[Out]

(2*x)/Sqrt[3] - (4*ArcTan[Sin[x]/(2 + Sqrt[3] + Cos[x])])/Sqrt[3]

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Rubi [A]  time = 0.0114407, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2657} \[ \frac{2 x}{\sqrt{3}}-\frac{4 \tan ^{-1}\left (\frac{\sin (x)}{\cos (x)+\sqrt{3}+2}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Cos[x]/2)^(-1),x]

[Out]

(2*x)/Sqrt[3] - (4*ArcTan[Sin[x]/(2 + Sqrt[3] + Cos[x])])/Sqrt[3]

Rule 2657

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{q = Rt[a^2 - b^2, 2]}, Simp[x/q, x] + Simp
[(2*ArcTan[(b*Cos[c + d*x])/(a + q + b*Sin[c + d*x])])/(d*q), x]] /; FreeQ[{a, b, c, d}, x] && GtQ[a^2 - b^2,
0] && PosQ[a]

Rubi steps

\begin{align*} \int \frac{1}{1+\frac{\cos (x)}{2}} \, dx &=\frac{2 x}{\sqrt{3}}-\frac{4 \tan ^{-1}\left (\frac{\sin (x)}{2+\sqrt{3}+\cos (x)}\right )}{\sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0099244, size = 20, normalized size = 0.65 \[ \frac{4 \tan ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right )}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Cos[x]/2)^(-1),x]

[Out]

(4*ArcTan[Tan[x/2]/Sqrt[3]])/Sqrt[3]

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Maple [A]  time = 0.013, size = 16, normalized size = 0.5 \begin{align*}{\frac{4\,\sqrt{3}}{3}\arctan \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+1/2*cos(x)),x)

[Out]

4/3*3^(1/2)*arctan(1/3*3^(1/2)*tan(1/2*x))

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Maxima [A]  time = 1.41696, size = 26, normalized size = 0.84 \begin{align*} \frac{4}{3} \, \sqrt{3} \arctan \left (\frac{\sqrt{3} \sin \left (x\right )}{3 \,{\left (\cos \left (x\right ) + 1\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*sqrt(3)*arctan(1/3*sqrt(3)*sin(x)/(cos(x) + 1))

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Fricas [A]  time = 1.18087, size = 84, normalized size = 2.71 \begin{align*} -\frac{2}{3} \, \sqrt{3} \arctan \left (\frac{2 \, \sqrt{3} \cos \left (x\right ) + \sqrt{3}}{3 \, \sin \left (x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.281181, size = 32, normalized size = 1.03 \begin{align*} \frac{4 \sqrt{3} \left (\operatorname{atan}{\left (\frac{\sqrt{3} \tan{\left (\frac{x}{2} \right )}}{3} \right )} + \pi \left \lfloor{\frac{\frac{x}{2} - \frac{\pi }{2}}{\pi }}\right \rfloor \right )}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*sqrt(3)*(atan(sqrt(3)*tan(x/2)/3) + pi*floor((x/2 - pi/2)/pi))/3

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Giac [A]  time = 1.08095, size = 54, normalized size = 1.74 \begin{align*} \frac{2}{3} \, \sqrt{3}{\left (x + 2 \, \arctan \left (-\frac{\sqrt{3} \sin \left (x\right ) - \sin \left (x\right )}{\sqrt{3} \cos \left (x\right ) + \sqrt{3} - \cos \left (x\right ) + 1}\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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