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]

2/3*x*3^(1/2)-4/3*arctan(sin(x)/(2+cos(x)+3^(1/2)))*3^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.43, size = 23, normalized size = 0.74 \[ -\frac {2}{3} \, \sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} \cos \relax (x) + \sqrt {3}}{3 \, \sin \relax (x)}\right ) \]

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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giac [A]  time = 0.01, size = 40, normalized size = 1.29 \[ \frac {2}{3} \, \sqrt {3} {\left (x + 2 \, \arctan \left (-\frac {\sqrt {3} \sin \relax (x) - \sin \relax (x)}{\sqrt {3} \cos \relax (x) + \sqrt {3} - \cos \relax (x) + 1}\right )\right )} \]

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)))

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maple [A]  time = 0.03, size = 16, normalized size = 0.52 \[ \frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \tan \left (\frac {x}{2}\right )}{3}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+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 = 1.30, size = 19, normalized size = 0.61 \[ \frac {4}{3} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} \sin \relax (x)}{3 \, {\left (\cos \relax (x) + 1\right )}}\right ) \]

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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mupad [B]  time = 0.22, size = 32, normalized size = 1.03 \[ \frac {4\,\sqrt {3}\,\left (\frac {x}{2}-\mathrm {atan}\left (\mathrm {tan}\left (\frac {x}{2}\right )\right )\right )}{3}+\frac {4\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,\mathrm {tan}\left (\frac {x}{2}\right )}{3}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(x)/2 + 1),x)

[Out]

(4*3^(1/2)*(x/2 - atan(tan(x/2))))/3 + (4*3^(1/2)*atan((3^(1/2)*tan(x/2))/3))/3

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sympy [A]  time = 0.29, size = 32, normalized size = 1.03 \[ \frac {4 \sqrt {3} \left (\operatorname {atan}{\left (\frac {\sqrt {3} \tan {\left (\frac {x}{2} \right )}}{3} \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{3} \]

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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