3.697 \(\int \frac {2-\sin (x)}{2+\sin (x)} \, dx\)

Optimal. Leaf size=34 \[ \frac {4 x}{\sqrt {3}}-x+\frac {8 \tan ^{-1}\left (\frac {\cos (x)}{\sin (x)+\sqrt {3}+2}\right )}{\sqrt {3}} \]

[Out]

-x+4/3*x*3^(1/2)+8/3*arctan(cos(x)/(2+sin(x)+3^(1/2)))*3^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2735, 2657} \[ \frac {4 x}{\sqrt {3}}-x+\frac {8 \tan ^{-1}\left (\frac {\cos (x)}{\sin (x)+\sqrt {3}+2}\right )}{\sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Int[(2 - Sin[x])/(2 + Sin[x]),x]

[Out]

-x + (4*x)/Sqrt[3] + (8*ArcTan[Cos[x]/(2 + Sqrt[3] + Sin[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]

Rule 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rubi steps

\begin {align*} \int \frac {2-\sin (x)}{2+\sin (x)} \, dx &=-x+4 \int \frac {1}{2+\sin (x)} \, dx\\ &=-x+\frac {4 x}{\sqrt {3}}+\frac {8 \tan ^{-1}\left (\frac {\cos (x)}{2+\sqrt {3}+\sin (x)}\right )}{\sqrt {3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 28, normalized size = 0.82 \[ \frac {8 \tan ^{-1}\left (\frac {2 \tan \left (\frac {x}{2}\right )+1}{\sqrt {3}}\right )}{\sqrt {3}}-x \]

Antiderivative was successfully verified.

[In]

Integrate[(2 - Sin[x])/(2 + Sin[x]),x]

[Out]

-x + (8*ArcTan[(1 + 2*Tan[x/2])/Sqrt[3]])/Sqrt[3]

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 27, normalized size = 0.79 \[ \frac {4}{3} \, \sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} \sin \relax (x) + \sqrt {3}}{3 \, \cos \relax (x)}\right ) - x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-sin(x))/(2+sin(x)),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.36, size = 51, normalized size = 1.50 \[ \frac {4}{3} \, \sqrt {3} {\left (x + 2 \, \arctan \left (-\frac {\sqrt {3} \sin \relax (x) - \cos \relax (x) - 2 \, \sin \relax (x) - 1}{\sqrt {3} \cos \relax (x) + \sqrt {3} - 2 \, \cos \relax (x) + \sin \relax (x) + 2}\right )\right )} - x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-sin(x))/(2+sin(x)),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.10, size = 24, normalized size = 0.71 \[ \frac {8 \sqrt {3}\, \arctan \left (\frac {\left (1+2 \tan \left (\frac {x}{2}\right )\right ) \sqrt {3}}{3}\right )}{3}-x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2-sin(x))/(2+sin(x)),x)

[Out]

8/3*3^(1/2)*arctan(1/3*(1+2*tan(1/2*x))*3^(1/2))-x

________________________________________________________________________________________

maxima [A]  time = 1.22, size = 36, normalized size = 1.06 \[ \frac {8}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, \sin \relax (x)}{\cos \relax (x) + 1} + 1\right )}\right ) - 2 \, \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-sin(x))/(2+sin(x)),x, algorithm="maxima")

[Out]

8/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*sin(x)/(cos(x) + 1) + 1)) - 2*arctan(sin(x)/(cos(x) + 1))

________________________________________________________________________________________

mupad [B]  time = 7.83, size = 36, normalized size = 1.06 \[ -x-\frac {8\,\sqrt {3}\,\mathrm {atan}\left (-\frac {\sqrt {3}\,\mathrm {tan}\left (\frac {x}{2}\right )-\sqrt {3}}{3\,\mathrm {tan}\left (\frac {x}{2}\right )+3}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(sin(x) - 2)/(sin(x) + 2),x)

[Out]

- x - (8*3^(1/2)*atan(-(3^(1/2)*tan(x/2) - 3^(1/2))/(3*tan(x/2) + 3)))/3

________________________________________________________________________________________

sympy [A]  time = 0.88, size = 42, normalized size = 1.24 \[ - x + \frac {8 \sqrt {3} \left (\operatorname {atan}{\left (\frac {2 \sqrt {3} \tan {\left (\frac {x}{2} \right )}}{3} + \frac {\sqrt {3}}{3} \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-sin(x))/(2+sin(x)),x)

[Out]

-x + 8*sqrt(3)*(atan(2*sqrt(3)*tan(x/2)/3 + sqrt(3)/3) + pi*floor((x/2 - pi/2)/pi))/3

________________________________________________________________________________________