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

________________________________________________________________________________________

Rubi [A]  time = 0.0327097, antiderivative size = 34, normalized size of antiderivative = 1., 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 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]

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{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 = 0.0282276, 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]

________________________________________________________________________________________

Maple [A]  time = 0.036, size = 24, normalized size = 0.7 \begin{align*}{\frac{8\,\sqrt{3}}{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 1+2\,\tan \left ( x/2 \right ) \right ) } \right ) }-x \end{align*}

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 = 1.70339, size = 49, normalized size = 1.44 \begin{align*} \frac{8}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (\frac{2 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right )}\right ) - 2 \, \arctan \left (\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((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))

________________________________________________________________________________________

Fricas [A]  time = 1.32982, size = 88, normalized size = 2.59 \begin{align*} \frac{4}{3} \, \sqrt{3} \arctan \left (\frac{2 \, \sqrt{3} \sin \left (x\right ) + \sqrt{3}}{3 \, \cos \left (x\right )}\right ) - x \end{align*}

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

________________________________________________________________________________________

Sympy [A]  time = 1.08916, size = 42, normalized size = 1.24 \begin{align*} - 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} \end{align*}

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

________________________________________________________________________________________

Giac [A]  time = 1.30076, size = 69, normalized size = 2.03 \begin{align*} \frac{4}{3} \, \sqrt{3}{\left (x + 2 \, \arctan \left (-\frac{\sqrt{3} \sin \left (x\right ) - \cos \left (x\right ) - 2 \, \sin \left (x\right ) - 1}{\sqrt{3} \cos \left (x\right ) + \sqrt{3} - 2 \, \cos \left (x\right ) + \sin \left (x\right ) + 2}\right )\right )} - x \end{align*}

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