3.793 \(\int \frac{3+\cos (c+d x)}{2-\cos (c+d x)} \, dx\)

Optimal. Leaf size=47 \[ \frac{10 \tan ^{-1}\left (\frac{\sin (c+d x)}{-\cos (c+d x)+\sqrt{3}+2}\right )}{\sqrt{3} d}+\frac{5 x}{\sqrt{3}}-x \]

[Out]

-x + (5*x)/Sqrt[3] + (10*ArcTan[Sin[c + d*x]/(2 + Sqrt[3] - Cos[c + d*x])])/(Sqrt[3]*d)

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Rubi [A]  time = 0.0670049, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2735, 2657} \[ \frac{10 \tan ^{-1}\left (\frac{\sin (c+d x)}{-\cos (c+d x)+\sqrt{3}+2}\right )}{\sqrt{3} d}+\frac{5 x}{\sqrt{3}}-x \]

Antiderivative was successfully verified.

[In]

Int[(3 + Cos[c + d*x])/(2 - Cos[c + d*x]),x]

[Out]

-x + (5*x)/Sqrt[3] + (10*ArcTan[Sin[c + d*x]/(2 + Sqrt[3] - Cos[c + d*x])])/(Sqrt[3]*d)

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{3+\cos (c+d x)}{2-\cos (c+d x)} \, dx &=-x+5 \int \frac{1}{2-\cos (c+d x)} \, dx\\ &=-x+\frac{5 x}{\sqrt{3}}+\frac{10 \tan ^{-1}\left (\frac{\sin (c+d x)}{2+\sqrt{3}-\cos (c+d x)}\right )}{\sqrt{3} d}\\ \end{align*}

Mathematica [A]  time = 0.0538518, size = 31, normalized size = 0.66 \[ \frac{10 \tan ^{-1}\left (\sqrt{3} \tan \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{3} d}-x \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + Cos[c + d*x])/(2 - Cos[c + d*x]),x]

[Out]

-x + (10*ArcTan[Sqrt[3]*Tan[(c + d*x)/2]])/(Sqrt[3]*d)

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Maple [A]  time = 0.093, size = 39, normalized size = 0.8 \begin{align*} -2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d}}+{\frac{10\,\sqrt{3}}{3\,d}\arctan \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sqrt{3} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+cos(d*x+c))/(2-cos(d*x+c)),x)

[Out]

-2/d*arctan(tan(1/2*d*x+1/2*c))+10/3/d*3^(1/2)*arctan(tan(1/2*d*x+1/2*c)*3^(1/2))

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Maxima [A]  time = 1.57573, size = 70, normalized size = 1.49 \begin{align*} \frac{2 \,{\left (5 \, \sqrt{3} \arctan \left (\frac{\sqrt{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right ) - 3 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )\right )}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+cos(d*x+c))/(2-cos(d*x+c)),x, algorithm="maxima")

[Out]

2/3*(5*sqrt(3)*arctan(sqrt(3)*sin(d*x + c)/(cos(d*x + c) + 1)) - 3*arctan(sin(d*x + c)/(cos(d*x + c) + 1)))/d

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Fricas [A]  time = 1.39132, size = 119, normalized size = 2.53 \begin{align*} -\frac{3 \, d x + 5 \, \sqrt{3} \arctan \left (\frac{2 \, \sqrt{3} \cos \left (d x + c\right ) - \sqrt{3}}{3 \, \sin \left (d x + c\right )}\right )}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+cos(d*x+c))/(2-cos(d*x+c)),x, algorithm="fricas")

[Out]

-1/3*(3*d*x + 5*sqrt(3)*arctan(1/3*(2*sqrt(3)*cos(d*x + c) - sqrt(3))/sin(d*x + c)))/d

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Sympy [A]  time = 7.03786, size = 56, normalized size = 1.19 \begin{align*} \begin{cases} - x + \frac{10 \sqrt{3} \left (\operatorname{atan}{\left (\sqrt{3} \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )} \right )} + \pi \left \lfloor{\frac{\frac{c}{2} + \frac{d x}{2} - \frac{\pi }{2}}{\pi }}\right \rfloor \right )}{3 d} & \text{for}\: d \neq 0 \\\frac{x \left (\cos{\left (c \right )} + 3\right )}{2 - \cos{\left (c \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+cos(d*x+c))/(2-cos(d*x+c)),x)

[Out]

Piecewise((-x + 10*sqrt(3)*(atan(sqrt(3)*tan(c/2 + d*x/2)) + pi*floor((c/2 + d*x/2 - pi/2)/pi))/(3*d), Ne(d, 0
)), (x*(cos(c) + 3)/(2 - cos(c)), True))

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Giac [A]  time = 1.28236, size = 97, normalized size = 2.06 \begin{align*} -\frac{3 \, d x - 5 \, \sqrt{3}{\left (d x + c + 2 \, \arctan \left (-\frac{\sqrt{3} \sin \left (d x + c\right ) - 3 \, \sin \left (d x + c\right )}{\sqrt{3} \cos \left (d x + c\right ) + \sqrt{3} - 3 \, \cos \left (d x + c\right ) + 3}\right )\right )} + 3 \, c}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+cos(d*x+c))/(2-cos(d*x+c)),x, algorithm="giac")

[Out]

-1/3*(3*d*x - 5*sqrt(3)*(d*x + c + 2*arctan(-(sqrt(3)*sin(d*x + c) - 3*sin(d*x + c))/(sqrt(3)*cos(d*x + c) + s
qrt(3) - 3*cos(d*x + c) + 3))) + 3*c)/d