3.1 \(\int \frac{3}{5-4 \cos (x)} \, dx\)

Optimal. Leaf size=16 \[ x+2 \tan ^{-1}\left (\frac{\sin (x)}{2-\cos (x)}\right ) \]

[Out]

x + 2*ArcTan[Sin[x]/(2 - Cos[x])]

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Rubi [A]  time = 0.013289, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {12, 2657} \[ x+2 \tan ^{-1}\left (\frac{\sin (x)}{2-\cos (x)}\right ) \]

Antiderivative was successfully verified.

[In]

Int[3/(5 - 4*Cos[x]),x]

[Out]

x + 2*ArcTan[Sin[x]/(2 - Cos[x])]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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}{5-4 \cos (x)} \, dx &=3 \int \frac{1}{5-4 \cos (x)} \, dx\\ &=x+2 \tan ^{-1}\left (\frac{\sin (x)}{2-\cos (x)}\right )\\ \end{align*}

Mathematica [A]  time = 0.0159833, size = 11, normalized size = 0.69 \[ 2 \tan ^{-1}\left (3 \tan \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[3/(5 - 4*Cos[x]),x]

[Out]

2*ArcTan[3*Tan[x/2]]

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Maple [A]  time = 0.008, size = 10, normalized size = 0.6 \begin{align*} 2\,\arctan \left ( 3\,\tan \left ( x/2 \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(3/(5-4*cos(x)),x)

[Out]

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

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Maxima [A]  time = 1.41024, size = 18, normalized size = 1.12 \begin{align*} 2 \, \arctan \left (\frac{3 \, \sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(3/(5-4*cos(x)),x, algorithm="maxima")

[Out]

2*arctan(3*sin(x)/(cos(x) + 1))

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Fricas [A]  time = 1.6948, size = 49, normalized size = 3.06 \begin{align*} -\arctan \left (\frac{5 \, \cos \left (x\right ) - 4}{3 \, \sin \left (x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(3/(5-4*cos(x)),x, algorithm="fricas")

[Out]

-arctan(1/3*(5*cos(x) - 4)/sin(x))

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Sympy [A]  time = 0.257417, size = 22, normalized size = 1.38 \begin{align*} 2 \operatorname{atan}{\left (3 \tan{\left (\frac{x}{2} \right )} \right )} + 2 \pi \left \lfloor{\frac{\frac{x}{2} - \frac{\pi }{2}}{\pi }}\right \rfloor \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(3/(5-4*cos(x)),x)

[Out]

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

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Giac [A]  time = 1.09064, size = 19, normalized size = 1.19 \begin{align*} x - 2 \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) - 2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(3/(5-4*cos(x)),x, algorithm="giac")

[Out]

x - 2*arctan(sin(x)/(cos(x) - 2))