3.1.1 \(\int \frac {3}{5-4 \cos (x)} \, dx\) [1]

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 = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {12, 2736} \begin {gather*} x+2 \tan ^{-1}\left (\frac {\sin (x)}{2-\cos (x)}\right ) \end {gather*}

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 2736

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{q = Rt[a^2 - b^2, 2]}, Simp[x/q, x] + Simp
[(2/(d*q))*ArcTan[b*(Cos[c + d*x]/(a + q + b*Sin[c + d*x]))], 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*}

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Mathematica [A]
time = 0.01, size = 11, normalized size = 0.69 \begin {gather*} 2 \tan ^{-1}\left (3 \tan \left (\frac {x}{2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[3/(5 - 4*Cos[x]),x]

[Out]

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

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 1.92, size = 22, normalized size = 1.38 \begin {gather*} 2 \text {Pi} \text {Floor}\left [-\frac {1}{2}+\frac {x}{2 \text {Pi}}\right ]+2 \text {ArcTan}\left [3 \text {Tan}\left [\frac {x}{2}\right ]\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[3/(5 - 4*Cos[x]),x]')

[Out]

2 Pi Floor[-1 / 2 + x / (2 Pi)] + 2 ArcTan[3 Tan[x / 2]]

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Maple [A]
time = 0.02, size = 10, normalized size = 0.62

method	result	size
default	\(2 \arctan \left (3 \tan \left (\frac {x}{2}\right )\right )\)	\(10\)
risch	\(-i \ln \left ({\mathrm e}^{i x}-\frac {1}{2}\right )+i \ln \left ({\mathrm e}^{i x}-2\right )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.36, size = 13, normalized size = 0.81 \begin {gather*} 2 \, \arctan \left (\frac {3 \, \sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \end {gather*}

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 = 0.31, size = 15, normalized size = 0.94 \begin {gather*} -\arctan \left (\frac {5 \, \cos \left (x\right ) - 4}{3 \, \sin \left (x\right )}\right ) \end {gather*}

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 = 0.12, size = 22, normalized size = 1.38 \begin {gather*} 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 {gather*}

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 = 0.00, size = 20, normalized size = 1.25 \begin {gather*} \frac {2}{3}\cdot 3 \left (\arctan \left (-\frac {\sin x}{\cos x-2}\right )+\frac {x}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 2*arctan(-sin(x)/(cos(x) - 2))

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Mupad [B]
time = 0.27, size = 18, normalized size = 1.12 \begin {gather*} x+2\,\mathrm {atan}\left (3\,\mathrm {tan}\left (\frac {x}{2}\right )\right )-2\,\mathrm {atan}\left (\mathrm {tan}\left (\frac {x}{2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-3/(4*cos(x) - 5),x)

[Out]

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

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