3.1 \(\int \frac {2}{3-\cos (4+6 x)} \, dx\)

Optimal. Leaf size=44 \[ \frac {x}{\sqrt {2}}+\frac {\tan ^{-1}\left (\frac {\sin (6 x+4)}{-\cos (6 x+4)+2 \sqrt {2}+3}\right )}{3 \sqrt {2}} \]

[Out]

1/2*x*2^(1/2)+1/6*arctan(sin(4+6*x)/(3-cos(4+6*x)+2*2^(1/2)))*2^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 2657} \[ \frac {x}{\sqrt {2}}+\frac {\tan ^{-1}\left (\frac {\sin (6 x+4)}{-\cos (6 x+4)+2 \sqrt {2}+3}\right )}{3 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Int[2/(3 - Cos[4 + 6*x]),x]

[Out]

x/Sqrt[2] + ArcTan[Sin[4 + 6*x]/(3 + 2*Sqrt[2] - Cos[4 + 6*x])]/(3*Sqrt[2])

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 {2}{3-\cos (4+6 x)} \, dx &=2 \int \frac {1}{3-\cos (4+6 x)} \, dx\\ &=\frac {x}{\sqrt {2}}+\frac {\tan ^{-1}\left (\frac {\sin (4+6 x)}{3+2 \sqrt {2}-\cos (4+6 x)}\right )}{3 \sqrt {2}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 22, normalized size = 0.50 \[ \frac {\tan ^{-1}\left (\sqrt {2} \tan (3 x+2)\right )}{3 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Integrate[2/(3 - Cos[4 + 6*x]),x]

[Out]

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

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fricas [A]  time = 0.65, size = 33, normalized size = 0.75 \[ -\frac {1}{12} \, \sqrt {2} \arctan \left (\frac {3 \, \sqrt {2} \cos \left (6 \, x + 4\right ) - \sqrt {2}}{4 \, \sin \left (6 \, x + 4\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/(3-cos(4+6*x)),x, algorithm="fricas")

[Out]

-1/12*sqrt(2)*arctan(1/4*(3*sqrt(2)*cos(6*x + 4) - sqrt(2))/sin(6*x + 4))

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giac [A]  time = 0.15, size = 57, normalized size = 1.30 \[ \frac {1}{6} \, \sqrt {2} {\left (3 \, x + \arctan \left (-\frac {\sqrt {2} \sin \left (6 \, x + 4\right ) - 2 \, \sin \left (6 \, x + 4\right )}{\sqrt {2} \cos \left (6 \, x + 4\right ) + \sqrt {2} - 2 \, \cos \left (6 \, x + 4\right ) + 2}\right ) + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/(3-cos(4+6*x)),x, algorithm="giac")

[Out]

1/6*sqrt(2)*(3*x + arctan(-(sqrt(2)*sin(6*x + 4) - 2*sin(6*x + 4))/(sqrt(2)*cos(6*x + 4) + sqrt(2) - 2*cos(6*x
+ 4) + 2)) + 2)

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maple [A]  time = 0.08, size = 17, normalized size = 0.39 \[ \frac {\sqrt {2}\, \arctan \left (\sqrt {2}\, \tan \left (2+3 x \right )\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2/(3-cos(4+6*x)),x)

[Out]

1/6*2^(1/2)*arctan(2^(1/2)*tan(2+3*x))

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maxima [A]  time = 0.41, size = 26, normalized size = 0.59 \[ \frac {1}{6} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sin \left (6 \, x + 4\right )}{\cos \left (6 \, x + 4\right ) + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/(3-cos(4+6*x)),x, algorithm="maxima")

[Out]

1/6*sqrt(2)*arctan(sqrt(2)*sin(6*x + 4)/(cos(6*x + 4) + 1))

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mupad [B]  time = 2.58, size = 35, normalized size = 0.80 \[ \frac {\sqrt {2}\,\left (3\,x-\mathrm {atan}\left (\mathrm {tan}\left (3\,x+2\right )\right )\right )}{6}+\frac {\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\mathrm {tan}\left (3\,x+2\right )\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-2/(cos(6*x + 4) - 3),x)

[Out]

(2^(1/2)*(3*x - atan(tan(3*x + 2))))/6 + (2^(1/2)*atan(2^(1/2)*tan(3*x + 2)))/6

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sympy [A]  time = 0.26, size = 32, normalized size = 0.73 \[ \frac {\sqrt {2} \left (\operatorname {atan}{\left (\sqrt {2} \tan {\left (3 x + 2 \right )} \right )} + \pi \left \lfloor {\frac {3 x - \frac {\pi }{2} + 2}{\pi }}\right \rfloor \right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/(3-cos(4+6*x)),x)

[Out]

sqrt(2)*(atan(sqrt(2)*tan(3*x + 2)) + pi*floor((3*x - pi/2 + 2)/pi))/6

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