\(\int \frac {2}{3+\cos (4+6 x)} \, dx\) [15]

Optimal result

Integrand size = 12, antiderivative size = 42 \[ \int \frac {2}{3+\cos (4+6 x)} \, dx=\frac {x}{\sqrt {2}}-\frac {\arctan \left (\frac {\sin (4+6 x)}{3+2 \sqrt {2}+\cos (4+6 x)}\right )}{3 \sqrt {2}} \]

[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)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {12, 2736} \[ \int \frac {2}{3+\cos (4+6 x)} \, dx=\frac {x}{\sqrt {2}}-\frac {\arctan \left (\frac {\sin (6 x+4)}{\cos (6 x+4)+2 \sqrt {2}+3}\right )}{3 \sqrt {2}} \]

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.52 \[ \int \frac {2}{3+\cos (4+6 x)} \, dx=\frac {\arctan \left (\frac {\tan (2+3 x)}{\sqrt {2}}\right )}{3 \sqrt {2}} \]

[In]

Integrate[2/(3 + Cos[4 + 6*x]),x]

[Out]

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

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.43

method	result	size
derivativedivides	\(\frac {\sqrt {2}\, \arctan \left (\frac {\tan \left (2+3 x \right ) \sqrt {2}}{2}\right )}{6}\)	\(18\)
default	\(\frac {\sqrt {2}\, \arctan \left (\frac {\tan \left (2+3 x \right ) \sqrt {2}}{2}\right )}{6}\)	\(18\)
risch	\(\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 i \left (2+3 x \right )}+3+2 \sqrt {2}\right )}{12}-\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 i \left (2+3 x \right )}+3-2 \sqrt {2}\right )}{12}\)	\(48\)

[In]

int(2/(3+cos(4+6*x)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.74 \[ \int \frac {2}{3+\cos (4+6 x)} \, dx=-\frac {1}{12} \, \sqrt {2} \arctan \left (\frac {3 \, \sqrt {2} \cos \left (6 \, x + 4\right ) + \sqrt {2}}{4 \, \sin \left (6 \, x + 4\right )}\right ) \]

[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))

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.81 \[ \int \frac {2}{3+\cos (4+6 x)} \, dx=\frac {\sqrt {2} \left (\operatorname {atan}{\left (\frac {\sqrt {2} \tan {\left (3 x + 2 \right )}}{2} \right )} + \pi \left \lfloor {\frac {3 x - \frac {\pi }{2} + 2}{\pi }}\right \rfloor \right )}{6} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.64 \[ \int \frac {2}{3+\cos (4+6 x)} \, dx=\frac {1}{6} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sin \left (6 \, x + 4\right )}{2 \, {\left (\cos \left (6 \, x + 4\right ) + 1\right )}}\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.36 \[ \int \frac {2}{3+\cos (4+6 x)} \, dx=\frac {1}{6} \, \sqrt {2} {\left (3 \, x + \arctan \left (-\frac {\sqrt {2} \sin \left (6 \, x + 4\right ) - \sin \left (6 \, x + 4\right )}{\sqrt {2} \cos \left (6 \, x + 4\right ) + \sqrt {2} - \cos \left (6 \, x + 4\right ) + 1}\right ) + 2\right )} \]

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

Mupad [B] (verification not implemented)

Time = 26.69 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.86 \[ \int \frac {2}{3+\cos (4+6 x)} \, dx=\frac {\sqrt {2}\,\left (3\,x-\mathrm {atan}\left (\mathrm {tan}\left (3\,x+2\right )\right )\right )}{6}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\mathrm {tan}\left (3\,x+2\right )}{2}\right )}{6} \]

[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))/2))/6