3.481 \(\int \frac{1}{4 \cos ^2(1+2 x)+3 \sin ^2(1+2 x)} \, dx\)

Optimal. Leaf size=53 \[ \frac{x}{2 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{\sin (2 x+1) \cos (2 x+1)}{\cos ^2(2 x+1)+2 \sqrt{3}+3}\right )}{4 \sqrt{3}} \]

[Out]

x/(2*Sqrt[3]) - ArcTan[(Cos[1 + 2*x]*Sin[1 + 2*x])/(3 + 2*Sqrt[3] + Cos[1 + 2*x]^2)]/(4*Sqrt[3])

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Rubi [A]  time = 0.0373063, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {203} \[ \frac{x}{2 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{\sin (2 x+1) \cos (2 x+1)}{\cos ^2(2 x+1)+2 \sqrt{3}+3}\right )}{4 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(4*Cos[1 + 2*x]^2 + 3*Sin[1 + 2*x]^2)^(-1),x]

[Out]

x/(2*Sqrt[3]) - ArcTan[(Cos[1 + 2*x]*Sin[1 + 2*x])/(3 + 2*Sqrt[3] + Cos[1 + 2*x]^2)]/(4*Sqrt[3])

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.042783, size = 25, normalized size = 0.47 \[ \frac{\tan ^{-1}\left (\frac{1}{2} \sqrt{3} \tan (2 x+1)\right )}{4 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(4*Cos[1 + 2*x]^2 + 3*Sin[1 + 2*x]^2)^(-1),x]

[Out]

ArcTan[(Sqrt[3]*Tan[1 + 2*x])/2]/(4*Sqrt[3])

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Maple [A]  time = 0.055, size = 18, normalized size = 0.3 \begin{align*}{\frac{\sqrt{3}}{12}\arctan \left ({\frac{\sqrt{3}\tan \left ( 1+2\,x \right ) }{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4*cos(1+2*x)^2+3*sin(1+2*x)^2),x)

[Out]

1/12*3^(1/2)*arctan(1/2*3^(1/2)*tan(1+2*x))

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Maxima [A]  time = 1.46692, size = 23, normalized size = 0.43 \begin{align*} \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{2} \, \sqrt{3} \tan \left (2 \, x + 1\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4*cos(1+2*x)^2+3*sin(1+2*x)^2),x, algorithm="maxima")

[Out]

1/12*sqrt(3)*arctan(1/2*sqrt(3)*tan(2*x + 1))

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Fricas [A]  time = 1.85086, size = 128, normalized size = 2.42 \begin{align*} -\frac{1}{24} \, \sqrt{3} \arctan \left (\frac{7 \, \sqrt{3} \cos \left (2 \, x + 1\right )^{2} - 3 \, \sqrt{3}}{12 \, \cos \left (2 \, x + 1\right ) \sin \left (2 \, x + 1\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4*cos(1+2*x)^2+3*sin(1+2*x)^2),x, algorithm="fricas")

[Out]

-1/24*sqrt(3)*arctan(1/12*(7*sqrt(3)*cos(2*x + 1)^2 - 3*sqrt(3))/(cos(2*x + 1)*sin(2*x + 1)))

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Sympy [A]  time = 1.29113, size = 87, normalized size = 1.64 \begin{align*} \frac{\sqrt{3} \left (\operatorname{atan}{\left (\frac{2 \sqrt{3} \tan{\left (x + \frac{1}{2} \right )}}{3} - \frac{\sqrt{3}}{3} \right )} + \pi \left \lfloor{\frac{x - \frac{\pi }{2} + \frac{1}{2}}{\pi }}\right \rfloor \right )}{12} + \frac{\sqrt{3} \left (\operatorname{atan}{\left (\frac{2 \sqrt{3} \tan{\left (x + \frac{1}{2} \right )}}{3} + \frac{\sqrt{3}}{3} \right )} + \pi \left \lfloor{\frac{x - \frac{\pi }{2} + \frac{1}{2}}{\pi }}\right \rfloor \right )}{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4*cos(1+2*x)**2+3*sin(1+2*x)**2),x)

[Out]

sqrt(3)*(atan(2*sqrt(3)*tan(x + 1/2)/3 - sqrt(3)/3) + pi*floor((x - pi/2 + 1/2)/pi))/12 + sqrt(3)*(atan(2*sqrt
(3)*tan(x + 1/2)/3 + sqrt(3)/3) + pi*floor((x - pi/2 + 1/2)/pi))/12

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Giac [A]  time = 1.14089, size = 82, normalized size = 1.55 \begin{align*} \frac{1}{12} \, \sqrt{3}{\left (2 \, x + \arctan \left (-\frac{2 \, \sqrt{3} \sin \left (4 \, x + 2\right ) - 3 \, \sin \left (4 \, x + 2\right )}{2 \, \sqrt{3} \cos \left (4 \, x + 2\right ) + 2 \, \sqrt{3} - 3 \, \cos \left (4 \, x + 2\right ) + 3}\right ) + 1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4*cos(1+2*x)^2+3*sin(1+2*x)^2),x, algorithm="giac")

[Out]

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