3.381 \(\int \frac{1}{(\cos (x)+2 \sec (x))^2} \, dx\)

Optimal. Leaf size=55 \[ \frac{x}{6 \sqrt{6}}+\frac{\tan (x)}{6 \left (2 \tan ^2(x)+3\right )}-\frac{\tan ^{-1}\left (\frac{\sin (x) \cos (x)}{\cos ^2(x)+\sqrt{6}+2}\right )}{6 \sqrt{6}} \]

[Out]

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

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Rubi [A]  time = 0.0315169, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {199, 203} \[ \frac{x}{6 \sqrt{6}}+\frac{\tan (x)}{6 \left (2 \tan ^2(x)+3\right )}-\frac{\tan ^{-1}\left (\frac{\sin (x) \cos (x)}{\cos ^2(x)+\sqrt{6}+2}\right )}{6 \sqrt{6}} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[x] + 2*Sec[x])^(-2),x]

[Out]

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

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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

Mathematica [A]  time = 0.0841971, size = 54, normalized size = 0.98 \[ \frac{(\cos (2 x)+5) \sec ^4(x) \left (6 \sin (2 x)+\sqrt{6} (\cos (2 x)+5) \tan ^{-1}\left (\sqrt{\frac{2}{3}} \tan (x)\right )\right )}{144 \left (2 \sec ^2(x)+1\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[x] + 2*Sec[x])^(-2),x]

[Out]

((5 + Cos[2*x])*Sec[x]^4*(Sqrt[6]*ArcTan[Sqrt[2/3]*Tan[x]]*(5 + Cos[2*x]) + 6*Sin[2*x]))/(144*(1 + 2*Sec[x]^2)
^2)

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Maple [A]  time = 0.038, size = 29, normalized size = 0.5 \begin{align*}{\frac{\tan \left ( x \right ) }{18+12\, \left ( \tan \left ( x \right ) \right ) ^{2}}}+{\frac{\sqrt{6}}{36}\arctan \left ({\frac{\tan \left ( x \right ) \sqrt{6}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(x)+2*sec(x))^2,x)

[Out]

1/6*tan(x)/(3+2*tan(x)^2)+1/36*6^(1/2)*arctan(1/3*tan(x)*6^(1/2))

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Maxima [A]  time = 1.41602, size = 38, normalized size = 0.69 \begin{align*} \frac{1}{36} \, \sqrt{6} \arctan \left (\frac{1}{3} \, \sqrt{6} \tan \left (x\right )\right ) + \frac{\tan \left (x\right )}{6 \,{\left (2 \, \tan \left (x\right )^{2} + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*sqrt(6)*arctan(1/3*sqrt(6)*tan(x)) + 1/6*tan(x)/(2*tan(x)^2 + 3)

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Fricas [A]  time = 2.12823, size = 184, normalized size = 3.35 \begin{align*} -\frac{{\left (\sqrt{6} \cos \left (x\right )^{2} + 2 \, \sqrt{6}\right )} \arctan \left (\frac{5 \, \sqrt{6} \cos \left (x\right )^{2} - 2 \, \sqrt{6}}{12 \, \cos \left (x\right ) \sin \left (x\right )}\right ) - 12 \, \cos \left (x\right ) \sin \left (x\right )}{72 \,{\left (\cos \left (x\right )^{2} + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/72*((sqrt(6)*cos(x)^2 + 2*sqrt(6))*arctan(1/12*(5*sqrt(6)*cos(x)^2 - 2*sqrt(6))/(cos(x)*sin(x))) - 12*cos(x
)*sin(x))/(cos(x)^2 + 2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (\cos{\left (x \right )} + 2 \sec{\left (x \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cos(x)+2*sec(x))**2,x)

[Out]

Integral((cos(x) + 2*sec(x))**(-2), x)

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Giac [A]  time = 1.08689, size = 82, normalized size = 1.49 \begin{align*} \frac{1}{36} \, \sqrt{6}{\left (x + \arctan \left (-\frac{\sqrt{6} \sin \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right )}{\sqrt{6} \cos \left (2 \, x\right ) + \sqrt{6} - 2 \, \cos \left (2 \, x\right ) + 2}\right )\right )} + \frac{\tan \left (x\right )}{6 \,{\left (2 \, \tan \left (x\right )^{2} + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*sqrt(6)*(x + arctan(-(sqrt(6)*sin(2*x) - 2*sin(2*x))/(sqrt(6)*cos(2*x) + sqrt(6) - 2*cos(2*x) + 2))) + 1/
6*tan(x)/(2*tan(x)^2 + 3)