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

Optimal. Leaf size=37 \[ \frac{x}{\sqrt{10}}-\frac{\tan ^{-1}\left (\frac{3 \sin (x) \cos (x)}{3 \cos ^2(x)+\sqrt{10}+2}\right )}{\sqrt{10}} \]

[Out]

x/Sqrt[10] - ArcTan[(3*Cos[x]*Sin[x])/(2 + Sqrt[10] + 3*Cos[x]^2)]/Sqrt[10]

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Rubi [A]  time = 0.0213491, antiderivative size = 48, normalized size of antiderivative = 1.3, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3181, 203} \[ \frac{x}{\sqrt{10}}-\frac{\tan ^{-1}\left (\frac{\left (\sqrt{\frac{5}{2}}-1\right ) \sin (x) \cos (x)}{\left (\sqrt{\frac{5}{2}}-1\right ) \cos ^2(x)+1}\right )}{\sqrt{10}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*Cos[x]^2)^(-1),x]

[Out]

x/Sqrt[10] - ArcTan[((-1 + Sqrt[5/2])*Cos[x]*Sin[x])/(1 + (-1 + Sqrt[5/2])*Cos[x]^2)]/Sqrt[10]

Rule 3181

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[1/(a + (a + b)*ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]

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}{2+3 \cos ^2(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{2+5 x^2} \, dx,x,\cot (x)\right )\\ &=\frac{x}{\sqrt{10}}-\frac{\tan ^{-1}\left (\frac{\left (-1+\sqrt{\frac{5}{2}}\right ) \cos (x) \sin (x)}{1+\left (-1+\sqrt{\frac{5}{2}}\right ) \cos ^2(x)}\right )}{\sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0487157, size = 17, normalized size = 0.46 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{5}} \tan (x)\right )}{\sqrt{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*Cos[x]^2)^(-1),x]

[Out]

ArcTan[Sqrt[2/5]*Tan[x]]/Sqrt[10]

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Maple [A]  time = 0.012, size = 14, normalized size = 0.4 \begin{align*}{\frac{\sqrt{10}}{10}\arctan \left ({\frac{\tan \left ( x \right ) \sqrt{10}}{5}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*10^(1/2)*arctan(1/5*tan(x)*10^(1/2))

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Maxima [A]  time = 1.43313, size = 18, normalized size = 0.49 \begin{align*} \frac{1}{10} \, \sqrt{10} \arctan \left (\frac{1}{5} \, \sqrt{10} \tan \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*sqrt(10)*arctan(1/5*sqrt(10)*tan(x))

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Fricas [A]  time = 1.96332, size = 108, normalized size = 2.92 \begin{align*} -\frac{1}{20} \, \sqrt{10} \arctan \left (\frac{7 \, \sqrt{10} \cos \left (x\right )^{2} - 2 \, \sqrt{10}}{20 \, \cos \left (x\right ) \sin \left (x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*sqrt(10)*arctan(1/20*(7*sqrt(10)*cos(x)^2 - 2*sqrt(10))/(cos(x)*sin(x)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{3 \cos ^{2}{\left (x \right )} + 2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(3*cos(x)**2 + 2), x)

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Giac [A]  time = 1.07651, size = 62, normalized size = 1.68 \begin{align*} \frac{1}{10} \, \sqrt{10}{\left (x + \arctan \left (-\frac{\sqrt{10} \sin \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right )}{\sqrt{10} \cos \left (2 \, x\right ) + \sqrt{10} - 2 \, \cos \left (2 \, x\right ) + 2}\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*sqrt(10)*(x + arctan(-(sqrt(10)*sin(2*x) - 2*sin(2*x))/(sqrt(10)*cos(2*x) + sqrt(10) - 2*cos(2*x) + 2)))