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 = 0.0498396, 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 \[ \frac{x}{\sqrt{10}}+\frac{\tan ^{-1}\left (\frac{\left (2-\sqrt{10}\right ) \sin (x) \cos (x)}{2-\left (2-\sqrt{10}\right ) \cos ^2(x)}\right )}{\sqrt{10}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*Cos[x]^2)^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 0.644473, size = 17, normalized size = 0.46 \[ \frac{\sqrt{10} \operatorname{atan}{\left (\frac{\sqrt{10} \tan{\left (x \right )}}{5} \right )}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2+3*cos(x)**2),x)

[Out]

sqrt(10)*atan(sqrt(10)*tan(x)/5)/10

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Mathematica [A]  time = 0.0213048, 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 = 0.02, size = 14, normalized size = 0.4 \[{\frac{\sqrt{10}}{10}\arctan \left ({\frac{\tan \left ( x \right ) \sqrt{10}}{5}} \right ) } \]

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 = 1.51015, size = 18, normalized size = 0.49 \[ \frac{1}{10} \, \sqrt{10} \arctan \left (\frac{1}{5} \, \sqrt{10} \tan \left (x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.231524, size = 42, normalized size = 1.14 \[ -\frac{1}{20} \, \sqrt{10} \arctan \left (\frac{7 \, \sqrt{10} \cos \left (x\right )^{2} - 2 \, \sqrt{10}}{20 \, \cos \left (x\right ) \sin \left (x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(3*cos(x)^2 + 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(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2+3*cos(x)**2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220057, size = 62, normalized size = 1.68 \[ \frac{1}{10} \, \sqrt{10}{\left (x + \arctan \left (-\frac{\sqrt{10} \sin \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right )}{\sqrt{10} \cos \left (2 \, x\right ) + \sqrt{10} - 2 \, \cos \left (2 \, x\right ) + 2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(3*cos(x)^2 + 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)))