3.102 \(\int \log \left (2+3 x^2\right ) \, dx\)

Optimal. Leaf size=33 \[ x \log \left (3 x^2+2\right )-2 x+2 \sqrt{\frac{2}{3}} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

[Out]

-2*x + 2*Sqrt[2/3]*ArcTan[Sqrt[3/2]*x] + x*Log[2 + 3*x^2]

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Rubi [A]  time = 0.0243126, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ x \log \left (3 x^2+2\right )-2 x+2 \sqrt{\frac{2}{3}} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

[In]  Int[Log[2 + 3*x^2],x]

[Out]

-2*x + 2*Sqrt[2/3]*ArcTan[Sqrt[3/2]*x] + x*Log[2 + 3*x^2]

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Rubi in Sympy [A]  time = 1.89674, size = 31, normalized size = 0.94 \[ x \log{\left (3 x^{2} + 2 \right )} - 2 x + \frac{2 \sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{6} x}{2} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(ln(3*x**2+2),x)

[Out]

x*log(3*x**2 + 2) - 2*x + 2*sqrt(6)*atan(sqrt(6)*x/2)/3

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Mathematica [A]  time = 0.0190889, size = 33, normalized size = 1. \[ x \log \left (3 x^2+2\right )-2 x+2 \sqrt{\frac{2}{3}} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Log[2 + 3*x^2],x]

[Out]

-2*x + 2*Sqrt[2/3]*ArcTan[Sqrt[3/2]*x] + x*Log[2 + 3*x^2]

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Maple [A]  time = 0., size = 27, normalized size = 0.8 \[ -2\,x+x\ln \left ( 3\,{x}^{2}+2 \right ) +{\frac{2\,\sqrt{6}}{3}\arctan \left ({\frac{x\sqrt{6}}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(ln(3*x^2+2),x)

[Out]

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

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Maxima [A]  time = 1.48589, size = 35, normalized size = 1.06 \[ x \log \left (3 \, x^{2} + 2\right ) + \frac{2}{3} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x\right ) - 2 \, x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*log(3*x^2 + 2) + 2/3*sqrt(6)*arctan(1/2*sqrt(6)*x) - 2*x

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Fricas [A]  time = 0.221527, size = 54, normalized size = 1.64 \[ \frac{1}{3} \, \sqrt{3}{\left (\sqrt{3} x \log \left (3 \, x^{2} + 2\right ) - 2 \, \sqrt{3} x + 2 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{3} \sqrt{2} x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(3*x^2 + 2),x, algorithm="fricas")

[Out]

1/3*sqrt(3)*(sqrt(3)*x*log(3*x^2 + 2) - 2*sqrt(3)*x + 2*sqrt(2)*arctan(1/2*sqrt(
3)*sqrt(2)*x))

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Sympy [A]  time = 0.139802, size = 31, normalized size = 0.94 \[ x \log{\left (3 x^{2} + 2 \right )} - 2 x + \frac{2 \sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{6} x}{2} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(ln(3*x**2+2),x)

[Out]

x*log(3*x**2 + 2) - 2*x + 2*sqrt(6)*atan(sqrt(6)*x/2)/3

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GIAC/XCAS [A]  time = 0.205687, size = 35, normalized size = 1.06 \[ x{\rm ln}\left (3 \, x^{2} + 2\right ) + \frac{2}{3} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x\right ) - 2 \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(3*x^2 + 2),x, algorithm="giac")

[Out]

x*ln(3*x^2 + 2) + 2/3*sqrt(6)*arctan(1/2*sqrt(6)*x) - 2*x