3.202 \(\int \frac{3+2 x}{3 x+x^3} \, dx\)

Optimal. Leaf size=28 \[ -\frac{1}{2} \log \left (x^2+3\right )+\log (x)+\frac{2 \tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0492847, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{1}{2} \log \left (x^2+3\right )+\log (x)+\frac{2 \tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(3 + 2*x)/(3*x + x^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 3.15747, size = 29, normalized size = 1.04 \[ \log{\left (x \right )} - \frac{\log{\left (x^{2} + 3 \right )}}{2} + \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} x}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0125286, size = 28, normalized size = 1. \[ -\frac{1}{2} \log \left (x^2+3\right )+\log (x)+\frac{2 \tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(3 + 2*x)/(3*x + x^3),x]

[Out]

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

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Maple [A]  time = 0.009, size = 24, normalized size = 0.9 \[ \ln \left ( x \right ) -{\frac{\ln \left ({x}^{2}+3 \right ) }{2}}+{\frac{2\,\sqrt{3}}{3}\arctan \left ({\frac{x\sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+2*x)/(x^3+3*x),x)

[Out]

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

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Maxima [A]  time = 1.5114, size = 31, normalized size = 1.11 \[ \frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} x\right ) - \frac{1}{2} \, \log \left (x^{2} + 3\right ) + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.204753, size = 43, normalized size = 1.54 \[ -\frac{1}{6} \, \sqrt{3}{\left (\sqrt{3} \log \left (x^{2} + 3\right ) - 2 \, \sqrt{3} \log \left (x\right ) - 4 \, \arctan \left (\frac{1}{3} \, \sqrt{3} x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*sqrt(3)*(sqrt(3)*log(x^2 + 3) - 2*sqrt(3)*log(x) - 4*arctan(1/3*sqrt(3)*x))

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Sympy [A]  time = 0.153047, size = 29, normalized size = 1.04 \[ \log{\left (x \right )} - \frac{\log{\left (x^{2} + 3 \right )}}{2} + \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} x}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3+2*x)/(x**3+3*x),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.204067, size = 32, normalized size = 1.14 \[ \frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} x\right ) - \frac{1}{2} \,{\rm ln}\left (x^{2} + 3\right ) +{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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