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

Optimal. Leaf size=12 \[ \frac{1}{3} \log \left (x^3+3 x\right ) \]

[Out]

Log[3*x + x^3]/3

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Rubi [A]  time = 0.00871026, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{1}{3} \log \left (x^3+3 x\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^2)/(3*x + x^3),x]

[Out]

Log[3*x + x^3]/3

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Rubi in Sympy [A]  time = 8.12128, size = 14, normalized size = 1.17 \[ \frac{\log{\left (x^{2} \right )}}{6} + \frac{\log{\left (x^{2} + 3 \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**2)/6 + log(x**2 + 3)/3

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Mathematica [A]  time = 0.00546691, size = 17, normalized size = 1.42 \[ \frac{1}{3} \log \left (x^2+3\right )+\frac{\log (x)}{3} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x^2)/(3*x + x^3),x]

[Out]

Log[x]/3 + Log[3 + x^2]/3

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Maple [A]  time = 0.002, size = 11, normalized size = 0.9 \[{\frac{\ln \left ( x \left ({x}^{2}+3 \right ) \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+1)/(x^3+3*x),x)

[Out]

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

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Maxima [A]  time = 0.800529, size = 14, normalized size = 1.17 \[ \frac{1}{3} \, \log \left (x^{3} + 3 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*log(x^3 + 3*x)

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Fricas [A]  time = 0.265983, size = 14, normalized size = 1.17 \[ \frac{1}{3} \, \log \left (x^{3} + 3 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*log(x^3 + 3*x)

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Sympy [A]  time = 0.159458, size = 8, normalized size = 0.67 \[ \frac{\log{\left (x^{3} + 3 x \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**3 + 3*x)/3

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GIAC/XCAS [A]  time = 0.261002, size = 20, normalized size = 1.67 \[ \frac{1}{3} \,{\rm ln}\left (x^{2} + 3\right ) + \frac{1}{6} \,{\rm ln}\left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*ln(x^2 + 3) + 1/6*ln(x^2)