3.690 \(\int \frac{1}{x \left (2+3 x^4\right )} \, dx\)

Optimal. Leaf size=19 \[ \frac{\log (x)}{2}-\frac{1}{8} \log \left (3 x^4+2\right ) \]

[Out]

Log[x]/2 - Log[2 + 3*x^4]/8

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Rubi [A]  time = 0.0250931, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{\log (x)}{2}-\frac{1}{8} \log \left (3 x^4+2\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(2 + 3*x^4)),x]

[Out]

Log[x]/2 - Log[2 + 3*x^4]/8

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Rubi in Sympy [A]  time = 3.90463, size = 15, normalized size = 0.79 \[ \frac{\log{\left (x^{4} \right )}}{8} - \frac{\log{\left (3 x^{4} + 2 \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(3*x**4+2),x)

[Out]

log(x**4)/8 - log(3*x**4 + 2)/8

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Mathematica [A]  time = 0.00625727, size = 19, normalized size = 1. \[ \frac{\log (x)}{2}-\frac{1}{8} \log \left (3 x^4+2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(2 + 3*x^4)),x]

[Out]

Log[x]/2 - Log[2 + 3*x^4]/8

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Maple [A]  time = 0.006, size = 16, normalized size = 0.8 \[{\frac{\ln \left ( x \right ) }{2}}-{\frac{\ln \left ( 3\,{x}^{4}+2 \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(3*x^4+2),x)

[Out]

1/2*ln(x)-1/8*ln(3*x^4+2)

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Maxima [A]  time = 1.41793, size = 23, normalized size = 1.21 \[ -\frac{1}{8} \, \log \left (3 \, x^{4} + 2\right ) + \frac{1}{8} \, \log \left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((3*x^4 + 2)*x),x, algorithm="maxima")

[Out]

-1/8*log(3*x^4 + 2) + 1/8*log(x^4)

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Fricas [A]  time = 0.223185, size = 20, normalized size = 1.05 \[ -\frac{1}{8} \, \log \left (3 \, x^{4} + 2\right ) + \frac{1}{2} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((3*x^4 + 2)*x),x, algorithm="fricas")

[Out]

-1/8*log(3*x^4 + 2) + 1/2*log(x)

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Sympy [A]  time = 0.195491, size = 14, normalized size = 0.74 \[ \frac{\log{\left (x \right )}}{2} - \frac{\log{\left (3 x^{4} + 2 \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(3*x**4+2),x)

[Out]

log(x)/2 - log(3*x**4 + 2)/8

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GIAC/XCAS [A]  time = 0.220728, size = 23, normalized size = 1.21 \[ -\frac{1}{8} \,{\rm ln}\left (3 \, x^{4} + 2\right ) + \frac{1}{8} \,{\rm ln}\left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((3*x^4 + 2)*x),x, algorithm="giac")

[Out]

-1/8*ln(3*x^4 + 2) + 1/8*ln(x^4)