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

Optimal. Leaf size=32 \[ \frac{1}{8 \left (3 x^4+2\right )}-\frac{1}{16} \log \left (3 x^4+2\right )+\frac{\log (x)}{4} \]

[Out]

1/(8*(2 + 3*x^4)) + Log[x]/4 - Log[2 + 3*x^4]/16

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Rubi [A]  time = 0.0453793, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{1}{8 \left (3 x^4+2\right )}-\frac{1}{16} \log \left (3 x^4+2\right )+\frac{\log (x)}{4} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(2 + 3*x^4)^2),x]

[Out]

1/(8*(2 + 3*x^4)) + Log[x]/4 - Log[2 + 3*x^4]/16

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Rubi in Sympy [A]  time = 5.38617, size = 24, normalized size = 0.75 \[ \frac{\log{\left (x^{4} \right )}}{16} - \frac{\log{\left (3 x^{4} + 2 \right )}}{16} + \frac{1}{8 \left (3 x^{4} + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**4)/16 - log(3*x**4 + 2)/16 + 1/(8*(3*x**4 + 2))

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Mathematica [A]  time = 0.015962, size = 32, normalized size = 1. \[ \frac{1}{8 \left (3 x^4+2\right )}-\frac{1}{16} \log \left (3 x^4+2\right )+\frac{\log (x)}{4} \]

Antiderivative was successfully verified.

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

[Out]

1/(8*(2 + 3*x^4)) + Log[x]/4 - Log[2 + 3*x^4]/16

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Maple [A]  time = 0.019, size = 27, normalized size = 0.8 \[{\frac{1}{24\,{x}^{4}+16}}+{\frac{\ln \left ( x \right ) }{4}}-{\frac{\ln \left ( 3\,{x}^{4}+2 \right ) }{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(3*x^4+2)^2,x)

[Out]

1/8/(3*x^4+2)+1/4*ln(x)-1/16*ln(3*x^4+2)

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Maxima [A]  time = 1.42523, size = 38, normalized size = 1.19 \[ \frac{1}{8 \,{\left (3 \, x^{4} + 2\right )}} - \frac{1}{16} \, \log \left (3 \, x^{4} + 2\right ) + \frac{1}{16} \, \log \left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8/(3*x^4 + 2) - 1/16*log(3*x^4 + 2) + 1/16*log(x^4)

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Fricas [A]  time = 0.229597, size = 54, normalized size = 1.69 \[ -\frac{{\left (3 \, x^{4} + 2\right )} \log \left (3 \, x^{4} + 2\right ) - 4 \,{\left (3 \, x^{4} + 2\right )} \log \left (x\right ) - 2}{16 \,{\left (3 \, x^{4} + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/16*((3*x^4 + 2)*log(3*x^4 + 2) - 4*(3*x^4 + 2)*log(x) - 2)/(3*x^4 + 2)

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Sympy [A]  time = 0.335434, size = 22, normalized size = 0.69 \[ \frac{\log{\left (x \right )}}{4} - \frac{\log{\left (3 x^{4} + 2 \right )}}{16} + \frac{1}{24 x^{4} + 16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(3*x**4+2)**2,x)

[Out]

log(x)/4 - log(3*x**4 + 2)/16 + 1/(24*x**4 + 16)

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GIAC/XCAS [A]  time = 0.226198, size = 47, normalized size = 1.47 \[ \frac{3 \, x^{4} + 4}{16 \,{\left (3 \, x^{4} + 2\right )}} - \frac{1}{16} \,{\rm ln}\left (3 \, x^{4} + 2\right ) + \frac{1}{16} \,{\rm ln}\left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*(3*x^4 + 4)/(3*x^4 + 2) - 1/16*ln(3*x^4 + 2) + 1/16*ln(x^4)