3.156 \(\int \frac{1}{x^4 \left (3+4 x^3+x^6\right )} \, dx\)

Optimal. Leaf size=34 \[ -\frac{1}{9 x^3}+\frac{1}{6} \log \left (x^3+1\right )-\frac{1}{54} \log \left (x^3+3\right )-\frac{4 \log (x)}{9} \]

[Out]

-1/(9*x^3) - (4*Log[x])/9 + Log[1 + x^3]/6 - Log[3 + x^3]/54

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Rubi [A]  time = 0.0757246, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ -\frac{1}{9 x^3}+\frac{1}{6} \log \left (x^3+1\right )-\frac{1}{54} \log \left (x^3+3\right )-\frac{4 \log (x)}{9} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(3 + 4*x^3 + x^6)),x]

[Out]

-1/(9*x^3) - (4*Log[x])/9 + Log[1 + x^3]/6 - Log[3 + x^3]/54

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Rubi in Sympy [A]  time = 16.4955, size = 31, normalized size = 0.91 \[ - \frac{4 \log{\left (x^{3} \right )}}{27} + \frac{\log{\left (x^{3} + 1 \right )}}{6} - \frac{\log{\left (x^{3} + 3 \right )}}{54} - \frac{1}{9 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(x**6+4*x**3+3),x)

[Out]

-4*log(x**3)/27 + log(x**3 + 1)/6 - log(x**3 + 3)/54 - 1/(9*x**3)

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Mathematica [A]  time = 0.00963341, size = 34, normalized size = 1. \[ -\frac{1}{9 x^3}+\frac{1}{6} \log \left (x^3+1\right )-\frac{1}{54} \log \left (x^3+3\right )-\frac{4 \log (x)}{9} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(3 + 4*x^3 + x^6)),x]

[Out]

-1/(9*x^3) - (4*Log[x])/9 + Log[1 + x^3]/6 - Log[3 + x^3]/54

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Maple [A]  time = 0.013, size = 36, normalized size = 1.1 \[ -{\frac{\ln \left ({x}^{3}+3 \right ) }{54}}+{\frac{\ln \left ( 1+x \right ) }{6}}-{\frac{1}{9\,{x}^{3}}}-{\frac{4\,\ln \left ( x \right ) }{9}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) }{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(x^6+4*x^3+3),x)

[Out]

-1/54*ln(x^3+3)+1/6*ln(1+x)-1/9/x^3-4/9*ln(x)+1/6*ln(x^2-x+1)

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Maxima [A]  time = 0.761698, size = 38, normalized size = 1.12 \[ -\frac{1}{9 \, x^{3}} - \frac{1}{54} \, \log \left (x^{3} + 3\right ) + \frac{1}{6} \, \log \left (x^{3} + 1\right ) - \frac{4}{27} \, \log \left (x^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 4*x^3 + 3)*x^4),x, algorithm="maxima")

[Out]

-1/9/x^3 - 1/54*log(x^3 + 3) + 1/6*log(x^3 + 1) - 4/27*log(x^3)

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Fricas [A]  time = 0.253134, size = 47, normalized size = 1.38 \[ -\frac{x^{3} \log \left (x^{3} + 3\right ) - 9 \, x^{3} \log \left (x^{3} + 1\right ) + 24 \, x^{3} \log \left (x\right ) + 6}{54 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 4*x^3 + 3)*x^4),x, algorithm="fricas")

[Out]

-1/54*(x^3*log(x^3 + 3) - 9*x^3*log(x^3 + 1) + 24*x^3*log(x) + 6)/x^3

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Sympy [A]  time = 0.421574, size = 29, normalized size = 0.85 \[ - \frac{4 \log{\left (x \right )}}{9} + \frac{\log{\left (x^{3} + 1 \right )}}{6} - \frac{\log{\left (x^{3} + 3 \right )}}{54} - \frac{1}{9 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(x**6+4*x**3+3),x)

[Out]

-4*log(x)/9 + log(x**3 + 1)/6 - log(x**3 + 3)/54 - 1/(9*x**3)

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GIAC/XCAS [A]  time = 0.283973, size = 49, normalized size = 1.44 \[ \frac{4 \, x^{3} - 3}{27 \, x^{3}} - \frac{1}{54} \,{\rm ln}\left ({\left | x^{3} + 3 \right |}\right ) + \frac{1}{6} \,{\rm ln}\left ({\left | x^{3} + 1 \right |}\right ) - \frac{4}{9} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 4*x^3 + 3)*x^4),x, algorithm="giac")

[Out]

1/27*(4*x^3 - 3)/x^3 - 1/54*ln(abs(x^3 + 3)) + 1/6*ln(abs(x^3 + 1)) - 4/9*ln(abs
(x))