3.157 \(\int \frac{1}{x^7 \left (3+4 x^3+x^6\right )} \, dx\)

Optimal. Leaf size=41 \[ -\frac{1}{18 x^6}+\frac{4}{27 x^3}-\frac{1}{6} \log \left (x^3+1\right )+\frac{1}{162} \log \left (x^3+3\right )+\frac{13 \log (x)}{27} \]

[Out]

-1/(18*x^6) + 4/(27*x^3) + (13*Log[x])/27 - Log[1 + x^3]/6 + Log[3 + x^3]/162

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Rubi [A]  time = 0.0852755, antiderivative size = 41, 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 \[ -\frac{1}{18 x^6}+\frac{4}{27 x^3}-\frac{1}{6} \log \left (x^3+1\right )+\frac{1}{162} \log \left (x^3+3\right )+\frac{13 \log (x)}{27} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^7*(3 + 4*x^3 + x^6)),x]

[Out]

-1/(18*x^6) + 4/(27*x^3) + (13*Log[x])/27 - Log[1 + x^3]/6 + Log[3 + x^3]/162

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Rubi in Sympy [A]  time = 15.1415, size = 37, normalized size = 0.9 \[ \frac{13 \log{\left (x^{3} \right )}}{81} - \frac{\log{\left (x^{3} + 1 \right )}}{6} + \frac{\log{\left (x^{3} + 3 \right )}}{162} + \frac{4}{27 x^{3}} - \frac{1}{18 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**7/(x**6+4*x**3+3),x)

[Out]

13*log(x**3)/81 - log(x**3 + 1)/6 + log(x**3 + 3)/162 + 4/(27*x**3) - 1/(18*x**6
)

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Mathematica [A]  time = 0.00949006, size = 41, normalized size = 1. \[ -\frac{1}{18 x^6}+\frac{4}{27 x^3}-\frac{1}{6} \log \left (x^3+1\right )+\frac{1}{162} \log \left (x^3+3\right )+\frac{13 \log (x)}{27} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^7*(3 + 4*x^3 + x^6)),x]

[Out]

-1/(18*x^6) + 4/(27*x^3) + (13*Log[x])/27 - Log[1 + x^3]/6 + Log[3 + x^3]/162

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Maple [A]  time = 0.013, size = 41, normalized size = 1. \[{\frac{\ln \left ({x}^{3}+3 \right ) }{162}}-{\frac{\ln \left ( 1+x \right ) }{6}}-{\frac{1}{18\,{x}^{6}}}+{\frac{4}{27\,{x}^{3}}}+{\frac{13\,\ln \left ( x \right ) }{27}}-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(x^6+4*x^3+3),x)

[Out]

1/162*ln(x^3+3)-1/6*ln(1+x)-1/18/x^6+4/27/x^3+13/27*ln(x)-1/6*ln(x^2-x+1)

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Maxima [A]  time = 0.772754, size = 47, normalized size = 1.15 \[ \frac{8 \, x^{3} - 3}{54 \, x^{6}} + \frac{1}{162} \, \log \left (x^{3} + 3\right ) - \frac{1}{6} \, \log \left (x^{3} + 1\right ) + \frac{13}{81} \, \log \left (x^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 4*x^3 + 3)*x^7),x, algorithm="maxima")

[Out]

1/54*(8*x^3 - 3)/x^6 + 1/162*log(x^3 + 3) - 1/6*log(x^3 + 1) + 13/81*log(x^3)

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Fricas [A]  time = 0.249315, size = 54, normalized size = 1.32 \[ \frac{x^{6} \log \left (x^{3} + 3\right ) - 27 \, x^{6} \log \left (x^{3} + 1\right ) + 78 \, x^{6} \log \left (x\right ) + 24 \, x^{3} - 9}{162 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 4*x^3 + 3)*x^7),x, algorithm="fricas")

[Out]

1/162*(x^6*log(x^3 + 3) - 27*x^6*log(x^3 + 1) + 78*x^6*log(x) + 24*x^3 - 9)/x^6

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Sympy [A]  time = 0.532243, size = 34, normalized size = 0.83 \[ \frac{13 \log{\left (x \right )}}{27} - \frac{\log{\left (x^{3} + 1 \right )}}{6} + \frac{\log{\left (x^{3} + 3 \right )}}{162} + \frac{8 x^{3} - 3}{54 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**7/(x**6+4*x**3+3),x)

[Out]

13*log(x)/27 - log(x**3 + 1)/6 + log(x**3 + 3)/162 + (8*x**3 - 3)/(54*x**6)

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GIAC/XCAS [A]  time = 0.303019, size = 55, normalized size = 1.34 \[ -\frac{13 \, x^{6} - 8 \, x^{3} + 3}{54 \, x^{6}} + \frac{1}{162} \,{\rm ln}\left ({\left | x^{3} + 3 \right |}\right ) - \frac{1}{6} \,{\rm ln}\left ({\left | x^{3} + 1 \right |}\right ) + \frac{13}{27} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 4*x^3 + 3)*x^7),x, algorithm="giac")

[Out]

-1/54*(13*x^6 - 8*x^3 + 3)/x^6 + 1/162*ln(abs(x^3 + 3)) - 1/6*ln(abs(x^3 + 1)) +
13/27*ln(abs(x))