3.1497 \(\int \frac{1}{x^9 \left (1+x^8\right )} \, dx\)

Optimal. Leaf size=22 \[ -\frac{1}{8 x^8}+\frac{1}{8} \log \left (x^8+1\right )-\log (x) \]

[Out]

-1/(8*x^8) - Log[x] + Log[1 + x^8]/8

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Rubi [A]  time = 0.0308534, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{1}{8 x^8}+\frac{1}{8} \log \left (x^8+1\right )-\log (x) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^9*(1 + x^8)),x]

[Out]

-1/(8*x^8) - Log[x] + Log[1 + x^8]/8

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Rubi in Sympy [A]  time = 3.97454, size = 20, normalized size = 0.91 \[ - \frac{\log{\left (x^{8} \right )}}{8} + \frac{\log{\left (x^{8} + 1 \right )}}{8} - \frac{1}{8 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**9/(x**8+1),x)

[Out]

-log(x**8)/8 + log(x**8 + 1)/8 - 1/(8*x**8)

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Mathematica [A]  time = 0.00643646, size = 22, normalized size = 1. \[ -\frac{1}{8 x^8}+\frac{1}{8} \log \left (x^8+1\right )-\log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^9*(1 + x^8)),x]

[Out]

-1/(8*x^8) - Log[x] + Log[1 + x^8]/8

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Maple [A]  time = 0.009, size = 19, normalized size = 0.9 \[ -{\frac{1}{8\,{x}^{8}}}-\ln \left ( x \right ) +{\frac{\ln \left ({x}^{8}+1 \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^9/(x^8+1),x)

[Out]

-1/8/x^8-ln(x)+1/8*ln(x^8+1)

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Maxima [A]  time = 1.42303, size = 27, normalized size = 1.23 \[ -\frac{1}{8 \, x^{8}} + \frac{1}{8} \, \log \left (x^{8} + 1\right ) - \frac{1}{8} \, \log \left (x^{8}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 1)*x^9),x, algorithm="maxima")

[Out]

-1/8/x^8 + 1/8*log(x^8 + 1) - 1/8*log(x^8)

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Fricas [A]  time = 0.213375, size = 32, normalized size = 1.45 \[ \frac{x^{8} \log \left (x^{8} + 1\right ) - 8 \, x^{8} \log \left (x\right ) - 1}{8 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 1)*x^9),x, algorithm="fricas")

[Out]

1/8*(x^8*log(x^8 + 1) - 8*x^8*log(x) - 1)/x^8

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Sympy [A]  time = 0.479366, size = 17, normalized size = 0.77 \[ - \log{\left (x \right )} + \frac{\log{\left (x^{8} + 1 \right )}}{8} - \frac{1}{8 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**9/(x**8+1),x)

[Out]

-log(x) + log(x**8 + 1)/8 - 1/(8*x**8)

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GIAC/XCAS [A]  time = 0.238277, size = 34, normalized size = 1.55 \[ \frac{x^{8} - 1}{8 \, x^{8}} + \frac{1}{8} \,{\rm ln}\left (x^{8} + 1\right ) - \frac{1}{8} \,{\rm ln}\left (x^{8}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 1)*x^9),x, algorithm="giac")

[Out]

1/8*(x^8 - 1)/x^8 + 1/8*ln(x^8 + 1) - 1/8*ln(x^8)