3.1519 \(\int \frac{1}{x^9 \sqrt{1+x^8}} \, dx\)

Optimal. Leaf size=31 \[ \frac{1}{8} \tanh ^{-1}\left (\sqrt{x^8+1}\right )-\frac{\sqrt{x^8+1}}{8 x^8} \]

[Out]

-Sqrt[1 + x^8]/(8*x^8) + ArcTanh[Sqrt[1 + x^8]]/8

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Rubi [A]  time = 0.0392923, antiderivative size = 31, 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 \[ \frac{1}{8} \tanh ^{-1}\left (\sqrt{x^8+1}\right )-\frac{\sqrt{x^8+1}}{8 x^8} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^9*Sqrt[1 + x^8]),x]

[Out]

-Sqrt[1 + x^8]/(8*x^8) + ArcTanh[Sqrt[1 + x^8]]/8

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Rubi in Sympy [A]  time = 4.19084, size = 24, normalized size = 0.77 \[ \frac{\operatorname{atanh}{\left (\sqrt{x^{8} + 1} \right )}}{8} - \frac{\sqrt{x^{8} + 1}}{8 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**9/(x**8+1)**(1/2),x)

[Out]

atanh(sqrt(x**8 + 1))/8 - sqrt(x**8 + 1)/(8*x**8)

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Mathematica [A]  time = 0.0311539, size = 31, normalized size = 1. \[ \frac{1}{8} \tanh ^{-1}\left (\sqrt{x^8+1}\right )-\frac{\sqrt{x^8+1}}{8 x^8} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^9*Sqrt[1 + x^8]),x]

[Out]

-Sqrt[1 + x^8]/(8*x^8) + ArcTanh[Sqrt[1 + x^8]]/8

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Maple [A]  time = 0.036, size = 32, normalized size = 1. \[ -{\frac{1}{8\,{x}^{8}}\sqrt{{x}^{8}+1}}-{\frac{1}{8}\ln \left ({1 \left ( \sqrt{{x}^{8}+1}-1 \right ){\frac{1}{\sqrt{{x}^{8}}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^9/(x^8+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.44261, size = 50, normalized size = 1.61 \[ -\frac{\sqrt{x^{8} + 1}}{8 \, x^{8}} + \frac{1}{16} \, \log \left (\sqrt{x^{8} + 1} + 1\right ) - \frac{1}{16} \, \log \left (\sqrt{x^{8} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*sqrt(x^8 + 1)/x^8 + 1/16*log(sqrt(x^8 + 1) + 1) - 1/16*log(sqrt(x^8 + 1) -
1)

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Fricas [A]  time = 0.239145, size = 59, normalized size = 1.9 \[ \frac{x^{8} \log \left (\sqrt{x^{8} + 1} + 1\right ) - x^{8} \log \left (\sqrt{x^{8} + 1} - 1\right ) - 2 \, \sqrt{x^{8} + 1}}{16 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 8.50369, size = 22, normalized size = 0.71 \[ \frac{\operatorname{asinh}{\left (\frac{1}{x^{4}} \right )}}{8} - \frac{\sqrt{1 + \frac{1}{x^{8}}}}{8 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**9/(x**8+1)**(1/2),x)

[Out]

asinh(x**(-4))/8 - sqrt(1 + x**(-8))/(8*x**4)

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GIAC/XCAS [A]  time = 0.225387, size = 50, normalized size = 1.61 \[ -\frac{\sqrt{x^{8} + 1}}{8 \, x^{8}} + \frac{1}{16} \,{\rm ln}\left (\sqrt{x^{8} + 1} + 1\right ) - \frac{1}{16} \,{\rm ln}\left (\sqrt{x^{8} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*sqrt(x^8 + 1)/x^8 + 1/16*ln(sqrt(x^8 + 1) + 1) - 1/16*ln(sqrt(x^8 + 1) - 1)