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

Optimal. Leaf size=14 \[ -\frac{1}{4} \tanh ^{-1}\left (\sqrt{x^8+1}\right ) \]

[Out]

-ArcTanh[Sqrt[1 + x^8]]/4

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Rubi [A]  time = 0.0191513, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{1}{4} \tanh ^{-1}\left (\sqrt{x^8+1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[1 + x^8]),x]

[Out]

-ArcTanh[Sqrt[1 + x^8]]/4

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Rubi in Sympy [A]  time = 1.51609, size = 12, normalized size = 0.86 \[ - \frac{\operatorname{atanh}{\left (\sqrt{x^{8} + 1} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-atanh(sqrt(x**8 + 1))/4

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Mathematica [A]  time = 0.0138991, size = 14, normalized size = 1. \[ -\frac{1}{4} \tanh ^{-1}\left (\sqrt{x^8+1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-ArcTanh[Sqrt[1 + x^8]]/4

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Maple [A]  time = 0., size = 19, normalized size = 1.4 \[{\frac{1}{4}\ln \left ({1 \left ( \sqrt{{x}^{8}+1}-1 \right ){\frac{1}{\sqrt{{x}^{8}}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^8+1)^(1/2)/x,x)

[Out]

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

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Maxima [A]  time = 1.42497, size = 34, normalized size = 2.43 \[ -\frac{1}{8} \, \log \left (\sqrt{x^{8} + 1} + 1\right ) + \frac{1}{8} \, \log \left (\sqrt{x^{8} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.203985, size = 34, normalized size = 2.43 \[ -\frac{1}{8} \, \log \left (\sqrt{x^{8} + 1} + 1\right ) + \frac{1}{8} \, \log \left (\sqrt{x^{8} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.5332, size = 8, normalized size = 0.57 \[ - \frac{\operatorname{asinh}{\left (\frac{1}{x^{4}} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-asinh(x**(-4))/4

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GIAC/XCAS [A]  time = 0.199463, size = 34, normalized size = 2.43 \[ -\frac{1}{8} \,{\rm ln}\left (\sqrt{x^{8} + 1} + 1\right ) + \frac{1}{8} \,{\rm ln}\left (\sqrt{x^{8} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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