3.677 \(\int \frac{1+2 x^8}{x \left (1+x^8\right )^{3/2}} \, dx\)

Optimal. Leaf size=28 \[ -\frac{1}{4 \sqrt{x^8+1}}-\frac{1}{4} \tanh ^{-1}\left (\sqrt{x^8+1}\right ) \]

[Out]

-1/(4*Sqrt[1 + x^8]) - ArcTanh[Sqrt[1 + x^8]]/4

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Rubi [A]  time = 0.054886, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{1}{4 \sqrt{x^8+1}}-\frac{1}{4} \tanh ^{-1}\left (\sqrt{x^8+1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + 2*x^8)/(x*(1 + x^8)^(3/2)),x]

[Out]

-1/(4*Sqrt[1 + x^8]) - ArcTanh[Sqrt[1 + x^8]]/4

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Rubi in Sympy [A]  time = 3.70643, size = 24, normalized size = 0.86 \[ - \frac{\operatorname{atanh}{\left (\sqrt{x^{8} + 1} \right )}}{4} - \frac{1}{4 \sqrt{x^{8} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*x**8+1)/x/(x**8+1)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0389295, size = 28, normalized size = 1. \[ -\frac{1}{4 \sqrt{x^8+1}}-\frac{1}{4} \tanh ^{-1}\left (\sqrt{x^8+1}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + 2*x^8)/(x*(1 + x^8)^(3/2)),x]

[Out]

-1/(4*Sqrt[1 + x^8]) - ArcTanh[Sqrt[1 + x^8]]/4

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Maple [A]  time = 0.04, size = 29, normalized size = 1. \[ -{\frac{1}{4}{\frac{1}{\sqrt{{x}^{8}+1}}}}+{\frac{1}{4}\ln \left ({1 \left ( \sqrt{{x}^{8}+1}-1 \right ){\frac{1}{\sqrt{{x}^{8}}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*x^8+1)/x/(x^8+1)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.781892, size = 46, normalized size = 1.64 \[ -\frac{1}{4 \, \sqrt{x^{8} + 1}} - \frac{1}{8} \, \log \left (\sqrt{x^{8} + 1} + 1\right ) + \frac{1}{8} \, \log \left (\sqrt{x^{8} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^8 + 1)/((x^8 + 1)^(3/2)*x),x, algorithm="maxima")

[Out]

-1/4/sqrt(x^8 + 1) - 1/8*log(sqrt(x^8 + 1) + 1) + 1/8*log(sqrt(x^8 + 1) - 1)

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Fricas [A]  time = 0.28572, size = 65, normalized size = 2.32 \[ -\frac{\sqrt{x^{8} + 1} \log \left (\sqrt{x^{8} + 1} + 1\right ) - \sqrt{x^{8} + 1} \log \left (\sqrt{x^{8} + 1} - 1\right ) + 2}{8 \, \sqrt{x^{8} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^8 + 1)/((x^8 + 1)^(3/2)*x),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x**8+1)/x/(x**8+1)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.264582, size = 46, normalized size = 1.64 \[ -\frac{1}{4 \, \sqrt{x^{8} + 1}} - \frac{1}{8} \,{\rm ln}\left (\sqrt{x^{8} + 1} + 1\right ) + \frac{1}{8} \,{\rm ln}\left (\sqrt{x^{8} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^8 + 1)/((x^8 + 1)^(3/2)*x),x, algorithm="giac")

[Out]

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