3.1509 \(\int \frac{\sqrt{1+x^8}}{x} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0367779, antiderivative size = 28, 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 \[ \frac{\sqrt{x^8+1}}{4}-\frac{1}{4} \tanh ^{-1}\left (\sqrt{x^8+1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 + x^8]/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 4.10398, size = 20, normalized size = 0.71 \[ \frac{\sqrt{x^{8} + 1}}{4} - \frac{\operatorname{atanh}{\left (\sqrt{x^{8} + 1} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0214165, size = 28, normalized size = 1. \[ \frac{\sqrt{x^8+1}}{4}-\frac{1}{4} \tanh ^{-1}\left (\sqrt{x^8+1}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 + x^8]/x,x]

[Out]

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

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Maple [B]  time = 0.055, size = 56, normalized size = 2. \[ -{\frac{1}{16\,\sqrt{\pi }} \left ( -2\, \left ( 2-2\,\ln \left ( 2 \right ) +8\,\ln \left ( x \right ) \right ) \sqrt{\pi }+4\,\sqrt{\pi }-4\,\sqrt{\pi }\sqrt{{x}^{8}+1}+4\,\sqrt{\pi }\ln \left ( 1/2+1/2\,\sqrt{{x}^{8}+1} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^8+1)^(1/2)/x,x)

[Out]

-1/16/Pi^(1/2)*(-2*(2-2*ln(2)+8*ln(x))*Pi^(1/2)+4*Pi^(1/2)-4*Pi^(1/2)*(x^8+1)^(1
/2)+4*Pi^(1/2)*ln(1/2+1/2*(x^8+1)^(1/2)))

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Maxima [A]  time = 1.45816, 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(sqrt(x^8 + 1)/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 = 0.225928, 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(sqrt(x^8 + 1)/x,x, algorithm="fricas")

[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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Sympy [A]  time = 4.08762, size = 39, normalized size = 1.39 \[ \frac{x^{4}}{4 \sqrt{1 + \frac{1}{x^{8}}}} - \frac{\operatorname{asinh}{\left (\frac{1}{x^{4}} \right )}}{4} + \frac{1}{4 x^{4} \sqrt{1 + \frac{1}{x^{8}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**8+1)**(1/2)/x,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.225921, 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(sqrt(x^8 + 1)/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)