3.1410 \(\int \frac{1}{x \left (2+x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=39 \[ \frac{1}{6 \sqrt{x^6+2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{6 \sqrt{2}} \]

[Out]

1/(6*Sqrt[2 + x^6]) - ArcTanh[Sqrt[2 + x^6]/Sqrt[2]]/(6*Sqrt[2])

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Rubi [A]  time = 0.0451154, antiderivative size = 39, 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 \[ \frac{1}{6 \sqrt{x^6+2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{6 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(2 + x^6)^(3/2)),x]

[Out]

1/(6*Sqrt[2 + x^6]) - ArcTanh[Sqrt[2 + x^6]/Sqrt[2]]/(6*Sqrt[2])

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Rubi in Sympy [A]  time = 4.36297, size = 34, normalized size = 0.87 \[ - \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{x^{6} + 2}}{2} \right )}}{12} + \frac{1}{6 \sqrt{x^{6} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(x**6+2)**(3/2),x)

[Out]

-sqrt(2)*atanh(sqrt(2)*sqrt(x**6 + 2)/2)/12 + 1/(6*sqrt(x**6 + 2))

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Mathematica [A]  time = 0.0454155, size = 39, normalized size = 1. \[ \frac{1}{6 \sqrt{x^6+2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{6 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(2 + x^6)^(3/2)),x]

[Out]

1/(6*Sqrt[2 + x^6]) - ArcTanh[Sqrt[2 + x^6]/Sqrt[2]]/(6*Sqrt[2])

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Maple [A]  time = 0.027, size = 36, normalized size = 0.9 \[{\frac{1}{6}{\frac{1}{\sqrt{{x}^{6}+2}}}}+{\frac{\sqrt{2}}{12}\ln \left ({1 \left ( \sqrt{{x}^{6}+2}-\sqrt{2} \right ){\frac{1}{\sqrt{{x}^{6}}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(x^6+2)^(3/2),x)

[Out]

1/6/(x^6+2)^(1/2)+1/12*2^(1/2)*ln(((x^6+2)^(1/2)-2^(1/2))/(x^6)^(1/2))

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Maxima [A]  time = 1.59787, size = 62, normalized size = 1.59 \[ \frac{1}{24} \, \sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2} - \sqrt{x^{6} + 2}\right )}}{2 \, \sqrt{2} + 2 \, \sqrt{x^{6} + 2}}\right ) + \frac{1}{6 \, \sqrt{x^{6} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 2)^(3/2)*x),x, algorithm="maxima")

[Out]

1/24*sqrt(2)*log(-2*(sqrt(2) - sqrt(x^6 + 2))/((2*sqrt(2)) + 2*sqrt(x^6 + 2))) +
1/6/sqrt(x^6 + 2)

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Fricas [A]  time = 0.221216, size = 68, normalized size = 1.74 \[ \frac{\sqrt{2}{\left (\sqrt{x^{6} + 2} \log \left (\frac{\sqrt{2}{\left (x^{6} + 4\right )} - 4 \, \sqrt{x^{6} + 2}}{x^{6}}\right ) + 2 \, \sqrt{2}\right )}}{24 \, \sqrt{x^{6} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 2)^(3/2)*x),x, algorithm="fricas")

[Out]

1/24*sqrt(2)*(sqrt(x^6 + 2)*log((sqrt(2)*(x^6 + 4) - 4*sqrt(x^6 + 2))/x^6) + 2*s
qrt(2))/sqrt(x^6 + 2)

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Sympy [A]  time = 4.97472, size = 194, normalized size = 4.97 \[ \frac{x^{6} \log{\left (x^{6} \right )}}{12 \sqrt{2} x^{6} + 24 \sqrt{2}} - \frac{2 x^{6} \log{\left (\sqrt{\frac{x^{6}}{2} + 1} + 1 \right )}}{12 \sqrt{2} x^{6} + 24 \sqrt{2}} - \frac{x^{6} \log{\left (2 \right )}}{12 \sqrt{2} x^{6} + 24 \sqrt{2}} + \frac{2 \sqrt{2} \sqrt{x^{6} + 2}}{12 \sqrt{2} x^{6} + 24 \sqrt{2}} + \frac{2 \log{\left (x^{6} \right )}}{12 \sqrt{2} x^{6} + 24 \sqrt{2}} - \frac{4 \log{\left (\sqrt{\frac{x^{6}}{2} + 1} + 1 \right )}}{12 \sqrt{2} x^{6} + 24 \sqrt{2}} - \frac{2 \log{\left (2 \right )}}{12 \sqrt{2} x^{6} + 24 \sqrt{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(x**6+2)**(3/2),x)

[Out]

x**6*log(x**6)/(12*sqrt(2)*x**6 + 24*sqrt(2)) - 2*x**6*log(sqrt(x**6/2 + 1) + 1)
/(12*sqrt(2)*x**6 + 24*sqrt(2)) - x**6*log(2)/(12*sqrt(2)*x**6 + 24*sqrt(2)) + 2
*sqrt(2)*sqrt(x**6 + 2)/(12*sqrt(2)*x**6 + 24*sqrt(2)) + 2*log(x**6)/(12*sqrt(2)
*x**6 + 24*sqrt(2)) - 4*log(sqrt(x**6/2 + 1) + 1)/(12*sqrt(2)*x**6 + 24*sqrt(2))
- 2*log(2)/(12*sqrt(2)*x**6 + 24*sqrt(2))

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GIAC/XCAS [A]  time = 0.221499, size = 59, normalized size = 1.51 \[ \frac{1}{24} \, \sqrt{2}{\rm ln}\left (-\frac{\sqrt{2} - \sqrt{x^{6} + 2}}{\sqrt{2} + \sqrt{x^{6} + 2}}\right ) + \frac{1}{6 \, \sqrt{x^{6} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 2)^(3/2)*x),x, algorithm="giac")

[Out]

1/24*sqrt(2)*ln(-(sqrt(2) - sqrt(x^6 + 2))/(sqrt(2) + sqrt(x^6 + 2))) + 1/6/sqrt
(x^6 + 2)