3.1385 \(\int \frac{1}{x \sqrt{2+x^6}} \, dx\)

Optimal. Leaf size=25 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{3 \sqrt{2}} \]

[Out]

-ArcTanh[Sqrt[2 + x^6]/Sqrt[2]]/(3*Sqrt[2])

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Rubi [A]  time = 0.0327199, antiderivative size = 25, 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 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{3 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[2 + x^6]),x]

[Out]

-ArcTanh[Sqrt[2 + x^6]/Sqrt[2]]/(3*Sqrt[2])

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Rubi in Sympy [A]  time = 3.5214, size = 24, normalized size = 0.96 \[ - \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{x^{6} + 2}}{2} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0235229, size = 25, normalized size = 1. \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{3 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

-ArcTanh[Sqrt[2 + x^6]/Sqrt[2]]/(3*Sqrt[2])

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Maple [A]  time = 0.008, size = 26, normalized size = 1. \[{\frac{\sqrt{2}}{6}\ln \left ({1 \left ( \sqrt{{x}^{6}+2}-\sqrt{2} \right ){\frac{1}{\sqrt{{x}^{6}}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(x^6+2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.58148, size = 49, normalized size = 1.96 \[ \frac{1}{12} \, \sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2} - \sqrt{x^{6} + 2}\right )}}{2 \, \sqrt{2} + 2 \, \sqrt{x^{6} + 2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.227217, size = 39, normalized size = 1.56 \[ \frac{1}{12} \, \sqrt{2} \log \left (\frac{\sqrt{2}{\left (x^{6} + 4\right )} - 4 \, \sqrt{x^{6} + 2}}{x^{6}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.38711, size = 17, normalized size = 0.68 \[ - \frac{\sqrt{2} \operatorname{asinh}{\left (\frac{\sqrt{2}}{x^{3}} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*asinh(sqrt(2)/x**3)/6

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GIAC/XCAS [A]  time = 0.223045, size = 50, normalized size = 2. \[ -\frac{1}{12} \, \sqrt{2}{\rm ln}\left (\sqrt{2} + \sqrt{x^{6} + 2}\right ) + \frac{1}{12} \, \sqrt{2}{\rm ln}\left (-\sqrt{2} + \sqrt{x^{6} + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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