3.1387 \(\int \frac{1}{x^{13} \sqrt{2+x^6}} \, dx\)

Optimal. Leaf size=58 \[ \frac{\sqrt{x^6+2}}{32 x^6}-\frac{\tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{32 \sqrt{2}}-\frac{\sqrt{x^6+2}}{24 x^{12}} \]

[Out]

-Sqrt[2 + x^6]/(24*x^12) + Sqrt[2 + x^6]/(32*x^6) - ArcTanh[Sqrt[2 + x^6]/Sqrt[2
]]/(32*Sqrt[2])

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Rubi [A]  time = 0.0588679, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{\sqrt{x^6+2}}{32 x^6}-\frac{\tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{32 \sqrt{2}}-\frac{\sqrt{x^6+2}}{24 x^{12}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^13*Sqrt[2 + x^6]),x]

[Out]

-Sqrt[2 + x^6]/(24*x^12) + Sqrt[2 + x^6]/(32*x^6) - ArcTanh[Sqrt[2 + x^6]/Sqrt[2
]]/(32*Sqrt[2])

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Rubi in Sympy [A]  time = 5.21242, size = 49, normalized size = 0.84 \[ - \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{x^{6} + 2}}{2} \right )}}{64} + \frac{\sqrt{x^{6} + 2}}{32 x^{6}} - \frac{\sqrt{x^{6} + 2}}{24 x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**13/(x**6+2)**(1/2),x)

[Out]

-sqrt(2)*atanh(sqrt(2)*sqrt(x**6 + 2)/2)/64 + sqrt(x**6 + 2)/(32*x**6) - sqrt(x*
*6 + 2)/(24*x**12)

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Mathematica [A]  time = 0.0498806, size = 51, normalized size = 0.88 \[ \left (\frac{1}{32 x^6}-\frac{1}{24 x^{12}}\right ) \sqrt{x^6+2}-\frac{\tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{32 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^13*Sqrt[2 + x^6]),x]

[Out]

(-1/(24*x^12) + 1/(32*x^6))*Sqrt[2 + x^6] - ArcTanh[Sqrt[2 + x^6]/Sqrt[2]]/(32*S
qrt[2])

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Maple [A]  time = 0.034, size = 51, normalized size = 0.9 \[{\frac{3\,{x}^{12}+2\,{x}^{6}-8}{96\,{x}^{12}}{\frac{1}{\sqrt{{x}^{6}+2}}}}+{\frac{\sqrt{2}}{64}\ln \left ({1 \left ( \sqrt{{x}^{6}+2}-\sqrt{2} \right ){\frac{1}{\sqrt{{x}^{6}}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^13/(x^6+2)^(1/2),x)

[Out]

1/96*(3*x^12+2*x^6-8)/x^12/(x^6+2)^(1/2)+1/64*2^(1/2)*ln(((x^6+2)^(1/2)-2^(1/2))
/(x^6)^(1/2))

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Maxima [A]  time = 1.59047, size = 103, normalized size = 1.78 \[ \frac{1}{128} \, \sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2} - \sqrt{x^{6} + 2}\right )}}{2 \, \sqrt{2} + 2 \, \sqrt{x^{6} + 2}}\right ) - \frac{3 \,{\left (x^{6} + 2\right )}^{\frac{3}{2}} - 10 \, \sqrt{x^{6} + 2}}{96 \,{\left (4 \, x^{6} -{\left (x^{6} + 2\right )}^{2} + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^6 + 2)*x^13),x, algorithm="maxima")

[Out]

1/128*sqrt(2)*log(-2*(sqrt(2) - sqrt(x^6 + 2))/((2*sqrt(2)) + 2*sqrt(x^6 + 2)))
- 1/96*(3*(x^6 + 2)^(3/2) - 10*sqrt(x^6 + 2))/(4*x^6 - (x^6 + 2)^2 + 4)

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Fricas [A]  time = 0.225567, size = 77, normalized size = 1.33 \[ \frac{\sqrt{2}{\left (3 \, x^{12} \log \left (\frac{\sqrt{2}{\left (x^{6} + 4\right )} - 4 \, \sqrt{x^{6} + 2}}{x^{6}}\right ) + 2 \, \sqrt{2}{\left (3 \, x^{6} - 4\right )} \sqrt{x^{6} + 2}\right )}}{384 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^6 + 2)*x^13),x, algorithm="fricas")

[Out]

1/384*sqrt(2)*(3*x^12*log((sqrt(2)*(x^6 + 4) - 4*sqrt(x^6 + 2))/x^6) + 2*sqrt(2)
*(3*x^6 - 4)*sqrt(x^6 + 2))/x^12

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Sympy [A]  time = 16.7862, size = 66, normalized size = 1.14 \[ - \frac{\sqrt{2} \operatorname{asinh}{\left (\frac{\sqrt{2}}{x^{3}} \right )}}{64} + \frac{1}{32 x^{3} \sqrt{1 + \frac{2}{x^{6}}}} + \frac{1}{48 x^{9} \sqrt{1 + \frac{2}{x^{6}}}} - \frac{1}{12 x^{15} \sqrt{1 + \frac{2}{x^{6}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**13/(x**6+2)**(1/2),x)

[Out]

-sqrt(2)*asinh(sqrt(2)/x**3)/64 + 1/(32*x**3*sqrt(1 + 2/x**6)) + 1/(48*x**9*sqrt
(1 + 2/x**6)) - 1/(12*x**15*sqrt(1 + 2/x**6))

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GIAC/XCAS [A]  time = 0.225791, size = 80, normalized size = 1.38 \[ \frac{1}{128} \, \sqrt{2}{\rm ln}\left (-\frac{\sqrt{2} - \sqrt{x^{6} + 2}}{\sqrt{2} + \sqrt{x^{6} + 2}}\right ) + \frac{3 \,{\left (x^{6} + 2\right )}^{\frac{3}{2}} - 10 \, \sqrt{x^{6} + 2}}{96 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^6 + 2)*x^13),x, algorithm="giac")

[Out]

1/128*sqrt(2)*ln(-(sqrt(2) - sqrt(x^6 + 2))/(sqrt(2) + sqrt(x^6 + 2))) + 1/96*(3
*(x^6 + 2)^(3/2) - 10*sqrt(x^6 + 2))/x^12