3.1386 \(\int \frac{1}{x^7 \sqrt{2+x^6}} \, dx\)

Optimal. Leaf size=42 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{12 \sqrt{2}}-\frac{\sqrt{x^6+2}}{12 x^6} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0454894, antiderivative size = 42, 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 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{12 \sqrt{2}}-\frac{\sqrt{x^6+2}}{12 x^6} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^7*Sqrt[2 + x^6]),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.25512, size = 36, normalized size = 0.86 \[ \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{x^{6} + 2}}{2} \right )}}{24} - \frac{\sqrt{x^{6} + 2}}{12 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**7/(x**6+2)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0365888, size = 42, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{12 \sqrt{2}}-\frac{\sqrt{x^6+2}}{12 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^7*Sqrt[2 + x^6]),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.035, size = 39, normalized size = 0.9 \[ -{\frac{1}{12\,{x}^{6}}\sqrt{{x}^{6}+2}}-{\frac{\sqrt{2}}{24}\ln \left ({1 \left ( \sqrt{{x}^{6}+2}-\sqrt{2} \right ){\frac{1}{\sqrt{{x}^{6}}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(x^6+2)^(1/2),x)

[Out]

-1/12*(x^6+2)^(1/2)/x^6-1/24*2^(1/2)*ln(((x^6+2)^(1/2)-2^(1/2))/(x^6)^(1/2))

_______________________________________________________________________________________

Maxima [A]  time = 1.58517, size = 66, normalized size = 1.57 \[ -\frac{1}{48} \, \sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2} - \sqrt{x^{6} + 2}\right )}}{2 \, \sqrt{2} + 2 \, \sqrt{x^{6} + 2}}\right ) - \frac{\sqrt{x^{6} + 2}}{12 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^6 + 2)*x^7),x, algorithm="maxima")

[Out]

-1/48*sqrt(2)*log(-2*(sqrt(2) - sqrt(x^6 + 2))/((2*sqrt(2)) + 2*sqrt(x^6 + 2)))
- 1/12*sqrt(x^6 + 2)/x^6

_______________________________________________________________________________________

Fricas [A]  time = 0.225992, size = 66, normalized size = 1.57 \[ \frac{\sqrt{2}{\left (x^{6} \log \left (\frac{\sqrt{2}{\left (x^{6} + 4\right )} + 4 \, \sqrt{x^{6} + 2}}{x^{6}}\right ) - 2 \, \sqrt{2} \sqrt{x^{6} + 2}\right )}}{48 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^6 + 2)*x^7),x, algorithm="fricas")

[Out]

1/48*sqrt(2)*(x^6*log((sqrt(2)*(x^6 + 4) + 4*sqrt(x^6 + 2))/x^6) - 2*sqrt(2)*sqr
t(x^6 + 2))/x^6

_______________________________________________________________________________________

Sympy [A]  time = 7.03888, size = 31, normalized size = 0.74 \[ \frac{\sqrt{2} \operatorname{asinh}{\left (\frac{\sqrt{2}}{x^{3}} \right )}}{24} - \frac{\sqrt{1 + \frac{2}{x^{6}}}}{12 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**7/(x**6+2)**(1/2),x)

[Out]

sqrt(2)*asinh(sqrt(2)/x**3)/24 - sqrt(1 + 2/x**6)/(12*x**3)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.228356, size = 63, normalized size = 1.5 \[ -\frac{1}{48} \, \sqrt{2}{\rm ln}\left (-\frac{\sqrt{2} - \sqrt{x^{6} + 2}}{\sqrt{2} + \sqrt{x^{6} + 2}}\right ) - \frac{\sqrt{x^{6} + 2}}{12 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^6 + 2)*x^7),x, algorithm="giac")

[Out]

-1/48*sqrt(2)*ln(-(sqrt(2) - sqrt(x^6 + 2))/(sqrt(2) + sqrt(x^6 + 2))) - 1/12*sq
rt(x^6 + 2)/x^6