3.1411 \(\int \frac{1}{x^7 \left (2+x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=58 \[ -\frac{\sqrt{x^6+2}}{8 x^6}+\frac{1}{6 x^6 \sqrt{x^6+2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{8 \sqrt{2}} \]

[Out]

1/(6*x^6*Sqrt[2 + x^6]) - Sqrt[2 + x^6]/(8*x^6) + ArcTanh[Sqrt[2 + x^6]/Sqrt[2]]
/(8*Sqrt[2])

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Rubi [A]  time = 0.0583377, 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 \[ -\frac{\sqrt{x^6+2}}{8 x^6}+\frac{1}{6 x^6 \sqrt{x^6+2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{8 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^7*(2 + x^6)^(3/2)),x]

[Out]

1/(6*x^6*Sqrt[2 + x^6]) - Sqrt[2 + x^6]/(8*x^6) + ArcTanh[Sqrt[2 + x^6]/Sqrt[2]]
/(8*Sqrt[2])

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Rubi in Sympy [A]  time = 5.23687, size = 51, normalized size = 0.88 \[ \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{x^{6} + 2}}{2} \right )}}{16} - \frac{\sqrt{x^{6} + 2}}{8 x^{6}} + \frac{1}{6 x^{6} \sqrt{x^{6} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**7/(x**6+2)**(3/2),x)

[Out]

sqrt(2)*atanh(sqrt(2)*sqrt(x**6 + 2)/2)/16 - sqrt(x**6 + 2)/(8*x**6) + 1/(6*x**6
*sqrt(x**6 + 2))

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Mathematica [A]  time = 0.072966, size = 49, normalized size = 0.84 \[ \frac{1}{48} \left (3 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )-\frac{2 \left (3 x^6+2\right )}{x^6 \sqrt{x^6+2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^7*(2 + x^6)^(3/2)),x]

[Out]

((-2*(2 + 3*x^6))/(x^6*Sqrt[2 + x^6]) + 3*Sqrt[2]*ArcTanh[Sqrt[2 + x^6]/Sqrt[2]]
)/48

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Maple [A]  time = 0.031, size = 46, normalized size = 0.8 \[ -{\frac{3\,{x}^{6}+2}{24\,{x}^{6}}{\frac{1}{\sqrt{{x}^{6}+2}}}}-{\frac{\sqrt{2}}{16}\ln \left ({1 \left ( \sqrt{{x}^{6}+2}-\sqrt{2} \right ){\frac{1}{\sqrt{{x}^{6}}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(x^6+2)^(3/2),x)

[Out]

-1/24*(3*x^6+2)/x^6/(x^6+2)^(1/2)-1/16*2^(1/2)*ln(((x^6+2)^(1/2)-2^(1/2))/(x^6)^
(1/2))

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Maxima [A]  time = 1.59194, size = 88, normalized size = 1.52 \[ -\frac{1}{32} \, \sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2} - \sqrt{x^{6} + 2}\right )}}{2 \, \sqrt{2} + 2 \, \sqrt{x^{6} + 2}}\right ) - \frac{3 \, x^{6} + 2}{24 \,{\left ({\left (x^{6} + 2\right )}^{\frac{3}{2}} - 2 \, \sqrt{x^{6} + 2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 2)^(3/2)*x^7),x, algorithm="maxima")

[Out]

-1/32*sqrt(2)*log(-2*(sqrt(2) - sqrt(x^6 + 2))/((2*sqrt(2)) + 2*sqrt(x^6 + 2)))
- 1/24*(3*x^6 + 2)/((x^6 + 2)^(3/2) - 2*sqrt(x^6 + 2))

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Fricas [A]  time = 0.220217, size = 86, normalized size = 1.48 \[ \frac{\sqrt{2}{\left (3 \, \sqrt{x^{6} + 2} x^{6} \log \left (\frac{\sqrt{2}{\left (x^{6} + 4\right )} + 4 \, \sqrt{x^{6} + 2}}{x^{6}}\right ) - 2 \, \sqrt{2}{\left (3 \, x^{6} + 2\right )}\right )}}{96 \, \sqrt{x^{6} + 2} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 2)^(3/2)*x^7),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 10.5817, size = 49, normalized size = 0.84 \[ \frac{\sqrt{2} \operatorname{asinh}{\left (\frac{\sqrt{2}}{x^{3}} \right )}}{16} - \frac{1}{8 x^{3} \sqrt{1 + \frac{2}{x^{6}}}} - \frac{1}{12 x^{9} \sqrt{1 + \frac{2}{x^{6}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**7/(x**6+2)**(3/2),x)

[Out]

sqrt(2)*asinh(sqrt(2)/x**3)/16 - 1/(8*x**3*sqrt(1 + 2/x**6)) - 1/(12*x**9*sqrt(1
+ 2/x**6))

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GIAC/XCAS [A]  time = 0.22471, size = 85, normalized size = 1.47 \[ -\frac{1}{32} \, \sqrt{2}{\rm ln}\left (-\frac{\sqrt{2} - \sqrt{x^{6} + 2}}{\sqrt{2} + \sqrt{x^{6} + 2}}\right ) - \frac{3 \, x^{6} + 2}{24 \,{\left ({\left (x^{6} + 2\right )}^{\frac{3}{2}} - 2 \, \sqrt{x^{6} + 2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 2)^(3/2)*x^7),x, algorithm="giac")

[Out]

-1/32*sqrt(2)*ln(-(sqrt(2) - sqrt(x^6 + 2))/(sqrt(2) + sqrt(x^6 + 2))) - 1/24*(3
*x^6 + 2)/((x^6 + 2)^(3/2) - 2*sqrt(x^6 + 2))