3.1412 \(\int \frac{1}{x^{13} \left (2+x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=74 \[ \frac{5 \sqrt{x^6+2}}{64 x^6}-\frac{5 \tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{64 \sqrt{2}}-\frac{5 \sqrt{x^6+2}}{48 x^{12}}+\frac{1}{6 x^{12} \sqrt{x^6+2}} \]

[Out]

1/(6*x^12*Sqrt[2 + x^6]) - (5*Sqrt[2 + x^6])/(48*x^12) + (5*Sqrt[2 + x^6])/(64*x
^6) - (5*ArcTanh[Sqrt[2 + x^6]/Sqrt[2]])/(64*Sqrt[2])

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Rubi [A]  time = 0.0742879, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{5 \sqrt{x^6+2}}{64 x^6}-\frac{5 \tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )}{64 \sqrt{2}}-\frac{5 \sqrt{x^6+2}}{48 x^{12}}+\frac{1}{6 x^{12} \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^13*(2 + x^6)^(3/2)),x]

[Out]

1/(6*x^12*Sqrt[2 + x^6]) - (5*Sqrt[2 + x^6])/(48*x^12) + (5*Sqrt[2 + x^6])/(64*x
^6) - (5*ArcTanh[Sqrt[2 + x^6]/Sqrt[2]])/(64*Sqrt[2])

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Rubi in Sympy [A]  time = 6.11277, size = 70, normalized size = 0.95 \[ - \frac{5 \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{x^{6} + 2}}{2} \right )}}{128} + \frac{5 \sqrt{x^{6} + 2}}{64 x^{6}} - \frac{5 \sqrt{x^{6} + 2}}{48 x^{12}} + \frac{1}{6 x^{12} \sqrt{x^{6} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**13/(x**6+2)**(3/2),x)

[Out]

-5*sqrt(2)*atanh(sqrt(2)*sqrt(x**6 + 2)/2)/128 + 5*sqrt(x**6 + 2)/(64*x**6) - 5*
sqrt(x**6 + 2)/(48*x**12) + 1/(6*x**12*sqrt(x**6 + 2))

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Mathematica [A]  time = 0.080524, size = 54, normalized size = 0.73 \[ \frac{1}{384} \left (\frac{2 \left (15 x^{12}+10 x^6-8\right )}{x^{12} \sqrt{x^6+2}}-15 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{x^6+2}}{\sqrt{2}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^13*(2 + x^6)^(3/2)),x]

[Out]

((2*(-8 + 10*x^6 + 15*x^12))/(x^12*Sqrt[2 + x^6]) - 15*Sqrt[2]*ArcTanh[Sqrt[2 +
x^6]/Sqrt[2]])/384

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Maple [A]  time = 0.037, size = 51, normalized size = 0.7 \[{\frac{15\,{x}^{12}+10\,{x}^{6}-8}{192\,{x}^{12}}{\frac{1}{\sqrt{{x}^{6}+2}}}}+{\frac{5\,\sqrt{2}}{128}\ln \left ({1 \left ( \sqrt{{x}^{6}+2}-\sqrt{2} \right ){\frac{1}{\sqrt{{x}^{6}}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^13/(x^6+2)^(3/2),x)

[Out]

1/192*(15*x^12+10*x^6-8)/x^12/(x^6+2)^(1/2)+5/128*2^(1/2)*ln(((x^6+2)^(1/2)-2^(1
/2))/(x^6)^(1/2))

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Maxima [A]  time = 1.58968, size = 112, normalized size = 1.51 \[ \frac{5}{256} \, \sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2} - \sqrt{x^{6} + 2}\right )}}{2 \, \sqrt{2} + 2 \, \sqrt{x^{6} + 2}}\right ) - \frac{50 \, x^{6} - 15 \,{\left (x^{6} + 2\right )}^{2} + 68}{192 \,{\left ({\left (x^{6} + 2\right )}^{\frac{5}{2}} - 4 \,{\left (x^{6} + 2\right )}^{\frac{3}{2}} + 4 \, \sqrt{x^{6} + 2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 2)^(3/2)*x^13),x, algorithm="maxima")

[Out]

5/256*sqrt(2)*log(-2*(sqrt(2) - sqrt(x^6 + 2))/((2*sqrt(2)) + 2*sqrt(x^6 + 2)))
- 1/192*(50*x^6 - 15*(x^6 + 2)^2 + 68)/((x^6 + 2)^(5/2) - 4*(x^6 + 2)^(3/2) + 4*
sqrt(x^6 + 2))

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Fricas [A]  time = 0.222825, size = 93, normalized size = 1.26 \[ \frac{\sqrt{2}{\left (15 \, \sqrt{x^{6} + 2} x^{12} \log \left (\frac{\sqrt{2}{\left (x^{6} + 4\right )} - 4 \, \sqrt{x^{6} + 2}}{x^{6}}\right ) + 2 \, \sqrt{2}{\left (15 \, x^{12} + 10 \, x^{6} - 8\right )}\right )}}{768 \, \sqrt{x^{6} + 2} x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 2)^(3/2)*x^13),x, algorithm="fricas")

[Out]

1/768*sqrt(2)*(15*sqrt(x^6 + 2)*x^12*log((sqrt(2)*(x^6 + 4) - 4*sqrt(x^6 + 2))/x
^6) + 2*sqrt(2)*(15*x^12 + 10*x^6 - 8))/(sqrt(x^6 + 2)*x^12)

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Sympy [A]  time = 25.5599, size = 68, normalized size = 0.92 \[ - \frac{5 \sqrt{2} \operatorname{asinh}{\left (\frac{\sqrt{2}}{x^{3}} \right )}}{128} + \frac{5}{64 x^{3} \sqrt{1 + \frac{2}{x^{6}}}} + \frac{5}{96 x^{9} \sqrt{1 + \frac{2}{x^{6}}}} - \frac{1}{24 x^{15} \sqrt{1 + \frac{2}{x^{6}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**13/(x**6+2)**(3/2),x)

[Out]

-5*sqrt(2)*asinh(sqrt(2)/x**3)/128 + 5/(64*x**3*sqrt(1 + 2/x**6)) + 5/(96*x**9*s
qrt(1 + 2/x**6)) - 1/(24*x**15*sqrt(1 + 2/x**6))

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GIAC/XCAS [A]  time = 0.223227, size = 92, normalized size = 1.24 \[ \frac{5}{256} \, \sqrt{2}{\rm ln}\left (-\frac{\sqrt{2} - \sqrt{x^{6} + 2}}{\sqrt{2} + \sqrt{x^{6} + 2}}\right ) + \frac{1}{24 \, \sqrt{x^{6} + 2}} + \frac{7 \,{\left (x^{6} + 2\right )}^{\frac{3}{2}} - 18 \, \sqrt{x^{6} + 2}}{192 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 2)^(3/2)*x^13),x, algorithm="giac")

[Out]

5/256*sqrt(2)*ln(-(sqrt(2) - sqrt(x^6 + 2))/(sqrt(2) + sqrt(x^6 + 2))) + 1/24/sq
rt(x^6 + 2) + 1/192*(7*(x^6 + 2)^(3/2) - 18*sqrt(x^6 + 2))/x^12