3.481 \(\int \frac{1}{x^{10} \sqrt{-1+x^3}} \, dx\)

Optimal. Leaf size=63 \[ \frac{5 \sqrt{x^3-1}}{24 x^3}+\frac{5}{24} \tan ^{-1}\left (\sqrt{x^3-1}\right )+\frac{\sqrt{x^3-1}}{9 x^9}+\frac{5 \sqrt{x^3-1}}{36 x^6} \]

[Out]

Sqrt[-1 + x^3]/(9*x^9) + (5*Sqrt[-1 + x^3])/(36*x^6) + (5*Sqrt[-1 + x^3])/(24*x^
3) + (5*ArcTan[Sqrt[-1 + x^3]])/24

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Rubi [A]  time = 0.0619961, antiderivative size = 63, 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 \[ \frac{5 \sqrt{x^3-1}}{24 x^3}+\frac{5}{24} \tan ^{-1}\left (\sqrt{x^3-1}\right )+\frac{\sqrt{x^3-1}}{9 x^9}+\frac{5 \sqrt{x^3-1}}{36 x^6} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^10*Sqrt[-1 + x^3]),x]

[Out]

Sqrt[-1 + x^3]/(9*x^9) + (5*Sqrt[-1 + x^3])/(36*x^6) + (5*Sqrt[-1 + x^3])/(24*x^
3) + (5*ArcTan[Sqrt[-1 + x^3]])/24

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Rubi in Sympy [A]  time = 5.62511, size = 56, normalized size = 0.89 \[ \frac{5 \operatorname{atan}{\left (\sqrt{x^{3} - 1} \right )}}{24} + \frac{5 \sqrt{x^{3} - 1}}{24 x^{3}} + \frac{5 \sqrt{x^{3} - 1}}{36 x^{6}} + \frac{\sqrt{x^{3} - 1}}{9 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**10/(x**3-1)**(1/2),x)

[Out]

5*atan(sqrt(x**3 - 1))/24 + 5*sqrt(x**3 - 1)/(24*x**3) + 5*sqrt(x**3 - 1)/(36*x*
*6) + sqrt(x**3 - 1)/(9*x**9)

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Mathematica [A]  time = 0.0646474, size = 55, normalized size = 0.87 \[ \frac{1}{72} \sqrt{x^3-1} \left (\frac{15 \tanh ^{-1}\left (\sqrt{1-x^3}\right )}{\sqrt{1-x^3}}+\frac{15 x^6+10 x^3+8}{x^9}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^10*Sqrt[-1 + x^3]),x]

[Out]

(Sqrt[-1 + x^3]*((8 + 10*x^3 + 15*x^6)/x^9 + (15*ArcTanh[Sqrt[1 - x^3]])/Sqrt[1
- x^3]))/72

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Maple [A]  time = 0.034, size = 48, normalized size = 0.8 \[{\frac{5}{24}\arctan \left ( \sqrt{{x}^{3}-1} \right ) }+{\frac{1}{9\,{x}^{9}}\sqrt{{x}^{3}-1}}+{\frac{5}{36\,{x}^{6}}\sqrt{{x}^{3}-1}}+{\frac{5}{24\,{x}^{3}}\sqrt{{x}^{3}-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^10/(x^3-1)^(1/2),x)

[Out]

5/24*arctan((x^3-1)^(1/2))+1/9*(x^3-1)^(1/2)/x^9+5/36*(x^3-1)^(1/2)/x^6+5/24*(x^
3-1)^(1/2)/x^3

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Maxima [A]  time = 1.59138, size = 89, normalized size = 1.41 \[ \frac{15 \,{\left (x^{3} - 1\right )}^{\frac{5}{2}} + 40 \,{\left (x^{3} - 1\right )}^{\frac{3}{2}} + 33 \, \sqrt{x^{3} - 1}}{72 \,{\left ({\left (x^{3} - 1\right )}^{3} + 3 \, x^{3} + 3 \,{\left (x^{3} - 1\right )}^{2} - 2\right )}} + \frac{5}{24} \, \arctan \left (\sqrt{x^{3} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^3 - 1)*x^10),x, algorithm="maxima")

[Out]

1/72*(15*(x^3 - 1)^(5/2) + 40*(x^3 - 1)^(3/2) + 33*sqrt(x^3 - 1))/((x^3 - 1)^3 +
3*x^3 + 3*(x^3 - 1)^2 - 2) + 5/24*arctan(sqrt(x^3 - 1))

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Fricas [A]  time = 0.233206, size = 53, normalized size = 0.84 \[ \frac{15 \, x^{9} \arctan \left (\sqrt{x^{3} - 1}\right ) +{\left (15 \, x^{6} + 10 \, x^{3} + 8\right )} \sqrt{x^{3} - 1}}{72 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^3 - 1)*x^10),x, algorithm="fricas")

[Out]

1/72*(15*x^9*arctan(sqrt(x^3 - 1)) + (15*x^6 + 10*x^3 + 8)*sqrt(x^3 - 1))/x^9

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Sympy [A]  time = 16.869, size = 182, normalized size = 2.89 \[ \begin{cases} \frac{5 i \operatorname{acosh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{24} - \frac{5 i}{24 x^{\frac{3}{2}} \sqrt{-1 + \frac{1}{x^{3}}}} + \frac{5 i}{72 x^{\frac{9}{2}} \sqrt{-1 + \frac{1}{x^{3}}}} + \frac{i}{36 x^{\frac{15}{2}} \sqrt{-1 + \frac{1}{x^{3}}}} + \frac{i}{9 x^{\frac{21}{2}} \sqrt{-1 + \frac{1}{x^{3}}}} & \text{for}\: \left |{\frac{1}{x^{3}}}\right | > 1 \\- \frac{5 \operatorname{asin}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{24} + \frac{5}{24 x^{\frac{3}{2}} \sqrt{1 - \frac{1}{x^{3}}}} - \frac{5}{72 x^{\frac{9}{2}} \sqrt{1 - \frac{1}{x^{3}}}} - \frac{1}{36 x^{\frac{15}{2}} \sqrt{1 - \frac{1}{x^{3}}}} - \frac{1}{9 x^{\frac{21}{2}} \sqrt{1 - \frac{1}{x^{3}}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**10/(x**3-1)**(1/2),x)

[Out]

Piecewise((5*I*acosh(x**(-3/2))/24 - 5*I/(24*x**(3/2)*sqrt(-1 + x**(-3))) + 5*I/
(72*x**(9/2)*sqrt(-1 + x**(-3))) + I/(36*x**(15/2)*sqrt(-1 + x**(-3))) + I/(9*x*
*(21/2)*sqrt(-1 + x**(-3))), Abs(x**(-3)) > 1), (-5*asin(x**(-3/2))/24 + 5/(24*x
**(3/2)*sqrt(1 - 1/x**3)) - 5/(72*x**(9/2)*sqrt(1 - 1/x**3)) - 1/(36*x**(15/2)*s
qrt(1 - 1/x**3)) - 1/(9*x**(21/2)*sqrt(1 - 1/x**3)), True))

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GIAC/XCAS [A]  time = 0.214667, size = 59, normalized size = 0.94 \[ \frac{15 \,{\left (x^{3} - 1\right )}^{\frac{5}{2}} + 40 \,{\left (x^{3} - 1\right )}^{\frac{3}{2}} + 33 \, \sqrt{x^{3} - 1}}{72 \, x^{9}} + \frac{5}{24} \, \arctan \left (\sqrt{x^{3} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^3 - 1)*x^10),x, algorithm="giac")

[Out]

1/72*(15*(x^3 - 1)^(5/2) + 40*(x^3 - 1)^(3/2) + 33*sqrt(x^3 - 1))/x^9 + 5/24*arc
tan(sqrt(x^3 - 1))