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

Optimal. Leaf size=71 \[ \frac{5 \sqrt{-x^3-1}}{24 x^3}-\frac{5}{24} \tan ^{-1}\left (\sqrt{-x^3-1}\right )+\frac{\sqrt{-x^3-1}}{9 x^9}-\frac{5 \sqrt{-x^3-1}}{36 x^6} \]

[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 = 0.0808786, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{5 \sqrt{-x^3-1}}{24 x^3}-\frac{5}{24} \tan ^{-1}\left (\sqrt{-x^3-1}\right )+\frac{\sqrt{-x^3-1}}{9 x^9}-\frac{5 \sqrt{-x^3-1}}{36 x^6} \]

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 = 7.17513, size = 63, 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)/(3
6*x**6) + sqrt(-x**3 - 1)/(9*x**9)

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Mathematica [A]  time = 0.0512101, size = 65, normalized size = 0.92 \[ -\frac{\sqrt{-x^3-1} \left (15 x^9 \tanh ^{-1}\left (\sqrt{x^3+1}\right )+\sqrt{x^3+1} \left (-15 x^6+10 x^3-8\right )\right )}{72 x^9 \sqrt{x^3+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.037, size = 56, 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 = 1.59212, size = 100, 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 = 0.229508, size = 59, normalized size = 0.83 \[ -\frac{15 \, x^{9} \arctan \left (\sqrt{-x^{3} - 1}\right ) -{\left (15 \, x^{6} - 10 \, x^{3} + 8\right )} \sqrt{-x^{3} - 1}}{72 \, x^{9}} \]

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 = 17.8202, size = 90, normalized size = 1.27 \[ - \frac{5 i \operatorname{asinh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{24} + \frac{5 i}{24 x^{\frac{3}{2}} \sqrt{1 + \frac{1}{x^{3}}}} + \frac{5 i}{72 x^{\frac{9}{2}} \sqrt{1 + \frac{1}{x^{3}}}} - \frac{i}{36 x^{\frac{15}{2}} \sqrt{1 + \frac{1}{x^{3}}}} + \frac{i}{9 x^{\frac{21}{2}} \sqrt{1 + \frac{1}{x^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*I*asinh(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)))

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GIAC/XCAS [A]  time = 0.218358, size = 80, normalized size = 1.13 \[ \frac{15 \,{\left (x^{3} + 1\right )}^{2} \sqrt{-x^{3} - 1} + 40 \,{\left (-x^{3} - 1\right )}^{\frac{3}{2}} + 33 \, \sqrt{-x^{3} - 1}}{72 \, x^{9}} - \frac{5}{24} \, \arctan \left (\sqrt{-x^{3} - 1}\right ) \]

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)^2*sqrt(-x^3 - 1) + 40*(-x^3 - 1)^(3/2) + 33*sqrt(-x^3 - 1))/x
^9 - 5/24*arctan(sqrt(-x^3 - 1))