3.498 \(\int \frac{1}{x^7 \sqrt{-1-x^3}} \, dx\)

Optimal. Leaf size=53 \[ -\frac{\sqrt{-x^3-1}}{4 x^3}+\frac{1}{4} \tan ^{-1}\left (\sqrt{-x^3-1}\right )+\frac{\sqrt{-x^3-1}}{6 x^6} \]

[Out]

Sqrt[-1 - x^3]/(6*x^6) - Sqrt[-1 - x^3]/(4*x^3) + ArcTan[Sqrt[-1 - x^3]]/4

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Rubi [A]  time = 0.0611939, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{\sqrt{-x^3-1}}{4 x^3}+\frac{1}{4} \tan ^{-1}\left (\sqrt{-x^3-1}\right )+\frac{\sqrt{-x^3-1}}{6 x^6} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^7*Sqrt[-1 - x^3]),x]

[Out]

Sqrt[-1 - x^3]/(6*x^6) - Sqrt[-1 - x^3]/(4*x^3) + ArcTan[Sqrt[-1 - x^3]]/4

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Rubi in Sympy [A]  time = 6.07326, size = 42, normalized size = 0.79 \[ \frac{\operatorname{atan}{\left (\sqrt{- x^{3} - 1} \right )}}{4} - \frac{\sqrt{- x^{3} - 1}}{4 x^{3}} + \frac{\sqrt{- x^{3} - 1}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(sqrt(-x**3 - 1))/4 - sqrt(-x**3 - 1)/(4*x**3) + sqrt(-x**3 - 1)/(6*x**6)

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Mathematica [A]  time = 0.044372, size = 62, normalized size = 1.17 \[ \frac{\sqrt{-x^3-1} \tanh ^{-1}\left (\sqrt{x^3+1}\right )}{4 \sqrt{x^3+1}}+\sqrt{-x^3-1} \left (\frac{1}{6 x^6}-\frac{1}{4 x^3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^7*Sqrt[-1 - x^3]),x]

[Out]

(1/(6*x^6) - 1/(4*x^3))*Sqrt[-1 - x^3] + (Sqrt[-1 - x^3]*ArcTanh[Sqrt[1 + x^3]])
/(4*Sqrt[1 + x^3])

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Maple [A]  time = 0.035, size = 42, normalized size = 0.8 \[{\frac{1}{4}\arctan \left ( \sqrt{-{x}^{3}-1} \right ) }+{\frac{1}{6\,{x}^{6}}\sqrt{-{x}^{3}-1}}-{\frac{1}{4\,{x}^{3}}\sqrt{-{x}^{3}-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(-x^3-1)^(1/2),x)

[Out]

1/4*arctan((-x^3-1)^(1/2))+1/6*(-x^3-1)^(1/2)/x^6-1/4*(-x^3-1)^(1/2)/x^3

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Maxima [A]  time = 1.57431, size = 76, normalized size = 1.43 \[ -\frac{3 \,{\left (-x^{3} - 1\right )}^{\frac{3}{2}} + 5 \, \sqrt{-x^{3} - 1}}{12 \,{\left (2 \, x^{3} -{\left (x^{3} + 1\right )}^{2} + 1\right )}} + \frac{1}{4} \, \arctan \left (\sqrt{-x^{3} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^3 - 1)*x^7),x, algorithm="maxima")

[Out]

-1/12*(3*(-x^3 - 1)^(3/2) + 5*sqrt(-x^3 - 1))/(2*x^3 - (x^3 + 1)^2 + 1) + 1/4*ar
ctan(sqrt(-x^3 - 1))

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Fricas [A]  time = 0.22738, size = 53, normalized size = 1. \[ \frac{3 \, x^{6} \arctan \left (\sqrt{-x^{3} - 1}\right ) -{\left (3 \, x^{3} - 2\right )} \sqrt{-x^{3} - 1}}{12 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^3 - 1)*x^7),x, algorithm="fricas")

[Out]

1/12*(3*x^6*arctan(sqrt(-x^3 - 1)) - (3*x^3 - 2)*sqrt(-x^3 - 1))/x^6

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Sympy [A]  time = 10.7799, size = 66, normalized size = 1.25 \[ \frac{i \operatorname{asinh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{4} - \frac{i}{4 x^{\frac{3}{2}} \sqrt{1 + \frac{1}{x^{3}}}} - \frac{i}{12 x^{\frac{9}{2}} \sqrt{1 + \frac{1}{x^{3}}}} + \frac{i}{6 x^{\frac{15}{2}} \sqrt{1 + \frac{1}{x^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

I*asinh(x**(-3/2))/4 - I/(4*x**(3/2)*sqrt(1 + x**(-3))) - I/(12*x**(9/2)*sqrt(1
+ x**(-3))) + I/(6*x**(15/2)*sqrt(1 + x**(-3)))

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GIAC/XCAS [A]  time = 0.213917, size = 55, normalized size = 1.04 \[ \frac{3 \,{\left (-x^{3} - 1\right )}^{\frac{3}{2}} + 5 \, \sqrt{-x^{3} - 1}}{12 \, x^{6}} + \frac{1}{4} \, \arctan \left (\sqrt{-x^{3} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^3 - 1)*x^7),x, algorithm="giac")

[Out]

1/12*(3*(-x^3 - 1)^(3/2) + 5*sqrt(-x^3 - 1))/x^6 + 1/4*arctan(sqrt(-x^3 - 1))