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

Optimal. Leaf size=47 \[ \frac{\sqrt{x^3-1}}{4 x^3}+\frac{1}{4} \tan ^{-1}\left (\sqrt{x^3-1}\right )+\frac{\sqrt{x^3-1}}{6 x^6} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0495756, antiderivative size = 47, 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 \[ \frac{\sqrt{x^3-1}}{4 x^3}+\frac{1}{4} \tan ^{-1}\left (\sqrt{x^3-1}\right )+\frac{\sqrt{x^3-1}}{6 x^6} \]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.84446, size = 37, 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)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0597511, size = 48, normalized size = 1.02 \[ \frac{1}{4} \sqrt{x^3-1} \left (\frac{2}{3 x^6}+\frac{1}{x^3}+\frac{\tanh ^{-1}\left (\sqrt{1-x^3}\right )}{\sqrt{1-x^3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[-1 + x^3]*(2/(3*x^6) + x^(-3) + ArcTanh[Sqrt[1 - x^3]]/Sqrt[1 - x^3]))/4

_______________________________________________________________________________________

Maple [A]  time = 0.03, size = 36, 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

_______________________________________________________________________________________

Maxima [A]  time = 1.58574, size = 65, normalized size = 1.38 \[ \frac{3 \,{\left (x^{3} - 1\right )}^{\frac{3}{2}} + 5 \, \sqrt{x^{3} - 1}}{12 \,{\left (2 \, x^{3} +{\left (x^{3} - 1\right )}^{2} - 1\right )}} + \frac{1}{4} \, \arctan \left (\sqrt{x^{3} - 1}\right ) \]

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*arcta
n(sqrt(x^3 - 1))

_______________________________________________________________________________________

Fricas [A]  time = 0.239076, size = 46, normalized size = 0.98 \[ \frac{3 \, x^{6} \arctan \left (\sqrt{x^{3} - 1}\right ) +{\left (3 \, x^{3} + 2\right )} \sqrt{x^{3} - 1}}{12 \, x^{6}} \]

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

_______________________________________________________________________________________

Sympy [A]  time = 10.6037, size = 138, normalized size = 2.94 \[ \begin{cases} \frac{i \operatorname{acosh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{4} - \frac{i}{4 x^{\frac{3}{2}} \sqrt{-1 + \frac{1}{x^{3}}}} + \frac{i}{12 x^{\frac{9}{2}} \sqrt{-1 + \frac{1}{x^{3}}}} + \frac{i}{6 x^{\frac{15}{2}} \sqrt{-1 + \frac{1}{x^{3}}}} & \text{for}\: \left |{\frac{1}{x^{3}}}\right | > 1 \\- \frac{\operatorname{asin}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{4} + \frac{1}{4 x^{\frac{3}{2}} \sqrt{1 - \frac{1}{x^{3}}}} - \frac{1}{12 x^{\frac{9}{2}} \sqrt{1 - \frac{1}{x^{3}}}} - \frac{1}{6 x^{\frac{15}{2}} \sqrt{1 - \frac{1}{x^{3}}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((I*acosh(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))), Abs(x**(-3)) > 1)
, (-asin(x**(-3/2))/4 + 1/(4*x**(3/2)*sqrt(1 - 1/x**3)) - 1/(12*x**(9/2)*sqrt(1
- 1/x**3)) - 1/(6*x**(15/2)*sqrt(1 - 1/x**3)), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.221712, size = 47, normalized size = 1. \[ \frac{3 \,{\left (x^{3} - 1\right )}^{\frac{3}{2}} + 5 \, \sqrt{x^{3} - 1}}{12 \, x^{6}} + \frac{1}{4} \, \arctan \left (\sqrt{x^{3} - 1}\right ) \]

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))