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

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0620185, 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 \[ -\frac{\sqrt{1-x^3}}{4 x^3}-\frac{1}{4} \tanh ^{-1}\left (\sqrt{1-x^3}\right )-\frac{\sqrt{1-x^3}}{6 x^6} \]

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) - ArcTanh[Sqrt[1 - x^3]]/4

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.91617, size = 39, normalized size = 0.74 \[ - \frac{\operatorname{atanh}{\left (\sqrt{- x^{3} + 1} \right )}}{4} - \frac{\sqrt{- x^{3} + 1}}{4 x^{3}} - \frac{\sqrt{- x^{3} + 1}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.045899, size = 44, normalized size = 0.83 \[ \left (-\frac{1}{6 x^6}-\frac{1}{4 x^3}\right ) \sqrt{1-x^3}-\frac{1}{4} \tanh ^{-1}\left (\sqrt{1-x^3}\right ) \]

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] - ArcTanh[Sqrt[1 - x^3]]/4

_______________________________________________________________________________________

Maple [A]  time = 0.036, size = 42, normalized size = 0.8 \[ -{\frac{1}{4}{\it Artanh} \left ( \sqrt{-{x}^{3}+1} \right ) }-{\frac{1}{6\,{x}^{6}}\sqrt{-{x}^{3}+1}}-{\frac{1}{4\,{x}^{3}}\sqrt{-{x}^{3}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(-x^3+1)^(1/2),x)

[Out]

-1/4*arctanh((-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 = 1.50524, size = 95, normalized size = 1.79 \[ \frac{3 \,{\left (-x^{3} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{-x^{3} + 1}}{12 \,{\left (2 \, x^{3} +{\left (x^{3} - 1\right )}^{2} - 1\right )}} - \frac{1}{8} \, \log \left (\sqrt{-x^{3} + 1} + 1\right ) + \frac{1}{8} \, \log \left (\sqrt{-x^{3} + 1} - 1\right ) \]

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) - 1/8*log
(sqrt(-x^3 + 1) + 1) + 1/8*log(sqrt(-x^3 + 1) - 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.235052, size = 78, normalized size = 1.47 \[ -\frac{3 \, x^{6} \log \left (\sqrt{-x^{3} + 1} + 1\right ) - 3 \, x^{6} \log \left (\sqrt{-x^{3} + 1} - 1\right ) + 2 \,{\left (3 \, x^{3} + 2\right )} \sqrt{-x^{3} + 1}}{24 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*(3*x^6*log(sqrt(-x^3 + 1) + 1) - 3*x^6*log(sqrt(-x^3 + 1) - 1) + 2*(3*x^3
+ 2)*sqrt(-x^3 + 1))/x^6

_______________________________________________________________________________________

Sympy [A]  time = 10.6667, size = 138, normalized size = 2.6 \[ \begin{cases} - \frac{\operatorname{acosh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{4} + \frac{1}{4 x^{\frac{3}{2}} \sqrt{-1 + \frac{1}{x^{3}}}} - \frac{1}{12 x^{\frac{9}{2}} \sqrt{-1 + \frac{1}{x^{3}}}} - \frac{1}{6 x^{\frac{15}{2}} \sqrt{-1 + \frac{1}{x^{3}}}} & \text{for}\: \left |{\frac{1}{x^{3}}}\right | > 1 \\\frac{i \operatorname{asin}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{4} - \frac{i}{4 x^{\frac{3}{2}} \sqrt{1 - \frac{1}{x^{3}}}} + \frac{i}{12 x^{\frac{9}{2}} \sqrt{1 - \frac{1}{x^{3}}}} + \frac{i}{6 x^{\frac{15}{2}} \sqrt{1 - \frac{1}{x^{3}}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.218147, size = 78, normalized size = 1.47 \[ \frac{3 \,{\left (-x^{3} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{-x^{3} + 1}}{12 \, x^{6}} - \frac{1}{8} \,{\rm ln}\left (\sqrt{-x^{3} + 1} + 1\right ) + \frac{1}{8} \,{\rm ln}\left ({\left | \sqrt{-x^{3} + 1} - 1 \right |}\right ) \]

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 - 1/8*ln(sqrt(-x^3 + 1) + 1) +
1/8*ln(abs(sqrt(-x^3 + 1) - 1))