3.901 \(\int \frac{1}{x^7 \left (1-x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=55 \[ -\frac{1}{6 \sqrt{1-x^4} x^6}+\frac{4 x^2}{3 \sqrt{1-x^4}}-\frac{2}{3 \sqrt{1-x^4} x^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0420547, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{1}{6 \sqrt{1-x^4} x^6}+\frac{4 x^2}{3 \sqrt{1-x^4}}-\frac{2}{3 \sqrt{1-x^4} x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^7*(1 - x^4)^(3/2)),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.29678, size = 44, normalized size = 0.8 \[ \frac{4 x^{2}}{3 \sqrt{- x^{4} + 1}} - \frac{2}{3 x^{2} \sqrt{- x^{4} + 1}} - \frac{1}{6 x^{6} \sqrt{- x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**7/(-x**4+1)**(3/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0223115, size = 30, normalized size = 0.55 \[ \frac{8 x^8-4 x^4-1}{6 x^6 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^7*(1 - x^4)^(3/2)),x]

[Out]

(-1 - 4*x^4 + 8*x^8)/(6*x^6*Sqrt[1 - x^4])

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 38, normalized size = 0.7 \[ -{\frac{ \left ( -1+x \right ) \left ( 1+x \right ) \left ({x}^{2}+1 \right ) \left ( 8\,{x}^{8}-4\,{x}^{4}-1 \right ) }{6\,{x}^{6}} \left ( -{x}^{4}+1 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(-x^4+1)^(3/2),x)

[Out]

-1/6*(-1+x)*(1+x)*(x^2+1)*(8*x^8-4*x^4-1)/x^6/(-x^4+1)^(3/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.44294, size = 58, normalized size = 1.05 \[ \frac{x^{2}}{2 \, \sqrt{-x^{4} + 1}} - \frac{\sqrt{-x^{4} + 1}}{x^{2}} - \frac{{\left (-x^{4} + 1\right )}^{\frac{3}{2}}}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-x^4 + 1)^(3/2)*x^7),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.288825, size = 128, normalized size = 2.33 \[ -\frac{8 \, x^{16} - 68 \, x^{12} + 95 \, x^{8} - 24 \, x^{4} + 4 \,{\left (8 \, x^{12} - 20 \, x^{8} + 7 \, x^{4} + 2\right )} \sqrt{-x^{4} + 1} - 8}{6 \,{\left (4 \, x^{14} - 12 \, x^{10} + 8 \, x^{6} -{\left (x^{14} - 8 \, x^{10} + 8 \, x^{6}\right )} \sqrt{-x^{4} + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-x^4 + 1)^(3/2)*x^7),x, algorithm="fricas")

[Out]

-1/6*(8*x^16 - 68*x^12 + 95*x^8 - 24*x^4 + 4*(8*x^12 - 20*x^8 + 7*x^4 + 2)*sqrt(
-x^4 + 1) - 8)/(4*x^14 - 12*x^10 + 8*x^6 - (x^14 - 8*x^10 + 8*x^6)*sqrt(-x^4 + 1
))

_______________________________________________________________________________________

Sympy [A]  time = 5.1927, size = 151, normalized size = 2.75 \[ \begin{cases} - \frac{8 x^{8} \sqrt{-1 + \frac{1}{x^{4}}}}{6 x^{8} - 6 x^{4}} + \frac{4 x^{4} \sqrt{-1 + \frac{1}{x^{4}}}}{6 x^{8} - 6 x^{4}} + \frac{\sqrt{-1 + \frac{1}{x^{4}}}}{6 x^{8} - 6 x^{4}} & \text{for}\: \left |{\frac{1}{x^{4}}}\right | > 1 \\- \frac{8 i x^{8} \sqrt{1 - \frac{1}{x^{4}}}}{6 x^{8} - 6 x^{4}} + \frac{4 i x^{4} \sqrt{1 - \frac{1}{x^{4}}}}{6 x^{8} - 6 x^{4}} + \frac{i \sqrt{1 - \frac{1}{x^{4}}}}{6 x^{8} - 6 x^{4}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**7/(-x**4+1)**(3/2),x)

[Out]

Piecewise((-8*x**8*sqrt(-1 + x**(-4))/(6*x**8 - 6*x**4) + 4*x**4*sqrt(-1 + x**(-
4))/(6*x**8 - 6*x**4) + sqrt(-1 + x**(-4))/(6*x**8 - 6*x**4), Abs(x**(-4)) > 1),
(-8*I*x**8*sqrt(1 - 1/x**4)/(6*x**8 - 6*x**4) + 4*I*x**4*sqrt(1 - 1/x**4)/(6*x*
*8 - 6*x**4) + I*sqrt(1 - 1/x**4)/(6*x**8 - 6*x**4), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.225852, size = 54, normalized size = 0.98 \[ -\frac{\sqrt{-x^{4} + 1} x^{2}}{2 \,{\left (x^{4} - 1\right )}} - \frac{1}{6} \,{\left (\frac{1}{x^{4}} - 1\right )}^{\frac{3}{2}} - \sqrt{\frac{1}{x^{4}} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-x^4 + 1)^(3/2)*x^7),x, algorithm="giac")

[Out]

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