3.881 \(\int \frac{1}{x^{11} \sqrt{1-x^4}} \, dx\)

Optimal. Leaf size=55 \[ -\frac{\sqrt{1-x^4}}{10 x^{10}}-\frac{2 \sqrt{1-x^4}}{15 x^6}-\frac{4 \sqrt{1-x^4}}{15 x^2} \]

[Out]

-Sqrt[1 - x^4]/(10*x^10) - (2*Sqrt[1 - x^4])/(15*x^6) - (4*Sqrt[1 - x^4])/(15*x^
2)

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Rubi [A]  time = 0.0459716, 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 \[ -\frac{\sqrt{1-x^4}}{10 x^{10}}-\frac{2 \sqrt{1-x^4}}{15 x^6}-\frac{4 \sqrt{1-x^4}}{15 x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^11*Sqrt[1 - x^4]),x]

[Out]

-Sqrt[1 - x^4]/(10*x^10) - (2*Sqrt[1 - x^4])/(15*x^6) - (4*Sqrt[1 - x^4])/(15*x^
2)

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Rubi in Sympy [A]  time = 4.6859, size = 44, normalized size = 0.8 \[ - \frac{4 \sqrt{- x^{4} + 1}}{15 x^{2}} - \frac{2 \sqrt{- x^{4} + 1}}{15 x^{6}} - \frac{\sqrt{- x^{4} + 1}}{10 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**11/(-x**4+1)**(1/2),x)

[Out]

-4*sqrt(-x**4 + 1)/(15*x**2) - 2*sqrt(-x**4 + 1)/(15*x**6) - sqrt(-x**4 + 1)/(10
*x**10)

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Mathematica [A]  time = 0.0152123, size = 30, normalized size = 0.55 \[ -\frac{\sqrt{1-x^4} \left (8 x^8+4 x^4+3\right )}{30 x^{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^11*Sqrt[1 - x^4]),x]

[Out]

-(Sqrt[1 - x^4]*(3 + 4*x^4 + 8*x^8))/(30*x^10)

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Maple [A]  time = 0.007, size = 38, normalized size = 0.7 \[{\frac{ \left ( -1+x \right ) \left ( 1+x \right ) \left ({x}^{2}+1 \right ) \left ( 8\,{x}^{8}+4\,{x}^{4}+3 \right ) }{30\,{x}^{10}}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^11/(-x^4+1)^(1/2),x)

[Out]

1/30*(-1+x)*(1+x)*(x^2+1)*(8*x^8+4*x^4+3)/x^10/(-x^4+1)^(1/2)

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Maxima [A]  time = 1.43514, size = 58, normalized size = 1.05 \[ -\frac{\sqrt{-x^{4} + 1}}{2 \, x^{2}} - \frac{{\left (-x^{4} + 1\right )}^{\frac{3}{2}}}{3 \, x^{6}} - \frac{{\left (-x^{4} + 1\right )}^{\frac{5}{2}}}{10 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + 1)*x^11),x, algorithm="maxima")

[Out]

-1/2*sqrt(-x^4 + 1)/x^2 - 1/3*(-x^4 + 1)^(3/2)/x^6 - 1/10*(-x^4 + 1)^(5/2)/x^10

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Fricas [A]  time = 0.252112, size = 140, normalized size = 2.55 \[ -\frac{8 \, x^{20} - 100 \, x^{16} + 175 \, x^{12} - 55 \, x^{8} + 20 \, x^{4} +{\left (40 \, x^{16} - 140 \, x^{12} + 63 \, x^{8} + 4 \, x^{4} + 48\right )} \sqrt{-x^{4} + 1} - 48}{30 \,{\left (5 \, x^{18} - 20 \, x^{14} + 16 \, x^{10} -{\left (x^{18} - 12 \, x^{14} + 16 \, x^{10}\right )} \sqrt{-x^{4} + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + 1)*x^11),x, algorithm="fricas")

[Out]

-1/30*(8*x^20 - 100*x^16 + 175*x^12 - 55*x^8 + 20*x^4 + (40*x^16 - 140*x^12 + 63
*x^8 + 4*x^4 + 48)*sqrt(-x^4 + 1) - 48)/(5*x^18 - 20*x^14 + 16*x^10 - (x^18 - 12
*x^14 + 16*x^10)*sqrt(-x^4 + 1))

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Sympy [A]  time = 6.4398, size = 104, normalized size = 1.89 \[ \begin{cases} - \frac{4 \sqrt{-1 + \frac{1}{x^{4}}}}{15} - \frac{2 \sqrt{-1 + \frac{1}{x^{4}}}}{15 x^{4}} - \frac{\sqrt{-1 + \frac{1}{x^{4}}}}{10 x^{8}} & \text{for}\: \left |{\frac{1}{x^{4}}}\right | > 1 \\- \frac{4 i \sqrt{1 - \frac{1}{x^{4}}}}{15} - \frac{2 i \sqrt{1 - \frac{1}{x^{4}}}}{15 x^{4}} - \frac{i \sqrt{1 - \frac{1}{x^{4}}}}{10 x^{8}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**11/(-x**4+1)**(1/2),x)

[Out]

Piecewise((-4*sqrt(-1 + x**(-4))/15 - 2*sqrt(-1 + x**(-4))/(15*x**4) - sqrt(-1 +
x**(-4))/(10*x**8), Abs(x**(-4)) > 1), (-4*I*sqrt(1 - 1/x**4)/15 - 2*I*sqrt(1 -
1/x**4)/(15*x**4) - I*sqrt(1 - 1/x**4)/(10*x**8), True))

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GIAC/XCAS [A]  time = 0.21538, size = 38, normalized size = 0.69 \[ -\frac{1}{10} \,{\left (\frac{1}{x^{4}} - 1\right )}^{\frac{5}{2}} - \frac{1}{3} \,{\left (\frac{1}{x^{4}} - 1\right )}^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{\frac{1}{x^{4}} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + 1)*x^11),x, algorithm="giac")

[Out]

-1/10*(1/x^4 - 1)^(5/2) - 1/3*(1/x^4 - 1)^(3/2) - 1/2*sqrt(1/x^4 - 1)