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

Optimal. Leaf size=46 \[ -\frac{1}{10} \left (1-x^4\right )^{5/2}+\frac{1}{3} \left (1-x^4\right )^{3/2}-\frac{\sqrt{1-x^4}}{2} \]

[Out]

-Sqrt[1 - x^4]/2 + (1 - x^4)^(3/2)/3 - (1 - x^4)^(5/2)/10

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Rubi [A]  time = 0.0531364, antiderivative size = 46, 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 \[ -\frac{1}{10} \left (1-x^4\right )^{5/2}+\frac{1}{3} \left (1-x^4\right )^{3/2}-\frac{\sqrt{1-x^4}}{2} \]

Antiderivative was successfully verified.

[In]  Int[x^11/Sqrt[1 - x^4],x]

[Out]

-Sqrt[1 - x^4]/2 + (1 - x^4)^(3/2)/3 - (1 - x^4)^(5/2)/10

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Rubi in Sympy [A]  time = 5.08847, size = 29, normalized size = 0.63 \[ - \frac{\left (- x^{4} + 1\right )^{\frac{5}{2}}}{10} + \frac{\left (- x^{4} + 1\right )^{\frac{3}{2}}}{3} - \frac{\sqrt{- x^{4} + 1}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0161124, size = 27, normalized size = 0.59 \[ -\frac{1}{30} \sqrt{1-x^4} \left (3 x^8+4 x^4+8\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/Sqrt[1 - x^4],x]

[Out]

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

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Maple [A]  time = 0.007, size = 35, normalized size = 0.8 \[{\frac{ \left ( -1+x \right ) \left ( 1+x \right ) \left ({x}^{2}+1 \right ) \left ( 3\,{x}^{8}+4\,{x}^{4}+8 \right ) }{30}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(-x^4+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.44005, size = 46, normalized size = 1. \[ -\frac{1}{10} \,{\left (-x^{4} + 1\right )}^{\frac{5}{2}} + \frac{1}{3} \,{\left (-x^{4} + 1\right )}^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{-x^{4} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.26617, size = 31, normalized size = 0.67 \[ -\frac{1}{30} \,{\left (3 \, x^{8} + 4 \, x^{4} + 8\right )} \sqrt{-x^{4} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.08342, size = 41, normalized size = 0.89 \[ - \frac{x^{8} \sqrt{- x^{4} + 1}}{10} - \frac{2 x^{4} \sqrt{- x^{4} + 1}}{15} - \frac{4 \sqrt{- x^{4} + 1}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**8*sqrt(-x**4 + 1)/10 - 2*x**4*sqrt(-x**4 + 1)/15 - 4*sqrt(-x**4 + 1)/15

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GIAC/XCAS [A]  time = 0.21244, size = 55, normalized size = 1.2 \[ -\frac{1}{10} \,{\left (x^{4} - 1\right )}^{2} \sqrt{-x^{4} + 1} + \frac{1}{3} \,{\left (-x^{4} + 1\right )}^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{-x^{4} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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