3.719 \(\int \frac{\sqrt{1-x^4}}{\sqrt{1-x^2}} \, dx\)

Optimal. Leaf size=21 \[ \frac{1}{2} \sqrt{x^2+1} x+\frac{1}{2} \sinh ^{-1}(x) \]

[Out]

(x*Sqrt[1 + x^2])/2 + ArcSinh[x]/2

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Rubi [A]  time = 0.0090648, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{1}{2} \sqrt{x^2+1} x+\frac{1}{2} \sinh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - x^4]/Sqrt[1 - x^2],x]

[Out]

(x*Sqrt[1 + x^2])/2 + ArcSinh[x]/2

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Rubi in Sympy [A]  time = 1.60775, size = 15, normalized size = 0.71 \[ \frac{x \sqrt{x^{2} + 1}}{2} + \frac{\operatorname{asinh}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(x**2 + 1)/2 + asinh(x)/2

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Mathematica [B]  time = 0.075347, size = 70, normalized size = 3.33 \[ \frac{1}{2} \left (\log \left (1-x^2\right )+\frac{\sqrt{1-x^4} x}{\sqrt{1-x^2}}-\log \left (x^3+\sqrt{1-x^2} \sqrt{1-x^4}-x\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 - x^4]/Sqrt[1 - x^2],x]

[Out]

((x*Sqrt[1 - x^4])/Sqrt[1 - x^2] + Log[1 - x^2] - Log[-x + x^3 + Sqrt[1 - x^2]*S
qrt[1 - x^4]])/2

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Maple [B]  time = 0.011, size = 47, normalized size = 2.2 \[ -{\frac{1}{2\,{x}^{2}-2}\sqrt{-{x}^{4}+1}\sqrt{-{x}^{2}+1} \left ( x\sqrt{{x}^{2}+1}+{\it Arcsinh} \left ( x \right ) \right ){\frac{1}{\sqrt{{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^4+1)^(1/2)/(-x^2+1)^(1/2),x)

[Out]

-1/2*(-x^4+1)^(1/2)*(-x^2+1)^(1/2)*(x*(x^2+1)^(1/2)+arcsinh(x))/(x^2-1)/(x^2+1)^
(1/2)

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Maxima [A]  time = 0.801589, size = 20, normalized size = 0.95 \[ \frac{1}{2} \, \sqrt{x^{2} + 1} x + \frac{1}{2} \, \operatorname{arsinh}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(x^2 + 1)*x + 1/2*arcsinh(x)

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Fricas [A]  time = 0.272544, size = 162, normalized size = 7.71 \[ -\frac{2 \, \sqrt{-x^{4} + 1} \sqrt{-x^{2} + 1} x +{\left (x^{2} - 1\right )} \log \left (\frac{x^{3} + \sqrt{-x^{4} + 1} \sqrt{-x^{2} + 1} - x}{x^{3} - x}\right ) -{\left (x^{2} - 1\right )} \log \left (-\frac{x^{3} - \sqrt{-x^{4} + 1} \sqrt{-x^{2} + 1} - x}{x^{3} - x}\right )}{4 \,{\left (x^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(2*sqrt(-x^4 + 1)*sqrt(-x^2 + 1)*x + (x^2 - 1)*log((x^3 + sqrt(-x^4 + 1)*sq
rt(-x^2 + 1) - x)/(x^3 - x)) - (x^2 - 1)*log(-(x^3 - sqrt(-x^4 + 1)*sqrt(-x^2 +
1) - x)/(x^3 - x)))/(x^2 - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}}{\sqrt{- \left (x - 1\right ) \left (x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(-(x - 1)*(x + 1)*(x**2 + 1))/sqrt(-(x - 1)*(x + 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-x^{4} + 1}}{\sqrt{-x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(-x^4 + 1)/sqrt(-x^2 + 1), x)