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

Optimal. Leaf size=23 \[ \frac{1}{2} \sqrt{1-x^2} x+\frac{1}{2} \sin ^{-1}(x) \]

[Out]

(x*Sqrt[1 - x^2])/2 + ArcSin[x]/2

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Rubi [A]  time = 0.011002, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{1}{2} \sqrt{1-x^2} x+\frac{1}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[1 - x^2])/2 + ArcSin[x]/2

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Rubi in Sympy [A]  time = 1.39287, size = 15, normalized size = 0.65 \[ \frac{x \sqrt{- x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left (x \right )}}{2} \]

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 + asin(x)/2

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Mathematica [B]  time = 0.0448667, size = 50, normalized size = 2.17 \[ \frac{1}{2} \left (\frac{\sqrt{1-x^4} x}{\sqrt{x^2+1}}+\tan ^{-1}\left (\frac{x \sqrt{x^2+1}}{\sqrt{1-x^4}}\right )\right ) \]

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

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Maple [B]  time = 0.011, size = 42, normalized size = 1.8 \[{\frac{1}{2}\sqrt{-{x}^{4}+1} \left ( x\sqrt{-{x}^{2}+1}+\arcsin \left ( x \right ) \right ){\frac{1}{\sqrt{{x}^{2}+1}}}{\frac{1}{\sqrt{-{x}^{2}+1}}}} \]

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)+arcsin(x))/(-x^2+1)^(1/2)

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Maxima [A]  time = 0.839288, size = 23, normalized size = 1. \[ \frac{1}{2} \, \sqrt{-x^{2} + 1} x + \frac{1}{2} \, \arcsin \left (x\right ) \]

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*arcsin(x)

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Fricas [A]  time = 0.268601, size = 81, normalized size = 3.52 \[ \frac{\sqrt{-x^{4} + 1} \sqrt{x^{2} + 1} x -{\left (x^{2} + 1\right )} \arctan \left (\frac{\sqrt{-x^{4} + 1} \sqrt{x^{2} + 1}}{x^{3} + x}\right )}{2 \,{\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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**2 + 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)