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

Optimal. Leaf size=33 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{\sqrt{x^4+1}-x^2}}\right )}{\sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.0914918, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{\sqrt{x^4+1}-x^2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 3.95058, size = 29, normalized size = 0.88 \[ \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{- x^{2} + \sqrt{x^{4} + 1}}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*atan(sqrt(2)*x/sqrt(-x**2 + sqrt(x**4 + 1)))/2

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Mathematica [B]  time = 1.43463, size = 162, normalized size = 4.91 \[ \frac{x \left (2 x^4-2 \sqrt{x^4+1} x^2+1\right )^2 \left (x^4-\sqrt{x^4+1} x^2+1\right ) \sin ^{-1}\left (x^2-\sqrt{x^4+1}\right )}{\sqrt{2} \sqrt{\sqrt{x^4+1}-x^2} \sqrt{x^2 \left (\sqrt{x^4+1}-x^2\right )} \left (-8 x^{10}-12 x^6+8 \sqrt{x^4+1} x^4+\sqrt{x^4+1}-4 x^2+8 \sqrt{x^4+1} x^8\right )} \]

Antiderivative was successfully verified.

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

[Out]

(x*(1 + 2*x^4 - 2*x^2*Sqrt[1 + x^4])^2*(1 + x^4 - x^2*Sqrt[1 + x^4])*ArcSin[x^2
- Sqrt[1 + x^4]])/(Sqrt[2]*Sqrt[-x^2 + Sqrt[1 + x^4]]*Sqrt[x^2*(-x^2 + Sqrt[1 +
x^4])]*(-4*x^2 - 12*x^6 - 8*x^10 + Sqrt[1 + x^4] + 8*x^4*Sqrt[1 + x^4] + 8*x^8*S
qrt[1 + x^4]))

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Maple [C]  time = 0.055, size = 22, normalized size = 0.7 \[ -{\frac{\sqrt{2}}{4\,{x}^{2}}{\mbox{$_3$F$_2$}({\frac{1}{2}},{\frac{3}{4}},{\frac{5}{4}};\,{\frac{3}{2}},{\frac{3}{2}};\,-{x}^{-4})}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*2^(1/2)/x^2*hypergeom([1/2,3/4,5/4],[3/2,3/2],-1/x^4)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.400327, size = 39, normalized size = 1.18 \[ -\frac{1}{2} \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-x^{2} + \sqrt{x^{4} + 1}}}{2 \, x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.16438, size = 15, normalized size = 0.45 \[ \frac{{G_{3, 3}^{2, 2}\left (\begin{matrix} \frac{1}{2}, 1 & 1 \\\frac{1}{4}, \frac{3}{4} & 0 \end{matrix} \middle |{x^{4}} \right )}}{4 \sqrt{\pi }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

meijerg(((1/2, 1), (1,)), ((1/4, 3/4), (0,)), x**4)/(4*sqrt(pi))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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