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

Optimal. Leaf size=22 \[ \tan ^{-1}\left (\frac{x}{\sqrt{\sqrt{x^4+1}-x^2}}\right ) \]

[Out]

ArcTan[x/Sqrt[-x^2 + Sqrt[1 + x^4]]]

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Rubi [A]  time = 0.101277, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ \tan ^{-1}\left (\frac{x}{\sqrt{\sqrt{x^4+1}-x^2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[x/Sqrt[-x^2 + Sqrt[1 + x^4]]]

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Rubi in Sympy [A]  time = 4.33165, size = 17, normalized size = 0.77 \[ \operatorname{atan}{\left (\frac{x}{\sqrt{- x^{2} + \sqrt{x^{4} + 1}}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(x/sqrt(-x**2 + sqrt(x**4 + 1)))

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Mathematica [A]  time = 1.37299, size = 24, normalized size = 1.09 \[ \cot ^{-1}\left (\frac{\sqrt{\sqrt{x^4+1}-x^2}}{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

ArcCot[Sqrt[-x^2 + Sqrt[1 + x^4]]/x]

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}+1}{\frac{1}{\sqrt{-{x}^{2}+\sqrt{{x}^{4}+1}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.787091, size = 77, normalized size = 3.5 \[ -\frac{1}{4} \, \arctan \left (\frac{4 \,{\left (2 \, x^{3} - \sqrt{x^{4} + 1} x\right )} \sqrt{-x^{2} + \sqrt{x^{4} + 1}}}{9 \, x^{4} - 8 \, \sqrt{x^{4} + 1} x^{2} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*arctan(4*(2*x^3 - sqrt(x^4 + 1)*x)*sqrt(-x^2 + sqrt(x^4 + 1))/(9*x^4 - 8*sq
rt(x^4 + 1)*x^2 + 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- x^{2} + \sqrt{x^{4} + 1}} \left (x^{4} + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(sqrt(-x**2 + sqrt(x**4 + 1))*(x**4 + 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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