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