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