∫ ∘ _________ x2 + √ ___

Optimal. Leaf size=57 \[ \frac {1}{2} \sqrt {\sqrt {x^4+1}+x^2} x+\frac {\tan ^{-1}\left (\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}\right )}{2 \sqrt {2}} \]

________________________________________________________________________________________

Rubi [F] time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \sqrt {x^2+\sqrt {1+x^4}} \, dx \end {gather*}

\begin {align*} \int \sqrt {x^2+\sqrt {1+x^4}} \, dx &=\int \sqrt {x^2+\sqrt {1+x^4}} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A] time = 0.18, size = 92, normalized size = 1.61 \begin {gather*} \frac {2 \left (\sqrt {x^4+1}+x^2\right ) x^2+\sqrt {2} \sqrt {x^2 \left (\sqrt {x^4+1}+x^2\right )} \tan ^{-1}\left (\sqrt {\left (\sqrt {x^4+1}+x^2\right )^2-1}\right )}{4 x \sqrt {\sqrt {x^4+1}+x^2}} \end {gather*}

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

________________________________________________________________________________________

IntegrateAlgebraic [A] time = 0.11, size = 57, normalized size = 1.00 \begin {gather*} \frac {1}{2} x \sqrt {x^2+\sqrt {1+x^4}}+\frac {\tan ^{-1}\left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}\right )}{2 \sqrt {2}} \end {gather*}

IntegrateAlgebraic[Sqrt[x^2 + Sqrt[1 + x^4]],x]

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

________________________________________________________________________________________

fricas [A] time = 0.98, size = 61, normalized size = 1.07 \begin {gather*} \frac {1}{2} \, \sqrt {x^{2} + \sqrt {x^{4} + 1}} x - \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {{\left (\sqrt {2} x^{2} - \sqrt {2} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{2 \, x}\right ) \end {gather*}

integrate((x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="fricas")

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

________________________________________________________________________________________

giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x^{2} + \sqrt {x^{4} + 1}}\,{d x} \end {gather*}

integrate((x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="giac")

________________________________________________________________________________________


method	result	size

meijerg	\(\frac {\sqrt {2}\, x^{2} \hypergeom \left (\left [-\frac {1}{2}, -\frac {1}{4}, \frac {1}{4}\right ], \left [\frac {1}{2}, \frac {1}{2}\right ], -\frac {1}{x^{4}}\right )}{2}\)	\(22\)

int((x^2+(x^4+1)^(1/2))^(1/2),x,method=_RETURNVERBOSE)

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

________________________________________________________________________________________

maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x^{2} + \sqrt {x^{4} + 1}}\,{d x} \end {gather*}

integrate((x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="maxima")

________________________________________________________________________________________

mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \sqrt {\sqrt {x^4+1}+x^2} \,d x \end {gather*}

________________________________________________________________________________________

sympy [A] time = 0.70, size = 17, normalized size = 0.30 \begin {gather*} - \frac {{G_{3, 3}^{2, 2}\left (\begin {matrix} \frac {3}{2}, 1 & 1 \\\frac {1}{4}, \frac {3}{4} & 0 \end {matrix} \middle | {x^{4}} \right )}}{16 \sqrt {\pi }} \end {gather*}

integrate((x**2+(x**4+1)**(1/2))**(1/2),x)

-meijerg(((3/2, 1), (1,)), ((1/4, 3/4), (0,)), x**4)/(16*sqrt(pi))

________________________________________________________________________________________

3.8.44 \(\int \sqrt {x^2+\sqrt {1+x^4}} \, dx\)