\(\int \sqrt {x^2+\sqrt {1+x^4}} \, dx\) [744]

Optimal result

Integrand size = 17, antiderivative size = 57 \[ \int \sqrt {x^2+\sqrt {1+x^4}} \, dx=\frac {1}{2} x \sqrt {x^2+\sqrt {1+x^4}}+\frac {\arctan \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}\right )}{2 \sqrt {2}} \]

[Out]

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

Rubi [F]

\[ \int \sqrt {x^2+\sqrt {1+x^4}} \, dx=\int \sqrt {x^2+\sqrt {1+x^4}} \, dx \]

[In]

Int[Sqrt[x^2 + Sqrt[1 + x^4]],x]

[Out]

Defer[Int][Sqrt[x^2 + Sqrt[1 + x^4]], x]

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00 \[ \int \sqrt {x^2+\sqrt {1+x^4}} \, dx=\frac {1}{2} x \sqrt {x^2+\sqrt {1+x^4}}+\frac {\arctan \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}\right )}{2 \sqrt {2}} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3.

Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.39

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.47 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.07 \[ \int \sqrt {x^2+\sqrt {1+x^4}} \, dx=\frac {1}{2} \, \sqrt {x^{2} + \sqrt {x^{4} + 1}} x - \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {{\left (\sqrt {2} x^{2} - \sqrt {2} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{2 \, x}\right ) \]

[In]

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

[Out]

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
t(x^4 + 1))/x)

Sympy [A] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.30 \[ \int \sqrt {x^2+\sqrt {1+x^4}} \, dx=- \frac {{G_{3, 3}^{2, 2}\left (\begin {matrix} \frac {3}{2}, 1 & 1 \\\frac {1}{4}, \frac {3}{4} & 0 \end {matrix} \middle | {x^{4}} \right )}}{16 \sqrt {\pi }} \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \sqrt {x^2+\sqrt {1+x^4}} \, dx=\int { \sqrt {x^{2} + \sqrt {x^{4} + 1}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \sqrt {x^2+\sqrt {1+x^4}} \, dx=\int { \sqrt {x^{2} + \sqrt {x^{4} + 1}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \sqrt {x^2+\sqrt {1+x^4}} \, dx=\int \sqrt {\sqrt {x^4+1}+x^2} \,d x \]

[In]

int(((x^4 + 1)^(1/2) + x^2)^(1/2),x)

[Out]

int(((x^4 + 1)^(1/2) + x^2)^(1/2), x)