\(\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^2} \, dx\) [930]

Optimal result

Integrand size = 21, antiderivative size = 70 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^2} \, dx=-\frac {\sqrt {x^2+\sqrt {1+x^4}}}{x}+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \]

[Out]

-(x^2+(x^4+1)^(1/2))^(1/2)/x+arctanh(2^(1/2)*x*(x^2+(x^4+1)^(1/2))^(1/2)/(1+x^2+(x^4+1)^(1/2)))*2^(1/2)

Rubi [F]

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

[In]

Int[Sqrt[x^2 + Sqrt[1 + x^4]]/x^2,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^2} \, dx=-\frac {\sqrt {x^2+\sqrt {1+x^4}}}{x}+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \]

[In]

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

[Out]

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

Maple [C] (verified)

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

Time = 0.16 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.73

method	result	size
meijerg	\(\frac {-\frac {\sqrt {\pi }\, \sqrt {2}\, \operatorname {hypergeom}\left (\left [\frac {3}{4}, 1, 1, \frac {5}{4}\right ], \left [\frac {3}{2}, 2, 2\right ], -\frac {1}{x^{4}}\right )}{2 x^{4}}-4 \left (-4 \ln \left (2\right )+4-4 \ln \left (x \right )\right ) \sqrt {\pi }\, \sqrt {2}}{16 \sqrt {\pi }}\)	\(51\)

[In]

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

[Out]

1/16/Pi^(1/2)*(-1/2*Pi^(1/2)*2^(1/2)/x^4*hypergeom([3/4,1,1,5/4],[3/2,2,2],-1/x^4)-4*(-4*ln(2)+4-4*ln(x))*Pi^(
1/2)*2^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.52 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.16 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^2} \, dx=\frac {\sqrt {2} x \log \left (4 \, x^{4} + 4 \, \sqrt {x^{4} + 1} x^{2} + 2 \, {\left (\sqrt {2} x^{3} + \sqrt {2} \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1\right ) - 4 \, \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{4 \, x} \]

[In]

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

[Out]

1/4*(sqrt(2)*x*log(4*x^4 + 4*sqrt(x^4 + 1)*x^2 + 2*(sqrt(2)*x^3 + sqrt(2)*sqrt(x^4 + 1)*x)*sqrt(x^2 + sqrt(x^4
+ 1)) + 1) - 4*sqrt(x^2 + sqrt(x^4 + 1)))/x

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.83 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{x^2} \, dx=- \frac {\log {\left (\frac {1}{x^{4}} \right )} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{4 \pi } - \frac {\Gamma \left (\frac {3}{4}\right ) \Gamma \left (\frac {5}{4}\right ) {{}_{4}F_{3}\left (\begin {matrix} \frac {3}{4}, 1, 1, \frac {5}{4} \\ \frac {3}{2}, 2, 2 \end {matrix}\middle | {\frac {e^{i \pi }}{x^{4}}} \right )}}{8 \pi x^{4}} \]

[In]

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

[Out]

-log(x**(-4))*gamma(1/4)*gamma(3/4)/(4*pi) - gamma(3/4)*gamma(5/4)*hyper((3/4, 1, 1, 5/4), (3/2, 2, 2), exp_po
lar(I*pi)/x**4)/(8*pi*x**4)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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