\(\int \frac {\sqrt {1+x^4}}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx\) [1059]

Optimal result

Integrand size = 27, antiderivative size = 79 \[ \int \frac {\sqrt {1+x^4}}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx=\frac {x}{8 \left (x^2+\sqrt {1+x^4}\right )^{3/2}}+\frac {1}{4} x \sqrt {x^2+\sqrt {1+x^4}}+\frac {5 \arctan \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}\right )}{8 \sqrt {2}} \]

[Out]

1/8*x/(x^2+(x^4+1)^(1/2))^(3/2)+1/4*x*(x^2+(x^4+1)^(1/2))^(1/2)+5/16*arctan(2^(1/2)*x*(x^2+(x^4+1)^(1/2))^(1/2
))*2^(1/2)

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {1+x^4}}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx=\frac {1}{16} \left (\frac {2 x \left (1+2 \left (x^2+\sqrt {1+x^4}\right )^2\right )}{\left (x^2+\sqrt {1+x^4}\right )^{3/2}}+5 \sqrt {2} \arctan \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}\right )\right ) \]

[In]

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

[Out]

((2*x*(1 + 2*(x^2 + Sqrt[1 + x^4])^2))/(x^2 + Sqrt[1 + x^4])^(3/2) + 5*Sqrt[2]*ArcTan[Sqrt[2]*x*Sqrt[x^2 + Sqr
t[1 + x^4]]])/16

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.52 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {1+x^4}}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx=\frac {1}{8} \, {\left (2 \, x^{5} - 2 \, \sqrt {x^{4} + 1} x^{3} + 3 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - \frac {5}{16} \, \sqrt {2} \arctan \left (-\frac {{\left (\sqrt {2} x^{2} - \sqrt {2} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{2 \, x}\right ) \]

[In]

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

[Out]

1/8*(2*x^5 - 2*sqrt(x^4 + 1)*x^3 + 3*x)*sqrt(x^2 + sqrt(x^4 + 1)) - 5/16*sqrt(2)*arctan(-1/2*(sqrt(2)*x^2 - sq
rt(2)*sqrt(x^4 + 1))*sqrt(x^2 + sqrt(x^4 + 1))/x)

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(x**4 + 1)/sqrt(x**2 + sqrt(x**4 + 1)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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