\(\int \frac {\sqrt {1+x^2}}{\sqrt {1+\sqrt {1+x^2}}} \, dx\) [1056]

Optimal result

Integrand size = 25, antiderivative size = 79 \[ \int \frac {\sqrt {1+x^2}}{\sqrt {1+\sqrt {1+x^2}}} \, dx=-\frac {2 x}{3 \sqrt {1+\sqrt {1+x^2}}}+\frac {2 x \sqrt {1+x^2}}{3 \sqrt {1+\sqrt {1+x^2}}}+\sqrt {2} \arctan \left (\frac {x}{\sqrt {2} \sqrt {1+\sqrt {1+x^2}}}\right ) \]

[Out]

-2/3*x/(1+(x^2+1)^(1/2))^(1/2)+2/3*x*(x^2+1)^(1/2)/(1+(x^2+1)^(1/2))^(1/2)+2^(1/2)*arctan(1/2*x*2^(1/2)/(1+(x^
2+1)^(1/2))^(1/2))

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {1+x^2}}{\sqrt {1+\sqrt {1+x^2}}} \, dx=\frac {2 x \left (-1+\sqrt {1+x^2}\right )}{3 \sqrt {1+\sqrt {1+x^2}}}+\sqrt {2} \arctan \left (\frac {x}{\sqrt {2} \sqrt {1+\sqrt {1+x^2}}}\right ) \]

[In]

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

[Out]

(2*x*(-1 + Sqrt[1 + x^2]))/(3*Sqrt[1 + Sqrt[1 + x^2]]) + Sqrt[2]*ArcTan[x/(Sqrt[2]*Sqrt[1 + Sqrt[1 + x^2]])]

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.68 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {1+x^2}}{\sqrt {1+\sqrt {1+x^2}}} \, dx=-\frac {3 \, \sqrt {2} x \arctan \left (\frac {\sqrt {2} \sqrt {\sqrt {x^{2} + 1} + 1}}{x}\right ) - 2 \, {\left (x^{2} - 2 \, \sqrt {x^{2} + 1} + 2\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{3 \, x} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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