\(\int \frac {\sqrt {1+\sqrt {1+x^2}}}{\sqrt {1+x^2}} \, dx\) [158]

Optimal result

Integrand size = 25, antiderivative size = 18 \[ \int \frac {\sqrt {1+\sqrt {1+x^2}}}{\sqrt {1+x^2}} \, dx=\frac {2 x}{\sqrt {1+\sqrt {1+x^2}}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+\sqrt {1+x^2}}}{\sqrt {1+x^2}} \, dx=\frac {2 x}{\sqrt {1+\sqrt {1+x^2}}} \]

[In]

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

[Out]

(2*x)/Sqrt[1 + Sqrt[1 + x^2]]

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.39 \[ \int \frac {\sqrt {1+\sqrt {1+x^2}}}{\sqrt {1+x^2}} \, dx=\frac {2 \, \sqrt {\sqrt {x^{2} + 1} + 1} {\left (\sqrt {x^{2} + 1} - 1\right )}}{x} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (15) = 30\).

Time = 0.45 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \frac {\sqrt {1+\sqrt {1+x^2}}}{\sqrt {1+x^2}} \, dx=\frac {\sqrt {2} x \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{\pi \sqrt {\sqrt {x^{2} + 1} + 1}} \]

[In]

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

[Out]

sqrt(2)*x*gamma(1/4)*gamma(3/4)/(pi*sqrt(sqrt(x**2 + 1) + 1))

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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