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