Integrand size = 19, antiderivative size = 54 \[ \int \frac {1}{x \sqrt {x+\sqrt {1+x^2}}} \, dx=\frac {2}{\sqrt {x+\sqrt {1+x^2}}}+2 \arctan \left (\sqrt {x+\sqrt {1+x^2}}\right )-2 \text {arctanh}\left (\sqrt {x+\sqrt {1+x^2}}\right ) \]
[Out]
Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2144, 464, 335, 304, 209, 212} \[ \int \frac {1}{x \sqrt {x+\sqrt {1+x^2}}} \, dx=2 \arctan \left (\sqrt {\sqrt {x^2+1}+x}\right )-2 \text {arctanh}\left (\sqrt {\sqrt {x^2+1}+x}\right )+\frac {2}{\sqrt {\sqrt {x^2+1}+x}} \]
[In]
[Out]
Rule 209
Rule 212
Rule 304
Rule 335
Rule 464
Rule 2144
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1+x^2}{x^{3/2} \left (-1+x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right ) \\ & = \frac {2}{\sqrt {x+\sqrt {1+x^2}}}+2 \text {Subst}\left (\int \frac {\sqrt {x}}{-1+x^2} \, dx,x,x+\sqrt {1+x^2}\right ) \\ & = \frac {2}{\sqrt {x+\sqrt {1+x^2}}}+4 \text {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right ) \\ & = \frac {2}{\sqrt {x+\sqrt {1+x^2}}}-2 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )+2 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right ) \\ & = \frac {2}{\sqrt {x+\sqrt {1+x^2}}}+2 \arctan \left (\sqrt {x+\sqrt {1+x^2}}\right )-2 \text {arctanh}\left (\sqrt {x+\sqrt {1+x^2}}\right ) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {x+\sqrt {1+x^2}}} \, dx=\frac {2}{\sqrt {x+\sqrt {1+x^2}}}+2 \arctan \left (\sqrt {x+\sqrt {1+x^2}}\right )-2 \text {arctanh}\left (\sqrt {x+\sqrt {1+x^2}}\right ) \]
[In]
[Out]
Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.41
method | result | size |
meijerg | \(-\frac {\sqrt {2}\, \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {1}{4}, \frac {3}{4}\right ], \left [\frac {5}{4}, \frac {3}{2}\right ], -\frac {1}{x^{2}}\right )}{\sqrt {x}}\) | \(22\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.28 \[ \int \frac {1}{x \sqrt {x+\sqrt {1+x^2}}} \, dx=-2 \, \sqrt {x + \sqrt {x^{2} + 1}} {\left (x - \sqrt {x^{2} + 1}\right )} + 2 \, \arctan \left (\sqrt {x + \sqrt {x^{2} + 1}}\right ) - \log \left (\sqrt {x + \sqrt {x^{2} + 1}} + 1\right ) + \log \left (\sqrt {x + \sqrt {x^{2} + 1}} - 1\right ) \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 1.78 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x \sqrt {x+\sqrt {1+x^2}}} \, dx=- \frac {\Gamma ^{2}\left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4}, \frac {3}{4} \\ \frac {5}{4}, \frac {3}{2} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{2}}} \right )}}{4 \pi \sqrt {x} \Gamma \left (\frac {5}{4}\right )} \]
[In]
[Out]
\[ \int \frac {1}{x \sqrt {x+\sqrt {1+x^2}}} \, dx=\int { \frac {1}{\sqrt {x + \sqrt {x^{2} + 1}} x} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{x \sqrt {x+\sqrt {1+x^2}}} \, dx=\int { \frac {1}{\sqrt {x + \sqrt {x^{2} + 1}} x} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{x \sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {1}{x\,\sqrt {x+\sqrt {x^2+1}}} \,d x \]
[In]
[Out]