Integrand size = 33, antiderivative size = 91 \[ \int \frac {\sqrt {x-\sqrt {1+x^2}}}{1-\sqrt {1+x^2}} \, dx=\frac {2 (-2-x) \sqrt {x-\sqrt {1+x^2}}+2 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1-x+\sqrt {1+x^2}}-2 \arctan \left (\sqrt {x-\sqrt {1+x^2}}\right ) \]
[Out]
Time = 0.24 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.91, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6874, 2144, 468, 335, 304, 209, 212, 2145, 474, 12, 327, 218} \[ \int \frac {\sqrt {x-\sqrt {1+x^2}}}{1-\sqrt {1+x^2}} \, dx=-2 \arctan \left (\sqrt {x-\sqrt {x^2+1}}\right )+\frac {\sqrt {x-\sqrt {x^2+1}}}{x}+2 \sqrt {x-\sqrt {x^2+1}}-\frac {1}{x \sqrt {x-\sqrt {x^2+1}}} \]
[In]
[Out]
Rule 12
Rule 209
Rule 212
Rule 218
Rule 304
Rule 327
Rule 335
Rule 468
Rule 474
Rule 2144
Rule 2145
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\sqrt {x-\sqrt {1+x^2}}}{x^2}-\frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{x^2}\right ) \, dx \\ & = -\int \frac {\sqrt {x-\sqrt {1+x^2}}}{x^2} \, dx-\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{x^2} \, dx \\ & = -\left (2 \text {Subst}\left (\int \frac {\sqrt {x} \left (1+x^2\right )}{\left (-1+x^2\right )^2} \, dx,x,x-\sqrt {1+x^2}\right )\right )+\text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{\sqrt {x} \left (-1+x^2\right )^2} \, dx,x,x-\sqrt {1+x^2}\right ) \\ & = \frac {\sqrt {x-\sqrt {1+x^2}}}{x}+\frac {\sqrt {x-\sqrt {1+x^2}}}{x \left (-x+\sqrt {1+x^2}\right )}+\frac {1}{2} \text {Subst}\left (\int \frac {2 x^{3/2}}{-1+x^2} \, dx,x,x-\sqrt {1+x^2}\right )-\text {Subst}\left (\int \frac {\sqrt {x}}{-1+x^2} \, dx,x,x-\sqrt {1+x^2}\right ) \\ & = \frac {\sqrt {x-\sqrt {1+x^2}}}{x}+\frac {\sqrt {x-\sqrt {1+x^2}}}{x \left (-x+\sqrt {1+x^2}\right )}-2 \text {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )+\text {Subst}\left (\int \frac {x^{3/2}}{-1+x^2} \, dx,x,x-\sqrt {1+x^2}\right ) \\ & = 2 \sqrt {x-\sqrt {1+x^2}}+\frac {\sqrt {x-\sqrt {1+x^2}}}{x}+\frac {\sqrt {x-\sqrt {1+x^2}}}{x \left (-x+\sqrt {1+x^2}\right )}+\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )+\text {Subst}\left (\int \frac {1}{\sqrt {x} \left (-1+x^2\right )} \, dx,x,x-\sqrt {1+x^2}\right )-\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right ) \\ & = 2 \sqrt {x-\sqrt {1+x^2}}+\frac {\sqrt {x-\sqrt {1+x^2}}}{x}+\frac {\sqrt {x-\sqrt {1+x^2}}}{x \left (-x+\sqrt {1+x^2}\right )}-\arctan \left (\sqrt {x-\sqrt {1+x^2}}\right )+\text {arctanh}\left (\sqrt {x-\sqrt {1+x^2}}\right )+2 \text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right ) \\ & = 2 \sqrt {x-\sqrt {1+x^2}}+\frac {\sqrt {x-\sqrt {1+x^2}}}{x}+\frac {\sqrt {x-\sqrt {1+x^2}}}{x \left (-x+\sqrt {1+x^2}\right )}-\arctan \left (\sqrt {x-\sqrt {1+x^2}}\right )+\text {arctanh}\left (\sqrt {x-\sqrt {1+x^2}}\right )-\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )-\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right ) \\ & = 2 \sqrt {x-\sqrt {1+x^2}}+\frac {\sqrt {x-\sqrt {1+x^2}}}{x}+\frac {\sqrt {x-\sqrt {1+x^2}}}{x \left (-x+\sqrt {1+x^2}\right )}-2 \arctan \left (\sqrt {x-\sqrt {1+x^2}}\right ) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {x-\sqrt {1+x^2}}}{1-\sqrt {1+x^2}} \, dx=\frac {2 \left (\sqrt {x-\sqrt {1+x^2}} \left (-2-x+\sqrt {1+x^2}\right )+\left (1+x-\sqrt {1+x^2}\right ) \arctan \left (\sqrt {x-\sqrt {1+x^2}}\right )\right )}{-1-x+\sqrt {1+x^2}} \]
[In]
[Out]
\[\int \frac {\sqrt {x -\sqrt {x^{2}+1}}}{1-\sqrt {x^{2}+1}}d x\]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.55 \[ \int \frac {\sqrt {x-\sqrt {1+x^2}}}{1-\sqrt {1+x^2}} \, dx=-\frac {2 \, x \arctan \left (\sqrt {x - \sqrt {x^{2} + 1}}\right ) - {\left (3 \, x + \sqrt {x^{2} + 1} + 1\right )} \sqrt {x - \sqrt {x^{2} + 1}}}{x} \]
[In]
[Out]
\[ \int \frac {\sqrt {x-\sqrt {1+x^2}}}{1-\sqrt {1+x^2}} \, dx=- \int \frac {\sqrt {x - \sqrt {x^{2} + 1}}}{\sqrt {x^{2} + 1} - 1}\, dx \]
[In]
[Out]
\[ \int \frac {\sqrt {x-\sqrt {1+x^2}}}{1-\sqrt {1+x^2}} \, dx=\int { -\frac {\sqrt {x - \sqrt {x^{2} + 1}}}{\sqrt {x^{2} + 1} - 1} \,d x } \]
[In]
[Out]
\[ \int \frac {\sqrt {x-\sqrt {1+x^2}}}{1-\sqrt {1+x^2}} \, dx=\int { -\frac {\sqrt {x - \sqrt {x^{2} + 1}}}{\sqrt {x^{2} + 1} - 1} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {x-\sqrt {1+x^2}}}{1-\sqrt {1+x^2}} \, dx=-\int \frac {\sqrt {x-\sqrt {x^2+1}}}{\sqrt {x^2+1}-1} \,d x \]
[In]
[Out]