Integrand size = 33, antiderivative size = 91 \[ \int \frac {\sqrt {x-\sqrt {1+x^2}}}{1-\sqrt {1+x^2}} \, dx=\frac {2 (-2-x) \sqrt {x-\sqrt {1+x^2}}+2 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1-x+\sqrt {1+x^2}}-2 \arctan \left (\sqrt {x-\sqrt {1+x^2}}\right ) \] Output:
(2*(-2-x)*(x-(x^2+1)^(1/2))^(1/2)+2*(x^2+1)^(1/2)*(x-(x^2+1)^(1/2))^(1/2)) /(-1-x+(x^2+1)^(1/2))-2*arctan((x-(x^2+1)^(1/2))^(1/2))
Time = 0.10 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {x-\sqrt {1+x^2}}}{1-\sqrt {1+x^2}} \, dx=\frac {2 \left (\sqrt {x-\sqrt {1+x^2}} \left (-2-x+\sqrt {1+x^2}\right )+\left (1+x-\sqrt {1+x^2}\right ) \arctan \left (\sqrt {x-\sqrt {1+x^2}}\right )\right )}{-1-x+\sqrt {1+x^2}} \] Input:
Integrate[Sqrt[x - Sqrt[1 + x^2]]/(1 - Sqrt[1 + x^2]),x]
Output:
(2*(Sqrt[x - Sqrt[1 + x^2]]*(-2 - x + Sqrt[1 + x^2]) + (1 + x - Sqrt[1 + x ^2])*ArcTan[Sqrt[x - Sqrt[1 + x^2]]]))/(-1 - x + Sqrt[1 + x^2])
Time = 0.58 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.91, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sqrt {x-\sqrt {x^2+1}}}{1-\sqrt {x^2+1}} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {\sqrt {x^2+1} \sqrt {x-\sqrt {x^2+1}}}{x^2}-\frac {\sqrt {x-\sqrt {x^2+1}}}{x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 \arctan \left (\sqrt {x-\sqrt {x^2+1}}\right )+\frac {\sqrt {x-\sqrt {x^2+1}}}{x}+2 \sqrt {x-\sqrt {x^2+1}}-\frac {1}{x \sqrt {x-\sqrt {x^2+1}}}\) |
Input:
Int[Sqrt[x - Sqrt[1 + x^2]]/(1 - Sqrt[1 + x^2]),x]
Output:
-(1/(x*Sqrt[x - Sqrt[1 + x^2]])) + 2*Sqrt[x - Sqrt[1 + x^2]] + Sqrt[x - Sq rt[1 + x^2]]/x - 2*ArcTan[Sqrt[x - Sqrt[1 + x^2]]]
\[\int \frac {\sqrt {x -\sqrt {x^{2}+1}}}{1-\sqrt {x^{2}+1}}d x\]
Input:
int((x-(x^2+1)^(1/2))^(1/2)/(1-(x^2+1)^(1/2)),x)
Output:
int((x-(x^2+1)^(1/2))^(1/2)/(1-(x^2+1)^(1/2)),x)
Time = 0.09 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.55 \[ \int \frac {\sqrt {x-\sqrt {1+x^2}}}{1-\sqrt {1+x^2}} \, dx=-\frac {2 \, x \arctan \left (\sqrt {x - \sqrt {x^{2} + 1}}\right ) - {\left (3 \, x + \sqrt {x^{2} + 1} + 1\right )} \sqrt {x - \sqrt {x^{2} + 1}}}{x} \] Input:
integrate((x-(x^2+1)^(1/2))^(1/2)/(1-(x^2+1)^(1/2)),x, algorithm="fricas")
Output:
-(2*x*arctan(sqrt(x - sqrt(x^2 + 1))) - (3*x + sqrt(x^2 + 1) + 1)*sqrt(x - sqrt(x^2 + 1)))/x
\[ \int \frac {\sqrt {x-\sqrt {1+x^2}}}{1-\sqrt {1+x^2}} \, dx=- \int \frac {\sqrt {x - \sqrt {x^{2} + 1}}}{\sqrt {x^{2} + 1} - 1}\, dx \] Input:
integrate((x-(x**2+1)**(1/2))**(1/2)/(1-(x**2+1)**(1/2)),x)
Output:
-Integral(sqrt(x - sqrt(x**2 + 1))/(sqrt(x**2 + 1) - 1), x)
\[ \int \frac {\sqrt {x-\sqrt {1+x^2}}}{1-\sqrt {1+x^2}} \, dx=\int { -\frac {\sqrt {x - \sqrt {x^{2} + 1}}}{\sqrt {x^{2} + 1} - 1} \,d x } \] Input:
integrate((x-(x^2+1)^(1/2))^(1/2)/(1-(x^2+1)^(1/2)),x, algorithm="maxima")
Output:
-integrate(sqrt(x - sqrt(x^2 + 1))/(sqrt(x^2 + 1) - 1), x)
\[ \int \frac {\sqrt {x-\sqrt {1+x^2}}}{1-\sqrt {1+x^2}} \, dx=\int { -\frac {\sqrt {x - \sqrt {x^{2} + 1}}}{\sqrt {x^{2} + 1} - 1} \,d x } \] Input:
integrate((x-(x^2+1)^(1/2))^(1/2)/(1-(x^2+1)^(1/2)),x, algorithm="giac")
Output:
integrate(-sqrt(x - sqrt(x^2 + 1))/(sqrt(x^2 + 1) - 1), x)
Timed out. \[ \int \frac {\sqrt {x-\sqrt {1+x^2}}}{1-\sqrt {1+x^2}} \, dx=-\int \frac {\sqrt {x-\sqrt {x^2+1}}}{\sqrt {x^2+1}-1} \,d x \] Input:
int(-(x - (x^2 + 1)^(1/2))^(1/2)/((x^2 + 1)^(1/2) - 1),x)
Output:
-int((x - (x^2 + 1)^(1/2))^(1/2)/((x^2 + 1)^(1/2) - 1), x)
\[ \int \frac {\sqrt {x-\sqrt {1+x^2}}}{1-\sqrt {1+x^2}} \, dx=-\mathit {atan} \left (\frac {\sqrt {-\sqrt {x^{2}+1}+x}\, \sqrt {x^{2}+1}}{2}+\frac {\sqrt {-\sqrt {x^{2}+1}+x}\, x}{2}+\frac {\sqrt {-\sqrt {x^{2}+1}+x}}{2}\right )+3 \sqrt {-\sqrt {x^{2}+1}+x}-\left (\int \frac {\sqrt {-\sqrt {x^{2}+1}+x}}{x^{4}+x^{2}}d x \right )-\frac {\left (\int \frac {\sqrt {-\sqrt {x^{2}+1}+x}}{x^{3}+x}d x \right )}{2}-\left (\int \frac {\sqrt {-\sqrt {x^{2}+1}+x}}{x^{2}+1}d x \right )-\left (\int \frac {\sqrt {-\sqrt {x^{2}+1}+x}\, \sqrt {x^{2}+1}}{x^{4}+x^{2}}d x \right )-\frac {\left (\int \frac {\sqrt {-\sqrt {x^{2}+1}+x}\, \sqrt {x^{2}+1}}{x^{3}+x}d x \right )}{2}-\frac {\left (\int \frac {\sqrt {-\sqrt {x^{2}+1}+x}\, x}{x^{2}+1}d x \right )}{2} \] Input:
int((x-(x^2+1)^(1/2))^(1/2)/(1-(x^2+1)^(1/2)),x)
Output:
( - 2*atan((sqrt( - sqrt(x**2 + 1) + x)*sqrt(x**2 + 1) + sqrt( - sqrt(x**2 + 1) + x)*x + sqrt( - sqrt(x**2 + 1) + x))/2) + 6*sqrt( - sqrt(x**2 + 1) + x) - 2*int(sqrt( - sqrt(x**2 + 1) + x)/(x**4 + x**2),x) - int(sqrt( - sq rt(x**2 + 1) + x)/(x**3 + x),x) - 2*int(sqrt( - sqrt(x**2 + 1) + x)/(x**2 + 1),x) - 2*int((sqrt( - sqrt(x**2 + 1) + x)*sqrt(x**2 + 1))/(x**4 + x**2) ,x) - int((sqrt( - sqrt(x**2 + 1) + x)*sqrt(x**2 + 1))/(x**3 + x),x) - int ((sqrt( - sqrt(x**2 + 1) + x)*x)/(x**2 + 1),x))/2