Integrand size = 22, antiderivative size = 16 \[ \int \frac {x}{1-x^2+\sqrt {1-x^2}} \, dx=-\log \left (1+\sqrt {1-x^2}\right ) \]
[Out]
Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2186, 31} \[ \int \frac {x}{1-x^2+\sqrt {1-x^2}} \, dx=-\log \left (\sqrt {1-x^2}+1\right ) \]
[In]
[Out]
Rule 31
Rule 2186
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{1+\sqrt {1-x}-x} \, dx,x,x^2\right ) \\ & = -\text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt {1-x^2}\right ) \\ & = -\log \left (1+\sqrt {1-x^2}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {x}{1-x^2+\sqrt {1-x^2}} \, dx=-\log \left (1+\sqrt {1-x^2}\right ) \]
[In]
[Out]
Time = 0.09 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94
method | result | size |
trager | \(-\ln \left (1+\sqrt {-x^{2}+1}\right )\) | \(15\) |
default | \(-\ln \left (x \right )+\sqrt {-x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right )-\frac {\sqrt {-\left (1+x \right )^{2}+2 x +2}}{2}-\frac {\sqrt {-\left (-1+x \right )^{2}-2 x +2}}{2}\) | \(59\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.31 \[ \int \frac {x}{1-x^2+\sqrt {1-x^2}} \, dx=-\log \left (x\right ) + \log \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (12) = 24\).
Time = 1.22 (sec) , antiderivative size = 48, normalized size of antiderivative = 3.00 \[ \int \frac {x}{1-x^2+\sqrt {1-x^2}} \, dx=\frac {\log {\left (2 \sqrt {1 - x^{2}} \right )}}{2} - \frac {\log {\left (2 \sqrt {1 - x^{2}} + 2 \right )}}{2} - \frac {\log {\left (2 x^{2} - 2 \sqrt {1 - x^{2}} - 2 \right )}}{2} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {x}{1-x^2+\sqrt {1-x^2}} \, dx=-\log \left (\sqrt {-x^{2} + 1} + 1\right ) \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {x}{1-x^2+\sqrt {1-x^2}} \, dx=-\log \left (\sqrt {-x^{2} + 1} + 1\right ) \]
[In]
[Out]
Time = 0.13 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.31 \[ \int \frac {x}{1-x^2+\sqrt {1-x^2}} \, dx=\ln \left (\sqrt {\frac {1}{x^2}-1}-\sqrt {\frac {1}{x^2}}\right )-\ln \left (x\right ) \]
[In]
[Out]