Optimal. Leaf size=51 \[ -\frac {1}{2} \sqrt {2 x-x^2}+\frac {1}{2} \tanh ^{-1}\left (\sqrt {2 x-x^2}\right )-\frac {x}{2}-\frac {1}{2} \log (1-x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6742, 43, 685, 688, 207} \[ -\frac {1}{2} \sqrt {2 x-x^2}+\frac {1}{2} \tanh ^{-1}\left (\sqrt {2 x-x^2}\right )-\frac {x}{2}-\frac {1}{2} \log (1-x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 207
Rule 685
Rule 688
Rule 6742
Rubi steps
\begin {align*} \int \frac {x+\sqrt {2 x-x^2}}{2-2 x} \, dx &=\int \left (-\frac {x}{2 (-1+x)}+\frac {\sqrt {2 x-x^2}}{2 (1-x)}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {x}{-1+x} \, dx\right )+\frac {1}{2} \int \frac {\sqrt {2 x-x^2}}{1-x} \, dx\\ &=-\frac {1}{2} \sqrt {2 x-x^2}-\frac {1}{2} \int \left (1+\frac {1}{-1+x}\right ) \, dx+\frac {1}{2} \int \frac {1}{(1-x) \sqrt {2 x-x^2}} \, dx\\ &=-\frac {x}{2}-\frac {1}{2} \sqrt {2 x-x^2}-\frac {1}{2} \log (1-x)-2 \operatorname {Subst}\left (\int \frac {1}{-4+4 x^2} \, dx,x,\sqrt {2 x-x^2}\right )\\ &=-\frac {x}{2}-\frac {1}{2} \sqrt {2 x-x^2}+\frac {1}{2} \tanh ^{-1}\left (\sqrt {2 x-x^2}\right )-\frac {1}{2} \log (1-x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 39, normalized size = 0.76 \[ \frac {1}{2} \left (-x-\sqrt {-((x-2) x)}-\log (1-x)+\tanh ^{-1}\left (\sqrt {-((x-2) x)}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 66, normalized size = 1.29 \[ -\frac {1}{2} \, x - \frac {1}{2} \, \sqrt {-x^{2} + 2 \, x} - \frac {1}{2} \, \log \left (x - 1\right ) + \frac {1}{2} \, \log \left (\frac {x + \sqrt {-x^{2} + 2 \, x}}{x}\right ) - \frac {1}{2} \, \log \left (-\frac {x - \sqrt {-x^{2} + 2 \, x}}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.48, size = 50, normalized size = 0.98 \[ -\frac {1}{2} \, x - \frac {1}{2} \, \sqrt {-x^{2} + 2 \, x} - \frac {1}{2} \, \log \left (-\frac {2 \, {\left (\sqrt {-x^{2} + 2 \, x} - 1\right )}}{{\left | -2 \, x + 2 \right |}}\right ) - \frac {1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 38, normalized size = 0.75 \[ -\frac {x}{2}+\frac {\arctanh \left (\frac {1}{\sqrt {-\left (x -1\right )^{2}+1}}\right )}{2}-\frac {\ln \left (x -1\right )}{2}-\frac {\sqrt {-\left (x -1\right )^{2}+1}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.98, size = 54, normalized size = 1.06 \[ -\frac {1}{2} \, x - \frac {1}{2} \, \sqrt {-x^{2} + 2 \, x} - \frac {1}{2} \, \log \left (x - 1\right ) + \frac {1}{2} \, \log \left (\frac {2 \, \sqrt {-x^{2} + 2 \, x}}{{\left | x - 1 \right |}} + \frac {2}{{\left | x - 1 \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int -\frac {x+\sqrt {2\,x-x^2}}{2\,x-2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {x}{x - 1}\, dx + \int \frac {\sqrt {- x^{2} + 2 x}}{x - 1}\, dx}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________