Optimal. Leaf size=39 \[ -\frac {1}{4} (1-2 x) \sqrt {-x+x^2}-\frac {1}{4} \tanh ^{-1}\left (\frac {x}{\sqrt {-x+x^2}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.00, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {626, 634, 212}
\begin {gather*} -\frac {1}{4} \sqrt {x^2-x} (1-2 x)-\frac {1}{4} \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-x}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 626
Rule 634
Rubi steps
\begin {align*} \int \sqrt {-x+x^2} \, dx &=-\frac {1}{4} (1-2 x) \sqrt {-x+x^2}-\frac {1}{8} \int \frac {1}{\sqrt {-x+x^2}} \, dx\\ &=-\frac {1}{4} (1-2 x) \sqrt {-x+x^2}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-x+x^2}}\right )\\ &=-\frac {1}{4} (1-2 x) \sqrt {-x+x^2}-\frac {1}{4} \tanh ^{-1}\left (\frac {x}{\sqrt {-x+x^2}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.08, size = 50, normalized size = 1.28 \begin {gather*} \frac {1}{4} \sqrt {(-1+x) x} \left (-1+2 x-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {-1+x}}{-1+\sqrt {x}}\right )}{\sqrt {-1+x} \sqrt {x}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.41, size = 33, normalized size = 0.85
method | result | size |
default | \(\frac {\left (2 x -1\right ) \sqrt {x^{2}-x}}{4}-\frac {\ln \left (x -\frac {1}{2}+\sqrt {x^{2}-x}\right )}{8}\) | \(33\) |
risch | \(\frac {\left (2 x -1\right ) x \left (x -1\right )}{4 \sqrt {x \left (x -1\right )}}-\frac {\ln \left (x -\frac {1}{2}+\sqrt {x^{2}-x}\right )}{8}\) | \(35\) |
trager | \(\left (\frac {x}{2}-\frac {1}{4}\right ) \sqrt {x^{2}-x}-\frac {\ln \left (2 x -1+2 \sqrt {x^{2}-x}\right )}{8}\) | \(36\) |
meijerg | \(-\frac {i \sqrt {\mathrm {signum}\left (x -1\right )}\, \left (-\frac {i \sqrt {\pi }\, \sqrt {x}\, \left (-6 x +3\right ) \sqrt {1-x}}{6}+\frac {i \sqrt {\pi }\, \arcsin \left (\sqrt {x}\right )}{2}\right )}{2 \sqrt {\pi }\, \sqrt {-\mathrm {signum}\left (x -1\right )}}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 43, normalized size = 1.10 \begin {gather*} \frac {1}{2} \, \sqrt {x^{2} - x} x - \frac {1}{4} \, \sqrt {x^{2} - x} - \frac {1}{8} \, \log \left (2 \, x + 2 \, \sqrt {x^{2} - x} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.60, size = 36, normalized size = 0.92 \begin {gather*} \frac {1}{4} \, \sqrt {x^{2} - x} {\left (2 \, x - 1\right )} + \frac {1}{8} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} - x} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x^{2} - x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 2.07, size = 37, normalized size = 0.95 \begin {gather*} \frac {1}{4} \, \sqrt {x^{2} - x} {\left (2 \, x - 1\right )} + \frac {1}{8} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} - x} + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.20, size = 29, normalized size = 0.74 \begin {gather*} \sqrt {x^2-x}\,\left (\frac {x}{2}-\frac {1}{4}\right )-\frac {\ln \left (x+\sqrt {x\,\left (x-1\right )}-\frac {1}{2}\right )}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________