Optimal. Leaf size=38 \[ -\frac{1}{4} \sqrt{-x^2+x+1} (1-2 x)-\frac{5}{8} \sin ^{-1}\left (\frac{1-2 x}{\sqrt{5}}\right ) \]
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Rubi [A] time = 0.0117996, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {612, 619, 216} \[ -\frac{1}{4} \sqrt{-x^2+x+1} (1-2 x)-\frac{5}{8} \sin ^{-1}\left (\frac{1-2 x}{\sqrt{5}}\right ) \]
Antiderivative was successfully verified.
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Rule 612
Rule 619
Rule 216
Rubi steps
\begin{align*} \int \sqrt{1+x-x^2} \, dx &=-\frac{1}{4} (1-2 x) \sqrt{1+x-x^2}+\frac{5}{8} \int \frac{1}{\sqrt{1+x-x^2}} \, dx\\ &=-\frac{1}{4} (1-2 x) \sqrt{1+x-x^2}-\frac{1}{8} \sqrt{5} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{5}}} \, dx,x,1-2 x\right )\\ &=-\frac{1}{4} (1-2 x) \sqrt{1+x-x^2}-\frac{5}{8} \sin ^{-1}\left (\frac{1-2 x}{\sqrt{5}}\right )\\ \end{align*}
Mathematica [A] time = 0.0146953, size = 39, normalized size = 1.03 \[ \left (\frac{x}{2}-\frac{1}{4}\right ) \sqrt{-x^2+x+1}-\frac{5}{8} \sin ^{-1}\left (\frac{1-2 x}{\sqrt{5}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 30, normalized size = 0.8 \begin{align*} -{\frac{1-2\,x}{4}\sqrt{-{x}^{2}+x+1}}+{\frac{5}{8}\arcsin \left ({\frac{2\,\sqrt{5}}{5} \left ( x-{\frac{1}{2}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41514, size = 53, normalized size = 1.39 \begin{align*} \frac{1}{2} \, \sqrt{-x^{2} + x + 1} x - \frac{1}{4} \, \sqrt{-x^{2} + x + 1} - \frac{5}{8} \, \arcsin \left (-\frac{1}{5} \, \sqrt{5}{\left (2 \, x - 1\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12894, size = 101, normalized size = 2.66 \begin{align*} \frac{1}{4} \, \sqrt{-x^{2} + x + 1}{\left (2 \, x - 1\right )} - \frac{5}{4} \, \arctan \left (\frac{\sqrt{-x^{2} + x + 1} - 1}{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{- x^{2} + x + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05773, size = 42, normalized size = 1.11 \begin{align*} \frac{1}{4} \, \sqrt{-x^{2} + x + 1}{\left (2 \, x - 1\right )} + \frac{5}{8} \, \arcsin \left (\frac{1}{5} \, \sqrt{5}{\left (2 \, x - 1\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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