Optimal. Leaf size=65 \[ -\sqrt {x^2-x-1}-\tan ^{-1}\left (\frac {3-x}{2 \sqrt {x^2-x-1}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {1-2 x}{2 \sqrt {x^2-x-1}}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {734, 843, 621, 206, 724, 204} \[ -\sqrt {x^2-x-1}-\tan ^{-1}\left (\frac {3-x}{2 \sqrt {x^2-x-1}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {1-2 x}{2 \sqrt {x^2-x-1}}\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 206
Rule 621
Rule 724
Rule 734
Rule 843
Rubi steps
\begin {align*} \int \frac {\sqrt {-1-x+x^2}}{1-x} \, dx &=-\sqrt {-1-x+x^2}+\frac {1}{2} \int \frac {-3+x}{(1-x) \sqrt {-1-x+x^2}} \, dx\\ &=-\sqrt {-1-x+x^2}-\frac {1}{2} \int \frac {1}{\sqrt {-1-x+x^2}} \, dx-\int \frac {1}{(1-x) \sqrt {-1-x+x^2}} \, dx\\ &=-\sqrt {-1-x+x^2}+2 \operatorname {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,\frac {3-x}{\sqrt {-1-x+x^2}}\right )-\operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-1+2 x}{\sqrt {-1-x+x^2}}\right )\\ &=-\sqrt {-1-x+x^2}-\tan ^{-1}\left (\frac {3-x}{2 \sqrt {-1-x+x^2}}\right )-\frac {1}{2} \tanh ^{-1}\left (\frac {-1+2 x}{2 \sqrt {-1-x+x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 65, normalized size = 1.00 \[ -\sqrt {x^2-x-1}-\tan ^{-1}\left (\frac {3-x}{2 \sqrt {x^2-x-1}}\right )-\frac {1}{2} \tanh ^{-1}\left (\frac {2 x-1}{2 \sqrt {x^2-x-1}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 51, normalized size = 0.78 \[ -\sqrt {x^{2} - x - 1} + 2 \, \arctan \left (-x + \sqrt {x^{2} - x - 1} + 1\right ) + \frac {1}{2} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} - x - 1} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 52, normalized size = 0.80 \[ -\sqrt {x^{2} - x - 1} + 2 \, \arctan \left (-x + \sqrt {x^{2} - x - 1} + 1\right ) + \frac {1}{2} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} - x - 1} + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 46, normalized size = 0.71 \[ \arctan \left (\frac {x -3}{2 \sqrt {x +\left (x -1\right )^{2}-2}}\right )-\frac {\ln \left (x -\frac {1}{2}+\sqrt {x +\left (x -1\right )^{2}-2}\right )}{2}-\sqrt {x +\left (x -1\right )^{2}-2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.00, size = 58, normalized size = 0.89 \[ -\sqrt {x^{2} - x - 1} + \arcsin \left (\frac {\sqrt {5} x}{5 \, {\left | x - 1 \right |}} - \frac {3 \, \sqrt {5}}{5 \, {\left | x - 1 \right |}}\right ) - \frac {1}{2} \, \log \left (2 \, x + 2 \, \sqrt {x^{2} - x - 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ -\int \frac {\sqrt {x^2-x-1}}{x-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\sqrt {x^{2} - x - 1}}{x - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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