Optimal. Leaf size=75 \[ -\frac {1}{2} \tan ^{-1}\left (\frac {3-x}{2 \sqrt {x^2-x-1}}\right )+\tanh ^{-1}\left (\frac {1-2 x}{2 \sqrt {x^2-x-1}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {3 x+1}{2 \sqrt {x^2-x-1}}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {990, 621, 206, 1033, 724, 204} \begin {gather*} -\frac {1}{2} \tan ^{-1}\left (\frac {3-x}{2 \sqrt {x^2-x-1}}\right )+\tanh ^{-1}\left (\frac {1-2 x}{2 \sqrt {x^2-x-1}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {3 x+1}{2 \sqrt {x^2-x-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 206
Rule 621
Rule 724
Rule 990
Rule 1033
Rubi steps
\begin {align*} \int \frac {\sqrt {-1-x+x^2}}{1-x^2} \, dx &=-\int \frac {1}{\sqrt {-1-x+x^2}} \, dx-\int \frac {x}{\left (1-x^2\right ) \sqrt {-1-x+x^2}} \, dx\\ &=-\left (\frac {1}{2} \int \frac {1}{(-1-x) \sqrt {-1-x+x^2}} \, dx\right )-\frac {1}{2} \int \frac {1}{(1-x) \sqrt {-1-x+x^2}} \, dx-2 \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-1+2 x}{\sqrt {-1-x+x^2}}\right )\\ &=\tanh ^{-1}\left (\frac {1-2 x}{2 \sqrt {-1-x+x^2}}\right )+\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+3 x}{\sqrt {-1-x+x^2}}\right )\\ &=-\frac {1}{2} \tan ^{-1}\left (\frac {3-x}{2 \sqrt {-1-x+x^2}}\right )+\tanh ^{-1}\left (\frac {1-2 x}{2 \sqrt {-1-x+x^2}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {1+3 x}{2 \sqrt {-1-x+x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 75, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \tan ^{-1}\left (\frac {3-x}{2 \sqrt {x^2-x-1}}\right )+\tanh ^{-1}\left (\frac {1-2 x}{2 \sqrt {x^2-x-1}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {3 x+1}{2 \sqrt {x^2-x-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 57, normalized size = 0.76 \begin {gather*} \log \left (2 \sqrt {x^2-x-1}-2 x+1\right )+\tan ^{-1}\left (\sqrt {x^2-x-1}-x+1\right )+\tanh ^{-1}\left (-\sqrt {x^2-x-1}+x+1\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 70, normalized size = 0.93 \begin {gather*} \arctan \left (-x + \sqrt {x^{2} - x - 1} + 1\right ) - \frac {1}{2} \, \log \left (-x + \sqrt {x^{2} - x - 1}\right ) + \frac {1}{2} \, \log \left (-x + \sqrt {x^{2} - x - 1} - 2\right ) + \log \left (-2 \, x + 2 \, \sqrt {x^{2} - x - 1} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 73, normalized size = 0.97 \begin {gather*} \arctan \left (-x + \sqrt {x^{2} - x - 1} + 1\right ) - \frac {1}{2} \, \log \left ({\left | -x + \sqrt {x^{2} - x - 1} \right |}\right ) + \frac {1}{2} \, \log \left ({\left | -x + \sqrt {x^{2} - x - 1} - 2 \right |}\right ) + \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} - x - 1} + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 102, normalized size = 1.36 \begin {gather*} -\frac {\arctanh \left (\frac {-3 x -1}{2 \sqrt {-3 x +\left (x +1\right )^{2}-2}}\right )}{2}+\frac {\arctan \left (\frac {x -3}{2 \sqrt {x +\left (x -1\right )^{2}-2}}\right )}{2}-\frac {3 \ln \left (x -\frac {1}{2}+\sqrt {-3 x +\left (x +1\right )^{2}-2}\right )}{4}-\frac {\ln \left (x -\frac {1}{2}+\sqrt {x +\left (x -1\right )^{2}-2}\right )}{4}-\frac {\sqrt {x +\left (x -1\right )^{2}-2}}{2}+\frac {\sqrt {-3 x +\left (x +1\right )^{2}-2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 83, normalized size = 1.11 \begin {gather*} \frac {1}{2} \, \arcsin \left (\frac {2 \, \sqrt {5} x}{5 \, {\left | 2 \, x - 2 \right |}} - \frac {6 \, \sqrt {5}}{5 \, {\left | 2 \, x - 2 \right |}}\right ) - \log \left (x + \sqrt {x^{2} - x - 1} - \frac {1}{2}\right ) - \frac {1}{2} \, \log \left (\frac {2 \, \sqrt {x^{2} - x - 1}}{{\left | 2 \, x + 2 \right |}} + \frac {2}{{\left | 2 \, x + 2 \right |}} - \frac {3}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {\sqrt {x^2-x-1}}{x^2-1} \,d x \end {gather*}
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 \begin {gather*} - \int \frac {\sqrt {x^{2} - x - 1}}{x^{2} - 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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