Optimal. Leaf size=65 \[ -\frac {2}{-\sqrt {x^2-2 x-3}-x+1}+2 \log \left (-\sqrt {x^2-2 x-3}-x+1\right )-\frac {3}{2} \log \left (\sqrt {x^2-2 x-3}+x\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2116, 893} \begin {gather*} -\frac {2}{-\sqrt {x^2-2 x-3}-x+1}+2 \log \left (-\sqrt {x^2-2 x-3}-x+1\right )-\frac {3}{2} \log \left (\sqrt {x^2-2 x-3}+x\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 893
Rule 2116
Rubi steps
\begin {align*} \int \frac {1}{x+\sqrt {-3-2 x+x^2}} \, dx &=2 \operatorname {Subst}\left (\int \frac {-3-2 x+x^2}{x (-2+2 x)^2} \, dx,x,x+\sqrt {-3-2 x+x^2}\right )\\ &=2 \operatorname {Subst}\left (\int \left (-\frac {1}{(-1+x)^2}+\frac {1}{-1+x}-\frac {3}{4 x}\right ) \, dx,x,x+\sqrt {-3-2 x+x^2}\right )\\ &=-\frac {2}{1-x-\sqrt {-3-2 x+x^2}}+2 \log \left (1-x-\sqrt {-3-2 x+x^2}\right )-\frac {3}{2} \log \left (x+\sqrt {-3-2 x+x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 59, normalized size = 0.91 \begin {gather*} 2 \left (\frac {1}{\sqrt {x^2-2 x-3}+x-1}+\log \left (-\sqrt {x^2-2 x-3}-x+1\right )-\frac {3}{4} \log \left (\sqrt {x^2-2 x-3}+x\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.31, size = 84, normalized size = 1.29 \begin {gather*} -\frac {1}{2} \sqrt {x^2-2 x-3}-\frac {1}{2} \log \left (\sqrt {x^2-2 x-3}-x-1\right )+2 \log \left (\sqrt {x^2-2 x-3}+x+1\right )-\frac {3}{2} \log \left (\sqrt {x^2-2 x-3}+3 x+3\right )+\frac {x}{2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 77, normalized size = 1.18 \begin {gather*} \frac {1}{2} \, x - \frac {1}{2} \, \sqrt {x^{2} - 2 \, x - 3} - \frac {3}{4} \, \log \left (2 \, x + 3\right ) - \frac {5}{4} \, \log \left (-x + \sqrt {x^{2} - 2 \, x - 3} + 1\right ) + \frac {3}{4} \, \log \left (-x + \sqrt {x^{2} - 2 \, x - 3}\right ) - \frac {3}{4} \, \log \left (-x + \sqrt {x^{2} - 2 \, x - 3} - 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 81, normalized size = 1.25 \begin {gather*} \frac {1}{2} \, x - \frac {1}{2} \, \sqrt {x^{2} - 2 \, x - 3} - \frac {3}{4} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac {5}{4} \, \log \left ({\left | -x + \sqrt {x^{2} - 2 \, x - 3} + 1 \right |}\right ) + \frac {3}{4} \, \log \left ({\left | -x + \sqrt {x^{2} - 2 \, x - 3} \right |}\right ) - \frac {3}{4} \, \log \left ({\left | -x + \sqrt {x^{2} - 2 \, x - 3} - 3 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 71, normalized size = 1.09 \begin {gather*} \frac {x}{2}+\frac {3 \arctanh \left (\frac {-\frac {10 x}{3}-2}{\sqrt {-20 x +4 \left (x +\frac {3}{2}\right )^{2}-21}}\right )}{4}-\frac {3 \ln \left (2 x +3\right )}{4}+\frac {5 \ln \left (x -1+\sqrt {-5 x +\left (x +\frac {3}{2}\right )^{2}-\frac {21}{4}}\right )}{4}-\frac {\sqrt {-20 x +4 \left (x +\frac {3}{2}\right )^{2}-21}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x + \sqrt {x^{2} - 2 \, x - 3}}\,{d x} \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 = -0.02 \begin {gather*} \frac {x}{2}-\frac {3\,\ln \left (x+\frac {3}{2}\right )}{4}-\int \frac {\sqrt {x^2-2\,x-3}}{2\,x+3} \,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 {1}{x + \sqrt {x^{2} - 2 x - 3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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