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 = 0.0300388, antiderivative size = 65, normalized size of antiderivative = 1., 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} \[ -\frac{2}{-\sqrt{x^2-2 x-3}-x+1}+2 \log \left (-\sqrt{x^2-2 x-3}-x+1\right )-\frac{3}{2} \log \left (\sqrt{x^2-2 x-3}+x\right ) \]
Antiderivative was successfully verified.
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Rule 2116
Rule 893
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*}
Mathematica [A] time = 0.0266081, size = 59, normalized size = 0.91 \[ 2 \left (\frac{1}{\sqrt{x^2-2 x-3}+x-1}+\log \left (-\sqrt{x^2-2 x-3}-x+1\right )-\frac{3}{4} \log \left (\sqrt{x^2-2 x-3}+x\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 71, normalized size = 1.1 \begin{align*} -{\frac{1}{4}\sqrt{4\, \left ( x+3/2 \right ) ^{2}-20\,x-21}}+{\frac{5}{4}\ln \left ( -1+x+\sqrt{ \left ( x+{\frac{3}{2}} \right ) ^{2}-5\,x-{\frac{21}{4}}} \right ) }+{\frac{3}{4}{\it Artanh} \left ({\frac{-6-10\,x}{3}{\frac{1}{\sqrt{4\, \left ( x+3/2 \right ) ^{2}-20\,x-21}}}} \right ) }+{\frac{x}{2}}-{\frac{3\,\ln \left ( 3+2\,x \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x + \sqrt{x^{2} - 2 \, x - 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77213, size = 227, normalized size = 3.49 \begin{align*} \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{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 \frac{1}{x + \sqrt{x^{2} - 2 x - 3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17132, size = 109, normalized size = 1.68 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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