Optimal. Leaf size=83 \[ -\frac {2}{-\sqrt {x^2-2 x-3}-x+1}+\frac {3}{2 \left (\sqrt {x^2-2 x-3}+x\right )}+4 \log \left (-\sqrt {x^2-2 x-3}-x+1\right )-4 \log \left (\sqrt {x^2-2 x-3}+x\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 83, 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} \[ -\frac {2}{-\sqrt {x^2-2 x-3}-x+1}+\frac {3}{2 \left (\sqrt {x^2-2 x-3}+x\right )}+4 \log \left (-\sqrt {x^2-2 x-3}-x+1\right )-4 \log \left (\sqrt {x^2-2 x-3}+x\right ) \]
Antiderivative was successfully verified.
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Rule 893
Rule 2116
Rubi steps
\begin {align*} \int \frac {1}{\left (x+\sqrt {-3-2 x+x^2}\right )^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {-3-2 x+x^2}{x^2 (-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 {2}{-1+x}-\frac {3}{4 x^2}-\frac {2}{x}\right ) \, dx,x,x+\sqrt {-3-2 x+x^2}\right )\\ &=-\frac {2}{1-x-\sqrt {-3-2 x+x^2}}+\frac {3}{2 \left (x+\sqrt {-3-2 x+x^2}\right )}+4 \log \left (1-x-\sqrt {-3-2 x+x^2}\right )-4 \log \left (x+\sqrt {-3-2 x+x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 79, normalized size = 0.95 \[ \frac {2}{\sqrt {x^2-2 x-3}+x-1}+\frac {3}{2 \left (\sqrt {x^2-2 x-3}+x\right )}+4 \log \left (-\sqrt {x^2-2 x-3}-x+1\right )-4 \log \left (\sqrt {x^2-2 x-3}+x\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 97, normalized size = 1.17 \[ \frac {4 \, x^{2} - 8 \, {\left (2 \, x + 3\right )} \log \left (x^{2} - \sqrt {x^{2} - 2 \, x - 3} {\left (x + 1\right )} - 3\right ) - 8 \, {\left (2 \, x + 3\right )} \log \left (2 \, x + 3\right ) + 8 \, {\left (2 \, x + 3\right )} \log \left (-x + \sqrt {x^{2} - 2 \, x - 3}\right ) - 4 \, \sqrt {x^{2} - 2 \, x - 3} {\left (x + 3\right )} + 2 \, x - 15}{4 \, {\left (2 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.51, size = 143, normalized size = 1.72 \[ \frac {1}{2} \, x - \frac {1}{2} \, \sqrt {x^{2} - 2 \, x - 3} - \frac {3 \, {\left (5 \, x - 5 \, \sqrt {x^{2} - 2 \, x - 3} + 3\right )}}{4 \, {\left ({\left (x - \sqrt {x^{2} - 2 \, x - 3}\right )}^{2} + 3 \, x - 3 \, \sqrt {x^{2} - 2 \, x - 3}\right )}} - \frac {9}{4 \, {\left (2 \, x + 3\right )}} - 2 \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - 2 \, \log \left ({\left | -x + \sqrt {x^{2} - 2 \, x - 3} + 1 \right |}\right ) + 2 \, \log \left ({\left | -x + \sqrt {x^{2} - 2 \, x - 3} \right |}\right ) - 2 \, \log \left ({\left | -x + \sqrt {x^{2} - 2 \, x - 3} - 3 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 118, normalized size = 1.42 \[ \frac {x}{2}+2 \arctanh \left (\frac {-\frac {10 x}{3}-2}{\sqrt {-20 x +4 \left (x +\frac {3}{2}\right )^{2}-21}}\right )-2 \ln \left (2 x +3\right )+2 \ln \left (x -1+\sqrt {-5 x +\left (x +\frac {3}{2}\right )^{2}-\frac {21}{4}}\right )-\frac {9}{4 \left (2 x +3\right )}-\frac {2 \sqrt {-20 x +4 \left (x +\frac {3}{2}\right )^{2}-21}}{3}-\frac {\left (-5 x +\left (x +\frac {3}{2}\right )^{2}-\frac {21}{4}\right )^{\frac {3}{2}}}{3 \left (x +\frac {3}{2}\right )}+\frac {\left (2 x -2\right ) \sqrt {-5 x +\left (x +\frac {3}{2}\right )^{2}-\frac {21}{4}}}{6} \]
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 \[ \int \frac {1}{{\left (x + \sqrt {x^{2} - 2 \, x - 3}\right )}^{2}}\,{d x} \]
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 \[ \int \frac {1}{{\left (x+\sqrt {x^2-2\,x-3}\right )}^2} \,d x \]
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 \[ \int \frac {1}{\left (x + \sqrt {x^{2} - 2 x - 3}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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