Optimal. Leaf size=101 \[ -\frac {2}{-\sqrt {x^2-2 x-3}-x+1}+\frac {4}{\sqrt {x^2-2 x-3}+x}+\frac {3}{4 \left (\sqrt {x^2-2 x-3}+x\right )^2}+6 \log \left (-\sqrt {x^2-2 x-3}-x+1\right )-6 \log \left (\sqrt {x^2-2 x-3}+x\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 101, 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}+\frac {4}{\sqrt {x^2-2 x-3}+x}+\frac {3}{4 \left (\sqrt {x^2-2 x-3}+x\right )^2}+6 \log \left (-\sqrt {x^2-2 x-3}-x+1\right )-6 \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}{\left (x+\sqrt {-3-2 x+x^2}\right )^3} \, dx &=2 \operatorname {Subst}\left (\int \frac {-3-2 x+x^2}{x^3 (-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 {3}{-1+x}-\frac {3}{4 x^3}-\frac {2}{x^2}-\frac {3}{x}\right ) \, dx,x,x+\sqrt {-3-2 x+x^2}\right )\\ &=-\frac {2}{1-x-\sqrt {-3-2 x+x^2}}+\frac {3}{4 \left (x+\sqrt {-3-2 x+x^2}\right )^2}+\frac {4}{x+\sqrt {-3-2 x+x^2}}+6 \log \left (1-x-\sqrt {-3-2 x+x^2}\right )-6 \log \left (x+\sqrt {-3-2 x+x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 97, normalized size = 0.96 \begin {gather*} \frac {2}{\sqrt {x^2-2 x-3}+x-1}+\frac {4}{\sqrt {x^2-2 x-3}+x}+\frac {3}{4 \left (\sqrt {x^2-2 x-3}+x\right )^2}+6 \log \left (-\sqrt {x^2-2 x-3}-x+1\right )-6 \log \left (\sqrt {x^2-2 x-3}+x\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.35, size = 86, normalized size = 0.85 \begin {gather*} \frac {\sqrt {x^2-2 x-3} \left (-4 x^2-31 x-33\right )}{2 (2 x+3)^2}-12 \tanh ^{-1}\left (\frac {x+1}{\sqrt {x^2-2 x-3}+2 x+2}\right )+\frac {16 x^3+48 x^2-108 x-189}{8 (2 x+3)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 129, normalized size = 1.28 \begin {gather*} \frac {8 \, x^{3} - 10 \, x^{2} - 12 \, {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (x^{2} - \sqrt {x^{2} - 2 \, x - 3} {\left (x + 1\right )} - 3\right ) - 12 \, {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (2 \, x + 3\right ) + 12 \, {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (-x + \sqrt {x^{2} - 2 \, x - 3}\right ) - 2 \, {\left (4 \, x^{2} + 31 \, x + 33\right )} \sqrt {x^{2} - 2 \, x - 3} - 156 \, x - 171}{4 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.42, size = 184, normalized size = 1.82 \begin {gather*} \frac {1}{2} \, x - \frac {1}{2} \, \sqrt {x^{2} - 2 \, x - 3} - \frac {104 \, {\left (x - \sqrt {x^{2} - 2 \, x - 3}\right )}^{3} + 315 \, {\left (x - \sqrt {x^{2} - 2 \, x - 3}\right )}^{2} + 162 \, x - 162 \, \sqrt {x^{2} - 2 \, x - 3} + 27}{8 \, {\left ({\left (x - \sqrt {x^{2} - 2 \, x - 3}\right )}^{2} + 3 \, x - 3 \, \sqrt {x^{2} - 2 \, x - 3}\right )}^{2}} - \frac {9 \, {\left (16 \, x + 21\right )}}{8 \, {\left (2 \, x + 3\right )}^{2}} - 3 \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - 3 \, \log \left ({\left | -x + \sqrt {x^{2} - 2 \, x - 3} + 1 \right |}\right ) + 3 \, \log \left ({\left | -x + \sqrt {x^{2} - 2 \, x - 3} \right |}\right ) - 3 \, \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 = 0.03, size = 146, normalized size = 1.45 \begin {gather*} \frac {x}{2}+3 \arctanh \left (\frac {-\frac {10 x}{3}-2}{\sqrt {-20 x +4 \left (x +\frac {3}{2}\right )^{2}-21}}\right )-3 \ln \left (2 x +3\right )+3 \ln \left (x -1+\sqrt {-5 x +\left (x +\frac {3}{2}\right )^{2}-\frac {21}{4}}\right )-\frac {9}{2 x +3}+\frac {27}{8 \left (2 x +3\right )^{2}}-\frac {\left (-5 x +\left (x +\frac {3}{2}\right )^{2}-\frac {21}{4}\right )^{\frac {3}{2}}}{2 \left (x +\frac {3}{2}\right )}-\sqrt {-20 x +4 \left (x +\frac {3}{2}\right )^{2}-21}+\frac {\left (2 x -2\right ) \sqrt {-5 x +\left (x +\frac {3}{2}\right )^{2}-\frac {21}{4}}}{4}+\frac {\left (-5 x +\left (x +\frac {3}{2}\right )^{2}-\frac {21}{4}\right )^{\frac {3}{2}}}{4 \left (x +\frac {3}{2}\right )^{2}} \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}{{\left (x + \sqrt {x^{2} - 2 \, x - 3}\right )}^{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 = -0.01 \begin {gather*} \int \frac {1}{{\left (x+\sqrt {x^2-2\,x-3}\right )}^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}{\left (x + \sqrt {x^{2} - 2 x - 3}\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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