Integrand size = 16, antiderivative size = 101 \[ \int \frac {1}{\left (x+\sqrt {-3-2 x+x^2}\right )^3} \, dx=-\frac {2}{1-x-\sqrt {-3-2 x+x^2}}+\frac {3}{4 \left (x+\sqrt {-3-2 x+x^2}\right )^2}+\frac {4}{x+\sqrt {-3-2 x+x^2}}+6 \log \left (1-x-\sqrt {-3-2 x+x^2}\right )-6 \log \left (x+\sqrt {-3-2 x+x^2}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2141, 907} \[ \int \frac {1}{\left (x+\sqrt {-3-2 x+x^2}\right )^3} \, dx=-\frac {2}{-\sqrt {x^2-2 x-3}-x+1}+\frac {4}{\sqrt {x^2-2 x-3}+x}+\frac {3}{4 \left (\sqrt {x^2-2 x-3}+x\right )^2}+6 \log \left (-\sqrt {x^2-2 x-3}-x+1\right )-6 \log \left (\sqrt {x^2-2 x-3}+x\right ) \]
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Rule 907
Rule 2141
Rubi steps \begin{align*} \text {integral}& = 2 \text {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 \text {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*}
Time = 0.34 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (x+\sqrt {-3-2 x+x^2}\right )^3} \, dx=-\frac {189+108 x-48 x^2-16 x^3+4 \sqrt {-3-2 x+x^2} \left (33+31 x+4 x^2\right )+96 (3+2 x)^2 \text {arctanh}\left (\frac {1+x}{2+2 x+\sqrt {-3-2 x+x^2}}\right )}{8 (3+2 x)^2} \]
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Time = 0.13 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.68
method | result | size |
trager | \(\frac {\left (4 x^{2}+33 x +36\right ) x}{2 \left (2 x +3\right )^{2}}-\frac {\left (4 x^{2}+31 x +33\right ) \sqrt {x^{2}-2 x -3}}{2 \left (2 x +3\right )^{2}}-6 \ln \left (x +3-\sqrt {x^{2}-2 x -3}\right )\) | \(69\) |
default | \(-\frac {9}{2 x +3}-3 \ln \left (2 x +3\right )+\frac {x}{2}+\frac {27}{8 \left (2 x +3\right )^{2}}-\frac {\left (\left (x +\frac {3}{2}\right )^{2}-5 x -\frac {21}{4}\right )^{\frac {3}{2}}}{2 \left (x +\frac {3}{2}\right )}-\sqrt {4 \left (x +\frac {3}{2}\right )^{2}-20 x -21}+3 \,\operatorname {arctanh}\left (\frac {-2-\frac {10 x}{3}}{\sqrt {4 \left (x +\frac {3}{2}\right )^{2}-20 x -21}}\right )+\frac {\left (2 x -2\right ) \sqrt {\left (x +\frac {3}{2}\right )^{2}-5 x -\frac {21}{4}}}{4}+3 \ln \left (-1+x +\sqrt {\left (x +\frac {3}{2}\right )^{2}-5 x -\frac {21}{4}}\right )+\frac {\left (\left (x +\frac {3}{2}\right )^{2}-5 x -\frac {21}{4}\right )^{\frac {3}{2}}}{4 \left (x +\frac {3}{2}\right )^{2}}\) | \(146\) |
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none
Time = 0.26 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.28 \[ \int \frac {1}{\left (x+\sqrt {-3-2 x+x^2}\right )^3} \, dx=\frac {8 \, x^{3} - 10 \, x^{2} - 12 \, {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (x^{2} - \sqrt {x^{2} - 2 \, x - 3} {\left (x + 1\right )} - 3\right ) - 12 \, {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (2 \, x + 3\right ) + 12 \, {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (-x + \sqrt {x^{2} - 2 \, x - 3}\right ) - 2 \, {\left (4 \, x^{2} + 31 \, x + 33\right )} \sqrt {x^{2} - 2 \, x - 3} - 156 \, x - 171}{4 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} \]
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\[ \int \frac {1}{\left (x+\sqrt {-3-2 x+x^2}\right )^3} \, dx=\int \frac {1}{\left (x + \sqrt {x^{2} - 2 x - 3}\right )^{3}}\, dx \]
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\[ \int \frac {1}{\left (x+\sqrt {-3-2 x+x^2}\right )^3} \, dx=\int { \frac {1}{{\left (x + \sqrt {x^{2} - 2 \, x - 3}\right )}^{3}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (85) = 170\).
Time = 0.34 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.82 \[ \int \frac {1}{\left (x+\sqrt {-3-2 x+x^2}\right )^3} \, dx=\frac {1}{2} \, x - \frac {1}{2} \, \sqrt {x^{2} - 2 \, x - 3} - \frac {104 \, {\left (x - \sqrt {x^{2} - 2 \, x - 3}\right )}^{3} + 315 \, {\left (x - \sqrt {x^{2} - 2 \, x - 3}\right )}^{2} + 162 \, x - 162 \, \sqrt {x^{2} - 2 \, x - 3} + 27}{8 \, {\left ({\left (x - \sqrt {x^{2} - 2 \, x - 3}\right )}^{2} + 3 \, x - 3 \, \sqrt {x^{2} - 2 \, x - 3}\right )}^{2}} - \frac {9 \, {\left (16 \, x + 21\right )}}{8 \, {\left (2 \, x + 3\right )}^{2}} - 3 \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - 3 \, \log \left ({\left | -x + \sqrt {x^{2} - 2 \, x - 3} + 1 \right |}\right ) + 3 \, \log \left ({\left | -x + \sqrt {x^{2} - 2 \, x - 3} \right |}\right ) - 3 \, \log \left ({\left | -x + \sqrt {x^{2} - 2 \, x - 3} - 3 \right |}\right ) \]
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Timed out. \[ \int \frac {1}{\left (x+\sqrt {-3-2 x+x^2}\right )^3} \, dx=\int \frac {1}{{\left (x+\sqrt {x^2-2\,x-3}\right )}^3} \,d x \]
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