Integrand size = 10, antiderivative size = 51 \[ \int \frac {1}{\left (13-4 x+x^2\right )^3} \, dx=-\frac {2-x}{36 \left (13-4 x+x^2\right )^2}-\frac {2-x}{216 \left (13-4 x+x^2\right )}+\frac {1}{648} \arctan \left (\frac {1}{3} (-2+x)\right ) \]
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Time = 0.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {628, 632, 210} \[ \int \frac {1}{\left (13-4 x+x^2\right )^3} \, dx=\frac {1}{648} \arctan \left (\frac {x-2}{3}\right )-\frac {2-x}{216 \left (x^2-4 x+13\right )}-\frac {2-x}{36 \left (x^2-4 x+13\right )^2} \]
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Rule 210
Rule 628
Rule 632
Rubi steps \begin{align*} \text {integral}& = -\frac {2-x}{36 \left (13-4 x+x^2\right )^2}+\frac {1}{12} \int \frac {1}{\left (13-4 x+x^2\right )^2} \, dx \\ & = -\frac {2-x}{36 \left (13-4 x+x^2\right )^2}-\frac {2-x}{216 \left (13-4 x+x^2\right )}+\frac {1}{216} \int \frac {1}{13-4 x+x^2} \, dx \\ & = -\frac {2-x}{36 \left (13-4 x+x^2\right )^2}-\frac {2-x}{216 \left (13-4 x+x^2\right )}-\frac {1}{108} \text {Subst}\left (\int \frac {1}{-36-x^2} \, dx,x,-4+2 x\right ) \\ & = -\frac {2-x}{36 \left (13-4 x+x^2\right )^2}-\frac {2-x}{216 \left (13-4 x+x^2\right )}+\frac {1}{648} \arctan \left (\frac {1}{3} (-2+x)\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\left (13-4 x+x^2\right )^3} \, dx=\frac {1}{648} \left (\frac {3 (-2+x) \left (19-4 x+x^2\right )}{\left (13-4 x+x^2\right )^2}+\arctan \left (\frac {1}{3} (-2+x)\right )\right ) \]
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Time = 0.55 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.71
method | result | size |
risch | \(\frac {\frac {1}{216} x^{3}-\frac {1}{36} x^{2}+\frac {1}{8} x -\frac {19}{108}}{\left (x^{2}-4 x +13\right )^{2}}+\frac {\arctan \left (-\frac {2}{3}+\frac {x}{3}\right )}{648}\) | \(36\) |
default | \(\frac {-4+2 x}{72 \left (x^{2}-4 x +13\right )^{2}}+\frac {-4+2 x}{432 x^{2}-1728 x +5616}+\frac {\arctan \left (-\frac {2}{3}+\frac {x}{3}\right )}{648}\) | \(44\) |
parallelrisch | \(-\frac {7098 i \ln \left (x -2-3 i\right ) x^{2}-28561 i \ln \left (x -2+3 i\right )+17576 i \ln \left (x -2+3 i\right ) x +169 i \ln \left (x -2-3 i\right ) x^{4}+1352 i \ln \left (x -2+3 i\right ) x^{3}-1352 i \ln \left (x -2-3 i\right ) x^{3}-228 x^{4}-7098 i \ln \left (x -2+3 i\right ) x^{2}-17576 i \ln \left (x -2-3 i\right ) x +810 x^{3}-169 i \ln \left (x -2+3 i\right ) x^{4}+28561 i \ln \left (x -2-3 i\right )-3492 x^{2}-3666 x}{219024 \left (x^{2}-4 x +13\right )^{2}}\) | \(142\) |
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Time = 0.24 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\left (13-4 x+x^2\right )^3} \, dx=\frac {3 \, x^{3} - 18 \, x^{2} + {\left (x^{4} - 8 \, x^{3} + 42 \, x^{2} - 104 \, x + 169\right )} \arctan \left (\frac {1}{3} \, x - \frac {2}{3}\right ) + 81 \, x - 114}{648 \, {\left (x^{4} - 8 \, x^{3} + 42 \, x^{2} - 104 \, x + 169\right )}} \]
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Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (13-4 x+x^2\right )^3} \, dx=\frac {x^{3} - 6 x^{2} + 27 x - 38}{216 x^{4} - 1728 x^{3} + 9072 x^{2} - 22464 x + 36504} + \frac {\operatorname {atan}{\left (\frac {x}{3} - \frac {2}{3} \right )}}{648} \]
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Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (13-4 x+x^2\right )^3} \, dx=\frac {x^{3} - 6 \, x^{2} + 27 \, x - 38}{216 \, {\left (x^{4} - 8 \, x^{3} + 42 \, x^{2} - 104 \, x + 169\right )}} + \frac {1}{648} \, \arctan \left (\frac {1}{3} \, x - \frac {2}{3}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (13-4 x+x^2\right )^3} \, dx=\frac {x^{3} - 6 \, x^{2} + 27 \, x - 38}{216 \, {\left (x^{2} - 4 \, x + 13\right )}^{2}} + \frac {1}{648} \, \arctan \left (\frac {1}{3} \, x - \frac {2}{3}\right ) \]
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Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (13-4 x+x^2\right )^3} \, dx=\frac {\mathrm {atan}\left (\frac {x}{3}-\frac {2}{3}\right )}{648}+6\,\left (x-2\right )\,\left (\frac {1}{1296\,\left (x^2-4\,x+13\right )}+\frac {1}{216\,{\left (x^2-4\,x+13\right )}^2}\right ) \]
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