Integrand size = 31, antiderivative size = 71 \[ \int \frac {(d+e x) \left (2-x-2 x^2+x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {d-2 e}{12 (2+x)}-\frac {1}{18} (d+e) \log (1-x)+\frac {1}{48} (d+2 e) \log (2-x)+\frac {1}{6} (d-e) \log (1+x)-\frac {1}{144} (19 d-26 e) \log (2+x) \]
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Time = 0.12 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1600, 6874} \[ \int \frac {(d+e x) \left (2-x-2 x^2+x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {d-2 e}{12 (x+2)}-\frac {1}{18} (d+e) \log (1-x)+\frac {1}{48} (d+2 e) \log (2-x)+\frac {1}{6} (d-e) \log (x+1)-\frac {1}{144} (19 d-26 e) \log (x+2) \]
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Rule 1600
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {d+e x}{(2+x)^2 \left (2-x-2 x^2+x^3\right )} \, dx \\ & = \int \left (\frac {d+2 e}{48 (-2+x)}+\frac {-d-e}{18 (-1+x)}+\frac {d-e}{6 (1+x)}+\frac {-d+2 e}{12 (2+x)^2}+\frac {-19 d+26 e}{144 (2+x)}\right ) \, dx \\ & = \frac {d-2 e}{12 (2+x)}-\frac {1}{18} (d+e) \log (1-x)+\frac {1}{48} (d+2 e) \log (2-x)+\frac {1}{6} (d-e) \log (1+x)-\frac {1}{144} (19 d-26 e) \log (2+x) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.93 \[ \int \frac {(d+e x) \left (2-x-2 x^2+x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {1}{144} \left (\frac {12 (d-2 e)}{2+x}+24 (d-e) \log (-1-x)-8 (d+e) \log (1-x)+3 (d+2 e) \log (2-x)+(-19 d+26 e) \log (2+x)\right ) \]
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Time = 0.07 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.90
method | result | size |
default | \(\left (-\frac {19 d}{144}+\frac {13 e}{72}\right ) \ln \left (x +2\right )-\frac {-\frac {d}{12}+\frac {e}{6}}{x +2}+\left (\frac {d}{6}-\frac {e}{6}\right ) \ln \left (x +1\right )+\left (-\frac {d}{18}-\frac {e}{18}\right ) \ln \left (x -1\right )+\left (\frac {d}{48}+\frac {e}{24}\right ) \ln \left (x -2\right )\) | \(64\) |
risch | \(\frac {d}{12 x +24}-\frac {e}{6 \left (x +2\right )}-\frac {\ln \left (x -1\right ) d}{18}-\frac {\ln \left (x -1\right ) e}{18}+\frac {\ln \left (x +1\right ) d}{6}-\frac {\ln \left (x +1\right ) e}{6}-\frac {19 \ln \left (-x -2\right ) d}{144}+\frac {13 \ln \left (-x -2\right ) e}{72}+\frac {\ln \left (2-x \right ) d}{48}+\frac {\ln \left (2-x \right ) e}{24}\) | \(82\) |
norman | \(\frac {\left (-\frac {d}{12}+\frac {e}{6}\right ) x +\left (\frac {d}{12}-\frac {e}{6}\right ) x^{3}+\left (-\frac {d}{6}+\frac {e}{3}\right ) x^{2}+\frac {d}{6}-\frac {e}{3}}{x^{4}-5 x^{2}+4}+\left (-\frac {19 d}{144}+\frac {13 e}{72}\right ) \ln \left (x +2\right )+\left (-\frac {d}{18}-\frac {e}{18}\right ) \ln \left (x -1\right )+\left (\frac {d}{6}-\frac {e}{6}\right ) \ln \left (x +1\right )+\left (\frac {d}{48}+\frac {e}{24}\right ) \ln \left (x -2\right )\) | \(101\) |
parallelrisch | \(\frac {3 \ln \left (x -2\right ) x d +6 \ln \left (x -2\right ) x e -8 \ln \left (x -1\right ) x d -8 \ln \left (x -1\right ) x e +24 \ln \left (x +1\right ) x d -24 \ln \left (x +1\right ) x e -19 \ln \left (x +2\right ) x d +26 \ln \left (x +2\right ) x e +6 \ln \left (x -2\right ) d +12 \ln \left (x -2\right ) e -16 \ln \left (x -1\right ) d -16 \ln \left (x -1\right ) e +48 \ln \left (x +1\right ) d -48 \ln \left (x +1\right ) e -38 \ln \left (x +2\right ) d +52 \ln \left (x +2\right ) e +12 d -24 e}{144 x +288}\) | \(135\) |
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Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.31 \[ \int \frac {(d+e x) \left (2-x-2 x^2+x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=-\frac {{\left ({\left (19 \, d - 26 \, e\right )} x + 38 \, d - 52 \, e\right )} \log \left (x + 2\right ) - 24 \, {\left ({\left (d - e\right )} x + 2 \, d - 2 \, e\right )} \log \left (x + 1\right ) + 8 \, {\left ({\left (d + e\right )} x + 2 \, d + 2 \, e\right )} \log \left (x - 1\right ) - 3 \, {\left ({\left (d + 2 \, e\right )} x + 2 \, d + 4 \, e\right )} \log \left (x - 2\right ) - 12 \, d + 24 \, e}{144 \, {\left (x + 2\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1188 vs. \(2 (60) = 120\).
Time = 7.10 (sec) , antiderivative size = 1188, normalized size of antiderivative = 16.73 \[ \int \frac {(d+e x) \left (2-x-2 x^2+x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=\text {Too large to display} \]
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Time = 0.18 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.80 \[ \int \frac {(d+e x) \left (2-x-2 x^2+x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=-\frac {1}{144} \, {\left (19 \, d - 26 \, e\right )} \log \left (x + 2\right ) + \frac {1}{6} \, {\left (d - e\right )} \log \left (x + 1\right ) - \frac {1}{18} \, {\left (d + e\right )} \log \left (x - 1\right ) + \frac {1}{48} \, {\left (d + 2 \, e\right )} \log \left (x - 2\right ) + \frac {d - 2 \, e}{12 \, {\left (x + 2\right )}} \]
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Time = 0.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.86 \[ \int \frac {(d+e x) \left (2-x-2 x^2+x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=-\frac {1}{144} \, {\left (19 \, d - 26 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{6} \, {\left (d - e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{18} \, {\left (d + e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{48} \, {\left (d + 2 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) + \frac {d - 2 \, e}{12 \, {\left (x + 2\right )}} \]
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Time = 7.95 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.90 \[ \int \frac {(d+e x) \left (2-x-2 x^2+x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {\frac {d}{12}-\frac {e}{6}}{x+2}+\ln \left (x+1\right )\,\left (\frac {d}{6}-\frac {e}{6}\right )-\ln \left (x-1\right )\,\left (\frac {d}{18}+\frac {e}{18}\right )+\ln \left (x-2\right )\,\left (\frac {d}{48}+\frac {e}{24}\right )-\ln \left (x+2\right )\,\left (\frac {19\,d}{144}-\frac {13\,e}{72}\right ) \]
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