Integrand size = 18, antiderivative size = 94 \[ \int \frac {d+e x}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {d x \left (17-5 x^2\right )}{72 \left (4-5 x^2+x^4\right )}+\frac {e \left (5-2 x^2\right )}{18 \left (4-5 x^2+x^4\right )}+\frac {19}{432} d \text {arctanh}\left (\frac {x}{2}\right )-\frac {1}{54} d \text {arctanh}(x)+\frac {1}{27} e \log \left (1-x^2\right )-\frac {1}{27} e \log \left (4-x^2\right ) \]
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Time = 0.04 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1687, 12, 1106, 1180, 213, 1121, 628, 630, 31} \[ \int \frac {d+e x}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {19}{432} d \text {arctanh}\left (\frac {x}{2}\right )-\frac {1}{54} d \text {arctanh}(x)+\frac {d x \left (17-5 x^2\right )}{72 \left (x^4-5 x^2+4\right )}+\frac {1}{27} e \log \left (1-x^2\right )-\frac {1}{27} e \log \left (4-x^2\right )+\frac {e \left (5-2 x^2\right )}{18 \left (x^4-5 x^2+4\right )} \]
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Rule 12
Rule 31
Rule 213
Rule 628
Rule 630
Rule 1106
Rule 1121
Rule 1180
Rule 1687
Rubi steps \begin{align*} \text {integral}& = \int \frac {d}{\left (4-5 x^2+x^4\right )^2} \, dx+\int \frac {e x}{\left (4-5 x^2+x^4\right )^2} \, dx \\ & = d \int \frac {1}{\left (4-5 x^2+x^4\right )^2} \, dx+e \int \frac {x}{\left (4-5 x^2+x^4\right )^2} \, dx \\ & = \frac {d x \left (17-5 x^2\right )}{72 \left (4-5 x^2+x^4\right )}-\frac {1}{72} d \int \frac {-1+5 x^2}{4-5 x^2+x^4} \, dx+\frac {1}{2} e \text {Subst}\left (\int \frac {1}{\left (4-5 x+x^2\right )^2} \, dx,x,x^2\right ) \\ & = \frac {d x \left (17-5 x^2\right )}{72 \left (4-5 x^2+x^4\right )}+\frac {e \left (5-2 x^2\right )}{18 \left (4-5 x^2+x^4\right )}+\frac {1}{54} d \int \frac {1}{-1+x^2} \, dx-\frac {1}{216} (19 d) \int \frac {1}{-4+x^2} \, dx-\frac {1}{9} e \text {Subst}\left (\int \frac {1}{4-5 x+x^2} \, dx,x,x^2\right ) \\ & = \frac {d x \left (17-5 x^2\right )}{72 \left (4-5 x^2+x^4\right )}+\frac {e \left (5-2 x^2\right )}{18 \left (4-5 x^2+x^4\right )}+\frac {19}{432} d \tanh ^{-1}\left (\frac {x}{2}\right )-\frac {1}{54} d \tanh ^{-1}(x)-\frac {1}{27} e \text {Subst}\left (\int \frac {1}{-4+x} \, dx,x,x^2\right )+\frac {1}{27} e \text {Subst}\left (\int \frac {1}{-1+x} \, dx,x,x^2\right ) \\ & = \frac {d x \left (17-5 x^2\right )}{72 \left (4-5 x^2+x^4\right )}+\frac {e \left (5-2 x^2\right )}{18 \left (4-5 x^2+x^4\right )}+\frac {19}{432} d \tanh ^{-1}\left (\frac {x}{2}\right )-\frac {1}{54} d \tanh ^{-1}(x)+\frac {1}{27} e \log \left (1-x^2\right )-\frac {1}{27} e \log \left (4-x^2\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.96 \[ \int \frac {d+e x}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {1}{864} \left (\frac {12 \left (e \left (20-8 x^2\right )+d x \left (17-5 x^2\right )\right )}{4-5 x^2+x^4}+8 (d+4 e) \log (1-x)-(19 d+32 e) \log (2-x)-8 (d-4 e) \log (1+x)+(19 d-32 e) \log (2+x)\right ) \]
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Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.88
method | result | size |
norman | \(\frac {-\frac {1}{9} e \,x^{2}+\frac {17}{72} d x -\frac {5}{72} x^{3} d +\frac {5}{18} e}{x^{4}-5 x^{2}+4}+\left (-\frac {19 d}{864}-\frac {e}{27}\right ) \ln \left (x -2\right )+\left (-\frac {d}{108}+\frac {e}{27}\right ) \ln \left (x +1\right )+\left (\frac {d}{108}+\frac {e}{27}\right ) \ln \left (x -1\right )+\left (\frac {19 d}{864}-\frac {e}{27}\right ) \ln \left (x +2\right )\) | \(83\) |
risch | \(\frac {-\frac {1}{9} e \,x^{2}+\frac {17}{72} d x -\frac {5}{72} x^{3} d +\frac {5}{18} e}{x^{4}-5 x^{2}+4}-\frac {\ln \left (x +1\right ) d}{108}+\frac {\ln \left (x +1\right ) e}{27}+\frac {\ln \left (1-x \right ) d}{108}+\frac {\ln \left (1-x \right ) e}{27}-\frac {19 \ln \left (2-x \right ) d}{864}-\frac {\ln \left (2-x \right ) e}{27}+\frac {19 \ln \left (x +2\right ) d}{864}-\frac {\ln \left (x +2\right ) e}{27}\) | \(99\) |
default | \(-\frac {\frac {d}{144}-\frac {e}{72}}{x +2}+\left (\frac {19 d}{864}-\frac {e}{27}\right ) \ln \left (x +2\right )+\left (-\frac {d}{108}+\frac {e}{27}\right ) \ln \left (x +1\right )-\frac {\frac {d}{36}-\frac {e}{36}}{x +1}-\frac {\frac {d}{36}+\frac {e}{36}}{x -1}+\left (\frac {d}{108}+\frac {e}{27}\right ) \ln \left (x -1\right )+\left (-\frac {19 d}{864}-\frac {e}{27}\right ) \ln \left (x -2\right )-\frac {\frac {d}{144}+\frac {e}{72}}{x -2}\) | \(106\) |
parallelrisch | \(-\frac {-240 e -204 d x +76 \ln \left (x -2\right ) d +128 \ln \left (x -2\right ) e -32 \ln \left (x -1\right ) d -128 \ln \left (x -1\right ) e +32 \ln \left (x -2\right ) x^{4} e +96 e \,x^{2}-160 \ln \left (x -2\right ) x^{2} e +40 \ln \left (x -1\right ) x^{2} d +160 \ln \left (x -1\right ) x^{2} e -40 \ln \left (x +1\right ) x^{2} d +160 \ln \left (x +1\right ) x^{2} e +95 \ln \left (x +2\right ) x^{2} d -160 \ln \left (x +2\right ) x^{2} e +19 \ln \left (x -2\right ) x^{4} d -76 \ln \left (x +2\right ) d +128 \ln \left (x +2\right ) e +32 \ln \left (x +1\right ) d -128 \ln \left (x +1\right ) e +60 x^{3} d -8 \ln \left (x -1\right ) x^{4} d -32 \ln \left (x -1\right ) x^{4} e +8 \ln \left (x +1\right ) x^{4} d -32 \ln \left (x +1\right ) x^{4} e -19 \ln \left (x +2\right ) x^{4} d +32 \ln \left (x +2\right ) x^{4} e -95 \ln \left (x -2\right ) x^{2} d}{864 \left (x^{4}-5 x^{2}+4\right )}\) | \(251\) |
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Leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (80) = 160\).
Time = 0.28 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.80 \[ \int \frac {d+e x}{\left (4-5 x^2+x^4\right )^2} \, dx=-\frac {60 \, d x^{3} + 96 \, e x^{2} - 204 \, d x - {\left ({\left (19 \, d - 32 \, e\right )} x^{4} - 5 \, {\left (19 \, d - 32 \, e\right )} x^{2} + 76 \, d - 128 \, e\right )} \log \left (x + 2\right ) + 8 \, {\left ({\left (d - 4 \, e\right )} x^{4} - 5 \, {\left (d - 4 \, e\right )} x^{2} + 4 \, d - 16 \, e\right )} \log \left (x + 1\right ) - 8 \, {\left ({\left (d + 4 \, e\right )} x^{4} - 5 \, {\left (d + 4 \, e\right )} x^{2} + 4 \, d + 16 \, e\right )} \log \left (x - 1\right ) + {\left ({\left (19 \, d + 32 \, e\right )} x^{4} - 5 \, {\left (19 \, d + 32 \, e\right )} x^{2} + 76 \, d + 128 \, e\right )} \log \left (x - 2\right ) - 240 \, e}{864 \, {\left (x^{4} - 5 \, x^{2} + 4\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 604 vs. \(2 (78) = 156\).
Time = 2.04 (sec) , antiderivative size = 604, normalized size of antiderivative = 6.43 \[ \int \frac {d+e x}{\left (4-5 x^2+x^4\right )^2} \, dx=- \frac {\left (d - 4 e\right ) \log {\left (x + \frac {- 6006260 d^{4} e + 2341251 d^{4} \left (d - 4 e\right ) - 18247680 d^{2} e^{3} + 24099840 d^{2} e^{2} \left (d - 4 e\right ) + 7387904 d^{2} e \left (d - 4 e\right )^{2} - 665280 d^{2} \left (d - 4 e\right )^{3} + 587202560 e^{5} - 12582912 e^{4} \left (d - 4 e\right ) - 36700160 e^{3} \left (d - 4 e\right )^{2} + 786432 e^{2} \left (d - 4 e\right )^{3}}{1675971 d^{5} - 66150400 d^{3} e^{2} + 318767104 d e^{4}} \right )}}{108} + \frac {\left (d + 4 e\right ) \log {\left (x + \frac {- 6006260 d^{4} e - 2341251 d^{4} \left (d + 4 e\right ) - 18247680 d^{2} e^{3} - 24099840 d^{2} e^{2} \left (d + 4 e\right ) + 7387904 d^{2} e \left (d + 4 e\right )^{2} + 665280 d^{2} \left (d + 4 e\right )^{3} + 587202560 e^{5} + 12582912 e^{4} \left (d + 4 e\right ) - 36700160 e^{3} \left (d + 4 e\right )^{2} - 786432 e^{2} \left (d + 4 e\right )^{3}}{1675971 d^{5} - 66150400 d^{3} e^{2} + 318767104 d e^{4}} \right )}}{108} + \frac {\left (19 d - 32 e\right ) \log {\left (x + \frac {- 6006260 d^{4} e - \frac {2341251 d^{4} \cdot \left (19 d - 32 e\right )}{8} - 18247680 d^{2} e^{3} - 3012480 d^{2} e^{2} \cdot \left (19 d - 32 e\right ) + 115436 d^{2} e \left (19 d - 32 e\right )^{2} + \frac {10395 d^{2} \left (19 d - 32 e\right )^{3}}{8} + 587202560 e^{5} + 1572864 e^{4} \cdot \left (19 d - 32 e\right ) - 573440 e^{3} \left (19 d - 32 e\right )^{2} - 1536 e^{2} \left (19 d - 32 e\right )^{3}}{1675971 d^{5} - 66150400 d^{3} e^{2} + 318767104 d e^{4}} \right )}}{864} - \frac {\left (19 d + 32 e\right ) \log {\left (x + \frac {- 6006260 d^{4} e + \frac {2341251 d^{4} \cdot \left (19 d + 32 e\right )}{8} - 18247680 d^{2} e^{3} + 3012480 d^{2} e^{2} \cdot \left (19 d + 32 e\right ) + 115436 d^{2} e \left (19 d + 32 e\right )^{2} - \frac {10395 d^{2} \left (19 d + 32 e\right )^{3}}{8} + 587202560 e^{5} - 1572864 e^{4} \cdot \left (19 d + 32 e\right ) - 573440 e^{3} \left (19 d + 32 e\right )^{2} + 1536 e^{2} \left (19 d + 32 e\right )^{3}}{1675971 d^{5} - 66150400 d^{3} e^{2} + 318767104 d e^{4}} \right )}}{864} + \frac {- 5 d x^{3} + 17 d x - 8 e x^{2} + 20 e}{72 x^{4} - 360 x^{2} + 288} \]
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Time = 0.18 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.88 \[ \int \frac {d+e x}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {1}{864} \, {\left (19 \, d - 32 \, e\right )} \log \left (x + 2\right ) - \frac {1}{108} \, {\left (d - 4 \, e\right )} \log \left (x + 1\right ) + \frac {1}{108} \, {\left (d + 4 \, e\right )} \log \left (x - 1\right ) - \frac {1}{864} \, {\left (19 \, d + 32 \, e\right )} \log \left (x - 2\right ) - \frac {5 \, d x^{3} + 8 \, e x^{2} - 17 \, d x - 20 \, e}{72 \, {\left (x^{4} - 5 \, x^{2} + 4\right )}} \]
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Time = 0.30 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.93 \[ \int \frac {d+e x}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {1}{864} \, {\left (19 \, d - 32 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) - \frac {1}{108} \, {\left (d - 4 \, e\right )} \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{108} \, {\left (d + 4 \, e\right )} \log \left ({\left | x - 1 \right |}\right ) - \frac {1}{864} \, {\left (19 \, d + 32 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) - \frac {5 \, d x^{3} + 8 \, e x^{2} - 17 \, d x - 20 \, e}{72 \, {\left (x^{4} - 5 \, x^{2} + 4\right )}} \]
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Time = 0.05 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.89 \[ \int \frac {d+e x}{\left (4-5 x^2+x^4\right )^2} \, dx=\ln \left (x-1\right )\,\left (\frac {d}{108}+\frac {e}{27}\right )-\ln \left (x+1\right )\,\left (\frac {d}{108}-\frac {e}{27}\right )-\ln \left (x-2\right )\,\left (\frac {19\,d}{864}+\frac {e}{27}\right )+\ln \left (x+2\right )\,\left (\frac {19\,d}{864}-\frac {e}{27}\right )+\frac {-\frac {5\,d\,x^3}{72}-\frac {e\,x^2}{9}+\frac {17\,d\,x}{72}+\frac {5\,e}{18}}{x^4-5\,x^2+4} \]
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