Integrand size = 28, antiderivative size = 138 \[ \int \frac {d+e x+f x^2+g x^3}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {x \left (17 d+20 f-(5 d+8 f) x^2\right )}{72 \left (4-5 x^2+x^4\right )}+\frac {5 e+8 g-(2 e+5 g) x^2}{18 \left (4-5 x^2+x^4\right )}+\frac {1}{432} (19 d+52 f) \text {arctanh}\left (\frac {x}{2}\right )-\frac {1}{54} (d+7 f) \text {arctanh}(x)+\frac {1}{54} (2 e+5 g) \log \left (1-x^2\right )-\frac {1}{54} (2 e+5 g) \log \left (4-x^2\right ) \]
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Time = 0.10 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1687, 1192, 1180, 213, 1261, 652, 630, 31} \[ \int \frac {d+e x+f x^2+g x^3}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {1}{432} \text {arctanh}\left (\frac {x}{2}\right ) (19 d+52 f)-\frac {1}{54} \text {arctanh}(x) (d+7 f)+\frac {x \left (-\left (x^2 (5 d+8 f)\right )+17 d+20 f\right )}{72 \left (x^4-5 x^2+4\right )}+\frac {1}{54} (2 e+5 g) \log \left (1-x^2\right )-\frac {1}{54} (2 e+5 g) \log \left (4-x^2\right )+\frac {-\left (x^2 (2 e+5 g)\right )+5 e+8 g}{18 \left (x^4-5 x^2+4\right )} \]
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Rule 31
Rule 213
Rule 630
Rule 652
Rule 1180
Rule 1192
Rule 1261
Rule 1687
Rubi steps \begin{align*} \text {integral}& = \int \frac {d+f x^2}{\left (4-5 x^2+x^4\right )^2} \, dx+\int \frac {x \left (e+g x^2\right )}{\left (4-5 x^2+x^4\right )^2} \, dx \\ & = \frac {x \left (17 d+20 f-(5 d+8 f) x^2\right )}{72 \left (4-5 x^2+x^4\right )}-\frac {1}{72} \int \frac {-d+20 f+(5 d+8 f) x^2}{4-5 x^2+x^4} \, dx+\frac {1}{2} \text {Subst}\left (\int \frac {e+g x}{\left (4-5 x+x^2\right )^2} \, dx,x,x^2\right ) \\ & = \frac {x \left (17 d+20 f-(5 d+8 f) x^2\right )}{72 \left (4-5 x^2+x^4\right )}+\frac {5 e+8 g-(2 e+5 g) x^2}{18 \left (4-5 x^2+x^4\right )}-\frac {1}{54} (-d-7 f) \int \frac {1}{-1+x^2} \, dx-\frac {1}{216} (19 d+52 f) \int \frac {1}{-4+x^2} \, dx+\frac {1}{18} (-2 e-5 g) \text {Subst}\left (\int \frac {1}{4-5 x+x^2} \, dx,x,x^2\right ) \\ & = \frac {x \left (17 d+20 f-(5 d+8 f) x^2\right )}{72 \left (4-5 x^2+x^4\right )}+\frac {5 e+8 g-(2 e+5 g) x^2}{18 \left (4-5 x^2+x^4\right )}+\frac {1}{432} (19 d+52 f) \tanh ^{-1}\left (\frac {x}{2}\right )-\frac {1}{54} (d+7 f) \tanh ^{-1}(x)+\frac {1}{54} (-2 e-5 g) \text {Subst}\left (\int \frac {1}{-4+x} \, dx,x,x^2\right )+\frac {1}{54} (2 e+5 g) \text {Subst}\left (\int \frac {1}{-1+x} \, dx,x,x^2\right ) \\ & = \frac {x \left (17 d+20 f-(5 d+8 f) x^2\right )}{72 \left (4-5 x^2+x^4\right )}+\frac {5 e+8 g-(2 e+5 g) x^2}{18 \left (4-5 x^2+x^4\right )}+\frac {1}{432} (19 d+52 f) \tanh ^{-1}\left (\frac {x}{2}\right )-\frac {1}{54} (d+7 f) \tanh ^{-1}(x)+\frac {1}{54} (2 e+5 g) \log \left (1-x^2\right )-\frac {1}{54} (2 e+5 g) \log \left (4-x^2\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.97 \[ \int \frac {d+e x+f x^2+g x^3}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {1}{864} \left (\frac {12 \left (17 d x+20 f x-5 d x^3-8 f x^3+e \left (20-8 x^2\right )-4 g \left (-8+5 x^2\right )\right )}{4-5 x^2+x^4}+8 (d+4 e+7 f+10 g) \log (1-x)-(19 d+32 e+52 f+80 g) \log (2-x)-8 (d-4 e+7 f-10 g) \log (1+x)+(19 d-32 e+52 f-80 g) \log (2+x)\right ) \]
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Time = 0.10 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.91
method | result | size |
norman | \(\frac {\left (-\frac {5 d}{72}-\frac {f}{9}\right ) x^{3}+\left (\frac {17 d}{72}+\frac {5 f}{18}\right ) x +\left (-\frac {e}{9}-\frac {5 g}{18}\right ) x^{2}+\frac {5 e}{18}+\frac {4 g}{9}}{x^{4}-5 x^{2}+4}+\left (-\frac {19 d}{864}-\frac {e}{27}-\frac {13 f}{216}-\frac {5 g}{54}\right ) \ln \left (x -2\right )+\left (-\frac {d}{108}+\frac {e}{27}-\frac {7 f}{108}+\frac {5 g}{54}\right ) \ln \left (x +1\right )+\left (\frac {d}{108}+\frac {e}{27}+\frac {7 f}{108}+\frac {5 g}{54}\right ) \ln \left (x -1\right )+\left (\frac {19 d}{864}-\frac {e}{27}+\frac {13 f}{216}-\frac {5 g}{54}\right ) \ln \left (x +2\right )\) | \(125\) |
default | \(-\frac {\frac {d}{144}-\frac {e}{72}+\frac {f}{36}-\frac {g}{18}}{x +2}+\left (\frac {19 d}{864}-\frac {e}{27}+\frac {13 f}{216}-\frac {5 g}{54}\right ) \ln \left (x +2\right )+\left (-\frac {d}{108}+\frac {e}{27}-\frac {7 f}{108}+\frac {5 g}{54}\right ) \ln \left (x +1\right )-\frac {\frac {d}{36}-\frac {e}{36}+\frac {f}{36}-\frac {g}{36}}{x +1}-\frac {\frac {d}{36}+\frac {e}{36}+\frac {f}{36}+\frac {g}{36}}{x -1}+\left (\frac {d}{108}+\frac {e}{27}+\frac {7 f}{108}+\frac {5 g}{54}\right ) \ln \left (x -1\right )+\left (-\frac {19 d}{864}-\frac {e}{27}-\frac {13 f}{216}-\frac {5 g}{54}\right ) \ln \left (x -2\right )-\frac {\frac {d}{144}+\frac {e}{72}+\frac {f}{36}+\frac {g}{18}}{x -2}\) | \(154\) |
risch | \(\frac {\left (-\frac {5 d}{72}-\frac {f}{9}\right ) x^{3}+\left (\frac {17 d}{72}+\frac {5 f}{18}\right ) x +\left (-\frac {e}{9}-\frac {5 g}{18}\right ) x^{2}+\frac {5 e}{18}+\frac {4 g}{9}}{x^{4}-5 x^{2}+4}+\frac {19 \ln \left (x +2\right ) d}{864}-\frac {\ln \left (x +2\right ) e}{27}+\frac {13 \ln \left (x +2\right ) f}{216}-\frac {5 \ln \left (x +2\right ) g}{54}+\frac {\ln \left (1-x \right ) d}{108}+\frac {\ln \left (1-x \right ) e}{27}+\frac {7 \ln \left (1-x \right ) f}{108}+\frac {5 \ln \left (1-x \right ) g}{54}-\frac {\ln \left (x +1\right ) d}{108}+\frac {\ln \left (x +1\right ) e}{27}-\frac {7 \ln \left (x +1\right ) f}{108}+\frac {5 \ln \left (x +1\right ) g}{54}-\frac {19 \ln \left (2-x \right ) d}{864}-\frac {\ln \left (2-x \right ) e}{27}-\frac {13 \ln \left (2-x \right ) f}{216}-\frac {5 \ln \left (2-x \right ) g}{54}\) | \(181\) |
parallelrisch | \(-\frac {-384 g -240 e +96 f \,x^{3}+240 g \,x^{2}-204 d x +76 \ln \left (x -2\right ) d +128 \ln \left (x -2\right ) e -32 \ln \left (x -1\right ) d -128 \ln \left (x -1\right ) e +32 \ln \left (x -2\right ) x^{4} e -208 \ln \left (x +2\right ) f +224 \ln \left (x +1\right ) f +96 e \,x^{2}-160 \ln \left (x -2\right ) x^{2} e +40 \ln \left (x -1\right ) x^{2} d +160 \ln \left (x -1\right ) x^{2} e -40 \ln \left (x +1\right ) x^{2} d +160 \ln \left (x +1\right ) x^{2} e +95 \ln \left (x +2\right ) x^{2} d -160 \ln \left (x +2\right ) x^{2} e +19 \ln \left (x -2\right ) x^{4} d -76 \ln \left (x +2\right ) d +128 \ln \left (x +2\right ) e +32 \ln \left (x +1\right ) d -128 \ln \left (x +1\right ) e +60 x^{3} d -56 \ln \left (x -1\right ) x^{4} f +56 \ln \left (x +1\right ) x^{4} f -52 \ln \left (x +2\right ) x^{4} f -320 \ln \left (x +1\right ) g +80 \ln \left (x -2\right ) x^{4} g -80 \ln \left (x -1\right ) x^{4} g -80 \ln \left (x +1\right ) x^{4} g +80 \ln \left (x +2\right ) x^{4} g +320 \ln \left (x +2\right ) g -400 \ln \left (x -2\right ) x^{2} g +400 \ln \left (x -1\right ) x^{2} g +400 \ln \left (x +1\right ) x^{2} g -400 \ln \left (x +2\right ) x^{2} g +320 \ln \left (x -2\right ) g -320 \ln \left (x -1\right ) g -260 \ln \left (x -2\right ) x^{2} f +280 \ln \left (x -1\right ) x^{2} f -280 \ln \left (x +1\right ) x^{2} f +260 \ln \left (x +2\right ) x^{2} f +52 \ln \left (x -2\right ) x^{4} f +208 \ln \left (x -2\right ) f -224 \ln \left (x -1\right ) f -8 \ln \left (x -1\right ) x^{4} d -32 \ln \left (x -1\right ) x^{4} e +8 \ln \left (x +1\right ) x^{4} d -32 \ln \left (x +1\right ) x^{4} e -19 \ln \left (x +2\right ) x^{4} d +32 \ln \left (x +2\right ) x^{4} e -95 \ln \left (x -2\right ) x^{2} d -240 f x}{864 \left (x^{4}-5 x^{2}+4\right )}\) | \(486\) |
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Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (122) = 244\).
Time = 0.49 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.90 \[ \int \frac {d+e x+f x^2+g x^3}{\left (4-5 x^2+x^4\right )^2} \, dx=-\frac {12 \, {\left (5 \, d + 8 \, f\right )} x^{3} + 48 \, {\left (2 \, e + 5 \, g\right )} x^{2} - 12 \, {\left (17 \, d + 20 \, f\right )} x - {\left ({\left (19 \, d - 32 \, e + 52 \, f - 80 \, g\right )} x^{4} - 5 \, {\left (19 \, d - 32 \, e + 52 \, f - 80 \, g\right )} x^{2} + 76 \, d - 128 \, e + 208 \, f - 320 \, g\right )} \log \left (x + 2\right ) + 8 \, {\left ({\left (d - 4 \, e + 7 \, f - 10 \, g\right )} x^{4} - 5 \, {\left (d - 4 \, e + 7 \, f - 10 \, g\right )} x^{2} + 4 \, d - 16 \, e + 28 \, f - 40 \, g\right )} \log \left (x + 1\right ) - 8 \, {\left ({\left (d + 4 \, e + 7 \, f + 10 \, g\right )} x^{4} - 5 \, {\left (d + 4 \, e + 7 \, f + 10 \, g\right )} x^{2} + 4 \, d + 16 \, e + 28 \, f + 40 \, g\right )} \log \left (x - 1\right ) + {\left ({\left (19 \, d + 32 \, e + 52 \, f + 80 \, g\right )} x^{4} - 5 \, {\left (19 \, d + 32 \, e + 52 \, f + 80 \, g\right )} x^{2} + 76 \, d + 128 \, e + 208 \, f + 320 \, g\right )} \log \left (x - 2\right ) - 240 \, e - 384 \, g}{864 \, {\left (x^{4} - 5 \, x^{2} + 4\right )}} \]
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Timed out. \[ \int \frac {d+e x+f x^2+g x^3}{\left (4-5 x^2+x^4\right )^2} \, dx=\text {Timed out} \]
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Time = 0.19 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.92 \[ \int \frac {d+e x+f x^2+g x^3}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {1}{864} \, {\left (19 \, d - 32 \, e + 52 \, f - 80 \, g\right )} \log \left (x + 2\right ) - \frac {1}{108} \, {\left (d - 4 \, e + 7 \, f - 10 \, g\right )} \log \left (x + 1\right ) + \frac {1}{108} \, {\left (d + 4 \, e + 7 \, f + 10 \, g\right )} \log \left (x - 1\right ) - \frac {1}{864} \, {\left (19 \, d + 32 \, e + 52 \, f + 80 \, g\right )} \log \left (x - 2\right ) - \frac {{\left (5 \, d + 8 \, f\right )} x^{3} + 4 \, {\left (2 \, e + 5 \, g\right )} x^{2} - {\left (17 \, d + 20 \, f\right )} x - 20 \, e - 32 \, g}{72 \, {\left (x^{4} - 5 \, x^{2} + 4\right )}} \]
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Time = 0.30 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.94 \[ \int \frac {d+e x+f x^2+g x^3}{\left (4-5 x^2+x^4\right )^2} \, dx=\frac {1}{864} \, {\left (19 \, d - 32 \, e + 52 \, f - 80 \, g\right )} \log \left ({\left | x + 2 \right |}\right ) - \frac {1}{108} \, {\left (d - 4 \, e + 7 \, f - 10 \, g\right )} \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{108} \, {\left (d + 4 \, e + 7 \, f + 10 \, g\right )} \log \left ({\left | x - 1 \right |}\right ) - \frac {1}{864} \, {\left (19 \, d + 32 \, e + 52 \, f + 80 \, g\right )} \log \left ({\left | x - 2 \right |}\right ) - \frac {5 \, d x^{3} + 8 \, f x^{3} + 8 \, e x^{2} + 20 \, g x^{2} - 17 \, d x - 20 \, f x - 20 \, e - 32 \, g}{72 \, {\left (x^{4} - 5 \, x^{2} + 4\right )}} \]
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Time = 7.88 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.93 \[ \int \frac {d+e x+f x^2+g x^3}{\left (4-5 x^2+x^4\right )^2} \, dx=\ln \left (x-1\right )\,\left (\frac {d}{108}+\frac {e}{27}+\frac {7\,f}{108}+\frac {5\,g}{54}\right )-\ln \left (x+1\right )\,\left (\frac {d}{108}-\frac {e}{27}+\frac {7\,f}{108}-\frac {5\,g}{54}\right )-\ln \left (x-2\right )\,\left (\frac {19\,d}{864}+\frac {e}{27}+\frac {13\,f}{216}+\frac {5\,g}{54}\right )+\ln \left (x+2\right )\,\left (\frac {19\,d}{864}-\frac {e}{27}+\frac {13\,f}{216}-\frac {5\,g}{54}\right )+\frac {\left (-\frac {5\,d}{72}-\frac {f}{9}\right )\,x^3+\left (-\frac {e}{9}-\frac {5\,g}{18}\right )\,x^2+\left (\frac {17\,d}{72}+\frac {5\,f}{18}\right )\,x+\frac {5\,e}{18}+\frac {4\,g}{9}}{x^4-5\,x^2+4} \]
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