Optimal. Leaf size=94 \[ \frac {d x \left (17-5 x^2\right )}{72 \left (x^4-5 x^2+4\right )}+\frac {19}{432} d \tanh ^{-1}\left (\frac {x}{2}\right )-\frac {1}{54} d \tanh ^{-1}(x)+\frac {1}{27} e \log \left (1-x^2\right )-\frac {1}{27} e \log \left (4-x^2\right )+\frac {e \left (5-2 x^2\right )}{18 \left (x^4-5 x^2+4\right )} \]
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Rubi [A] time = 0.05, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1673, 12, 1092, 1166, 207, 1107, 614, 616, 31} \[ \frac {d x \left (17-5 x^2\right )}{72 \left (x^4-5 x^2+4\right )}+\frac {19}{432} d \tanh ^{-1}\left (\frac {x}{2}\right )-\frac {1}{54} d \tanh ^{-1}(x)+\frac {e \left (5-2 x^2\right )}{18 \left (x^4-5 x^2+4\right )}+\frac {1}{27} e \log \left (1-x^2\right )-\frac {1}{27} e \log \left (4-x^2\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 207
Rule 614
Rule 616
Rule 1092
Rule 1107
Rule 1166
Rule 1673
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (4-5 x^2+x^4\right )^2} \, dx &=\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 \operatorname {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 \operatorname {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 \operatorname {Subst}\left (\int \frac {1}{-4+x} \, dx,x,x^2\right )+\frac {1}{27} e \operatorname {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*}
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Mathematica [A] time = 0.05, size = 90, normalized size = 0.96 \[ \frac {1}{864} \left (\frac {12 \left (d x \left (17-5 x^2\right )+e \left (20-8 x^2\right )\right )}{x^4-5 x^2+4}+8 (d+4 e) \log (1-x)-(19 d+32 e) \log (2-x)-8 (d-4 e) \log (x+1)+(19 d-32 e) \log (x+2)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.32, size = 169, normalized size = 1.80 \[ -\frac {60 \, d x^{3} + 96 \, e x^{2} - 204 \, d x - {\left ({\left (19 \, d - 32 \, e\right )} x^{4} - 5 \, {\left (19 \, d - 32 \, e\right )} x^{2} + 76 \, d - 128 \, e\right )} \log \left (x + 2\right ) + 8 \, {\left ({\left (d - 4 \, e\right )} x^{4} - 5 \, {\left (d - 4 \, e\right )} x^{2} + 4 \, d - 16 \, e\right )} \log \left (x + 1\right ) - 8 \, {\left ({\left (d + 4 \, e\right )} x^{4} - 5 \, {\left (d + 4 \, e\right )} x^{2} + 4 \, d + 16 \, e\right )} \log \left (x - 1\right ) + {\left ({\left (19 \, d + 32 \, e\right )} x^{4} - 5 \, {\left (19 \, d + 32 \, e\right )} x^{2} + 76 \, d + 128 \, e\right )} \log \left (x - 2\right ) - 240 \, e}{864 \, {\left (x^{4} - 5 \, x^{2} + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 93, normalized size = 0.99 \[ \frac {1}{864} \, {\left (19 \, d - 32 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) - \frac {1}{108} \, {\left (d - 4 \, e\right )} \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{108} \, {\left (d + 4 \, e\right )} \log \left ({\left | x - 1 \right |}\right ) - \frac {1}{864} \, {\left (19 \, d + 32 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) - \frac {5 \, d x^{3} + 8 \, x^{2} e - 17 \, d x - 20 \, e}{72 \, {\left (x^{4} - 5 \, x^{2} + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 122, normalized size = 1.30 \[ \frac {19 d \ln \left (x +2\right )}{864}-\frac {19 d \ln \left (x -2\right )}{864}+\frac {d \ln \left (x -1\right )}{108}-\frac {d \ln \left (x +1\right )}{108}-\frac {e \ln \left (x +2\right )}{27}-\frac {e \ln \left (x -2\right )}{27}+\frac {e \ln \left (x -1\right )}{27}+\frac {e \ln \left (x +1\right )}{27}-\frac {d}{144 \left (x -2\right )}-\frac {d}{36 \left (x +1\right )}-\frac {d}{36 \left (x -1\right )}-\frac {d}{144 \left (x +2\right )}-\frac {e}{72 \left (x -2\right )}+\frac {e}{36 x +36}-\frac {e}{36 \left (x -1\right )}+\frac {e}{72 x +144} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.68, size = 83, normalized size = 0.88 \[ \frac {1}{864} \, {\left (19 \, d - 32 \, e\right )} \log \left (x + 2\right ) - \frac {1}{108} \, {\left (d - 4 \, e\right )} \log \left (x + 1\right ) + \frac {1}{108} \, {\left (d + 4 \, e\right )} \log \left (x - 1\right ) - \frac {1}{864} \, {\left (19 \, d + 32 \, e\right )} \log \left (x - 2\right ) - \frac {5 \, d x^{3} + 8 \, e x^{2} - 17 \, d x - 20 \, e}{72 \, {\left (x^{4} - 5 \, x^{2} + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 84, normalized size = 0.89 \[ \ln \left (x-1\right )\,\left (\frac {d}{108}+\frac {e}{27}\right )-\ln \left (x+1\right )\,\left (\frac {d}{108}-\frac {e}{27}\right )-\ln \left (x-2\right )\,\left (\frac {19\,d}{864}+\frac {e}{27}\right )+\ln \left (x+2\right )\,\left (\frac {19\,d}{864}-\frac {e}{27}\right )+\frac {-\frac {5\,d\,x^3}{72}-\frac {e\,x^2}{9}+\frac {17\,d\,x}{72}+\frac {5\,e}{18}}{x^4-5\,x^2+4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.57, size = 604, normalized size = 6.43 \[ - \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} \left (19 d - 32 e\right )}{8} - 18247680 d^{2} e^{3} - 3012480 d^{2} e^{2} \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} \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} \left (19 d + 32 e\right )}{8} - 18247680 d^{2} e^{3} + 3012480 d^{2} e^{2} \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} \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} \]
Verification of antiderivative is not currently implemented for this CAS.
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