Optimal. Leaf size=82 \[ \frac {d-2 e+4 f}{12 (x+2)}-\frac {1}{18} \log (1-x) (d+e+f)+\frac {1}{48} \log (2-x) (d+2 e+4 f)+\frac {1}{6} \log (x+1) (d-e+f)-\frac {1}{144} \log (x+2) (19 d-26 e+28 f) \]
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Rubi [A] time = 0.20, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1586, 6742} \[ \frac {d-2 e+4 f}{12 (x+2)}-\frac {1}{18} \log (1-x) (d+e+f)+\frac {1}{48} \log (2-x) (d+2 e+4 f)+\frac {1}{6} \log (x+1) (d-e+f)-\frac {1}{144} \log (x+2) (19 d-26 e+28 f) \]
Antiderivative was successfully verified.
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Rule 1586
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (d+e x+f x^2\right ) \left (2-x-2 x^2+x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx &=\int \frac {d+e x+f x^2}{(2+x)^2 \left (2-x-2 x^2+x^3\right )} \, dx\\ &=\int \left (\frac {d+2 e+4 f}{48 (-2+x)}+\frac {-d-e-f}{18 (-1+x)}+\frac {d-e+f}{6 (1+x)}+\frac {-d+2 e-4 f}{12 (2+x)^2}+\frac {-19 d+26 e-28 f}{144 (2+x)}\right ) \, dx\\ &=\frac {d-2 e+4 f}{12 (2+x)}-\frac {1}{18} (d+e+f) \log (1-x)+\frac {1}{48} (d+2 e+4 f) \log (2-x)+\frac {1}{6} (d-e+f) \log (1+x)-\frac {1}{144} (19 d-26 e+28 f) \log (2+x)\\ \end {align*}
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Mathematica [A] time = 0.06, size = 77, normalized size = 0.94 \[ \frac {1}{144} \left (\frac {12 (d-2 e+4 f)}{x+2}+24 \log (-x-1) (d-e+f)-8 \log (1-x) (d+e+f)+3 \log (2-x) (d+2 e+4 f)+\log (x+2) (-19 d+26 e-28 f)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 116, normalized size = 1.41 \[ -\frac {{\left ({\left (19 \, d - 26 \, e + 28 \, f\right )} x + 38 \, d - 52 \, e + 56 \, f\right )} \log \left (x + 2\right ) - 24 \, {\left ({\left (d - e + f\right )} x + 2 \, d - 2 \, e + 2 \, f\right )} \log \left (x + 1\right ) + 8 \, {\left ({\left (d + e + f\right )} x + 2 \, d + 2 \, e + 2 \, f\right )} \log \left (x - 1\right ) - 3 \, {\left ({\left (d + 2 \, e + 4 \, f\right )} x + 2 \, d + 4 \, e + 8 \, f\right )} \log \left (x - 2\right ) - 12 \, d + 24 \, e - 48 \, f}{144 \, {\left (x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 77, normalized size = 0.94 \[ -\frac {1}{144} \, {\left (19 \, d + 28 \, f - 26 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{6} \, {\left (d + f - e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{18} \, {\left (d + f + e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{48} \, {\left (d + 4 \, f + 2 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) + \frac {d + 4 \, f - 2 \, e}{12 \, {\left (x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 110, normalized size = 1.34 \[ -\frac {19 d \ln \left (x +2\right )}{144}+\frac {d \ln \left (x -2\right )}{48}-\frac {d \ln \left (x -1\right )}{18}+\frac {d \ln \left (x +1\right )}{6}+\frac {13 e \ln \left (x +2\right )}{72}+\frac {e \ln \left (x -2\right )}{24}-\frac {e \ln \left (x -1\right )}{18}-\frac {e \ln \left (x +1\right )}{6}-\frac {7 f \ln \left (x +2\right )}{36}+\frac {f \ln \left (x -2\right )}{12}-\frac {f \ln \left (x -1\right )}{18}+\frac {f \ln \left (x +1\right )}{6}+\frac {d}{12 x +24}-\frac {e}{6 \left (x +2\right )}+\frac {f}{3 x +6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 68, normalized size = 0.83 \[ -\frac {1}{144} \, {\left (19 \, d - 26 \, e + 28 \, f\right )} \log \left (x + 2\right ) + \frac {1}{6} \, {\left (d - e + f\right )} \log \left (x + 1\right ) - \frac {1}{18} \, {\left (d + e + f\right )} \log \left (x - 1\right ) + \frac {1}{48} \, {\left (d + 2 \, e + 4 \, f\right )} \log \left (x - 2\right ) + \frac {d - 2 \, e + 4 \, f}{12 \, {\left (x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.84, size = 79, normalized size = 0.96 \[ \frac {\frac {d}{12}-\frac {e}{6}+\frac {f}{3}}{x+2}+\ln \left (x+1\right )\,\left (\frac {d}{6}-\frac {e}{6}+\frac {f}{6}\right )-\ln \left (x-1\right )\,\left (\frac {d}{18}+\frac {e}{18}+\frac {f}{18}\right )+\ln \left (x-2\right )\,\left (\frac {d}{48}+\frac {e}{24}+\frac {f}{12}\right )-\ln \left (x+2\right )\,\left (\frac {19\,d}{144}-\frac {13\,e}{72}+\frac {7\,f}{36}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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