Optimal. Leaf size=95 \[ \frac {d-2 e+4 f-8 g}{12 (x+2)}-\frac {1}{18} \log (1-x) (d+e+f+g)+\frac {1}{48} \log (2-x) (d+2 e+4 f+8 g)+\frac {1}{6} \log (x+1) (d-e+f-g)-\frac {1}{144} \log (x+2) (19 d-26 e+28 f-8 g) \]
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Rubi [A] time = 0.22, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {1586, 6742} \begin {gather*} \frac {d-2 e+4 f-8 g}{12 (x+2)}-\frac {1}{18} \log (1-x) (d+e+f+g)+\frac {1}{48} \log (2-x) (d+2 e+4 f+8 g)+\frac {1}{6} \log (x+1) (d-e+f-g)-\frac {1}{144} \log (x+2) (19 d-26 e+28 f-8 g) \end {gather*}
Antiderivative was successfully verified.
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Rule 1586
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx &=\int \frac {d+e x+f x^2+g x^3}{(2+x)^2 \left (2-x-2 x^2+x^3\right )} \, dx\\ &=\int \left (\frac {d+2 e+4 f+8 g}{48 (-2+x)}+\frac {-d-e-f-g}{18 (-1+x)}+\frac {d-e+f-g}{6 (1+x)}+\frac {-d+2 e-4 f+8 g}{12 (2+x)^2}+\frac {-19 d+26 e-28 f+8 g}{144 (2+x)}\right ) \, dx\\ &=\frac {d-2 e+4 f-8 g}{12 (2+x)}-\frac {1}{18} (d+e+f+g) \log (1-x)+\frac {1}{48} (d+2 e+4 f+8 g) \log (2-x)+\frac {1}{6} (d-e+f-g) \log (1+x)-\frac {1}{144} (19 d-26 e+28 f-8 g) \log (2+x)\\ \end {align*}
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Mathematica [A] time = 0.05, size = 90, normalized size = 0.95 \begin {gather*} \frac {1}{144} \left (\frac {12 (d-2 e+4 f-8 g)}{x+2}+24 \log (-x-1) (d-e+f-g)-8 \log (1-x) (d+e+f+g)+3 \log (2-x) (d+2 e+4 f+8 g)+\log (x+2) (-19 d+26 e-28 f+8 g)\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 2.74, size = 141, normalized size = 1.48 \begin {gather*} -\frac {{\left ({\left (19 \, d - 26 \, e + 28 \, f - 8 \, g\right )} x + 38 \, d - 52 \, e + 56 \, f - 16 \, g\right )} \log \left (x + 2\right ) - 24 \, {\left ({\left (d - e + f - g\right )} x + 2 \, d - 2 \, e + 2 \, f - 2 \, g\right )} \log \left (x + 1\right ) + 8 \, {\left ({\left (d + e + f + g\right )} x + 2 \, d + 2 \, e + 2 \, f + 2 \, g\right )} \log \left (x - 1\right ) - 3 \, {\left ({\left (d + 2 \, e + 4 \, f + 8 \, g\right )} x + 2 \, d + 4 \, e + 8 \, f + 16 \, g\right )} \log \left (x - 2\right ) - 12 \, d + 24 \, e - 48 \, f + 96 \, g}{144 \, {\left (x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 90, normalized size = 0.95 \begin {gather*} -\frac {1}{144} \, {\left (19 \, d + 28 \, f - 8 \, g - 26 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{6} \, {\left (d + f - g - e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{18} \, {\left (d + f + g + e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{48} \, {\left (d + 4 \, f + 8 \, g + 2 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) + \frac {d + 4 \, f - 8 \, g - 2 \, e}{12 \, {\left (x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 146, normalized size = 1.54 \begin {gather*} -\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 {g \ln \left (x +2\right )}{18}+\frac {g \ln \left (x -2\right )}{6}-\frac {g \ln \left (x -1\right )}{18}-\frac {g \ln \left (x +1\right )}{6}+\frac {d}{12 x +24}-\frac {e}{6 \left (x +2\right )}+\frac {f}{3 x +6}-\frac {2 g}{3 \left (x +2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 81, normalized size = 0.85 \begin {gather*} -\frac {1}{144} \, {\left (19 \, d - 26 \, e + 28 \, f - 8 \, g\right )} \log \left (x + 2\right ) + \frac {1}{6} \, {\left (d - e + f - g\right )} \log \left (x + 1\right ) - \frac {1}{18} \, {\left (d + e + f + g\right )} \log \left (x - 1\right ) + \frac {1}{48} \, {\left (d + 2 \, e + 4 \, f + 8 \, g\right )} \log \left (x - 2\right ) + \frac {d - 2 \, e + 4 \, f - 8 \, g}{12 \, {\left (x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.88, size = 94, normalized size = 0.99 \begin {gather*} \frac {\frac {d}{12}-\frac {e}{6}+\frac {f}{3}-\frac {2\,g}{3}}{x+2}+\ln \left (x+1\right )\,\left (\frac {d}{6}-\frac {e}{6}+\frac {f}{6}-\frac {g}{6}\right )-\ln \left (x-1\right )\,\left (\frac {d}{18}+\frac {e}{18}+\frac {f}{18}+\frac {g}{18}\right )+\ln \left (x-2\right )\,\left (\frac {d}{48}+\frac {e}{24}+\frac {f}{12}+\frac {g}{6}\right )-\ln \left (x+2\right )\,\left (\frac {19\,d}{144}-\frac {13\,e}{72}+\frac {7\,f}{36}-\frac {g}{18}\right ) \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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