Optimal. Leaf size=71 \[ \frac {d-2 e}{12 (2+x)}-\frac {1}{18} (d+e) \log (1-x)+\frac {1}{48} (d+2 e) \log (2-x)+\frac {1}{6} (d-e) \log (1+x)-\frac {1}{144} (19 d-26 e) \log (2+x) \]
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Rubi [A]
time = 0.12, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1600, 6874}
\begin {gather*} \frac {d-2 e}{12 (x+2)}-\frac {1}{18} (d+e) \log (1-x)+\frac {1}{48} (d+2 e) \log (2-x)+\frac {1}{6} (d-e) \log (x+1)-\frac {1}{144} (19 d-26 e) \log (x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 1600
Rule 6874
Rubi steps
\begin {align*} \int \frac {(d+e x) \left (2-x-2 x^2+x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx &=\int \frac {d+e x}{(2+x)^2 \left (2-x-2 x^2+x^3\right )} \, dx\\ &=\int \left (\frac {d+2 e}{48 (-2+x)}+\frac {-d-e}{18 (-1+x)}+\frac {d-e}{6 (1+x)}+\frac {-d+2 e}{12 (2+x)^2}+\frac {-19 d+26 e}{144 (2+x)}\right ) \, dx\\ &=\frac {d-2 e}{12 (2+x)}-\frac {1}{18} (d+e) \log (1-x)+\frac {1}{48} (d+2 e) \log (2-x)+\frac {1}{6} (d-e) \log (1+x)-\frac {1}{144} (19 d-26 e) \log (2+x)\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 66, normalized size = 0.93 \begin {gather*} \frac {1}{144} \left (\frac {12 (d-2 e)}{2+x}+24 (d-e) \log (-1-x)-8 (d+e) \log (1-x)+3 (d+2 e) \log (2-x)+(-19 d+26 e) \log (2+x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 64, normalized size = 0.90
method | result | size |
default | \(\left (-\frac {19 d}{144}+\frac {13 e}{72}\right ) \ln \left (x +2\right )-\frac {-\frac {d}{12}+\frac {e}{6}}{x +2}+\left (\frac {d}{48}+\frac {e}{24}\right ) \ln \left (x -2\right )+\left (-\frac {d}{18}-\frac {e}{18}\right ) \ln \left (-1+x \right )+\left (\frac {d}{6}-\frac {e}{6}\right ) \ln \left (1+x \right )\) | \(64\) |
risch | \(\frac {d}{12 x +24}-\frac {e}{6 \left (x +2\right )}+\frac {\ln \left (2-x \right ) d}{48}+\frac {\ln \left (2-x \right ) e}{24}-\frac {\ln \left (-1+x \right ) d}{18}-\frac {\ln \left (-1+x \right ) e}{18}+\frac {\ln \left (1+x \right ) d}{6}-\frac {\ln \left (1+x \right ) e}{6}-\frac {19 \ln \left (-x -2\right ) d}{144}+\frac {13 \ln \left (-x -2\right ) e}{72}\) | \(82\) |
norman | \(\frac {\left (-\frac {d}{12}+\frac {e}{6}\right ) x +\left (\frac {d}{12}-\frac {e}{6}\right ) x^{3}+\left (-\frac {d}{6}+\frac {e}{3}\right ) x^{2}+\frac {d}{6}-\frac {e}{3}}{x^{4}-5 x^{2}+4}+\left (-\frac {19 d}{144}+\frac {13 e}{72}\right ) \ln \left (x +2\right )+\left (-\frac {d}{18}-\frac {e}{18}\right ) \ln \left (-1+x \right )+\left (\frac {d}{6}-\frac {e}{6}\right ) \ln \left (1+x \right )+\left (\frac {d}{48}+\frac {e}{24}\right ) \ln \left (x -2\right )\) | \(101\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 62, normalized size = 0.87 \begin {gather*} -\frac {1}{144} \, {\left (19 \, d - 26 \, e\right )} \log \left (x + 2\right ) + \frac {1}{6} \, {\left (d - e\right )} \log \left (x + 1\right ) - \frac {1}{18} \, {\left (d + e\right )} \log \left (x - 1\right ) + \frac {1}{48} \, {\left (d + 2 \, e\right )} \log \left (x - 2\right ) + \frac {d - 2 \, e}{12 \, {\left (x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 93, normalized size = 1.31 \begin {gather*} -\frac {{\left ({\left (19 \, d - 26 \, e\right )} x + 38 \, d - 52 \, e\right )} \log \left (x + 2\right ) - 24 \, {\left ({\left (d - e\right )} x + 2 \, d - 2 \, e\right )} \log \left (x + 1\right ) + 8 \, {\left ({\left (d + e\right )} x + 2 \, d + 2 \, e\right )} \log \left (x - 1\right ) - 3 \, {\left ({\left (d + 2 \, e\right )} x + 2 \, d + 4 \, e\right )} \log \left (x - 2\right ) - 12 \, d + 24 \, e}{144 \, {\left (x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1188 vs.
\(2 (60) = 120\).
time = 7.55, size = 1188, normalized size = 16.73 \begin {gather*} \frac {d - 2 e}{12 x + 24} + \frac {\left (d - e\right ) \log {\left (x + \frac {- 1534775 d^{6} + 8032360 d^{5} e - 984027 d^{5} \left (d - e\right ) - 12991180 d^{4} e^{2} + 11797266 d^{4} e \left (d - e\right ) + 3567168 d^{4} \left (d - e\right )^{2} + 1075200 d^{3} e^{3} - 32721528 d^{3} e^{2} \left (d - e\right ) - 8725248 d^{3} e \left (d - e\right )^{2} - 247104 d^{3} \left (d - e\right )^{3} + 16959280 d^{2} e^{4} + 38977296 d^{2} e^{3} \left (d - e\right ) - 2820096 d^{2} e^{2} \left (d - e\right )^{2} - 10357632 d^{2} e \left (d - e\right )^{3} - 15836800 d e^{5} - 21294960 d e^{4} \left (d - e\right ) + 15436800 d e^{3} \left (d - e\right )^{2} + 16277760 d e^{2} \left (d - e\right )^{3} + 4283840 e^{6} + 3876000 e^{5} \left (d - e\right ) - 6865920 e^{4} \left (d - e\right )^{2} - 4078080 e^{3} \left (d - e\right )^{3}}{801262 d^{6} - 4662251 d^{5} e + 7296938 d^{4} e^{2} + 1388616 d^{3} e^{3} - 12447440 d^{2} e^{4} + 9990800 d e^{5} - 2380000 e^{6}} \right )}}{6} - \frac {\left (d + e\right ) \log {\left (x + \frac {- 1534775 d^{6} + 8032360 d^{5} e + 328009 d^{5} \left (d + e\right ) - 12991180 d^{4} e^{2} - 3932422 d^{4} e \left (d + e\right ) + 396352 d^{4} \left (d + e\right )^{2} + 1075200 d^{3} e^{3} + 10907176 d^{3} e^{2} \left (d + e\right ) - 969472 d^{3} e \left (d + e\right )^{2} + 9152 d^{3} \left (d + e\right )^{3} + 16959280 d^{2} e^{4} - 12992432 d^{2} e^{3} \left (d + e\right ) - 313344 d^{2} e^{2} \left (d + e\right )^{2} + 383616 d^{2} e \left (d + e\right )^{3} - 15836800 d e^{5} + 7098320 d e^{4} \left (d + e\right ) + 1715200 d e^{3} \left (d + e\right )^{2} - 602880 d e^{2} \left (d + e\right )^{3} + 4283840 e^{6} - 1292000 e^{5} \left (d + e\right ) - 762880 e^{4} \left (d + e\right )^{2} + 151040 e^{3} \left (d + e\right )^{3}}{801262 d^{6} - 4662251 d^{5} e + 7296938 d^{4} e^{2} + 1388616 d^{3} e^{3} - 12447440 d^{2} e^{4} + 9990800 d e^{5} - 2380000 e^{6}} \right )}}{18} + \frac {\left (d + 2 e\right ) \log {\left (x + \frac {- 1534775 d^{6} + 8032360 d^{5} e - \frac {984027 d^{5} \left (d + 2 e\right )}{8} - 12991180 d^{4} e^{2} + \frac {5898633 d^{4} e \left (d + 2 e\right )}{4} + 55737 d^{4} \left (d + 2 e\right )^{2} + 1075200 d^{3} e^{3} - 4090191 d^{3} e^{2} \left (d + 2 e\right ) - 136332 d^{3} e \left (d + 2 e\right )^{2} - \frac {3861 d^{3} \left (d + 2 e\right )^{3}}{8} + 16959280 d^{2} e^{4} + 4872162 d^{2} e^{3} \left (d + 2 e\right ) - 44064 d^{2} e^{2} \left (d + 2 e\right )^{2} - \frac {80919 d^{2} e \left (d + 2 e\right )^{3}}{4} - 15836800 d e^{5} - 2661870 d e^{4} \left (d + 2 e\right ) + 241200 d e^{3} \left (d + 2 e\right )^{2} + \frac {63585 d e^{2} \left (d + 2 e\right )^{3}}{2} + 4283840 e^{6} + 484500 e^{5} \left (d + 2 e\right ) - 107280 e^{4} \left (d + 2 e\right )^{2} - 7965 e^{3} \left (d + 2 e\right )^{3}}{801262 d^{6} - 4662251 d^{5} e + 7296938 d^{4} e^{2} + 1388616 d^{3} e^{3} - 12447440 d^{2} e^{4} + 9990800 d e^{5} - 2380000 e^{6}} \right )}}{48} - \frac {\left (19 d - 26 e\right ) \log {\left (x + \frac {- 1534775 d^{6} + 8032360 d^{5} e + \frac {328009 d^{5} \cdot \left (19 d - 26 e\right )}{8} - 12991180 d^{4} e^{2} - \frac {1966211 d^{4} e \left (19 d - 26 e\right )}{4} + 6193 d^{4} \left (19 d - 26 e\right )^{2} + 1075200 d^{3} e^{3} + 1363397 d^{3} e^{2} \cdot \left (19 d - 26 e\right ) - 15148 d^{3} e \left (19 d - 26 e\right )^{2} + \frac {143 d^{3} \left (19 d - 26 e\right )^{3}}{8} + 16959280 d^{2} e^{4} - 1624054 d^{2} e^{3} \cdot \left (19 d - 26 e\right ) - 4896 d^{2} e^{2} \left (19 d - 26 e\right )^{2} + \frac {2997 d^{2} e \left (19 d - 26 e\right )^{3}}{4} - 15836800 d e^{5} + 887290 d e^{4} \cdot \left (19 d - 26 e\right ) + 26800 d e^{3} \left (19 d - 26 e\right )^{2} - \frac {2355 d e^{2} \left (19 d - 26 e\right )^{3}}{2} + 4283840 e^{6} - 161500 e^{5} \cdot \left (19 d - 26 e\right ) - 11920 e^{4} \left (19 d - 26 e\right )^{2} + 295 e^{3} \left (19 d - 26 e\right )^{3}}{801262 d^{6} - 4662251 d^{5} e + 7296938 d^{4} e^{2} + 1388616 d^{3} e^{3} - 12447440 d^{2} e^{4} + 9990800 d e^{5} - 2380000 e^{6}} \right )}}{144} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.77, size = 66, normalized size = 0.93 \begin {gather*} -\frac {1}{144} \, {\left (19 \, d - 26 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{6} \, {\left (d - e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{18} \, {\left (d + e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{48} \, {\left (d + 2 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) + \frac {d - 2 \, e}{12 \, {\left (x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.81, size = 64, normalized size = 0.90 \begin {gather*} \frac {\frac {d}{12}-\frac {e}{6}}{x+2}+\ln \left (x+1\right )\,\left (\frac {d}{6}-\frac {e}{6}\right )-\ln \left (x-1\right )\,\left (\frac {d}{18}+\frac {e}{18}\right )+\ln \left (x-2\right )\,\left (\frac {d}{48}+\frac {e}{24}\right )-\ln \left (x+2\right )\,\left (\frac {19\,d}{144}-\frac {13\,e}{72}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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