Optimal. Leaf size=47 \[ -\frac {1}{2} \log (1-x) (d+e+f)+\frac {1}{3} \log (2-x) (d+2 e+4 f)+\frac {1}{6} \log (x+1) (d-e+f) \]
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Rubi [A] time = 0.06, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1586, 2074} \begin {gather*} -\frac {1}{2} \log (1-x) (d+e+f)+\frac {1}{3} \log (2-x) (d+2 e+4 f)+\frac {1}{6} \log (x+1) (d-e+f) \end {gather*}
Antiderivative was successfully verified.
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Rule 1586
Rule 2074
Rubi steps
\begin {align*} \int \frac {(2+x) \left (d+e x+f x^2\right )}{4-5 x^2+x^4} \, dx &=\int \frac {d+e x+f x^2}{2-x-2 x^2+x^3} \, dx\\ &=\int \left (\frac {d+2 e+4 f}{3 (-2+x)}+\frac {-d-e-f}{2 (-1+x)}+\frac {d-e+f}{6 (1+x)}\right ) \, dx\\ &=-\frac {1}{2} (d+e+f) \log (1-x)+\frac {1}{3} (d+2 e+4 f) \log (2-x)+\frac {1}{6} (d-e+f) \log (1+x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 0.94 \begin {gather*} \frac {1}{6} (-3 \log (1-x) (d+e+f)+2 \log (2-x) (d+2 e+4 f)+\log (x+1) (d-e+f)) \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 {(2+x) \left (d+e x+f x^2\right )}{4-5 x^2+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.18, size = 37, normalized size = 0.79 \begin {gather*} \frac {1}{6} \, {\left (d - e + f\right )} \log \left (x + 1\right ) - \frac {1}{2} \, {\left (d + e + f\right )} \log \left (x - 1\right ) + \frac {1}{3} \, {\left (d + 2 \, e + 4 \, f\right )} \log \left (x - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 43, normalized size = 0.91 \begin {gather*} \frac {1}{6} \, {\left (d + f - e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{2} \, {\left (d + f + e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{3} \, {\left (d + 4 \, f + 2 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 65, normalized size = 1.38 \begin {gather*} \frac {d \ln \left (x -2\right )}{3}-\frac {d \ln \left (x -1\right )}{2}+\frac {d \ln \left (x +1\right )}{6}+\frac {2 e \ln \left (x -2\right )}{3}-\frac {e \ln \left (x -1\right )}{2}-\frac {e \ln \left (x +1\right )}{6}+\frac {4 f \ln \left (x -2\right )}{3}-\frac {f \ln \left (x -1\right )}{2}+\frac {f \ln \left (x +1\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 37, normalized size = 0.79 \begin {gather*} \frac {1}{6} \, {\left (d - e + f\right )} \log \left (x + 1\right ) - \frac {1}{2} \, {\left (d + e + f\right )} \log \left (x - 1\right ) + \frac {1}{3} \, {\left (d + 2 \, e + 4 \, f\right )} \log \left (x - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 47, normalized size = 1.00 \begin {gather*} \ln \left (x-2\right )\,\left (\frac {d}{3}+\frac {2\,e}{3}+\frac {4\,f}{3}\right )-\ln \left (x-1\right )\,\left (\frac {d}{2}+\frac {e}{2}+\frac {f}{2}\right )+\ln \left (x+1\right )\,\left (\frac {d}{6}-\frac {e}{6}+\frac {f}{6}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 12.72, size = 716, normalized size = 15.23 \begin {gather*} \frac {\left (d - e + f\right ) \log {\left (x + \frac {26 d^{3} + 66 d^{2} e + 132 d^{2} f - 9 d^{2} \left (d - e + f\right ) + 78 d e^{2} + 276 d e f - 12 d e \left (d - e + f\right ) + 222 d f^{2} + 6 d f \left (d - e + f\right ) - 7 d \left (d - e + f\right )^{2} + 46 e^{3} + 204 e^{2} f + 3 e^{2} \left (d - e + f\right ) + 282 e f^{2} + 36 e f \left (d - e + f\right ) - 8 e \left (d - e + f\right )^{2} + 116 f^{3} + 51 f^{2} \left (d - e + f\right ) - 13 f \left (d - e + f\right )^{2}}{10 d^{3} + 69 d^{2} e + 102 d^{2} f + 102 d e^{2} + 318 d e f + 246 d f^{2} + 35 e^{3} + 174 e^{2} f + 285 e f^{2} + 154 f^{3}} \right )}}{6} - \frac {\left (d + e + f\right ) \log {\left (x + \frac {26 d^{3} + 66 d^{2} e + 132 d^{2} f + 27 d^{2} \left (d + e + f\right ) + 78 d e^{2} + 276 d e f + 36 d e \left (d + e + f\right ) + 222 d f^{2} - 18 d f \left (d + e + f\right ) - 63 d \left (d + e + f\right )^{2} + 46 e^{3} + 204 e^{2} f - 9 e^{2} \left (d + e + f\right ) + 282 e f^{2} - 108 e f \left (d + e + f\right ) - 72 e \left (d + e + f\right )^{2} + 116 f^{3} - 153 f^{2} \left (d + e + f\right ) - 117 f \left (d + e + f\right )^{2}}{10 d^{3} + 69 d^{2} e + 102 d^{2} f + 102 d e^{2} + 318 d e f + 246 d f^{2} + 35 e^{3} + 174 e^{2} f + 285 e f^{2} + 154 f^{3}} \right )}}{2} + \frac {\left (d + 2 e + 4 f\right ) \log {\left (x + \frac {26 d^{3} + 66 d^{2} e + 132 d^{2} f - 18 d^{2} \left (d + 2 e + 4 f\right ) + 78 d e^{2} + 276 d e f - 24 d e \left (d + 2 e + 4 f\right ) + 222 d f^{2} + 12 d f \left (d + 2 e + 4 f\right ) - 28 d \left (d + 2 e + 4 f\right )^{2} + 46 e^{3} + 204 e^{2} f + 6 e^{2} \left (d + 2 e + 4 f\right ) + 282 e f^{2} + 72 e f \left (d + 2 e + 4 f\right ) - 32 e \left (d + 2 e + 4 f\right )^{2} + 116 f^{3} + 102 f^{2} \left (d + 2 e + 4 f\right ) - 52 f \left (d + 2 e + 4 f\right )^{2}}{10 d^{3} + 69 d^{2} e + 102 d^{2} f + 102 d e^{2} + 318 d e f + 246 d f^{2} + 35 e^{3} + 174 e^{2} f + 285 e f^{2} + 154 f^{3}} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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