Optimal. Leaf size=74 \[ -\frac {1}{2} \log (1-x) (d+e+f+g+h)+\frac {1}{3} \log (2-x) (d+2 e+4 f+8 g+16 h)+\frac {1}{6} \log (x+1) (d-e+f-g+h)+x (g+2 h)+\frac {h x^2}{2} \]
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Rubi [A] time = 0.11, antiderivative size = 74, 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, 2074} \begin {gather*} -\frac {1}{2} \log (1-x) (d+e+f+g+h)+\frac {1}{3} \log (2-x) (d+2 e+4 f+8 g+16 h)+\frac {1}{6} \log (x+1) (d-e+f-g+h)+x (g+2 h)+\frac {h x^2}{2} \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+g x^3+h x^4\right )}{4-5 x^2+x^4} \, dx &=\int \frac {d+e x+f x^2+g x^3+h x^4}{2-x-2 x^2+x^3} \, dx\\ &=\int \left (g \left (1+\frac {2 h}{g}\right )+\frac {d+2 e+4 f+8 g+16 h}{3 (-2+x)}+\frac {-d-e-f-g-h}{2 (-1+x)}+h x+\frac {d-e+f-g+h}{6 (1+x)}\right ) \, dx\\ &=(g+2 h) x+\frac {h x^2}{2}-\frac {1}{2} (d+e+f+g+h) \log (1-x)+\frac {1}{3} (d+2 e+4 f+8 g+16 h) \log (2-x)+\frac {1}{6} (d-e+f-g+h) \log (1+x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 71, normalized size = 0.96 \begin {gather*} \frac {1}{6} \left (-3 \log (1-x) (d+e+f+g+h)+2 \log (2-x) (d+2 (e+2 f+4 g+8 h))+\log (x+1) (d-e+f-g+h)+6 x (g+2 h)+3 h x^2\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 {(2+x) \left (d+e x+f x^2+g x^3+h x^4\right )}{4-5 x^2+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.74, size = 62, normalized size = 0.84 \begin {gather*} \frac {1}{2} \, h x^{2} + {\left (g + 2 \, h\right )} x + \frac {1}{6} \, {\left (d - e + f - g + h\right )} \log \left (x + 1\right ) - \frac {1}{2} \, {\left (d + e + f + g + h\right )} \log \left (x - 1\right ) + \frac {1}{3} \, {\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h\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 = 0.33, size = 68, normalized size = 0.92 \begin {gather*} \frac {1}{2} \, h x^{2} + g x + 2 \, h x + \frac {1}{6} \, {\left (d + f - g + h - e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{2} \, {\left (d + f + g + h + e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{3} \, {\left (d + 4 \, f + 8 \, g + 16 \, h + 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 = 120, normalized size = 1.62 \begin {gather*} \frac {h \,x^{2}}{2}+\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}+g x +\frac {8 g \ln \left (x -2\right )}{3}-\frac {g \ln \left (x -1\right )}{2}-\frac {g \ln \left (x +1\right )}{6}+2 h x +\frac {16 h \ln \left (x -2\right )}{3}-\frac {h \ln \left (x -1\right )}{2}+\frac {h \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 = 0.45, size = 62, normalized size = 0.84 \begin {gather*} \frac {1}{2} \, h x^{2} + {\left (g + 2 \, h\right )} x + \frac {1}{6} \, {\left (d - e + f - g + h\right )} \log \left (x + 1\right ) - \frac {1}{2} \, {\left (d + e + f + g + h\right )} \log \left (x - 1\right ) + \frac {1}{3} \, {\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h\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 = 0.88, size = 78, normalized size = 1.05 \begin {gather*} x\,\left (g+2\,h\right )+\frac {h\,x^2}{2}-\ln \left (x-1\right )\,\left (\frac {d}{2}+\frac {e}{2}+\frac {f}{2}+\frac {g}{2}+\frac {h}{2}\right )+\ln \left (x+1\right )\,\left (\frac {d}{6}-\frac {e}{6}+\frac {f}{6}-\frac {g}{6}+\frac {h}{6}\right )+\ln \left (x-2\right )\,\left (\frac {d}{3}+\frac {2\,e}{3}+\frac {4\,f}{3}+\frac {8\,g}{3}+\frac {16\,h}{3}\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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