Optimal. Leaf size=25 \[ 4+x^2-\log \left (\frac {5 \left (-3+e^4+\frac {x}{2}\right )}{3 x}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 22, normalized size of antiderivative = 0.88, number of steps used = 4, number of rules used = 3, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {6, 1593, 1620} \begin {gather*} x^2-\log \left (2 \left (3-e^4\right )-x\right )+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 1593
Rule 1620
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-6-12 x^2+2 x^3+e^4 \left (2+4 x^2\right )}{\left (-6+2 e^4\right ) x+x^2} \, dx\\ &=\int \frac {-6-12 x^2+2 x^3+e^4 \left (2+4 x^2\right )}{x \left (-6+2 e^4+x\right )} \, dx\\ &=\int \left (\frac {1}{6-2 e^4-x}+\frac {1}{x}+2 x\right ) \, dx\\ &=x^2-\log \left (2 \left (3-e^4\right )-x\right )+\log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 1.24 \begin {gather*} 2 \left (\frac {x^2}{2}-\frac {1}{2} \log \left (6-2 e^4-x\right )+\frac {\log (x)}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 16, normalized size = 0.64 \begin {gather*} x^{2} - \log \left (x + 2 \, e^{4} - 6\right ) + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 18, normalized size = 0.72 \begin {gather*} x^{2} - \log \left ({\left | x + 2 \, e^{4} - 6 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 17, normalized size = 0.68
method | result | size |
default | \(x^{2}-\ln \left (2 \,{\mathrm e}^{4}+x -6\right )+\ln \relax (x )\) | \(17\) |
norman | \(x^{2}-\ln \left (2 \,{\mathrm e}^{4}+x -6\right )+\ln \relax (x )\) | \(17\) |
risch | \(x^{2}-\ln \left (2 \,{\mathrm e}^{4}+x -6\right )+\ln \relax (x )\) | \(17\) |
meijerg | \(\left (4 \,{\mathrm e}^{4}-12\right ) \left (2 \,{\mathrm e}^{4}-6\right ) \left (\frac {x}{2 \,{\mathrm e}^{4}-6}-\ln \left (1+\frac {x}{2 \,{\mathrm e}^{4}-6}\right )\right )+8 \left ({\mathrm e}^{4}-3\right )^{2} \left (-\frac {x \left (-\frac {3 x}{2 \left ({\mathrm e}^{4}-3\right )}+6\right )}{12 \left ({\mathrm e}^{4}-3\right )}+\ln \left (1+\frac {x}{2 \,{\mathrm e}^{4}-6}\right )\right )+\frac {{\mathrm e}^{4} \left (-\ln \left (1+\frac {x}{2 \,{\mathrm e}^{4}-6}\right )+\ln \relax (x )-\ln \relax (2)-\ln \left ({\mathrm e}^{4}-3\right )\right )}{{\mathrm e}^{4}-3}-\frac {3 \left (-\ln \left (1+\frac {x}{2 \,{\mathrm e}^{4}-6}\right )+\ln \relax (x )-\ln \relax (2)-\ln \left ({\mathrm e}^{4}-3\right )\right )}{{\mathrm e}^{4}-3}\) | \(153\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 16, normalized size = 0.64 \begin {gather*} x^{2} - \log \left (x + 2 \, e^{4} - 6\right ) + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 20, normalized size = 0.80 \begin {gather*} x^2-2\,\mathrm {atanh}\left (\frac {4\,x}{4\,{\mathrm {e}}^4-12}+1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 15, normalized size = 0.60 \begin {gather*} x^{2} + \log {\relax (x )} - \log {\left (x - 6 + 2 e^{4} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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