Optimal. Leaf size=21 \[ \log \left (4-x+\frac {2 e^3 \left (\frac {1}{10}+x^2\right )}{x^2}\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 25, normalized size of antiderivative = 1.19, 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 = {2074, 1587} \begin {gather*} \log \left (-5 x^3+10 \left (2+e^3\right ) x^2+e^3\right )-2 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 1587
Rule 2074
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2}{x}+\frac {5 \left (8+4 e^3-3 x\right ) x}{e^3+10 \left (2+e^3\right ) x^2-5 x^3}\right ) \, dx\\ &=-2 \log (x)+5 \int \frac {\left (8+4 e^3-3 x\right ) x}{e^3+10 \left (2+e^3\right ) x^2-5 x^3} \, dx\\ &=-2 \log (x)+\log \left (e^3+10 \left (2+e^3\right ) x^2-5 x^3\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 28, normalized size = 1.33 \begin {gather*} -2 \log (x)+\log \left (e^3+20 x^2+10 e^3 x^2-5 x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 28, normalized size = 1.33 \begin {gather*} \log \left (5 \, x^{3} - 20 \, x^{2} - {\left (10 \, x^{2} + 1\right )} e^{3}\right ) - 2 \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 30, normalized size = 1.43 \begin {gather*} \log \left ({\left | 5 \, x^{3} - 10 \, x^{2} e^{3} - 20 \, x^{2} - e^{3} \right |}\right ) - 2 \, \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 = 0.06, size = 27, normalized size = 1.29
method | result | size |
norman | \(-2 \ln \relax (x )+\ln \left (10 x^{2} {\mathrm e}^{3}-5 x^{3}+20 x^{2}+{\mathrm e}^{3}\right )\) | \(27\) |
default | \(-2 \ln \relax (x )+\ln \left (-10 x^{2} {\mathrm e}^{3}+5 x^{3}-20 x^{2}-{\mathrm e}^{3}\right )\) | \(29\) |
risch | \(-2 \ln \left (-x \right )+\ln \left (5 x^{3}+\left (-10 \,{\mathrm e}^{3}-20\right ) x^{2}-{\mathrm e}^{3}\right )\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 25, normalized size = 1.19 \begin {gather*} \log \left (5 \, x^{3} - 10 \, x^{2} {\left (e^{3} + 2\right )} - e^{3}\right ) - 2 \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 26, normalized size = 1.24 \begin {gather*} \ln \left (x^3-2\,x^2\,{\mathrm {e}}^3-4\,x^2-\frac {{\mathrm {e}}^3}{5}\right )-2\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.06, size = 26, normalized size = 1.24 \begin {gather*} - 2 \log {\relax (x )} + \log {\left (x^{3} + x^{2} \left (- 2 e^{3} - 4\right ) - \frac {e^{3}}{5} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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