Optimal. Leaf size=20 \[ e^x-2 x \log (-3+x)-\frac {\log (x)}{e^{3/2}} \]
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Rubi [A] time = 0.93, antiderivative size = 32, normalized size of antiderivative = 1.60, number of steps used = 11, number of rules used = 7, integrand size = 64, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {12, 1593, 6742, 2194, 893, 2389, 2295} \begin {gather*} e^x-6 \log (3-x)+2 (3-x) \log (x-3)-\frac {\log (x)}{e^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 893
Rule 1593
Rule 2194
Rule 2295
Rule 2389
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {3-x-2 e^{3/2} x^2+e^{\frac {3}{2}+x} \left (-3 x+x^2\right )+e^{3/2} \left (6 x-2 x^2\right ) \log (-3+x)}{-3 x+x^2} \, dx}{e^{3/2}}\\ &=\frac {\int \frac {3-x-2 e^{3/2} x^2+e^{\frac {3}{2}+x} \left (-3 x+x^2\right )+e^{3/2} \left (6 x-2 x^2\right ) \log (-3+x)}{(-3+x) x} \, dx}{e^{3/2}}\\ &=\frac {\int \left (e^{\frac {3}{2}+x}+\frac {3-x-2 e^{3/2} x^2+6 e^{3/2} x \log (-3+x)-2 e^{3/2} x^2 \log (-3+x)}{(-3+x) x}\right ) \, dx}{e^{3/2}}\\ &=\frac {\int e^{\frac {3}{2}+x} \, dx}{e^{3/2}}+\frac {\int \frac {3-x-2 e^{3/2} x^2+6 e^{3/2} x \log (-3+x)-2 e^{3/2} x^2 \log (-3+x)}{(-3+x) x} \, dx}{e^{3/2}}\\ &=e^x+\frac {\int \left (\frac {3-x-2 e^{3/2} x^2}{(-3+x) x}-2 e^{3/2} \log (-3+x)\right ) \, dx}{e^{3/2}}\\ &=e^x-2 \int \log (-3+x) \, dx+\frac {\int \frac {3-x-2 e^{3/2} x^2}{(-3+x) x} \, dx}{e^{3/2}}\\ &=e^x-2 \operatorname {Subst}(\int \log (x) \, dx,x,-3+x)+\frac {\int \left (-2 e^{3/2}-\frac {6 e^{3/2}}{-3+x}-\frac {1}{x}\right ) \, dx}{e^{3/2}}\\ &=e^x-6 \log (3-x)+2 (3-x) \log (-3+x)-\frac {\log (x)}{e^{3/2}}\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.22, size = 47, normalized size = 2.35 \begin {gather*} \frac {e^{\frac {3}{2}+x}-6 e^{3/2} \log (3-x)+2 e^{3/2} (3-x) \log (-3+x)-\log (x)}{e^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 22, normalized size = 1.10 \begin {gather*} -{\left (2 \, x e^{\frac {3}{2}} \log \left (x - 3\right ) - e^{\left (x + \frac {3}{2}\right )} + \log \relax (x)\right )} e^{\left (-\frac {3}{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 22, normalized size = 1.10 \begin {gather*} -{\left (2 \, x e^{\frac {3}{2}} \log \left (x - 3\right ) - e^{\left (x + \frac {3}{2}\right )} + \log \relax (x)\right )} e^{\left (-\frac {3}{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 17, normalized size = 0.85
method | result | size |
risch | \({\mathrm e}^{x}-\ln \relax (x ) {\mathrm e}^{-\frac {3}{2}}-2 \ln \left (x -3\right ) x\) | \(17\) |
norman | \({\mathrm e}^{x}-\ln \relax (x ) {\mathrm e}^{-\frac {3}{2}}-2 \ln \left (x -3\right ) x\) | \(19\) |
default | \({\mathrm e}^{-\frac {3}{2}} \left ({\mathrm e}^{\frac {3}{2}} {\mathrm e}^{x}-2 \,{\mathrm e}^{\frac {3}{2}} \ln \left (x -3\right ) x -6 \,{\mathrm e}^{\frac {3}{2}}-\ln \relax (x )\right )\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -{\left (2 \, {\left (x + 3 \, \log \left (x - 3\right )\right )} e^{\frac {3}{2}} \log \left (x - 3\right ) - 3 \, e^{\frac {3}{2}} \log \left (x - 3\right )^{2} - {\left (3 \, \log \left (x - 3\right )^{2} + 2 \, x + 6 \, \log \left (x - 3\right )\right )} e^{\frac {3}{2}} + 2 \, {\left (x + 3 \, \log \left (x - 3\right )\right )} e^{\frac {3}{2}} - 3 \, e^{\frac {9}{2}} E_{1}\left (-x + 3\right ) - \frac {x e^{\left (x + \frac {3}{2}\right )}}{x - 3} - 3 \, \int \frac {e^{\left (x + \frac {3}{2}\right )}}{x^{2} - 6 \, x + 9}\,{d x} + \log \relax (x)\right )} e^{\left (-\frac {3}{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 16, normalized size = 0.80 \begin {gather*} {\mathrm {e}}^x-2\,x\,\ln \left (x-3\right )-{\mathrm {e}}^{-\frac {3}{2}}\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.36, size = 19, normalized size = 0.95 \begin {gather*} - 2 x \log {\left (x - 3 \right )} + e^{x} - \frac {\log {\relax (x )}}{e^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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