Optimal. Leaf size=21 \[ -1+\frac {e^x}{2}+\log \left (\frac {-e^3+x}{x}\right ) \]
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Rubi [A] time = 0.25, antiderivative size = 20, normalized size of antiderivative = 0.95, number of steps used = 7, number of rules used = 6, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1593, 6742, 2194, 36, 31, 29} \begin {gather*} \frac {e^x}{2}+\log \left (e^3-x\right )-\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 1593
Rule 2194
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2 e^3+e^x \left (e^3 x-x^2\right )}{\left (2 e^3-2 x\right ) x} \, dx\\ &=\int \left (\frac {e^x}{2}-\frac {e^3}{\left (e^3-x\right ) x}\right ) \, dx\\ &=\frac {\int e^x \, dx}{2}-e^3 \int \frac {1}{\left (e^3-x\right ) x} \, dx\\ &=\frac {e^x}{2}-\int \frac {1}{e^3-x} \, dx-\int \frac {1}{x} \, dx\\ &=\frac {e^x}{2}+\log \left (e^3-x\right )-\log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.04, size = 20, normalized size = 0.95 \begin {gather*} \frac {e^x}{2}+\log \left (e^3-x\right )-\log (x) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 16, normalized size = 0.76 \begin {gather*} \frac {1}{2} \, e^{x} + \log \left (x - e^{3}\right ) - \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 16, normalized size = 0.76 \begin {gather*} \frac {1}{2} \, e^{x} + \log \left (x - e^{3}\right ) - \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 17, normalized size = 0.81
method | result | size |
default | \(-\ln \relax (x )+\ln \left (x -{\mathrm e}^{3}\right )+\frac {{\mathrm e}^{x}}{2}\) | \(17\) |
norman | \(\frac {{\mathrm e}^{x}}{2}-\ln \relax (x )+\ln \left (-x +{\mathrm e}^{3}\right )\) | \(17\) |
risch | \(-\ln \relax (x )+\ln \left (x -{\mathrm e}^{3}\right )+\frac {{\mathrm e}^{x}}{2}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 25, normalized size = 1.19 \begin {gather*} {\left (e^{\left (-3\right )} \log \left (x - e^{3}\right ) - e^{\left (-3\right )} \log \relax (x)\right )} e^{3} + \frac {1}{2} \, e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 16, normalized size = 0.76 \begin {gather*} \ln \left (x-{\mathrm {e}}^3\right )+\frac {{\mathrm {e}}^x}{2}-\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 24, normalized size = 1.14 \begin {gather*} \left (- \frac {\log {\relax (x )}}{e^{3}} + \frac {\log {\left (x - e^{3} \right )}}{e^{3}}\right ) e^{3} + \frac {e^{x}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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