Optimal. Leaf size=33 \[ x+\frac {1}{3} e \left (2-e^x-x^2+e^2 \left (5+x \log \left (\frac {x}{4}\right )\right )\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.52, number of steps
used = 4, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {12, 2225, 2332}
\begin {gather*} -\frac {e x^2}{3}+\frac {1}{3} \left (3+e^3\right ) x-\frac {e^3 x}{3}-\frac {e^{x+1}}{3}+\frac {1}{3} e^3 x \log \left (\frac {x}{4}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2225
Rule 2332
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \left (3+e^3-e^{1+x}-2 e x+e^3 \log \left (\frac {x}{4}\right )\right ) \, dx\\ &=\frac {1}{3} \left (3+e^3\right ) x-\frac {e x^2}{3}-\frac {1}{3} \int e^{1+x} \, dx+\frac {1}{3} e^3 \int \log \left (\frac {x}{4}\right ) \, dx\\ &=-\frac {e^{1+x}}{3}-\frac {e^3 x}{3}+\frac {1}{3} \left (3+e^3\right ) x-\frac {e x^2}{3}+\frac {1}{3} e^3 x \log \left (\frac {x}{4}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 32, normalized size = 0.97 \begin {gather*} \frac {1}{3} \left (-e^{1+x}+3 x-e x^2+e^3 x \log \left (\frac {x}{4}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 27, normalized size = 0.82
method | result | size |
risch | \(\frac {{\mathrm e}^{3} x \ln \left (\frac {x}{4}\right )}{3}-\frac {{\mathrm e}^{x +1}}{3}-\frac {x^{2} {\mathrm e}}{3}+x\) | \(25\) |
default | \(x -\frac {x^{2} {\mathrm e}}{3}-\frac {{\mathrm e} \,{\mathrm e}^{x}}{3}+\frac {{\mathrm e}^{2} {\mathrm e} \ln \left (\frac {x}{4}\right ) x}{3}\) | \(27\) |
norman | \(x -\frac {x^{2} {\mathrm e}}{3}-\frac {{\mathrm e} \,{\mathrm e}^{x}}{3}+\frac {{\mathrm e}^{2} {\mathrm e} \ln \left (\frac {x}{4}\right ) x}{3}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 34, normalized size = 1.03 \begin {gather*} -\frac {1}{3} \, x^{2} e + \frac {1}{3} \, {\left (x \log \left (\frac {1}{4} \, x\right ) - x\right )} e^{3} + \frac {1}{3} \, x e^{3} + x - \frac {1}{3} \, e^{\left (x + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 24, normalized size = 0.73 \begin {gather*} -\frac {1}{3} \, x^{2} e + \frac {1}{3} \, x e^{3} \log \left (\frac {1}{4} \, x\right ) + x - \frac {1}{3} \, e^{\left (x + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 29, normalized size = 0.88 \begin {gather*} - \frac {e x^{2}}{3} + \frac {x e^{3} \log {\left (\frac {x}{4} \right )}}{3} + x - \frac {e e^{x}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 34, normalized size = 1.03 \begin {gather*} -\frac {1}{3} \, x^{2} e + \frac {1}{3} \, {\left (x \log \left (\frac {1}{4} \, x\right ) - x\right )} e^{3} + \frac {1}{3} \, x e^{3} + x - \frac {1}{3} \, e^{\left (x + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.08, size = 29, normalized size = 0.88 \begin {gather*} x-\frac {{\mathrm {e}}^{x+1}}{3}-\frac {x^2\,\mathrm {e}}{3}+\frac {x\,{\mathrm {e}}^3\,\ln \left (x\right )}{3}-\frac {2\,x\,{\mathrm {e}}^3\,\ln \left (2\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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