Optimal. Leaf size=24 \[ -x-\frac {1}{3} \left (\left (4+2 e^x\right )^2-x\right ) x+\log (x) \]
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Rubi [B] time = 0.05, antiderivative size = 55, normalized size of antiderivative = 2.29, number of steps used = 9, number of rules used = 4, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {12, 14, 2176, 2194} \begin {gather*} \frac {x^2}{3}-\frac {19 x}{3}+\frac {16 e^x}{3}+\frac {2 e^{2 x}}{3}-\frac {16}{3} e^x (x+1)-\frac {2}{3} e^{2 x} (2 x+1)+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2176
Rule 2194
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {3-19 x+2 x^2+e^x \left (-16 x-16 x^2\right )+e^{2 x} \left (-4 x-8 x^2\right )}{x} \, dx\\ &=\frac {1}{3} \int \left (-16 e^x (1+x)-4 e^{2 x} (1+2 x)+\frac {3-19 x+2 x^2}{x}\right ) \, dx\\ &=\frac {1}{3} \int \frac {3-19 x+2 x^2}{x} \, dx-\frac {4}{3} \int e^{2 x} (1+2 x) \, dx-\frac {16}{3} \int e^x (1+x) \, dx\\ &=-\frac {16}{3} e^x (1+x)-\frac {2}{3} e^{2 x} (1+2 x)+\frac {1}{3} \int \left (-19+\frac {3}{x}+2 x\right ) \, dx+\frac {4}{3} \int e^{2 x} \, dx+\frac {16 \int e^x \, dx}{3}\\ &=\frac {16 e^x}{3}+\frac {2 e^{2 x}}{3}-\frac {19 x}{3}+\frac {x^2}{3}-\frac {16}{3} e^x (1+x)-\frac {2}{3} e^{2 x} (1+2 x)+\log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.03, size = 29, normalized size = 1.21 \begin {gather*} \frac {1}{3} \left (-19 x-16 e^x x-4 e^{2 x} x+x^2+3 \log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.39, size = 23, normalized size = 0.96 \begin {gather*} \frac {1}{3} \, x^{2} - \frac {4}{3} \, x e^{\left (2 \, x\right )} - \frac {16}{3} \, x e^{x} - \frac {19}{3} \, x + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 23, normalized size = 0.96 \begin {gather*} \frac {1}{3} \, x^{2} - \frac {4}{3} \, x e^{\left (2 \, x\right )} - \frac {16}{3} \, x e^{x} - \frac {19}{3} \, x + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 24, normalized size = 1.00
method | result | size |
default | \(\frac {x^{2}}{3}-\frac {19 x}{3}+\ln \relax (x )-\frac {4 x \,{\mathrm e}^{2 x}}{3}-\frac {16 \,{\mathrm e}^{x} x}{3}\) | \(24\) |
norman | \(\frac {x^{2}}{3}-\frac {19 x}{3}+\ln \relax (x )-\frac {4 x \,{\mathrm e}^{2 x}}{3}-\frac {16 \,{\mathrm e}^{x} x}{3}\) | \(24\) |
risch | \(\frac {x^{2}}{3}-\frac {19 x}{3}+\ln \relax (x )-\frac {4 x \,{\mathrm e}^{2 x}}{3}-\frac {16 \,{\mathrm e}^{x} x}{3}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 39, normalized size = 1.62 \begin {gather*} \frac {1}{3} \, x^{2} - \frac {2}{3} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} - \frac {16}{3} \, {\left (x - 1\right )} e^{x} - \frac {19}{3} \, x - \frac {2}{3} \, e^{\left (2 \, x\right )} - \frac {16}{3} \, e^{x} + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 23, normalized size = 0.96 \begin {gather*} \ln \relax (x)-\frac {19\,x}{3}-\frac {4\,x\,{\mathrm {e}}^{2\,x}}{3}-\frac {16\,x\,{\mathrm {e}}^x}{3}+\frac {x^2}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 31, normalized size = 1.29 \begin {gather*} \frac {x^{2}}{3} - \frac {4 x e^{2 x}}{3} - \frac {16 x e^{x}}{3} - \frac {19 x}{3} + \log {\relax (x )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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