Optimal. Leaf size=23 \[ 2 e^{2 x}+\frac {\log \left (e^x+3 x\right )}{3 x} \]
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Rubi [A] time = 0.94, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, integrand size = 64, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {6741, 12, 6742, 2194, 14, 2551} \begin {gather*} 2 e^{2 x}+\frac {\log \left (3 x+e^x\right )}{3 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2194
Rule 2551
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 x+e^x x+e^{2 x} \left (12 e^x x^2+36 x^3\right )+\left (-e^x-3 x\right ) \log \left (e^x+3 x\right )}{3 x^2 \left (e^x+3 x\right )} \, dx\\ &=\frac {1}{3} \int \frac {3 x+e^x x+e^{2 x} \left (12 e^x x^2+36 x^3\right )+\left (-e^x-3 x\right ) \log \left (e^x+3 x\right )}{x^2 \left (e^x+3 x\right )} \, dx\\ &=\frac {1}{3} \int \left (12 e^{2 x}-\frac {3 (-1+x)}{x \left (e^x+3 x\right )}+\frac {x-\log \left (e^x+3 x\right )}{x^2}\right ) \, dx\\ &=\frac {1}{3} \int \frac {x-\log \left (e^x+3 x\right )}{x^2} \, dx+4 \int e^{2 x} \, dx-\int \frac {-1+x}{x \left (e^x+3 x\right )} \, dx\\ &=2 e^{2 x}+\frac {1}{3} \int \left (\frac {1}{x}-\frac {\log \left (e^x+3 x\right )}{x^2}\right ) \, dx-\int \left (\frac {1}{e^x+3 x}-\frac {1}{x \left (e^x+3 x\right )}\right ) \, dx\\ &=2 e^{2 x}+\frac {\log (x)}{3}-\frac {1}{3} \int \frac {\log \left (e^x+3 x\right )}{x^2} \, dx-\int \frac {1}{e^x+3 x} \, dx+\int \frac {1}{x \left (e^x+3 x\right )} \, dx\\ &=2 e^{2 x}+\frac {\log (x)}{3}+\frac {\log \left (e^x+3 x\right )}{3 x}-\frac {1}{3} \int \frac {3+e^x}{e^x x+3 x^2} \, dx-\int \frac {1}{e^x+3 x} \, dx+\int \frac {1}{x \left (e^x+3 x\right )} \, dx\\ &=2 e^{2 x}+\frac {\log (x)}{3}+\frac {\log \left (e^x+3 x\right )}{3 x}-\frac {1}{3} \int \left (\frac {1}{x}-\frac {3 (-1+x)}{x \left (e^x+3 x\right )}\right ) \, dx-\int \frac {1}{e^x+3 x} \, dx+\int \frac {1}{x \left (e^x+3 x\right )} \, dx\\ &=2 e^{2 x}+\frac {\log \left (e^x+3 x\right )}{3 x}-\int \frac {1}{e^x+3 x} \, dx+\int \frac {1}{x \left (e^x+3 x\right )} \, dx+\int \frac {-1+x}{x \left (e^x+3 x\right )} \, dx\\ &=2 e^{2 x}+\frac {\log \left (e^x+3 x\right )}{3 x}-\int \frac {1}{e^x+3 x} \, dx+\int \frac {1}{x \left (e^x+3 x\right )} \, dx+\int \left (\frac {1}{e^x+3 x}-\frac {1}{x \left (e^x+3 x\right )}\right ) \, dx\\ &=2 e^{2 x}+\frac {\log \left (e^x+3 x\right )}{3 x}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.12, size = 24, normalized size = 1.04 \begin {gather*} \frac {1}{3} \left (6 e^{2 x}+\frac {\log \left (e^x+3 x\right )}{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 20, normalized size = 0.87 \begin {gather*} \frac {6 \, x e^{\left (2 \, x\right )} + \log \left (3 \, x + e^{x}\right )}{3 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 20, normalized size = 0.87 \begin {gather*} \frac {6 \, x e^{\left (2 \, x\right )} + \log \left (3 \, x + e^{x}\right )}{3 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 20, normalized size = 0.87
method | result | size |
risch | \(\frac {\ln \left (3 x +{\mathrm e}^{x}\right )}{3 x}+2 \,{\mathrm e}^{2 x}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 20, normalized size = 0.87 \begin {gather*} \frac {6 \, x e^{\left (2 \, x\right )} + \log \left (3 \, x + e^{x}\right )}{3 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.63, size = 19, normalized size = 0.83 \begin {gather*} 2\,{\mathrm {e}}^{2\,x}+\frac {\ln \left (3\,x+{\mathrm {e}}^x\right )}{3\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 17, normalized size = 0.74 \begin {gather*} 2 e^{2 x} + \frac {\log {\left (3 x + e^{x} \right )}}{3 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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