Optimal. Leaf size=40 \[ \frac {2 x-\frac {2-x \left (\frac {3}{x}-\frac {e^x x}{3 \log (x)}\right )}{4+3 x}}{x} \]
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Rubi [A] time = 0.92, antiderivative size = 43, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, integrand size = 71, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.085, Rules used = {1594, 27, 12, 6742, 74, 2288} \begin {gather*} \frac {1}{x (3 x+4)}-\frac {e^x \left (3 x^2 \log (x)+4 x \log (x)\right )}{3 (3 x+4)^2 \log ^2(x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 74
Rule 1594
Rule 2288
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x \left (4 x^2+3 x^3\right )+e^x \left (-4 x^2-4 x^3-3 x^4\right ) \log (x)+(-12-18 x) \log ^2(x)}{x^2 \left (48+72 x+27 x^2\right ) \log ^2(x)} \, dx\\ &=\int \frac {e^x \left (4 x^2+3 x^3\right )+e^x \left (-4 x^2-4 x^3-3 x^4\right ) \log (x)+(-12-18 x) \log ^2(x)}{3 x^2 (4+3 x)^2 \log ^2(x)} \, dx\\ &=\frac {1}{3} \int \frac {e^x \left (4 x^2+3 x^3\right )+e^x \left (-4 x^2-4 x^3-3 x^4\right ) \log (x)+(-12-18 x) \log ^2(x)}{x^2 (4+3 x)^2 \log ^2(x)} \, dx\\ &=\frac {1}{3} \int \left (-\frac {6 (2+3 x)}{x^2 (4+3 x)^2}-\frac {e^x \left (-4-3 x+4 \log (x)+4 x \log (x)+3 x^2 \log (x)\right )}{(4+3 x)^2 \log ^2(x)}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {e^x \left (-4-3 x+4 \log (x)+4 x \log (x)+3 x^2 \log (x)\right )}{(4+3 x)^2 \log ^2(x)} \, dx\right )-2 \int \frac {2+3 x}{x^2 (4+3 x)^2} \, dx\\ &=\frac {1}{x (4+3 x)}-\frac {e^x \left (4 x \log (x)+3 x^2 \log (x)\right )}{3 (4+3 x)^2 \log ^2(x)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.17, size = 31, normalized size = 0.78 \begin {gather*} -\frac {e^x x^2-3 \log (x)}{3 \left (4 x \log (x)+3 x^2 \log (x)\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 28, normalized size = 0.70 \begin {gather*} -\frac {x^{2} e^{x} - 3 \, \log \relax (x)}{3 \, {\left (3 \, x^{2} + 4 \, x\right )} \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 28, normalized size = 0.70 \begin {gather*} -\frac {x^{2} e^{x} - 3 \, \log \relax (x)}{3 \, {\left (3 \, x^{2} \log \relax (x) + 4 \, x \log \relax (x)\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 29, normalized size = 0.72
method | result | size |
risch | \(\frac {1}{\left (4+3 x \right ) x}-\frac {x \,{\mathrm e}^{x}}{3 \left (4+3 x \right ) \ln \relax (x )}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 28, normalized size = 0.70 \begin {gather*} -\frac {x^{2} e^{x} - 3 \, \log \relax (x)}{3 \, {\left (3 \, x^{2} + 4 \, x\right )} \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.40, size = 28, normalized size = 0.70 \begin {gather*} \frac {3\,\ln \relax (x)-x^2\,{\mathrm {e}}^x}{3\,x\,\ln \relax (x)\,\left (3\,x+4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.35, size = 26, normalized size = 0.65 \begin {gather*} - \frac {x e^{x}}{9 x \log {\relax (x )} + 12 \log {\relax (x )}} + \frac {1}{3 x^{2} + 4 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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