Optimal. Leaf size=32 \[ \frac {e^{5+x \left (x-\left (3-e^3\right ) x-\frac {5 x^2}{9}\right )}+x}{\log (x)} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(78\) vs. \(2(32)=64\).
time = 1.09, antiderivative size = 78, normalized size of antiderivative = 2.44, number of steps
used = 9, number of rules used = 6, integrand size = 96, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {12, 6874, 2407,
2334, 2335, 2326} \begin {gather*} \frac {e^{-\frac {5 x^3}{9}-\left (2-e^3\right ) x^2+5} \left (5 x^3 \log (x)+6 \left (2-e^3\right ) x^2 \log (x)\right )}{\left (5 x^2+6 \left (2-e^3\right ) x\right ) x \log ^2(x)}+\frac {x}{\log (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2326
Rule 2334
Rule 2335
Rule 2407
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {-3 e^{\frac {1}{9} \left (45-18 x^2+9 e^3 x^2-5 x^3\right )}-3 x+\left (3 x+e^{\frac {1}{9} \left (45-18 x^2+9 e^3 x^2-5 x^3\right )} \left (-12 x^2+6 e^3 x^2-5 x^3\right )\right ) \log (x)}{x \log ^2(x)} \, dx\\ &=\frac {1}{3} \int \left (\frac {3 (-1+\log (x))}{\log ^2(x)}+\frac {e^{5-\left (2-e^3\right ) x^2-\frac {5 x^3}{9}} \left (-3-12 \left (1-\frac {e^3}{2}\right ) x^2 \log (x)-5 x^3 \log (x)\right )}{x \log ^2(x)}\right ) \, dx\\ &=\frac {1}{3} \int \frac {e^{5-\left (2-e^3\right ) x^2-\frac {5 x^3}{9}} \left (-3-12 \left (1-\frac {e^3}{2}\right ) x^2 \log (x)-5 x^3 \log (x)\right )}{x \log ^2(x)} \, dx+\int \frac {-1+\log (x)}{\log ^2(x)} \, dx\\ &=\frac {e^{5-\left (2-e^3\right ) x^2-\frac {5 x^3}{9}} \left (6 \left (2-e^3\right ) x^2 \log (x)+5 x^3 \log (x)\right )}{x \left (6 \left (2-e^3\right ) x+5 x^2\right ) \log ^2(x)}+\int \left (-\frac {1}{\log ^2(x)}+\frac {1}{\log (x)}\right ) \, dx\\ &=\frac {e^{5-\left (2-e^3\right ) x^2-\frac {5 x^3}{9}} \left (6 \left (2-e^3\right ) x^2 \log (x)+5 x^3 \log (x)\right )}{x \left (6 \left (2-e^3\right ) x+5 x^2\right ) \log ^2(x)}-\int \frac {1}{\log ^2(x)} \, dx+\int \frac {1}{\log (x)} \, dx\\ &=\frac {x}{\log (x)}+\frac {e^{5-\left (2-e^3\right ) x^2-\frac {5 x^3}{9}} \left (6 \left (2-e^3\right ) x^2 \log (x)+5 x^3 \log (x)\right )}{x \left (6 \left (2-e^3\right ) x+5 x^2\right ) \log ^2(x)}+\text {li}(x)-\int \frac {1}{\log (x)} \, dx\\ &=\frac {x}{\log (x)}+\frac {e^{5-\left (2-e^3\right ) x^2-\frac {5 x^3}{9}} \left (6 \left (2-e^3\right ) x^2 \log (x)+5 x^3 \log (x)\right )}{x \left (6 \left (2-e^3\right ) x+5 x^2\right ) \log ^2(x)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.52, size = 27, normalized size = 0.84 \begin {gather*} \frac {e^{5+\left (-2+e^3\right ) x^2-\frac {5 x^3}{9}}+x}{\log (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.64, size = 27, normalized size = 0.84
method | result | size |
risch | \(\frac {{\mathrm e}^{x^{2} {\mathrm e}^{3}-\frac {5 x^{3}}{9}-2 x^{2}+5}+x}{\ln \left (x \right )}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 26, normalized size = 0.81 \begin {gather*} \frac {x + e^{\left (-\frac {5}{9} \, x^{3} + x^{2} e^{3} - 2 \, x^{2} + 5\right )}}{\log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 29, normalized size = 0.91 \begin {gather*} \frac {x}{\log {\left (x \right )}} + \frac {e^{- \frac {5 x^{3}}{9} - 2 x^{2} + x^{2} e^{3} + 5}}{\log {\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.74, size = 31, normalized size = 0.97 \begin {gather*} \frac {x}{\ln \left (x\right )}+\frac {{\mathrm {e}}^{x^2\,{\mathrm {e}}^3-2\,x^2-\frac {5\,x^3}{9}+5}}{\ln \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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