Optimal. Leaf size=21 \[ \frac {1}{2} \left (\frac {1}{e^3}-x\right ) \left (-x+\frac {x}{\log (x)}\right ) \]
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Rubi [A]
time = 0.19, antiderivative size = 36, normalized size of antiderivative = 1.71, number of steps
used = 15, number of rules used = 7, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {12, 6874,
2357, 2367, 2335, 2346, 2209} \begin {gather*} \frac {x^2}{2}-\frac {x}{2 e^3}+\frac {\left (1-e^3 x\right ) x}{2 e^3 \log (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2209
Rule 2335
Rule 2346
Rule 2357
Rule 2367
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-1+e^3 x+\left (1-2 e^3 x\right ) \log (x)+\left (-1+2 e^3 x\right ) \log ^2(x)}{\log ^2(x)} \, dx}{2 e^3}\\ &=\frac {\int \left (-1+2 e^3 x+\frac {-1+e^3 x}{\log ^2(x)}+\frac {1-2 e^3 x}{\log (x)}\right ) \, dx}{2 e^3}\\ &=-\frac {x}{2 e^3}+\frac {x^2}{2}+\frac {\int \frac {-1+e^3 x}{\log ^2(x)} \, dx}{2 e^3}+\frac {\int \frac {1-2 e^3 x}{\log (x)} \, dx}{2 e^3}\\ &=-\frac {x}{2 e^3}+\frac {x^2}{2}+\frac {x \left (1-e^3 x\right )}{2 e^3 \log (x)}+\frac {\int \left (\frac {1}{\log (x)}-\frac {2 e^3 x}{\log (x)}\right ) \, dx}{2 e^3}+\frac {\int \frac {1}{\log (x)} \, dx}{2 e^3}+\frac {\int \frac {-1+e^3 x}{\log (x)} \, dx}{e^3}\\ &=-\frac {x}{2 e^3}+\frac {x^2}{2}+\frac {x \left (1-e^3 x\right )}{2 e^3 \log (x)}+\frac {\text {li}(x)}{2 e^3}+\frac {\int \frac {1}{\log (x)} \, dx}{2 e^3}+\frac {\int \left (-\frac {1}{\log (x)}+\frac {e^3 x}{\log (x)}\right ) \, dx}{e^3}-\int \frac {x}{\log (x)} \, dx\\ &=-\frac {x}{2 e^3}+\frac {x^2}{2}+\frac {x \left (1-e^3 x\right )}{2 e^3 \log (x)}+\frac {\text {li}(x)}{e^3}-\frac {\int \frac {1}{\log (x)} \, dx}{e^3}+\int \frac {x}{\log (x)} \, dx-\text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )\\ &=-\frac {x}{2 e^3}+\frac {x^2}{2}-\text {Ei}(2 \log (x))+\frac {x \left (1-e^3 x\right )}{2 e^3 \log (x)}+\text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )\\ &=-\frac {x}{2 e^3}+\frac {x^2}{2}+\frac {x \left (1-e^3 x\right )}{2 e^3 \log (x)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.04, size = 23, normalized size = 1.10 \begin {gather*} \frac {x \left (-1+e^3 x\right ) (-1+\log (x))}{2 e^3 \log (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.46, size = 54, normalized size = 2.57
method | result | size |
risch | \(\frac {x^{2}}{2}-\frac {{\mathrm e}^{-3} x}{2}-\frac {{\mathrm e}^{-3} x \left (x \,{\mathrm e}^{3}-1\right )}{2 \ln \left (x \right )}\) | \(27\) |
norman | \(\frac {-\frac {x^{2}}{2}+\frac {{\mathrm e}^{-3} x}{2}+\frac {x^{2} \ln \left (x \right )}{2}-\frac {x \,{\mathrm e}^{-3} \ln \left (x \right )}{2}}{\ln \left (x \right )}\) | \(35\) |
default | \(\frac {{\mathrm e}^{-3} \left (x^{2} {\mathrm e}^{3}+2 \,{\mathrm e}^{3} \expIntegral \left (1, -2 \ln \left (x \right )\right )-x +{\mathrm e}^{3} \left (-\frac {x^{2}}{\ln \left (x \right )}-2 \expIntegral \left (1, -2 \ln \left (x \right )\right )\right )+\frac {x}{\ln \left (x \right )}\right )}{2}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.32, size = 44, normalized size = 2.10 \begin {gather*} \frac {1}{2} \, {\left (x^{2} e^{3} - 2 \, {\rm Ei}\left (2 \, \log \left (x\right )\right ) e^{3} + 2 \, e^{3} \Gamma \left (-1, -2 \, \log \left (x\right )\right ) - x + {\rm Ei}\left (\log \left (x\right )\right ) - \Gamma \left (-1, -\log \left (x\right )\right )\right )} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 32, normalized size = 1.52 \begin {gather*} -\frac {{\left (x^{2} e^{3} - {\left (x^{2} e^{3} - x\right )} \log \left (x\right ) - x\right )} e^{\left (-3\right )}}{2 \, \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 27, normalized size = 1.29 \begin {gather*} \frac {x^{2}}{2} - \frac {x}{2 e^{3}} + \frac {- x^{2} e^{3} + x}{2 e^{3} \log {\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 31, normalized size = 1.48 \begin {gather*} \frac {1}{2} \, {\left (x^{2} e^{3} - \frac {x^{2} e^{3}}{\log \left (x\right )} - x + \frac {x}{\log \left (x\right )}\right )} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.19, size = 19, normalized size = 0.90 \begin {gather*} \frac {x\,{\mathrm {e}}^{-3}\,\left (\ln \left (x\right )-1\right )\,\left (x\,{\mathrm {e}}^3-1\right )}{2\,\ln \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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