Optimal. Leaf size=23 \[ \frac {\log \left (x \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )\right )}{x} \]
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Rubi [A]
time = 5.88, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 24, number of rules used = 7, integrand size = 156, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6874, 6873,
12, 6820, 14, 30, 2635} \begin {gather*} \frac {\log \left (x \log \left (\frac {9 x}{4}+e^{x-2}-\log (x)\right )\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 30
Rule 2635
Rule 6820
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {e^2 \left (4-9 x+9 x^2-4 x \log (x)\right )}{x^2 \left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )}+\frac {x+\log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )-\log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right ) \log \left (x \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )\right )}{x^2 \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )}\right ) \, dx\\ &=-\left (e^2 \int \frac {4-9 x+9 x^2-4 x \log (x)}{x^2 \left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )} \, dx\right )+\int \frac {x+\log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )-\log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right ) \log \left (x \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )\right )}{x^2 \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )} \, dx\\ &=-\left (e^2 \int \left (\frac {9}{\left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )}+\frac {4}{x^2 \left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )}-\frac {9}{x \left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )}-\frac {4 \log (x)}{x \left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )}\right ) \, dx\right )+\int \frac {1+\frac {x}{\log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )}-\log \left (x \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )\right )}{x^2} \, dx\\ &=-\left (\left (4 e^2\right ) \int \frac {1}{x^2 \left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )} \, dx\right )+\left (4 e^2\right ) \int \frac {\log (x)}{x \left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )} \, dx-\left (9 e^2\right ) \int \frac {1}{\left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )} \, dx+\left (9 e^2\right ) \int \frac {1}{x \left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )} \, dx+\int \left (\frac {x+\log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )}{x^2 \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )}-\frac {\log \left (x \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )\right )}{x^2}\right ) \, dx\\ &=-\left (\left (4 e^2\right ) \int \frac {1}{x^2 \left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )} \, dx\right )+\left (4 e^2\right ) \int \frac {\log (x)}{x \left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )} \, dx+\left (9 e^2\right ) \int \frac {1}{x \left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )} \, dx-\left (36 e^2\right ) \text {Subst}\left (\int \frac {1}{\left (4 e^{4 x}+36 e^2 x-4 e^2 \log (4 x)\right ) \log \left (e^{-2+4 x}+9 x-\log (4 x)\right )} \, dx,x,\frac {x}{4}\right )+\int \frac {x+\log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )}{x^2 \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )} \, dx-\int \frac {\log \left (x \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )\right )}{x^2} \, dx\\ &=\frac {\log \left (x \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )\right )}{x}-\left (4 e^2\right ) \int \frac {1}{x^2 \left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )} \, dx+\left (4 e^2\right ) \int \frac {\log (x)}{x \left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )} \, dx+\left (9 e^2\right ) \int \frac {1}{x \left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )} \, dx-\left (36 e^2\right ) \text {Subst}\left (\int \frac {1}{4 \left (e^{4 x}+9 e^2 x-e^2 \log (4 x)\right ) \log \left (e^{-2+4 x}+9 x-\log (4 x)\right )} \, dx,x,\frac {x}{4}\right )+\int \frac {1+\frac {x}{\log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )}}{x^2} \, dx-\int \frac {\frac {1}{x}+\frac {\frac {9}{4}+e^{-2+x}-\frac {1}{x}}{\left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )}}{x} \, dx\\ &=\frac {\log \left (x \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )\right )}{x}-\left (4 e^2\right ) \int \frac {1}{x^2 \left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )} \, dx+\left (4 e^2\right ) \int \frac {\log (x)}{x \left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )} \, dx+\left (9 e^2\right ) \int \frac {1}{x \left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )} \, dx-\left (9 e^2\right ) \text {Subst}\left (\int \frac {1}{\left (e^{4 x}+9 e^2 x-e^2 \log (4 x)\right ) \log \left (e^{-2+4 x}+9 x-\log (4 x)\right )} \, dx,x,\frac {x}{4}\right )+\int \left (\frac {1}{x^2}+\frac {1}{x \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )}\right ) \, dx-\int \left (-\frac {e^2 \left (4-9 x+9 x^2-4 x \log (x)\right )}{x^2 \left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )}+\frac {x+\log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )}{x^2 \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )}\right ) \, dx\\ &=-\frac {1}{x}+\frac {\log \left (x \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )\right )}{x}+e^2 \int \frac {4-9 x+9 x^2-4 x \log (x)}{x^2 \left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )} \, dx-\left (4 e^2\right ) \int \frac {1}{x^2 \left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )} \, dx+\left (4 e^2\right ) \int \frac {\log (x)}{x \left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )} \, dx+\left (9 e^2\right ) \int \frac {1}{x \left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )} \, dx-\left (9 e^2\right ) \text {Subst}\left (\int \frac {1}{\left (e^{4 x}+9 e^2 x-e^2 \log (4 x)\right ) \log \left (e^{-2+4 x}+9 x-\log (4 x)\right )} \, dx,x,\frac {x}{4}\right )+\int \frac {1}{x \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )} \, dx-\int \frac {x+\log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )}{x^2 \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )} \, dx\\ &=-\frac {1}{x}+\frac {\log \left (x \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )\right )}{x}+e^2 \int \left (\frac {9}{\left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )}+\frac {4}{x^2 \left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )}-\frac {9}{x \left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )}-\frac {4 \log (x)}{x \left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )}\right ) \, dx-\left (4 e^2\right ) \int \frac {1}{x^2 \left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )} \, dx+\left (4 e^2\right ) \int \frac {\log (x)}{x \left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )} \, dx+\left (9 e^2\right ) \int \frac {1}{x \left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )} \, dx-\left (9 e^2\right ) \text {Subst}\left (\int \frac {1}{\left (e^{4 x}+9 e^2 x-e^2 \log (4 x)\right ) \log \left (e^{-2+4 x}+9 x-\log (4 x)\right )} \, dx,x,\frac {x}{4}\right )-\int \frac {1+\frac {x}{\log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )}}{x^2} \, dx+\int \frac {1}{x \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )} \, dx\\ &=-\frac {1}{x}+\frac {\log \left (x \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )\right )}{x}+\left (9 e^2\right ) \int \frac {1}{\left (4 e^x+9 e^2 x-4 e^2 \log (x)\right ) \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )} \, dx-\left (9 e^2\right ) \text {Subst}\left (\int \frac {1}{\left (e^{4 x}+9 e^2 x-e^2 \log (4 x)\right ) \log \left (e^{-2+4 x}+9 x-\log (4 x)\right )} \, dx,x,\frac {x}{4}\right )-\int \left (\frac {1}{x^2}+\frac {1}{x \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )}\right ) \, dx+\int \frac {1}{x \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )} \, dx\\ &=\frac {\log \left (x \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )\right )}{x}-\left (9 e^2\right ) \text {Subst}\left (\int \frac {1}{\left (e^{4 x}+9 e^2 x-e^2 \log (4 x)\right ) \log \left (e^{-2+4 x}+9 x-\log (4 x)\right )} \, dx,x,\frac {x}{4}\right )+\left (36 e^2\right ) \text {Subst}\left (\int \frac {1}{\left (4 e^{4 x}+36 e^2 x-4 e^2 \log (4 x)\right ) \log \left (e^{-2+4 x}+9 x-\log (4 x)\right )} \, dx,x,\frac {x}{4}\right )\\ &=\frac {\log \left (x \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )\right )}{x}-\left (9 e^2\right ) \text {Subst}\left (\int \frac {1}{\left (e^{4 x}+9 e^2 x-e^2 \log (4 x)\right ) \log \left (e^{-2+4 x}+9 x-\log (4 x)\right )} \, dx,x,\frac {x}{4}\right )+\left (36 e^2\right ) \text {Subst}\left (\int \frac {1}{4 \left (e^{4 x}+9 e^2 x-e^2 \log (4 x)\right ) \log \left (e^{-2+4 x}+9 x-\log (4 x)\right )} \, dx,x,\frac {x}{4}\right )\\ &=\frac {\log \left (x \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )\right )}{x}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.16, size = 23, normalized size = 1.00 \begin {gather*} \frac {\log \left (x \log \left (e^{-2+x}+\frac {9 x}{4}-\log (x)\right )\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.20, size = 168, normalized size = 7.30
method | result | size |
risch | \(\frac {\ln \left (\ln \left (-\ln \left (x \right )+{\mathrm e}^{x -2}+\frac {9 x}{4}\right )\right )}{x}+\frac {-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \left (-\ln \left (x \right )+{\mathrm e}^{x -2}+\frac {9 x}{4}\right )\right ) \mathrm {csgn}\left (i x \ln \left (-\ln \left (x \right )+{\mathrm e}^{x -2}+\frac {9 x}{4}\right )\right )+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \left (-\ln \left (x \right )+{\mathrm e}^{x -2}+\frac {9 x}{4}\right )\right )^{2}+i \pi \,\mathrm {csgn}\left (i \ln \left (-\ln \left (x \right )+{\mathrm e}^{x -2}+\frac {9 x}{4}\right )\right ) \mathrm {csgn}\left (i x \ln \left (-\ln \left (x \right )+{\mathrm e}^{x -2}+\frac {9 x}{4}\right )\right )^{2}-i \pi \mathrm {csgn}\left (i x \ln \left (-\ln \left (x \right )+{\mathrm e}^{x -2}+\frac {9 x}{4}\right )\right )^{3}+2 \ln \left (x \right )}{2 x}\) | \(168\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 31, normalized size = 1.35 \begin {gather*} \frac {\log \left (x\right ) + \log \left (-2 \, \log \left (2\right ) + \log \left (9 \, x e^{2} - 4 \, e^{2} \log \left (x\right ) + 4 \, e^{x}\right ) - 2\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 20, normalized size = 0.87 \begin {gather*} \frac {\log \left (x \log \left (\frac {9}{4} \, x + e^{\left (x - 2\right )} - \log \left (x\right )\right )\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 42.77, size = 19, normalized size = 0.83 \begin {gather*} \frac {\log {\left (x \log {\left (\frac {9 x}{4} + e^{x - 2} - \log {\left (x \right )} \right )} \right )}}{x} \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 = 4.99, size = 21, normalized size = 0.91 \begin {gather*} \frac {\ln \left (x\,\ln \left (\frac {9\,x}{4}-\ln \left (x\right )+{\mathrm {e}}^{-2}\,{\mathrm {e}}^x\right )\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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