Optimal. Leaf size=25 \[ \log \left (1+x \log \left (9 e^{2 x}\right )+4 \left (20+\frac {\log (2 x)}{x}\right )\right ) \]
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Rubi [A]
time = 0.24, antiderivative size = 28, normalized size of antiderivative = 1.12, number of steps
used = 3, number of rules used = 2, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {6874, 6816}
\begin {gather*} \log \left (x^2 \log \left (9 e^{2 x}\right )+81 x+4 \log (2 x)\right )-\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 6816
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {1}{x}+\frac {4+81 x+2 x^3+2 x^2 \log \left (9 e^{2 x}\right )}{x \left (81 x+x^2 \log \left (9 e^{2 x}\right )+4 \log (2 x)\right )}\right ) \, dx\\ &=-\log (x)+\int \frac {4+81 x+2 x^3+2 x^2 \log \left (9 e^{2 x}\right )}{x \left (81 x+x^2 \log \left (9 e^{2 x}\right )+4 \log (2 x)\right )} \, dx\\ &=-\log (x)+\log \left (81 x+x^2 \log \left (9 e^{2 x}\right )+4 \log (2 x)\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.08, size = 28, normalized size = 1.12 \begin {gather*} -\log (x)+\log \left (81 x+x^2 \log \left (9 e^{2 x}\right )+4 \log (2 x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(53\) vs.
\(2(21)=42\).
time = 1.73, size = 54, normalized size = 2.16
method | result | size |
default | \(-\ln \left (2 x \right )+\ln \left (8 x^{3}+8 x^{2} \left (\ln \left ({\mathrm e}^{x}\right )-x \right )+4 x^{2} \left (\ln \left (9 \,{\mathrm e}^{2 x}\right )-2 \ln \left ({\mathrm e}^{x}\right )\right )+16 \ln \left (2 x \right )+324 x \right )\) | \(54\) |
risch | \(\ln \left (x \right )+\ln \left (\ln \left ({\mathrm e}^{x}\right )-\frac {i \left (x^{2} \pi \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{3}-2 x^{2} \pi \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right )+x^{2} \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{2}+4 i \ln \left (3\right ) x^{2}+162 i x +8 i \ln \left (2 x \right )\right )}{4 x^{2}}\right )\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 26, normalized size = 1.04 \begin {gather*} \log \left (\frac {1}{2} \, x^{3} + \frac {1}{2} \, x^{2} \log \left (3\right ) + \frac {81}{4} \, x + \log \left (2\right ) + \log \left (x\right )\right ) - \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 30, normalized size = 1.20 \begin {gather*} \log \left (2 \, x^{3} + 2 \, x^{2} \log \left (3\right ) + 81 \, x + 4 \, \log \left (2 \, x\right )\right ) - \log \left (2 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 27, normalized size = 1.08 \begin {gather*} - \log {\left (x \right )} + \log {\left (\frac {x^{3}}{2} + \frac {x^{2} \log {\left (3 \right )}}{2} + \frac {81 x}{4} + \log {\left (2 x \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 30, normalized size = 1.20 \begin {gather*} \log \left (2 \, x^{3} + 2 \, x^{2} \log \left (3\right ) + 81 \, x + 4 \, \log \left (2\right ) + 4 \, \log \left (x\right )\right ) - \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x^2\,\ln \left (9\,{\mathrm {e}}^{2\,x}\right )-4\,\ln \left (2\,x\right )+2\,x^3+4}{4\,x\,\ln \left (2\,x\right )+x^3\,\ln \left (9\,{\mathrm {e}}^{2\,x}\right )+81\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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