Optimal. Leaf size=28 \[ \log \left (-1+\frac {x-\log (x)}{e^4+\log ^2\left (12 \left (4+9 x^2\right )\right )}\right ) \]
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Rubi [A] time = 3.87, antiderivative size = 42, normalized size of antiderivative = 1.50, number of steps used = 5, number of rules used = 5, integrand size = 182, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {6688, 6725, 6684, 6696, 203} \begin {gather*} \log \left (\log ^2\left (12 \left (9 x^2+4\right )\right )-x+\log (x)+e^4\right )-\log \left (\log ^2\left (12 \left (9 x^2+4\right )\right )+e^4\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 6684
Rule 6688
Rule 6696
Rule 6725
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-e^4 \left (-4+4 x-9 x^2+9 x^3\right )+36 x^2 (x-\log (x)) \log \left (12 \left (4+9 x^2\right )\right )-\left (-4+4 x-9 x^2+9 x^3\right ) \log ^2\left (12 \left (4+9 x^2\right )\right )}{x \left (4+9 x^2\right ) \left (e^4+\log ^2\left (12 \left (4+9 x^2\right )\right )\right ) \left (e^4-x+\log (x)+\log ^2\left (12 \left (4+9 x^2\right )\right )\right )} \, dx\\ &=\int \left (\frac {-4+4 x-9 x^2+9 x^3-36 x^2 \log \left (12 \left (4+9 x^2\right )\right )}{x \left (4+9 x^2\right ) \left (-e^4+x-\log (x)-\log ^2\left (12 \left (4+9 x^2\right )\right )\right )}+\frac {36 x \log \left (48+108 x^2\right )}{\left (-4-9 x^2\right ) \left (e^4+\log ^2\left (12 \left (4+9 x^2\right )\right )\right )}\right ) \, dx\\ &=36 \int \frac {x \log \left (48+108 x^2\right )}{\left (-4-9 x^2\right ) \left (e^4+\log ^2\left (12 \left (4+9 x^2\right )\right )\right )} \, dx+\int \frac {-4+4 x-9 x^2+9 x^3-36 x^2 \log \left (12 \left (4+9 x^2\right )\right )}{x \left (4+9 x^2\right ) \left (-e^4+x-\log (x)-\log ^2\left (12 \left (4+9 x^2\right )\right )\right )} \, dx\\ &=-\log \left (e^4+\log ^2\left (12 \left (4+9 x^2\right )\right )\right )+\log \left (e^4-x+\log (x)+\log ^2\left (12 \left (4+9 x^2\right )\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.11, size = 42, normalized size = 1.50 \begin {gather*} -\log \left (e^4+\log ^2\left (12 \left (4+9 x^2\right )\right )\right )+\log \left (e^4-x+\log (x)+\log ^2\left (12 \left (4+9 x^2\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 2.46, size = 36, normalized size = 1.29 \begin {gather*} \log \left (\log \left (108 \, x^{2} + 48\right )^{2} - x + e^{4} + \log \relax (x)\right ) - \log \left (\log \left (108 \, x^{2} + 48\right )^{2} + e^{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 29.61, size = 36, normalized size = 1.29 \begin {gather*} \log \left (\log \left (108 \, x^{2} + 48\right )^{2} - x + e^{4} + \log \relax (x)\right ) - \log \left (\log \left (108 \, x^{2} + 48\right )^{2} + e^{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 37, normalized size = 1.32
method | result | size |
risch | \(\ln \left (\ln \left (108 x^{2}+48\right )^{2}+{\mathrm e}^{4}-x +\ln \relax (x )\right )-\ln \left (\ln \left (108 x^{2}+48\right )^{2}+{\mathrm e}^{4}\right )\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.60, size = 102, normalized size = 3.64 \begin {gather*} \log \left (\log \relax (3)^{2} + 4 \, \log \relax (3) \log \relax (2) + 4 \, \log \relax (2)^{2} + 2 \, {\left (\log \relax (3) + 2 \, \log \relax (2)\right )} \log \left (9 \, x^{2} + 4\right ) + \log \left (9 \, x^{2} + 4\right )^{2} - x + e^{4} + \log \relax (x)\right ) - \log \left (\log \relax (3)^{2} + 4 \, \log \relax (3) \log \relax (2) + 4 \, \log \relax (2)^{2} + 2 \, {\left (\log \relax (3) + 2 \, \log \relax (2)\right )} \log \left (9 \, x^{2} + 4\right ) + \log \left (9 \, x^{2} + 4\right )^{2} + e^{4}\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 {\left (9\,x^3-9\,x^2+4\,x-4\right )\,{\ln \left (108\,x^2+48\right )}^2+\left (36\,x^2\,\ln \relax (x)-36\,x^3\right )\,\ln \left (108\,x^2+48\right )+{\mathrm {e}}^4\,\left (9\,x^3-9\,x^2+4\,x-4\right )}{\left (9\,x^3+4\,x\right )\,{\ln \left (108\,x^2+48\right )}^4+\left ({\mathrm {e}}^4\,\left (18\,x^3+8\,x\right )+\ln \relax (x)\,\left (9\,x^3+4\,x\right )-4\,x^2-9\,x^4\right )\,{\ln \left (108\,x^2+48\right )}^2+{\mathrm {e}}^8\,\left (9\,x^3+4\,x\right )-{\mathrm {e}}^4\,\left (9\,x^4+4\,x^2\right )+{\mathrm {e}}^4\,\ln \relax (x)\,\left (9\,x^3+4\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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