Optimal. Leaf size=21 \[ \left (4+\frac {3}{x}\right ) \left (4+x^{\frac {6 (5+\log (x))}{e^4}}\right ) \]
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Rubi [A] time = 0.18, antiderivative size = 41, normalized size of antiderivative = 1.95, number of steps used = 4, number of rules used = 3, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.060, Rules used = {12, 14, 2288} \begin {gather*} \frac {x^{\frac {30}{e^4}-1} e^{\frac {6 \log ^2(x)}{e^4}} (4 x \log (x)+3 \log (x))}{\log (x)}+\frac {12}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2288
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-12 e^4+e^{\frac {2 \left (15 \log (x)+3 \log ^2(x)\right )}{e^4}} \left (90-3 e^4+120 x+(36+48 x) \log (x)\right )}{x^2} \, dx}{e^4}\\ &=\frac {\int \left (-\frac {12 e^4}{x^2}+3 e^{\frac {6 \log ^2(x)}{e^4}} x^{-2+\frac {30}{e^4}} \left (30 \left (1-\frac {e^4}{30}\right )+40 x+12 \log (x)+16 x \log (x)\right )\right ) \, dx}{e^4}\\ &=\frac {12}{x}+\frac {3 \int e^{\frac {6 \log ^2(x)}{e^4}} x^{-2+\frac {30}{e^4}} \left (30 \left (1-\frac {e^4}{30}\right )+40 x+12 \log (x)+16 x \log (x)\right ) \, dx}{e^4}\\ &=\frac {12}{x}+\frac {e^{\frac {6 \log ^2(x)}{e^4}} x^{-1+\frac {30}{e^4}} (3 \log (x)+4 x \log (x))}{\log (x)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.23, size = 32, normalized size = 1.52 \begin {gather*} \frac {12}{x}+e^{\frac {6 \log ^2(x)}{e^4}} x^{-1+\frac {30}{e^4}} (3+4 x) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.21, size = 26, normalized size = 1.24 \begin {gather*} \frac {{\left (4 \, x + 3\right )} e^{\left (6 \, {\left (\log \relax (x)^{2} + 5 \, \log \relax (x)\right )} e^{\left (-4\right )}\right )} + 12}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 48, normalized size = 2.29 \begin {gather*} \frac {{\left (4 \, x e^{\left (6 \, {\left (\log \relax (x)^{2} + 5 \, \log \relax (x)\right )} e^{\left (-4\right )} + 4\right )} + 12 \, e^{4} + 3 \, e^{\left (6 \, {\left (\log \relax (x)^{2} + 5 \, \log \relax (x)\right )} e^{\left (-4\right )} + 4\right )}\right )} e^{\left (-4\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 28, normalized size = 1.33
method | result | size |
risch | \(\frac {12}{x}+\frac {\left (3+4 x \right ) x^{2 \left (15+3 \ln \relax (x )\right ) {\mathrm e}^{-4}}}{x}\) | \(28\) |
norman | \(\frac {12+3 \,{\mathrm e}^{2 \left (3 \ln \relax (x )^{2}+15 \ln \relax (x )\right ) {\mathrm e}^{-4}}+4 x \,{\mathrm e}^{2 \left (3 \ln \relax (x )^{2}+15 \ln \relax (x )\right ) {\mathrm e}^{-4}}}{x}\) | \(50\) |
default | \({\mathrm e}^{-4} \left (\frac {3 \,{\mathrm e}^{4} {\mathrm e}^{2 \left (3 \ln \relax (x )^{2}+15 \ln \relax (x )\right ) {\mathrm e}^{-4}}+4 x \,{\mathrm e}^{4} {\mathrm e}^{2 \left (3 \ln \relax (x )^{2}+15 \ln \relax (x )\right ) {\mathrm e}^{-4}}}{x}+\frac {12 \,{\mathrm e}^{4}}{x}\right )\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.53, size = 361, normalized size = 17.19 \begin {gather*} -\frac {1}{12} \, {\left (-3 i \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{12} i \, \sqrt {6} {\left (e^{4} - 30\right )} e^{\left (-2\right )} + i \, \sqrt {6} e^{\left (-2\right )} \log \relax (x)\right ) e^{\left (-\frac {1}{24} \, {\left (e^{4} - 30\right )}^{2} e^{\left (-4\right )} + 6\right )} + 90 i \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{12} i \, \sqrt {6} {\left (e^{4} - 30\right )} e^{\left (-2\right )} + i \, \sqrt {6} e^{\left (-2\right )} \log \relax (x)\right ) e^{\left (-\frac {1}{24} \, {\left (e^{4} - 30\right )}^{2} e^{\left (-4\right )} + 2\right )} + 120 i \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (i \, \sqrt {6} e^{\left (-2\right )} \log \relax (x) + \frac {5}{2} i \, \sqrt {6} e^{\left (-2\right )}\right ) e^{\left (-\frac {75}{2} \, e^{\left (-4\right )} + 2\right )} + 3 \, \sqrt {6} {\left (\frac {\sqrt {6} \sqrt {\frac {1}{6}} \sqrt {\pi } {\left ({\left (e^{4} - 30\right )} e^{\left (-4\right )} - 12 \, e^{\left (-4\right )} \log \relax (x)\right )} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {\frac {1}{6}} \sqrt {-{\left ({\left (e^{4} - 30\right )} e^{\left (-4\right )} - 12 \, e^{\left (-4\right )} \log \relax (x)\right )}^{2} e^{4}}\right ) - 1\right )} {\left (e^{4} - 30\right )} e^{2}}{\sqrt {-{\left ({\left (e^{4} - 30\right )} e^{\left (-4\right )} - 12 \, e^{\left (-4\right )} \log \relax (x)\right )}^{2} e^{4}}} - 2 \, \sqrt {6} e^{\left (\frac {1}{24} \, {\left ({\left (e^{4} - 30\right )} e^{\left (-4\right )} - 12 \, e^{\left (-4\right )} \log \relax (x)\right )}^{2} e^{4} + 2\right )}\right )} e^{\left (-\frac {1}{24} \, {\left (e^{4} - 30\right )}^{2} e^{\left (-4\right )} + 2\right )} + 8 \, \sqrt {6} {\left (\frac {15 \, \sqrt {\pi } {\left (2 \, e^{\left (-4\right )} \log \relax (x) + 5 \, e^{\left (-4\right )}\right )} {\left (\operatorname {erf}\left (\sqrt {\frac {3}{2}} \sqrt {-{\left (2 \, e^{\left (-4\right )} \log \relax (x) + 5 \, e^{\left (-4\right )}\right )}^{2} e^{4}}\right ) - 1\right )} e^{2}}{\sqrt {-{\left (2 \, e^{\left (-4\right )} \log \relax (x) + 5 \, e^{\left (-4\right )}\right )}^{2} e^{4}}} - \sqrt {6} e^{\left (\frac {3}{2} \, {\left (2 \, e^{\left (-4\right )} \log \relax (x) + 5 \, e^{\left (-4\right )}\right )}^{2} e^{4} + 2\right )}\right )} e^{\left (-\frac {75}{2} \, e^{\left (-4\right )} + 2\right )} - \frac {144 \, e^{4}}{x}\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.28, size = 41, normalized size = 1.95 \begin {gather*} \frac {3\,x^{30\,{\mathrm {e}}^{-4}}\,{\mathrm {e}}^{6\,{\mathrm {e}}^{-4}\,{\ln \relax (x)}^2}+12}{x}+4\,x^{30\,{\mathrm {e}}^{-4}}\,{\mathrm {e}}^{6\,{\mathrm {e}}^{-4}\,{\ln \relax (x)}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 26, normalized size = 1.24 \begin {gather*} \frac {\left (4 x + 3\right ) e^{\frac {2 \left (3 \log {\relax (x )}^{2} + 15 \log {\relax (x )}\right )}{e^{4}}}}{x} + \frac {12}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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