Integrand size = 37, antiderivative size = 22 \[ \int \frac {4+e^{2+\left (4+4 x+x^2\right ) \log ^2(3)} \left (-4 x-2 x^2\right ) \log ^2(3)}{x} \, dx=1-e^{2+(2+x)^2 \log ^2(3)}+\log \left (x^4\right ) \]
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Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {14, 2268} \[ \int \frac {4+e^{2+\left (4+4 x+x^2\right ) \log ^2(3)} \left (-4 x-2 x^2\right ) \log ^2(3)}{x} \, dx=4 \log (x)-\exp \left (x^2 \log ^2(3)+4 x \log ^2(3)+2 \left (1+2 \log ^2(3)\right )\right ) \]
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Rule 14
Rule 2268
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {4}{x}+2 \exp \left (4 x \log ^2(3)+x^2 \log ^2(3)+2 \left (1+2 \log ^2(3)\right )\right ) (-2-x) \log ^2(3)\right ) \, dx \\ & = 4 \log (x)+\left (2 \log ^2(3)\right ) \int \exp \left (4 x \log ^2(3)+x^2 \log ^2(3)+2 \left (1+2 \log ^2(3)\right )\right ) (-2-x) \, dx \\ & = -\exp \left (4 x \log ^2(3)+x^2 \log ^2(3)+2 \left (1+2 \log ^2(3)\right )\right )+4 \log (x) \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {4+e^{2+\left (4+4 x+x^2\right ) \log ^2(3)} \left (-4 x-2 x^2\right ) \log ^2(3)}{x} \, dx=-e^{2+(2+x)^2 \log ^2(3)}+4 \log (x) \]
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Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09
method | result | size |
norman | \(-{\mathrm e}^{\left (x^{2}+4 x +4\right ) \ln \left (3\right )^{2}+2}+4 \ln \left (x \right )\) | \(24\) |
parallelrisch | \(-{\mathrm e}^{\left (x^{2}+4 x +4\right ) \ln \left (3\right )^{2}+2}+4 \ln \left (x \right )\) | \(24\) |
risch | \(-{\mathrm e}^{x^{2} \ln \left (3\right )^{2}+4 x \ln \left (3\right )^{2}+4 \ln \left (3\right )^{2}+2}+4 \ln \left (x \right )\) | \(32\) |
parts | \(4 \ln \left (x \right )-{\mathrm e}^{4 \ln \left (3\right )^{2}} {\mathrm e}^{2} {\mathrm e}^{x^{2} \ln \left (3\right )^{2}+4 x \ln \left (3\right )^{2}}\) | \(34\) |
default | \(4 \ln \left (x \right )+2 i \ln \left (3\right ) {\mathrm e}^{4 \ln \left (3\right )^{2}} {\mathrm e}^{2} \sqrt {\pi }\, {\mathrm e}^{-4 \ln \left (3\right )^{2}} \operatorname {erf}\left (i \ln \left (3\right ) x +2 i \ln \left (3\right )\right )-2 \ln \left (3\right )^{2} {\mathrm e}^{4 \ln \left (3\right )^{2}} {\mathrm e}^{2} \left (\frac {{\mathrm e}^{x^{2} \ln \left (3\right )^{2}+4 x \ln \left (3\right )^{2}}}{2 \ln \left (3\right )^{2}}+\frac {i \sqrt {\pi }\, {\mathrm e}^{-4 \ln \left (3\right )^{2}} \operatorname {erf}\left (i \ln \left (3\right ) x +2 i \ln \left (3\right )\right )}{\ln \left (3\right )}\right )\) | \(112\) |
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Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {4+e^{2+\left (4+4 x+x^2\right ) \log ^2(3)} \left (-4 x-2 x^2\right ) \log ^2(3)}{x} \, dx=-e^{\left ({\left (x^{2} + 4 \, x + 4\right )} \log \left (3\right )^{2} + 2\right )} + 4 \, \log \left (x\right ) \]
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Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {4+e^{2+\left (4+4 x+x^2\right ) \log ^2(3)} \left (-4 x-2 x^2\right ) \log ^2(3)}{x} \, dx=- e^{\left (x^{2} + 4 x + 4\right ) \log {\left (3 \right )}^{2} + 2} + 4 \log {\left (x \right )} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.33 (sec) , antiderivative size = 129, normalized size of antiderivative = 5.86 \[ \int \frac {4+e^{2+\left (4+4 x+x^2\right ) \log ^2(3)} \left (-4 x-2 x^2\right ) \log ^2(3)}{x} \, dx=2 i \, \sqrt {\pi } \operatorname {erf}\left (i \, x \log \left (3\right ) + 2 i \, \log \left (3\right )\right ) e^{2} \log \left (3\right ) + {\left (\frac {2 \, \sqrt {\pi } {\left (x \log \left (3\right )^{2} + 2 \, \log \left (3\right )^{2}\right )} {\left (\operatorname {erf}\left (\sqrt {-\frac {{\left (x \log \left (3\right )^{2} + 2 \, \log \left (3\right )^{2}\right )}^{2}}{\log \left (3\right )^{2}}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (x \log \left (3\right )^{2} + 2 \, \log \left (3\right )^{2}\right )}^{2}}{\log \left (3\right )^{2}}} \log \left (3\right )} - \frac {e^{\left (\frac {{\left (x \log \left (3\right )^{2} + 2 \, \log \left (3\right )^{2}\right )}^{2}}{\log \left (3\right )^{2}}\right )}}{\log \left (3\right )}\right )} e^{2} \log \left (3\right ) + 4 \, \log \left (x\right ) \]
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Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.41 \[ \int \frac {4+e^{2+\left (4+4 x+x^2\right ) \log ^2(3)} \left (-4 x-2 x^2\right ) \log ^2(3)}{x} \, dx=-e^{\left (x^{2} \log \left (3\right )^{2} + 4 \, x \log \left (3\right )^{2} + 4 \, \log \left (3\right )^{2} + 2\right )} + 4 \, \log \left (x\right ) \]
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Time = 0.13 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {4+e^{2+\left (4+4 x+x^2\right ) \log ^2(3)} \left (-4 x-2 x^2\right ) \log ^2(3)}{x} \, dx=4\,\ln \left (x\right )-{\mathrm {e}}^{4\,x\,{\ln \left (3\right )}^2}\,{\mathrm {e}}^2\,{\mathrm {e}}^{x^2\,{\ln \left (3\right )}^2}\,{\mathrm {e}}^{4\,{\ln \left (3\right )}^2} \]
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