Integrand size = 101, antiderivative size = 24 \[ \int \frac {e^x x \log (x)+\left (2 e^{2 x} x+e^x \left (1+2 x^2\right )\right ) \log ^2(x)+\left (-e^x x+e^x \left (x+x^2\right ) \log (x)+\left (e^{2 x} \left (2 x+4 x^2\right )+e^x \left (x+4 x^2+2 x^3\right )\right ) \log ^2(x)\right ) \log (25 x)}{x \log ^2(x)} \, dx=e^x \left (1+2 x \left (e^x+x\right )+\frac {x}{\log (x)}\right ) \log (25 x) \]
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Leaf count is larger than twice the leaf count of optimal. \(58\) vs. \(2(24)=48\).
Time = 1.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.42, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {6820, 6874, 2326} \[ \int \frac {e^x x \log (x)+\left (2 e^{2 x} x+e^x \left (1+2 x^2\right )\right ) \log ^2(x)+\left (-e^x x+e^x \left (x+x^2\right ) \log (x)+\left (e^{2 x} \left (2 x+4 x^2\right )+e^x \left (x+4 x^2+2 x^3\right )\right ) \log ^2(x)\right ) \log (25 x)}{x \log ^2(x)} \, dx=\frac {e^x \left (2 x^3 \log ^2(x) \log (25 x)+x^2 \log (x) \log (25 x)+x \log ^2(x) \log (25 x)\right )}{x \log ^2(x)}+2 e^{2 x} x \log (25 x) \]
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Rule 2326
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^x \left (-x \log (25 x)+\log (x) (x+x (1+x) \log (25 x))+\log ^2(x) \left (1+2 e^x x+2 x^2+x \left (1+4 x+2 x^2+e^x (2+4 x)\right ) \log (25 x)\right )\right )}{x \log ^2(x)} \, dx \\ & = \int \left (2 e^{2 x} (1+\log (25 x)+2 x \log (25 x))+\frac {e^x \left (x \log (x)+\log ^2(x)+2 x^2 \log ^2(x)-x \log (25 x)+x \log (x) \log (25 x)+x^2 \log (x) \log (25 x)+x \log ^2(x) \log (25 x)+4 x^2 \log ^2(x) \log (25 x)+2 x^3 \log ^2(x) \log (25 x)\right )}{x \log ^2(x)}\right ) \, dx \\ & = 2 \int e^{2 x} (1+\log (25 x)+2 x \log (25 x)) \, dx+\int \frac {e^x \left (x \log (x)+\log ^2(x)+2 x^2 \log ^2(x)-x \log (25 x)+x \log (x) \log (25 x)+x^2 \log (x) \log (25 x)+x \log ^2(x) \log (25 x)+4 x^2 \log ^2(x) \log (25 x)+2 x^3 \log ^2(x) \log (25 x)\right )}{x \log ^2(x)} \, dx \\ & = 2 e^{2 x} x \log (25 x)+\frac {e^x \left (x^2 \log (x) \log (25 x)+x \log ^2(x) \log (25 x)+2 x^3 \log ^2(x) \log (25 x)\right )}{x \log ^2(x)} \\ \end{align*}
Time = 0.69 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25 \[ \int \frac {e^x x \log (x)+\left (2 e^{2 x} x+e^x \left (1+2 x^2\right )\right ) \log ^2(x)+\left (-e^x x+e^x \left (x+x^2\right ) \log (x)+\left (e^{2 x} \left (2 x+4 x^2\right )+e^x \left (x+4 x^2+2 x^3\right )\right ) \log ^2(x)\right ) \log (25 x)}{x \log ^2(x)} \, dx=\frac {e^x \left (x+\left (1+2 e^x x+2 x^2\right ) \log (x)\right ) \log (25 x)}{\log (x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(52\) vs. \(2(22)=44\).
Time = 0.44 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.21
method | result | size |
parallelrisch | \(\frac {120 \ln \left (x \right ) {\mathrm e}^{x} \ln \left (25 x \right ) x^{2}+60 \,{\mathrm e}^{x} \ln \left (25 x \right ) x +120 \ln \left (25 x \right ) {\mathrm e}^{2 x} \ln \left (x \right ) x +60 \ln \left (x \right ) {\mathrm e}^{x} \ln \left (25 x \right )}{60 \ln \left (x \right )}\) | \(53\) |
risch | \(\left (2 \,{\mathrm e}^{x} x^{2}+2 x \,{\mathrm e}^{2 x}+{\mathrm e}^{x}\right ) \ln \left (x \right )+4 x \ln \left (5\right ) {\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} \ln \left (5\right )+{\mathrm e}^{x} x +4 x^{2} \ln \left (5\right ) {\mathrm e}^{x}+\frac {2 x \,{\mathrm e}^{x} \ln \left (5\right )}{\ln \left (x \right )}\) | \(61\) |
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (22) = 44\).
Time = 0.25 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.71 \[ \int \frac {e^x x \log (x)+\left (2 e^{2 x} x+e^x \left (1+2 x^2\right )\right ) \log ^2(x)+\left (-e^x x+e^x \left (x+x^2\right ) \log (x)+\left (e^{2 x} \left (2 x+4 x^2\right )+e^x \left (x+4 x^2+2 x^3\right )\right ) \log ^2(x)\right ) \log (25 x)}{x \log ^2(x)} \, dx=\frac {2 \, x e^{x} \log \left (5\right ) + {\left (2 \, x e^{\left (2 \, x\right )} + {\left (2 \, x^{2} + 1\right )} e^{x}\right )} \log \left (x\right )^{2} + {\left (4 \, x e^{\left (2 \, x\right )} \log \left (5\right ) + {\left (2 \, {\left (2 \, x^{2} + 1\right )} \log \left (5\right ) + x\right )} e^{x}\right )} \log \left (x\right )}{\log \left (x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (24) = 48\).
Time = 0.15 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.17 \[ \int \frac {e^x x \log (x)+\left (2 e^{2 x} x+e^x \left (1+2 x^2\right )\right ) \log ^2(x)+\left (-e^x x+e^x \left (x+x^2\right ) \log (x)+\left (e^{2 x} \left (2 x+4 x^2\right )+e^x \left (x+4 x^2+2 x^3\right )\right ) \log ^2(x)\right ) \log (25 x)}{x \log ^2(x)} \, dx=\frac {\left (2 x \log {\left (x \right )}^{2} + 4 x \log {\left (5 \right )} \log {\left (x \right )}\right ) e^{2 x} + \left (2 x^{2} \log {\left (x \right )}^{2} + 4 x^{2} \log {\left (5 \right )} \log {\left (x \right )} + x \log {\left (x \right )} + 2 x \log {\left (5 \right )} + \log {\left (x \right )}^{2} + 2 \log {\left (5 \right )} \log {\left (x \right )}\right ) e^{x}}{\log {\left (x \right )}} \]
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\[ \int \frac {e^x x \log (x)+\left (2 e^{2 x} x+e^x \left (1+2 x^2\right )\right ) \log ^2(x)+\left (-e^x x+e^x \left (x+x^2\right ) \log (x)+\left (e^{2 x} \left (2 x+4 x^2\right )+e^x \left (x+4 x^2+2 x^3\right )\right ) \log ^2(x)\right ) \log (25 x)}{x \log ^2(x)} \, dx=\int { \frac {x e^{x} \log \left (x\right ) + {\left (2 \, x e^{\left (2 \, x\right )} + {\left (2 \, x^{2} + 1\right )} e^{x}\right )} \log \left (x\right )^{2} + {\left ({\left (x^{2} + x\right )} e^{x} \log \left (x\right ) + {\left (2 \, {\left (2 \, x^{2} + x\right )} e^{\left (2 \, x\right )} + {\left (2 \, x^{3} + 4 \, x^{2} + x\right )} e^{x}\right )} \log \left (x\right )^{2} - x e^{x}\right )} \log \left (25 \, x\right )}{x \log \left (x\right )^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (22) = 44\).
Time = 0.27 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.25 \[ \int \frac {e^x x \log (x)+\left (2 e^{2 x} x+e^x \left (1+2 x^2\right )\right ) \log ^2(x)+\left (-e^x x+e^x \left (x+x^2\right ) \log (x)+\left (e^{2 x} \left (2 x+4 x^2\right )+e^x \left (x+4 x^2+2 x^3\right )\right ) \log ^2(x)\right ) \log (25 x)}{x \log ^2(x)} \, dx=\frac {4 \, x^{2} e^{x} \log \left (5\right ) \log \left (x\right ) + 2 \, x^{2} e^{x} \log \left (x\right )^{2} + 4 \, x e^{\left (2 \, x\right )} \log \left (5\right ) \log \left (x\right ) + 2 \, x e^{\left (2 \, x\right )} \log \left (x\right )^{2} + 2 \, x e^{x} \log \left (5\right ) + x e^{x} \log \left (x\right ) + 2 \, e^{x} \log \left (5\right ) \log \left (x\right ) + e^{x} \log \left (x\right )^{2}}{\log \left (x\right )} \]
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Time = 13.83 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.62 \[ \int \frac {e^x x \log (x)+\left (2 e^{2 x} x+e^x \left (1+2 x^2\right )\right ) \log ^2(x)+\left (-e^x x+e^x \left (x+x^2\right ) \log (x)+\left (e^{2 x} \left (2 x+4 x^2\right )+e^x \left (x+4 x^2+2 x^3\right )\right ) \log ^2(x)\right ) \log (25 x)}{x \log ^2(x)} \, dx={\mathrm {e}}^x\,\ln \left (x\right )+2\,{\mathrm {e}}^x\,\ln \left (5\right )+x\,{\mathrm {e}}^x+4\,x\,{\mathrm {e}}^{2\,x}\,\ln \left (5\right )+4\,x^2\,{\mathrm {e}}^x\,\ln \left (5\right )+2\,x\,{\mathrm {e}}^{2\,x}\,\ln \left (x\right )+2\,x^2\,{\mathrm {e}}^x\,\ln \left (x\right )+\frac {2\,x\,{\mathrm {e}}^x\,\ln \left (5\right )}{\ln \left (x\right )} \]
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