Integrand size = 45, antiderivative size = 23 \[ \int \frac {1}{2} \left (30+9 x+e^x \left (108+66 x+9 x^2\right ) \log (19)+e^{2 x} \left (54+45 x+9 x^2\right ) \log ^2(19)\right ) \, dx=\left (2+\frac {3 (2+x) \left (x+e^x x \log (19)\right )}{2 x}\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(77\) vs. \(2(23)=46\).
Time = 0.07 (sec) , antiderivative size = 77, normalized size of antiderivative = 3.35, number of steps used = 18, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {12, 2227, 2225, 2207} \[ \int \frac {1}{2} \left (30+9 x+e^x \left (108+66 x+9 x^2\right ) \log (19)+e^{2 x} \left (54+45 x+9 x^2\right ) \log ^2(19)\right ) \, dx=\frac {9 x^2}{4}+\frac {9}{4} e^{2 x} x^2 \log ^2(19)+\frac {9}{2} e^x x^2 \log (19)+15 x+9 e^{2 x} x \log ^2(19)+9 e^{2 x} \log ^2(19)+24 e^x x \log (19)+30 e^x \log (19) \]
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Rule 12
Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \left (30+9 x+e^x \left (108+66 x+9 x^2\right ) \log (19)+e^{2 x} \left (54+45 x+9 x^2\right ) \log ^2(19)\right ) \, dx \\ & = 15 x+\frac {9 x^2}{4}+\frac {1}{2} \log (19) \int e^x \left (108+66 x+9 x^2\right ) \, dx+\frac {1}{2} \log ^2(19) \int e^{2 x} \left (54+45 x+9 x^2\right ) \, dx \\ & = 15 x+\frac {9 x^2}{4}+\frac {1}{2} \log (19) \int \left (108 e^x+66 e^x x+9 e^x x^2\right ) \, dx+\frac {1}{2} \log ^2(19) \int \left (54 e^{2 x}+45 e^{2 x} x+9 e^{2 x} x^2\right ) \, dx \\ & = 15 x+\frac {9 x^2}{4}+\frac {1}{2} (9 \log (19)) \int e^x x^2 \, dx+(33 \log (19)) \int e^x x \, dx+(54 \log (19)) \int e^x \, dx+\frac {1}{2} \left (9 \log ^2(19)\right ) \int e^{2 x} x^2 \, dx+\frac {1}{2} \left (45 \log ^2(19)\right ) \int e^{2 x} x \, dx+\left (27 \log ^2(19)\right ) \int e^{2 x} \, dx \\ & = 15 x+\frac {9 x^2}{4}+54 e^x \log (19)+33 e^x x \log (19)+\frac {9}{2} e^x x^2 \log (19)+\frac {27}{2} e^{2 x} \log ^2(19)+\frac {45}{4} e^{2 x} x \log ^2(19)+\frac {9}{4} e^{2 x} x^2 \log ^2(19)-(9 \log (19)) \int e^x x \, dx-(33 \log (19)) \int e^x \, dx-\frac {1}{2} \left (9 \log ^2(19)\right ) \int e^{2 x} x \, dx-\frac {1}{4} \left (45 \log ^2(19)\right ) \int e^{2 x} \, dx \\ & = 15 x+\frac {9 x^2}{4}+21 e^x \log (19)+24 e^x x \log (19)+\frac {9}{2} e^x x^2 \log (19)+\frac {63}{8} e^{2 x} \log ^2(19)+9 e^{2 x} x \log ^2(19)+\frac {9}{4} e^{2 x} x^2 \log ^2(19)+(9 \log (19)) \int e^x \, dx+\frac {1}{4} \left (9 \log ^2(19)\right ) \int e^{2 x} \, dx \\ & = 15 x+\frac {9 x^2}{4}+30 e^x \log (19)+24 e^x x \log (19)+\frac {9}{2} e^x x^2 \log (19)+9 e^{2 x} \log ^2(19)+9 e^{2 x} x \log ^2(19)+\frac {9}{4} e^{2 x} x^2 \log ^2(19) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(55\) vs. \(2(23)=46\).
Time = 0.20 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.39 \[ \int \frac {1}{2} \left (30+9 x+e^x \left (108+66 x+9 x^2\right ) \log (19)+e^{2 x} \left (54+45 x+9 x^2\right ) \log ^2(19)\right ) \, dx=\frac {1}{2} \left (30 x+\frac {9 x^2}{2}+3 e^x \left (20+16 x+3 x^2\right ) \log (19)+9 e^{2 x} \left (2+2 x+\frac {x^2}{2}\right ) \log ^2(19)\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(45\) vs. \(2(20)=40\).
Time = 0.15 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.00
method | result | size |
risch | \(\frac {\ln \left (19\right )^{2} \left (18+18 x +\frac {9}{2} x^{2}\right ) {\mathrm e}^{2 x}}{2}+\frac {\ln \left (19\right ) \left (9 x^{2}+48 x +60\right ) {\mathrm e}^{x}}{2}+\frac {9 x^{2}}{4}+15 x\) | \(46\) |
default | \(15 x +\frac {9 \ln \left (19\right )^{2} \left (\frac {{\mathrm e}^{2 x} x^{2}}{2}+2 x \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{2 x}\right )}{2}+\frac {3 \ln \left (19\right ) \left (16 \,{\mathrm e}^{x} x +20 \,{\mathrm e}^{x}+3 \,{\mathrm e}^{x} x^{2}\right )}{2}+\frac {9 x^{2}}{4}\) | \(60\) |
parts | \(15 x +\frac {9 \ln \left (19\right )^{2} \left (\frac {{\mathrm e}^{2 x} x^{2}}{2}+2 x \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{2 x}\right )}{2}+\frac {3 \ln \left (19\right ) \left (16 \,{\mathrm e}^{x} x +20 \,{\mathrm e}^{x}+3 \,{\mathrm e}^{x} x^{2}\right )}{2}+\frac {9 x^{2}}{4}\) | \(60\) |
norman | \(9 \ln \left (19\right )^{2} {\mathrm e}^{2 x}+9 \ln \left (19\right )^{2} {\mathrm e}^{2 x} x +\frac {9 \ln \left (19\right )^{2} {\mathrm e}^{2 x} x^{2}}{4}+\frac {9 \,{\mathrm e}^{x} \ln \left (19\right ) x^{2}}{2}+24 x \,{\mathrm e}^{x} \ln \left (19\right )+30 \ln \left (19\right ) {\mathrm e}^{x}+\frac {9 x^{2}}{4}+15 x\) | \(66\) |
parallelrisch | \(9 \ln \left (19\right )^{2} {\mathrm e}^{2 x}+9 \ln \left (19\right )^{2} {\mathrm e}^{2 x} x +\frac {9 \ln \left (19\right )^{2} {\mathrm e}^{2 x} x^{2}}{4}+\frac {9 \,{\mathrm e}^{x} \ln \left (19\right ) x^{2}}{2}+24 x \,{\mathrm e}^{x} \ln \left (19\right )+30 \ln \left (19\right ) {\mathrm e}^{x}+\frac {9 x^{2}}{4}+15 x\) | \(66\) |
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Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.87 \[ \int \frac {1}{2} \left (30+9 x+e^x \left (108+66 x+9 x^2\right ) \log (19)+e^{2 x} \left (54+45 x+9 x^2\right ) \log ^2(19)\right ) \, dx=\frac {9}{4} \, {\left (x^{2} + 4 \, x + 4\right )} e^{\left (2 \, x\right )} \log \left (19\right )^{2} + \frac {3}{2} \, {\left (3 \, x^{2} + 16 \, x + 20\right )} e^{x} \log \left (19\right ) + \frac {9}{4} \, x^{2} + 15 \, x \]
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Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (20) = 40\).
Time = 0.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.87 \[ \int \frac {1}{2} \left (30+9 x+e^x \left (108+66 x+9 x^2\right ) \log (19)+e^{2 x} \left (54+45 x+9 x^2\right ) \log ^2(19)\right ) \, dx=\frac {9 x^{2}}{4} + 15 x + \frac {\left (36 x^{2} \log {\left (19 \right )} + 192 x \log {\left (19 \right )} + 240 \log {\left (19 \right )}\right ) e^{x}}{8} + \frac {\left (18 x^{2} \log {\left (19 \right )}^{2} + 72 x \log {\left (19 \right )}^{2} + 72 \log {\left (19 \right )}^{2}\right ) e^{2 x}}{8} \]
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Time = 0.20 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.87 \[ \int \frac {1}{2} \left (30+9 x+e^x \left (108+66 x+9 x^2\right ) \log (19)+e^{2 x} \left (54+45 x+9 x^2\right ) \log ^2(19)\right ) \, dx=\frac {9}{4} \, {\left (x^{2} + 4 \, x + 4\right )} e^{\left (2 \, x\right )} \log \left (19\right )^{2} + \frac {3}{2} \, {\left (3 \, x^{2} + 16 \, x + 20\right )} e^{x} \log \left (19\right ) + \frac {9}{4} \, x^{2} + 15 \, x \]
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Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.87 \[ \int \frac {1}{2} \left (30+9 x+e^x \left (108+66 x+9 x^2\right ) \log (19)+e^{2 x} \left (54+45 x+9 x^2\right ) \log ^2(19)\right ) \, dx=\frac {9}{4} \, {\left (x^{2} + 4 \, x + 4\right )} e^{\left (2 \, x\right )} \log \left (19\right )^{2} + \frac {3}{2} \, {\left (3 \, x^{2} + 16 \, x + 20\right )} e^{x} \log \left (19\right ) + \frac {9}{4} \, x^{2} + 15 \, x \]
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Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.83 \[ \int \frac {1}{2} \left (30+9 x+e^x \left (108+66 x+9 x^2\right ) \log (19)+e^{2 x} \left (54+45 x+9 x^2\right ) \log ^2(19)\right ) \, dx=15\,x+9\,{\mathrm {e}}^{2\,x}\,{\ln \left (19\right )}^2+30\,{\mathrm {e}}^x\,\ln \left (19\right )+\frac {9\,x^2}{4}+24\,x\,{\mathrm {e}}^x\,\ln \left (19\right )+\frac {9\,x^2\,{\mathrm {e}}^{2\,x}\,{\ln \left (19\right )}^2}{4}+\frac {9\,x^2\,{\mathrm {e}}^x\,\ln \left (19\right )}{2}+9\,x\,{\mathrm {e}}^{2\,x}\,{\ln \left (19\right )}^2 \]
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