Integrand size = 85, antiderivative size = 22 \[ \int \frac {16 x+x^4+5 x^5+e^{2 x} \left (1+x+2 x^2\right )+e^x \left (-2 x^2-6 x^3-2 x^4\right )+\left (2 e^{2 x} x+4 x^4+e^x \left (-4 x^2-2 x^3\right )\right ) \log (x)}{16 x} \, dx=-4+x+\frac {1}{16} \left (e^x-x^2\right )^2 (x+\log (x)) \]
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Leaf count is larger than twice the leaf count of optimal. \(54\) vs. \(2(22)=44\).
Time = 0.14 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.45, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {12, 14, 2341, 2326} \[ \int \frac {16 x+x^4+5 x^5+e^{2 x} \left (1+x+2 x^2\right )+e^x \left (-2 x^2-6 x^3-2 x^4\right )+\left (2 e^{2 x} x+4 x^4+e^x \left (-4 x^2-2 x^3\right )\right ) \log (x)}{16 x} \, dx=\frac {x^5}{16}+\frac {1}{16} x^4 \log (x)-\frac {1}{8} e^x x \left (x^2+x \log (x)\right )+\frac {e^{2 x} \left (x^2+x \log (x)\right )}{16 x}+x \]
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Rule 12
Rule 14
Rule 2326
Rule 2341
Rubi steps \begin{align*} \text {integral}& = \frac {1}{16} \int \frac {16 x+x^4+5 x^5+e^{2 x} \left (1+x+2 x^2\right )+e^x \left (-2 x^2-6 x^3-2 x^4\right )+\left (2 e^{2 x} x+4 x^4+e^x \left (-4 x^2-2 x^3\right )\right ) \log (x)}{x} \, dx \\ & = \frac {1}{16} \int \left (16+x^3+5 x^4+4 x^3 \log (x)-2 e^x x \left (1+3 x+x^2+2 \log (x)+x \log (x)\right )+\frac {e^{2 x} \left (1+x+2 x^2+2 x \log (x)\right )}{x}\right ) \, dx \\ & = x+\frac {x^4}{64}+\frac {x^5}{16}+\frac {1}{16} \int \frac {e^{2 x} \left (1+x+2 x^2+2 x \log (x)\right )}{x} \, dx-\frac {1}{8} \int e^x x \left (1+3 x+x^2+2 \log (x)+x \log (x)\right ) \, dx+\frac {1}{4} \int x^3 \log (x) \, dx \\ & = x+\frac {x^5}{16}+\frac {1}{16} x^4 \log (x)+\frac {e^{2 x} \left (x^2+x \log (x)\right )}{16 x}-\frac {1}{8} e^x x \left (x^2+x \log (x)\right ) \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.77 \[ \int \frac {16 x+x^4+5 x^5+e^{2 x} \left (1+x+2 x^2\right )+e^x \left (-2 x^2-6 x^3-2 x^4\right )+\left (2 e^{2 x} x+4 x^4+e^x \left (-4 x^2-2 x^3\right )\right ) \log (x)}{16 x} \, dx=\frac {1}{16} \left (x \left (16+e^{2 x}-2 e^x x^2+x^4\right )+\left (e^x-x^2\right )^2 \log (x)\right ) \]
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Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.86
method | result | size |
risch | \(\frac {\left ({\mathrm e}^{2 x}-2 \,{\mathrm e}^{x} x^{2}+x^{4}\right ) \ln \left (x \right )}{16}+\frac {x^{5}}{16}-\frac {{\mathrm e}^{x} x^{3}}{8}+\frac {x \,{\mathrm e}^{2 x}}{16}+x\) | \(41\) |
default | \(\frac {x^{4} \ln \left (x \right )}{16}-\frac {x^{2} {\mathrm e}^{x} \ln \left (x \right )}{8}+\frac {\ln \left (x \right ) {\mathrm e}^{2 x}}{16}+\frac {x^{5}}{16}-\frac {{\mathrm e}^{x} x^{3}}{8}+\frac {x \,{\mathrm e}^{2 x}}{16}+x\) | \(46\) |
parallelrisch | \(\frac {x^{4} \ln \left (x \right )}{16}-\frac {x^{2} {\mathrm e}^{x} \ln \left (x \right )}{8}+\frac {\ln \left (x \right ) {\mathrm e}^{2 x}}{16}+\frac {x^{5}}{16}-\frac {{\mathrm e}^{x} x^{3}}{8}+\frac {x \,{\mathrm e}^{2 x}}{16}+x\) | \(46\) |
parts | \(\frac {x^{4} \ln \left (x \right )}{16}-\frac {x^{2} {\mathrm e}^{x} \ln \left (x \right )}{8}+\frac {\ln \left (x \right ) {\mathrm e}^{2 x}}{16}+\frac {x^{5}}{16}-\frac {{\mathrm e}^{x} x^{3}}{8}+\frac {x \,{\mathrm e}^{2 x}}{16}+x\) | \(46\) |
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Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (19) = 38\).
Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.82 \[ \int \frac {16 x+x^4+5 x^5+e^{2 x} \left (1+x+2 x^2\right )+e^x \left (-2 x^2-6 x^3-2 x^4\right )+\left (2 e^{2 x} x+4 x^4+e^x \left (-4 x^2-2 x^3\right )\right ) \log (x)}{16 x} \, dx=\frac {1}{16} \, x^{5} - \frac {1}{8} \, x^{3} e^{x} + \frac {1}{16} \, x e^{\left (2 \, x\right )} + \frac {1}{16} \, {\left (x^{4} - 2 \, x^{2} e^{x} + e^{\left (2 \, x\right )}\right )} \log \left (x\right ) + x \]
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Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (19) = 38\).
Time = 0.18 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.23 \[ \int \frac {16 x+x^4+5 x^5+e^{2 x} \left (1+x+2 x^2\right )+e^x \left (-2 x^2-6 x^3-2 x^4\right )+\left (2 e^{2 x} x+4 x^4+e^x \left (-4 x^2-2 x^3\right )\right ) \log (x)}{16 x} \, dx=\frac {x^{5}}{16} + \frac {x^{4} \log {\left (x \right )}}{16} + x + \frac {\left (8 x + 8 \log {\left (x \right )}\right ) e^{2 x}}{128} + \frac {\left (- 16 x^{3} - 16 x^{2} \log {\left (x \right )}\right ) e^{x}}{128} \]
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\[ \int \frac {16 x+x^4+5 x^5+e^{2 x} \left (1+x+2 x^2\right )+e^x \left (-2 x^2-6 x^3-2 x^4\right )+\left (2 e^{2 x} x+4 x^4+e^x \left (-4 x^2-2 x^3\right )\right ) \log (x)}{16 x} \, dx=\int { \frac {5 \, x^{5} + x^{4} + {\left (2 \, x^{2} + x + 1\right )} e^{\left (2 \, x\right )} - 2 \, {\left (x^{4} + 3 \, x^{3} + x^{2}\right )} e^{x} + 2 \, {\left (2 \, x^{4} + x e^{\left (2 \, x\right )} - {\left (x^{3} + 2 \, x^{2}\right )} e^{x}\right )} \log \left (x\right ) + 16 \, x}{16 \, x} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (19) = 38\).
Time = 0.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.05 \[ \int \frac {16 x+x^4+5 x^5+e^{2 x} \left (1+x+2 x^2\right )+e^x \left (-2 x^2-6 x^3-2 x^4\right )+\left (2 e^{2 x} x+4 x^4+e^x \left (-4 x^2-2 x^3\right )\right ) \log (x)}{16 x} \, dx=\frac {1}{16} \, x^{5} + \frac {1}{16} \, x^{4} \log \left (x\right ) - \frac {1}{8} \, x^{3} e^{x} - \frac {1}{8} \, x^{2} e^{x} \log \left (x\right ) + \frac {1}{16} \, x e^{\left (2 \, x\right )} + \frac {1}{16} \, e^{\left (2 \, x\right )} \log \left (x\right ) + x \]
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Time = 7.80 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.05 \[ \int \frac {16 x+x^4+5 x^5+e^{2 x} \left (1+x+2 x^2\right )+e^x \left (-2 x^2-6 x^3-2 x^4\right )+\left (2 e^{2 x} x+4 x^4+e^x \left (-4 x^2-2 x^3\right )\right ) \log (x)}{16 x} \, dx=x+\frac {x\,{\mathrm {e}}^{2\,x}}{16}-\frac {x^3\,{\mathrm {e}}^x}{8}+\frac {x^4\,\ln \left (x\right )}{16}+\frac {{\mathrm {e}}^{2\,x}\,\ln \left (x\right )}{16}+\frac {x^5}{16}-\frac {x^2\,{\mathrm {e}}^x\,\ln \left (x\right )}{8} \]
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