Integrand size = 107, antiderivative size = 26 \[ \int \frac {-10 x-6 x^2+28 x^3+38 x^4+18 x^5+4 x^6+e^{2 x} \left (2 x^4+2 x^5\right )+e^x \left (-2 x^2+10 x^3+22 x^4+12 x^5+2 x^6\right )+\left (2+10 x-2 x^3-2 e^x x^3\right ) \log (x)-2 \log ^2(x)}{x^3} \, dx=7+\left (6+3 x+e^x x+x^2-\frac {x+\log (x)}{x}\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(90\) vs. \(2(26)=52\).
Time = 0.32 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.46, number of steps used = 38, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.131, Rules used = {14, 2227, 2207, 2225, 6874, 2230, 2209, 2634, 1597, 1634, 2404, 2332, 2341, 2342} \[ \int \frac {-10 x-6 x^2+28 x^3+38 x^4+18 x^5+4 x^6+e^{2 x} \left (2 x^4+2 x^5\right )+e^x \left (-2 x^2+10 x^3+22 x^4+12 x^5+2 x^6\right )+\left (2+10 x-2 x^3-2 e^x x^3\right ) \log (x)-2 \log ^2(x)}{x^3} \, dx=x^4+2 e^x x^3+\frac {8 x^3}{3}+6 e^x x^2+e^{2 x} x^2+9 x^2+\frac {\log ^2(x)}{x^2}+10 e^x x+20 x+\frac {10}{3} (x+1)^3-2 x \log (x)-2 e^x \log (x)-6 \log (x)-\frac {10 \log (x)}{x} \]
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Rule 14
Rule 1597
Rule 1634
Rule 2207
Rule 2209
Rule 2225
Rule 2227
Rule 2230
Rule 2332
Rule 2341
Rule 2342
Rule 2404
Rule 2634
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (2 e^{2 x} x (1+x)+\frac {2 e^x \left (-1+5 x+11 x^2+6 x^3+x^4-x \log (x)\right )}{x}+\frac {2 \left (-5 x-3 x^2+14 x^3+19 x^4+9 x^5+2 x^6+\log (x)+5 x \log (x)-x^3 \log (x)-\log ^2(x)\right )}{x^3}\right ) \, dx \\ & = 2 \int e^{2 x} x (1+x) \, dx+2 \int \frac {e^x \left (-1+5 x+11 x^2+6 x^3+x^4-x \log (x)\right )}{x} \, dx+2 \int \frac {-5 x-3 x^2+14 x^3+19 x^4+9 x^5+2 x^6+\log (x)+5 x \log (x)-x^3 \log (x)-\log ^2(x)}{x^3} \, dx \\ & = 2 \int \left (e^{2 x} x+e^{2 x} x^2\right ) \, dx+2 \int \left (\frac {e^x \left (-1+5 x+11 x^2+6 x^3+x^4\right )}{x}-e^x \log (x)\right ) \, dx+2 \int \left (\frac {(1+x)^2 \left (-5+7 x+5 x^2+2 x^3\right )}{x^2}-\frac {\left (-1-5 x+x^3\right ) \log (x)}{x^3}-\frac {\log ^2(x)}{x^3}\right ) \, dx \\ & = 2 \int e^{2 x} x \, dx+2 \int e^{2 x} x^2 \, dx+2 \int \frac {(1+x)^2 \left (-5+7 x+5 x^2+2 x^3\right )}{x^2} \, dx+2 \int \frac {e^x \left (-1+5 x+11 x^2+6 x^3+x^4\right )}{x} \, dx-2 \int e^x \log (x) \, dx-2 \int \frac {\left (-1-5 x+x^3\right ) \log (x)}{x^3} \, dx-2 \int \frac {\log ^2(x)}{x^3} \, dx \\ & = e^{2 x} x+e^{2 x} x^2+\frac {10}{3} (1+x)^3-2 e^x \log (x)+\frac {\log ^2(x)}{x^2}+2 \int \frac {e^x}{x} \, dx-2 \int e^{2 x} x \, dx+2 \int \frac {(1+x)^2 \left (-5+7 x+2 x^3\right )}{x^2} \, dx+2 \int \left (5 e^x-\frac {e^x}{x}+11 e^x x+6 e^x x^2+e^x x^3\right ) \, dx-2 \int \frac {\log (x)}{x^3} \, dx-2 \int \left (\log (x)-\frac {\log (x)}{x^3}-\frac {5 \log (x)}{x^2}\right ) \, dx-\int e^{2 x} \, dx \\ & = -\frac {e^{2 x}}{2}+\frac {1}{2 x^2}+e^{2 x} x^2+\frac {10}{3} (1+x)^3+2 \text {Ei}(x)-2 e^x \log (x)+\frac {\log (x)}{x^2}+\frac {\log ^2(x)}{x^2}-2 \int \frac {e^x}{x} \, dx+2 \int e^x x^3 \, dx+2 \int \left (9-\frac {5}{x^2}-\frac {3}{x}+9 x+4 x^2+2 x^3\right ) \, dx-2 \int \log (x) \, dx+2 \int \frac {\log (x)}{x^3} \, dx+10 \int e^x \, dx+10 \int \frac {\log (x)}{x^2} \, dx+12 \int e^x x^2 \, dx+22 \int e^x x \, dx+\int e^{2 x} \, dx \\ & = 10 e^x+20 x+22 e^x x+9 x^2+12 e^x x^2+e^{2 x} x^2+\frac {8 x^3}{3}+2 e^x x^3+x^4+\frac {10}{3} (1+x)^3-6 \log (x)-2 e^x \log (x)-\frac {10 \log (x)}{x}-2 x \log (x)+\frac {\log ^2(x)}{x^2}-6 \int e^x x^2 \, dx-22 \int e^x \, dx-24 \int e^x x \, dx \\ & = -12 e^x+20 x-2 e^x x+9 x^2+6 e^x x^2+e^{2 x} x^2+\frac {8 x^3}{3}+2 e^x x^3+x^4+\frac {10}{3} (1+x)^3-6 \log (x)-2 e^x \log (x)-\frac {10 \log (x)}{x}-2 x \log (x)+\frac {\log ^2(x)}{x^2}+12 \int e^x x \, dx+24 \int e^x \, dx \\ & = 12 e^x+20 x+10 e^x x+9 x^2+6 e^x x^2+e^{2 x} x^2+\frac {8 x^3}{3}+2 e^x x^3+x^4+\frac {10}{3} (1+x)^3-6 \log (x)-2 e^x \log (x)-\frac {10 \log (x)}{x}-2 x \log (x)+\frac {\log ^2(x)}{x^2}-12 \int e^x \, dx \\ & = 20 x+10 e^x x+9 x^2+6 e^x x^2+e^{2 x} x^2+\frac {8 x^3}{3}+2 e^x x^3+x^4+\frac {10}{3} (1+x)^3-6 \log (x)-2 e^x \log (x)-\frac {10 \log (x)}{x}-2 x \log (x)+\frac {\log ^2(x)}{x^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(63\) vs. \(2(26)=52\).
Time = 0.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.42 \[ \int \frac {-10 x-6 x^2+28 x^3+38 x^4+18 x^5+4 x^6+e^{2 x} \left (2 x^4+2 x^5\right )+e^x \left (-2 x^2+10 x^3+22 x^4+12 x^5+2 x^6\right )+\left (2+10 x-2 x^3-2 e^x x^3\right ) \log (x)-2 \log ^2(x)}{x^3} \, dx=\frac {x^3 \left (30+19 x+e^{2 x} x+6 x^2+x^3+2 e^x \left (5+3 x+x^2\right )\right )-2 x \left (5+\left (3+e^x\right ) x+x^2\right ) \log (x)+\log ^2(x)}{x^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(73\) vs. \(2(25)=50\).
Time = 0.14 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.85
method | result | size |
risch | \(\frac {\ln \left (x \right )^{2}}{x^{2}}-\frac {2 \left (x^{2}+{\mathrm e}^{x} x +5\right ) \ln \left (x \right )}{x}+x^{4}+2 \,{\mathrm e}^{x} x^{3}+{\mathrm e}^{2 x} x^{2}+6 x^{3}+6 \,{\mathrm e}^{x} x^{2}+19 x^{2}+10 \,{\mathrm e}^{x} x -6 \ln \left (x \right )+30 x +25\) | \(74\) |
default | \(10 \,{\mathrm e}^{x} x +6 \,{\mathrm e}^{x} x^{2}+2 \,{\mathrm e}^{x} x^{3}-2 \,{\mathrm e}^{x} \ln \left (x \right )+\frac {\ln \left (x \right )^{2}}{x^{2}}+x^{4}+6 x^{3}+19 x^{2}+30 x -6 \ln \left (x \right )+{\mathrm e}^{2 x} x^{2}-2 x \ln \left (x \right )-\frac {10 \ln \left (x \right )}{x}\) | \(75\) |
parts | \(10 \,{\mathrm e}^{x} x +6 \,{\mathrm e}^{x} x^{2}+2 \,{\mathrm e}^{x} x^{3}-2 \,{\mathrm e}^{x} \ln \left (x \right )+\frac {\ln \left (x \right )^{2}}{x^{2}}+x^{4}+6 x^{3}+19 x^{2}+30 x -6 \ln \left (x \right )+{\mathrm e}^{2 x} x^{2}-2 x \ln \left (x \right )-\frac {10 \ln \left (x \right )}{x}\) | \(75\) |
parallelrisch | \(-\frac {-x^{6}-2 x^{5} {\mathrm e}^{x}-{\mathrm e}^{2 x} x^{4}-6 x^{5}-6 \,{\mathrm e}^{x} x^{4}-19 x^{4}+2 x^{3} \ln \left (x \right )-10 \,{\mathrm e}^{x} x^{3}+2 x^{2} {\mathrm e}^{x} \ln \left (x \right )+6 x^{2} \ln \left (x \right )-30 x^{3}+10 x \ln \left (x \right )-\ln \left (x \right )^{2}}{x^{2}}\) | \(91\) |
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Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (25) = 50\).
Time = 0.26 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.88 \[ \int \frac {-10 x-6 x^2+28 x^3+38 x^4+18 x^5+4 x^6+e^{2 x} \left (2 x^4+2 x^5\right )+e^x \left (-2 x^2+10 x^3+22 x^4+12 x^5+2 x^6\right )+\left (2+10 x-2 x^3-2 e^x x^3\right ) \log (x)-2 \log ^2(x)}{x^3} \, dx=\frac {x^{6} + 6 \, x^{5} + x^{4} e^{\left (2 \, x\right )} + 19 \, x^{4} + 30 \, x^{3} + 2 \, {\left (x^{5} + 3 \, x^{4} + 5 \, x^{3}\right )} e^{x} - 2 \, {\left (x^{3} + x^{2} e^{x} + 3 \, x^{2} + 5 \, x\right )} \log \left (x\right ) + \log \left (x\right )^{2}}{x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (22) = 44\).
Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.81 \[ \int \frac {-10 x-6 x^2+28 x^3+38 x^4+18 x^5+4 x^6+e^{2 x} \left (2 x^4+2 x^5\right )+e^x \left (-2 x^2+10 x^3+22 x^4+12 x^5+2 x^6\right )+\left (2+10 x-2 x^3-2 e^x x^3\right ) \log (x)-2 \log ^2(x)}{x^3} \, dx=x^{4} + 6 x^{3} + x^{2} e^{2 x} + 19 x^{2} + 30 x + \left (2 x^{3} + 6 x^{2} + 10 x - 2 \log {\left (x \right )}\right ) e^{x} - 6 \log {\left (x \right )} + \frac {\left (- 2 x^{2} - 10\right ) \log {\left (x \right )}}{x} + \frac {\log {\left (x \right )}^{2}}{x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 135 vs. \(2 (25) = 50\).
Time = 0.23 (sec) , antiderivative size = 135, normalized size of antiderivative = 5.19 \[ \int \frac {-10 x-6 x^2+28 x^3+38 x^4+18 x^5+4 x^6+e^{2 x} \left (2 x^4+2 x^5\right )+e^x \left (-2 x^2+10 x^3+22 x^4+12 x^5+2 x^6\right )+\left (2+10 x-2 x^3-2 e^x x^3\right ) \log (x)-2 \log ^2(x)}{x^3} \, dx=x^{4} + 6 \, x^{3} + 19 \, x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} + \frac {1}{2} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{3} - 3 \, x^{2} + 6 \, x - 6\right )} e^{x} + 12 \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} + 22 \, {\left (x - 1\right )} e^{x} - 2 \, x \log \left (x\right ) - 2 \, e^{x} \log \left (x\right ) + 30 \, x - \frac {10 \, \log \left (x\right )}{x} + \frac {2 \, \log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 1}{2 \, x^{2}} - \frac {\log \left (x\right )}{x^{2}} - \frac {1}{2 \, x^{2}} + 10 \, e^{x} - 6 \, \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (25) = 50\).
Time = 0.26 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.23 \[ \int \frac {-10 x-6 x^2+28 x^3+38 x^4+18 x^5+4 x^6+e^{2 x} \left (2 x^4+2 x^5\right )+e^x \left (-2 x^2+10 x^3+22 x^4+12 x^5+2 x^6\right )+\left (2+10 x-2 x^3-2 e^x x^3\right ) \log (x)-2 \log ^2(x)}{x^3} \, dx=\frac {x^{6} + 2 \, x^{5} e^{x} + 6 \, x^{5} + x^{4} e^{\left (2 \, x\right )} + 6 \, x^{4} e^{x} + 19 \, x^{4} + 10 \, x^{3} e^{x} - 2 \, x^{3} \log \left (x\right ) - 2 \, x^{2} e^{x} \log \left (x\right ) + 30 \, x^{3} - 6 \, x^{2} \log \left (x\right ) - 10 \, x \log \left (x\right ) + \log \left (x\right )^{2}}{x^{2}} \]
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Time = 9.82 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.00 \[ \int \frac {-10 x-6 x^2+28 x^3+38 x^4+18 x^5+4 x^6+e^{2 x} \left (2 x^4+2 x^5\right )+e^x \left (-2 x^2+10 x^3+22 x^4+12 x^5+2 x^6\right )+\left (2+10 x-2 x^3-2 e^x x^3\right ) \log (x)-2 \log ^2(x)}{x^3} \, dx=30\,x-6\,\ln \left (x\right )+x^2\,{\mathrm {e}}^{2\,x}+\frac {{\ln \left (x\right )}^2}{x^2}-\ln \left (x\right )\,\left (4\,x+2\,{\mathrm {e}}^x-\frac {2\,x^2-10}{x}\right )+19\,x^2+6\,x^3+x^4+{\mathrm {e}}^x\,\left (2\,x^3+6\,x^2+10\,x\right ) \]
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