Integrand size = 94, antiderivative size = 30 \[ \int \frac {1}{5} \left (5-10 x+30 x^2-25 x^4+\left (40 x-40 x^3\right ) \log (4)+\left (10-15 x^2\right ) \log ^2(4)+e^{x/5} \left (-5-x-20 x^3-x^4+\left (-30 x^2-2 x^3\right ) \log (4)+\left (-10 x-x^2\right ) \log ^2(4)\right )\right ) \, dx=5+x-x \left (e^{x/5}-\frac {2}{x}+x\right ) \left (1+x (x+\log (4))^2\right ) \]
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Leaf count is larger than twice the leaf count of optimal. \(99\) vs. \(2(30)=60\).
Time = 0.19 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.30, number of steps used = 34, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {12, 2227, 2225, 2207} \[ \int \frac {1}{5} \left (5-10 x+30 x^2-25 x^4+\left (40 x-40 x^3\right ) \log (4)+\left (10-15 x^2\right ) \log ^2(4)+e^{x/5} \left (-5-x-20 x^3-x^4+\left (-30 x^2-2 x^3\right ) \log (4)+\left (-10 x-x^2\right ) \log ^2(4)\right )\right ) \, dx=-x^5-e^{x/5} x^4-2 x^4 \log (4)+2 x^3-x^3 \log ^2(4)-2 e^{x/5} x^3 \log (4)-x^2-e^{x/5} x^2 \log ^2(4)+4 x^2 \log (4)-e^{x/5} x+x+2 x \log ^2(4) \]
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Rule 12
Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \int \left (5-10 x+30 x^2-25 x^4+\left (40 x-40 x^3\right ) \log (4)+\left (10-15 x^2\right ) \log ^2(4)+e^{x/5} \left (-5-x-20 x^3-x^4+\left (-30 x^2-2 x^3\right ) \log (4)+\left (-10 x-x^2\right ) \log ^2(4)\right )\right ) \, dx \\ & = x-x^2+2 x^3-x^5+\frac {1}{5} \int e^{x/5} \left (-5-x-20 x^3-x^4+\left (-30 x^2-2 x^3\right ) \log (4)+\left (-10 x-x^2\right ) \log ^2(4)\right ) \, dx+\frac {1}{5} \log (4) \int \left (40 x-40 x^3\right ) \, dx+\frac {1}{5} \log ^2(4) \int \left (10-15 x^2\right ) \, dx \\ & = x-x^2+2 x^3-x^5+4 x^2 \log (4)-2 x^4 \log (4)+2 x \log ^2(4)-x^3 \log ^2(4)+\frac {1}{5} \int \left (-5 e^{x/5}-e^{x/5} x-20 e^{x/5} x^3-e^{x/5} x^4-2 e^{x/5} x^2 (15+x) \log (4)-e^{x/5} x (10+x) \log ^2(4)\right ) \, dx \\ & = x-x^2+2 x^3-x^5+4 x^2 \log (4)-2 x^4 \log (4)+2 x \log ^2(4)-x^3 \log ^2(4)-\frac {1}{5} \int e^{x/5} x \, dx-\frac {1}{5} \int e^{x/5} x^4 \, dx-4 \int e^{x/5} x^3 \, dx-\frac {1}{5} (2 \log (4)) \int e^{x/5} x^2 (15+x) \, dx-\frac {1}{5} \log ^2(4) \int e^{x/5} x (10+x) \, dx-\int e^{x/5} \, dx \\ & = -5 e^{x/5}+x-e^{x/5} x-x^2+2 x^3-20 e^{x/5} x^3-e^{x/5} x^4-x^5+4 x^2 \log (4)-2 x^4 \log (4)+2 x \log ^2(4)-x^3 \log ^2(4)+4 \int e^{x/5} x^3 \, dx+60 \int e^{x/5} x^2 \, dx-\frac {1}{5} (2 \log (4)) \int \left (15 e^{x/5} x^2+e^{x/5} x^3\right ) \, dx-\frac {1}{5} \log ^2(4) \int \left (10 e^{x/5} x+e^{x/5} x^2\right ) \, dx+\int e^{x/5} \, dx \\ & = x-e^{x/5} x-x^2+300 e^{x/5} x^2+2 x^3-e^{x/5} x^4-x^5+4 x^2 \log (4)-2 x^4 \log (4)+2 x \log ^2(4)-x^3 \log ^2(4)-60 \int e^{x/5} x^2 \, dx-600 \int e^{x/5} x \, dx-\frac {1}{5} (2 \log (4)) \int e^{x/5} x^3 \, dx-(6 \log (4)) \int e^{x/5} x^2 \, dx-\frac {1}{5} \log ^2(4) \int e^{x/5} x^2 \, dx-\left (2 \log ^2(4)\right ) \int e^{x/5} x \, dx \\ & = x-3001 e^{x/5} x-x^2+2 x^3-e^{x/5} x^4-x^5+4 x^2 \log (4)-30 e^{x/5} x^2 \log (4)-2 e^{x/5} x^3 \log (4)-2 x^4 \log (4)+2 x \log ^2(4)-10 e^{x/5} x \log ^2(4)-e^{x/5} x^2 \log ^2(4)-x^3 \log ^2(4)+600 \int e^{x/5} x \, dx+3000 \int e^{x/5} \, dx+(6 \log (4)) \int e^{x/5} x^2 \, dx+(60 \log (4)) \int e^{x/5} x \, dx+\left (2 \log ^2(4)\right ) \int e^{x/5} x \, dx+\left (10 \log ^2(4)\right ) \int e^{x/5} \, dx \\ & = 15000 e^{x/5}+x-e^{x/5} x-x^2+2 x^3-e^{x/5} x^4-x^5+300 e^{x/5} x \log (4)+4 x^2 \log (4)-2 e^{x/5} x^3 \log (4)-2 x^4 \log (4)+50 e^{x/5} \log ^2(4)+2 x \log ^2(4)-e^{x/5} x^2 \log ^2(4)-x^3 \log ^2(4)-3000 \int e^{x/5} \, dx-(60 \log (4)) \int e^{x/5} x \, dx-(300 \log (4)) \int e^{x/5} \, dx-\left (10 \log ^2(4)\right ) \int e^{x/5} \, dx \\ & = x-e^{x/5} x-x^2+2 x^3-e^{x/5} x^4-x^5-1500 e^{x/5} \log (4)+4 x^2 \log (4)-2 e^{x/5} x^3 \log (4)-2 x^4 \log (4)+2 x \log ^2(4)-e^{x/5} x^2 \log ^2(4)-x^3 \log ^2(4)+(300 \log (4)) \int e^{x/5} \, dx \\ & = x-e^{x/5} x-x^2+2 x^3-e^{x/5} x^4-x^5+4 x^2 \log (4)-2 e^{x/5} x^3 \log (4)-2 x^4 \log (4)+2 x \log ^2(4)-e^{x/5} x^2 \log ^2(4)-x^3 \log ^2(4) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(69\) vs. \(2(30)=60\).
Time = 0.26 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.30 \[ \int \frac {1}{5} \left (5-10 x+30 x^2-25 x^4+\left (40 x-40 x^3\right ) \log (4)+\left (10-15 x^2\right ) \log ^2(4)+e^{x/5} \left (-5-x-20 x^3-x^4+\left (-30 x^2-2 x^3\right ) \log (4)+\left (-10 x-x^2\right ) \log ^2(4)\right )\right ) \, dx=x-x^5-2 x^4 \log (4)+2 x \log ^2(4)-x^3 \left (-2+\log ^2(4)\right )-e^{x/5} \left (x+x^4+2 x^3 \log (4)+x^2 \log ^2(4)\right )+x^2 (-1+\log (256)) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(82\) vs. \(2(29)=58\).
Time = 0.18 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.77
method | result | size |
risch | \(\frac {\left (-20 x^{2} \ln \left (2\right )^{2}-20 x^{3} \ln \left (2\right )-5 x^{4}-5 x \right ) {\mathrm e}^{\frac {x}{5}}}{5}-4 x^{3} \ln \left (2\right )^{2}+8 x \ln \left (2\right )^{2}-4 x^{4} \ln \left (2\right )+8 x^{2} \ln \left (2\right )-4 \ln \left (2\right )-x^{5}+2 x^{3}-x^{2}+x\) | \(83\) |
norman | \(\left (2-4 \ln \left (2\right )^{2}\right ) x^{3}+\left (8 \ln \left (2\right )^{2}+1\right ) x +\left (8 \ln \left (2\right )-1\right ) x^{2}-x^{5}-x \,{\mathrm e}^{\frac {x}{5}}-x^{4} {\mathrm e}^{\frac {x}{5}}-4 x^{4} \ln \left (2\right )-4 \,{\mathrm e}^{\frac {x}{5}} \ln \left (2\right ) x^{3}-4 \,{\mathrm e}^{\frac {x}{5}} \ln \left (2\right )^{2} x^{2}\) | \(86\) |
derivativedivides | \(x -x \,{\mathrm e}^{\frac {x}{5}}-x^{4} {\mathrm e}^{\frac {x}{5}}-4 \,{\mathrm e}^{\frac {x}{5}} \ln \left (2\right ) x^{3}-4 \,{\mathrm e}^{\frac {x}{5}} \ln \left (2\right )^{2} x^{2}-x^{2}+2 x^{3}-x^{5}+20 \ln \left (2\right )^{2} \left (-\frac {1}{5} x^{3}+\frac {2}{5} x \right )-4 x^{4} \ln \left (2\right )+8 x^{2} \ln \left (2\right )\) | \(87\) |
default | \(x -x^{4} {\mathrm e}^{\frac {x}{5}}-x \,{\mathrm e}^{\frac {x}{5}}-4 \,{\mathrm e}^{\frac {x}{5}} \ln \left (2\right )^{2} x^{2}-4 \,{\mathrm e}^{\frac {x}{5}} \ln \left (2\right ) x^{3}-x^{2}+2 x^{3}-x^{5}+4 \ln \left (2\right )^{2} \left (-x^{3}+2 x \right )-4 x^{4} \ln \left (2\right )+8 x^{2} \ln \left (2\right )\) | \(87\) |
parallelrisch | \(8 x \ln \left (2\right )^{2}+8 x^{2} \ln \left (2\right )-x^{2}-4 x^{3} \ln \left (2\right )^{2}+2 x^{3}-4 x^{4} \ln \left (2\right )-x^{5}-x^{4} {\mathrm e}^{\frac {x}{5}}-4 \,{\mathrm e}^{\frac {x}{5}} \ln \left (2\right )^{2} x^{2}-x \,{\mathrm e}^{\frac {x}{5}}-4 \,{\mathrm e}^{\frac {x}{5}} \ln \left (2\right ) x^{3}+x\) | \(88\) |
parts | \(8 x \ln \left (2\right )^{2}+8 x^{2} \ln \left (2\right )-x^{2}-4 x^{3} \ln \left (2\right )^{2}+2 x^{3}-4 x^{4} \ln \left (2\right )-x^{5}-x^{4} {\mathrm e}^{\frac {x}{5}}-4 \,{\mathrm e}^{\frac {x}{5}} \ln \left (2\right )^{2} x^{2}-x \,{\mathrm e}^{\frac {x}{5}}-4 \,{\mathrm e}^{\frac {x}{5}} \ln \left (2\right ) x^{3}+x\) | \(88\) |
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (29) = 58\).
Time = 0.24 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.33 \[ \int \frac {1}{5} \left (5-10 x+30 x^2-25 x^4+\left (40 x-40 x^3\right ) \log (4)+\left (10-15 x^2\right ) \log ^2(4)+e^{x/5} \left (-5-x-20 x^3-x^4+\left (-30 x^2-2 x^3\right ) \log (4)+\left (-10 x-x^2\right ) \log ^2(4)\right )\right ) \, dx=-x^{5} + 2 \, x^{3} - 4 \, {\left (x^{3} - 2 \, x\right )} \log \left (2\right )^{2} - x^{2} - {\left (x^{4} + 4 \, x^{3} \log \left (2\right ) + 4 \, x^{2} \log \left (2\right )^{2} + x\right )} e^{\left (\frac {1}{5} \, x\right )} - 4 \, {\left (x^{4} - 2 \, x^{2}\right )} \log \left (2\right ) + x \]
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Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (26) = 52\).
Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.43 \[ \int \frac {1}{5} \left (5-10 x+30 x^2-25 x^4+\left (40 x-40 x^3\right ) \log (4)+\left (10-15 x^2\right ) \log ^2(4)+e^{x/5} \left (-5-x-20 x^3-x^4+\left (-30 x^2-2 x^3\right ) \log (4)+\left (-10 x-x^2\right ) \log ^2(4)\right )\right ) \, dx=- x^{5} - 4 x^{4} \log {\left (2 \right )} + x^{3} \cdot \left (2 - 4 \log {\left (2 \right )}^{2}\right ) + x^{2} \left (-1 + 8 \log {\left (2 \right )}\right ) + x \left (1 + 8 \log {\left (2 \right )}^{2}\right ) + \left (- x^{4} - 4 x^{3} \log {\left (2 \right )} - 4 x^{2} \log {\left (2 \right )}^{2} - x\right ) e^{\frac {x}{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (29) = 58\).
Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.33 \[ \int \frac {1}{5} \left (5-10 x+30 x^2-25 x^4+\left (40 x-40 x^3\right ) \log (4)+\left (10-15 x^2\right ) \log ^2(4)+e^{x/5} \left (-5-x-20 x^3-x^4+\left (-30 x^2-2 x^3\right ) \log (4)+\left (-10 x-x^2\right ) \log ^2(4)\right )\right ) \, dx=-x^{5} + 2 \, x^{3} - 4 \, {\left (x^{3} - 2 \, x\right )} \log \left (2\right )^{2} - x^{2} - {\left (x^{4} + 4 \, x^{3} \log \left (2\right ) + 4 \, x^{2} \log \left (2\right )^{2} + x\right )} e^{\left (\frac {1}{5} \, x\right )} - 4 \, {\left (x^{4} - 2 \, x^{2}\right )} \log \left (2\right ) + x \]
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (29) = 58\).
Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.33 \[ \int \frac {1}{5} \left (5-10 x+30 x^2-25 x^4+\left (40 x-40 x^3\right ) \log (4)+\left (10-15 x^2\right ) \log ^2(4)+e^{x/5} \left (-5-x-20 x^3-x^4+\left (-30 x^2-2 x^3\right ) \log (4)+\left (-10 x-x^2\right ) \log ^2(4)\right )\right ) \, dx=-x^{5} + 2 \, x^{3} - 4 \, {\left (x^{3} - 2 \, x\right )} \log \left (2\right )^{2} - x^{2} - {\left (x^{4} + 4 \, x^{3} \log \left (2\right ) + 4 \, x^{2} \log \left (2\right )^{2} + x\right )} e^{\left (\frac {1}{5} \, x\right )} - 4 \, {\left (x^{4} - 2 \, x^{2}\right )} \log \left (2\right ) + x \]
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Time = 12.76 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.87 \[ \int \frac {1}{5} \left (5-10 x+30 x^2-25 x^4+\left (40 x-40 x^3\right ) \log (4)+\left (10-15 x^2\right ) \log ^2(4)+e^{x/5} \left (-5-x-20 x^3-x^4+\left (-30 x^2-2 x^3\right ) \log (4)+\left (-10 x-x^2\right ) \log ^2(4)\right )\right ) \, dx=x\,\left (8\,{\ln \left (2\right )}^2+1\right )-x\,{\mathrm {e}}^{x/5}+x^2\,\left (8\,\ln \left (2\right )-1\right )-x^4\,{\mathrm {e}}^{x/5}-4\,x^4\,\ln \left (2\right )-x^3\,\left (4\,{\ln \left (2\right )}^2-2\right )-x^5-4\,x^2\,{\mathrm {e}}^{x/5}\,{\ln \left (2\right )}^2-4\,x^3\,{\mathrm {e}}^{x/5}\,\ln \left (2\right ) \]
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