Integrand size = 81, antiderivative size = 26 \[ \int \frac {e^{15} (250-1000 x)+e^{20} \left (-x+4 x^2\right )+\left (e^{10} (127500-10000 x)-500 e^{15} x\right ) \log \left (x^2\right )+\left (1875000 e^5-2500 e^{10} x\right ) \log ^2\left (x^2\right )+6250000 \log ^3\left (x^2\right )}{32 e^{20} x} \, dx=\frac {1}{16} \left (6+x-\left (\frac {5}{2}+\frac {25 \log \left (x^2\right )}{e^5}\right )^2\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(131\) vs. \(2(26)=52\).
Time = 0.10 (sec) , antiderivative size = 131, normalized size of antiderivative = 5.04, number of steps used = 15, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {12, 14, 77, 2388, 2338, 2332, 2339, 30, 2333} \[ \int \frac {e^{15} (250-1000 x)+e^{20} \left (-x+4 x^2\right )+\left (e^{10} (127500-10000 x)-500 e^{15} x\right ) \log \left (x^2\right )+\left (1875000 e^5-2500 e^{10} x\right ) \log ^2\left (x^2\right )+6250000 \log ^3\left (x^2\right )}{32 e^{20} x} \, dx=\frac {x^2}{16}+\frac {390625 \log ^4\left (x^2\right )}{16 e^{20}}+\frac {78125 \log ^3\left (x^2\right )}{8 e^{15}}-\frac {625 x \log ^2\left (x^2\right )}{8 e^{10}}+\frac {31875 \log ^2\left (x^2\right )}{32 e^{10}}-\frac {125 \left (20+e^5\right ) x \log \left (x^2\right )}{8 e^{10}}+\frac {625 x \log \left (x^2\right )}{2 e^{10}}-\frac {\left (1000+e^5\right ) x}{32 e^5}+\frac {125 \left (20+e^5\right ) x}{4 e^{10}}-\frac {625 x}{e^{10}}+\frac {125 \log (x)}{16 e^5} \]
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Rule 12
Rule 14
Rule 30
Rule 77
Rule 2332
Rule 2333
Rule 2338
Rule 2339
Rule 2388
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {e^{15} (250-1000 x)+e^{20} \left (-x+4 x^2\right )+\left (e^{10} (127500-10000 x)-500 e^{15} x\right ) \log \left (x^2\right )+\left (1875000 e^5-2500 e^{10} x\right ) \log ^2\left (x^2\right )+6250000 \log ^3\left (x^2\right )}{x} \, dx}{32 e^{20}} \\ & = \frac {\int \left (\frac {e^{15} (-1+4 x) \left (-250+e^5 x\right )}{x}+\frac {500 e^{10} \left (255-\left (20+e^5\right ) x\right ) \log \left (x^2\right )}{x}-\frac {2500 e^5 \left (-750+e^5 x\right ) \log ^2\left (x^2\right )}{x}+\frac {6250000 \log ^3\left (x^2\right )}{x}\right ) \, dx}{32 e^{20}} \\ & = \frac {390625 \int \frac {\log ^3\left (x^2\right )}{x} \, dx}{2 e^{20}}-\frac {625 \int \frac {\left (-750+e^5 x\right ) \log ^2\left (x^2\right )}{x} \, dx}{8 e^{15}}+\frac {125 \int \frac {\left (255+\left (-20-e^5\right ) x\right ) \log \left (x^2\right )}{x} \, dx}{8 e^{10}}+\frac {\int \frac {(-1+4 x) \left (-250+e^5 x\right )}{x} \, dx}{32 e^5} \\ & = \frac {390625 \text {Subst}\left (\int x^3 \, dx,x,\log \left (x^2\right )\right )}{4 e^{20}}+\frac {234375 \int \frac {\log ^2\left (x^2\right )}{x} \, dx}{4 e^{15}}-\frac {625 \int \log ^2\left (x^2\right ) \, dx}{8 e^{10}}+\frac {31875 \int \frac {\log \left (x^2\right )}{x} \, dx}{8 e^{10}}+\frac {\int \left (-1000-e^5+\frac {250}{x}+4 e^5 x\right ) \, dx}{32 e^5}-\frac {\left (125 \left (20+e^5\right )\right ) \int \log \left (x^2\right ) \, dx}{8 e^{10}} \\ & = \frac {125 \left (20+e^5\right ) x}{4 e^{10}}-\frac {\left (1000+e^5\right ) x}{32 e^5}+\frac {x^2}{16}+\frac {125 \log (x)}{16 e^5}-\frac {125 \left (20+e^5\right ) x \log \left (x^2\right )}{8 e^{10}}+\frac {31875 \log ^2\left (x^2\right )}{32 e^{10}}-\frac {625 x \log ^2\left (x^2\right )}{8 e^{10}}+\frac {390625 \log ^4\left (x^2\right )}{16 e^{20}}+\frac {234375 \text {Subst}\left (\int x^2 \, dx,x,\log \left (x^2\right )\right )}{8 e^{15}}+\frac {625 \int \log \left (x^2\right ) \, dx}{2 e^{10}} \\ & = -\frac {625 x}{e^{10}}+\frac {125 \left (20+e^5\right ) x}{4 e^{10}}-\frac {\left (1000+e^5\right ) x}{32 e^5}+\frac {x^2}{16}+\frac {125 \log (x)}{16 e^5}+\frac {625 x \log \left (x^2\right )}{2 e^{10}}-\frac {125 \left (20+e^5\right ) x \log \left (x^2\right )}{8 e^{10}}+\frac {31875 \log ^2\left (x^2\right )}{32 e^{10}}-\frac {625 x \log ^2\left (x^2\right )}{8 e^{10}}+\frac {78125 \log ^3\left (x^2\right )}{8 e^{15}}+\frac {390625 \log ^4\left (x^2\right )}{16 e^{20}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(81\) vs. \(2(26)=52\).
Time = 0.07 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.12 \[ \int \frac {e^{15} (250-1000 x)+e^{20} \left (-x+4 x^2\right )+\left (e^{10} (127500-10000 x)-500 e^{15} x\right ) \log \left (x^2\right )+\left (1875000 e^5-2500 e^{10} x\right ) \log ^2\left (x^2\right )+6250000 \log ^3\left (x^2\right )}{32 e^{20} x} \, dx=\frac {-e^{20} x+2 e^{20} x^2+250 e^{15} \log (x)-500 e^{15} x \log \left (x^2\right )+31875 e^{10} \log ^2\left (x^2\right )-2500 e^{10} x \log ^2\left (x^2\right )+312500 e^5 \log ^3\left (x^2\right )+781250 \log ^4\left (x^2\right )}{32 e^{20}} \]
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Time = 0.09 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.85
method | result | size |
risch | \(\frac {x^{2}}{16}-\frac {x}{32}+\frac {125 \ln \left (x \right ) {\mathrm e}^{-5}}{16}+\frac {390625 \,{\mathrm e}^{-20} \ln \left (x^{2}\right )^{4}}{16}+\frac {78125 \,{\mathrm e}^{-15} \ln \left (x^{2}\right )^{3}}{8}-\frac {625 \,{\mathrm e}^{-10} \ln \left (x^{2}\right )^{2} x}{8}-\frac {125 x \ln \left (x^{2}\right ) {\mathrm e}^{-5}}{8}-\frac {31875 \ln \left (x \right )^{2} {\mathrm e}^{-10}}{8}+\frac {31875 \ln \left (x \right ) \ln \left (x^{2}\right ) {\mathrm e}^{-10}}{8}\) | \(74\) |
norman | \(\left (\frac {125 \,{\mathrm e}^{10} \ln \left (x^{2}\right )}{32}+\frac {78125 \ln \left (x^{2}\right )^{3}}{8}-\frac {x \,{\mathrm e}^{15}}{32}+\frac {x^{2} {\mathrm e}^{15}}{16}+\frac {390625 \,{\mathrm e}^{-5} \ln \left (x^{2}\right )^{4}}{16}+\frac {31875 \,{\mathrm e}^{5} \ln \left (x^{2}\right )^{2}}{32}-\frac {625 x \,{\mathrm e}^{5} \ln \left (x^{2}\right )^{2}}{8}-\frac {125 \,{\mathrm e}^{10} \ln \left (x^{2}\right ) x}{8}\right ) {\mathrm e}^{-15}\) | \(85\) |
parts | \(\frac {390625 \,{\mathrm e}^{-20} \ln \left (x^{2}\right )^{4}}{16}+\frac {x^{2}}{16}-\frac {x}{32}+\frac {125 \ln \left (x \right ) {\mathrm e}^{-5}}{16}-\frac {125 \,{\mathrm e}^{-5} x}{4}+\left (-625 x +\frac {625 x \ln \left (x^{2}\right )}{2}+\frac {78125 \,{\mathrm e}^{-5} \ln \left (x^{2}\right )^{3}}{8}-\frac {625 x \ln \left (x^{2}\right )^{2}}{8}\right ) {\mathrm e}^{-10}-\frac {125 \,{\mathrm e}^{-10} \left (x \,{\mathrm e}^{5} \ln \left (x^{2}\right )-2 x \,{\mathrm e}^{5}+20 x \ln \left (x^{2}\right )-40 x -255 \ln \left (x \right ) \ln \left (x^{2}\right )+255 \ln \left (x \right )^{2}\right )}{8}\) | \(118\) |
default | \(\frac {{\mathrm e}^{-20} \left ({\mathrm e}^{15} \left (2 x^{2} {\mathrm e}^{5}-x \,{\mathrm e}^{5}-1000 x +250 \ln \left (x \right )\right )+781250 \ln \left (x^{2}\right )^{4}+312500 \,{\mathrm e}^{5} \ln \left (x^{2}\right )^{3}-20000 x \,{\mathrm e}^{10}+10000 \,{\mathrm e}^{10} \ln \left (x^{2}\right ) x -2500 \,{\mathrm e}^{10} \ln \left (x^{2}\right )^{2} x -500 \,{\mathrm e}^{10} \left (x \,{\mathrm e}^{5} \ln \left (x^{2}\right )-2 x \,{\mathrm e}^{5}+20 x \ln \left (x^{2}\right )-40 x -255 \ln \left (x \right ) \ln \left (x^{2}\right )+255 \ln \left (x \right )^{2}\right )\right )}{32}\) | \(126\) |
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Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (23) = 46\).
Time = 0.25 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.46 \[ \int \frac {e^{15} (250-1000 x)+e^{20} \left (-x+4 x^2\right )+\left (e^{10} (127500-10000 x)-500 e^{15} x\right ) \log \left (x^2\right )+\left (1875000 e^5-2500 e^{10} x\right ) \log ^2\left (x^2\right )+6250000 \log ^3\left (x^2\right )}{32 e^{20} x} \, dx=-\frac {1}{32} \, {\left (625 \, {\left (4 \, x - 51\right )} e^{10} \log \left (x^{2}\right )^{2} - 312500 \, e^{5} \log \left (x^{2}\right )^{3} - 781250 \, \log \left (x^{2}\right )^{4} + 125 \, {\left (4 \, x - 1\right )} e^{15} \log \left (x^{2}\right ) - {\left (2 \, x^{2} - x\right )} e^{20}\right )} e^{\left (-20\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (24) = 48\).
Time = 0.15 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.08 \[ \int \frac {e^{15} (250-1000 x)+e^{20} \left (-x+4 x^2\right )+\left (e^{10} (127500-10000 x)-500 e^{15} x\right ) \log \left (x^2\right )+\left (1875000 e^5-2500 e^{10} x\right ) \log ^2\left (x^2\right )+6250000 \log ^3\left (x^2\right )}{32 e^{20} x} \, dx=- \frac {125 x \log {\left (x^{2} \right )}}{8 e^{5}} + \frac {\left (31875 - 2500 x\right ) \log {\left (x^{2} \right )}^{2}}{32 e^{10}} + \frac {2 x^{2} e^{5} - x e^{5} + 250 \log {\left (x \right )}}{32 e^{5}} + \frac {390625 \log {\left (x^{2} \right )}^{4}}{16 e^{20}} + \frac {78125 \log {\left (x^{2} \right )}^{3}}{8 e^{15}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (23) = 46\).
Time = 0.20 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.08 \[ \int \frac {e^{15} (250-1000 x)+e^{20} \left (-x+4 x^2\right )+\left (e^{10} (127500-10000 x)-500 e^{15} x\right ) \log \left (x^2\right )+\left (1875000 e^5-2500 e^{10} x\right ) \log ^2\left (x^2\right )+6250000 \log ^3\left (x^2\right )}{32 e^{20} x} \, dx=-\frac {1}{32} \, {\left (2500 \, x e^{10} \log \left (x^{2}\right )^{2} - 312500 \, e^{5} \log \left (x^{2}\right )^{3} - 781250 \, \log \left (x^{2}\right )^{4} - 2 \, x^{2} e^{20} - 31875 \, e^{10} \log \left (x^{2}\right )^{2} + x e^{20} + 500 \, {\left (x \log \left (x^{2}\right ) - 2 \, x\right )} e^{15} + 1000 \, x e^{15} - 250 \, e^{15} \log \left (x\right )\right )} e^{\left (-20\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (23) = 46\).
Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.69 \[ \int \frac {e^{15} (250-1000 x)+e^{20} \left (-x+4 x^2\right )+\left (e^{10} (127500-10000 x)-500 e^{15} x\right ) \log \left (x^2\right )+\left (1875000 e^5-2500 e^{10} x\right ) \log ^2\left (x^2\right )+6250000 \log ^3\left (x^2\right )}{32 e^{20} x} \, dx=-\frac {1}{32} \, {\left (2500 \, x e^{10} \log \left (x^{2}\right )^{2} - 312500 \, e^{5} \log \left (x^{2}\right )^{3} - 781250 \, \log \left (x^{2}\right )^{4} - 2 \, x^{2} e^{20} + 500 \, x e^{15} \log \left (x^{2}\right ) - 31875 \, e^{10} \log \left (x^{2}\right )^{2} + x e^{20} - 250 \, e^{15} \log \left (x\right )\right )} e^{\left (-20\right )} \]
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Time = 9.39 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.58 \[ \int \frac {e^{15} (250-1000 x)+e^{20} \left (-x+4 x^2\right )+\left (e^{10} (127500-10000 x)-500 e^{15} x\right ) \log \left (x^2\right )+\left (1875000 e^5-2500 e^{10} x\right ) \log ^2\left (x^2\right )+6250000 \log ^3\left (x^2\right )}{32 e^{20} x} \, dx=\frac {x^2}{16}-\frac {625\,{\mathrm {e}}^{-10}\,x\,{\ln \left (x^2\right )}^2}{8}-\frac {125\,{\mathrm {e}}^{-5}\,x\,\ln \left (x^2\right )}{8}-\frac {x}{32}+\frac {390625\,{\mathrm {e}}^{-20}\,{\ln \left (x^2\right )}^4}{16}+\frac {78125\,{\mathrm {e}}^{-15}\,{\ln \left (x^2\right )}^3}{8}+\frac {31875\,{\mathrm {e}}^{-10}\,{\ln \left (x^2\right )}^2}{32}+\frac {125\,{\mathrm {e}}^{-5}\,\ln \left (x^2\right )}{32} \]
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