Integrand size = 156, antiderivative size = 24 \[ \int \frac {-9 x \log (x)-90 e^{5 e^{2 x}+2 x} x \log (x)+\left (9 x+x^2+e^{10 e^{2 x}} (1-\log (x))+\left (-9 x-x^2\right ) \log (x)+e^{5 e^{2 x}} (9+2 x+(-9-2 x) \log (x))\right ) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{e^{10 e^{2 x}} x^2+9 x^3+x^4+e^{5 e^{2 x}} \left (9 x^2+2 x^3\right )} \, dx=\frac {\log (x) \log \left (1+\frac {9}{e^{5 e^{2 x}}+x}\right )}{x} \]
[Out]
Time = 7.72 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38, number of steps used = 39, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.064, Rules used = {6874, 2634, 12, 14, 6820, 2340, 2631, 2341, 30, 2637} \[ \int \frac {-9 x \log (x)-90 e^{5 e^{2 x}+2 x} x \log (x)+\left (9 x+x^2+e^{10 e^{2 x}} (1-\log (x))+\left (-9 x-x^2\right ) \log (x)+e^{5 e^{2 x}} (9+2 x+(-9-2 x) \log (x))\right ) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{e^{10 e^{2 x}} x^2+9 x^3+x^4+e^{5 e^{2 x}} \left (9 x^2+2 x^3\right )} \, dx=\frac {\log (x) \log \left (\frac {x+e^{5 e^{2 x}}+9}{x+e^{5 e^{2 x}}}\right )}{x} \]
[In]
[Out]
Rule 12
Rule 14
Rule 30
Rule 2340
Rule 2341
Rule 2631
Rule 2634
Rule 2637
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {90 e^{5 e^{2 x}+2 x} \log (x)}{x \left (e^{5 e^{2 x}}+x\right ) \left (9+e^{5 e^{2 x}}+x\right )}-\frac {9 x \log (x)-9 e^{5 e^{2 x}} \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )-e^{10 e^{2 x}} \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )-9 x \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )-2 e^{5 e^{2 x}} x \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )-x^2 \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )+9 e^{5 e^{2 x}} \log (x) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )+e^{10 e^{2 x}} \log (x) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )+9 x \log (x) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )+2 e^{5 e^{2 x}} x \log (x) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )+x^2 \log (x) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{x^2 \left (e^{5 e^{2 x}}+x\right ) \left (9+e^{5 e^{2 x}}+x\right )}\right ) \, dx \\ & = -\left (90 \int \frac {e^{5 e^{2 x}+2 x} \log (x)}{x \left (e^{5 e^{2 x}}+x\right ) \left (9+e^{5 e^{2 x}}+x\right )} \, dx\right )-\int \frac {9 x \log (x)-9 e^{5 e^{2 x}} \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )-e^{10 e^{2 x}} \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )-9 x \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )-2 e^{5 e^{2 x}} x \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )-x^2 \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )+9 e^{5 e^{2 x}} \log (x) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )+e^{10 e^{2 x}} \log (x) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )+9 x \log (x) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )+2 e^{5 e^{2 x}} x \log (x) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )+x^2 \log (x) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{x^2 \left (e^{5 e^{2 x}}+x\right ) \left (9+e^{5 e^{2 x}}+x\right )} \, dx \\ & = 90 \int \frac {\int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx-\int \frac {e^{5 e^{2 x}+2 x}}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{9 x} \, dx-(10 \log (x)) \int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx+(10 \log (x)) \int \frac {e^{5 e^{2 x}+2 x}}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx-\int \frac {-\left (\left (e^{10 e^{2 x}}+x (9+x)+e^{5 e^{2 x}} (9+2 x)\right ) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )\right )+\log (x) \left (9 x+\left (e^{10 e^{2 x}}+x (9+x)+e^{5 e^{2 x}} (9+2 x)\right ) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )\right )}{x^2 \left (e^{5 e^{2 x}}+x\right ) \left (9+e^{5 e^{2 x}}+x\right )} \, dx \\ & = 10 \int \frac {\int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx-\int \frac {e^{5 e^{2 x}+2 x}}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx-(10 \log (x)) \int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx+(10 \log (x)) \int \frac {e^{5 e^{2 x}+2 x}}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx-\int \left (\frac {\log (x)}{x \left (e^{5 e^{2 x}}+x\right )}-\frac {\log (x)}{x \left (9+e^{5 e^{2 x}}+x\right )}+\frac {(-1+\log (x)) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{x^2}\right ) \, dx \\ & = 10 \int \left (\frac {\int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx}{x}-\frac {\int \frac {e^{5 e^{2 x}+2 x}}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{x}\right ) \, dx-(10 \log (x)) \int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx+(10 \log (x)) \int \frac {e^{5 e^{2 x}+2 x}}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx-\int \frac {\log (x)}{x \left (e^{5 e^{2 x}}+x\right )} \, dx+\int \frac {\log (x)}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx-\int \frac {(-1+\log (x)) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{x^2} \, dx \\ & = 10 \int \frac {\int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx-10 \int \frac {\int \frac {e^{5 e^{2 x}+2 x}}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx-\log (x) \int \frac {1}{x \left (e^{5 e^{2 x}}+x\right )} \, dx+\log (x) \int \frac {1}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx-(10 \log (x)) \int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx+(10 \log (x)) \int \frac {e^{5 e^{2 x}+2 x}}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx-\int \left (-\frac {\log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{x^2}+\frac {\log (x) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{x^2}\right ) \, dx-\int \frac {\int \frac {1}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx+\int \frac {\int \frac {1}{e^{5 e^{2 x}} x+x^2} \, dx}{x} \, dx \\ & = 10 \int \frac {\int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx-10 \int \frac {\int \frac {e^{5 e^{2 x}+2 x}}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx-\log (x) \int \frac {1}{x \left (e^{5 e^{2 x}}+x\right )} \, dx+\log (x) \int \frac {1}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx-(10 \log (x)) \int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx+(10 \log (x)) \int \frac {e^{5 e^{2 x}+2 x}}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx+\int \frac {\log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{x^2} \, dx-\int \frac {\log (x) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{x^2} \, dx-\int \frac {\int \frac {1}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx+\int \frac {\int \frac {1}{e^{5 e^{2 x}} x+x^2} \, dx}{x} \, dx \\ & = -\frac {\log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{x}+\frac {\log (x) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{x}+10 \int \frac {\int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx-10 \int \frac {\int \frac {e^{5 e^{2 x}+2 x}}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx-\log (x) \int \frac {1}{x \left (e^{5 e^{2 x}}+x\right )} \, dx+\log (x) \int \frac {1}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx-(10 \log (x)) \int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx+(10 \log (x)) \int \frac {e^{5 e^{2 x}+2 x}}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx+\int \frac {9 \left (-1-10 e^{5 e^{2 x}+2 x}\right )}{x \left (e^{5 e^{2 x}}+x\right ) \left (9+e^{5 e^{2 x}}+x\right )} \, dx+\int \frac {9 \left (1+10 e^{5 e^{2 x}+2 x}\right ) \log (x)}{x \left (e^{5 e^{2 x}}+x\right ) \left (9+e^{5 e^{2 x}}+x\right )} \, dx-\int \frac {\log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{x^2} \, dx-\int \frac {\int \frac {1}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx+\int \frac {\int \frac {1}{e^{5 e^{2 x}} x+x^2} \, dx}{x} \, dx \\ & = \frac {\log (x) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{x}+9 \int \frac {-1-10 e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right ) \left (9+e^{5 e^{2 x}}+x\right )} \, dx+9 \int \frac {\left (1+10 e^{5 e^{2 x}+2 x}\right ) \log (x)}{x \left (e^{5 e^{2 x}}+x\right ) \left (9+e^{5 e^{2 x}}+x\right )} \, dx+10 \int \frac {\int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx-10 \int \frac {\int \frac {e^{5 e^{2 x}+2 x}}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx-\log (x) \int \frac {1}{x \left (e^{5 e^{2 x}}+x\right )} \, dx+\log (x) \int \frac {1}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx-(10 \log (x)) \int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx+(10 \log (x)) \int \frac {e^{5 e^{2 x}+2 x}}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx-\int \frac {9 \left (-1-10 e^{5 e^{2 x}+2 x}\right )}{x \left (e^{5 e^{2 x}}+x\right ) \left (9+e^{5 e^{2 x}}+x\right )} \, dx-\int \frac {\int \frac {1}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx+\int \frac {\int \frac {1}{e^{5 e^{2 x}} x+x^2} \, dx}{x} \, dx \\ & = \frac {\log (x) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{x}-9 \int \frac {-1-10 e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right ) \left (9+e^{5 e^{2 x}}+x\right )} \, dx+9 \int \left (-\frac {1}{x \left (e^{5 e^{2 x}}+x\right ) \left (9+e^{5 e^{2 x}}+x\right )}-\frac {10 e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right ) \left (9+e^{5 e^{2 x}}+x\right )}\right ) \, dx-9 \int \frac {10 \int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx-\int \frac {1}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx-10 \int \frac {e^{5 e^{2 x}+2 x}}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx+\int \frac {1}{e^{5 e^{2 x}} x+x^2} \, dx}{9 x} \, dx+10 \int \frac {\int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx-10 \int \frac {\int \frac {e^{5 e^{2 x}+2 x}}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx-\int \frac {\int \frac {1}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx+\int \frac {\int \frac {1}{e^{5 e^{2 x}} x+x^2} \, dx}{x} \, dx \\ & = \frac {\log (x) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{x}-9 \int \frac {1}{x \left (e^{5 e^{2 x}}+x\right ) \left (9+e^{5 e^{2 x}}+x\right )} \, dx-9 \int \left (-\frac {1}{x \left (e^{5 e^{2 x}}+x\right ) \left (9+e^{5 e^{2 x}}+x\right )}-\frac {10 e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right ) \left (9+e^{5 e^{2 x}}+x\right )}\right ) \, dx+10 \int \frac {\int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx-10 \int \frac {\int \frac {e^{5 e^{2 x}+2 x}}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx-90 \int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right ) \left (9+e^{5 e^{2 x}}+x\right )} \, dx-\int \frac {\int \frac {1}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx+\int \frac {\int \frac {1}{e^{5 e^{2 x}} x+x^2} \, dx}{x} \, dx-\int \frac {10 \int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx-\int \frac {1}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx-10 \int \frac {e^{5 e^{2 x}+2 x}}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx+\int \frac {1}{e^{5 e^{2 x}} x+x^2} \, dx}{x} \, dx \\ & = \frac {\log (x) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{x}+9 \int \frac {1}{x \left (e^{5 e^{2 x}}+x\right ) \left (9+e^{5 e^{2 x}}+x\right )} \, dx-9 \int \left (\frac {1}{9 x \left (e^{5 e^{2 x}}+x\right )}-\frac {1}{9 x \left (9+e^{5 e^{2 x}}+x\right )}\right ) \, dx+10 \int \frac {\int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx-10 \int \frac {\int \frac {e^{5 e^{2 x}+2 x}}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx+90 \int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right ) \left (9+e^{5 e^{2 x}}+x\right )} \, dx-90 \int \left (\frac {e^{5 e^{2 x}+2 x}}{9 x \left (e^{5 e^{2 x}}+x\right )}-\frac {e^{5 e^{2 x}+2 x}}{9 x \left (9+e^{5 e^{2 x}}+x\right )}\right ) \, dx-\int \frac {\int \frac {1}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx+\int \frac {\int \frac {1}{e^{5 e^{2 x}} x+x^2} \, dx}{x} \, dx-\int \left (\frac {10 \int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx-\int \frac {1}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx-10 \int \frac {e^{5 e^{2 x}+2 x}}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{x}+\frac {\int \frac {1}{e^{5 e^{2 x}} x+x^2} \, dx}{x}\right ) \, dx \\ & = \frac {\log (x) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{x}+9 \int \left (\frac {1}{9 x \left (e^{5 e^{2 x}}+x\right )}-\frac {1}{9 x \left (9+e^{5 e^{2 x}}+x\right )}\right ) \, dx-10 \int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx+10 \int \frac {e^{5 e^{2 x}+2 x}}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx+10 \int \frac {\int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx-10 \int \frac {\int \frac {e^{5 e^{2 x}+2 x}}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx+90 \int \left (\frac {e^{5 e^{2 x}+2 x}}{9 x \left (e^{5 e^{2 x}}+x\right )}-\frac {e^{5 e^{2 x}+2 x}}{9 x \left (9+e^{5 e^{2 x}}+x\right )}\right ) \, dx-\int \frac {1}{x \left (e^{5 e^{2 x}}+x\right )} \, dx+\int \frac {1}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx-\int \frac {\int \frac {1}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx-\int \frac {10 \int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx-\int \frac {1}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx-10 \int \frac {e^{5 e^{2 x}+2 x}}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx \\ & = \frac {\log (x) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{x}+10 \int \frac {\int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx-10 \int \frac {\int \frac {e^{5 e^{2 x}+2 x}}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx-\int \frac {\int \frac {1}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx-\int \left (\frac {10 \int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx-\int \frac {1}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{x}-\frac {10 \int \frac {e^{5 e^{2 x}+2 x}}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{x}\right ) \, dx \\ & = \frac {\log (x) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{x}+10 \int \frac {\int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx-\int \frac {10 \int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx-\int \frac {1}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx-\int \frac {\int \frac {1}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx \\ & = \frac {\log (x) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{x}+10 \int \frac {\int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx-\int \frac {\int \frac {1}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{x} \, dx-\int \left (\frac {10 \int \frac {e^{5 e^{2 x}+2 x}}{x \left (e^{5 e^{2 x}}+x\right )} \, dx}{x}-\frac {\int \frac {1}{x \left (9+e^{5 e^{2 x}}+x\right )} \, dx}{x}\right ) \, dx \\ & = \frac {\log (x) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{x} \\ \end{align*}
Time = 0.50 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38 \[ \int \frac {-9 x \log (x)-90 e^{5 e^{2 x}+2 x} x \log (x)+\left (9 x+x^2+e^{10 e^{2 x}} (1-\log (x))+\left (-9 x-x^2\right ) \log (x)+e^{5 e^{2 x}} (9+2 x+(-9-2 x) \log (x))\right ) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{e^{10 e^{2 x}} x^2+9 x^3+x^4+e^{5 e^{2 x}} \left (9 x^2+2 x^3\right )} \, dx=\frac {\log (x) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{x} \]
[In]
[Out]
Time = 49.65 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25
method | result | size |
parallelrisch | \(\frac {\ln \left (\frac {{\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x +9}{{\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x}\right ) \ln \left (x \right )}{x}\) | \(30\) |
risch | \(\frac {\ln \left (x \right ) \ln \left ({\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x +9\right )}{x}-\frac {\ln \left (x \right ) \left (i \pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x}\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x +9\right )\right ) \operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x +9\right )}{{\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x}\right )-i \pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x}\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x +9\right )}{{\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x}\right )}^{2}-i \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x +9\right )\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x +9\right )}{{\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x}\right )}^{2}+i \pi {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x +9\right )}{{\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x}\right )}^{3}+2 \ln \left ({\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x \right )\right )}{2 x}\) | \(219\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (22) = 44\).
Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.00 \[ \int \frac {-9 x \log (x)-90 e^{5 e^{2 x}+2 x} x \log (x)+\left (9 x+x^2+e^{10 e^{2 x}} (1-\log (x))+\left (-9 x-x^2\right ) \log (x)+e^{5 e^{2 x}} (9+2 x+(-9-2 x) \log (x))\right ) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{e^{10 e^{2 x}} x^2+9 x^3+x^4+e^{5 e^{2 x}} \left (9 x^2+2 x^3\right )} \, dx=\frac {\log \left (x\right ) \log \left (\frac {{\left (x + 9\right )} e^{\left (2 \, x\right )} + e^{\left (2 \, x + 5 \, e^{\left (2 \, x\right )}\right )}}{x e^{\left (2 \, x\right )} + e^{\left (2 \, x + 5 \, e^{\left (2 \, x\right )}\right )}}\right )}{x} \]
[In]
[Out]
Time = 2.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {-9 x \log (x)-90 e^{5 e^{2 x}+2 x} x \log (x)+\left (9 x+x^2+e^{10 e^{2 x}} (1-\log (x))+\left (-9 x-x^2\right ) \log (x)+e^{5 e^{2 x}} (9+2 x+(-9-2 x) \log (x))\right ) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{e^{10 e^{2 x}} x^2+9 x^3+x^4+e^{5 e^{2 x}} \left (9 x^2+2 x^3\right )} \, dx=\frac {\log {\left (x \right )} \log {\left (\frac {x + e^{5 e^{2 x}} + 9}{x + e^{5 e^{2 x}}} \right )}}{x} \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38 \[ \int \frac {-9 x \log (x)-90 e^{5 e^{2 x}+2 x} x \log (x)+\left (9 x+x^2+e^{10 e^{2 x}} (1-\log (x))+\left (-9 x-x^2\right ) \log (x)+e^{5 e^{2 x}} (9+2 x+(-9-2 x) \log (x))\right ) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{e^{10 e^{2 x}} x^2+9 x^3+x^4+e^{5 e^{2 x}} \left (9 x^2+2 x^3\right )} \, dx=\frac {\log \left (x + e^{\left (5 \, e^{\left (2 \, x\right )}\right )} + 9\right ) \log \left (x\right ) - \log \left (x + e^{\left (5 \, e^{\left (2 \, x\right )}\right )}\right ) \log \left (x\right )}{x} \]
[In]
[Out]
none
Time = 0.49 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38 \[ \int \frac {-9 x \log (x)-90 e^{5 e^{2 x}+2 x} x \log (x)+\left (9 x+x^2+e^{10 e^{2 x}} (1-\log (x))+\left (-9 x-x^2\right ) \log (x)+e^{5 e^{2 x}} (9+2 x+(-9-2 x) \log (x))\right ) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{e^{10 e^{2 x}} x^2+9 x^3+x^4+e^{5 e^{2 x}} \left (9 x^2+2 x^3\right )} \, dx=\frac {\log \left (x + e^{\left (5 \, e^{\left (2 \, x\right )}\right )} + 9\right ) \log \left (x\right ) - \log \left (x + e^{\left (5 \, e^{\left (2 \, x\right )}\right )}\right ) \log \left (x\right )}{x} \]
[In]
[Out]
Time = 14.75 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {-9 x \log (x)-90 e^{5 e^{2 x}+2 x} x \log (x)+\left (9 x+x^2+e^{10 e^{2 x}} (1-\log (x))+\left (-9 x-x^2\right ) \log (x)+e^{5 e^{2 x}} (9+2 x+(-9-2 x) \log (x))\right ) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{e^{10 e^{2 x}} x^2+9 x^3+x^4+e^{5 e^{2 x}} \left (9 x^2+2 x^3\right )} \, dx=\frac {\ln \left (\frac {x+{\mathrm {e}}^{5\,{\mathrm {e}}^{2\,x}}+9}{x+{\mathrm {e}}^{5\,{\mathrm {e}}^{2\,x}}}\right )\,\ln \left (x\right )}{x} \]
[In]
[Out]