Optimal. Leaf size=24 \[ \frac {\log (x) \log \left (1+\frac {9}{e^{5 e^{2 x}}+x}\right )}{x} \]
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Rubi [A] time = 12.30, antiderivative size = 33, normalized size of antiderivative = 1.38, number of steps used = 39, number of rules used = 10, integrand size = 156, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.064, Rules used = {6742, 2554, 12, 14, 6688, 2303, 2551, 2304, 30, 2557} \begin {gather*} \frac {\log (x) \log \left (\frac {x+e^{5 e^{2 x}}+9}{x+e^{5 e^{2 x}}}\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 30
Rule 2303
Rule 2304
Rule 2551
Rule 2554
Rule 2557
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \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 {aligned} \end {gather*}
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Mathematica [A] time = 0.99, size = 33, normalized size = 1.38 \begin {gather*} \frac {\log (x) \log \left (\frac {9+e^{5 e^{2 x}}+x}{e^{5 e^{2 x}}+x}\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.66, size = 48, normalized size = 2.00 \begin {gather*} \frac {\log \relax (x) \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.58, size = 33, normalized size = 1.38 \begin {gather*} \frac {\log \left (x + e^{\left (5 \, e^{\left (2 \, x\right )}\right )} + 9\right ) \log \relax (x) - \log \left (x + e^{\left (5 \, e^{\left (2 \, x\right )}\right )}\right ) \log \relax (x)}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.25, size = 219, normalized size = 9.12
method | result | size |
risch | \(\frac {\ln \relax (x ) \ln \left ({\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x +9\right )}{x}-\frac {\ln \relax (x ) \left (i \pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x +9\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x +9\right )}{{\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x}\right )-i \pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x}\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x +9\right )}{{\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x}\right )^{2}-i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x +9\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x +9\right )}{{\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x +9\right )}{{\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x}\right )^{3}+2 \ln \left ({\mathrm e}^{5 \,{\mathrm e}^{2 x}}+x \right )\right )}{2 x}\) | \(219\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 33, normalized size = 1.38 \begin {gather*} \frac {\log \left (x + e^{\left (5 \, e^{\left (2 \, x\right )}\right )} + 9\right ) \log \relax (x) - \log \left (x + e^{\left (5 \, e^{\left (2 \, x\right )}\right )}\right ) \log \relax (x)}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.35, size = 29, normalized size = 1.21 \begin {gather*} \frac {\ln \left (\frac {x+{\mathrm {e}}^{5\,{\mathrm {e}}^{2\,x}}+9}{x+{\mathrm {e}}^{5\,{\mathrm {e}}^{2\,x}}}\right )\,\ln \relax (x)}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.86, size = 27, normalized size = 1.12 \begin {gather*} \frac {\log {\relax (x )} \log {\left (\frac {x + e^{5 e^{2 x}} + 9}{x + e^{5 e^{2 x}}} \right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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