Integrand size = 131, antiderivative size = 23 \[ \int \frac {25^{-\frac {1}{\log \left (\frac {2+x}{2}\right )}} \left (25 x+e^{2 x} x\right )^{\frac {1}{\log \left (\frac {2+x}{2}\right )}} \left (\left (50+25 x+e^{2 x} \left (2+5 x+2 x^2\right )\right ) \log \left (\frac {2+x}{2}\right )+\left (-25 x-e^{2 x} x\right ) \log \left (\frac {1}{25} \left (25 x+e^{2 x} x\right )\right )\right )}{\left (50 x+25 x^2+e^{2 x} \left (2 x+x^2\right )\right ) \log ^2\left (\frac {2+x}{2}\right )} \, dx=\left (x+\frac {1}{25} e^{2 x} x\right )^{\frac {1}{\log \left (\frac {2+x}{2}\right )}} \]
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\[ \int \frac {25^{-\frac {1}{\log \left (\frac {2+x}{2}\right )}} \left (25 x+e^{2 x} x\right )^{\frac {1}{\log \left (\frac {2+x}{2}\right )}} \left (\left (50+25 x+e^{2 x} \left (2+5 x+2 x^2\right )\right ) \log \left (\frac {2+x}{2}\right )+\left (-25 x-e^{2 x} x\right ) \log \left (\frac {1}{25} \left (25 x+e^{2 x} x\right )\right )\right )}{\left (50 x+25 x^2+e^{2 x} \left (2 x+x^2\right )\right ) \log ^2\left (\frac {2+x}{2}\right )} \, dx=\int \frac {25^{-\frac {1}{\log \left (\frac {2+x}{2}\right )}} \left (25 x+e^{2 x} x\right )^{\frac {1}{\log \left (\frac {2+x}{2}\right )}} \left (\left (50+25 x+e^{2 x} \left (2+5 x+2 x^2\right )\right ) \log \left (\frac {2+x}{2}\right )+\left (-25 x-e^{2 x} x\right ) \log \left (\frac {1}{25} \left (25 x+e^{2 x} x\right )\right )\right )}{\left (50 x+25 x^2+e^{2 x} \left (2 x+x^2\right )\right ) \log ^2\left (\frac {2+x}{2}\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {25^{-\frac {1}{\log \left (1+\frac {x}{2}\right )}} \left (25 x+e^{2 x} x\right )^{\frac {1}{\log \left (1+\frac {x}{2}\right )}} \left (\left (50+25 x+e^{2 x} \left (2+5 x+2 x^2\right )\right ) \log \left (\frac {2+x}{2}\right )+\left (-25 x-e^{2 x} x\right ) \log \left (\frac {1}{25} \left (25 x+e^{2 x} x\right )\right )\right )}{\left (25+e^{2 x}\right ) x (2+x) \log ^2\left (1+\frac {x}{2}\right )} \, dx \\ & = \int \left (\frac {2\ 25^{1-\frac {1}{\log \left (1+\frac {x}{2}\right )}} \left (25 x+e^{2 x} x\right )^{\frac {1}{\log \left (1+\frac {x}{2}\right )}}}{\left (-25-e^{2 x}\right ) \log \left (1+\frac {x}{2}\right )}+\frac {25^{-\frac {1}{\log \left (1+\frac {x}{2}\right )}} \left (25 x+e^{2 x} x\right )^{\frac {1}{\log \left (1+\frac {x}{2}\right )}} \left (2 \log \left (\frac {2+x}{2}\right )+5 x \log \left (\frac {2+x}{2}\right )+2 x^2 \log \left (\frac {2+x}{2}\right )-x \log \left (x+\frac {1}{25} e^{2 x} x\right )\right )}{x (2+x) \log ^2\left (1+\frac {x}{2}\right )}\right ) \, dx \\ & = 2 \int \frac {25^{1-\frac {1}{\log \left (1+\frac {x}{2}\right )}} \left (25 x+e^{2 x} x\right )^{\frac {1}{\log \left (1+\frac {x}{2}\right )}}}{\left (-25-e^{2 x}\right ) \log \left (1+\frac {x}{2}\right )} \, dx+\int \frac {25^{-\frac {1}{\log \left (1+\frac {x}{2}\right )}} \left (25 x+e^{2 x} x\right )^{\frac {1}{\log \left (1+\frac {x}{2}\right )}} \left (2 \log \left (\frac {2+x}{2}\right )+5 x \log \left (\frac {2+x}{2}\right )+2 x^2 \log \left (\frac {2+x}{2}\right )-x \log \left (x+\frac {1}{25} e^{2 x} x\right )\right )}{x (2+x) \log ^2\left (1+\frac {x}{2}\right )} \, dx \\ & = 2 \int \frac {25^{1-\frac {1}{\log \left (1+\frac {x}{2}\right )}} \left (25 x+e^{2 x} x\right )^{\frac {1}{\log \left (1+\frac {x}{2}\right )}}}{\left (-25-e^{2 x}\right ) \log \left (1+\frac {x}{2}\right )} \, dx+\int \left (\frac {25^{-\frac {1}{\log \left (1+\frac {x}{2}\right )}} (1+2 x) \left (25 x+e^{2 x} x\right )^{\frac {1}{\log \left (1+\frac {x}{2}\right )}}}{x \log \left (1+\frac {x}{2}\right )}+\frac {25^{-\frac {1}{\log \left (1+\frac {x}{2}\right )}} \left (25 x+e^{2 x} x\right )^{\frac {1}{\log \left (1+\frac {x}{2}\right )}} \log \left (\frac {1}{25} \left (25+e^{2 x}\right ) x\right )}{(-2-x) \log ^2\left (1+\frac {x}{2}\right )}\right ) \, dx \\ & = 2 \int \frac {25^{1-\frac {1}{\log \left (1+\frac {x}{2}\right )}} \left (25 x+e^{2 x} x\right )^{\frac {1}{\log \left (1+\frac {x}{2}\right )}}}{\left (-25-e^{2 x}\right ) \log \left (1+\frac {x}{2}\right )} \, dx+\int \frac {25^{-\frac {1}{\log \left (1+\frac {x}{2}\right )}} (1+2 x) \left (25 x+e^{2 x} x\right )^{\frac {1}{\log \left (1+\frac {x}{2}\right )}}}{x \log \left (1+\frac {x}{2}\right )} \, dx+\int \frac {25^{-\frac {1}{\log \left (1+\frac {x}{2}\right )}} \left (25 x+e^{2 x} x\right )^{\frac {1}{\log \left (1+\frac {x}{2}\right )}} \log \left (\frac {1}{25} \left (25+e^{2 x}\right ) x\right )}{(-2-x) \log ^2\left (1+\frac {x}{2}\right )} \, dx \\ & = 2 \int \frac {25^{1-\frac {1}{\log \left (1+\frac {x}{2}\right )}} \left (25 x+e^{2 x} x\right )^{\frac {1}{\log \left (1+\frac {x}{2}\right )}}}{\left (-25-e^{2 x}\right ) \log \left (1+\frac {x}{2}\right )} \, dx+\int \left (\frac {2\ 25^{-\frac {1}{\log \left (1+\frac {x}{2}\right )}} \left (25 x+e^{2 x} x\right )^{\frac {1}{\log \left (1+\frac {x}{2}\right )}}}{\log \left (1+\frac {x}{2}\right )}+\frac {25^{-\frac {1}{\log \left (1+\frac {x}{2}\right )}} \left (25 x+e^{2 x} x\right )^{\frac {1}{\log \left (1+\frac {x}{2}\right )}}}{x \log \left (1+\frac {x}{2}\right )}\right ) \, dx+\int \frac {25^{-\frac {1}{\log \left (1+\frac {x}{2}\right )}} \left (25 x+e^{2 x} x\right )^{\frac {1}{\log \left (1+\frac {x}{2}\right )}} \log \left (\frac {1}{25} \left (25+e^{2 x}\right ) x\right )}{(-2-x) \log ^2\left (1+\frac {x}{2}\right )} \, dx \\ & = 2 \int \frac {25^{-\frac {1}{\log \left (1+\frac {x}{2}\right )}} \left (25 x+e^{2 x} x\right )^{\frac {1}{\log \left (1+\frac {x}{2}\right )}}}{\log \left (1+\frac {x}{2}\right )} \, dx+2 \int \frac {25^{1-\frac {1}{\log \left (1+\frac {x}{2}\right )}} \left (25 x+e^{2 x} x\right )^{\frac {1}{\log \left (1+\frac {x}{2}\right )}}}{\left (-25-e^{2 x}\right ) \log \left (1+\frac {x}{2}\right )} \, dx+\int \frac {25^{-\frac {1}{\log \left (1+\frac {x}{2}\right )}} \left (25 x+e^{2 x} x\right )^{\frac {1}{\log \left (1+\frac {x}{2}\right )}}}{x \log \left (1+\frac {x}{2}\right )} \, dx+\int \frac {25^{-\frac {1}{\log \left (1+\frac {x}{2}\right )}} \left (25 x+e^{2 x} x\right )^{\frac {1}{\log \left (1+\frac {x}{2}\right )}} \log \left (\frac {1}{25} \left (25+e^{2 x}\right ) x\right )}{(-2-x) \log ^2\left (1+\frac {x}{2}\right )} \, dx \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {25^{-\frac {1}{\log \left (\frac {2+x}{2}\right )}} \left (25 x+e^{2 x} x\right )^{\frac {1}{\log \left (\frac {2+x}{2}\right )}} \left (\left (50+25 x+e^{2 x} \left (2+5 x+2 x^2\right )\right ) \log \left (\frac {2+x}{2}\right )+\left (-25 x-e^{2 x} x\right ) \log \left (\frac {1}{25} \left (25 x+e^{2 x} x\right )\right )\right )}{\left (50 x+25 x^2+e^{2 x} \left (2 x+x^2\right )\right ) \log ^2\left (\frac {2+x}{2}\right )} \, dx=\left (x+\frac {1}{25} e^{2 x} x\right )^{\frac {1}{\log \left (\frac {2+x}{2}\right )}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.26 (sec) , antiderivative size = 126, normalized size of antiderivative = 5.48
\[{\mathrm e}^{-\frac {-i \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{2 x}+25\right )\right ) {\operatorname {csgn}\left (i x \left ({\mathrm e}^{2 x}+25\right )\right )}^{2}+i \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{2 x}+25\right )\right ) \operatorname {csgn}\left (i x \left ({\mathrm e}^{2 x}+25\right )\right ) \operatorname {csgn}\left (i x \right )+i \pi {\operatorname {csgn}\left (i x \left ({\mathrm e}^{2 x}+25\right )\right )}^{3}-i \pi {\operatorname {csgn}\left (i x \left ({\mathrm e}^{2 x}+25\right )\right )}^{2} \operatorname {csgn}\left (i x \right )+4 \ln \left (5\right )-2 \ln \left (x \right )-2 \ln \left ({\mathrm e}^{2 x}+25\right )}{2 \ln \left (1+\frac {x}{2}\right )}}\]
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Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \frac {25^{-\frac {1}{\log \left (\frac {2+x}{2}\right )}} \left (25 x+e^{2 x} x\right )^{\frac {1}{\log \left (\frac {2+x}{2}\right )}} \left (\left (50+25 x+e^{2 x} \left (2+5 x+2 x^2\right )\right ) \log \left (\frac {2+x}{2}\right )+\left (-25 x-e^{2 x} x\right ) \log \left (\frac {1}{25} \left (25 x+e^{2 x} x\right )\right )\right )}{\left (50 x+25 x^2+e^{2 x} \left (2 x+x^2\right )\right ) \log ^2\left (\frac {2+x}{2}\right )} \, dx={\left (\frac {1}{25} \, x e^{\left (2 \, x\right )} + x\right )}^{\left (\frac {1}{\log \left (\frac {1}{2} \, x + 1\right )}\right )} \]
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Timed out. \[ \int \frac {25^{-\frac {1}{\log \left (\frac {2+x}{2}\right )}} \left (25 x+e^{2 x} x\right )^{\frac {1}{\log \left (\frac {2+x}{2}\right )}} \left (\left (50+25 x+e^{2 x} \left (2+5 x+2 x^2\right )\right ) \log \left (\frac {2+x}{2}\right )+\left (-25 x-e^{2 x} x\right ) \log \left (\frac {1}{25} \left (25 x+e^{2 x} x\right )\right )\right )}{\left (50 x+25 x^2+e^{2 x} \left (2 x+x^2\right )\right ) \log ^2\left (\frac {2+x}{2}\right )} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (18) = 36\).
Time = 0.42 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.26 \[ \int \frac {25^{-\frac {1}{\log \left (\frac {2+x}{2}\right )}} \left (25 x+e^{2 x} x\right )^{\frac {1}{\log \left (\frac {2+x}{2}\right )}} \left (\left (50+25 x+e^{2 x} \left (2+5 x+2 x^2\right )\right ) \log \left (\frac {2+x}{2}\right )+\left (-25 x-e^{2 x} x\right ) \log \left (\frac {1}{25} \left (25 x+e^{2 x} x\right )\right )\right )}{\left (50 x+25 x^2+e^{2 x} \left (2 x+x^2\right )\right ) \log ^2\left (\frac {2+x}{2}\right )} \, dx=e^{\left (\frac {2 \, \log \left (5\right )}{\log \left (2\right ) - \log \left (x + 2\right )} - \frac {\log \left (x\right )}{\log \left (2\right ) - \log \left (x + 2\right )} - \frac {\log \left (e^{\left (2 \, x\right )} + 25\right )}{\log \left (2\right ) - \log \left (x + 2\right )}\right )} \]
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\[ \int \frac {25^{-\frac {1}{\log \left (\frac {2+x}{2}\right )}} \left (25 x+e^{2 x} x\right )^{\frac {1}{\log \left (\frac {2+x}{2}\right )}} \left (\left (50+25 x+e^{2 x} \left (2+5 x+2 x^2\right )\right ) \log \left (\frac {2+x}{2}\right )+\left (-25 x-e^{2 x} x\right ) \log \left (\frac {1}{25} \left (25 x+e^{2 x} x\right )\right )\right )}{\left (50 x+25 x^2+e^{2 x} \left (2 x+x^2\right )\right ) \log ^2\left (\frac {2+x}{2}\right )} \, dx=\int { -\frac {{\left ({\left (x e^{\left (2 \, x\right )} + 25 \, x\right )} \log \left (\frac {1}{25} \, x e^{\left (2 \, x\right )} + x\right ) - {\left ({\left (2 \, x^{2} + 5 \, x + 2\right )} e^{\left (2 \, x\right )} + 25 \, x + 50\right )} \log \left (\frac {1}{2} \, x + 1\right )\right )} {\left (\frac {1}{25} \, x e^{\left (2 \, x\right )} + x\right )}^{\left (\frac {1}{\log \left (\frac {1}{2} \, x + 1\right )}\right )}}{{\left (25 \, x^{2} + {\left (x^{2} + 2 \, x\right )} e^{\left (2 \, x\right )} + 50 \, x\right )} \log \left (\frac {1}{2} \, x + 1\right )^{2}} \,d x } \]
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Time = 12.89 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \frac {25^{-\frac {1}{\log \left (\frac {2+x}{2}\right )}} \left (25 x+e^{2 x} x\right )^{\frac {1}{\log \left (\frac {2+x}{2}\right )}} \left (\left (50+25 x+e^{2 x} \left (2+5 x+2 x^2\right )\right ) \log \left (\frac {2+x}{2}\right )+\left (-25 x-e^{2 x} x\right ) \log \left (\frac {1}{25} \left (25 x+e^{2 x} x\right )\right )\right )}{\left (50 x+25 x^2+e^{2 x} \left (2 x+x^2\right )\right ) \log ^2\left (\frac {2+x}{2}\right )} \, dx={\left (x+\frac {x\,{\mathrm {e}}^{2\,x}}{25}\right )}^{\frac {1}{\ln \left (\frac {x}{2}+1\right )}} \]
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