Integrand size = 118, antiderivative size = 25 \[ \int \frac {e^{\frac {1}{16} \left (1+8 \log \left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )+16 \log ^2\left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )} \left (-e^5+e^x (-1+x)+\left (-4 e^5+e^x (-4+4 x)\right ) \log \left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )}{\left (2 e^5 x+2 e^x x\right ) \log \left (\frac {e^5+e^x}{x}\right )} \, dx=e^{\left (-\frac {1}{4}-\log \left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )^2} \]
[Out]
Leaf count is larger than twice the leaf count of optimal. \(107\) vs. \(2(25)=50\).
Time = 0.52 (sec) , antiderivative size = 107, normalized size of antiderivative = 4.28, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {2306, 12, 2326} \[ \int \frac {e^{\frac {1}{16} \left (1+8 \log \left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )+16 \log ^2\left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )} \left (-e^5+e^x (-1+x)+\left (-4 e^5+e^x (-4+4 x)\right ) \log \left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )}{\left (2 e^5 x+2 e^x x\right ) \log \left (\frac {e^5+e^x}{x}\right )} \, dx=\frac {\sqrt {2} \left (e^x+e^5\right ) \left (e^x (1-x)+e^5\right ) e^{\frac {1}{16} \left (16 \log ^2\left (2 \log \left (\frac {e^x+e^5}{x}\right )\right )+1\right )} \sqrt {\log \left (\frac {e^x+e^5}{x}\right )}}{\left (\frac {e^x+e^5}{x^2}-\frac {e^x}{x}\right ) x \left (e^x x+e^5 x\right )} \]
[In]
[Out]
Rule 12
Rule 2306
Rule 2326
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {2} e^{\frac {1}{16} \left (1+16 \log ^2\left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )} \left (-e^5+e^x (-1+x)+\left (-4 e^5+e^x (-4+4 x)\right ) \log \left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )}{\left (2 e^5 x+2 e^x x\right ) \sqrt {\log \left (\frac {e^5+e^x}{x}\right )}} \, dx \\ & = \sqrt {2} \int \frac {e^{\frac {1}{16} \left (1+16 \log ^2\left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )} \left (-e^5+e^x (-1+x)+\left (-4 e^5+e^x (-4+4 x)\right ) \log \left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )}{\left (2 e^5 x+2 e^x x\right ) \sqrt {\log \left (\frac {e^5+e^x}{x}\right )}} \, dx \\ & = \frac {\sqrt {2} e^{\frac {1}{16} \left (1+16 \log ^2\left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )} \left (e^5+e^x\right ) \left (e^5+e^x (1-x)\right ) \sqrt {\log \left (\frac {e^5+e^x}{x}\right )}}{\left (\frac {e^5+e^x}{x^2}-\frac {e^x}{x}\right ) x \left (e^5 x+e^x x\right )} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.80 \[ \int \frac {e^{\frac {1}{16} \left (1+8 \log \left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )+16 \log ^2\left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )} \left (-e^5+e^x (-1+x)+\left (-4 e^5+e^x (-4+4 x)\right ) \log \left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )}{\left (2 e^5 x+2 e^x x\right ) \log \left (\frac {e^5+e^x}{x}\right )} \, dx=\sqrt {2} e^{\frac {1}{16}+\log ^2\left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )} \sqrt {\log \left (\frac {e^5+e^x}{x}\right )} \]
[In]
[Out]
Time = 44.64 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36
method | result | size |
parallelrisch | \({\mathrm e}^{{\ln \left (2 \ln \left (\frac {{\mathrm e}^{5}+{\mathrm e}^{x}}{x}\right )\right )}^{2}+\frac {\ln \left (2 \ln \left (\frac {{\mathrm e}^{5}+{\mathrm e}^{x}}{x}\right )\right )}{2}+\frac {1}{16}}\) | \(34\) |
risch | \(\sqrt {-2 \ln \left (x \right )+2 \ln \left ({\mathrm e}^{5}+{\mathrm e}^{x}\right )-i \pi \,\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{5}+{\mathrm e}^{x}\right )}{x}\right ) \left (-\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{5}+{\mathrm e}^{x}\right )}{x}\right )+\operatorname {csgn}\left (\frac {i}{x}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{5}+{\mathrm e}^{x}\right )}{x}\right )+\operatorname {csgn}\left (i \left ({\mathrm e}^{5}+{\mathrm e}^{x}\right )\right )\right )}\, {\mathrm e}^{{\ln \left (-2 \ln \left (x \right )+2 \ln \left ({\mathrm e}^{5}+{\mathrm e}^{x}\right )-i \pi \,\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{5}+{\mathrm e}^{x}\right )}{x}\right ) \left (-\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{5}+{\mathrm e}^{x}\right )}{x}\right )+\operatorname {csgn}\left (\frac {i}{x}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{5}+{\mathrm e}^{x}\right )}{x}\right )+\operatorname {csgn}\left (i \left ({\mathrm e}^{5}+{\mathrm e}^{x}\right )\right )\right )\right )}^{2}+\frac {1}{16}}\) | \(160\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {e^{\frac {1}{16} \left (1+8 \log \left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )+16 \log ^2\left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )} \left (-e^5+e^x (-1+x)+\left (-4 e^5+e^x (-4+4 x)\right ) \log \left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )}{\left (2 e^5 x+2 e^x x\right ) \log \left (\frac {e^5+e^x}{x}\right )} \, dx=e^{\left (\log \left (2 \, \log \left (\frac {e^{5} + e^{x}}{x}\right )\right )^{2} + \frac {1}{2} \, \log \left (2 \, \log \left (\frac {e^{5} + e^{x}}{x}\right )\right ) + \frac {1}{16}\right )} \]
[In]
[Out]
Timed out. \[ \int \frac {e^{\frac {1}{16} \left (1+8 \log \left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )+16 \log ^2\left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )} \left (-e^5+e^x (-1+x)+\left (-4 e^5+e^x (-4+4 x)\right ) \log \left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )}{\left (2 e^5 x+2 e^x x\right ) \log \left (\frac {e^5+e^x}{x}\right )} \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (22) = 44\).
Time = 0.57 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.16 \[ \int \frac {e^{\frac {1}{16} \left (1+8 \log \left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )+16 \log ^2\left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )} \left (-e^5+e^x (-1+x)+\left (-4 e^5+e^x (-4+4 x)\right ) \log \left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )}{\left (2 e^5 x+2 e^x x\right ) \log \left (\frac {e^5+e^x}{x}\right )} \, dx=\sqrt {2} \sqrt {-\log \left (x\right ) + \log \left (e^{5} + e^{x}\right )} e^{\left (\log \left (2\right )^{2} + 2 \, \log \left (2\right ) \log \left (-\log \left (x\right ) + \log \left (e^{5} + e^{x}\right )\right ) + \log \left (-\log \left (x\right ) + \log \left (e^{5} + e^{x}\right )\right )^{2} + \frac {1}{16}\right )} \]
[In]
[Out]
\[ \int \frac {e^{\frac {1}{16} \left (1+8 \log \left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )+16 \log ^2\left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )} \left (-e^5+e^x (-1+x)+\left (-4 e^5+e^x (-4+4 x)\right ) \log \left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )}{\left (2 e^5 x+2 e^x x\right ) \log \left (\frac {e^5+e^x}{x}\right )} \, dx=\int { \frac {{\left ({\left (x - 1\right )} e^{x} + 4 \, {\left ({\left (x - 1\right )} e^{x} - e^{5}\right )} \log \left (2 \, \log \left (\frac {e^{5} + e^{x}}{x}\right )\right ) - e^{5}\right )} e^{\left (\log \left (2 \, \log \left (\frac {e^{5} + e^{x}}{x}\right )\right )^{2} + \frac {1}{2} \, \log \left (2 \, \log \left (\frac {e^{5} + e^{x}}{x}\right )\right ) + \frac {1}{16}\right )}}{2 \, {\left (x e^{5} + x e^{x}\right )} \log \left (\frac {e^{5} + e^{x}}{x}\right )} \,d x } \]
[In]
[Out]
Time = 13.48 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56 \[ \int \frac {e^{\frac {1}{16} \left (1+8 \log \left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )+16 \log ^2\left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )} \left (-e^5+e^x (-1+x)+\left (-4 e^5+e^x (-4+4 x)\right ) \log \left (2 \log \left (\frac {e^5+e^x}{x}\right )\right )\right )}{\left (2 e^5 x+2 e^x x\right ) \log \left (\frac {e^5+e^x}{x}\right )} \, dx={\mathrm {e}}^{1/16}\,{\mathrm {e}}^{{\ln \left (2\,\ln \left (\frac {1}{x}\right )+\ln \left ({\left ({\mathrm {e}}^5+{\mathrm {e}}^x\right )}^2\right )\right )}^2}\,\sqrt {2\,\ln \left (\frac {1}{x}\right )+\ln \left ({\left ({\mathrm {e}}^5+{\mathrm {e}}^x\right )}^2\right )} \]
[In]
[Out]