Integrand size = 108, antiderivative size = 23 \[ \int \frac {e^x \left (1-2 x-4 x^2\right )+e^x (-4-5 x) \log (x)+\left (e^x \left (x^2+x^3\right )+e^x \left (x+x^2\right ) \log (x)\right ) \log \left (\frac {10 x^4+10 x^5}{x+\log (x)}\right )}{\left (x^2+x^3+\left (x+x^2\right ) \log (x)\right ) \log ^2\left (\frac {10 x^4+10 x^5}{x+\log (x)}\right )} \, dx=\frac {e^x}{\log \left (\frac {10 x^3 \left (x+x^2\right )}{x+\log (x)}\right )} \]
[Out]
\[ \int \frac {e^x \left (1-2 x-4 x^2\right )+e^x (-4-5 x) \log (x)+\left (e^x \left (x^2+x^3\right )+e^x \left (x+x^2\right ) \log (x)\right ) \log \left (\frac {10 x^4+10 x^5}{x+\log (x)}\right )}{\left (x^2+x^3+\left (x+x^2\right ) \log (x)\right ) \log ^2\left (\frac {10 x^4+10 x^5}{x+\log (x)}\right )} \, dx=\int \frac {e^x \left (1-2 x-4 x^2\right )+e^x (-4-5 x) \log (x)+\left (e^x \left (x^2+x^3\right )+e^x \left (x+x^2\right ) \log (x)\right ) \log \left (\frac {10 x^4+10 x^5}{x+\log (x)}\right )}{\left (x^2+x^3+\left (x+x^2\right ) \log (x)\right ) \log ^2\left (\frac {10 x^4+10 x^5}{x+\log (x)}\right )} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^x \left (1-2 x-4 x^2\right )+e^x (-4-5 x) \log (x)+\left (e^x \left (x^2+x^3\right )+e^x \left (x+x^2\right ) \log (x)\right ) \log \left (\frac {10 x^4+10 x^5}{x+\log (x)}\right )}{x (1+x) (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx \\ & = \int \left (\frac {e^x \left (1-2 x-4 x^2-4 \log (x)-5 x \log (x)+x^2 \log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )+x^3 \log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )+x \log (x) \log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )+x^2 \log (x) \log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )\right )}{x (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )}-\frac {e^x \left (1-2 x-4 x^2-4 \log (x)-5 x \log (x)+x^2 \log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )+x^3 \log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )+x \log (x) \log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )+x^2 \log (x) \log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )\right )}{(1+x) (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )}\right ) \, dx \\ & = \int \frac {e^x \left (1-2 x-4 x^2-4 \log (x)-5 x \log (x)+x^2 \log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )+x^3 \log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )+x \log (x) \log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )+x^2 \log (x) \log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )\right )}{x (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx-\int \frac {e^x \left (1-2 x-4 x^2-4 \log (x)-5 x \log (x)+x^2 \log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )+x^3 \log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )+x \log (x) \log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )+x^2 \log (x) \log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )\right )}{(1+x) (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx \\ & = \int \frac {e^x \left (1-2 x-4 x^2+x^2 (1+x) \log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )+\log (x) \left (-4-5 x+x (1+x) \log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )\right )\right )}{x (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx-\int \frac {e^x \left (1-2 x-4 x^2+x^2 (1+x) \log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )+\log (x) \left (-4-5 x+x (1+x) \log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )\right )\right )}{(1+x) (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx \\ & = -\int \left (\frac {e^x \left (1-2 x-4 x^2-4 \log (x)-5 x \log (x)\right )}{(1+x) (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )}+\frac {e^x x}{\log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )}\right ) \, dx+\int \left (\frac {e^x \left (1-2 x-4 x^2-4 \log (x)-5 x \log (x)\right )}{x (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )}+\frac {e^x (1+x)}{\log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )}\right ) \, dx \\ & = \int \frac {e^x \left (1-2 x-4 x^2-4 \log (x)-5 x \log (x)\right )}{x (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx-\int \frac {e^x \left (1-2 x-4 x^2-4 \log (x)-5 x \log (x)\right )}{(1+x) (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx-\int \frac {e^x x}{\log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx+\int \frac {e^x (1+x)}{\log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx \\ & = \int \left (-\frac {2 e^x}{(x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )}+\frac {e^x}{x (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )}-\frac {4 e^x x}{(x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )}-\frac {5 e^x \log (x)}{(x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )}-\frac {4 e^x \log (x)}{x (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )}\right ) \, dx-\int \left (\frac {e^x}{(1+x) (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )}-\frac {2 e^x x}{(1+x) (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )}-\frac {4 e^x x^2}{(1+x) (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )}-\frac {4 e^x \log (x)}{(1+x) (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )}-\frac {5 e^x x \log (x)}{(1+x) (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )}\right ) \, dx+\int \left (\frac {e^x}{\log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )}+\frac {e^x x}{\log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )}\right ) \, dx-\int \frac {e^x x}{\log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx \\ & = -\left (2 \int \frac {e^x}{(x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx\right )+2 \int \frac {e^x x}{(1+x) (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx-4 \int \frac {e^x x}{(x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx+4 \int \frac {e^x x^2}{(1+x) (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx-4 \int \frac {e^x \log (x)}{x (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx+4 \int \frac {e^x \log (x)}{(1+x) (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx-5 \int \frac {e^x \log (x)}{(x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx+5 \int \frac {e^x x \log (x)}{(1+x) (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx+\int \frac {e^x}{x (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx-\int \frac {e^x}{(1+x) (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx+\int \frac {e^x}{\log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx \\ & = 2 \int \left (\frac {e^x}{(x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )}-\frac {e^x}{(1+x) (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )}\right ) \, dx-2 \int \frac {e^x}{(x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx+4 \int \left (-\frac {e^x}{(x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )}+\frac {e^x x}{(x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )}+\frac {e^x}{(1+x) (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )}\right ) \, dx-4 \int \frac {e^x x}{(x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx-4 \int \frac {e^x \log (x)}{x (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx+4 \int \frac {e^x \log (x)}{(1+x) (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx+5 \int \left (\frac {e^x \log (x)}{(x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )}-\frac {e^x \log (x)}{(1+x) (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )}\right ) \, dx-5 \int \frac {e^x \log (x)}{(x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx+\int \frac {e^x}{x (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx-\int \frac {e^x}{(1+x) (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx+\int \frac {e^x}{\log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx \\ & = -\left (2 \int \frac {e^x}{(1+x) (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx\right )-4 \int \frac {e^x}{(x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx+4 \int \frac {e^x}{(1+x) (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx-4 \int \frac {e^x \log (x)}{x (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx+4 \int \frac {e^x \log (x)}{(1+x) (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx-5 \int \frac {e^x \log (x)}{(1+x) (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx+\int \frac {e^x}{x (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx-\int \frac {e^x}{(1+x) (x+\log (x)) \log ^2\left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx+\int \frac {e^x}{\log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \, dx \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {e^x \left (1-2 x-4 x^2\right )+e^x (-4-5 x) \log (x)+\left (e^x \left (x^2+x^3\right )+e^x \left (x+x^2\right ) \log (x)\right ) \log \left (\frac {10 x^4+10 x^5}{x+\log (x)}\right )}{\left (x^2+x^3+\left (x+x^2\right ) \log (x)\right ) \log ^2\left (\frac {10 x^4+10 x^5}{x+\log (x)}\right )} \, dx=\frac {e^x}{\log \left (\frac {10 x^4 (1+x)}{x+\log (x)}\right )} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 51.94 (sec) , antiderivative size = 441, normalized size of antiderivative = 19.17
method | result | size |
risch | \(\frac {2 i {\mathrm e}^{x}}{\pi \,\operatorname {csgn}\left (i x^{3}\right ) \operatorname {csgn}\left (i x^{4}\right ) \operatorname {csgn}\left (i x \right )+2 i \ln \left (1+x \right )+2 i \ln \left (2\right )+2 i \ln \left (5\right )+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+\pi \,\operatorname {csgn}\left (i x^{4}\right ) \operatorname {csgn}\left (\frac {i \left (1+x \right )}{x +\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i x^{4} \left (1+x \right )}{x +\ln \left (x \right )}\right )+\pi \,\operatorname {csgn}\left (\frac {i}{x +\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i \left (1+x \right )}{x +\ln \left (x \right )}\right ) \operatorname {csgn}\left (i \left (1+x \right )\right )-\pi \,\operatorname {csgn}\left (i x^{3}\right ) \operatorname {csgn}\left (i x^{4}\right )^{2}-\pi \operatorname {csgn}\left (i x^{4}\right )^{2} \operatorname {csgn}\left (i x \right )+8 i \ln \left (x \right )+\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{4}\right )^{3}+\pi \operatorname {csgn}\left (\frac {i x^{4} \left (1+x \right )}{x +\ln \left (x \right )}\right )^{3}-\pi \,\operatorname {csgn}\left (\frac {i}{x +\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i \left (1+x \right )}{x +\ln \left (x \right )}\right )^{2}-\pi \operatorname {csgn}\left (\frac {i \left (1+x \right )}{x +\ln \left (x \right )}\right )^{2} \operatorname {csgn}\left (i \left (1+x \right )\right )-\pi \,\operatorname {csgn}\left (i x^{4}\right ) \operatorname {csgn}\left (\frac {i x^{4} \left (1+x \right )}{x +\ln \left (x \right )}\right )^{2}-\pi \,\operatorname {csgn}\left (\frac {i \left (1+x \right )}{x +\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i x^{4} \left (1+x \right )}{x +\ln \left (x \right )}\right )^{2}-\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right )^{2}-\pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )^{2}+\pi \operatorname {csgn}\left (\frac {i \left (1+x \right )}{x +\ln \left (x \right )}\right )^{3}+\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )-2 i \ln \left (x +\ln \left (x \right )\right )+\pi \operatorname {csgn}\left (i x^{3}\right )^{3}}\) | \(441\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {e^x \left (1-2 x-4 x^2\right )+e^x (-4-5 x) \log (x)+\left (e^x \left (x^2+x^3\right )+e^x \left (x+x^2\right ) \log (x)\right ) \log \left (\frac {10 x^4+10 x^5}{x+\log (x)}\right )}{\left (x^2+x^3+\left (x+x^2\right ) \log (x)\right ) \log ^2\left (\frac {10 x^4+10 x^5}{x+\log (x)}\right )} \, dx=\frac {e^{x}}{\log \left (\frac {10 \, {\left (x^{5} + x^{4}\right )}}{x + \log \left (x\right )}\right )} \]
[In]
[Out]
Time = 0.37 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {e^x \left (1-2 x-4 x^2\right )+e^x (-4-5 x) \log (x)+\left (e^x \left (x^2+x^3\right )+e^x \left (x+x^2\right ) \log (x)\right ) \log \left (\frac {10 x^4+10 x^5}{x+\log (x)}\right )}{\left (x^2+x^3+\left (x+x^2\right ) \log (x)\right ) \log ^2\left (\frac {10 x^4+10 x^5}{x+\log (x)}\right )} \, dx=\frac {e^{x}}{\log {\left (\frac {10 x^{5} + 10 x^{4}}{x + \log {\left (x \right )}} \right )}} \]
[In]
[Out]
none
Time = 0.35 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {e^x \left (1-2 x-4 x^2\right )+e^x (-4-5 x) \log (x)+\left (e^x \left (x^2+x^3\right )+e^x \left (x+x^2\right ) \log (x)\right ) \log \left (\frac {10 x^4+10 x^5}{x+\log (x)}\right )}{\left (x^2+x^3+\left (x+x^2\right ) \log (x)\right ) \log ^2\left (\frac {10 x^4+10 x^5}{x+\log (x)}\right )} \, dx=\frac {e^{x}}{\log \left (5\right ) + \log \left (2\right ) - \log \left (x + \log \left (x\right )\right ) + \log \left (x + 1\right ) + 4 \, \log \left (x\right )} \]
[In]
[Out]
none
Time = 0.42 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {e^x \left (1-2 x-4 x^2\right )+e^x (-4-5 x) \log (x)+\left (e^x \left (x^2+x^3\right )+e^x \left (x+x^2\right ) \log (x)\right ) \log \left (\frac {10 x^4+10 x^5}{x+\log (x)}\right )}{\left (x^2+x^3+\left (x+x^2\right ) \log (x)\right ) \log ^2\left (\frac {10 x^4+10 x^5}{x+\log (x)}\right )} \, dx=\frac {e^{x}}{\log \left (10 \, x + 10\right ) - \log \left (x + \log \left (x\right )\right ) + 4 \, \log \left (x\right )} \]
[In]
[Out]
Time = 17.33 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {e^x \left (1-2 x-4 x^2\right )+e^x (-4-5 x) \log (x)+\left (e^x \left (x^2+x^3\right )+e^x \left (x+x^2\right ) \log (x)\right ) \log \left (\frac {10 x^4+10 x^5}{x+\log (x)}\right )}{\left (x^2+x^3+\left (x+x^2\right ) \log (x)\right ) \log ^2\left (\frac {10 x^4+10 x^5}{x+\log (x)}\right )} \, dx=\frac {{\mathrm {e}}^x}{\ln \left (\frac {10\,x^5+10\,x^4}{x+\ln \left (x\right )}\right )} \]
[In]
[Out]