Integrand size = 177, antiderivative size = 24 \[ \int \frac {e^{\frac {36+48 x-7 x^2-7 x^3+x^4+\left (-3 x^2+x^3\right ) \log (5)}{-12-20 x-4 x^2+x^3+x^2 \log (5)}} \left (144+456 x+476 x^2+136 x^3-25 x^4-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \log ^2(5)\right )}{144+480 x+496 x^2+136 x^3-24 x^4-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \log ^2(5)} \, dx=e^{-3+x+\frac {1}{x-\frac {4 \left (5+\frac {3}{x}+x\right )}{x}+\log (5)}} \]
[Out]
\[ \int \frac {e^{\frac {36+48 x-7 x^2-7 x^3+x^4+\left (-3 x^2+x^3\right ) \log (5)}{-12-20 x-4 x^2+x^3+x^2 \log (5)}} \left (144+456 x+476 x^2+136 x^3-25 x^4-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \log ^2(5)\right )}{144+480 x+496 x^2+136 x^3-24 x^4-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \log ^2(5)} \, dx=\int \frac {\exp \left (\frac {36+48 x-7 x^2-7 x^3+x^4+\left (-3 x^2+x^3\right ) \log (5)}{-12-20 x-4 x^2+x^3+x^2 \log (5)}\right ) \left (144+456 x+476 x^2+136 x^3-25 x^4-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \log ^2(5)\right )}{144+480 x+496 x^2+136 x^3-24 x^4-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \log ^2(5)} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {36+48 x-7 x^2-7 x^3+x^4+\left (-3 x^2+x^3\right ) \log (5)}{-12-20 x-4 x^2+x^3+x^2 \log (5)}\right ) \left (144+456 x+476 x^2+136 x^3-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \left (-25+\log ^2(5)\right )\right )}{144+480 x+496 x^2+136 x^3-24 x^4-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \log ^2(5)} \, dx \\ & = \int \frac {\exp \left (\frac {36+48 x-7 x^2-7 x^3+x^4+\left (-3 x^2+x^3\right ) \log (5)}{-12-20 x-4 x^2+x^3+x^2 \log (5)}\right ) \left (144+456 x+476 x^2+136 x^3-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \left (-25+\log ^2(5)\right )\right )}{144+480 x+496 x^2+136 x^3-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \left (-24+\log ^2(5)\right )} \, dx \\ & = \int \frac {\exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) \left (144+456 x+476 x^2+136 x^3-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \left (-25+\log ^2(5)\right )\right )}{\left (12+20 x-x^3+x^2 (4-\log (5))\right )^2} \, dx \\ & = \int \left (\exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right )+\frac {\exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) (4+x-\log (5))}{12+20 x-x^3+x^2 (4-\log (5))}+\frac {\exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) \left (-4 x (29-5 \log (5))-12 (4-\log (5))-x^2 \left (56-8 \log (5)+\log ^2(5)\right )\right )}{\left (12+20 x-x^3+x^2 (4-\log (5))\right )^2}\right ) \, dx \\ & = \int \exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) \, dx+\int \frac {\exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) (4+x-\log (5))}{12+20 x-x^3+x^2 (4-\log (5))} \, dx+\int \frac {\exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) \left (-4 x (29-5 \log (5))-12 (4-\log (5))-x^2 \left (56-8 \log (5)+\log ^2(5)\right )\right )}{\left (12+20 x-x^3+x^2 (4-\log (5))\right )^2} \, dx \\ & = \int \exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) \, dx+\int \left (\frac {\exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) x}{12+20 x-x^3+x^2 (4-\log (5))}+\frac {4 \exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) \left (1-\frac {\log (5)}{4}\right )}{12+20 x-x^3+x^2 (4-\log (5))}\right ) \, dx+\int \left (\frac {12 \exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) (-4+\log (5))}{\left (12+20 x-x^3+x^2 (4-\log (5))\right )^2}+\frac {4 \exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) x (-29+5 \log (5))}{\left (12+20 x-x^3+x^2 (4-\log (5))\right )^2}+\frac {\exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) x^2 \left (-56+8 \log (5)-\log ^2(5)\right )}{\left (12+20 x-x^3+x^2 (4-\log (5))\right )^2}\right ) \, dx \\ & = -\left ((4 (29-5 \log (5))) \int \frac {\exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) x}{\left (12+20 x-x^3+x^2 (4-\log (5))\right )^2} \, dx\right )+(4-\log (5)) \int \frac {\exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right )}{12+20 x-x^3+x^2 (4-\log (5))} \, dx-(12 (4-\log (5))) \int \frac {\exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right )}{\left (12+20 x-x^3+x^2 (4-\log (5))\right )^2} \, dx+\left (-56+8 \log (5)-\log ^2(5)\right ) \int \frac {\exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) x^2}{\left (12+20 x-x^3+x^2 (4-\log (5))\right )^2} \, dx+\int \exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) \, dx+\int \frac {\exp \left (\frac {36+48 x+x^4-x^3 (7-\log (5))-x^2 (7+\log (125))}{-12-20 x+x^3-x^2 (4-\log (5))}\right ) x}{12+20 x-x^3+x^2 (4-\log (5))} \, dx \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(79\) vs. \(2(24)=48\).
Time = 0.19 (sec) , antiderivative size = 79, normalized size of antiderivative = 3.29 \[ \int \frac {e^{\frac {36+48 x-7 x^2-7 x^3+x^4+\left (-3 x^2+x^3\right ) \log (5)}{-12-20 x-4 x^2+x^3+x^2 \log (5)}} \left (144+456 x+476 x^2+136 x^3-25 x^4-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \log ^2(5)\right )}{144+480 x+496 x^2+136 x^3-24 x^4-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \log ^2(5)} \, dx=5^{-\frac {x^2 \log (5)}{-12-20 x+x^3+x^2 (-4+\log (5))}} e^{\frac {36+48 x+x^4+x^3 (-7+\log (5))+x^2 \left (-7-3 \log (5)+\log ^2(5)\right )}{-12-20 x+x^3+x^2 (-4+\log (5))}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(53\) vs. \(2(23)=46\).
Time = 1.00 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.25
method | result | size |
parallelrisch | \({\mathrm e}^{\frac {\left (x^{3}-3 x^{2}\right ) \ln \left (5\right )+x^{4}-7 x^{3}-7 x^{2}+48 x +36}{x^{2} \ln \left (5\right )+x^{3}-4 x^{2}-20 x -12}}\) | \(54\) |
gosper | \({\mathrm e}^{\frac {x^{3} \ln \left (5\right )+x^{4}-3 x^{2} \ln \left (5\right )-7 x^{3}-7 x^{2}+48 x +36}{x^{2} \ln \left (5\right )+x^{3}-4 x^{2}-20 x -12}}\) | \(55\) |
risch | \({\mathrm e}^{\frac {x^{3} \ln \left (5\right )+x^{4}-3 x^{2} \ln \left (5\right )-7 x^{3}-7 x^{2}+48 x +36}{x^{2} \ln \left (5\right )+x^{3}-4 x^{2}-20 x -12}}\) | \(55\) |
norman | \(\frac {x^{3} {\mathrm e}^{\frac {\left (x^{3}-3 x^{2}\right ) \ln \left (5\right )+x^{4}-7 x^{3}-7 x^{2}+48 x +36}{x^{2} \ln \left (5\right )+x^{3}-4 x^{2}-20 x -12}}+\left (\ln \left (5\right )-4\right ) x^{2} {\mathrm e}^{\frac {\left (x^{3}-3 x^{2}\right ) \ln \left (5\right )+x^{4}-7 x^{3}-7 x^{2}+48 x +36}{x^{2} \ln \left (5\right )+x^{3}-4 x^{2}-20 x -12}}-20 x \,{\mathrm e}^{\frac {\left (x^{3}-3 x^{2}\right ) \ln \left (5\right )+x^{4}-7 x^{3}-7 x^{2}+48 x +36}{x^{2} \ln \left (5\right )+x^{3}-4 x^{2}-20 x -12}}-12 \,{\mathrm e}^{\frac {\left (x^{3}-3 x^{2}\right ) \ln \left (5\right )+x^{4}-7 x^{3}-7 x^{2}+48 x +36}{x^{2} \ln \left (5\right )+x^{3}-4 x^{2}-20 x -12}}}{x^{2} \ln \left (5\right )+x^{3}-4 x^{2}-20 x -12}\) | \(253\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (23) = 46\).
Time = 0.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.21 \[ \int \frac {e^{\frac {36+48 x-7 x^2-7 x^3+x^4+\left (-3 x^2+x^3\right ) \log (5)}{-12-20 x-4 x^2+x^3+x^2 \log (5)}} \left (144+456 x+476 x^2+136 x^3-25 x^4-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \log ^2(5)\right )}{144+480 x+496 x^2+136 x^3-24 x^4-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \log ^2(5)} \, dx=e^{\left (\frac {x^{4} - 7 \, x^{3} - 7 \, x^{2} + {\left (x^{3} - 3 \, x^{2}\right )} \log \left (5\right ) + 48 \, x + 36}{x^{3} + x^{2} \log \left (5\right ) - 4 \, x^{2} - 20 \, x - 12}\right )} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (20) = 40\).
Time = 1.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.12 \[ \int \frac {e^{\frac {36+48 x-7 x^2-7 x^3+x^4+\left (-3 x^2+x^3\right ) \log (5)}{-12-20 x-4 x^2+x^3+x^2 \log (5)}} \left (144+456 x+476 x^2+136 x^3-25 x^4-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \log ^2(5)\right )}{144+480 x+496 x^2+136 x^3-24 x^4-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \log ^2(5)} \, dx=e^{\frac {x^{4} - 7 x^{3} - 7 x^{2} + 48 x + \left (x^{3} - 3 x^{2}\right ) \log {\left (5 \right )} + 36}{x^{3} - 4 x^{2} + x^{2} \log {\left (5 \right )} - 20 x - 12}} \]
[In]
[Out]
none
Time = 0.53 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {e^{\frac {36+48 x-7 x^2-7 x^3+x^4+\left (-3 x^2+x^3\right ) \log (5)}{-12-20 x-4 x^2+x^3+x^2 \log (5)}} \left (144+456 x+476 x^2+136 x^3-25 x^4-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \log ^2(5)\right )}{144+480 x+496 x^2+136 x^3-24 x^4-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \log ^2(5)} \, dx=e^{\left (x + \frac {x^{2}}{x^{3} + x^{2} {\left (\log \left (5\right ) - 4\right )} - 20 \, x - 12} - 3\right )} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 181 vs. \(2 (23) = 46\).
Time = 0.32 (sec) , antiderivative size = 181, normalized size of antiderivative = 7.54 \[ \int \frac {e^{\frac {36+48 x-7 x^2-7 x^3+x^4+\left (-3 x^2+x^3\right ) \log (5)}{-12-20 x-4 x^2+x^3+x^2 \log (5)}} \left (144+456 x+476 x^2+136 x^3-25 x^4-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \log ^2(5)\right )}{144+480 x+496 x^2+136 x^3-24 x^4-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \log ^2(5)} \, dx=e^{\left (\frac {x^{4}}{x^{3} + x^{2} \log \left (5\right ) - 4 \, x^{2} - 20 \, x - 12} + \frac {x^{3} \log \left (5\right )}{x^{3} + x^{2} \log \left (5\right ) - 4 \, x^{2} - 20 \, x - 12} - \frac {7 \, x^{3}}{x^{3} + x^{2} \log \left (5\right ) - 4 \, x^{2} - 20 \, x - 12} - \frac {3 \, x^{2} \log \left (5\right )}{x^{3} + x^{2} \log \left (5\right ) - 4 \, x^{2} - 20 \, x - 12} - \frac {7 \, x^{2}}{x^{3} + x^{2} \log \left (5\right ) - 4 \, x^{2} - 20 \, x - 12} + \frac {48 \, x}{x^{3} + x^{2} \log \left (5\right ) - 4 \, x^{2} - 20 \, x - 12} + \frac {36}{x^{3} + x^{2} \log \left (5\right ) - 4 \, x^{2} - 20 \, x - 12}\right )} \]
[In]
[Out]
Time = 13.62 (sec) , antiderivative size = 184, normalized size of antiderivative = 7.67 \[ \int \frac {e^{\frac {36+48 x-7 x^2-7 x^3+x^4+\left (-3 x^2+x^3\right ) \log (5)}{-12-20 x-4 x^2+x^3+x^2 \log (5)}} \left (144+456 x+476 x^2+136 x^3-25 x^4-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \log ^2(5)\right )}{144+480 x+496 x^2+136 x^3-24 x^4-8 x^5+x^6+\left (-24 x^2-40 x^3-8 x^4+2 x^5\right ) \log (5)+x^4 \log ^2(5)} \, dx=5^{\frac {3\,x^2-x^3}{20\,x-x^2\,\ln \left (5\right )+4\,x^2-x^3+12}}\,{\mathrm {e}}^{-\frac {x^4}{20\,x-x^2\,\ln \left (5\right )+4\,x^2-x^3+12}}\,{\mathrm {e}}^{\frac {7\,x^2}{20\,x-x^2\,\ln \left (5\right )+4\,x^2-x^3+12}}\,{\mathrm {e}}^{\frac {7\,x^3}{20\,x-x^2\,\ln \left (5\right )+4\,x^2-x^3+12}}\,{\mathrm {e}}^{-\frac {36}{20\,x-x^2\,\ln \left (5\right )+4\,x^2-x^3+12}}\,{\mathrm {e}}^{-\frac {48\,x}{20\,x-x^2\,\ln \left (5\right )+4\,x^2-x^3+12}} \]
[In]
[Out]