Gosper - Maple Help

DEtools

 Gosper
 perform indefinite hyperexponential integration

 Calling Sequence Gosper(T, x)

Parameters

 T - hyperexponential function of x x - variable

Description

 • The Gosper(T,x) command solves the problem of indefinite hyperexponential integration, that is, for the input hyperexponential function T of x, it constructs another hyperexponential function G of x such that $T\left(x\right)=\frac{{ⅆ}}{{ⅆ}x}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}G\left(x\right)$, provided that such a G exists. Otherwise, the function returns the error message no polynomial solution found''.

Examples

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > $T≔\frac{-21{x}^{2}+116x-94}{-73-35{x}^{4}-14{x}^{3}-9{x}^{2}-51x}\mathrm{exp}\left(-\frac{91}{-86+5x}\right)-\frac{-68-7{x}^{3}+58{x}^{2}-94x}{{\left(-73-35{x}^{4}-14{x}^{3}-9{x}^{2}-51x\right)}^{2}}\mathrm{exp}\left(-\frac{91}{-86+5x}\right)\left(-140{x}^{3}-42{x}^{2}-18x-51\right)+\frac{455\left(-68-7{x}^{3}+58{x}^{2}-94x\right)}{\left(-73-35{x}^{4}-14{x}^{3}-9{x}^{2}-51x\right){\left(-86+5x\right)}^{2}}\mathrm{exp}\left(-\frac{91}{-86+5x}\right)$
 ${T}{≔}\frac{\left({-}{21}{}{{x}}^{{2}}{+}{116}{}{x}{-}{94}\right){}{{ⅇ}}^{{-}\frac{{91}}{{-}{86}{+}{5}{}{x}}}}{{-}{35}{}{{x}}^{{4}}{-}{14}{}{{x}}^{{3}}{-}{9}{}{{x}}^{{2}}{-}{51}{}{x}{-}{73}}{-}\frac{\left({-}{7}{}{{x}}^{{3}}{+}{58}{}{{x}}^{{2}}{-}{94}{}{x}{-}{68}\right){}{{ⅇ}}^{{-}\frac{{91}}{{-}{86}{+}{5}{}{x}}}{}\left({-}{140}{}{{x}}^{{3}}{-}{42}{}{{x}}^{{2}}{-}{18}{}{x}{-}{51}\right)}{{\left({-}{35}{}{{x}}^{{4}}{-}{14}{}{{x}}^{{3}}{-}{9}{}{{x}}^{{2}}{-}{51}{}{x}{-}{73}\right)}^{{2}}}{+}\frac{{455}{}\left({-}{7}{}{{x}}^{{3}}{+}{58}{}{{x}}^{{2}}{-}{94}{}{x}{-}{68}\right){}{{ⅇ}}^{{-}\frac{{91}}{{-}{86}{+}{5}{}{x}}}}{\left({-}{35}{}{{x}}^{{4}}{-}{14}{}{{x}}^{{3}}{-}{9}{}{{x}}^{{2}}{-}{51}{}{x}{-}{73}\right){}{\left({-}{86}{+}{5}{}{x}\right)}^{{2}}}$ (1)
 > $\mathrm{Int}\left(T,x\right)=\mathrm{Gosper}\left(T,x\right)$
 ${\int }\left(\frac{\left({-}{21}{}{{x}}^{{2}}{+}{116}{}{x}{-}{94}\right){}{{ⅇ}}^{{-}\frac{{91}}{{-}{86}{+}{5}{}{x}}}}{{-}{35}{}{{x}}^{{4}}{-}{14}{}{{x}}^{{3}}{-}{9}{}{{x}}^{{2}}{-}{51}{}{x}{-}{73}}{-}\frac{\left({-}{7}{}{{x}}^{{3}}{+}{58}{}{{x}}^{{2}}{-}{94}{}{x}{-}{68}\right){}{{ⅇ}}^{{-}\frac{{91}}{{-}{86}{+}{5}{}{x}}}{}\left({-}{140}{}{{x}}^{{3}}{-}{42}{}{{x}}^{{2}}{-}{18}{}{x}{-}{51}\right)}{{\left({-}{35}{}{{x}}^{{4}}{-}{14}{}{{x}}^{{3}}{-}{9}{}{{x}}^{{2}}{-}{51}{}{x}{-}{73}\right)}^{{2}}}{+}\frac{{455}{}\left({-}{7}{}{{x}}^{{3}}{+}{58}{}{{x}}^{{2}}{-}{94}{}{x}{-}{68}\right){}{{ⅇ}}^{{-}\frac{{91}}{{-}{86}{+}{5}{}{x}}}}{\left({-}{35}{}{{x}}^{{4}}{-}{14}{}{{x}}^{{3}}{-}{9}{}{{x}}^{{2}}{-}{51}{}{x}{-}{73}\right){}{\left({-}{86}{+}{5}{}{x}\right)}^{{2}}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{=}\frac{\left({7}{}{{x}}^{{3}}{-}{58}{}{{x}}^{{2}}{+}{94}{}{x}{+}{68}\right){}{{ⅇ}}^{{-}\frac{{91}}{{-}{86}{+}{5}{}{x}}}}{{35}{}{{x}}^{{4}}{+}{14}{}{{x}}^{{3}}{+}{9}{}{{x}}^{{2}}{+}{51}{}{x}{+}{73}}$ (2)

References

 Almkvist, G, and Zeilberger, D. "The method of differentiating under the integral sign." Journal of Symbolic Computation. Vol. 10 (1990): 571-591.