$\mathrm{int}\left(\cos \left(x+1\right)\,x\,\mathrm{method}=\mathrm{meijerg\_raw}\right)$

$\cos \left(1\right)x\mathrm{hypergeom}\left(\left[\right]\,\left[\frac{3}{2}\right]\,-\frac{x^2}{4}\right)-\frac{\sin \left(1\right)x^2\mathrm{hypergeom}\left(\left[1\right]\,\left[\frac{3}{2}\,2\right]\,-\frac{x^2}{4}\right)}{2}$

$\mathrm{int}\left(\cos \left(x+1\right)\,x\,\mathrm{method}=\mathrm{risch}\right)$

$\sin \left(x+1\right)$

$\mathrm{int}\left(\cos \left(x+1\right)\,x\,\mathrm{method}=\mathrm{\_RETURNVERBOSE}\right)$

$\left[''derivativedivides''=\sin \left(x+1\right)\,''default''=\sin \left(x+1\right)\,''norman''=\frac{2\tan \left(\frac{x}{2}+\frac{1}{2}\right)}{1+{\tan \left(\frac{x}{2}+\frac{1}{2}\right)}^2}\,''meijerg''=\cos \left(1\right)\sin \left(x\right)-\sin \left(1\right)\sqrt{\mathrm{\pi }}\left(\frac{1}{\sqrt{\mathrm{\pi }}}-\frac{\cos \left(x\right)}{\sqrt{\mathrm{\pi }}}\right)\,''risch''=\sin \left(x+1\right)\,\mathrm{FAILS}=\left(''gosper''\,''lookup''\,''trager''\,''elliptic''\right)\right]$

$\mathrm{int}\left(\frac{1}{\sqrt{\left(-t^2+1\right)\left(-2t^2+1\right)}}\,t=0..1\,\mathrm{method}=\mathrm{\_RETURNVERBOSE}\right)$

$\left[\mathrm{ftoc}=\mathrm{EllipticK}\left(\sqrt{2}\right)\,\mathrm{elliptic}=-\frac{\mathrm{I}\sqrt{2}\mathrm{EllipticK}\left(\frac{\sqrt{2}}{2}\right)}{2}+\frac{\sqrt{2}\mathrm{EllipticK}\left(\frac{\sqrt{2}}{2}\right)}{2}\,\mathrm{ftocms}=\mathrm{EllipticK}\left(\sqrt{2}\right)\,\mathrm{FAILS}=\left(\mathrm{distribution}\,\mathrm{piecewise}\,\mathrm{series}\,o\,\mathrm{polynomial}\,\ln \,\mathrm{lookup}\,\mathrm{cook}\,\mathrm{ratpoly}\,\mathrm{elliptictrig}\,\mathrm{meijergspecial}\,\mathrm{improper}\,\mathrm{asymptotic}\,\mathrm{meijerg}\,\mathrm{contour}\right)\right]$

$\int \frac{\left(2+3x^5\right){\left(-1+x^2+x^5\right)}^{\frac{1}{2}}}{{\left(-1+x^5\right)}^2}\ⅆx$

$-\frac{x\sqrt{x^5+x^2-1}}{x^5-1}+\frac{\ln \left(-\frac{-x^5+2\sqrt{x^5+x^2-1}x-2x^2+1}{\left(x-1\right)\left(x^4+x^3+x^2+x+1\right)}\right)}{2}$

$g ≔ x^2-x\cdot y\+y^2\+z^4\:$

$\mathrm{PolynomialTools}:-\mathrm{IsHomogeneous}\left(g\right)$

$\mathrm{false}$

$\mathrm{PolynomialTools}:-\mathrm{IsHomogeneous}\left(g\,\left[x\,y\,z\right]\,\left[2\,2\,1\right]\right)$

$\mathrm{true}$

$$\begin{gathered} \mathrm{RC}≔\mathrm{RegularChains}\: R≔\mathrm{RC}:-\mathrm{PolynomialRing}\left(\left[x\,y\,z\right]\right)\: \\ \mathrm{RC}:-\mathrm{Display}\left(\mathrm{RC}:-\mathrm{RealTriangularize}\left(\left[g\right]\,R\right)\,R\right) \end{gathered}$$

$\left[\left\{\begin{array}{cc}x=0& \\ y=0& \\ z=0& \end{array}\right.\right]$

$\mathrm{limit}\left(\frac{x\cdot y\cdot z}{g}\,\left\{x\,y\,z\right\}\=~0\right)$

$0$

$\mathrm{limit}\left(\frac{g\+x\cdot y\cdot z}{g\+x\cdot y^3-y\cdot z^3}\,\left\{x\,y\,z\right\}\=~0\right)$

$1$

$\mathrm{limit}\left(\frac{g\+x\cdot z^2}{g\+y\cdot z^2}\,\left\{x\,y\,z\right\}\=~0\right)$

$\mathrm{undefined}$

$\mathrm{limit}\left(\frac{\sin \left(x\+y\right)\+z^2}{g}\,\left\{x\,y\,z\right\}\=~0\right)$

$\mathrm{undefined}$

$\mathrm{limit}\left(\frac{g}{{\left(x-z^2\right)}^4}\,\left\{x\,y\,z\right\}\=~0\right)$

$\mathrm{\infty }$

$\left[\mathrm{Add}\,\mathrm{ApproximatelyEqual}\,\mathrm{ApproximatelyZero}\,\mathrm{Copy}\,\mathrm{Degree}\,\mathrm{Display}\,\mathrm{Divide}\,\mathrm{EvaluateAtOrigin}\,\mathrm{Exponentiate}\,\mathrm{GeometricSeries}\,\mathrm{GetAnalyticExpression}\,\mathrm{GetCoefficient}\,\mathrm{HenselFactorize}\,\mathrm{HomogeneousPart}\,\mathrm{Inverse}\,\mathrm{IsUnit}\,\mathrm{MainVariable}\,\mathrm{Multiply}\,\mathrm{Negate}\,\mathrm{PowerSeries}\,\mathrm{Precision}\,\mathrm{SetDefaultDisplayStyle}\,\mathrm{SetDisplayStyle}\,\mathrm{Subtract}\,\mathrm{SumOfAllMonomials}\,\mathrm{TaylorShift}\,\mathrm{Truncate}\,\mathrm{UnivariatePolynomialOverPowerSeries}\,\mathrm{UpdatePrecision}\,\mathrm{Variables}\,\mathrm{WeierstrassPreparation}\right]$

$a≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\frac{1}{1-x-y}\mathrm{:}1+x+y+\mathrm{\dots }\right]$

$b≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\frac{1}{\left(1-x\right)\left(1-y\right)}\mathrm{:}1+x+y+\mathrm{\dots }\right]$

$c≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\frac{1}{1-x-y}-\frac{1}{\left(1-x\right)\left(1-y\right)}\mathrm{:}0+\mathrm{\dots }\right]$

$99x^{99}y+4949x^{98}{y}^2+161699x^{97}{y}^3+3921224x^{96}{y}^4+75287519x^{95}{y}^5+1192052399x^{94}{y}^6+16007560799x^{93}{y}^7+186087894299x^{92}{y}^8+1902231808399x^{91}{y}^9+17310309456439x^{90}{y}^{10}+141629804643599x^{89}{y}^{11}+1050421051106699x^{88}{y}^{12}+7110542499799199x^{87}{y}^{13}+44186942677323599x^{86}{y}^{14}+253338471349988639x^{85}{y}^{15}+1345860629046814649x^{84}{y}^{16}+6650134872937201799x^{83}{y}^{17}+30664510802988208299x^{82}{y}^{18}+132341572939212267399x^{81}{y}^{19}+535983370403809682969x^{80}{y}^{20}+2041841411062132125599x^{79}{y}^{21}+7332066885177656269199x^{78}{y}^{22}+24865270306254660391199x^{77}{y}^{23}+79776075565900368755099x^{76}{y}^{24}+242519269720337121015503x^{75}{y}^{25}+699574816500972464467799x^{74}{y}^{26}+1917353200780443050763599x^{73}{y}^{27}+4998813702034726525205099x^{72}{y}^{28}+12410847811948286545336799x^{71}{y}^{29}+29372339821610944823963759x^{70}{y}^{30}+66324638306863423796047199x^{69}{y}^{31}+143012501349174257560226774x^{68}{y}^{32}+294692427022540894366527899x^{67}{y}^{33}+580717429720889409486981449x^{66}{y}^{34}+1095067153187962886461165019x^{65}{y}^{35}+1977204582144932989443770174x^{64}{y}^{36}+3420029547493938143902737599x^{63}{y}^{37}+5670048986634686922786117599x^{62}{y}^{38}+9013924030034630492634340799x^{61}{y}^{39}+13746234145802811501267369719x^{60}{y}^{40}+20116440213369968050635175199x^{59}{y}^{41}+28258808871162574166368460399x^{58}{y}^{42}+38116532895986727945334202399x^{57}{y}^{43}+49378235797073715747364762199x^{56}{y}^{44}+61448471214136179596720592959x^{55}{y}^{45}+73470998190814997343905056799x^{54}{y}^{46}+84413487283064039501507937599x^{53}{y}^{47}+93206558875049876949581681099x^{52}{y}^{48}+98913082887808032681188722799x^{51}{y}^{49}+100891344545564193334812497255x^{50}{y}^{50}+98913082887808032681188722799x^{49}{y}^{51}+93206558875049876949581681099x^{48}{y}^{52}+84413487283064039501507937599x^{47}{y}^{53}+73470998190814997343905056799x^{46}{y}^{54}+61448471214136179596720592959x^{45}{y}^{55}+49378235797073715747364762199x^{44}{y}^{56}+38116532895986727945334202399x^{43}{y}^{57}+28258808871162574166368460399x^{42}{y}^{58}+20116440213369968050635175199x^{41}{y}^{59}+13746234145802811501267369719x^{40}{y}^{60}+9013924030034630492634340799x^{39}{y}^{61}+5670048986634686922786117599x^{38}{y}^{62}+3420029547493938143902737599x^{37}{y}^{63}+1977204582144932989443770174x^{36}{y}^{64}+1095067153187962886461165019x^{35}{y}^{65}+580717429720889409486981449x^{34}{y}^{66}+294692427022540894366527899x^{33}{y}^{67}+143012501349174257560226774x^{32}{y}^{68}+66324638306863423796047199x^{31}{y}^{69}+29372339821610944823963759x^{30}{y}^{70}+12410847811948286545336799x^{29}{y}^{71}+4998813702034726525205099x^{28}{y}^{72}+1917353200780443050763599x^{27}{y}^{73}+699574816500972464467799x^{26}{y}^{74}+242519269720337121015503x^{25}{y}^{75}+79776075565900368755099x^{24}{y}^{76}+24865270306254660391199x^{23}{y}^{77}+7332066885177656269199x^{22}{y}^{78}+2041841411062132125599x^{21}{y}^{79}+535983370403809682969x^{20}{y}^{80}+132341572939212267399x^{19}{y}^{81}+30664510802988208299x^{18}{y}^{82}+6650134872937201799x^{17}{y}^{83}+1345860629046814649x^{16}{y}^{84}+253338471349988639x^{15}{y}^{85}+44186942677323599x^{14}{y}^{86}+7110542499799199x^{13}{y}^{87}+1050421051106699x^{12}{y}^{88}+141629804643599x^{11}{y}^{89}+17310309456439x^{10}{y}^{90}+1902231808399x^9{y}^{91}+186087894299x^8{y}^{92}+16007560799x^7{y}^{93}+1192052399x^6{y}^{94}+75287519x^5{y}^{95}+3921224x^4{y}^{96}+161699x^3{y}^{97}+4949x^2{y}^{98}+99x{y}^{99}$

$d≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}{\ⅇ}^{x+y}\mathrm{:}1+\mathrm{\dots }\right]$

$e≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\sin \left(x\right)\mathrm{:}0+\mathrm{\dots }\right]$

$u≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(4+\mathrm{\dots }\right)\mathrm{+}\left(0+\mathrm{\dots }\right)z\mathrm{+}\left(\mathrm{-3}+\mathrm{\dots }\right)z^2\mathrm{+}\left(1\right)z^3\right]$

$\mathrm{factorization}≔\left[\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(1+\mathrm{\dots }\right)\mathrm{+}\left(1\right)z\right]\,\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(4+\mathrm{\dots }\right)\mathrm{+}\left(\mathrm{-4}+\mathrm{\dots }\right)z\mathrm{+}\left(1\right)z^2\right]\right]$

$\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(4+4x+\frac{32y}{9}+\frac{22x^2}{9}+\frac{388xy}{81}+\frac{578y^2}{243}+\frac{140x^3}{81}+\frac{3968x^{2}y}{729}+\frac{3974x{y}^2}{729}+\frac{11876y^3}{6561}+\frac{2291x^4}{1458}+\frac{41530x^{3}y}{6561}+\frac{62197x^2{y}^2}{6561}+\frac{124138x{y}^3}{19683}+\frac{185939y^4}{118098}+\left(\frac{47494x^5}{32805}+\frac{424718x^{4}y}{59049}+\frac{2546836x^3{y}^2}{177147}+\frac{7642522x^2{y}^3}{531441}+\frac{22924393x{y}^4}{3188646}+\mathrm{\dots }\right)+\mathrm{\dots }\right)\mathrm{+}\mathrm{\dots }\right]$

$\mathrm{simplify}\left(W\left( \frac{5 \sqrt[4]{2} \ln \left(4\right)}{4}\right)\right)$

$\frac{5\ln \left(2\right)}{4}$

$\mathrm{simplify}\left(W\left(\frac{2\cdot {617673396283947}^{\frac{14}{15}}\cdot {\left({14}^{\frac{2}{9}}\cdot 3^{\frac{7}{9}}\right)}^{\frac{1}{15}}\cdot \ln \left(\frac{14}{3}\right)}{9265100944259205}\right)\right)$

$\frac{2\ln \left(2\right)}{135}+\frac{2\ln \left(7\right)}{135}-\frac{2\ln \left(3\right)}{135}$

$\mathrm{simplify}\left(W\left(-1\,-\frac{3^4 \ln \left(\frac{4}{3}\right)}{4^3}\right)\right)$

$4\ln \left(3\right)-8\ln \left(2\right)$

$\mathrm{simplify}\left(W\left(-1\, -\frac{5}{104} \ln \left(\frac{13}{5}\right) {13}^{\frac{3}{8}} 5^{\frac{5}{8}}\right)\right)$

$-\frac{13\ln \left(13\right)}{8}+\frac{13\ln \left(5\right)}{8}$

$\mathrm{simplify}\left(W\left(\frac{528309}{168976} \ln \left(\frac{9}{4}\right) 2^{\frac{3110}{10561}} 3^{\frac{7451}{10561}}\right)\right)$

$\frac{39134\ln \left(3\right)}{10561}-\frac{39134\ln \left(2\right)}{10561}$

$\mathrm{dEqs}≔\left[\begin{array}{c}\frac{\ⅆ}{\ⅆt}\mathrm{A\_\_1}\left(t\right)\\ \frac{\ⅆ}{\ⅆt}\mathrm{A\_\_2}\left(t\right)\end{array}\right]=\left[\begin{array}{c}\mathrm{B\_\_1}\left(t\right)\\ \mathrm{B\_\_2}\left(t\right)\end{array}\right],\left[\begin{array}{c}\frac{\ⅆ}{\ⅆt}\mathrm{B\_\_1}\left(t\right)\\ \frac{\ⅆ}{\ⅆt}\mathrm{B\_\_2}\left(t\right)\end{array}\right]=\left[\begin{array}{c}1.2\mathrm{A\_\_1}\left(t\right)\\ 2.3\mathrm{A\_\_2}\left(t\right)\end{array}\right]\:$

$\mathrm{InEqs}≔\left[\begin{array}{c}\mathrm{A\_\_1}\left(0\right)\\ \mathrm{A\_\_2}\left(0\right)\end{array}\right]=\left[\begin{array}{c}1.1\\ 0.3\end{array}\right],\left[\begin{array}{c}\mathrm{B\_\_1}\left(0\right)\\ \mathrm{B\_\_2}\left(0\right)\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]\:$

$\mathrm{sol}≔ \mathrm{dsolve}\left(\left\{\mathrm{dEqs}\, \mathrm{InEqs}\right\}\,\mathrm{numeric}\right)$

$\mathrm{sol}≔\mathbf{proc}\left(\mathrm{x\_rkf45}\right)...\mathbf{end\; proc}$

$\mathrm{plots}:-\mathrm{odeplot}\left(\mathrm{sol}\,\left[\left[t\,\mathrm{A\_\_1}\left(t\right)\right]\,\left[t\,\mathrm{A\_\_2}\left(t\right)\right]\,\left[t\,\mathrm{B\_\_1}\left(t\right)\right]\,\left[t\,\mathrm{B\_\_2}\left(t\right)\right]\right]\,0..5\,\mathrm{legend}=\left[\mathrm{A\_\_1}\left(t\right)\,\mathrm{A\_\_2}\left(t\right)\,\mathrm{B\_\_1}\left(t\right)\,\mathrm{B\_\_2}\left(t\right)\right]\right)$

$\mathrm{with}\left(\mathrm{PolyhedralSets}\right)$

$\left[\mathrm{AffineHull}\,\mathrm{Area}\,\mathrm{CharacteristicCone}\,\mathrm{ConvexHull}\,\mathrm{Coordinates}\,\mathrm{Dimension}\,\mathrm{Display}\,\mathrm{DualSet}\,\mathrm{Edges}\,\mathrm{Equal}\,\mathrm{ExampleSets}\,\mathrm{Faces}\,\mathrm{Facets}\,\mathrm{Graph}\,\mathrm{ID}\,\mathrm{InteriorPoint}\,\mathrm{IsBounded}\,\mathrm{IsEmpty}\,\mathrm{IsFace}\,\mathrm{IsInInterior}\,\mathrm{IsUniversalSet}\,\mathrm{Length}\,\mathrm{LinearTransformation}\,\mathrm{LinearitySpace}\,\mathrm{LocatePoint}\,\mathrm{Plot}\,\mathrm{PolyhedralSet}\,\mathrm{PrintLevel}\,\mathrm{Project}\,\mathrm{Relations}\,\mathrm{SplitIntoSimplices}\,\mathrm{Translate}\,\mathrm{Vertices}\,\mathrm{VerticesAndRays}\,\mathrm{Volume}\,\mathrm{in}\,\mathrm{intersect}\,\mathrm{subset}\right]$

$$\begin{gathered} l≔\left[-65 \mathrm{x\_\_1}-63\mathrm{x\_\_2}-87\mathrm{x\_\_3}+66\mathrm{x\_\_4}+10\mathrm{x\_\_5}\le 77\, -81 \mathrm{x\_\_1}+\mathrm{x\_\_2}+6\mathrm{x\_\_3}+23\mathrm{x\_\_4}-92\mathrm{x\_\_5}\le -17\, \\ -99 \mathrm{x\_\_1}+43\mathrm{x\_\_2}+40\mathrm{x\_\_3}-66\mathrm{x\_\_4}-21\mathrm{x\_\_5}\le -56\, -3 \mathrm{x\_\_1}+63\mathrm{x\_\_2}-22\mathrm{x\_\_3}-82\mathrm{x\_\_4}+85\mathrm{x\_\_5}\le 70\, \\ 37 \mathrm{x\_\_1}-64\mathrm{x\_\_2}+88\mathrm{x\_\_3}+29\mathrm{x\_\_4}-43\mathrm{x\_\_5}\le -98\, 41 \mathrm{x\_\_1}-38\mathrm{x\_\_2}+66\mathrm{x\_\_3}+38\mathrm{x\_\_4}-28\mathrm{x\_\_5}\le 10\, \\ 71 \mathrm{x\_\_1}-52\mathrm{x\_\_2}-34\mathrm{x\_\_3}+80\mathrm{x\_\_4}+38\mathrm{x\_\_5}\le 75\, 25 \mathrm{x\_\_1}-9\mathrm{x\_\_2}-5\mathrm{x\_\_3}-77\mathrm{x\_\_4}+99\mathrm{x\_\_5}\le 91\, \\ 33 \mathrm{x\_\_1}-5\mathrm{x\_\_2}+30\mathrm{x\_\_3}-82\mathrm{x\_\_4}+97\mathrm{x\_\_5}\le -16\, 56 \mathrm{x\_\_1}+77\mathrm{x\_\_2}+48\mathrm{x\_\_3}-11\mathrm{x\_\_4}+60\mathrm{x\_\_5}\le -58\right]\: \end{gathered}$$

$P≔\mathrm{PolyhedralSet}\left(l\right)$

$P≔\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[\mathrm{x\_\_1}\,\mathrm{x\_\_2}\,\mathrm{x\_\_3}\,\mathrm{x\_\_4}\,\mathrm{x\_\_5}\right]\\ \mathrm{Relations}& \:& \left[-\mathrm{x\_\_1}-\frac{63\mathrm{x\_\_2}}{65}-\frac{87\mathrm{x\_\_3}}{65}+\frac{66\mathrm{x\_\_4}}{65}+\frac{2\mathrm{x\_\_5}}{13}\le \frac{77}{65}\,-\mathrm{x\_\_1}+\frac{\mathrm{x\_\_2}}{81}+\frac{2\mathrm{x\_\_3}}{27}+\frac{23\mathrm{x\_\_4}}{81}-\frac{92\mathrm{x\_\_5}}{81}\le -\frac{17}{81}\,-\mathrm{x\_\_1}+\frac{43\mathrm{x\_\_2}}{99}+\frac{40\mathrm{x\_\_3}}{99}-\frac{2\mathrm{x\_\_4}}{3}-\frac{7\mathrm{x\_\_5}}{33}\le -\frac{56}{99}\,-\mathrm{x\_\_1}+21\mathrm{x\_\_2}-\frac{22\mathrm{x\_\_3}}{3}-\frac{82\mathrm{x\_\_4}}{3}+\frac{85\mathrm{x\_\_5}}{3}\le \frac{70}{3}\,\mathrm{x\_\_1}-\frac{64\mathrm{x\_\_2}}{37}+\frac{88\mathrm{x\_\_3}}{37}+\frac{29\mathrm{x\_\_4}}{37}-\frac{43\mathrm{x\_\_5}}{37}\le -\frac{98}{37}\,\mathrm{x\_\_1}-\frac{38\mathrm{x\_\_2}}{41}+\frac{66\mathrm{x\_\_3}}{41}+\frac{38\mathrm{x\_\_4}}{41}-\frac{28\mathrm{x\_\_5}}{41}\le \frac{10}{41}\,\mathrm{x\_\_1}-\frac{52\mathrm{x\_\_2}}{71}-\frac{34\mathrm{x\_\_3}}{71}+\frac{80\mathrm{x\_\_4}}{71}+\frac{38\mathrm{x\_\_5}}{71}\le \frac{75}{71}\,\mathrm{x\_\_1}-\frac{9\mathrm{x\_\_2}}{25}-\frac{\mathrm{x\_\_3}}{5}-\frac{77\mathrm{x\_\_4}}{25}+\frac{99\mathrm{x\_\_5}}{25}\le \frac{91}{25}\,\mathrm{x\_\_1}-\frac{5\mathrm{x\_\_2}}{33}+\frac{10\mathrm{x\_\_3}}{11}-\frac{82\mathrm{x\_\_4}}{33}+\frac{97\mathrm{x\_\_5}}{33}\le -\frac{16}{33}\,\mathrm{x\_\_1}+\frac{11\mathrm{x\_\_2}}{8}+\frac{6\mathrm{x\_\_3}}{7}-\frac{11\mathrm{x\_\_4}}{56}+\frac{15\mathrm{x\_\_5}}{14}\le -\frac{29}{28}\right]\end{array}$

$\mathrm{CodeTools}:-\mathrm{Usage}\left(\mathrm{Project}\left(P\,\left[\mathrm{x\_\_5}\right]\right)\right)$

memory used=261.74MiB, alloc change=64.00MiB, cpu time=1.80s, real time=1.72s, gc time=187.50ms

$\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[\mathrm{x\_\_1}\,\mathrm{x\_\_2}\,\mathrm{x\_\_3}\,\mathrm{x\_\_4}\,\mathrm{x\_\_5}\right]\\ \mathrm{Relations}& \:& \left[\mathrm{x\_\_5}\le -\frac{161907933}{1240163204}\,\mathrm{x\_\_4}=0\,\mathrm{x\_\_3}=0\,\mathrm{x\_\_2}=0\,\mathrm{x\_\_1}=0\right]\end{array}$

$\mathrm{asympt}\left(\mathrm{Li}\left(x\right)\,x\right)$

$\left(\frac{1}{\ln \left(x\right)}+\frac{1}{{\ln \left(x\right)}^2}+\frac{2}{{\ln \left(x\right)}^3}+\frac{6}{{\ln \left(x\right)}^4}+\frac{24}{{\ln \left(x\right)}^5}+\mathrm{O}\left(\frac{1}{{\ln \left(x\right)}^6}\right)\right)x$

$\mathrm{asympt}\left(\mathrm{Li}\left(\frac{1}{x}\right)\,x\right)$

$\frac{-\frac{1}{\ln \left(x\right)}+\frac{1}{{\ln \left(x\right)}^2}-\frac{2}{{\ln \left(x\right)}^3}+\frac{6}{{\ln \left(x\right)}^4}-\frac{24}{{\ln \left(x\right)}^5}+\mathrm{O}\left(\frac{1}{{\ln \left(x\right)}^6}\right)}{x}$

$\mathrm{F8}≔\mathrm{GF}\left(2\, 3\, t^3\+t\+1\right)$

$\mathrm{F8}≔{𝔽}_8$

$$\begin{gathered} a≔\mathrm{F8}\left(t^2\+1\right)\; \\ b≔\mathrm{F8}\left(t^2\+t\right) \end{gathered}$$

$a≔\left(t^2+1\right)\mathbf{mod}2$

$b≔\left(t^2+t\right)\mathbf{mod}2$

$$\begin{gathered} \mathrm{F8}\left[\mathrm{`+`}\right]\left(a\,b\right)\; \\ \mathrm{F8}\left[\mathrm{`*`}\right]\left(a\,b\right) \end{gathered}$$

$\left(t+1\right)\mathbf{mod}2$

$\left(t+1\right)\mathbf{mod}2$

$\mathrm{unassign}\left('f'\,'x'\,'y'\right)$

$\mathrm{ie1}≔{\int }_0^{\mathrm{\infty }}\frac{f\left(y\right)}{5+\sin \left(y\right)}\cos \left(xy\right)\ⅆy={\ⅇ}^{-x}$

$\mathrm{ie1}≔{\int }_0^{\mathrm{\infty }}\frac{f\left(y\right)\cos \left(xy\right)}{5+\sin \left(y\right)}\ⅆy={\ⅇ}^{-x}$

$\mathrm{intsolve}\left(\mathrm{ie1}\,f\left(x\right)\right)$

$f\left(x\right)=\frac{2\left(5+\sin \left(x\right)\right)}{\mathrm{\pi }\left(x^2+1\right)}$

$\mathrm{unassign}\left('g'\,'k'\,'u'\,'v'\right)$

$k≔\left(u\&comma;v\right)\mapsto \left\{\begin{array}{cc}u\cdot \left(2-v\right)& u<v\\ v\cdot \left(2-u\right)& \mathrm{otherwise}\end{array}\right.$

$k≔\left(u\&comma;v\right)\mapsto \left\{\begin{array}{cc}u\cdot \left(2-v\right)& u<v\\ v\cdot \left(2-u\right)& \mathrm{otherwise}\end{array}\right.$

$\mathrm{ie2}≔g\left(u\right)+{\int }_0^{2}g\left(v\right)'k'\left(u\&comma;v\right)\&DifferentialD;v+u^2=0$

$\mathrm{ie2}≔g\left(u\right)+{\int }_0^{2}g\left(v\right)k\left(u\&comma;v\right)\&DifferentialD;v+u^2=0$

$\mathrm{intsolve}\left(\mathrm{ie2}\&comma;g\left(u\right)\&comma;'\mathrm{method}'='\mathrm{collocation}'\&comma;'\mathrm{order}'=5\right)$

$g\left(u\right)=-\frac{1875}{35993}{u}^5+\frac{34265625}{4018042562}{u}^4+\frac{1240000}{2009021281}{u}^3-\frac{1813412625}{2009021281}{u}^2+\frac{1141270188}{2009021281}u$

$\mathrm{with}\left(\mathrm{LREtools}\right)$

$\left[\mathrm{AnalyticityConditions}\&comma;\mathrm{GCRD}\&comma;\mathrm{GeneralizedExponents}\&comma;\mathrm{GuessRecurrence}\&comma;\mathrm{HypergeometricTerm}\&comma;\mathrm{IsDesingularizable}\&comma;\mathrm{LCLM}\&comma;\mathrm{MinimalRecurrence}\&comma;\mathrm{MultiplyOperators}\&comma;\mathrm{OperatorToRecurrence}\&comma;\mathrm{REcontent}\&comma;\mathrm{REcreate}\&comma;\mathrm{REplot}\&comma;\mathrm{REprimpart}\&comma;\mathrm{REreduceorder}\&comma;\mathrm{REtoDE}\&comma;\mathrm{REtodelta}\&comma;\mathrm{REtoproc}\&comma;\mathrm{RecurrenceToOperator}\&comma;\mathrm{RightDivision}\&comma;\mathrm{RightFactors}\&comma;\mathrm{SumDecompose}\&comma;\mathrm{ValuesAtPoint}\&comma;\mathrm{autodispersion}\&comma;\mathrm{constcoeffsol}\&comma;\mathrm{dAlembertiansols}\&comma;\mathrm{\delta }\&comma;\mathrm{dispersion}\&comma;\mathrm{divconq}\&comma;\mathrm{firstlin}\&comma;\mathrm{hypergeomsols}\&comma;\mathrm{polysols}\&comma;\mathrm{ratpolysols}\&comma;\mathrm{riccati}\&comma;\mathrm{shift}\right]$

$$\begin{gathered} \mathrm{fib} ≔ \mathbf{proc}\left(n\colon\colon \mathrm{nonnegint}\right) \mathbf{if} n\&equals;0 \mathbf{then} 0\&semi; \mathbf{elif} n\&equals;1 \mathbf{then} 1\&semi; \mathbf{else} \mathrm{fib}\left(n-1\right)\&plus;\mathrm{fib}\left(n-2\right)\&semi; \mathbf{end} \mathbf{if}\&colon; \mathbf{end} \mathbf{proc}\&colon; \\ \mathrm{vals} ≔ \left[\mathrm{seq}\left(\mathrm{fib}\left(i\right)\&comma;i\&equals;0..20\right)\right]\&semi; \end{gathered}$$

$\mathrm{vals}≔\left[0\&comma;1\&comma;1\&comma;2\&comma;3\&comma;5\&comma;8\&comma;13\&comma;21\&comma;34\&comma;55\&comma;89\&comma;144\&comma;233\&comma;377\&comma;610\&comma;987\&comma;1597\&comma;2584\&comma;4181\&comma;6765\right]$

$\mathrm{GuessRecurrence}\left(\mathrm{vals}\&comma;F\left(i\right)\&comma; 0\right)\&semi;$

$F\left(i+2\right)-F\left(i+1\right)-F\left(i\right)=0$

$\mathrm{\_Env\_LRE\_tau}≔E\&colon;$$\mathrm{\_Env\_LRE\_x}≔x\&colon;$

$L≔E^3-3E^2+3E-1$

$L≔E^3-3E^2+3E-1$

$\mathrm{RightFactors}\left(L\&comma;1\right)$

$\left\{E-1\&comma;\left[\left(x+{\mathrm{\_Z}}_1\right)E-x-{\mathrm{\_Z}}_1-1\&comma;''a\; family\; of\; dimension''\&comma;1\&comma;''with\; equations''\&comma;\varnothing \right]\&comma;\left[\left(x^2+{\mathrm{\_Z}}_{2}x+{\mathrm{\_Z}}_1\right)E-x^2-{\mathrm{\_Z}}_{2}x-2x-{\mathrm{\_Z}}_1-{\mathrm{\_Z}}_2-1\&comma;''a\; family\; of\; dimension''\&comma;2\&comma;''with\; equations''\&comma;\varnothing \right]\right\}$

$\mathrm{RightFactors}\left(L\&comma;2\right)$

$\left\{E^2-2E+1\&comma;\left[\left(x+{\mathrm{\_Z}}_1\right)E^2+\left(-2x-2{\mathrm{\_Z}}_1-1\right)E+x+{\mathrm{\_Z}}_1+1\&comma;''a\; family\; of\; dimension''\&comma;1\&comma;''with\; equations''\&comma;\varnothing \right]\&comma;\left[\left(x^2+{\mathrm{\_Z}}_{2}x+{\mathrm{\_Z}}_1\right)E^2+\left(-2x^2-2{\mathrm{\_Z}}_{2}x-2x-2{\mathrm{\_Z}}_1-{\mathrm{\_Z}}_2+1\right)E+x^2+{\mathrm{\_Z}}_{2}x+2x+{\mathrm{\_Z}}_1+{\mathrm{\_Z}}_2+1\&comma;''a\; family\; of\; dimension''\&comma;2\&comma;''with\; equations''\&comma;\varnothing \right]\right\}$