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

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

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

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

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

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

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

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

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

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

$g ≔ x^2-x\cdot y\+y^2\+z^4\:$

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

$\mathrm{false}$

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

$\mathrm{true}$

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

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

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

$0$

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

$1$

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

$\mathrm{undefined}$

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

$\mathrm{undefined}$

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

$\mathrm{\infty }$

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

$\mathrm{Object}\left(\mathrm{PowerSeriesObject}\,\mathrm{Array(0..1,\; \{(1)\; =\; 1\})}\,1\,\mathrm{geo\_gen}\,\left\{x\,y\right\}\,\mathrm{ancestors}\right)$

$\mathrm{Object}\left(\mathrm{PowerSeriesObject}\,\mathrm{Array(0..1,\; \{(1)\; =\; 1\})}\,1\,\mathrm{sam\_gen}\,\left\{x\,y\right\}\,\mathrm{ancestors}\right)$

$\mathrm{Object}\left(\mathrm{PowerSeriesObject}\,\mathrm{Array(0..0,\; \{\})}\,0\,\mathrm{add\_gen}\,\left\{x\,y\right\}\,\mathrm{ancestors}\right)$

$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}$

$\mathrm{Object}\left(\mathrm{PowerSeriesObject}\,\mathrm{Array(0..0,\; \{\})}\,0\,\mathrm{proc\_gen}\,\left\{x\,y\right\}\,\mathrm{ancestors}\right)$

$\mathrm{Object}\left(\mathrm{PowerSeriesObject}\,\mathrm{Array(0..0,\; \{\})}\,0\,\mathrm{proc\_gen}\,\left\{x\right\}\,\mathrm{ancestors}\right)$

$\mathrm{Object}\left(\mathrm{UnivariatePolynomialOverPowerSeriesObject}\,\mathrm{Array(0..3,\; \{(1)\; =\; module\; PowerSeriesObject\; ()\; local\; hpoly::Array,\; deg::nonnegint,\; gen::procedure,\; vars::(set(name)),\; ancestors::record,\; algexpr::algebraic,\; dstyle::(list(:-identical(:-maxterms,\; :-precision)\; =\; \{:-nonnegint,\; :-identical(:-infinity)\}));\; option\; object;\; end\; module,\; (2)\; =\; module\; PowerSeriesObject\; ()\; local\; hpoly::Array,\; deg::nonnegint,\; gen::procedure,\; vars::(set(name)),\; ancestors::record,\; algexpr::algebraic,\; dstyle::(list(:-identical(:-maxterms,\; :-precision)\; =\; \{:-nonnegint,\; :-identical(:-infinity)\}));\; option\; object;\; end\; module,\; (3)\; =\; module\; PowerSeriesObject\; ()\; local\; hpoly::Array,\; deg::nonnegint,\; gen::procedure,\; vars::(set(name)),\; ancestors::record,\; algexpr::algebraic,\; dstyle::(list(:-identical(:-maxterms,\; :-precision)\; =\; \{:-nonnegint,\; :-identical(:-infinity)\}));\; option\; object;\; end\; module\})}\right)$

$\left[\mathbf{module}\mathrm{UnivariatePolynomialOverPowerSeriesObject}\left(\right)\mathbf{option}\mathrm{object}\;\mathbf{local}\mathrm{upoly}::\mathrm{Array}\,\mathrm{vname}::\mathrm{name}\,\mathrm{dstyle}::\left(\mathrm{list}\left(\left\{:-\mathrm{identical}\left(:-\mathrm{maxdegree}\right)\=\left\{:-\mathrm{nonnegint}\,:-\mathrm{identical}\left(:-\mathrm{\∞}\right)\right\}\,:-\mathrm{identical}\left(:-\mathrm{maxterms}\,:-\mathrm{precision}\right)\=\left\{:-\mathrm{nonnegint}\,:-\mathrm{identical}\left(:-\mathrm{\∞}\right)\right\}\right\}\right)\right)\;\mathbf{end\; module}\,\mathbf{module}\mathrm{UnivariatePolynomialOverPowerSeriesObject}\left(\right)\mathbf{option}\mathrm{object}\;\mathbf{local}\mathrm{upoly}::\mathrm{Array}\,\mathrm{vname}::\mathrm{name}\,\mathrm{dstyle}::\left(\mathrm{list}\left(\left\{:-\mathrm{identical}\left(:-\mathrm{maxdegree}\right)\=\left\{:-\mathrm{nonnegint}\,:-\mathrm{identical}\left(:-\mathrm{\∞}\right)\right\}\,:-\mathrm{identical}\left(:-\mathrm{maxterms}\,:-\mathrm{precision}\right)\=\left\{:-\mathrm{nonnegint}\,:-\mathrm{identical}\left(:-\mathrm{\∞}\right)\right\}\right\}\right)\right)\;\mathbf{end\; module}\right]$

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

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

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

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

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

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

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

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

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

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

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

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