$\mathrm{convert}\left(\mathrm{laplace}\left(f\left(t\right)\,t\,s\right)\,\mathrm{int}\right)$

${\∫}_0^{\mathrm{\∞}}f\left(t\right){\ⅇ}^{-ts}\ⅆt$

$f:=\mathrm{randpoly}\left(\left[t\+a\,t\right]\,\mathrm{terms}\=3\right)$

$f:=87t^2-56{\left(t\+\mathrm{a~}\right)}^2{t}^2$

$\mathrm{laplace}\left(f\,t\,p\right)$

$\frac{174}{p^3}-\frac{1344}{p^5}-\frac{672\mathrm{a~}}{p^4}-\frac{112{\mathrm{a~}}^2}{p^3}$

$\mathrm{invlaplace}\left(f\,t\,p\right)$

$-112\mathrm{a~}\mathrm{Dirac}\left(3\,p\right)-56\mathrm{Dirac}\left(4\,p\right)-\mathrm{Dirac}\left(2\,p\right)\left(-87\+56{\mathrm{a~}}^2\right)$

$\mathrm{laplace}\left({\ⅇ}^t\,t\,p\right)$

$\frac{1}{p-1}$

$\mathrm{invlaplace}\left({\ⅇ}^{-at}\,t\,p\right)$

$\mathrm{Dirac}\left(p-\mathrm{a~}\right)$

$\mathrm{laplace}\left(\ln \left(t\right)\,t\,p\right)$

$-\frac{\mathrm{\γ}\+\ln \left(p\right)}{p}$

$\mathrm{invlaplace}\left(\ln \left(t\+1\right)-\ln \left(t\right)\,t\,p\right)$

$\frac{-{\ⅇ}^{-p}\+1}{p}$

$\mathrm{laplace}\left(\frac{1\sin \left(3\sqrt{t}\right)}{t^{\frac{1}{4}}}\,t\,p\right)$

$\frac{3}{8}\frac{\mathrm{\π}\sqrt{3}\sqrt{2}{\ⅇ}^{-\frac{9}{8p}}\left(-\mathrm{BesselI}\left(\frac{3}{4}\,\frac{9}{8p}\right)\+\mathrm{BesselI}\left(-\frac{1}{4}\,\frac{9}{8p}\right)\right)}{p^{3\/2}}$

$\mathrm{invlaplace}\left(\frac{1\sin \left(\frac{\mathrm{\α}}{t}\right)}{t}\,t\,p\right)$

$\mathrm{KelvinBei}\left(0\,2\sqrt{\mathrm{\α}p}\right)$

$\mathrm{laplace}\left(\arctan \left(t\right)\,t\,p\right)$

$\frac{\mathrm{Ci}\left(p\right)\sin \left(p\right)-\mathrm{Ssi}\left(p\right)\cos \left(p\right)}{p}$

$\mathrm{invlaplace}\left(\arctan \left(\frac{1}{t}\right)\,t\,p\right)$

$\frac{\sin \left(p\right)}{p}$

$\mathrm{laplace}\left(\tanh \left(t\right)\,t\,p\right)$

$\frac{1}{2}\mathrm{\Ψ}\left(\frac{1}{2}\+\frac{p}{4}\right)-\frac{1}{2}\mathrm{\Ψ}\left(\frac{p}{4}\right)-\frac{1}{p}$

$\mathrm{invlaplace}\left(\frac{1\sinh \left(\frac{\mathrm{\α}}{t}\right)}{\sqrt{t}}\,t\,p\right)$

$\frac{1}{2}\frac{\cosh \left(2\sqrt{\mathrm{\α}p}\right)-\cos \left(2\sqrt{\mathrm{\α}p}\right)}{\sqrt{\mathrm{\π}p}}$

$\mathrm{laplace}\left(\mathrm{arcsinh}\left(t\right)\,t\,p\right)$

$\frac{1}{2}\frac{\mathrm{\π}\left(\mathrm{StruveH}\left(0\,p\right)-\mathrm{BesselY}\left(0\,p\right)\right)}{p}$

$\mathrm{invlaplace}\left(\frac{1\mathrm{arcsinh}\left(t\right)}{t}\,t\,p\right)$

$-\left({\∫}_p^{\mathrm{\∞}}\frac{\mathrm{BesselJ}\left(0\,\mathrm{\_U}\right)}{\mathrm{\_U}}\ⅆ\mathrm{\_U}\right)$

$\mathrm{laplace}\left(\mathrm{FresnelC}\left(t\right)\,t\,p\right)$

$\frac{1}{4}\mathrm{AngerJ}\left(\frac{1}{2}\,\frac{p^2}{2\mathrm{\π}}\right)-\frac{1}{4}\mathrm{WeberE}\left(\frac{1}{2}\,\frac{p^2}{2\mathrm{\π}}\right)\+\frac{1}{2}\frac{-\sin \left(\frac{p^2}{2\mathrm{\π}}\right)\+\cos \left(\frac{p^2}{2\mathrm{\π}}\right)}{p}$

$\mathrm{invlaplace}\left(\frac{1\left({\left(\frac{1}{2}-\mathrm{FresnelC}\left(t^2\right)\right)}^2\+{\left(\frac{1}{2}-\mathrm{FresnelS}\left(t^2\right)\right)}^2\right)}{t}\,t\,p\right)$

$\frac{\mathrm{Si}\left(\frac{p^2}{4}\right)}{\mathrm{\π}}$

$\mathrm{laplace}\left(\mathrm{FresnelS}\left(t\right)\,t\,p\right)$

$\frac{-1\+\mathrm{LommelS2}\left(1\,\frac{1}{2}\,\frac{p^2}{2\mathrm{\π}}\right)}{\mathrm{\π}}$

$\mathrm{laplace}\left(\mathrm{Ei}\left(t\right)\,t\,p\right)$

$\frac{-\ln \left(-p\+1\right)\+\ln \left(-p\right)-\ln \left(p\right)}{p}$

$\mathrm{invlaplace}\left(\mathrm{Ei}\left(t\right)\,t\,p\right)$

$-\frac{1}{p}$

$\mathrm{laplace}\left(\mathrm{Si}\left(t\right)\,t\,p\right)$

$\frac{\mathrm{arccot}\left(p\right)}{p}$

$\mathrm{laplace}\left(\mathrm{Ci}\left(t\right)\,t\,p\right)$

$-\frac{1}{2}\frac{\ln \left(p^2\+1\right)}{p}$

$\mathrm{laplace}\left(\mathrm{erf}\left(t\right)\,t\,p\right)$

$\frac{{\ⅇ}^{\frac{p^2}{4}}\mathrm{erfc}\left(\frac{p}{2}\right)}{p}$

$\mathrm{invlaplace}\left(\frac{1\mathrm{erf}\left(\sqrt{at}\right)}{\sqrt{t}}\,t\,p\right)$

$\frac{\mathrm{Heaviside}\left(-p\+\mathrm{a~}\right)}{\sqrt{\mathrm{\π}p}}$

$\mathrm{laplace}\left(\mathrm{erfc}\left(t\right)\,t\,p\right)$

$\frac{1-{\ⅇ}^{\frac{p^2}{4}}\mathrm{erfc}\left(\frac{p}{2}\right)}{p}$

$\mathrm{invlaplace}\left(\frac{1{\ⅇ}^{t^2}\mathrm{erfc}\left(t\+a\right)}{t}\,t\,p\right)$

$\mathrm{Heaviside}\left(p-2\mathrm{a~}\right)\left(\mathrm{erf}\left(\frac{p}{2}\right)-\mathrm{erf}\left(\mathrm{a~}\right)\right)$

$\mathrm{laplace}\left(\mathrm{HankelH1}\left(\frac{1}{2}\,\mathrm{\β}t\right)\,t\,p\right)$

$-\frac{\mathrm{I}\sqrt{2}}{\sqrt{\mathrm{\β}}\sqrt{p-\mathrm{I}\mathrm{\β}}}$

$\mathrm{invlaplace}\left(\mathrm{HankelH1}\left(\frac{1}{2}\,It\right)\,t\,p\right)$

$\frac{\left(-1-\mathrm{I}\right)\mathrm{Heaviside}\left(p-1\right)}{\mathrm{\π}\sqrt{p-1}}$

$\mathrm{laplace}\left(\mathrm{HankelH2}\left(\frac{1}{2}\,\mathrm{\δ}t\right)\,t\,p\right)$

$\frac{\sqrt{2}\mathrm{I}}{\sqrt{\mathrm{\δ}}\sqrt{p\+\mathrm{\δ}\mathrm{I}}}$

$\mathrm{invlaplace}\left(\mathrm{HankelH2}\left(\frac{1}{2}\,-It\right)\,t\,p\right)$

$\frac{\left(-1\+\mathrm{I}\right)\mathrm{Heaviside}\left(p-1\right)}{\mathrm{\π}\sqrt{p-1}}$

$\mathrm{laplace}\left(\mathrm{BesselJ}\left(0\,t\right)\mathrm{BesselJ}\left(1\,t\right)\,t\,p\right)$

$\frac{1}{2}-\frac{p\mathrm{EllipticK}\left(\frac{2}{\sqrt{p^2\+4}}\right)}{\mathrm{\π}\sqrt{p^2\+4}}$

$\mathrm{invlaplace}\left(\frac{1{\ⅇ}^{-\frac{\mathrm{\α}}{t}}\mathrm{BesselJ}\left(0\,\frac{\mathrm{\β}}{t}\right)}{t}\,t\,p\right)$

$\mathrm{BesselI}\left(0\,\sqrt{2}\sqrt{\left(\sqrt{{\mathrm{\α}}^2\+{\mathrm{\β}}^2}-\mathrm{\α}\right)p}\right)\mathrm{BesselJ}\left(0\,\sqrt{2}\sqrt{\left(\sqrt{{\mathrm{\α}}^2\+{\mathrm{\β}}^2}\+\mathrm{\α}\right)p}\right)$

$\mathrm{laplace}\left(\mathrm{BesselK}\left(0\,t\right)\,t\,p\right)$

$\frac{\arccos \left(p\right)}{\sqrt{1-p^2}}$

$\mathrm{invlaplace}\left(\mathrm{BesselK}\left(0\,\mathrm{\β}t\right)\,t\,p\right)$

$\frac{\mathrm{Heaviside}\left(p-\mathrm{\β}\right)}{\sqrt{p^2-{\mathrm{\β}}^2}}$

$\mathrm{laplace}\left(\mathrm{BesselY}\left(0\,t\right)\,t\,p\right)$

$-\frac{2\ln \left(\sqrt{p^2\+1}\+p\right)}{\mathrm{\π}\sqrt{p^2\+1}}$

$\mathrm{invlaplace}\left(\frac{1{\ⅇ}^{-\frac{\mathrm{\μ}}{t}}\mathrm{BesselY}\left(0\,\frac{\mathrm{\ν}}{t}\right)}{t}\,t\,p\right)$

$\mathrm{BesselI}\left(0\,\sqrt{2}\sqrt{\left(\sqrt{{\mathrm{\μ}}^2\+{\mathrm{\ν}}^2}-\mathrm{\μ}\right)p}\right)\mathrm{BesselY}\left(0\,\sqrt{2}\sqrt{\left(\sqrt{{\mathrm{\μ}}^2\+{\mathrm{\ν}}^2}\+\mathrm{\μ}\right)p}\right)-\frac{2\mathrm{BesselJ}\left(0\,\sqrt{2}\sqrt{\left(\sqrt{{\mathrm{\μ}}^2\+{\mathrm{\ν}}^2}\+\mathrm{\μ}\right)p}\right)\mathrm{BesselK}\left(0\,\sqrt{2}\sqrt{\left(\sqrt{{\mathrm{\μ}}^2\+{\mathrm{\ν}}^2}-\mathrm{\μ}\right)p}\right)}{\mathrm{\π}}$

$\mathrm{laplace}\left(\mathrm{BesselI}\left(0\,t\right)\,t\,p\right)$

$\frac{1}{\sqrt{-1\+p^2}}$

$\mathrm{invlaplace}\left({\ⅇ}^{-at}\mathrm{BesselI}\left(0\,at\right)\,t\,p\right)$

$\frac{\mathrm{Heaviside}\left(-p\+2\mathrm{a~}\right)}{\sqrt{-p\left(p-2\mathrm{a~}\right)}\mathrm{\π}}$

$\mathrm{laplace}\left(\mathrm{AngerJ}\left(0\,t\right)\,t\,p\right)$

$\frac{1}{\sqrt{p^2\+1}}$

$\mathrm{invlaplace}\left(\mathrm{AngerJ}\left(\mathrm{\ν}\,t\right)-\mathrm{BesselJ}\left(\mathrm{\ν}\,t\right)\,t\,p\right)$

$\frac{\sin \left(\mathrm{\ν}\mathrm{\π}\right){\left(\sqrt{p^2\+1}-p\right)}^{\mathrm{\ν}}}{\mathrm{\π}\sqrt{p^2\+1}}$

$\mathrm{invlaplace}\left(\frac{1\mathrm{\Γ}\left(\mathrm{\ν}\,\mathrm{\α}t\right)}{t^{\mathrm{\ν}}}\,t\,p\right)$

$\mathrm{Heaviside}\left(p-\mathrm{\α}\right)p^{\mathrm{\ν}-1}$

$\mathrm{invlaplace}\left(\mathrm{\Ψ}\left(t\right)\,t\,p\right)$

$-\frac{1}{2}\coth \left(\frac{p}{2}\right)-\frac{1}{2}$

$\mathrm{de1}:=\frac{{\ⅆ}^2}{\ⅆt^2}y\left(t\right)\+5\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)\+6y\left(t\right)\=0$

$\mathrm{de1}:=\frac{{\ⅆ}^2}{\ⅆt^2}y\left(t\right)\+5\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)\+6y\left(t\right)\=0$

$\mathrm{de2}:=\frac{{\ⅆ}^2}{\ⅆt^2}y\left(t\right)\+5\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)\+6y\left(t\right)\=5$

$\mathrm{de2}:=\frac{{\ⅆ}^2}{\ⅆt^2}y\left(t\right)\+5\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)\+6y\left(t\right)\=5$

$\mathrm{de3}:=\frac{{\ⅆ}^2}{\ⅆt^2}y\left(t\right)\+5\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)\+6y\left(t\right)\+{\ⅇ}^t\=\sin \left(t\right)$

$\mathrm{de3}:=\frac{{\ⅆ}^2}{\ⅆt^2}y\left(t\right)\+5\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)\+6y\left(t\right)\+{\ⅇ}^t\=\sin \left(t\right)$

$\mathrm{de5}:=\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)-y\left(x\right)\=\sin \left(x\right)$

$\mathrm{de5}:=\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)-y\left(x\right)\=\sin \left(x\right)$

$\mathrm{de6}:=\frac{\ⅆ}{\ⅆt}v\left(t\right)\+2t\=0$

$\mathrm{de6}:=\frac{\ⅆ}{\ⅆt}v\left(t\right)\+2t\=0$

$\mathrm{de7}:=\frac{\ⅆ}{\ⅆx}y\left(x\right)-\mathrm{\α}y\left(x\right)\=0$

$\mathrm{de7}:=\frac{\ⅆ}{\ⅆx}y\left(x\right)-\mathrm{\α}y\left(x\right)\=0$

$\mathrm{de8}:=\frac{\ⅆ}{\ⅆx}y\left(x\right)\=z\left(x\right)-y\left(x\right)-x\,\frac{\ⅆ}{\ⅆx}z\left(x\right)\=y\left(x\right)\;$$\mathrm{fcns}:=\left\{y\left(x\right)\,z\left(x\right)\right\}$

$\mathrm{de8}:=\frac{\ⅆ}{\ⅆx}y\left(x\right)\=z\left(x\right)-y\left(x\right)-x\,\frac{\ⅆ}{\ⅆx}z\left(x\right)\=y\left(x\right)$

$\mathrm{fcns}:=\left\{y\left(x\right)\,z\left(x\right)\right\}$

$\mathrm{dsolve}\left(\left\{\mathrm{de1}\,y\left(0\right)\=0\,\mathrm{D}\left(y\right)\left(0\right)\=1\right\}\,y\left(t\right)\,\mathrm{method}\=\mathrm{laplace}\right)$

$y\left(t\right)\={\ⅇ}^{-2t}-{\ⅇ}^{-3t}$

$\mathrm{dsolve}\left(\left\{\mathrm{de2}\,y\left(0\right)\=0\,\mathrm{D}\left(y\right)\left(0\right)\=1\right\}\,y\left(t\right)\,\mathrm{method}\=\mathrm{laplace}\right)$

$y\left(t\right)\=-\frac{3}{2}{\ⅇ}^{-2t}\+\frac{5}{6}\+\frac{2}{3}{\ⅇ}^{-3t}$

$\mathrm{dsolve}\left(\left\{\mathrm{de3}\,y\left(0\right)\=0\,\mathrm{D}\left(y\right)\left(0\right)\=1\right\}\,y\left(t\right)\,\mathrm{method}\=\mathrm{laplace}\right)$

$y\left(t\right)\=-\frac{1}{10}\cos \left(t\right)\+\frac{1}{10}\sin \left(t\right)\+\frac{23}{15}{\ⅇ}^{-2t}-\frac{27}{20}{\ⅇ}^{-3t}-\frac{1}{12}{\ⅇ}^t$

$\mathrm{dsolve}\left(\left\{\mathrm{de5}\,y\left(0\right)\=0\,\mathrm{D}\left(y\right)\left(0\right)\=0\right\}\,y\left(x\right)\,\mathrm{method}\=\mathrm{laplace}\right)$

$y\left(x\right)\=-\frac{1}{2}\sin \left(x\right)\+\frac{1}{2}\sinh \left(x\right)$

$\mathrm{dsolve}\left(\left\{\mathrm{de6}\,v\left(1\right)\=\mathrm{\α}\right\}\,v\left(t\right)\,\mathrm{method}\=\mathrm{laplace}\right)$

$v\left(t\right)\=\mathrm{\α}-{\left(t-1\right)}^2-2t\+2$

$\mathrm{dsolve}\left(\left\{\mathrm{de7}\,y\left(0\right)\=\mathrm{\β}\right\}\,y\left(x\right)\,\mathrm{method}\=\mathrm{laplace}\right)$

$y\left(x\right)\=\mathrm{\β}{\ⅇ}^{\mathrm{\α}x}$

$\mathrm{dsolve}\left(\left\{\mathrm{de8}\,y\left(0\right)\=0\,z\left(0\right)\=1\right\}\,\mathrm{fcns}\,\mathrm{method}\=\mathrm{laplace}\right)$

$\left\{z\left(x\right)\=1\+x-\frac{2}{5}{\ⅇ}^{-\frac{x}{2}}\sqrt{5}\sinh \left(\frac{x\sqrt{5}}{2}\right)\,y\left(x\right)\=1\+\frac{1}{5}{\ⅇ}^{-\frac{x}{2}}\left(-5\cosh \left(\frac{x\sqrt{5}}{2}\right)\+\sqrt{5}\sinh \left(\frac{x\sqrt{5}}{2}\right)\right)\right\}$

$\mathrm{dsolve}\left(\left\{\mathrm{de8}\,y\left(0\right)\=1\,z\left(0\right)\=0\right\}\,\mathrm{fcns}\,\mathrm{method}\=\mathrm{laplace}\right)$

$\left\{z\left(x\right)\=1\+x-\frac{1}{5}{\ⅇ}^{-\frac{x}{2}}\left(5\cosh \left(\frac{x\sqrt{5}}{2}\right)\+\sqrt{5}\sinh \left(\frac{x\sqrt{5}}{2}\right)\right)\,y\left(x\right)\=1-\frac{2}{5}{\ⅇ}^{-\frac{x}{2}}\sqrt{5}\sinh \left(\frac{x\sqrt{5}}{2}\right)\right\}$