The FunctionAdvisor(integral_form, math_function) command returns the integral form for the function, if it exists.

$\mathrm{FunctionAdvisor}\left(\mathrm{integral\_form}\,\sin \right)$

$\left[\sin \left(z\right)=\frac{z{\int }_0^1{\ⅇ}^{2\mathrm{I}\mathrm{\_t1}z}\ⅆ\mathrm{\_t1}}{{\ⅇ}^{\mathrm{I}z}}\,\mathrm{with\; no\; restrictions\; on}\left(z\right)\right]$

$\mathrm{FunctionAdvisor}\left(\mathrm{integral\_form}\,\mathrm{{\rm B}}\left(a\,z\right)\right)$

$\left[\mathrm{{\rm B}}\left(a\&comma;z\right)={\int }_0^1{\mathrm{\_k1}}^{a-1}{\left(1-\mathrm{\_k1}\right)}^{z-1}\&DifferentialD;\mathrm{\_k1}\&comma;0<\mathrm{\Re }\left(a\right)\wedge 0<\mathrm{\Re }\left(z\right)\right]$

$\mathrm{FunctionAdvisor}\left(\mathrm{describe}\&comma;\mathrm{EllipticE}\right)$

$\mathrm{EllipticE}=\mathrm{incomplete\; or\; complete\; elliptic\; integral\; of\; the\; second\; kind}$

$\mathrm{FunctionAdvisor}\left(\mathrm{integral}\&comma;\mathrm{EllipticE}\right)$

* Partial match of "integral" against topic "integral_form".

$\left[\mathrm{EllipticE}\left(k\right)={\int }_0^1\frac{\sqrt{-k^2{\mathrm{\_\&alpha;1}}^2+1}}{\sqrt{-{\mathrm{\_\&alpha;1}}^2+1}}\&DifferentialD;\mathrm{\_\&alpha;1}\&comma;\mathrm{with\; no\; restrictions\; on}\left(k\right)\right],\left[\mathrm{EllipticE}\left(z\&comma;k\right)={\int }_0^z\frac{\sqrt{-k^2{\mathrm{\_\&alpha;1}}^2+1}}{\sqrt{-{\mathrm{\_\&alpha;1}}^2+1}}\&DifferentialD;\mathrm{\_\&alpha;1}\&comma;\mathrm{with\; no\; restrictions\; on}\left(z\&comma;k\right)\right]$

$\mathrm{ex1}≔\left[\mathrm{FunctionAdvisor}\left(\mathrm{integral}\&comma;\mathrm{BesselJ}\right)\right]$

* Partial match of "integral" against topic "integral_form".

$\mathrm{ex1}≔\left[\left[\mathrm{BesselJ}\left(a\&comma;z\right)={\int }_{-\mathrm{\pi }}^{\mathrm{\pi }}\frac{1}{2\mathrm{\pi }{\&ExponentialE;}^{\mathrm{I}\left(a\mathrm{\_k1}-z\sin \left(\mathrm{\_k1}\right)\right)}}\&DifferentialD;\mathrm{\_k1}\&comma;a::\mathbb{Z}\right]\&comma;\left[\mathrm{BesselJ}\left(a\&comma;z\right)={\int }_0^{\mathrm{\infty }}-\frac{2\sin \left(-z\cosh \left(\mathrm{\_k1}\right)+\frac{\mathrm{\pi }a}{2}\right)\cosh \left(a\mathrm{\_k1}\right)}{\mathrm{\pi }}\&DifferentialD;\mathrm{\_k1}\&comma;z::\mathrm{real}\right]\&comma;\left[\mathrm{BesselJ}\left(a\&comma;z\right)={\int }_0^{\mathrm{\pi }}\frac{\cos \left(a\mathrm{\_k1}-z\sin \left(\mathrm{\_k1}\right)\right)}{\mathrm{\pi }}\&DifferentialD;\mathrm{\_k1}-\frac{\sin \left(\mathrm{\pi }a\right){\int }_0^{\mathrm{\infty }}\frac{1}{{\&ExponentialE;}^{\mathrm{I}\mathrm{\_k1}+z\sinh \left(\mathrm{\_k1}\right)}}\&DifferentialD;\mathrm{\_k1}}{\mathrm{\pi }}\&comma;0<\mathrm{\Re }\left(z\right)\right]\&comma;\left[\mathrm{BesselJ}\left(a\&comma;z\right)=\frac{z^a{\int }_0^1{\&ExponentialE;}^{2\mathrm{I}\mathrm{\_t1}z}{\mathrm{\_t1}}^{-\frac{1}{2}+a}{\left(1-\mathrm{\_t1}\right)}^{-\frac{1}{2}+a}\&DifferentialD;\mathrm{\_t1}{2}^{1+2a}}{22^a\mathrm{\Gamma }\left(\frac{1}{2}+a\right){\&ExponentialE;}^{\mathrm{I}z}\sqrt{\mathrm{\pi }}}\&comma;0<\frac{1}{2}+\mathrm{\Re }\left(a\right)\right]\right]$

$\mathrm{depends}\left(\mathrm{ex1}\&comma;a\right),\mathrm{depends}\left(\mathrm{ex1}\&comma;z\right)$

$\mathrm{false},\mathrm{false}$

$\mathrm{FunctionAdvisor}\left(\mathrm{calling}\&comma;\mathrm{EllipticF}\right)$

* Partial match of "calling" against topic "calling_sequence".

$\mathrm{EllipticF}\left(z\&comma;k\right)$

$\mathrm{ex2}≔\mathrm{FunctionAdvisor}\left(\mathrm{integral}\&comma;\mathrm{EllipticF}\left(a\&comma;z\right)\right)$

* Partial match of "integral" against topic "integral_form".

$\mathrm{ex2}≔\left[\mathrm{EllipticF}\left(a\&comma;z\right)={\int }_0^a\frac{1}{\sqrt{-{\mathrm{\_\&alpha;1}}^2+1}\sqrt{-z^2{\mathrm{\_\&alpha;1}}^2+1}}\&DifferentialD;\mathrm{\_\&alpha;1}\&comma;\mathrm{with\; no\; restrictions\; on}\left(a\&comma;z\right)\right]$

$\mathrm{depends}\left(\mathrm{ex2}\&comma;a\right),\mathrm{depends}\left(\mathrm{ex2}\&comma;z\right)$

$\mathrm{true},\mathrm{true}$