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 CancelInverses
 normalize expression by canceling functions inverse to each other

 Calling Sequence CancelInverses(expr) CancelInverses(expr,'safe')

Parameters

 expr - expression 'safe' - 'safe'

Description

 • The CancelInverses command performs cancellation of the functions that are inverse to each other.
 • The following simplifications are made:

$\mathrm{trig}\left(\mathrm{arctrig}\left(\mathrm{subexpr}\right)\right)\to \mathrm{subexpr}$

$\mathrm{arctrig}\left(\mathrm{trig}\left(\mathrm{subexpr}\right)\right)\to \mathrm{subexpr}$

$\mathrm{ln}\left({ⅇ}^{\mathrm{subexpr}}\right)\to \mathrm{subexpr}$

${ⅇ}^{\mathrm{ln}\left(\mathrm{subexpr}\right)}\to \mathrm{subexpr}$

$\mathrm{ln}\left(\mathrm{LambertW}\left({ⅇ}^{\mathrm{subexpr}}\right)\right)\to \mathrm{subexpr}-\mathrm{LambertW}\left({ⅇ}^{\mathrm{subexpr}}\right)$

${ⅇ}^{\mathrm{integer}\mathrm{LambertW}\left(\mathrm{subexpr}\right)}\to \frac{{\mathrm{subexpr}}^{\mathrm{integer}}}{\mathrm{LambertW}{\left(\mathrm{subexpr}\right)}^{\mathrm{integer}}}$

$1-\mathrm{sin}{\left(\mathrm{subexpr}\right)}^{2}\to \mathrm{cos}{\left(\mathrm{subexpr}\right)}^{2}$

$1-\mathrm{cos}{\left(\mathrm{subexpr}\right)}^{2}\to \mathrm{sin}{\left(\mathrm{subexpr}\right)}^{2}$

${\mathrm{subexpr}}^{\frac{1}{2}}\to \sqrt{\mathrm{subexpr}}$

${\left({\mathrm{subexpr}}^{\mathrm{rational}}\right)}^{\mathrm{integer}}\to {\mathrm{subexpr}}^{\mathrm{rational}\mathrm{integer}}$

 Any combinations of the previous simplifications are made.  Also, arctan of two arguments is simplified if it contains sin(subexpr).
 Note: Not all simplifications are valid everywhere. You should be aware of this when calling CancelInverses, if option 'safe' is given, then only valid simplifications are applied.

Examples

 > with(SolveTools):
 > CancelInverses(1+exp(-x^2));
 ${1}{+}\frac{{1}}{{{ⅇ}}^{{{x}}^{{2}}}}$ (1)
 > CancelInverses(arccos(cos(x-y*2)));
 ${x}{-}{2}{}{y}$ (2)
 > CancelInverses(tan(arctan(x-y*2)));
 ${x}{-}{2}{}{y}$ (3)
 > CancelInverses(ln(exp(x-y*2)));
 ${x}{-}{2}{}{y}$ (4)
 > CancelInverses(exp(ln(x-y*2)));
 ${x}{-}{2}{}{y}$ (5)
 > CancelInverses(ln(LambertW(exp(a+b^2+c^3))));
 ${{c}}^{{3}}{+}{{b}}^{{2}}{+}{a}{-}{\mathrm{LambertW}}{}\left({{ⅇ}}^{{{c}}^{{3}}{+}{{b}}^{{2}}{+}{a}}\right)$ (6)
 > CancelInverses(exp(LambertW(a+b^2+c^3)));
 $\frac{{{c}}^{{3}}{+}{{b}}^{{2}}{+}{a}}{{\mathrm{LambertW}}{}\left({{c}}^{{3}}{+}{{b}}^{{2}}{+}{a}\right)}$ (7)
 > CancelInverses(g(1-cos(x+y+z)^2));
 ${g}{}\left({{\mathrm{sin}}{}\left({x}{+}{y}{+}{z}\right)}^{{2}}\right)$ (8)
 > CancelInverses((-5)^(1/2));
 ${I}{}\sqrt{{5}}$ (9)
 > CancelInverses(((x*sin(x))^(1/2))^3);
 ${\left({x}{}{\mathrm{sin}}{}\left({x}\right)\right)}^{{3}}{{2}}}$ (10)
 > CancelInverses(exp(ln(x)+ln(y+z)+t));
 ${{ⅇ}}^{{t}}{}\left({y}{+}{z}\right){}{x}$ (11)
 > CancelInverses(exp(2*ln(x)+3*ln(y+z)-4*t));
 ${{ⅇ}}^{{-}{4}{}{t}}{}{\left({y}{+}{z}\right)}^{{3}}{}{{x}}^{{2}}$ (12)
 > CancelInverses(exp(2*ln(sin(x))));
 ${{\mathrm{sin}}{}\left({x}\right)}^{{2}}$ (13)
 > CancelInverses(ln(exp(c*k)^z*exp(b*t)));
 ${z}{}{c}{}{k}{+}{b}{}{t}$ (14)
 > CancelInverses(ln(z^(3/2)*x));
 ${\mathrm{ln}}{}\left({x}\right){+}\frac{{3}{}{\mathrm{ln}}{}\left({z}\right)}{{2}}$ (15)
 > CancelInverses(ln((z^(3/2))^5*x));
 ${\mathrm{ln}}{}\left({x}\right){+}\frac{{15}{}{\mathrm{ln}}{}\left({z}\right)}{{2}}$ (16)
 > CancelInverses(ln(exp(z)^cos(x)));
 ${\mathrm{cos}}{}\left({x}\right){}{z}$ (17)
 > CancelInverses(ln((s^f(x))^g(z)));
 ${g}{}\left({z}\right){}{f}{}\left({x}\right){}{\mathrm{ln}}{}\left({s}\right)$ (18)
 > CancelInverses(x^((x+1)/ln(x)+(x-1)/ln(x)));
 ${{ⅇ}}^{{2}{}{x}}$ (19)
 > CancelInverses(arctan(-sin(b), cos(b)));
 ${-}{b}$ (20)
 > CancelInverses(arctan(cos(b), sin(b)));
 $\frac{{\mathrm{\pi }}}{{2}}{-}{b}$ (21)
 > CancelInverses(ln(exp(x))+cos(arccos(x)));
 ${2}{}{x}$ (22)
 > CancelInverses(ln(exp(x))+cos(arccos(x)),'safe');
 ${\mathrm{ln}}{}\left({{ⅇ}}^{{x}}\right){+}{x}$ (23)