\begin{patch}{RationatFunctionPagePatch1} \begin{paste}{RationatFunctionPageFull1}{RationatFunctionPageEmpty1} \pastebutton{RationatFunctionPageFull1}{\hidepaste} \tab{5}\spadcommand{f := (x - y) / (x + y)\bound{f }} \indentrel{3}\begin{verbatim} - y + x (1) ������� y + x Type: Fraction Polynomial Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RationatFunctionPageEmpty1} \begin{paste}{RationatFunctionPageEmpty1}{RationatFunctionPagePatch1} \pastebutton{RationatFunctionPageEmpty1}{\showpaste} \tab{5}\spadcommand{f := (x - y) / (x + y)\bound{f }} \end{paste}\end{patch} \begin{patch}{RationatFunctionPagePatch2} \begin{paste}{RationatFunctionPageFull2}{RationatFunctionPageEmpty2} \pastebutton{RationatFunctionPageFull2}{\hidepaste} \tab{5}\spadcommand{numer f\free{f }} \indentrel{3}\begin{verbatim} (2) - y + x Type: Polynomial Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RationatFunctionPageEmpty2} \begin{paste}{RationatFunctionPageEmpty2}{RationatFunctionPagePatch2} \pastebutton{RationatFunctionPageEmpty2}{\showpaste} \tab{5}\spadcommand{numer f\free{f }} \end{paste}\end{patch} \begin{patch}{RationatFunctionPagePatch3} \begin{paste}{RationatFunctionPageFull3}{RationatFunctionPageEmpty3} \pastebutton{RationatFunctionPageFull3}{\hidepaste} \tab{5}\spadcommand{denom f\free{f }} \indentrel{3}\begin{verbatim} (3) y + x Type: Polynomial Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RationatFunctionPageEmpty3} \begin{paste}{RationatFunctionPageEmpty3}{RationatFunctionPagePatch3} \pastebutton{RationatFunctionPageEmpty3}{\showpaste} \tab{5}\spadcommand{denom f\free{f }} \end{paste}\end{patch} \begin{patch}{RationatFunctionPagePatch4} \begin{paste}{RationatFunctionPageFull4}{RationatFunctionPageEmpty4} \pastebutton{RationatFunctionPageFull4}{\hidepaste} \tab{5}\spadcommand{eval(f, x = 1/x)\free{f }} \indentrel{3}\begin{verbatim} - x y + 1 (4) ��������� x y + 1 Type: Fraction Polynomial Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RationatFunctionPageEmpty4} \begin{paste}{RationatFunctionPageEmpty4}{RationatFunctionPagePatch4} \pastebutton{RationatFunctionPageEmpty4}{\showpaste} \tab{5}\spadcommand{eval(f, x = 1/x)\free{f }} \end{paste}\end{patch} \begin{patch}{RationatFunctionPagePatch5} \begin{paste}{RationatFunctionPageFull5}{RationatFunctionPageEmpty5} \pastebutton{RationatFunctionPageFull5}{\hidepaste} \tab{5}\spadcommand{eval(f, [x = y, y = x])\free{f }} \indentrel{3}\begin{verbatim} y - x (5) ����� y + x Type: Fraction Polynomial Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RationatFunctionPageEmpty5} \begin{paste}{RationatFunctionPageEmpty5}{RationatFunctionPagePatch5} \pastebutton{RationatFunctionPageEmpty5}{\showpaste} \tab{5}\spadcommand{eval(f, [x = y, y = x])\free{f }} \end{paste}\end{patch} \begin{patch}{AlgebraicFunctionPagePatch1} \begin{paste}{AlgebraicFunctionPageFull1}{AlgebraicFunctionPageEmpty1} \pastebutton{AlgebraicFunctionPageFull1}{\hidepaste} \tab{5}\spadcommand{f := sqrt(1 + x ** (1/3))\bound{f }} \indentrel{3}\begin{verbatim} ��������Ŀ �3�Ŀ (1) \�\�x + 1 Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{AlgebraicFunctionPageEmpty1} \begin{paste}{AlgebraicFunctionPageEmpty1}{AlgebraicFunctionPagePatch1} \pastebutton{AlgebraicFunctionPageEmpty1}{\showpaste} \tab{5}\spadcommand{f := sqrt(1 + x ** (1/3))\bound{f }} \end{paste}\end{patch} \begin{patch}{AlgebraicFunctionPagePatch2} \begin{paste}{AlgebraicFunctionPageFull2}{AlgebraicFunctionPageEmpty2} \pastebutton{AlgebraicFunctionPageFull2}{\hidepaste} \tab{5}\spadcommand{y := rootOf(y**3 + y**2 - x*y + x**3 - 1, y)\bound{y }} \indentrel{3}\begin{verbatim} (2) y Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{AlgebraicFunctionPageEmpty2} \begin{paste}{AlgebraicFunctionPageEmpty2}{AlgebraicFunctionPagePatch2} \pastebutton{AlgebraicFunctionPageEmpty2}{\showpaste} \tab{5}\spadcommand{y := rootOf(y**3 + y**2 - x*y + x**3 - 1, y)\bound{y }} \end{paste}\end{patch} \begin{patch}{AlgebraicFunctionPagePatch3} \begin{paste}{AlgebraicFunctionPageFull3}{AlgebraicFunctionPageEmpty3} \pastebutton{AlgebraicFunctionPageFull3}{\hidepaste} \tab{5}\spadcommand{differentiate(y, x)\free{y }} \indentrel{3}\begin{verbatim} 2 y - 3x (3) ������������ 2 3y + 2y - x Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{AlgebraicFunctionPageEmpty3} \begin{paste}{AlgebraicFunctionPageEmpty3}{AlgebraicFunctionPagePatch3} \pastebutton{AlgebraicFunctionPageEmpty3}{\showpaste} \tab{5}\spadcommand{differentiate(y, x)\free{y }} \end{paste}\end{patch} \begin{patch}{AlgebraicFunctionPagePatch4} \begin{paste}{AlgebraicFunctionPageFull4}{AlgebraicFunctionPageEmpty4} \pastebutton{AlgebraicFunctionPageFull4}{\hidepaste} \tab{5}\spadcommand{(y + 1) ** 3\free{y }} \indentrel{3}\begin{verbatim} 2 3 (4) 2y + (x + 3)y - x + 2 Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{AlgebraicFunctionPageEmpty4} \begin{paste}{AlgebraicFunctionPageEmpty4}{AlgebraicFunctionPagePatch4} \pastebutton{AlgebraicFunctionPageEmpty4}{\showpaste} \tab{5}\spadcommand{(y + 1) ** 3\free{y }} \end{paste}\end{patch} \begin{patch}{AlgebraicFunctionPagePatch5} \begin{paste}{AlgebraicFunctionPageFull5}{AlgebraicFunctionPageEmpty5} \pastebutton{AlgebraicFunctionPageFull5}{\hidepaste} \tab{5}\spadcommand{g := inv f\bound{g }\free{y }} \indentrel{3}\begin{verbatim} 1 (5) ����������� ��������Ŀ �3�Ŀ \�\�x + 1 Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{AlgebraicFunctionPageEmpty5} \begin{paste}{AlgebraicFunctionPageEmpty5}{AlgebraicFunctionPagePatch5} \pastebutton{AlgebraicFunctionPageEmpty5}{\showpaste} \tab{5}\spadcommand{g := inv f\bound{g }\free{y }} \end{paste}\end{patch} \begin{patch}{AlgebraicFunctionPagePatch6} \begin{paste}{AlgebraicFunctionPageFull6}{AlgebraicFunctionPageEmpty6} \pastebutton{AlgebraicFunctionPageFull6}{\hidepaste} \tab{5}\spadcommand{ratDenom g\free{g }} \indentrel{3}\begin{verbatim} ��������Ŀ 3�Ŀ2 3�Ŀ �3�Ŀ (\�x - \�x + 1)\�\�x + 1 (6) ����������������������������� x + 1 Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{AlgebraicFunctionPageEmpty6} \begin{paste}{AlgebraicFunctionPageEmpty6}{AlgebraicFunctionPagePatch6} \pastebutton{AlgebraicFunctionPageEmpty6}{\showpaste} \tab{5}\spadcommand{ratDenom g\free{g }} \end{paste}\end{patch} \begin{patch}{ElementaryFunctionPagePatch1} \begin{paste}{ElementaryFunctionPageFull1}{ElementaryFunctionPageEmpty1} \pastebutton{ElementaryFunctionPageFull1}{\hidepaste} \tab{5}\spadcommand{f := x * log y * sin(1/(x+y))\bound{f }} \indentrel{3}\begin{verbatim} 1 (1) x log(y)sin(�����) y + x Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ElementaryFunctionPageEmpty1} \begin{paste}{ElementaryFunctionPageEmpty1}{ElementaryFunctionPagePatch1} \pastebutton{ElementaryFunctionPageEmpty1}{\showpaste} \tab{5}\spadcommand{f := x * log y * sin(1/(x+y))\bound{f }} \end{paste}\end{patch} \begin{patch}{ElementaryFunctionPagePatch2} \begin{paste}{ElementaryFunctionPageFull2}{ElementaryFunctionPageEmpty2} \pastebutton{ElementaryFunctionPageFull2}{\hidepaste} \tab{5}\spadcommand{eval(f, [x = y, y = x])\free{f }} \indentrel{3}\begin{verbatim} 1 (2) y log(x)sin(�����) y + x Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ElementaryFunctionPageEmpty2} \begin{paste}{ElementaryFunctionPageEmpty2}{ElementaryFunctionPagePatch2} \pastebutton{ElementaryFunctionPageEmpty2}{\showpaste} \tab{5}\spadcommand{eval(f, [x = y, y = x])\free{f }} \end{paste}\end{patch} \begin{patch}{ElementaryFunctionPagePatch3} \begin{paste}{ElementaryFunctionPageFull3}{ElementaryFunctionPageEmpty3} \pastebutton{ElementaryFunctionPageFull3}{\hidepaste} \tab{5}\spadcommand{eval(f, log y = acosh(x + sqrt y))\free{f }} \indentrel{3}\begin{verbatim} 1 �Ŀ (3) x sin(�����)acosh(\�y + x) y + x Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ElementaryFunctionPageEmpty3} \begin{paste}{ElementaryFunctionPageEmpty3}{ElementaryFunctionPagePatch3} \pastebutton{ElementaryFunctionPageEmpty3}{\showpaste} \tab{5}\spadcommand{eval(f, log y = acosh(x + sqrt y))\free{f }} \end{paste}\end{patch} \begin{patch}{FunctionSimplificationPagePatch1} \begin{paste}{FunctionSimplificationPageFull1}{FunctionSimplificationPageEmpty1} \pastebutton{FunctionSimplificationPageFull1}{\hidepaste} \tab{5}\spadcommand{f := cos(x)/sec(x) * log(sin(x)**2/(cos(x)**2+sin(x)**2))\bound{f }} \indentrel{3}\begin{verbatim} 2 sin(x) cos(x)log(�����������������) 2 2 sin(x) + cos(x) (1) ���������������������������� sec(x) Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{FunctionSimplificationPageEmpty1} \begin{paste}{FunctionSimplificationPageEmpty1}{FunctionSimplificationPagePatch1} \pastebutton{FunctionSimplificationPageEmpty1}{\showpaste} \tab{5}\spadcommand{f := cos(x)/sec(x) * log(sin(x)**2/(cos(x)**2+sin(x)**2))\bound{f }} \end{paste}\end{patch} \begin{patch}{FunctionSimplificationPagePatch2} \begin{paste}{FunctionSimplificationPageFull2}{FunctionSimplificationPageEmpty2} \pastebutton{FunctionSimplificationPageFull2}{\hidepaste} \tab{5}\spadcommand{g := simplify f\bound{g }\free{f }} \indentrel{3}\begin{verbatim} 2 2 (2) cos(x) log(- cos(x) + 1) Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{FunctionSimplificationPageEmpty2} \begin{paste}{FunctionSimplificationPageEmpty2}{FunctionSimplificationPagePatch2} \pastebutton{FunctionSimplificationPageEmpty2}{\showpaste} \tab{5}\spadcommand{g := simplify f\bound{g }\free{f }} \end{paste}\end{patch} \begin{patch}{FunctionSimplificationPagePatch3} \begin{paste}{FunctionSimplificationPageFull3}{FunctionSimplificationPageEmpty3} \pastebutton{FunctionSimplificationPageFull3}{\hidepaste} \tab{5}\spadcommand{h := sin2csc cos2sec g\bound{h }\free{g }} \indentrel{3}\begin{verbatim} 2 sec(x) - 1 log(�����������) 2 sec(x) (3) ���������������� 2 sec(x) Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{FunctionSimplificationPageEmpty3} \begin{paste}{FunctionSimplificationPageEmpty3}{FunctionSimplificationPagePatch3} \pastebutton{FunctionSimplificationPageEmpty3}{\showpaste} \tab{5}\spadcommand{h := sin2csc cos2sec g\bound{h }\free{g }} \end{paste}\end{patch} \begin{patch}{FunctionSimplificationPagePatch4} \begin{paste}{FunctionSimplificationPageFull4}{FunctionSimplificationPageEmpty4} \pastebutton{FunctionSimplificationPageFull4}{\hidepaste} \tab{5}\spadcommand{expandLog h\free{h }} \indentrel{3}\begin{verbatim} 2 log(sec(x) - 1) - 2log(sec(x)) (4) ������������������������������� 2 sec(x) Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{FunctionSimplificationPageEmpty4} \begin{paste}{FunctionSimplificationPageEmpty4}{FunctionSimplificationPagePatch4} \pastebutton{FunctionSimplificationPageEmpty4}{\showpaste} \tab{5}\spadcommand{expandLog h\free{h }} \end{paste}\end{patch} \begin{patch}{FunctionSimplificationPagePatch5} \begin{paste}{FunctionSimplificationPageFull5}{FunctionSimplificationPageEmpty5} \pastebutton{FunctionSimplificationPageFull5}{\hidepaste} \tab{5}\spadcommand{f1 := sqrt((x+1)**3)\bound{f1 }} \indentrel{3}\begin{verbatim} �����������������Ŀ � 3 2 (5) \�x + 3x + 3x + 1 Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{FunctionSimplificationPageEmpty5} \begin{paste}{FunctionSimplificationPageEmpty5}{FunctionSimplificationPagePatch5} \pastebutton{FunctionSimplificationPageEmpty5}{\showpaste} \tab{5}\spadcommand{f1 := sqrt((x+1)**3)\bound{f1 }} \end{paste}\end{patch} \begin{patch}{FunctionSimplificationPagePatch6} \begin{paste}{FunctionSimplificationPageFull6}{FunctionSimplificationPageEmpty6} \pastebutton{FunctionSimplificationPageFull6}{\hidepaste} \tab{5}\spadcommand{rootSimp f1\free{f1 }} \indentrel{3}\begin{verbatim} �����Ŀ (6) (x + 1)\�x + 1 Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{FunctionSimplificationPageEmpty6} \begin{paste}{FunctionSimplificationPageEmpty6}{FunctionSimplificationPagePatch6} \pastebutton{FunctionSimplificationPageEmpty6}{\showpaste} \tab{5}\spadcommand{rootSimp f1\free{f1 }} \end{paste}\end{patch} \begin{patch}{FunctionSimplificationPagePatch7} \begin{paste}{FunctionSimplificationPageFull7}{FunctionSimplificationPageEmpty7} \pastebutton{FunctionSimplificationPageFull7}{\hidepaste} \tab{5}\spadcommand{g1 := sin(x + cos x)\bound{g1 }} \indentrel{3}\begin{verbatim} (7) sin(cos(x) + x) Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{FunctionSimplificationPageEmpty7} \begin{paste}{FunctionSimplificationPageEmpty7}{FunctionSimplificationPagePatch7} \pastebutton{FunctionSimplificationPageEmpty7}{\showpaste} \tab{5}\spadcommand{g1 := sin(x + cos x)\bound{g1 }} \end{paste}\end{patch} \begin{patch}{FunctionSimplificationPagePatch8} \begin{paste}{FunctionSimplificationPageFull8}{FunctionSimplificationPageEmpty8} \pastebutton{FunctionSimplificationPageFull8}{\hidepaste} \tab{5}\spadcommand{g2 := complexElementary g1\bound{g2 }\free{g1 }} \indentrel{3}\begin{verbatim} (8) - ���Ŀ \�- 1 * %e ** ���Ŀ 2 ���Ŀ ���Ŀ x\�- 1 ���Ŀ x\�- 1 \�- 1 (%e ) + 2x\�- 1 %e + ���Ŀ \�- 1 / ���Ŀ x\�- 1 2%e ** 2 + ���Ŀ \�- 1 / ���Ŀ 2 ���Ŀ ���Ŀ x\�- 1 ���Ŀ x\�- 1 ���Ŀ \�- 1 (%e ) + 2x\�- 1 %e + \�- 1 ����������������������������������������������� ���Ŀ x\�- 1 2%e 2%e Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{FunctionSimplificationPageEmpty8} \begin{paste}{FunctionSimplificationPageEmpty8}{FunctionSimplificationPagePatch8} \pastebutton{FunctionSimplificationPageEmpty8}{\showpaste} \tab{5}\spadcommand{g2 := complexElementary g1\bound{g2 }\free{g1 }} \end{paste}\end{patch} \begin{patch}{FunctionSimplificationPagePatch9} \begin{paste}{FunctionSimplificationPageFull9}{FunctionSimplificationPageEmpty9} \pastebutton{FunctionSimplificationPageFull9}{\hidepaste} \tab{5}\spadcommand{trigs g2\free{g2 }} \indentrel{3}\begin{verbatim} (9) sin(cos(x) + x) Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{FunctionSimplificationPageEmpty9} \begin{paste}{FunctionSimplificationPageEmpty9}{FunctionSimplificationPagePatch9} \pastebutton{FunctionSimplificationPageEmpty9}{\showpaste} \tab{5}\spadcommand{trigs g2\free{g2 }} \end{paste}\end{patch} \begin{patch}{FunctionSimplificationPagePatch10} \begin{paste}{FunctionSimplificationPageFull10}{FunctionSimplificationPageEmpty10} \pastebutton{FunctionSimplificationPageFull10}{\hidepaste} \tab{5}\spadcommand{h1 := sinh(x + cosh x)\bound{h1 }} \indentrel{3}\begin{verbatim} (10) sinh(cosh(x) + x) Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{FunctionSimplificationPageEmpty10} \begin{paste}{FunctionSimplificationPageEmpty10}{FunctionSimplificationPagePatch10} \pastebutton{FunctionSimplificationPageEmpty10}{\showpaste} \tab{5}\spadcommand{h1 := sinh(x + cosh x)\bound{h1 }} \end{paste}\end{patch} \begin{patch}{FunctionSimplificationPagePatch11} \begin{paste}{FunctionSimplificationPageFull11}{FunctionSimplificationPageEmpty11} \pastebutton{FunctionSimplificationPageFull11}{\hidepaste} \tab{5}\spadcommand{h2 := realElementary h1\bound{h2 }\free{h1 }} \indentrel{3}\begin{verbatim} x 2 x 2 (%e ) + 2x %e + 1 ������������������� x 2%e (%e ) - 1 (11) ���������������������������� x 2 x (%e ) + 2x %e + 1 ������������������� x 2%e 2%e Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{FunctionSimplificationPageEmpty11} \begin{paste}{FunctionSimplificationPageEmpty11}{FunctionSimplificationPagePatch11} \pastebutton{FunctionSimplificationPageEmpty11}{\showpaste} \tab{5}\spadcommand{h2 := realElementary h1\bound{h2 }\free{h1 }} \end{paste}\end{patch} \begin{patch}{FunctionSimplificationPagePatch12} \begin{paste}{FunctionSimplificationPageFull12}{FunctionSimplificationPageEmpty12} \pastebutton{FunctionSimplificationPageFull12}{\hidepaste} \tab{5}\spadcommand{htrigs h2\free{h2 }} \indentrel{3}\begin{verbatim} (12) sinh(cosh(x) + x) Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{FunctionSimplificationPageEmpty12} \begin{paste}{FunctionSimplificationPageEmpty12}{FunctionSimplificationPagePatch12} \pastebutton{FunctionSimplificationPageEmpty12}{\showpaste} \tab{5}\spadcommand{htrigs h2\free{h2 }} \end{paste}\end{patch}