\begin{patch}{ExIntRationalWithRealParameterPatch1} \begin{paste}{ExIntRationalWithRealParameterFull1}{ExIntRationalWithRealParameterEmpty1} \pastebutton{ExIntRationalWithRealParameterFull1}{\hidepaste} \tab{5}\spadcommand{integrate(1/(x**2 + a),x)} \indentrel{3}\begin{verbatim} 2 ���Ŀ (x - a)\�- a + 2a x �Ŀ log(���������������������) x\�a 2 atan(�����) x + a a (1) [��������������������������,�����������] ���Ŀ �Ŀ 2\�- a \�a Type: Union(List Expression Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ExIntRationalWithRealParameterEmpty1} \begin{paste}{ExIntRationalWithRealParameterEmpty1}{ExIntRationalWithRealParameterPatch1} \pastebutton{ExIntRationalWithRealParameterEmpty1}{\showpaste} \tab{5}\spadcommand{integrate(1/(x**2 + a),x)} \end{paste}\end{patch} \begin{patch}{ExIntRationalFunctionPatch1} \begin{paste}{ExIntRationalFunctionFull1}{ExIntRationalFunctionEmpty1} \pastebutton{ExIntRationalFunctionFull1}{\hidepaste} \tab{5}\spadcommand{integrate((x**2+2*x+1)/((x+1)**6+1),x)} \indentrel{3}\begin{verbatim} 3 2 atan(x + 3x + 3x + 1) (1) ����������������������� 3 Type: Union(Expression Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ExIntRationalFunctionEmpty1} \begin{paste}{ExIntRationalFunctionEmpty1}{ExIntRationalFunctionPatch1} \pastebutton{ExIntRationalFunctionEmpty1}{\showpaste} \tab{5}\spadcommand{integrate((x**2+2*x+1)/((x+1)**6+1),x)} \end{paste}\end{patch} \begin{patch}{ExIntRationalFunctionPatch2} \begin{paste}{ExIntRationalFunctionFull2}{ExIntRationalFunctionEmpty2} \pastebutton{ExIntRationalFunctionFull2}{\hidepaste} \tab{5}\spadcommand{integrate(1/(x**3+x+1),x)\bound{i }} \indentrel{3}\begin{verbatim} (2) ��������������Ŀ � 2 �- 93%%H0 + 12 ( ��������������� - %%H0) \� 31 * log ��������������Ŀ � 2 �- 93%%H0 + 12 2 (62%%H0 + 31) ��������������� + 62%%H0 \� 31 + - 31%%H0 + 18x - 4 + ��������������Ŀ � 2 �- 93%%H0 + 12 (- ��������������� - %%H0) \� 31 * log ��������������Ŀ � 2 �- 93%%H0 + 12 2 (- 62%%H0 - 31) ��������������� + 62%%H0 \� 31 + - 31%%H0 + 18x - 4 + 2 2%%H0 log(- 62%%H0 + 31%%H0 + 9x + 4) / 2 Type: Union(Expression Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ExIntRationalFunctionEmpty2} \begin{paste}{ExIntRationalFunctionEmpty2}{ExIntRationalFunctionPatch2} \pastebutton{ExIntRationalFunctionEmpty2}{\showpaste} \tab{5}\spadcommand{integrate(1/(x**3+x+1),x)\bound{i }} \end{paste}\end{patch} \begin{patch}{ExIntRationalFunctionPatch3} \begin{paste}{ExIntRationalFunctionFull3}{ExIntRationalFunctionEmpty3} \pastebutton{ExIntRationalFunctionFull3}{\hidepaste} \tab{5}\spadcommand{definingPolynomial(tower(\%).2::EXPR INT)\free{i }} \indentrel{3}\begin{verbatim} 3 31%%H0 - 3%%H0 - 1 (3) ������������������� 31 Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ExIntRationalFunctionEmpty3} \begin{paste}{ExIntRationalFunctionEmpty3}{ExIntRationalFunctionPatch3} \pastebutton{ExIntRationalFunctionEmpty3}{\showpaste} \tab{5}\spadcommand{definingPolynomial(tower(\%).2::EXPR INT)\free{i }} \end{paste}\end{patch} \begin{patch}{ExIntAlgebraicRelationPatch1} \begin{paste}{ExIntAlgebraicRelationFull1}{ExIntAlgebraicRelationEmpty1} \pastebutton{ExIntAlgebraicRelationFull1}{\hidepaste} \tab{5}\spadcommand{integrate(tan(atan(x)/3),x)} \indentrel{3}\begin{verbatim} (1) atan(x) 2 atan(x) 2 8log(3tan(�������) - 1) - 3tan(�������) 3 3 + atan(x) 18x tan(�������) 3 / 18 Type: Union(Expression Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ExIntAlgebraicRelationEmpty1} \begin{paste}{ExIntAlgebraicRelationEmpty1}{ExIntAlgebraicRelationPatch1} \pastebutton{ExIntAlgebraicRelationEmpty1}{\showpaste} \tab{5}\spadcommand{integrate(tan(atan(x)/3),x)} \end{paste}\end{patch} \begin{patch}{ExIntRationalWithComplexParameterPatch1} \begin{paste}{ExIntRationalWithComplexParameterFull1}{ExIntRationalWithComplexParameterEmpty1} \pastebutton{ExIntRationalWithComplexParameterFull1}{\hidepaste} \tab{5}\spadcommand{complexIntegrate(1/(x**2 + a),x)} \indentrel{3}\begin{verbatim} ���Ŀ ���Ŀ x\�- a + a x\�- a - a log(�����������) - log(�����������) ���Ŀ ���Ŀ \�- a \�- a (1) ����������������������������������� ���Ŀ 2\�- a Type: Expression Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ExIntRationalWithComplexParameterEmpty1} \begin{paste}{ExIntRationalWithComplexParameterEmpty1}{ExIntRationalWithComplexParameterPatch1} \pastebutton{ExIntRationalWithComplexParameterEmpty1}{\showpaste} \tab{5}\spadcommand{complexIntegrate(1/(x**2 + a),x)} \end{paste}\end{patch} \begin{patch}{ExIntNoSolutionPatch1} \begin{paste}{ExIntNoSolutionFull1}{ExIntNoSolutionEmpty1} \pastebutton{ExIntNoSolutionFull1}{\hidepaste} \tab{5}\spadcommand{integrate(log(1 + sqrt(a*x + b)) / x,x)} \indentrel{3}\begin{verbatim} x ��������Ŀ ڿ log(\�b + %T a + 1) (1) � �������������������� d%T �� %T Type: Union(Expression Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ExIntNoSolutionEmpty1} \begin{paste}{ExIntNoSolutionEmpty1}{ExIntNoSolutionPatch1} \pastebutton{ExIntNoSolutionEmpty1}{\showpaste} \tab{5}\spadcommand{integrate(log(1 + sqrt(a*x + b)) / x,x)} \end{paste}\end{patch} \begin{patch}{ExIntTwoSimilarIntegrandsPatch1} \begin{paste}{ExIntTwoSimilarIntegrandsFull1}{ExIntTwoSimilarIntegrandsEmpty1} \pastebutton{ExIntTwoSimilarIntegrandsFull1}{\hidepaste} \tab{5}\spadcommand{integrate(x**3 / (a+b*x)**(1/3),x)} \indentrel{3}\begin{verbatim} (1) 3 3 2 2 2 3 3�������Ŀ2 (120b x - 135a b x + 162a b x - 243a )\�b x + a ��������������������������������������������������� 4 440b Type: Union(Expression Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ExIntTwoSimilarIntegrandsEmpty1} \begin{paste}{ExIntTwoSimilarIntegrandsEmpty1}{ExIntTwoSimilarIntegrandsPatch1} \pastebutton{ExIntTwoSimilarIntegrandsEmpty1}{\showpaste} \tab{5}\spadcommand{integrate(x**3 / (a+b*x)**(1/3),x)} \end{paste}\end{patch} \begin{patch}{ExIntTwoSimilarIntegrandsPatch2} \begin{paste}{ExIntTwoSimilarIntegrandsFull2}{ExIntTwoSimilarIntegrandsEmpty2} \pastebutton{ExIntTwoSimilarIntegrandsFull2}{\hidepaste} \tab{5}\spadcommand{integrate(1 / (x**3 * (a+b*x)**(1/3)),x)} \indentrel{3}\begin{verbatim} (2) - 2 2 �Ŀ 2b x \�3 * 3�Ŀ3�������Ŀ2 3�Ŀ2 3�������Ŀ log(\�a \�b x + a + \�a \�b x + a + a) + 2 2 �Ŀ 3�Ŀ2 3�������Ŀ 4b x \�3 log(\�a \�b x + a - a) + �Ŀ3�Ŀ2 3�������Ŀ �Ŀ 2 2 2\�3 \�a \�b x + a + a\�3 12b x atan(����������������������������) 3a + �Ŀ3�Ŀ3�������Ŀ2 (12b x - 9a)\�3 \�a \�b x + a / 2 2 �Ŀ3�Ŀ 18a x \�3 \�a Type: Union(Expression Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ExIntTwoSimilarIntegrandsEmpty2} \begin{paste}{ExIntTwoSimilarIntegrandsEmpty2}{ExIntTwoSimilarIntegrandsPatch2} \pastebutton{ExIntTwoSimilarIntegrandsEmpty2}{\showpaste} \tab{5}\spadcommand{integrate(1 / (x**3 * (a+b*x)**(1/3)),x)} \end{paste}\end{patch} \begin{patch}{ExIntRadicalOfTranscendentalPatch1} \begin{paste}{ExIntRadicalOfTranscendentalFull1}{ExIntRadicalOfTranscendentalEmpty1} \pastebutton{ExIntRadicalOfTranscendentalFull1}{\hidepaste} \tab{5}\spadcommand{integrate((x + 1) / (x * (x + log x)**(3/2)),x)} \indentrel{3}\begin{verbatim} ����������Ŀ 2\�log(x) + x (1) - �������������� log(x) + x Type: Union(Expression Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ExIntRadicalOfTranscendentalEmpty1} \begin{paste}{ExIntRadicalOfTranscendentalEmpty1}{ExIntRadicalOfTranscendentalPatch1} \pastebutton{ExIntRadicalOfTranscendentalEmpty1}{\showpaste} \tab{5}\spadcommand{integrate((x + 1) / (x * (x + log x)**(3/2)),x)} \end{paste}\end{patch} \begin{patch}{ExIntNonElementaryPatch1} \begin{paste}{ExIntNonElementaryFull1}{ExIntNonElementaryEmpty1} \pastebutton{ExIntNonElementaryFull1}{\hidepaste} \tab{5}\spadcommand{integrate(exp(-x**2) * erf(x) / (erf(x)**3 - erf(x)**2 - erf(x) + 1),x)} \indentrel{3}\begin{verbatim} ���Ŀ erf(x) - 1 ���Ŀ (erf(x) - 1)\�%pi log(����������) - 2\�%pi erf(x) + 1 (1) ������������������������������������������� 8erf(x) - 8 Type: Union(Expression Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ExIntNonElementaryEmpty1} \begin{paste}{ExIntNonElementaryEmpty1}{ExIntNonElementaryPatch1} \pastebutton{ExIntNonElementaryEmpty1}{\showpaste} \tab{5}\spadcommand{integrate(exp(-x**2) * erf(x) / (erf(x)**3 - erf(x)**2 - erf(x) + 1),x)} \end{paste}\end{patch} \begin{patch}{ExIntTrigPatch1} \begin{paste}{ExIntTrigFull1}{ExIntTrigEmpty1} \pastebutton{ExIntTrigFull1}{\hidepaste} \tab{5}\spadcommand{integrate((sinh(1+sqrt(x+b))+2*sqrt(x+b))/(sqrt(x+b)*(x+cosh(1+sqrt(x+b)))),x)} \indentrel{3}\begin{verbatim} (1) �����Ŀ - 2cosh(\�x + b + 1) - 2x 2log(���������������������������������������) �����Ŀ �����Ŀ sinh(\�x + b + 1) - cosh(\�x + b + 1) + �����Ŀ - 2\�x + b Type: Union(Expression Integer,...) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ExIntTrigEmpty1} \begin{paste}{ExIntTrigEmpty1}{ExIntTrigPatch1} \pastebutton{ExIntTrigEmpty1}{\showpaste} \tab{5}\spadcommand{integrate((sinh(1+sqrt(x+b))+2*sqrt(x+b))/(sqrt(x+b)*(x+cosh(1+sqrt(x+b)))),x)} \end{paste}\end{patch}