︠e0f05b16-93ee-47be-9e25-9a1b18871aa8i︠ %md Based on http://axiom-wiki.newsynthesis.org/ExampleIntegration ︡55131260-690d-4817-803a-374f3fe0267a︡{"md":"Based on http://axiom-wiki.newsynthesis.org/ExampleIntegration\n"}︡ ︠5cf58055-ba0c-4838-864e-58ddcb2ca0db︠ %axiom f := x/(x^3-x+1) ︡fc33fc9f-a617-4b11-b4c3-e8cd1570b78a︡{"stdout":" x\r\n ----------\r\n 3\r\n x - x + 1\r\n Type: Fraction(Polynomial(Integer))\r\n"}︡ ︠ff82fdf9-7a2a-4d3f-b2d7-936732ae779b︠ %axiom )set output tex off ︡6845bb90-0e8a-4bfb-bc32-fd822d6ab8be︡ ︠41164c9e-2b73-4107-91fb-e40d117a5bad︠ ︠be09b8c4-14ce-4aae-9c70-51aeeadde3ef︠ %axiom )set output algebra on i := integrate(f, x) ︡59b6136f-5678-499a-a5ee-22e5251a6dab︡{"stdout":"\n +-----------+\r\n +-----------+ +--+ +--+ | 2\r\n | 2 (9\\|23 %%F0 + \\|23 )\\|69%%F0 + 4 +--+ 2 +--+ 2 2\r\n - 2\\|69%%F0 + 4 atan(----------------------------------) + 2\\|23 %%F0 log(207%%F0 - 23%%F0 + 25x + 6) - \\|23 %%F0 log((- 207x + 46)%%F0 + (23x - 69)%%F0 + 25x - 6x - 7)\r\n 2\r\n 207%%F0 - 23%%F0 - 50x + 6\r\n ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------\r\n +--+\r\n 2\\|23\r\n Type: Union(Expression(Integer),...)\r\n"}︡ ︠468fa5f2-d2e5-4d5d-8cf4-630e0f67a1f4︠ %axiom )set output tex on )set output algebra off ︡bd0604d6-b9a6-4a03-b757-921150ba1e64︡ ︠53dd8ca8-9b91-4f90-bf03-7c9536dfcf28︠ %axiom D(i, x) ︡54b4c935-5f47-4810-813a-ed696963b506︡{"stdout":"$\r\nfrac{x}{{{x^3} -x+1}} \r\nleqno(13)\r\n$\r\n\n Type: Expression(Integer)\r\n"}︡ ︠abd8d6f2-2de7-498e-a7af-597da3845c3a︠ show(axiom('D(i,x)')) ︡e1a7b6bf-996f-4626-914b-85a7d4a2d5e5︡{"html":"
$\\displaystyle \\frac{x}{{{x^3} -x+1}} $
"}︡ ︠9382da25-f3b1-4553-9d63-a19b6dd90a7e︠ show(axiom('ks := kernels i')) ︡dffe948b-07cd-44ac-b049-f7daeb76c0f0︡{"html":"
$\\displaystyle {\\tan^{-1} \\left( {{ \\frac{{{\\left( {9 \\ {\\sqrt {{23}}} \\ \\%\\%F0}+{\\sqrt {{23}}} \\right)} \\ {\\sqrt {{{{69} \\ { \\%\\%F0^2}}+4}}}}}{{{{207} \\ { \\%\\%F0^2}} -{{23} \\ \\%\\%F0} -{{50} \\ x}+6}}}} \\right)} \\ {\\sqrt {{{{69} \\ { \\%\\%F0^2}}+4}}} \\ {\\log \\left( {{{{207} \\ { \\%\\%F0^2}} -{{23} \\ \\%\\%F0}+{{25} \\ x}+6}} \\right)} \\ {\\log \\left( {{{{\\left( -{{207} \\ x}+{46} \\right)} \\ { \\%\\%F0^2}}+{{\\left( {{23} \\ x} -{69} \\right)} \\ \\%\\%F0}+{{25} \\ {x^2}} -{6 \\ x} -7}} \\right)} \\ \\%\\%F0 \\ {\\sqrt {{23}}} $
"}︡ ︠90604cb1-e9c5-4b5d-b252-52998f60e587︠ show(axiom("[definingPolynomial(k::Expression(Integer)) for k in ks | name operator k = 'rootOf]")) ︡c2e3a3c8-992f-4f33-834a-2d92697c658f︡{"html":"
$\\displaystyle { \\frac{{{{23} \\ { \\%\\%F0^3}}+ \\%\\%F0+1}}{{23}}} $
"}︡ ︠c4afead7-b880-47d5-b294-4423b142925a︠ show(axiom("definingPolynomial(%%F0)")) ︡97e074eb-28ef-4555-8447-abb376574fb8︡{"html":"
$\\displaystyle \\frac{{{{23} \\ { \\%\\%F0^3}}+ \\%\\%F0+1}}{{23}} $
"}︡ ︠dbf51fd8-b181-4fd7-aa4e-7ebf4cd63a1b︠ show(axiom("s:=radicalSolve(definingPolynomial(%%F0)::Fraction Polynomial Integer)")) ︡3b1c0c11-a6fd-4499-aa93-26a42ebea7f5︡{"html":"
$\\displaystyle { \\%\\%F0={ \\frac{{{{\\left( -{{69} \\ {\\sqrt {-3}}}+{69} \\right)} \\ {{\\root {3} \\of {{ \\frac{{-{3 \\ {\\sqrt {{69}}}}+{25}}}{{{138} \\ {\\sqrt {{69}}}}}}}}^2}}+2}}{{{\\left( {{69} \\ {\\sqrt {-3}}}+{69} \\right)} \\ {\\root {3} \\of {{ \\frac{{-{3 \\ {\\sqrt {{69}}}}+{25}}}{{{138} \\ {\\sqrt {{69}}}}}}}}}}}} \\ { \\%\\%F0={ \\frac{{{{\\left( -{{69} \\ {\\sqrt {-3}}} -{69} \\right)} \\ {{\\root {3} \\of {{ \\frac{{-{3 \\ {\\sqrt {{69}}}}+{25}}}{{{138} \\ {\\sqrt {{69}}}}}}}}^2}} -2}}{{{\\left( {{69} \\ {\\sqrt {-3}}} -{69} \\right)} \\ {\\root {3} \\of {{ \\frac{{-{3 \\ {\\sqrt {{69}}}}+{25}}}{{{138} \\ {\\sqrt {{69}}}}}}}}}}}} \\ { \\%\\%F0={ \\frac{{{{69} \\ {{\\root {3} \\of {{ \\frac{{-{3 \\ {\\sqrt {{69}}}}+{25}}}{{{138} \\ {\\sqrt {{69}}}}}}}}^2}} -1}}{{{69} \\ {\\root {3} \\of {{ \\frac{{-{3 \\ {\\sqrt {{69}}}}+{25}}}{{{138} \\ {\\sqrt {{69}}}}}}}}}}}} $
"}︡ ︠9f888dfc-0dc3-4184-b4c6-60a027ad8be4︠ show(axiom("i0 := integrate(f, x=1..2)")) ︡724a348c-b9b4-422a-99b0-eb475d85fd81︡{"html":"
$\\displaystyle \\frac{{-{4 \\ {\\sqrt {{{{69} \\ { \\%\\%O0^2}}+4}}} \\ {\\tan^{-1} \\left( {{ \\frac{{{\\left( {9 \\ {\\sqrt {{23}}} \\ \\%\\%O0}+{\\sqrt {{23}}} \\right)} \\ {\\sqrt {{{{69} \\ { \\%\\%O0^2}}+4}}}}}{{{{207} \\ { \\%\\%O0^2}} -{{23} \\ \\%\\%O0} -{94}}}}} \\right)}}+{4 \\ {\\sqrt {{{{69} \\ { \\%\\%O0^2}}+4}}} \\ {\\tan^{-1} \\left( {{ \\frac{{{\\left( {9 \\ {\\sqrt {{23}}} \\ \\%\\%O0}+{\\sqrt {{23}}} \\right)} \\ {\\sqrt {{{{69} \\ { \\%\\%O0^2}}+4}}}}}{{{{207} \\ { \\%\\%O0^2}} -{{23} \\ \\%\\%O0} -{44}}}}} \\right)}}+{2 \\ {\\sqrt {{23}}} \\ \\%\\%O0 \\ {\\log \\left( {{{{21850} \\ { \\%\\%O0^2}} -{{4025} \\ \\%\\%O0}+{3550}}} \\right)}} -{2 \\ {\\sqrt {{23}}} \\ \\%\\%O0 \\ {\\log \\left( {{{{11500} \\ { \\%\\%O0^2}} -{{2875} \\ \\%\\%O0}+{1375}}} \\right)}}+{{\\sqrt {{23}}} \\ \\%\\%O0 \\ {\\log \\left( {{-{{2875} \\ { \\%\\%O0^2}} -{{2875} \\ \\%\\%O0} -{500}}} \\right)}} -{{\\sqrt {{23}}} \\ \\%\\%O0 \\ {\\log \\left( {{-{{64975} \\ { \\%\\%O0^2}} -{{10350} \\ \\%\\%O0}+{5825}}} \\right)}}}}{{4 \\ {\\sqrt {{23}}}}} $
"}︡ ︠9f732103-09dd-4057-9913-7ffa20273582︠ show(axiom("eval(i0,%%F0=rhs(s(1)));")) ︡cdd98409-550a-44f8-9104-08678874b133︡{"html":"
$\\displaystyle \\frac{{-{4 \\ {\\sqrt {{{{69} \\ { \\%\\%O0^2}}+4}}} \\ {\\tan^{-1} \\left( {{ \\frac{{{\\left( {9 \\ {\\sqrt {{23}}} \\ \\%\\%O0}+{\\sqrt {{23}}} \\right)} \\ {\\sqrt {{{{69} \\ { \\%\\%O0^2}}+4}}}}}{{{{207} \\ { \\%\\%O0^2}} -{{23} \\ \\%\\%O0} -{94}}}}} \\right)}}+{4 \\ {\\sqrt {{{{69} \\ { \\%\\%O0^2}}+4}}} \\ {\\tan^{-1} \\left( {{ \\frac{{{\\left( {9 \\ {\\sqrt {{23}}} \\ \\%\\%O0}+{\\sqrt {{23}}} \\right)} \\ {\\sqrt {{{{69} \\ { \\%\\%O0^2}}+4}}}}}{{{{207} \\ { \\%\\%O0^2}} -{{23} \\ \\%\\%O0} -{44}}}}} \\right)}}+{2 \\ {\\sqrt {{23}}} \\ \\%\\%O0 \\ {\\log \\left( {{{{21850} \\ { \\%\\%O0^2}} -{{4025} \\ \\%\\%O0}+{3550}}} \\right)}} -{2 \\ {\\sqrt {{23}}} \\ \\%\\%O0 \\ {\\log \\left( {{{{11500} \\ { \\%\\%O0^2}} -{{2875} \\ \\%\\%O0}+{1375}}} \\right)}}+{{\\sqrt {{23}}} \\ \\%\\%O0 \\ {\\log \\left( {{-{{2875} \\ { \\%\\%O0^2}} -{{2875} \\ \\%\\%O0} -{500}}} \\right)}} -{{\\sqrt {{23}}} \\ \\%\\%O0 \\ {\\log \\left( {{-{{64975} \\ { \\%\\%O0^2}} -{{10350} \\ \\%\\%O0}+{5825}}} \\right)}}}}{{4 \\ {\\sqrt {{23}}}}} $
"}︡ ︠e51188d2-0d12-4cb2-ba51-597445bcb3dd︠ %axiom )set output algebra on )set output tex off ︡3a488d6d-cd33-4679-8102-bcb2bc7bcecf︡{"stdout":"\n"}︡ ︠388a798f-b82d-4a51-b900-5e8e06cc1174︠ axiom("eval(i0,%%F0=rhs(s(1)));") ︡bda07d02-240c-425c-8c74-11986cbe5a67︡{"stdout":" +-----------+ +-----------+\r\n +-----------+ +--+ +--+ | 2 +-----------+ +--+ +--+ | 2\r\n | 2 (9\\|23 %%O0 + \\|23 )\\|69%%O0 + 4 | 2 (9\\|23 %%O0 + \\|23 )\\|69%%O0 + 4 +--+ 2 +--+ 2\r\n - 4\\|69%%O0 + 4 atan(----------------------------------) + 4\\|69%%O0 + 4 atan(----------------------------------) + 2\\|23 %%O0 log(21850%%O0 - 4025%%O0 + 3550) - 2\\|23 %%O0 log(11500%%O0 - 2875%%O0 + 1375)\r\n 2 2\r\n 207%%O0 - 23%%O0 - 94 207%%O0 - 23%%O0 - 44\r\n + \r\n +--+ 2 +--+ 2\r\n \\|23 %%O0 log(- 2875%%O0 - 2875%%O0 - 500) - \\|23 %%O0 log(- 64975%%O0 - 10350%%O0 + 5825)\r\n /\r\n +--+\r\n 4\\|23\n"}︡ ︠24079d17-435f-4439-b925-1c3cf98bb9cb︠ axiom("definingPolynomial(%%O0)") ︡8fe87382-3983-4c5b-9f33-93c388d7ffcf︡{"stdout":" 3\r\n 23%%O0 + %%O0 + 1\r\n ------------------\r\n 23\n"}︡ ︠caef36d7-94f1-46c0-ab39-6de937006ba0︠