View source for 4.23

{{DISPLAYTITLE:Elementary Functions - 4.23 Inverse Trigonometric Functions}}

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! scope="col" style="position: sticky; top: 0;" | DLMF 
! scope="col" style="position: sticky; top: 0;" | Formula 
! scope="col" style="position: sticky; top: 0;" | Constraints 
! scope="col" style="position: sticky; top: 0;" | Maple
! scope="col" style="position: sticky; top: 0;" | Mathematica
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/4.23.E1 4.23.E1] || <math qid="Q1753">\Asin@@{z} = \int_{0}^{z}\frac{\diff{t}}{(1-t^{2})^{1/2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Asin@@{z} = \int_{0}^{z}\frac{\diff{t}}{(1-t^{2})^{1/2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcSin[z] == Integrate[Divide[1,(1 - (t)^(2))^(1/2)], {t, 0, z}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 7]
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| [https://dlmf.nist.gov/4.23.E2 4.23.E2] || <math qid="Q1754">\Acos@@{z} = \int_{z}^{1}\frac{\diff{t}}{(1-t^{2})^{1/2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Acos@@{z} = \int_{z}^{1}\frac{\diff{t}}{(1-t^{2})^{1/2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcCos[z] == Integrate[Divide[1,(1 - (t)^(2))^(1/2)], {t, z, 1}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 7]
|- 
| [https://dlmf.nist.gov/4.23.E3 4.23.E3] || <math qid="Q1755">\Atan@@{z} = \int_{0}^{z}\frac{\diff{t}}{1+t^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Atan@@{z} = \int_{0}^{z}\frac{\diff{t}}{1+t^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcTan[z] == Integrate[Divide[1,1 + (t)^(2)], {t, 0, z}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/4.23.E4 4.23.E4] || <math qid="Q1756">\Acsc@@{z} = \Asin@{1/z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Acsc@@{z} = \Asin@{1/z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcCsc[z] == ArcSin[1/z]</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 7]
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| [https://dlmf.nist.gov/4.23.E5 4.23.E5] || <math qid="Q1757">\Asec@@{z} = \Acos@{1/z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Asec@@{z} = \Acos@{1/z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcSec[z] == ArcCos[1/z]</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 7]
|- 
| [https://dlmf.nist.gov/4.23.E6 4.23.E6] || <math qid="Q1758">\Acot@@{z} = \Atan@{1/z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Acot@@{z} = \Atan@{1/z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcCot[z] == ArcTan[1/z]</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 7]
|- 
| [https://dlmf.nist.gov/4.23.E7 4.23.E7] || <math qid="Q1759">\acsc@@{z} = \asin@{1/z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\acsc@@{z} = \asin@{1/z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arccsc(z) = arcsin(1/z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcCsc[z] == ArcSin[1/z]</syntaxhighlight> || Failure || Successful || Successful [Tested: 7] || Successful [Tested: 7]
|- 
| [https://dlmf.nist.gov/4.23.E8 4.23.E8] || <math qid="Q1760">\asec@@{z} = \acos@{1/z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\asec@@{z} = \acos@{1/z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arcsec(z) = arccos(1/z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcSec[z] == ArcCos[1/z]</syntaxhighlight> || Failure || Successful || Successful [Tested: 7] || Successful [Tested: 7]
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| [https://dlmf.nist.gov/4.23.E9 4.23.E9] || <math qid="Q1761">\acot@@{z} = \atan@{1/z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\acot@@{z} = \atan@{1/z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arccot(z) = arctan(1/z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcCot[z] == ArcTan[1/z]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 3.141592654+0.*I
Test Values: {z = -1/2+1/2*I*3^(1/2), z = 1/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 3.141592654+0.*I
Test Values: {z = -1/2*3^(1/2)-1/2*I, z = 1/2}</syntaxhighlight><br></div></div> || Successful [Tested: 1]
|- 
| [https://dlmf.nist.gov/4.23.E10 4.23.E10] || <math qid="Q1762">\asin@{-z} = -\asin@@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\asin@{-z} = -\asin@@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arcsin(- z) = - arcsin(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcSin[- z] == - ArcSin[z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
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| [https://dlmf.nist.gov/4.23.E11 4.23.E11] || <math qid="Q1763">\acos@{-z} = \pi-\acos@@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\acos@{-z} = \pi-\acos@@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arccos(- z) = Pi - arccos(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcCos[- z] == Pi - ArcCos[z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
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| [https://dlmf.nist.gov/4.23.E12 4.23.E12] || <math qid="Q1764">\atan@{-z} = -\atan@@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\atan@{-z} = -\atan@@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arctan(- z) = - arctan(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcTan[- z] == - ArcTan[z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/4.23.E13 4.23.E13] || <math qid="Q1765">\acsc@{-z} = -\acsc@@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\acsc@{-z} = -\acsc@@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arccsc(- z) = - arccsc(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcCsc[- z] == - ArcCsc[z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
|- 
| [https://dlmf.nist.gov/4.23.E14 4.23.E14] || <math qid="Q1766">\asec@{-z} = \pi-\asec@@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\asec@{-z} = \pi-\asec@@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arcsec(- z) = Pi - arcsec(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcSec[- z] == Pi - ArcSec[z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
|- 
| [https://dlmf.nist.gov/4.23.E15 4.23.E15] || <math qid="Q1767">\acot@{-z} = -\acot@@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\acot@{-z} = -\acot@@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arccot(- z) = - arccot(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcCot[- z] == - ArcCot[z]</syntaxhighlight> || Failure || Successful || Skip - No test values generated || Successful [Tested: 1]
|- 
| [https://dlmf.nist.gov/4.23.E16 4.23.E16] || <math qid="Q1768">\acos@@{z} = \tfrac{1}{2}\pi-\asin@@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\acos@@{z} = \tfrac{1}{2}\pi-\asin@@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arccos(z) = (1)/(2)*Pi - arcsin(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcCos[z] == Divide[1,2]*Pi - ArcSin[z]</syntaxhighlight> || Failure || Successful || Successful [Tested: 7] || Successful [Tested: 7]
|- 
| [https://dlmf.nist.gov/4.23.E17 4.23.E17] || <math qid="Q1769">\asec@@{z} = \tfrac{1}{2}\pi-\acsc@@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\asec@@{z} = \tfrac{1}{2}\pi-\acsc@@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arcsec(z) = (1)/(2)*Pi - arccsc(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcSec[z] == Divide[1,2]*Pi - ArcCsc[z]</syntaxhighlight> || Failure || Successful || Successful [Tested: 7] || Successful [Tested: 7]
|- 
| [https://dlmf.nist.gov/4.23.E18 4.23.E18] || <math qid="Q1770">\acot@@{z} = +\tfrac{1}{2}\pi-\atan@@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\acot@@{z} = +\tfrac{1}{2}\pi-\atan@@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arccot(z) = +(1)/(2)*Pi - arctan(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcCot[z] == +Divide[1,2]*Pi - ArcTan[z]</syntaxhighlight> || Successful || Failure || Skip - symbolical successful subtest || Successful [Tested: 1]
|- 
| [https://dlmf.nist.gov/4.23.E18 4.23.E18] || <math qid="Q1770">\acot@@{z} = -\tfrac{1}{2}\pi-\atan@@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\acot@@{z} = -\tfrac{1}{2}\pi-\atan@@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arccot(z) = -(1)/(2)*Pi - arctan(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcCot[z] == -Divide[1,2]*Pi - ArcTan[z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 3.141592654+0.*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, z = 1/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 3.141592654+0.*I
Test Values: {z = -1/2+1/2*I*3^(1/2), z = 1/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 3.141592654+0.*I
Test Values: {z = 1/2-1/2*I*3^(1/2), z = 1/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 3.141592654+0.*I
Test Values: {z = -1/2*3^(1/2)-1/2*I, z = 1/2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 1]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 3.141592653589793
Test Values: {Rule[z, Rational[1, 2]]}</syntaxhighlight><br></div></div>
|- 
| [https://dlmf.nist.gov/4.23.E19 4.23.E19] || <math qid="Q1771">\asin@@{z} = -i\ln@{(1-z^{2})^{1/2}+iz}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\asin@@{z} = -i\ln@{(1-z^{2})^{1/2}+iz}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arcsin(z) = - I*ln((1 - (z)^(2))^(1/2)+ I*z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcSin[z] == - I*Log[(1 - (z)^(2))^(1/2)+ I*z]</syntaxhighlight> || Failure || Successful || Successful [Tested: 7] || Successful [Tested: 7]
|- 
| [https://dlmf.nist.gov/4.23.E20 4.23.E20] || <math qid="Q1772">\asin@@{x} = \tfrac{1}{2}\pi+ i\ln@{(x^{2}-1)^{1/2}+x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\asin@@{x} = \tfrac{1}{2}\pi+ i\ln@{(x^{2}-1)^{1/2}+x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arcsin(x) = (1)/(2)*Pi + I*ln(((x)^(2)- 1)^(1/2)+ x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcSin[x] == Divide[1,2]*Pi + I*Log[((x)^(2)- 1)^(1/2)+ x]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 0.-1.924847300*I
Test Values: {x = 1.5, x = 3/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 0.-2.633915794*I
Test Values: {x = 2, x = 3/2}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 1]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.0, -1.9248473002384139]
Test Values: {Rule[x, Rational[3, 2]]}</syntaxhighlight><br></div></div>
|- 
| [https://dlmf.nist.gov/4.23.E20 4.23.E20] || <math qid="Q1772">\asin@@{x} = \tfrac{1}{2}\pi- i\ln@{(x^{2}-1)^{1/2}+x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\asin@@{x} = \tfrac{1}{2}\pi- i\ln@{(x^{2}-1)^{1/2}+x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arcsin(x) = (1)/(2)*Pi - I*ln(((x)^(2)- 1)^(1/2)+ x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcSin[x] == Divide[1,2]*Pi - I*Log[((x)^(2)- 1)^(1/2)+ x]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -2.094395102+.1347500000e-10*I
Test Values: {x = .5, x = 3/2}</syntaxhighlight><br></div></div> || Successful [Tested: 1]
|- 
| [https://dlmf.nist.gov/4.23.E21 4.23.E21] || <math qid="Q1773">\asin@@{x} = -\tfrac{1}{2}\pi+ i\ln@{(x^{2}-1)^{1/2}-x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\asin@@{x} = -\tfrac{1}{2}\pi+ i\ln@{(x^{2}-1)^{1/2}-x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arcsin(x) = -(1)/(2)*Pi + I*ln(((x)^(2)- 1)^(1/2)- x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcSin[x] == -Divide[1,2]*Pi + I*Log[((x)^(2)- 1)^(1/2)- x]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 6.283185308+.7e-9*I
Test Values: {x = 1.5, x = -2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 4.188790205-.1347500000e-10*I
Test Values: {x = .5, x = -2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 6.283185308+.2e-8*I
Test Values: {x = 2, x = -2}</syntaxhighlight><br></div></div> || Successful [Tested: 1]
|- 
| [https://dlmf.nist.gov/4.23.E21 4.23.E21] || <math qid="Q1773">\asin@@{x} = -\tfrac{1}{2}\pi- i\ln@{(x^{2}-1)^{1/2}-x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\asin@@{x} = -\tfrac{1}{2}\pi- i\ln@{(x^{2}-1)^{1/2}-x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arcsin(x) = -(1)/(2)*Pi - I*ln(((x)^(2)- 1)^(1/2)- x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcSin[x] == -Divide[1,2]*Pi - I*Log[((x)^(2)- 1)^(1/2)- x]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 0.-1.924847301*I
Test Values: {x = 1.5, x = -2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 0.-2.633915796*I
Test Values: {x = 2, x = -2}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 1]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.0, 2.633915793849633]
Test Values: {Rule[x, -2]}</syntaxhighlight><br></div></div>
|- 
| [https://dlmf.nist.gov/4.23.E22 4.23.E22] || <math qid="Q1774">\acos@@{z} = \tfrac{1}{2}\pi+i\ln@{(1-z^{2})^{1/2}+iz}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\acos@@{z} = \tfrac{1}{2}\pi+i\ln@{(1-z^{2})^{1/2}+iz}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arccos(z) = (1)/(2)*Pi + I*ln((1 - (z)^(2))^(1/2)+ I*z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcCos[z] == Divide[1,2]*Pi + I*Log[(1 - (z)^(2))^(1/2)+ I*z]</syntaxhighlight> || Failure || Successful || Successful [Tested: 7] || Successful [Tested: 7]
|- 
| [https://dlmf.nist.gov/4.23.E23 4.23.E23] || <math qid="Q1775">\acos@@{z} = -2i\ln@{\left(\frac{1+z}{2}\right)^{1/2}+i\left(\frac{1-z}{2}\right)^{1/2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\acos@@{z} = -2i\ln@{\left(\frac{1+z}{2}\right)^{1/2}+i\left(\frac{1-z}{2}\right)^{1/2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arccos(z) = - 2*I*ln(((1 + z)/(2))^(1/2)+ I*((1 - z)/(2))^(1/2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcCos[z] == - 2*I*Log[(Divide[1 + z,2])^(1/2)+ I*(Divide[1 - z,2])^(1/2)]</syntaxhighlight> || Failure || Failure || Successful [Tested: 7] || Successful [Tested: 7]
|- 
| [https://dlmf.nist.gov/4.23.E24 4.23.E24] || <math qid="Q1776">\acos@@{x} = - i\ln@{(x^{2}-1)^{1/2}+x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\acos@@{x} = - i\ln@{(x^{2}-1)^{1/2}+x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arccos(x) = - I*ln(((x)^(2)- 1)^(1/2)+ x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcCos[x] == - I*Log[((x)^(2)- 1)^(1/2)+ x]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.924847300*I
Test Values: {x = 1.5, x = 3/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 2.633915794*I
Test Values: {x = 2, x = 3/2}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 1]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.0, 1.9248473002384139]
Test Values: {Rule[x, Rational[3, 2]]}</syntaxhighlight><br></div></div>
|- 
| [https://dlmf.nist.gov/4.23.E24 4.23.E24] || <math qid="Q1776">\acos@@{x} = + i\ln@{(x^{2}-1)^{1/2}+x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\acos@@{x} = + i\ln@{(x^{2}-1)^{1/2}+x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arccos(x) = + I*ln(((x)^(2)- 1)^(1/2)+ x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcCos[x] == + I*Log[((x)^(2)- 1)^(1/2)+ x]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 2.094395102-.1347500000e-10*I
Test Values: {x = .5, x = 3/2}</syntaxhighlight><br></div></div> || Successful [Tested: 1]
|- 
| [https://dlmf.nist.gov/4.23.E25 4.23.E25] || <math qid="Q1777">\acos@@{x} = \pi- i\ln@{(x^{2}-1)^{1/2}-x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\acos@@{x} = \pi- i\ln@{(x^{2}-1)^{1/2}-x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arccos(x) = Pi - I*ln(((x)^(2)- 1)^(1/2)- x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcCos[x] == Pi - I*Log[((x)^(2)- 1)^(1/2)- x]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -6.283185308-.7e-9*I
Test Values: {x = 1.5, x = -2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -4.188790205+.1347500000e-10*I
Test Values: {x = .5, x = -2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -6.283185308-.2e-8*I
Test Values: {x = 2, x = -2}</syntaxhighlight><br></div></div> || Successful [Tested: 1]
|- 
| [https://dlmf.nist.gov/4.23.E25 4.23.E25] || <math qid="Q1777">\acos@@{x} = \pi+ i\ln@{(x^{2}-1)^{1/2}-x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\acos@@{x} = \pi+ i\ln@{(x^{2}-1)^{1/2}-x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arccos(x) = Pi + I*ln(((x)^(2)- 1)^(1/2)- x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcCos[x] == Pi + I*Log[((x)^(2)- 1)^(1/2)- x]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 0.+1.924847301*I
Test Values: {x = 1.5, x = -2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 0.+2.633915796*I
Test Values: {x = 2, x = -2}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 1]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.0, -2.633915793849633]
Test Values: {Rule[x, -2]}</syntaxhighlight><br></div></div>
|- 
| [https://dlmf.nist.gov/4.23.E26 4.23.E26] || <math qid="Q1778">\atan@@{z} = \frac{i}{2}\ln@{\frac{i+z}{i-z}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\atan@@{z} = \frac{i}{2}\ln@{\frac{i+z}{i-z}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arctan(z) = (I)/(2)*ln((I + z)/(I - z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcTan[z] == Divide[I,2]*Log[Divide[I + z,I - z]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 7] || Successful [Tested: 7]
|- 
| [https://dlmf.nist.gov/4.23.E27 4.23.E27] || <math qid="Q1779">\atan@{iy} = +\frac{1}{2}\pi+\frac{i}{2}\ln@{\frac{y+1}{y-1}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\atan@{iy} = +\frac{1}{2}\pi+\frac{i}{2}\ln@{\frac{y+1}{y-1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arctan(I*y) = +(1)/(2)*Pi +(I)/(2)*ln((y + 1)/(y - 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcTan[I*y] == +Divide[1,2]*Pi +Divide[I,2]*Log[Divide[y + 1,y - 1]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 6]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -3.141592654-.2e-9*I
Test Values: {y = -1.5, y = -3/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -3.141592654+.2e-9*I
Test Values: {y = -2, y = -3/2}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 1]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-3.141592653589793, -1.1102230246251565*^-16]
Test Values: {Rule[y, Rational[-3, 2]]}</syntaxhighlight><br></div></div>
|- 
| [https://dlmf.nist.gov/4.23.E27 4.23.E27] || <math qid="Q1779">\atan@{iy} = -\frac{1}{2}\pi+\frac{i}{2}\ln@{\frac{y+1}{y-1}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\atan@{iy} = -\frac{1}{2}\pi+\frac{i}{2}\ln@{\frac{y+1}{y-1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arctan(I*y) = -(1)/(2)*Pi +(I)/(2)*ln((y + 1)/(y - 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcTan[I*y] == -Divide[1,2]*Pi +Divide[I,2]*Log[Divide[y + 1,y - 1]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [4 / 6]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 3.141592654+.2e-9*I
Test Values: {y = 1.5, y = -3/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 3.141592654+.2e-9*I
Test Values: {y = -.5, y = -3/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 3.141592654-.2e-9*I
Test Values: {y = .5, y = -3/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 3.141592654-.2e-9*I
Test Values: {y = 2, y = -3/2}</syntaxhighlight><br></div></div> || Successful [Tested: 1]
|- 
| [https://dlmf.nist.gov/4.23.E28 4.23.E28] || <math qid="Q1780">z = \sin@@{w}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z = \sin@@{w}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>z = sin(w)</syntaxhighlight> || <syntaxhighlight lang=mathematica>z == Sin[w]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .70450695e-2+.1624035369*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.358980334+.5284289409*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.3589803345-1.203621867*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.725005738-.8375964631*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.007045069484300837, 0.16240353677712993]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-1.3589803343001376, 0.5284289405615687]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/4.23.E29 4.23.E29] || <math qid="Q1781">z = \cos@@{w}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z = \cos@@{w}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>z = cos(w)</syntaxhighlight> || <syntaxhighlight lang=mathematica>z == Cos[w]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .1354823851+.8969495503*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.230543019+1.262974954*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.2305430189-.4690758537*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.596568423-.1030504497*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.13548238472721352, 0.8969495502290324]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-1.230543019057225, 1.2629749540134712]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/4.23.E30 4.23.E30] || <math qid="Q1782">z = \tan@@{w}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z = \tan@@{w}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>z = tan(w)</syntaxhighlight> || <syntaxhighlight lang=mathematica>z == Tan[w]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .1520945236-.3500402975*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.213930880+.159851065e-1*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.2139308804-1.716065702*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.579956284-1.350040298*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.1520945235384168, -0.3500402971922752]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-1.2139308802460218, 0.015985106592163567]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/4.23.E31 4.23.E31] || <math qid="Q1783">w = \Asin@@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>w = \Asin@@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>w == ArcSin[z]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.0806272403869902, -0.15847894846240845]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.2407598364931787, -0.3314429455293106]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/4.23.E31 4.23.E31] || <math qid="Q1783">\Asin@@{z} = (-1)^{k}\asin@@{z}+k\pi</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Asin@@{z} = (-1)^{k}\asin@@{z}+k\pi</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcSin[z] == (- 1)^(k)* ArcSin[z]+ k*Pi</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.5707963267948961, 1.3169578969248168]
Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -6.283185307179586
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/4.23.E32 4.23.E32] || <math qid="Q1784">w = \Acos@@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>w = \Acos@@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>w == ArcCos[z]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.08062724038699065, 1.1584789484624083]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-1.0795053557191978, 1.3314429455293104]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/4.23.E32 4.23.E32] || <math qid="Q1784">\Acos@@{z} = +\acos@@{z}+2k\pi</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Acos@@{z} = +\acos@@{z}+2k\pi</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcCos[z] == + ArcCos[z]+ 2*k*Pi</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -6.283185307179586
Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -12.566370614359172
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/4.23.E32 4.23.E32] || <math qid="Q1784">\Acos@@{z} = -\acos@@{z}+2k\pi</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Acos@@{z} = -\acos@@{z}+2k\pi</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcCos[z] == - ArcCos[z]+ 2*k*Pi</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-4.71238898038469, -1.3169578969248168]
Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-10.995574287564276, -1.3169578969248168]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/4.23.E33 4.23.E33] || <math qid="Q1785">w = \Atan@@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>w = \Atan@@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>w == ArcTan[z]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.4023777947836326, 0.49999999999999994]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Rational[1, 2]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.9636476090008059, 0.8660254037844387]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[z, Rational[1, 2]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/4.23.E33 4.23.E33] || <math qid="Q1785">\Atan@@{z} = \atan@@{z}+k\pi</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Atan@@{z} = \atan@@{z}+k\pi</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcTan[z] == ArcTan[z]+ k*Pi</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -3.141592653589793
Test Values: {Rule[k, 1], Rule[z, Rational[1, 2]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -6.283185307179586
Test Values: {Rule[k, 2], Rule[z, Rational[1, 2]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/4.23.E34 4.23.E34] || <math qid="Q1786">\asin@@{z} = \asin@@{\beta}+\iunit\sign@{y}\ln@{\alpha+(\alpha^{2}-1)^{1/2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\asin@@{z} = \asin@@{\beta}+\iunit\sign@{y}\ln@{\alpha+(\alpha^{2}-1)^{1/2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arcsin(x + y*I) = arcsin((1)/(2)*((x + 1)^(2)+ (y)^(2))^(1/2)-(1)/(2)*((x - 1)^(2)+ (y)^(2))^(1/2))+ I*signum(y)*ln(((1)/(2)*((x + 1)^(2)+ (y)^(2))^(1/2)+(1)/(2)*((x - 1)^(2)+ (y)^(2))^(1/2))+(((1)/(2)*((x + 1)^(2)+ (y)^(2))^(1/2)+(1)/(2)*((x - 1)^(2)+ (y)^(2))^(1/2))^(2)- 1)^(1/2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcSin[x + y*I] == ArcSin[Divide[1,2]*((x + 1)^(2)+ (y)^(2))^(1/2)-Divide[1,2]*((x - 1)^(2)+ (y)^(2))^(1/2)]+ I*Sign[y]*Log[(Divide[1,2]*((x + 1)^(2)+ (y)^(2))^(1/2)+Divide[1,2]*((x - 1)^(2)+ (y)^(2))^(1/2))+((Divide[1,2]*((x + 1)^(2)+ (y)^(2))^(1/2)+Divide[1,2]*((x - 1)^(2)+ (y)^(2))^(1/2))^(2)- 1)^(1/2)]</syntaxhighlight> || Failure || Failure || Successful [Tested: 18] || Successful [Tested: 18]
|- 
| [https://dlmf.nist.gov/4.23.E35 4.23.E35] || <math qid="Q1787">\acos@@{z} = \acos@@{\beta}-\iunit\sign@{y}\ln@{\alpha+(\alpha^{2}-1)^{1/2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\acos@@{z} = \acos@@{\beta}-\iunit\sign@{y}\ln@{\alpha+(\alpha^{2}-1)^{1/2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arccos(x + y*I) = arccos((1)/(2)*((x + 1)^(2)+ (y)^(2))^(1/2)-(1)/(2)*((x - 1)^(2)+ (y)^(2))^(1/2))- I*signum(y)*ln(((1)/(2)*((x + 1)^(2)+ (y)^(2))^(1/2)+(1)/(2)*((x - 1)^(2)+ (y)^(2))^(1/2))+(((1)/(2)*((x + 1)^(2)+ (y)^(2))^(1/2)+(1)/(2)*((x - 1)^(2)+ (y)^(2))^(1/2))^(2)- 1)^(1/2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcCos[x + y*I] == ArcCos[Divide[1,2]*((x + 1)^(2)+ (y)^(2))^(1/2)-Divide[1,2]*((x - 1)^(2)+ (y)^(2))^(1/2)]- I*Sign[y]*Log[(Divide[1,2]*((x + 1)^(2)+ (y)^(2))^(1/2)+Divide[1,2]*((x - 1)^(2)+ (y)^(2))^(1/2))+((Divide[1,2]*((x + 1)^(2)+ (y)^(2))^(1/2)+Divide[1,2]*((x - 1)^(2)+ (y)^(2))^(1/2))^(2)- 1)^(1/2)]</syntaxhighlight> || Failure || Failure || Successful [Tested: 18] || Successful [Tested: 18]
|- 
| [https://dlmf.nist.gov/4.23.E36 4.23.E36] || <math qid="Q1788">\atan@@{z} = \tfrac{1}{2}\atan@{\frac{2x}{1-x^{2}-y^{2}}}+\tfrac{1}{4}i\ln@{\frac{x^{2}+(y+1)^{2}}{x^{2}+(y-1)^{2}}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\atan@@{z} = \tfrac{1}{2}\atan@{\frac{2x}{1-x^{2}-y^{2}}}+\tfrac{1}{4}i\ln@{\frac{x^{2}+(y+1)^{2}}{x^{2}+(y-1)^{2}}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>arctan(x + y*I) = (1)/(2)*arctan((2*x)/(1 - (x)^(2)- (y)^(2)))+(1)/(4)*I*ln(((x)^(2)+(y + 1)^(2))/((x)^(2)+(y - 1)^(2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcTan[x + y*I] == Divide[1,2]*ArcTan[Divide[2*x,1 - (x)^(2)- (y)^(2)]]+Divide[1,4]*I*Log[Divide[(x)^(2)+(y + 1)^(2),(x)^(2)+(y - 1)^(2)]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [16 / 18]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.570796327-.1e-9*I
Test Values: {x = 1.5, y = -1.5}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.570796327-.1e-9*I
Test Values: {x = 1.5, y = 1.5}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.570796327+0.*I
Test Values: {x = 1.5, y = -.5}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.570796327+0.*I
Test Values: {x = 1.5, y = .5}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [16 / 18]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.5707963267948968, 1.1102230246251565*^-16]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.5707963267948968, -1.6653345369377348*^-16]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/4.23.E39 4.23.E39] || <math qid="Q1791">\Gudermannian@{x} = \int_{0}^{x}\sech@@{t}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Gudermannian@{x} = \int_{0}^{x}\sech@@{t}\diff{t}</syntaxhighlight> || <math>-\infty < x, x < \infty</math> || <syntaxhighlight lang=mathematica>arctan(sinh(x)) = int(sech(t), t = 0..x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gudermannian[x] == Integrate[Sech[t], {t, 0, x}, GenerateConditions->None]</syntaxhighlight> || Successful || Aborted || - || Successful [Tested: 3]
|- 
| [https://dlmf.nist.gov/4.23.E40 4.23.E40] || <math qid="Q1792">\Gudermannian@{x} = 2\atan@{e^{x}}-\tfrac{1}{2}\pi\\</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Gudermannian@{x} = 2\atan@{e^{x}}-\tfrac{1}{2}\pi\\</syntaxhighlight> || <math>-\infty < x, x < \infty</math> || <syntaxhighlight lang=mathematica>arctan(sinh(x)) = 2*arctan(exp(x))-(1)/(2)*Pi</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gudermannian[x] == 2*ArcTan[Exp[x]]-Divide[1,2]*Pi</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
|- 
| [https://dlmf.nist.gov/4.23.E40 4.23.E40] || <math qid="Q1792">2\atan@{e^{x}}-\tfrac{1}{2}\pi\\ = \asin@{\tanh@@{x}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>2\atan@{e^{x}}-\tfrac{1}{2}\pi\\ = \asin@{\tanh@@{x}}</syntaxhighlight> || <math>-\infty < x, x < \infty</math> || <syntaxhighlight lang=mathematica>2*arctan(exp(x))-(1)/(2)*Pi = arcsin(tanh(x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>2*ArcTan[Exp[x]]-Divide[1,2]*Pi == ArcSin[Tanh[x]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| [https://dlmf.nist.gov/4.23.E40 4.23.E40] || <math qid="Q1792">\asin@{\tanh@@{x}} = \acsc@{\coth@@{x}}\\</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\asin@{\tanh@@{x}} = \acsc@{\coth@@{x}}\\</syntaxhighlight> || <math>-\infty < x, x < \infty</math> || <syntaxhighlight lang=mathematica>arcsin(tanh(x)) = arccsc(coth(x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcSin[Tanh[x]] == ArcCsc[Coth[x]]</syntaxhighlight> || Failure || Successful || Successful [Tested: 3] || Successful [Tested: 3]
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| [https://dlmf.nist.gov/4.23.E40 4.23.E40] || <math qid="Q1792">\acsc@{\coth@@{x}}\\ = \acos@{\sech@@{x}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\acsc@{\coth@@{x}}\\ = \acos@{\sech@@{x}}</syntaxhighlight> || <math>-\infty < x, x < \infty</math> || <syntaxhighlight lang=mathematica>arccsc(coth(x)) = arccos(sech(x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcCsc[Coth[x]] == ArcCos[Sech[x]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| [https://dlmf.nist.gov/4.23.E40 4.23.E40] || <math qid="Q1792">\acos@{\sech@@{x}} = \asec@{\cosh@@{x}}\\</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\acos@{\sech@@{x}} = \asec@{\cosh@@{x}}\\</syntaxhighlight> || <math>-\infty < x, x < \infty</math> || <syntaxhighlight lang=mathematica>arccos(sech(x)) = arcsec(cosh(x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcCos[Sech[x]] == ArcSec[Cosh[x]]</syntaxhighlight> || Failure || Successful || Successful [Tested: 3] || Successful [Tested: 3]
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| [https://dlmf.nist.gov/4.23.E40 4.23.E40] || <math qid="Q1792">\asec@{\cosh@@{x}}\\ = \atan@{\sinh@@{x}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\asec@{\cosh@@{x}}\\ = \atan@{\sinh@@{x}}</syntaxhighlight> || <math>-\infty < x, x < \infty</math> || <syntaxhighlight lang=mathematica>arcsec(cosh(x)) = arctan(sinh(x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcSec[Cosh[x]] == ArcTan[Sinh[x]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| [https://dlmf.nist.gov/4.23.E40 4.23.E40] || <math qid="Q1792">\atan@{\sinh@@{x}} = \acot@{\csch@@{x}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\atan@{\sinh@@{x}} = \acot@{\csch@@{x}}</syntaxhighlight> || <math>-\infty < x, x < \infty</math> || <syntaxhighlight lang=mathematica>arctan(sinh(x)) = arccot(csch(x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcTan[Sinh[x]] == ArcCot[Csch[x]]</syntaxhighlight> || Failure || Successful || Successful [Tested: 3] || Successful [Tested: 3]
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| [https://dlmf.nist.gov/4.23.E41 4.23.E41] || <math qid="Q1793">\aGudermannian@{x} = \int_{0}^{x}\sec@@{t}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\aGudermannian@{x} = \int_{0}^{x}\sec@@{t}\diff{t}</syntaxhighlight> || <math>-\frac{1}{2}\pi < x, x < \frac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>arctanh(sin(x)) = int(sec(t), t = 0..x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>InverseGudermannian[x] == Integrate[Sec[t], {t, 0, x}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Successful [Tested: 2] || Successful [Tested: 2]
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| [https://dlmf.nist.gov/4.23.E42 4.23.E42] || <math qid="Q1794">\aGudermannian@{x} = \ln@@{\tan@{\tfrac{1}{2}x+\tfrac{1}{4}\pi}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\aGudermannian@{x} = \ln@@{\tan@{\tfrac{1}{2}x+\tfrac{1}{4}\pi}}</syntaxhighlight> || <math>-\frac{1}{2}\pi < x, x < \frac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>arctanh(sin(x)) = ln(tan((1)/(2)*x +(1)/(4)*Pi))</syntaxhighlight> || <syntaxhighlight lang=mathematica>InverseGudermannian[x] == Log[Tan[Divide[1,2]*x +Divide[1,4]*Pi]]</syntaxhighlight> || Failure || Successful || Successful [Tested: 2] || Successful [Tested: 2]
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| [https://dlmf.nist.gov/4.23.E42 4.23.E42] || <math qid="Q1794">\ln@@{\tan@{\tfrac{1}{2}x+\tfrac{1}{4}\pi}} = \ln@{\sec@@{x}+\tan@@{x}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ln@@{\tan@{\tfrac{1}{2}x+\tfrac{1}{4}\pi}} = \ln@{\sec@@{x}+\tan@@{x}}</syntaxhighlight> || <math>-\frac{1}{2}\pi < x, x < \frac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>ln(tan((1)/(2)*x +(1)/(4)*Pi)) = ln(sec(x)+ tan(x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Log[Tan[Divide[1,2]*x +Divide[1,4]*Pi]] == Log[Sec[x]+ Tan[x]]</syntaxhighlight> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 2]
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| [https://dlmf.nist.gov/4.23.E42 4.23.E42] || <math qid="Q1794">\ln@{\sec@@{x}+\tan@@{x}} = \asinh@{\tan@@{x}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ln@{\sec@@{x}+\tan@@{x}} = \asinh@{\tan@@{x}}</syntaxhighlight> || <math>-\frac{1}{2}\pi < x, x < \frac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>ln(sec(x)+ tan(x)) = arcsinh(tan(x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Log[Sec[x]+ Tan[x]] == ArcSinh[Tan[x]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 2] || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[3.046904887125347, 3.141592653589793]
Test Values: {Rule[x, 2]}</syntaxhighlight><br></div></div>
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| [https://dlmf.nist.gov/4.23.E42 4.23.E42] || <math qid="Q1794">\asinh@{\tan@@{x}} = \acsch@{\cot@@{x}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\asinh@{\tan@@{x}} = \acsch@{\cot@@{x}}</syntaxhighlight> || <math>-\frac{1}{2}\pi < x, x < \frac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>arcsinh(tan(x)) = arccsch(cot(x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcSinh[Tan[x]] == ArcCsch[Cot[x]]</syntaxhighlight> || Failure || Successful || Successful [Tested: 2] || Successful [Tested: 2]
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| [https://dlmf.nist.gov/4.23.E42 4.23.E42] || <math qid="Q1794">\acsch@{\cot@@{x}} = \acosh@{\sec@@{x}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\acsch@{\cot@@{x}} = \acosh@{\sec@@{x}}</syntaxhighlight> || <math>-\frac{1}{2}\pi < x, x < \frac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>arccsch(cot(x)) = arccosh(sec(x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcCsch[Cot[x]] == ArcCosh[Sec[x]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 2] || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-3.046904887125347, -3.141592653589793]
Test Values: {Rule[x, 2]}</syntaxhighlight><br></div></div>
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| [https://dlmf.nist.gov/4.23.E42 4.23.E42] || <math qid="Q1794">\acosh@{\sec@@{x}} = \asech@{\cos@@{x}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\acosh@{\sec@@{x}} = \asech@{\cos@@{x}}</syntaxhighlight> || <math>-\frac{1}{2}\pi < x, x < \frac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>arccosh(sec(x)) = arcsech(cos(x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcCosh[Sec[x]] == ArcSech[Cos[x]]</syntaxhighlight> || Failure || Successful || Successful [Tested: 2] || Successful [Tested: 2]
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| [https://dlmf.nist.gov/4.23.E42 4.23.E42] || <math qid="Q1794">\asech@{\cos@@{x}} = \atanh@{\sin@@{x}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\asech@{\cos@@{x}} = \atanh@{\sin@@{x}}</syntaxhighlight> || <math>-\frac{1}{2}\pi < x, x < \frac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>arcsech(cos(x)) = arctanh(sin(x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcSech[Cos[x]] == ArcTanh[Sin[x]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 2] || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.0, 3.141592653589793]
Test Values: {Rule[x, 2]}</syntaxhighlight><br></div></div>
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| [https://dlmf.nist.gov/4.23.E42 4.23.E42] || <math qid="Q1794">\atanh@{\sin@@{x}} = \acoth@{\csc@@{x}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\atanh@{\sin@@{x}} = \acoth@{\csc@@{x}}</syntaxhighlight> || <math>-\frac{1}{2}\pi < x, x < \frac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>arctanh(sin(x)) = arccoth(csc(x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ArcTanh[Sin[x]] == ArcCoth[Csc[x]]</syntaxhighlight> || Failure || Successful || Successful [Tested: 2] || Successful [Tested: 2]
|}
</div>