Elementary Functions - 4.23 Inverse Trigonometric Functions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
4.23.E1 Arcsin ⁑ z = ∫ 0 z d t ( 1 - t 2 ) 1 / 2 multivalued-inverse-sine 𝑧 superscript subscript 0 𝑧 𝑑 superscript 1 superscript 𝑑 2 1 2 {\displaystyle{\displaystyle\operatorname{Arcsin}z=\int_{0}^{z}\frac{\mathrm{d% }t}{(1-t^{2})^{1/2}}}}
\Asin@@{z} = \int_{0}^{z}\frac{\diff{t}}{(1-t^{2})^{1/2}}		 

Error

ArcSin[z] == Integrate[Divide[1,(1 - (t)^(2))^(1/2)], {t, 0, z}, GenerateConditions->None]
Missing Macro Error Successful - Successful [Tested: 7]
4.23.E2 Arccos ⁑ z = ∫ z 1 d t ( 1 - t 2 ) 1 / 2 multivalued-inverse-cosine 𝑧 superscript subscript 𝑧 1 𝑑 superscript 1 superscript 𝑑 2 1 2 {\displaystyle{\displaystyle\operatorname{Arccos}z=\int_{z}^{1}\frac{\mathrm{d% }t}{(1-t^{2})^{1/2}}}}
\Acos@@{z} = \int_{z}^{1}\frac{\diff{t}}{(1-t^{2})^{1/2}}		 

Error

ArcCos[z] == Integrate[Divide[1,(1 - (t)^(2))^(1/2)], {t, z, 1}, GenerateConditions->None]
Missing Macro Error Successful - Successful [Tested: 7]
4.23.E3 Arctan ⁑ z = ∫ 0 z d t 1 + t 2 multivalued-inverse-tangent 𝑧 superscript subscript 0 𝑧 𝑑 1 superscript 𝑑 2 {\displaystyle{\displaystyle\operatorname{Arctan}z=\int_{0}^{z}\frac{\mathrm{d% }t}{1+t^{2}}}}
\Atan@@{z} = \int_{0}^{z}\frac{\diff{t}}{1+t^{2}}		 

Error

ArcTan[z] == Integrate[Divide[1,1 + (t)^(2)], {t, 0, z}, GenerateConditions->None]
Missing Macro Error Successful - Successful [Tested: 1]
4.23.E4 Arccsc ⁑ z = Arcsin ⁑ ( 1 / z ) multivalued-inverse-cosecant 𝑧 multivalued-inverse-sine 1 𝑧 {\displaystyle{\displaystyle\operatorname{Arccsc}z=\operatorname{Arcsin}\left(% 1/z\right)}}
\Acsc@@{z} = \Asin@{1/z}		 

Error

ArcCsc[z] == ArcSin[1/z]
Missing Macro Error Successful - Successful [Tested: 7]
4.23.E5 Arcsec ⁑ z = Arccos ⁑ ( 1 / z ) multivalued-inverse-secant 𝑧 multivalued-inverse-cosine 1 𝑧 {\displaystyle{\displaystyle\operatorname{Arcsec}z=\operatorname{Arccos}\left(% 1/z\right)}}
\Asec@@{z} = \Acos@{1/z}		 

Error

ArcSec[z] == ArcCos[1/z]
Missing Macro Error Successful - Successful [Tested: 7]
4.23.E6 Arccot ⁑ z = Arctan ⁑ ( 1 / z ) multivalued-inverse-cotangent 𝑧 multivalued-inverse-tangent 1 𝑧 {\displaystyle{\displaystyle\operatorname{Arccot}z=\operatorname{Arctan}\left(% 1/z\right)}}
\Acot@@{z} = \Atan@{1/z}		 

Error

ArcCot[z] == ArcTan[1/z]
Missing Macro Error Successful - Successful [Tested: 7]
4.23.E7 arccsc ⁑ z = arcsin ⁑ ( 1 / z ) 𝑧 1 𝑧 {\displaystyle{\displaystyle\operatorname{arccsc}z=\operatorname{arcsin}\left(% 1/z\right)}}
\acsc@@{z} = \asin@{1/z}		 

arccsc(z) = arcsin(1/z)

ArcCsc[z] == ArcSin[1/z]
Failure Successful Successful [Tested: 7] Successful [Tested: 7]
4.23.E8 arcsec ⁑ z = arccos ⁑ ( 1 / z ) 𝑧 1 𝑧 {\displaystyle{\displaystyle\operatorname{arcsec}z=\operatorname{arccos}\left(% 1/z\right)}}
\asec@@{z} = \acos@{1/z}		 

arcsec(z) = arccos(1/z)

ArcSec[z] == ArcCos[1/z]
Failure Successful Successful [Tested: 7] Successful [Tested: 7]
4.23.E9 arccot ⁑ z = arctan ⁑ ( 1 / z ) 𝑧 1 𝑧 {\displaystyle{\displaystyle\operatorname{arccot}z=\operatorname{arctan}\left(% 1/z\right)}}
\acot@@{z} = \atan@{1/z}		 

arccot(z) = arctan(1/z)

ArcCot[z] == ArcTan[1/z]
Failure Successful
Failed [2 / 7]
Result: 3.141592654+0.*I
Test Values: {z = -1/2+1/2*I*3^(1/2), z = 1/2}


Result: 3.141592654+0.*I
Test Values: {z = -1/2*3^(1/2)-1/2*I, z = 1/2}

Successful [Tested: 1]
4.23.E10 arcsin ⁑ ( - z ) = - arcsin ⁑ z 𝑧 𝑧 {\displaystyle{\displaystyle\operatorname{arcsin}\left(-z\right)=-% \operatorname{arcsin}z}}
\asin@{-z} = -\asin@@{z}		 

arcsin(- z) = - arcsin(z)

ArcSin[- z] == - ArcSin[z]
Successful Successful - Successful [Tested: 7]
4.23.E11 arccos ⁑ ( - z ) = Ο€ - arccos ⁑ z 𝑧 πœ‹ 𝑧 {\displaystyle{\displaystyle\operatorname{arccos}\left(-z\right)=\pi-% \operatorname{arccos}z}}
\acos@{-z} = \pi-\acos@@{z}		 

arccos(- z) = Pi - arccos(z)

ArcCos[- z] == Pi - ArcCos[z]
Successful Successful - Successful [Tested: 7]
4.23.E12 arctan ⁑ ( - z ) = - arctan ⁑ z 𝑧 𝑧 {\displaystyle{\displaystyle\operatorname{arctan}\left(-z\right)=-% \operatorname{arctan}z}}
\atan@{-z} = -\atan@@{z}		 

arctan(- z) = - arctan(z)

ArcTan[- z] == - ArcTan[z]
Successful Successful - Successful [Tested: 1]
4.23.E13 arccsc ⁑ ( - z ) = - arccsc ⁑ z 𝑧 𝑧 {\displaystyle{\displaystyle\operatorname{arccsc}\left(-z\right)=-% \operatorname{arccsc}z}}
\acsc@{-z} = -\acsc@@{z}		 

arccsc(- z) = - arccsc(z)

ArcCsc[- z] == - ArcCsc[z]
Successful Successful - Successful [Tested: 7]
4.23.E14 arcsec ⁑ ( - z ) = Ο€ - arcsec ⁑ z 𝑧 πœ‹ 𝑧 {\displaystyle{\displaystyle\operatorname{arcsec}\left(-z\right)=\pi-% \operatorname{arcsec}z}}
\asec@{-z} = \pi-\asec@@{z}		 

arcsec(- z) = Pi - arcsec(z)

ArcSec[- z] == Pi - ArcSec[z]
Successful Successful - Successful [Tested: 7]
4.23.E15 arccot ⁑ ( - z ) = - arccot ⁑ z 𝑧 𝑧 {\displaystyle{\displaystyle\operatorname{arccot}\left(-z\right)=-% \operatorname{arccot}z}}
\acot@{-z} = -\acot@@{z}		 

arccot(- z) = - arccot(z)

ArcCot[- z] == - ArcCot[z]
Failure Successful Skip - No test values generated Successful [Tested: 1]
4.23.E16 arccos ⁑ z = 1 2 ⁒ Ο€ - arcsin ⁑ z 𝑧 1 2 πœ‹ 𝑧 {\displaystyle{\displaystyle\operatorname{arccos}z=\tfrac{1}{2}\pi-% \operatorname{arcsin}z}}
\acos@@{z} = \tfrac{1}{2}\pi-\asin@@{z}		 

arccos(z) = (1)/(2)*Pi - arcsin(z)

ArcCos[z] == Divide[1,2]*Pi - ArcSin[z]
Failure Successful Successful [Tested: 7] Successful [Tested: 7]
4.23.E17 arcsec ⁑ z = 1 2 ⁒ Ο€ - arccsc ⁑ z 𝑧 1 2 πœ‹ 𝑧 {\displaystyle{\displaystyle\operatorname{arcsec}z=\tfrac{1}{2}\pi-% \operatorname{arccsc}z}}
\asec@@{z} = \tfrac{1}{2}\pi-\acsc@@{z}		 

arcsec(z) = (1)/(2)*Pi - arccsc(z)

ArcSec[z] == Divide[1,2]*Pi - ArcCsc[z]
Failure Successful Successful [Tested: 7] Successful [Tested: 7]
4.23.E18 arccot ⁑ z = + 1 2 ⁒ Ο€ - arctan ⁑ z 𝑧 1 2 πœ‹ 𝑧 {\displaystyle{\displaystyle\operatorname{arccot}z=+\tfrac{1}{2}\pi-% \operatorname{arctan}z}}
\acot@@{z} = +\tfrac{1}{2}\pi-\atan@@{z}		 

arccot(z) = +(1)/(2)*Pi - arctan(z)

ArcCot[z] == +Divide[1,2]*Pi - ArcTan[z]
Successful Failure Skip - symbolical successful subtest Successful [Tested: 1]
4.23.E18 arccot ⁑ z = - 1 2 ⁒ Ο€ - arctan ⁑ z 𝑧 1 2 πœ‹ 𝑧 {\displaystyle{\displaystyle\operatorname{arccot}z=-\tfrac{1}{2}\pi-% \operatorname{arctan}z}}
\acot@@{z} = -\tfrac{1}{2}\pi-\atan@@{z}		 

arccot(z) = -(1)/(2)*Pi - arctan(z)

ArcCot[z] == -Divide[1,2]*Pi - ArcTan[z]
Failure Failure
Failed [7 / 7]
Result: 3.141592654+0.*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, z = 1/2}


Result: 3.141592654+0.*I
Test Values: {z = -1/2+1/2*I*3^(1/2), z = 1/2}


Result: 3.141592654+0.*I
Test Values: {z = 1/2-1/2*I*3^(1/2), z = 1/2}


Result: 3.141592654+0.*I
Test Values: {z = -1/2*3^(1/2)-1/2*I, z = 1/2}

... skip entries to safe data
Failed [1 / 1]
Result: 3.141592653589793
Test Values: {Rule[z, Rational[1, 2]]}

4.23.E19 arcsin ⁑ z = - i ⁒ ln ⁑ ( ( 1 - z 2 ) 1 / 2 + i ⁒ z ) 𝑧 𝑖 superscript 1 superscript 𝑧 2 1 2 𝑖 𝑧 {\displaystyle{\displaystyle\operatorname{arcsin}z=-i\ln\left((1-z^{2})^{1/2}+% iz\right)}}
\asin@@{z} = -i\ln@{(1-z^{2})^{1/2}+iz}		 

arcsin(z) = - I*ln((1 - (z)^(2))^(1/2)+ I*z)

ArcSin[z] == - I*Log[(1 - (z)^(2))^(1/2)+ I*z]
Failure Successful Successful [Tested: 7] Successful [Tested: 7]
4.23.E20 arcsin ⁑ x = 1 2 ⁒ Ο€ + i ⁒ ln ⁑ ( ( x 2 - 1 ) 1 / 2 + x ) π‘₯ 1 2 πœ‹ 𝑖 superscript superscript π‘₯ 2 1 1 2 π‘₯ {\displaystyle{\displaystyle\operatorname{arcsin}x=\tfrac{1}{2}\pi+i\ln\left((% x^{2}-1)^{1/2}+x\right)}}
\asin@@{x} = \tfrac{1}{2}\pi+ i\ln@{(x^{2}-1)^{1/2}+x}		 

arcsin(x) = (1)/(2)*Pi + I*ln(((x)^(2)- 1)^(1/2)+ x)

ArcSin[x] == Divide[1,2]*Pi + I*Log[((x)^(2)- 1)^(1/2)+ x]
Failure Failure
Failed [2 / 3]
Result: 0.-1.924847300*I
Test Values: {x = 1.5, x = 3/2}


Result: 0.-2.633915794*I
Test Values: {x = 2, x = 3/2}

Failed [1 / 1]
Result: Complex[0.0, -1.9248473002384139]
Test Values: {Rule[x, Rational[3, 2]]}

4.23.E20 arcsin ⁑ x = 1 2 ⁒ Ο€ - i ⁒ ln ⁑ ( ( x 2 - 1 ) 1 / 2 + x ) π‘₯ 1 2 πœ‹ 𝑖 superscript superscript π‘₯ 2 1 1 2 π‘₯ {\displaystyle{\displaystyle\operatorname{arcsin}x=\tfrac{1}{2}\pi-i\ln\left((% x^{2}-1)^{1/2}+x\right)}}
\asin@@{x} = \tfrac{1}{2}\pi- i\ln@{(x^{2}-1)^{1/2}+x}		 

arcsin(x) = (1)/(2)*Pi - I*ln(((x)^(2)- 1)^(1/2)+ x)

ArcSin[x] == Divide[1,2]*Pi - I*Log[((x)^(2)- 1)^(1/2)+ x]
Failure Failure
Failed [1 / 3]
Result: -2.094395102+.1347500000e-10*I
Test Values: {x = .5, x = 3/2}

Successful [Tested: 1]
4.23.E21 arcsin ⁑ x = - 1 2 ⁒ Ο€ + i ⁒ ln ⁑ ( ( x 2 - 1 ) 1 / 2 - x ) π‘₯ 1 2 πœ‹ 𝑖 superscript superscript π‘₯ 2 1 1 2 π‘₯ {\displaystyle{\displaystyle\operatorname{arcsin}x=-\tfrac{1}{2}\pi+i\ln\left(% (x^{2}-1)^{1/2}-x\right)}}
\asin@@{x} = -\tfrac{1}{2}\pi+ i\ln@{(x^{2}-1)^{1/2}-x}		 

arcsin(x) = -(1)/(2)*Pi + I*ln(((x)^(2)- 1)^(1/2)- x)

ArcSin[x] == -Divide[1,2]*Pi + I*Log[((x)^(2)- 1)^(1/2)- x]
Failure Failure
Failed [3 / 3]
Result: 6.283185308+.7e-9*I
Test Values: {x = 1.5, x = -2}


Result: 4.188790205-.1347500000e-10*I
Test Values: {x = .5, x = -2}


Result: 6.283185308+.2e-8*I
Test Values: {x = 2, x = -2}

Successful [Tested: 1]
4.23.E21 arcsin ⁑ x = - 1 2 ⁒ Ο€ - i ⁒ ln ⁑ ( ( x 2 - 1 ) 1 / 2 - x ) π‘₯ 1 2 πœ‹ 𝑖 superscript superscript π‘₯ 2 1 1 2 π‘₯ {\displaystyle{\displaystyle\operatorname{arcsin}x=-\tfrac{1}{2}\pi-i\ln\left(% (x^{2}-1)^{1/2}-x\right)}}
\asin@@{x} = -\tfrac{1}{2}\pi- i\ln@{(x^{2}-1)^{1/2}-x}		 

arcsin(x) = -(1)/(2)*Pi - I*ln(((x)^(2)- 1)^(1/2)- x)

ArcSin[x] == -Divide[1,2]*Pi - I*Log[((x)^(2)- 1)^(1/2)- x]
Failure Failure
Failed [2 / 3]
Result: 0.-1.924847301*I
Test Values: {x = 1.5, x = -2}


Result: 0.-2.633915796*I
Test Values: {x = 2, x = -2}

Failed [1 / 1]
Result: Complex[0.0, 2.633915793849633]
Test Values: {Rule[x, -2]}

4.23.E22 arccos ⁑ z = 1 2 ⁒ Ο€ + i ⁒ ln ⁑ ( ( 1 - z 2 ) 1 / 2 + i ⁒ z ) 𝑧 1 2 πœ‹ 𝑖 superscript 1 superscript 𝑧 2 1 2 𝑖 𝑧 {\displaystyle{\displaystyle\operatorname{arccos}z=\tfrac{1}{2}\pi+i\ln\left((% 1-z^{2})^{1/2}+iz\right)}}
\acos@@{z} = \tfrac{1}{2}\pi+i\ln@{(1-z^{2})^{1/2}+iz}		 

arccos(z) = (1)/(2)*Pi + I*ln((1 - (z)^(2))^(1/2)+ I*z)

ArcCos[z] == Divide[1,2]*Pi + I*Log[(1 - (z)^(2))^(1/2)+ I*z]
Failure Successful Successful [Tested: 7] Successful [Tested: 7]
4.23.E23 arccos ⁑ z = - 2 ⁒ i ⁒ ln ⁑ ( ( 1 + z 2 ) 1 / 2 + i ⁒ ( 1 - z 2 ) 1 / 2 ) 𝑧 2 𝑖 superscript 1 𝑧 2 1 2 𝑖 superscript 1 𝑧 2 1 2 {\displaystyle{\displaystyle\operatorname{arccos}z=-2i\ln\left(\left(\frac{1+z% }{2}\right)^{1/2}+i\left(\frac{1-z}{2}\right)^{1/2}\right)}}
\acos@@{z} = -2i\ln@{\left(\frac{1+z}{2}\right)^{1/2}+i\left(\frac{1-z}{2}\right)^{1/2}}		 

arccos(z) = - 2*I*ln(((1 + z)/(2))^(1/2)+ I*((1 - z)/(2))^(1/2))

ArcCos[z] == - 2*I*Log[(Divide[1 + z,2])^(1/2)+ I*(Divide[1 - z,2])^(1/2)]
Failure Failure Successful [Tested: 7] Successful [Tested: 7]
4.23.E24 arccos ⁑ x = - i ⁒ ln ⁑ ( ( x 2 - 1 ) 1 / 2 + x ) π‘₯ 𝑖 superscript superscript π‘₯ 2 1 1 2 π‘₯ {\displaystyle{\displaystyle\operatorname{arccos}x=-i\ln\left((x^{2}-1)^{1/2}+% x\right)}}
\acos@@{x} = - i\ln@{(x^{2}-1)^{1/2}+x}		 

arccos(x) = - I*ln(((x)^(2)- 1)^(1/2)+ x)

ArcCos[x] == - I*Log[((x)^(2)- 1)^(1/2)+ x]
Failure Failure
Failed [2 / 3]
Result: 1.924847300*I
Test Values: {x = 1.5, x = 3/2}


Result: 2.633915794*I
Test Values: {x = 2, x = 3/2}

Failed [1 / 1]
Result: Complex[0.0, 1.9248473002384139]
Test Values: {Rule[x, Rational[3, 2]]}

4.23.E24 arccos ⁑ x = + i ⁒ ln ⁑ ( ( x 2 - 1 ) 1 / 2 + x ) π‘₯ 𝑖 superscript superscript π‘₯ 2 1 1 2 π‘₯ {\displaystyle{\displaystyle\operatorname{arccos}x=+i\ln\left((x^{2}-1)^{1/2}+% x\right)}}
\acos@@{x} = + i\ln@{(x^{2}-1)^{1/2}+x}		 

arccos(x) = + I*ln(((x)^(2)- 1)^(1/2)+ x)
 
ArcCos[x] == + I*Log[((x)^(2)- 1)^(1/2)+ x]
 Failure Failure
Failed [1 / 3]
Result: 2.094395102-.1347500000e-10*I
Test Values: {x = .5, x = 3/2}

 Successful [Tested: 1]
4.23.E25 arccos ⁑ x = Ο€ - i ⁒ ln ⁑ ( ( x 2 - 1 ) 1 / 2 - x ) π‘₯ πœ‹ 𝑖 superscript superscript π‘₯ 2 1 1 2 π‘₯ {\displaystyle{\displaystyle\operatorname{arccos}x=\pi-i\ln\left((x^{2}-1)^{1/% 2}-x\right)}}
\acos@@{x} = \pi- i\ln@{(x^{2}-1)^{1/2}-x}		 

arccos(x) = Pi - I*ln(((x)^(2)- 1)^(1/2)- x)
 
ArcCos[x] == Pi - I*Log[((x)^(2)- 1)^(1/2)- x]
 Failure Failure
Failed [3 / 3]
Result: -6.283185308-.7e-9*I
Test Values: {x = 1.5, x = -2}

Result: -4.188790205+.1347500000e-10*I
Test Values: {x = .5, x = -2}

Result: -6.283185308-.2e-8*I
Test Values: {x = 2, x = -2}

 Successful [Tested: 1]
4.23.E25 arccos ⁑ x = Ο€ + i ⁒ ln ⁑ ( ( x 2 - 1 ) 1 / 2 - x ) π‘₯ πœ‹ 𝑖 superscript superscript π‘₯ 2 1 1 2 π‘₯ {\displaystyle{\displaystyle\operatorname{arccos}x=\pi+i\ln\left((x^{2}-1)^{1/% 2}-x\right)}}
\acos@@{x} = \pi+ i\ln@{(x^{2}-1)^{1/2}-x}		 

arccos(x) = Pi + I*ln(((x)^(2)- 1)^(1/2)- x)
 
ArcCos[x] == Pi + I*Log[((x)^(2)- 1)^(1/2)- x]
 Failure Failure
Failed [2 / 3]
Result: 0.+1.924847301*I
Test Values: {x = 1.5, x = -2}

Result: 0.+2.633915796*I
Test Values: {x = 2, x = -2}
Failed [1 / 1]
Result: Complex[0.0, -2.633915793849633]
Test Values: {Rule[x, -2]}

4.23.E26 arctan ⁑ z = i 2 ⁒ ln ⁑ ( i + z i - z ) 𝑧 𝑖 2 𝑖 𝑧 𝑖 𝑧 {\displaystyle{\displaystyle\operatorname{arctan}z=\frac{i}{2}\ln\left(\frac{i% +z}{i-z}\right)}}
\atan@@{z} = \frac{i}{2}\ln@{\frac{i+z}{i-z}}		 

arctan(z) = (I)/(2)*ln((I + z)/(I - z))
 
ArcTan[z] == Divide[I,2]*Log[Divide[I + z,I - z]]
 Failure Failure Successful [Tested: 7] Successful [Tested: 7]
4.23.E27 arctan ⁑ ( i ⁒ y ) = + 1 2 ⁒ Ο€ + i 2 ⁒ ln ⁑ ( y + 1 y - 1 ) 𝑖 𝑦 1 2 πœ‹ 𝑖 2 𝑦 1 𝑦 1 {\displaystyle{\displaystyle\operatorname{arctan}\left(iy\right)=+\frac{1}{2}% \pi+\frac{i}{2}\ln\left(\frac{y+1}{y-1}\right)}}
\atan@{iy} = +\frac{1}{2}\pi+\frac{i}{2}\ln@{\frac{y+1}{y-1}}		 

arctan(I*y) = +(1)/(2)*Pi +(I)/(2)*ln((y + 1)/(y - 1))
 
ArcTan[I*y] == +Divide[1,2]*Pi +Divide[I,2]*Log[Divide[y + 1,y - 1]]
 Failure Failure
Failed [2 / 6]
Result: -3.141592654-.2e-9*I
Test Values: {y = -1.5, y = -3/2}

Result: -3.141592654+.2e-9*I
Test Values: {y = -2, y = -3/2}
Failed [1 / 1]
Result: Complex[-3.141592653589793, -1.1102230246251565*^-16]
Test Values: {Rule[y, Rational[-3, 2]]}

4.23.E27 arctan ⁑ ( i ⁒ y ) = - 1 2 ⁒ Ο€ + i 2 ⁒ ln ⁑ ( y + 1 y - 1 ) 𝑖 𝑦 1 2 πœ‹ 𝑖 2 𝑦 1 𝑦 1 {\displaystyle{\displaystyle\operatorname{arctan}\left(iy\right)=-\frac{1}{2}% \pi+\frac{i}{2}\ln\left(\frac{y+1}{y-1}\right)}}
\atan@{iy} = -\frac{1}{2}\pi+\frac{i}{2}\ln@{\frac{y+1}{y-1}}		 

arctan(I*y) = -(1)/(2)*Pi +(I)/(2)*ln((y + 1)/(y - 1))
 
ArcTan[I*y] == -Divide[1,2]*Pi +Divide[I,2]*Log[Divide[y + 1,y - 1]]
 Failure Failure
Failed [4 / 6]
Result: 3.141592654+.2e-9*I
Test Values: {y = 1.5, y = -3/2}

Result: 3.141592654+.2e-9*I
Test Values: {y = -.5, y = -3/2}

Result: 3.141592654-.2e-9*I
Test Values: {y = .5, y = -3/2}

Result: 3.141592654-.2e-9*I
Test Values: {y = 2, y = -3/2}

 Successful [Tested: 1]
4.23.E28 z = sin ⁑ w 𝑧 𝑀 {\displaystyle{\displaystyle z=\sin w}}
z = \sin@@{w}		 

z = sin(w)
 
z == Sin[w]
 Failure Failure
Failed [70 / 70]
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}

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)}

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)}

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}

... skip entries to safe data
Failed [70 / 70]
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]]]}

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]]]}

... skip entries to safe data
4.23.E29 z = cos ⁑ w 𝑧 𝑀 {\displaystyle{\displaystyle z=\cos w}}
z = \cos@@{w}		 

z = cos(w)
 
z == Cos[w]
 Failure Failure
Failed [70 / 70]
Result: .1354823851+.8969495503*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}

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)}

Result: -.2305430189-.4690758537*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)}

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}

... skip entries to safe data
Failed [70 / 70]
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]]]}

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]]]}

... skip entries to safe data
4.23.E30 z = tan ⁑ w 𝑧 𝑀 {\displaystyle{\displaystyle z=\tan w}}
z = \tan@@{w}		 

z = tan(w)
 
z == Tan[w]
 Failure Failure
Failed [70 / 70]
Result: .1520945236-.3500402975*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}

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)}

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)}

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}

... skip entries to safe data
Failed [70 / 70]
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]]]}

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]]]}

... skip entries to safe data
4.23.E31 w = Arcsin ⁑ z 𝑀 multivalued-inverse-sine 𝑧 {\displaystyle{\displaystyle w=\operatorname{Arcsin}z}}
w = \Asin@@{z}		 

Error
 
w == ArcSin[z]
 Missing Macro Error Failure -
Failed [70 / 70]
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]]]}

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]]]}

... skip entries to safe data
4.23.E31 Arcsin ⁑ z = ( - 1 ) k ⁒ arcsin ⁑ z + k ⁒ Ο€ multivalued-inverse-sine 𝑧 superscript 1 π‘˜ 𝑧 π‘˜ πœ‹ {\displaystyle{\displaystyle\operatorname{Arcsin}z=(-1)^{k}\operatorname{% arcsin}z+k\pi}}
\Asin@@{z} = (-1)^{k}\asin@@{z}+k\pi		 

Error
 
ArcSin[z] == (- 1)^(k)* ArcSin[z]+ k*Pi
 Missing Macro Error Failure -
Failed [21 / 21]
Result: Complex[-1.5707963267948961, 1.3169578969248168]
Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: -6.283185307179586
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
4.23.E32 w = Arccos ⁑ z 𝑀 multivalued-inverse-cosine 𝑧 {\displaystyle{\displaystyle w=\operatorname{Arccos}z}}
w = \Acos@@{z}		 

Error
 
w == ArcCos[z]
 Missing Macro Error Failure -
Failed [70 / 70]
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]]]}

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]]]}

... skip entries to safe data
4.23.E32 Arccos ⁑ z = + arccos ⁑ z + 2 ⁒ k ⁒ Ο€ multivalued-inverse-cosine 𝑧 𝑧 2 π‘˜ πœ‹ {\displaystyle{\displaystyle\operatorname{Arccos}z=+\operatorname{arccos}z+2k% \pi}}
\Acos@@{z} = +\acos@@{z}+2k\pi		 

Error
 
ArcCos[z] == + ArcCos[z]+ 2*k*Pi
 Missing Macro Error Failure -
Failed [21 / 21]
Result: -6.283185307179586
Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: -12.566370614359172
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
4.23.E32 Arccos ⁑ z = - arccos ⁑ z + 2 ⁒ k ⁒ Ο€ multivalued-inverse-cosine 𝑧 𝑧 2 π‘˜ πœ‹ {\displaystyle{\displaystyle\operatorname{Arccos}z=-\operatorname{arccos}z+2k% \pi}}
\Acos@@{z} = -\acos@@{z}+2k\pi		 

Error
 
ArcCos[z] == - ArcCos[z]+ 2*k*Pi
 Missing Macro Error Failure -
Failed [21 / 21]
Result: Complex[-4.71238898038469, -1.3169578969248168]
Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[-10.995574287564276, -1.3169578969248168]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
4.23.E33 w = Arctan ⁑ z 𝑀 multivalued-inverse-tangent 𝑧 {\displaystyle{\displaystyle w=\operatorname{Arctan}z}}
w = \Atan@@{z}		 

Error
 
w == ArcTan[z]
 Missing Macro Error Failure -
Failed [10 / 10]
Result: Complex[0.4023777947836326, 0.49999999999999994]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Rational[1, 2]]}

Result: Complex[-0.9636476090008059, 0.8660254037844387]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[z, Rational[1, 2]]}

... skip entries to safe data
4.23.E33 Arctan ⁑ z = arctan ⁑ z + k ⁒ Ο€ multivalued-inverse-tangent 𝑧 𝑧 π‘˜ πœ‹ {\displaystyle{\displaystyle\operatorname{Arctan}z=\operatorname{arctan}z+k\pi}}
\Atan@@{z} = \atan@@{z}+k\pi		 

Error
 
ArcTan[z] == ArcTan[z]+ k*Pi
 Missing Macro Error Failure -
Failed [3 / 3]
Result: -3.141592653589793
Test Values: {Rule[k, 1], Rule[z, Rational[1, 2]]}

Result: -6.283185307179586
Test Values: {Rule[k, 2], Rule[z, Rational[1, 2]]}

... skip entries to safe data
4.23.E34 arcsin ⁑ z = arcsin ⁑ Ξ² + i ⁒ sign ⁑ ( y ) ⁒ ln ⁑ ( Ξ± + ( Ξ± 2 - 1 ) 1 / 2 ) 𝑧 𝛽 imaginary-unit sign 𝑦 𝛼 superscript superscript 𝛼 2 1 1 2 {\displaystyle{\displaystyle\operatorname{arcsin}z=\operatorname{arcsin}\beta+% \mathrm{i}\operatorname{sign}\left(y\right)\ln\left(\alpha+(\alpha^{2}-1)^{1/2% }\right)}}
\asin@@{z} = \asin@@{\beta}+\iunit\sign@{y}\ln@{\alpha+(\alpha^{2}-1)^{1/2}}		 

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))
 
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)]
 Failure Failure Successful [Tested: 18] Successful [Tested: 18]
4.23.E35 arccos ⁑ z = arccos ⁑ Ξ² - i ⁒ sign ⁑ ( y ) ⁒ ln ⁑ ( Ξ± + ( Ξ± 2 - 1 ) 1 / 2 ) 𝑧 𝛽 imaginary-unit sign 𝑦 𝛼 superscript superscript 𝛼 2 1 1 2 {\displaystyle{\displaystyle\operatorname{arccos}z=\operatorname{arccos}\beta-% \mathrm{i}\operatorname{sign}\left(y\right)\ln\left(\alpha+(\alpha^{2}-1)^{1/2% }\right)}}
\acos@@{z} = \acos@@{\beta}-\iunit\sign@{y}\ln@{\alpha+(\alpha^{2}-1)^{1/2}}		 

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))
 
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)]
 Failure Failure Successful [Tested: 18] Successful [Tested: 18]
4.23.E36 arctan ⁑ z = 1 2 ⁒ arctan ⁑ ( 2 ⁒ x 1 - x 2 - y 2 ) + 1 4 ⁒ i ⁒ ln ⁑ ( x 2 + ( y + 1 ) 2 x 2 + ( y - 1 ) 2 ) 𝑧 1 2 2 π‘₯ 1 superscript π‘₯ 2 superscript 𝑦 2 1 4 𝑖 superscript π‘₯ 2 superscript 𝑦 1 2 superscript π‘₯ 2 superscript 𝑦 1 2 {\displaystyle{\displaystyle\operatorname{arctan}z=\tfrac{1}{2}\operatorname{% arctan}\left(\frac{2x}{1-x^{2}-y^{2}}\right)+\tfrac{1}{4}i\ln\left(\frac{x^{2}% +(y+1)^{2}}{x^{2}+(y-1)^{2}}\right)}}
\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}}}		 

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)))
 
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)]]
 Failure Failure
Failed [16 / 18]
Result: 1.570796327-.1e-9*I
Test Values: {x = 1.5, y = -1.5}

Result: 1.570796327-.1e-9*I
Test Values: {x = 1.5, y = 1.5}

Result: 1.570796327+0.*I
Test Values: {x = 1.5, y = -.5}

Result: 1.570796327+0.*I
Test Values: {x = 1.5, y = .5}

... skip entries to safe data
Failed [16 / 18]
Result: Complex[1.5707963267948968, 1.1102230246251565*^-16]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}

Result: Complex[1.5707963267948968, -1.6653345369377348*^-16]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
4.23.E39 gd ⁑ ( x ) = ∫ 0 x sech ⁑ t ⁒ d t Gudermannian π‘₯ superscript subscript 0 π‘₯ 𝑑 𝑑 {\displaystyle{\displaystyle\operatorname{gd}\left(x\right)=\int_{0}^{x}% \operatorname{sech}t\mathrm{d}t}}
\Gudermannian@{x} = \int_{0}^{x}\sech@@{t}\diff{t}		 - ∞ < x , x < ∞ formulae-sequence π‘₯ π‘₯ {\displaystyle{\displaystyle-\infty<x,x<\infty}} 
arctan(sinh(x)) = int(sech(t), t = 0..x)
 
Gudermannian[x] == Integrate[Sech[t], {t, 0, x}, GenerateConditions->None]
 Successful Aborted - Successful [Tested: 3]
4.23.E40 gd ⁑ ( x ) = 2 ⁒ arctan ⁑ ( e x ) - 1 2 ⁒ Ο€ Gudermannian π‘₯ 2 superscript 𝑒 π‘₯ 1 2 πœ‹ {\displaystyle{\displaystyle\operatorname{gd}\left(x\right)=2\operatorname{% arctan}\left(e^{x}\right)-\tfrac{1}{2}\pi\\ }}
\Gudermannian@{x} = 2\atan@{e^{x}}-\tfrac{1}{2}\pi\\		 - ∞ < x , x < ∞ formulae-sequence π‘₯ π‘₯ {\displaystyle{\displaystyle-\infty<x,x<\infty}} 
arctan(sinh(x)) = 2*arctan(exp(x))-(1)/(2)*Pi
 
Gudermannian[x] == 2*ArcTan[Exp[x]]-Divide[1,2]*Pi
 Failure Failure Successful [Tested: 3] Successful [Tested: 3]
4.23.E40 2 ⁒ arctan ⁑ ( e x ) - 1 2 ⁒ Ο€ = arcsin ⁑ ( tanh ⁑ x ) 2 superscript 𝑒 π‘₯ 1 2 πœ‹ π‘₯ {\displaystyle{\displaystyle 2\operatorname{arctan}\left(e^{x}\right)-\tfrac{1% }{2}\pi\\ =\operatorname{arcsin}\left(\tanh x\right)}}
2\atan@{e^{x}}-\tfrac{1}{2}\pi\\ = \asin@{\tanh@@{x}}		 - ∞ < x , x < ∞ formulae-sequence π‘₯ π‘₯ {\displaystyle{\displaystyle-\infty<x,x<\infty}} 
2*arctan(exp(x))-(1)/(2)*Pi = arcsin(tanh(x))
 
2*ArcTan[Exp[x]]-Divide[1,2]*Pi == ArcSin[Tanh[x]]
 Failure Failure Successful [Tested: 3] Successful [Tested: 3]
4.23.E40 arcsin ⁑ ( tanh ⁑ x ) = arccsc ⁑ ( coth ⁑ x ) π‘₯ hyperbolic-cotangent π‘₯ {\displaystyle{\displaystyle\operatorname{arcsin}\left(\tanh x\right)=% \operatorname{arccsc}\left(\coth x\right)\\ }}
\asin@{\tanh@@{x}} = \acsc@{\coth@@{x}}\\		 - ∞ < x , x < ∞ formulae-sequence π‘₯ π‘₯ {\displaystyle{\displaystyle-\infty<x,x<\infty}} 
arcsin(tanh(x)) = arccsc(coth(x))
 
ArcSin[Tanh[x]] == ArcCsc[Coth[x]]
 Failure Successful Successful [Tested: 3] Successful [Tested: 3]
4.23.E40 arccsc ⁑ ( coth ⁑ x ) = arccos ⁑ ( sech ⁑ x ) hyperbolic-cotangent π‘₯ π‘₯ {\displaystyle{\displaystyle\operatorname{arccsc}\left(\coth x\right)\\ =\operatorname{arccos}\left(\operatorname{sech}x\right)}}
\acsc@{\coth@@{x}}\\ = \acos@{\sech@@{x}}		 - ∞ < x , x < ∞ formulae-sequence π‘₯ π‘₯ {\displaystyle{\displaystyle-\infty<x,x<\infty}} 
arccsc(coth(x)) = arccos(sech(x))
 
ArcCsc[Coth[x]] == ArcCos[Sech[x]]
 Failure Failure Successful [Tested: 3] Successful [Tested: 3]
4.23.E40 arccos ⁑ ( sech ⁑ x ) = arcsec ⁑ ( cosh ⁑ x ) π‘₯ π‘₯ {\displaystyle{\displaystyle\operatorname{arccos}\left(\operatorname{sech}x% \right)=\operatorname{arcsec}\left(\cosh x\right)\\ }}
\acos@{\sech@@{x}} = \asec@{\cosh@@{x}}\\		 - ∞ < x , x < ∞ formulae-sequence π‘₯ π‘₯ {\displaystyle{\displaystyle-\infty<x,x<\infty}} 
arccos(sech(x)) = arcsec(cosh(x))
 
ArcCos[Sech[x]] == ArcSec[Cosh[x]]
 Failure Successful Successful [Tested: 3] Successful [Tested: 3]
4.23.E40 arcsec ⁑ ( cosh ⁑ x ) = arctan ⁑ ( sinh ⁑ x ) π‘₯ π‘₯ {\displaystyle{\displaystyle\operatorname{arcsec}\left(\cosh x\right)\\ =\operatorname{arctan}\left(\sinh x\right)}}
\asec@{\cosh@@{x}}\\ = \atan@{\sinh@@{x}}		 - ∞ < x , x < ∞ formulae-sequence π‘₯ π‘₯ {\displaystyle{\displaystyle-\infty<x,x<\infty}} 
arcsec(cosh(x)) = arctan(sinh(x))
 
ArcSec[Cosh[x]] == ArcTan[Sinh[x]]
 Failure Failure Successful [Tested: 3] Successful [Tested: 3]
4.23.E40 arctan ⁑ ( sinh ⁑ x ) = arccot ⁑ ( csch ⁑ x ) π‘₯ π‘₯ {\displaystyle{\displaystyle\operatorname{arctan}\left(\sinh x\right)=% \operatorname{arccot}\left(\operatorname{csch}x\right)}}
\atan@{\sinh@@{x}} = \acot@{\csch@@{x}}		 - ∞ < x , x < ∞ formulae-sequence π‘₯ π‘₯ {\displaystyle{\displaystyle-\infty<x,x<\infty}} 
arctan(sinh(x)) = arccot(csch(x))
 
ArcTan[Sinh[x]] == ArcCot[Csch[x]]
 Failure Successful Successful [Tested: 3] Successful [Tested: 3]
4.23.E41 gd - 1 ⁑ ( x ) = ∫ 0 x sec ⁑ t ⁒ d t inverse-Gudermannian π‘₯ superscript subscript 0 π‘₯ 𝑑 𝑑 {\displaystyle{\displaystyle{\operatorname{gd}^{-1}}\left(x\right)=\int_{0}^{x% }\sec t\mathrm{d}t}}
\aGudermannian@{x} = \int_{0}^{x}\sec@@{t}\diff{t}		 - 1 2 ⁒ Ο€ < x , x < 1 2 ⁒ Ο€ formulae-sequence 1 2 πœ‹ π‘₯ π‘₯ 1 2 πœ‹ {\displaystyle{\displaystyle-\frac{1}{2}\pi<x,x<\frac{1}{2}\pi}} 
arctanh(sin(x)) = int(sec(t), t = 0..x)
 
InverseGudermannian[x] == Integrate[Sec[t], {t, 0, x}, GenerateConditions->None]
 Failure Aborted Successful [Tested: 2] Successful [Tested: 2]
4.23.E42 gd - 1 ⁑ ( x ) = ln ⁑ tan ⁑ ( 1 2 ⁒ x + 1 4 ⁒ Ο€ ) inverse-Gudermannian π‘₯ 1 2 π‘₯ 1 4 πœ‹ {\displaystyle{\displaystyle{\operatorname{gd}^{-1}}\left(x\right)=\ln\tan% \left(\tfrac{1}{2}x+\tfrac{1}{4}\pi\right)}}
\aGudermannian@{x} = \ln@@{\tan@{\tfrac{1}{2}x+\tfrac{1}{4}\pi}}		 - 1 2 ⁒ Ο€ < x , x < 1 2 ⁒ Ο€ formulae-sequence 1 2 πœ‹ π‘₯ π‘₯ 1 2 πœ‹ {\displaystyle{\displaystyle-\frac{1}{2}\pi<x,x<\frac{1}{2}\pi}} 
arctanh(sin(x)) = ln(tan((1)/(2)*x +(1)/(4)*Pi))
 
InverseGudermannian[x] == Log[Tan[Divide[1,2]*x +Divide[1,4]*Pi]]
 Failure Successful Successful [Tested: 2] Successful [Tested: 2]
4.23.E42 ln ⁑ tan ⁑ ( 1 2 ⁒ x + 1 4 ⁒ Ο€ ) = ln ⁑ ( sec ⁑ x + tan ⁑ x ) 1 2 π‘₯ 1 4 πœ‹ π‘₯ π‘₯ {\displaystyle{\displaystyle\ln\tan\left(\tfrac{1}{2}x+\tfrac{1}{4}\pi\right)=% \ln\left(\sec x+\tan x\right)}}
\ln@@{\tan@{\tfrac{1}{2}x+\tfrac{1}{4}\pi}} = \ln@{\sec@@{x}+\tan@@{x}}		 - 1 2 ⁒ Ο€ < x , x < 1 2 ⁒ Ο€ formulae-sequence 1 2 πœ‹ π‘₯ π‘₯ 1 2 πœ‹ {\displaystyle{\displaystyle-\frac{1}{2}\pi<x,x<\frac{1}{2}\pi}} 
ln(tan((1)/(2)*x +(1)/(4)*Pi)) = ln(sec(x)+ tan(x))
 
Log[Tan[Divide[1,2]*x +Divide[1,4]*Pi]] == Log[Sec[x]+ Tan[x]]
 Successful Successful Skip - symbolical successful subtest Successful [Tested: 2]
4.23.E42 ln ⁑ ( sec ⁑ x + tan ⁑ x ) = arcsinh ⁑ ( tan ⁑ x ) π‘₯ π‘₯ hyperbolic-inverse-sine π‘₯ {\displaystyle{\displaystyle\ln\left(\sec x+\tan x\right)=\operatorname{% arcsinh}\left(\tan x\right)}}
\ln@{\sec@@{x}+\tan@@{x}} = \asinh@{\tan@@{x}}		 - 1 2 ⁒ Ο€ < x , x < 1 2 ⁒ Ο€ formulae-sequence 1 2 πœ‹ π‘₯ π‘₯ 1 2 πœ‹ {\displaystyle{\displaystyle-\frac{1}{2}\pi<x,x<\frac{1}{2}\pi}} 
ln(sec(x)+ tan(x)) = arcsinh(tan(x))
 
Log[Sec[x]+ Tan[x]] == ArcSinh[Tan[x]]
 Failure Failure Successful [Tested: 2]
Failed [1 / 3]
Result: Complex[3.046904887125347, 3.141592653589793]
Test Values: {Rule[x, 2]}

4.23.E42 arcsinh ⁑ ( tan ⁑ x ) = arccsch ⁑ ( cot ⁑ x ) hyperbolic-inverse-sine π‘₯ hyperbolic-inverse-cosecant π‘₯ {\displaystyle{\displaystyle\operatorname{arcsinh}\left(\tan x\right)=% \operatorname{arccsch}\left(\cot x\right)}}
\asinh@{\tan@@{x}} = \acsch@{\cot@@{x}}		 - 1 2 ⁒ Ο€ < x , x < 1 2 ⁒ Ο€ formulae-sequence 1 2 πœ‹ π‘₯ π‘₯ 1 2 πœ‹ {\displaystyle{\displaystyle-\frac{1}{2}\pi<x,x<\frac{1}{2}\pi}} 
arcsinh(tan(x)) = arccsch(cot(x))
 
ArcSinh[Tan[x]] == ArcCsch[Cot[x]]
 Failure Successful Successful [Tested: 2] Successful [Tested: 2]
4.23.E42 arccsch ⁑ ( cot ⁑ x ) = arccosh ⁑ ( sec ⁑ x ) hyperbolic-inverse-cosecant π‘₯ hyperbolic-inverse-cosine π‘₯ {\displaystyle{\displaystyle\operatorname{arccsch}\left(\cot x\right)=% \operatorname{arccosh}\left(\sec x\right)}}
\acsch@{\cot@@{x}} = \acosh@{\sec@@{x}}		 - 1 2 ⁒ Ο€ < x , x < 1 2 ⁒ Ο€ formulae-sequence 1 2 πœ‹ π‘₯ π‘₯ 1 2 πœ‹ {\displaystyle{\displaystyle-\frac{1}{2}\pi<x,x<\frac{1}{2}\pi}} 
arccsch(cot(x)) = arccosh(sec(x))
 
ArcCsch[Cot[x]] == ArcCosh[Sec[x]]
 Failure Failure Successful [Tested: 2]
Failed [1 / 3]
Result: Complex[-3.046904887125347, -3.141592653589793]
Test Values: {Rule[x, 2]}

4.23.E42 arccosh ⁑ ( sec ⁑ x ) = arcsech ⁑ ( cos ⁑ x ) hyperbolic-inverse-cosine π‘₯ hyperbolic-inverse-secant π‘₯ {\displaystyle{\displaystyle\operatorname{arccosh}\left(\sec x\right)=% \operatorname{arcsech}\left(\cos x\right)}}
\acosh@{\sec@@{x}} = \asech@{\cos@@{x}}		 - 1 2 ⁒ Ο€ < x , x < 1 2 ⁒ Ο€ formulae-sequence 1 2 πœ‹ π‘₯ π‘₯ 1 2 πœ‹ {\displaystyle{\displaystyle-\frac{1}{2}\pi<x,x<\frac{1}{2}\pi}} 
arccosh(sec(x)) = arcsech(cos(x))
 
ArcCosh[Sec[x]] == ArcSech[Cos[x]]
 Failure Successful Successful [Tested: 2] Successful [Tested: 2]
4.23.E42 arcsech ⁑ ( cos ⁑ x ) = arctanh ⁑ ( sin ⁑ x ) hyperbolic-inverse-secant π‘₯ hyperbolic-inverse-tangent π‘₯ {\displaystyle{\displaystyle\operatorname{arcsech}\left(\cos x\right)=% \operatorname{arctanh}\left(\sin x\right)}}
\asech@{\cos@@{x}} = \atanh@{\sin@@{x}}		 - 1 2 ⁒ Ο€ < x , x < 1 2 ⁒ Ο€ formulae-sequence 1 2 πœ‹ π‘₯ π‘₯ 1 2 πœ‹ {\displaystyle{\displaystyle-\frac{1}{2}\pi<x,x<\frac{1}{2}\pi}} 
arcsech(cos(x)) = arctanh(sin(x))
 
ArcSech[Cos[x]] == ArcTanh[Sin[x]]
 Failure Failure Successful [Tested: 2]
Failed [1 / 3]
Result: Complex[0.0, 3.141592653589793]
Test Values: {Rule[x, 2]}

4.23.E42 arctanh ⁑ ( sin ⁑ x ) = arccoth ⁑ ( csc ⁑ x ) hyperbolic-inverse-tangent π‘₯ hyperbolic-inverse-cotangent π‘₯ {\displaystyle{\displaystyle\operatorname{arctanh}\left(\sin x\right)=% \operatorname{arccoth}\left(\csc x\right)}}
\atanh@{\sin@@{x}} = \acoth@{\csc@@{x}}		 - 1 2 ⁒ Ο€ < x , x < 1 2 ⁒ Ο€ formulae-sequence 1 2 πœ‹ π‘₯ π‘₯ 1 2 πœ‹ {\displaystyle{\displaystyle-\frac{1}{2}\pi<x,x<\frac{1}{2}\pi}} 
arctanh(sin(x)) = arccoth(csc(x))
 
ArcTanh[Sin[x]] == ArcCoth[Csc[x]]
 Failure Successful Successful [Tested: 2] Successful [Tested: 2]
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