Elementary Functions - 4.38 Inverse Hyperbolic Functions: Further Properties

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
4.38.E4 arccosh ⁡ z = ( 2 ⁢ ( z - 1 ) ) 1 / 2 ⁢ ( 1 + ∑ n = 1 ∞ ( - 1 ) n ⁢ 1 ⋅ 3 ⋅ 5 ⁢ ⋯ ⁢ ( 2 ⁢ n - 1 ) 2 2 ⁢ n ⁢ n ! ⁢ ( 2 ⁢ n + 1 ) ⁢ ( z - 1 ) n ) hyperbolic-inverse-cosine 𝑧 superscript 2 𝑧 1 1 2 1 superscript subscript 𝑛 1 superscript 1 𝑛 ⋅ 1 3 5 ⋯ 2 𝑛 1 superscript 2 2 𝑛 𝑛 2 𝑛 1 superscript 𝑧 1 𝑛 {\displaystyle{\displaystyle\operatorname{arccosh}z=(2(z-1))^{1/2}\*{\left(1+% \sum_{n=1}^{\infty}(-1)^{n}\frac{1\cdot 3\cdot 5\cdots(2n-1)}{2^{2n}n!(2n+1)}(% z-1)^{n}\right)}}}
\acosh@@{z} = (2(z-1))^{1/2}\*{\left(1+\sum_{n=1}^{\infty}(-1)^{n}\frac{1\cdot 3\cdot 5\cdots(2n-1)}{2^{2n}n!(2n+1)}(z-1)^{n}\right)}		 ℜ ⁡ z > 0 , | z - 1 | ≤ 2 formulae-sequence 𝑧 0 𝑧 1 2 {\displaystyle{\displaystyle\Re z>0,|z-1|\leq 2}} 
arccosh(z) = (2*(z - 1))^(1/2)*(1 + sum((- 1)^(n)*(1 * 3 * 5*(2*n - 1))/((2)^(2*n)* factorial(n)*(2*n + 1))*(z - 1)^(n), n = 1..infinity))

ArcCosh[z] == (2*(z - 1))^(1/2)*(1 + Sum[(- 1)^(n)*Divide[1 * 3 * 5*(2*n - 1),(2)^(2*n)* (n)!*(2*n + 1)]*(z - 1)^(n), {n, 1, Infinity}, GenerateConditions->None])
Failure Failure
Failed [5 / 5]
Result: -.5552108774+.3065228369*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}


Result: -1.819822265-.3498215011*I
Test Values: {z = 1/2-1/2*I*3^(1/2)}


Result: .5204832489
Test Values: {z = 1.5}


Result: -.651724541*I
Test Values: {z = .5}

... skip entries to safe data
Failed [5 / 5]
Result: Complex[-0.5552108781095244, 0.30652283644847583]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-1.8198222655846492, -0.34982149976378074]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
4.38.E8 x 2 - y 2 = 1 2 superscript 𝑥 2 superscript 𝑦 2 1 2 {\displaystyle{\displaystyle x^{2}-y^{2}=\tfrac{1}{2}}}
x^{2}-y^{2} = \tfrac{1}{2}		 

(x)^(2)- (y)^(2) = (1)/(2)
 
(x)^(2)- (y)^(2) == Divide[1,2]
 Skipped - no semantic math Skipped - no semantic math - -
4.38.E9 d d z ⁡ arcsinh ⁡ z = ( 1 + z 2 ) - 1 / 2 derivative 𝑧 hyperbolic-inverse-sine 𝑧 superscript 1 superscript 𝑧 2 1 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{% arcsinh}z=(1+z^{2})^{-1/2}}}
\deriv{}{z}\asinh@@{z} = (1+z^{2})^{-1/2}		 

diff(arcsinh(z), z) = (1 + (z)^(2))^(- 1/2)

D[ArcSinh[z], z] == (1 + (z)^(2))^(- 1/2)
Successful Successful - Successful [Tested: 7]
4.38.E10 d d z ⁡ arccosh ⁡ z = + ( z 2 - 1 ) - 1 / 2 derivative 𝑧 hyperbolic-inverse-cosine 𝑧 superscript superscript 𝑧 2 1 1 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{% arccosh}z=+(z^{2}-1)^{-1/2}}}
\deriv{}{z}\acosh@@{z} = +(z^{2}-1)^{-1/2}		 

diff(arccosh(z), z) = +((z)^(2)- 1)^(- 1/2)

D[ArcCosh[z], z] == +((z)^(2)- 1)^(- 1/2)
Failure Failure
Failed [2 / 7]
Result: -.3933198932-1.467889825*I
Test Values: {z = -1/2+1/2*I*3^(1/2), z = 1/2}


Result: -1.000000000+1.732050808*I
Test Values: {z = -1/2*3^(1/2)-1/2*I, z = 1/2}

Successful [Tested: 1]
4.38.E10 d d z ⁡ arccosh ⁡ z = - ( z 2 - 1 ) - 1 / 2 derivative 𝑧 hyperbolic-inverse-cosine 𝑧 superscript superscript 𝑧 2 1 1 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{% arccosh}z=-(z^{2}-1)^{-1/2}}}
\deriv{}{z}\acosh@@{z} = -(z^{2}-1)^{-1/2}		 

diff(arccosh(z), z) = -((z)^(2)- 1)^(- 1/2)

D[ArcCosh[z], z] == -((z)^(2)- 1)^(- 1/2)
Failure Failure
Failed [5 / 7]
Result: 1.000000000-1.732050808*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, z = 1/2}


Result: .3933198932+1.467889825*I
Test Values: {z = 1/2-1/2*I*3^(1/2), z = 1/2}


Result: 1.788854382
Test Values: {z = 1.5, z = 1/2}


Result: -2.309401076*I
Test Values: {z = .5, z = 1/2}

... skip entries to safe data
Failed [1 / 1]
Result: Complex[0.0, -2.3094010767585034]
Test Values: {Rule[z, Rational[1, 2]]}

4.38.E11 d d z ⁡ arctanh ⁡ z = 1 1 - z 2 derivative 𝑧 hyperbolic-inverse-tangent 𝑧 1 1 superscript 𝑧 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{% arctanh}z=\frac{1}{1-z^{2}}}}
\deriv{}{z}\atanh@@{z} = \frac{1}{1-z^{2}}		 

diff(arctanh(z), z) = (1)/(1 - (z)^(2))

D[ArcTanh[z], z] == Divide[1,1 - (z)^(2)]
Successful Successful - Successful [Tested: 7]
4.38.E12 d d z ⁡ arccsch ⁡ z = - 1 z ⁢ ( 1 + z 2 ) 1 / 2 derivative 𝑧 hyperbolic-inverse-cosecant 𝑧 1 𝑧 superscript 1 superscript 𝑧 2 1 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{% arccsch}z=-\frac{1}{z(1+z^{2})^{1/2}}}}
\deriv{}{z}\acsch@@{z} = -\frac{1}{z(1+z^{2})^{1/2}}		 

diff(arccsch(z), z) = -(1)/(z*(1 + (z)^(2))^(1/2))

D[ArcCsch[z], z] == -Divide[1,z*(1 + (z)^(2))^(1/2)]
Failure Failure
Failed [2 / 7]
Result: .6696152420e-9-2.000000000*I
Test Values: {z = -1/2+1/2*I*3^(1/2), z = 1/2}


Result: -1.074569932+1.074569932*I
Test Values: {z = -1/2*3^(1/2)-1/2*I, z = 1/2}

Successful [Tested: 1]
4.38.E12 d d z ⁡ arccsch ⁡ z = + 1 z ⁢ ( 1 + z 2 ) 1 / 2 derivative 𝑧 hyperbolic-inverse-cosecant 𝑧 1 𝑧 superscript 1 superscript 𝑧 2 1 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{% arccsch}z=+\frac{1}{z(1+z^{2})^{1/2}}}}
\deriv{}{z}\acsch@@{z} = +\frac{1}{z(1+z^{2})^{1/2}}		 

diff(arccsch(z), z) = +(1)/(z*(1 + (z)^(2))^(1/2))

D[ArcCsch[z], z] == +Divide[1,z*(1 + (z)^(2))^(1/2)]
Failure Failure
Failed [5 / 7]
Result: -1.074569932+1.074569932*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, z = 1/2}


Result: .6696152420e-9-2.000000000*I
Test Values: {z = 1/2-1/2*I*3^(1/2), z = 1/2}


Result: -.7396002616
Test Values: {z = 1.5, z = 1/2}


Result: -3.577708764
Test Values: {z = .5, z = 1/2}

... skip entries to safe data
Failed [1 / 1]
Result: -3.5777087639996634
Test Values: {Rule[z, Rational[1, 2]]}

4.38.E13 d d z ⁡ arcsech ⁡ z = - 1 z ⁢ ( 1 - z 2 ) 1 / 2 derivative 𝑧 hyperbolic-inverse-secant 𝑧 1 𝑧 superscript 1 superscript 𝑧 2 1 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{% arcsech}z=-\frac{1}{z(1-z^{2})^{1/2}}}}
\deriv{}{z}\asech@@{z} = -\frac{1}{z(1-z^{2})^{1/2}}		 

diff(arcsech(z), z) = -(1)/(z*(1 - (z)^(2))^(1/2))

D[ArcSech[z], z] == -Divide[1,z*(1 - (z)^(2))^(1/2)]
Failure Failure Successful [Tested: 7] Successful [Tested: 7]
4.38.E14 d d z ⁡ arccoth ⁡ z = 1 1 - z 2 derivative 𝑧 hyperbolic-inverse-cotangent 𝑧 1 1 superscript 𝑧 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{% arccoth}z=\frac{1}{1-z^{2}}}}
\deriv{}{z}\acoth@@{z} = \frac{1}{1-z^{2}}		 

diff(arccoth(z), z) = (1)/(1 - (z)^(2))

D[ArcCoth[z], z] == Divide[1,1 - (z)^(2)]
Successful Successful - Successful [Tested: 7]
4.38.E15 Arcsinh ⁡ u + Arcsinh ⁡ v = Arcsinh ⁡ ( u ⁢ ( 1 + v 2 ) 1 / 2 + v ⁢ ( 1 + u 2 ) 1 / 2 ) multivalued-hyperbolic-inverse-sine 𝑢 multivalued-hyperbolic-inverse-sine 𝑣 multivalued-hyperbolic-inverse-sine 𝑢 superscript 1 superscript 𝑣 2 1 2 𝑣 superscript 1 superscript 𝑢 2 1 2 {\displaystyle{\displaystyle\operatorname{Arcsinh}u+\operatorname{Arcsinh}v=% \operatorname{Arcsinh}\left(u(1+v^{2})^{1/2}+v(1+u^{2})^{1/2}\right)}}
\Asinh@@{u}+\Asinh@@{v} = \Asinh@{u(1+v^{2})^{1/2}+ v(1+u^{2})^{1/2}}		 

Error

ArcSinh[u]+ ArcSinh[v] == ArcSinh[u*(1 + (v)^(2))^(1/2)+ v*(1 + (u)^(2))^(1/2)]
Missing Macro Error Failure -
Failed [1 / 100]
Result: Complex[-2.633915793849633, 4.440892098500626*^-16]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

4.38.E15 Arcsinh ⁡ u - Arcsinh ⁡ v = Arcsinh ⁡ ( u ⁢ ( 1 + v 2 ) 1 / 2 - v ⁢ ( 1 + u 2 ) 1 / 2 ) multivalued-hyperbolic-inverse-sine 𝑢 multivalued-hyperbolic-inverse-sine 𝑣 multivalued-hyperbolic-inverse-sine 𝑢 superscript 1 superscript 𝑣 2 1 2 𝑣 superscript 1 superscript 𝑢 2 1 2 {\displaystyle{\displaystyle\operatorname{Arcsinh}u-\operatorname{Arcsinh}v=% \operatorname{Arcsinh}\left(u(1+v^{2})^{1/2}-v(1+u^{2})^{1/2}\right)}}
\Asinh@@{u}-\Asinh@@{v} = \Asinh@{u(1+v^{2})^{1/2}- v(1+u^{2})^{1/2}}		 

Error

ArcSinh[u]- ArcSinh[v] == ArcSinh[u*(1 + (v)^(2))^(1/2)- v*(1 + (u)^(2))^(1/2)]
Missing Macro Error Failure - Successful [Tested: 100]
4.38.E16 Arccosh ⁡ u + Arccosh ⁡ v = Arccosh ⁡ ( u ⁢ v + ( ( u 2 - 1 ) ⁢ ( v 2 - 1 ) ) 1 / 2 ) multivalued-hyperbolic-inverse-cosine 𝑢 multivalued-hyperbolic-inverse-cosine 𝑣 multivalued-hyperbolic-inverse-cosine 𝑢 𝑣 superscript superscript 𝑢 2 1 superscript 𝑣 2 1 1 2 {\displaystyle{\displaystyle\operatorname{Arccosh}u+\operatorname{Arccosh}v=% \operatorname{Arccosh}\left(uv+((u^{2}-1)(v^{2}-1))^{1/2}\right)}}
\Acosh@@{u}+\Acosh@@{v} = \Acosh@{uv+((u^{2}-1)(v^{2}-1))^{1/2}}		 

Error

ArcCosh[u]+ ArcCosh[v] == ArcCosh[u*v +(((u)^(2)- 1)*((v)^(2)- 1))^(1/2)]
Missing Macro Error Failure -
Failed [60 / 100]
Result: Complex[1.3169578969248166, 1.5707963267948966]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.316957896924817, 1.5707963267948966]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
4.38.E16 Arccosh ⁡ u - Arccosh ⁡ v = Arccosh ⁡ ( u ⁢ v - ( ( u 2 - 1 ) ⁢ ( v 2 - 1 ) ) 1 / 2 ) multivalued-hyperbolic-inverse-cosine 𝑢 multivalued-hyperbolic-inverse-cosine 𝑣 multivalued-hyperbolic-inverse-cosine 𝑢 𝑣 superscript superscript 𝑢 2 1 superscript 𝑣 2 1 1 2 {\displaystyle{\displaystyle\operatorname{Arccosh}u-\operatorname{Arccosh}v=% \operatorname{Arccosh}\left(uv-((u^{2}-1)(v^{2}-1))^{1/2}\right)}}
\Acosh@@{u}-\Acosh@@{v} = \Acosh@{uv-((u^{2}-1)(v^{2}-1))^{1/2}}		 

Error

ArcCosh[u]- ArcCosh[v] == ArcCosh[u*v -(((u)^(2)- 1)*((v)^(2)- 1))^(1/2)]
Missing Macro Error Failure -
Failed [80 / 100]
Result: Complex[-1.3169578969248166, -1.5707963267948966]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-1.6628858910586213, -3.8910615190072733]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
4.38.E17 Arctanh ⁡ u + Arctanh ⁡ v = Arctanh ⁡ ( u + v 1 + u ⁢ v ) multivalued-hyperbolic-inverse-tangent 𝑢 multivalued-hyperbolic-inverse-tangent 𝑣 multivalued-hyperbolic-inverse-tangent 𝑢 𝑣 1 𝑢 𝑣 {\displaystyle{\displaystyle\operatorname{Arctanh}u+\operatorname{Arctanh}v=% \operatorname{Arctanh}\left(\frac{u+v}{1+uv}\right)}}
\Atanh@@{u}+\Atanh@@{v} = \Atanh@{\frac{u+ v}{1+ uv}}		 

Error

ArcTanh[u]+ ArcTanh[v] == ArcTanh[Divide[u + v,1 + u*v]]
Missing Macro Error Failure - Successful [Tested: 1]
4.38.E17 Arctanh ⁡ u - Arctanh ⁡ v = Arctanh ⁡ ( u - v 1 - u ⁢ v ) multivalued-hyperbolic-inverse-tangent 𝑢 multivalued-hyperbolic-inverse-tangent 𝑣 multivalued-hyperbolic-inverse-tangent 𝑢 𝑣 1 𝑢 𝑣 {\displaystyle{\displaystyle\operatorname{Arctanh}u-\operatorname{Arctanh}v=% \operatorname{Arctanh}\left(\frac{u-v}{1-uv}\right)}}
\Atanh@@{u}-\Atanh@@{v} = \Atanh@{\frac{u- v}{1- uv}}		 

Error

ArcTanh[u]- ArcTanh[v] == ArcTanh[Divide[u - v,1 - u*v]]
Missing Macro Error Failure - Successful [Tested: 1]
4.38.E18 Arcsinh ⁡ u + Arccosh ⁡ v = Arcsinh ⁡ ( u ⁢ v + ( ( 1 + u 2 ) ⁢ ( v 2 - 1 ) ) 1 / 2 ) multivalued-hyperbolic-inverse-sine 𝑢 multivalued-hyperbolic-inverse-cosine 𝑣 multivalued-hyperbolic-inverse-sine 𝑢 𝑣 superscript 1 superscript 𝑢 2 superscript 𝑣 2 1 1 2 {\displaystyle{\displaystyle\operatorname{Arcsinh}u+\operatorname{Arccosh}v=% \operatorname{Arcsinh}\left(uv+((1+u^{2})(v^{2}-1))^{1/2}\right)}}
\Asinh@@{u}+\Acosh@@{v} = \Asinh@{uv+((1+u^{2})(v^{2}-1))^{1/2}}		 

Error

ArcSinh[u]+ ArcCosh[v] == ArcSinh[u*v +((1 + (u)^(2))*((v)^(2)- 1))^(1/2)]
Missing Macro Error Failure -
Failed [53 / 100]
Result: Complex[1.66288587615746, 3.891061504106112]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[1.6628858910586204, -2.3921237881723125]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

... skip entries to safe data
4.38.E18 Arcsinh ⁡ u - Arccosh ⁡ v = Arcsinh ⁡ ( u ⁢ v - ( ( 1 + u 2 ) ⁢ ( v 2 - 1 ) ) 1 / 2 ) multivalued-hyperbolic-inverse-sine 𝑢 multivalued-hyperbolic-inverse-cosine 𝑣 multivalued-hyperbolic-inverse-sine 𝑢 𝑣 superscript 1 superscript 𝑢 2 superscript 𝑣 2 1 1 2 {\displaystyle{\displaystyle\operatorname{Arcsinh}u-\operatorname{Arccosh}v=% \operatorname{Arcsinh}\left(uv-((1+u^{2})(v^{2}-1))^{1/2}\right)}}
\Asinh@@{u}-\Acosh@@{v} = \Asinh@{uv-((1+u^{2})(v^{2}-1))^{1/2}}		 

Error

ArcSinh[u]- ArcCosh[v] == ArcSinh[u*v -((1 + (u)^(2))*((v)^(2)- 1))^(1/2)]
Missing Macro Error Failure -
Failed [53 / 100]
Result: Complex[1.6628858910586208, -2.392123788172313]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[1.6628858910586208, 3.8910615190072733]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

... skip entries to safe data
4.38.E18 Arcsinh ⁡ ( u ⁢ v + ( ( 1 + u 2 ) ⁢ ( v 2 - 1 ) ) 1 / 2 ) = Arccosh ⁡ ( v ⁢ ( 1 + u 2 ) 1 / 2 + u ⁢ ( v 2 - 1 ) 1 / 2 ) multivalued-hyperbolic-inverse-sine 𝑢 𝑣 superscript 1 superscript 𝑢 2 superscript 𝑣 2 1 1 2 multivalued-hyperbolic-inverse-cosine 𝑣 superscript 1 superscript 𝑢 2 1 2 𝑢 superscript superscript 𝑣 2 1 1 2 {\displaystyle{\displaystyle\operatorname{Arcsinh}\left(uv+((1+u^{2})(v^{2}-1)% )^{1/2}\right)=\operatorname{Arccosh}\left(v(1+u^{2})^{1/2}+u(v^{2}-1)^{1/2}% \right)}}
\Asinh@{uv+((1+u^{2})(v^{2}-1))^{1/2}} = \Acosh@{v(1+u^{2})^{1/2}+ u(v^{2}-1)^{1/2}}		 

Error

ArcSinh[u*v +((1 + (u)^(2))*((v)^(2)- 1))^(1/2)] == ArcCosh[v*(1 + (u)^(2))^(1/2)+ u*((v)^(2)- 1)^(1/2)]
Missing Macro Error Failure -
Failed [65 / 100]
Result: Complex[1.4901161193847656*^-8, -3.141592638688632]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[-0.34592799413380415, -2.320265192212377]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

... skip entries to safe data
4.38.E18 Arcsinh ⁡ ( u ⁢ v - ( ( 1 + u 2 ) ⁢ ( v 2 - 1 ) ) 1 / 2 ) = Arccosh ⁡ ( v ⁢ ( 1 + u 2 ) 1 / 2 - u ⁢ ( v 2 - 1 ) 1 / 2 ) multivalued-hyperbolic-inverse-sine 𝑢 𝑣 superscript 1 superscript 𝑢 2 superscript 𝑣 2 1 1 2 multivalued-hyperbolic-inverse-cosine 𝑣 superscript 1 superscript 𝑢 2 1 2 𝑢 superscript superscript 𝑣 2 1 1 2 {\displaystyle{\displaystyle\operatorname{Arcsinh}\left(uv-((1+u^{2})(v^{2}-1)% )^{1/2}\right)=\operatorname{Arccosh}\left(v(1+u^{2})^{1/2}-u(v^{2}-1)^{1/2}% \right)}}
\Asinh@{uv-((1+u^{2})(v^{2}-1))^{1/2}} = \Acosh@{v(1+u^{2})^{1/2}- u(v^{2}-1)^{1/2}}		 

Error

ArcSinh[u*v -((1 + (u)^(2))*((v)^(2)- 1))^(1/2)] == ArcCosh[v*(1 + (u)^(2))^(1/2)- u*((v)^(2)- 1)^(1/2)]
Missing Macro Error Failure -
Failed [86 / 100]
Result: Complex[-3.325771782117242, -1.4989377308349603]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[-2.9798437879834374, 0.8213274613774169]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

... skip entries to safe data
4.38.E19 Arctanh ⁡ u + Arccoth ⁡ v = Arctanh ⁡ ( u ⁢ v + 1 v + u ) multivalued-hyperbolic-inverse-tangent 𝑢 multivalued-hyperbolic-inverse-cotangent 𝑣 multivalued-hyperbolic-inverse-tangent 𝑢 𝑣 1 𝑣 𝑢 {\displaystyle{\displaystyle\operatorname{Arctanh}u+\operatorname{Arccoth}v=% \operatorname{Arctanh}\left(\frac{uv+1}{v+u}\right)}}
\Atanh@@{u}+\Acoth@@{v} = \Atanh@{\frac{uv+ 1}{v+ u}}		 

Error

ArcTanh[u]+ ArcCoth[v] == ArcTanh[Divide[u*v + 1,v + u]]
Missing Macro Error Failure -
Failed [1 / 10]
Result: Indeterminate
Test Values: {Rule[u, Rational[1, 2]], Rule[v, -0.5]}

4.38.E19 Arctanh ⁡ u - Arccoth ⁡ v = Arctanh ⁡ ( u ⁢ v - 1 v - u ) multivalued-hyperbolic-inverse-tangent 𝑢 multivalued-hyperbolic-inverse-cotangent 𝑣 multivalued-hyperbolic-inverse-tangent 𝑢 𝑣 1 𝑣 𝑢 {\displaystyle{\displaystyle\operatorname{Arctanh}u-\operatorname{Arccoth}v=% \operatorname{Arctanh}\left(\frac{uv-1}{v-u}\right)}}
\Atanh@@{u}-\Acoth@@{v} = \Atanh@{\frac{uv- 1}{v- u}}		 

Error

ArcTanh[u]- ArcCoth[v] == ArcTanh[Divide[u*v - 1,v - u]]
Missing Macro Error Failure -
Failed [1 / 10]
Result: Indeterminate
Test Values: {Rule[u, Rational[1, 2]], Rule[v, 0.5]}

4.38.E19 Arctanh ⁡ ( u ⁢ v + 1 v + u ) = Arccoth ⁡ ( v + u u ⁢ v + 1 ) multivalued-hyperbolic-inverse-tangent 𝑢 𝑣 1 𝑣 𝑢 multivalued-hyperbolic-inverse-cotangent 𝑣 𝑢 𝑢 𝑣 1 {\displaystyle{\displaystyle\operatorname{Arctanh}\left(\frac{uv+1}{v+u}\right% )=\operatorname{Arccoth}\left(\frac{v+u}{uv+1}\right)}}
\Atanh@{\frac{uv+ 1}{v+ u}} = \Acoth@{\frac{v+ u}{uv+ 1}}		 

Error
 
ArcTanh[Divide[u*v + 1,v + u]] == ArcCoth[Divide[v + u,u*v + 1]]
 Missing Macro Error Successful -
Failed [8 / 100]
Result: Indeterminate
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

Result: Indeterminate
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
4.38.E19 Arctanh ⁡ ( u ⁢ v - 1 v - u ) = Arccoth ⁡ ( v - u u ⁢ v - 1 ) multivalued-hyperbolic-inverse-tangent 𝑢 𝑣 1 𝑣 𝑢 multivalued-hyperbolic-inverse-cotangent 𝑣 𝑢 𝑢 𝑣 1 {\displaystyle{\displaystyle\operatorname{Arctanh}\left(\frac{uv-1}{v-u}\right% )=\operatorname{Arccoth}\left(\frac{v-u}{uv-1}\right)}}
\Atanh@{\frac{uv- 1}{v- u}} = \Acoth@{\frac{v- u}{uv- 1}}		 

Error
 
ArcTanh[Divide[u*v - 1,v - u]] == ArcCoth[Divide[v - u,u*v - 1]]
 Missing Macro Error Successful -
Failed [10 / 100]
Result: Indeterminate
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Indeterminate
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
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