Jacobian Elliptic Functions - 22.15 Inverse Functions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
22.15.E1 sn ⁑ ( ΞΎ , k ) = x Jacobi-elliptic-sn πœ‰ π‘˜ π‘₯ {\displaystyle{\displaystyle\operatorname{sn}\left(\xi,k\right)=x}}
\Jacobiellsnk@{\xi}{k} = x		 - 1 ≀ x , x ≀ 1 formulae-sequence 1 π‘₯ π‘₯ 1 {\displaystyle{\displaystyle-1\leq x,x\leq 1}} 
JacobiSN(xi, k) = x

JacobiSN[\[Xi], (k)^2] == x
Failure Failure
Failed [30 / 30]
Result: .2924027565+.2435601371*I
Test Values: {x = 1/2, xi = 1/2*3^(1/2)+1/2*I, k = 1}


Result: .1797898601-.1565493762e-1*I
Test Values: {x = 1/2, xi = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[0.29240275641803626, 0.2435601371571337]
Test Values: {Rule[k, 1], Rule[x, 0.5], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.17978986006074704, -0.015654937469336286]
Test Values: {Rule[k, 2], Rule[x, 0.5], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.15.E2 cn ⁑ ( Ξ· , k ) = x Jacobi-elliptic-cn πœ‚ π‘˜ π‘₯ {\displaystyle{\displaystyle\operatorname{cn}\left(\eta,k\right)=x}}
\Jacobiellcnk@{\eta}{k} = x		 - 1 ≀ x , x ≀ 1 formulae-sequence 1 π‘₯ π‘₯ 1 {\displaystyle{\displaystyle-1\leq x,x\leq 1}} 
JacobiCN(eta, k) = x

JacobiCN[\[Eta], (k)^2] == x
Failure Failure
Failed [30 / 30]
Result: .2107428373-.2715436778*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, x = 1/2, k = 1}


Result: .2337173832+.1450431473e-1*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, x = 1/2, k = 2}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[0.21074283744314704, -0.27154367778248023]
Test Values: {Rule[k, 1], Rule[x, 0.5], Rule[Ξ·, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.23371738317128377, 0.01450431459800293]
Test Values: {Rule[k, 2], Rule[x, 0.5], Rule[Ξ·, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.15.E3 dn ⁑ ( ΞΆ , k ) = x Jacobi-elliptic-dn 𝜁 π‘˜ π‘₯ {\displaystyle{\displaystyle\operatorname{dn}\left(\zeta,k\right)=x}}
\Jacobielldnk@{\zeta}{k} = x		 k β€² ≀ x , x ≀ 1 formulae-sequence superscript π‘˜ β€² π‘₯ π‘₯ 1 {\displaystyle{\displaystyle k^{\prime}\leq x,x\leq 1}} 
JacobiDN(InverseJacobiDN(x, k), k) = x

JacobiDN[InverseJacobiDN[x, (k)^2], (k)^2] == x
Successful Successful - Successful [Tested: 1]
22.15.E5 - K ≀ arcsn ⁑ ( x , k ) 𝐾 inverse-Jacobi-elliptic-sn π‘₯ π‘˜ {\displaystyle{\displaystyle-K\leq\operatorname{arcsn}\left(x,k\right)}}
-K \leq \aJacobiellsnk@{x}{k}		 

- EllipticK(k) <= InverseJacobiSN(x, k)

- EllipticK[(k)^2] <= InverseJacobiSN[x, (k)^2]
Failure Failure Error
Failed [9 / 9]
Result: LessEqual[DirectedInfinity[], Complex[0.8047189562170503, -1.5707963267948966]]
Test Values: {Rule[k, 1], Rule[x, 1.5]}


Result: LessEqual[Complex[-0.8428751774062981, 1.0782578237498217], Complex[0.372543189356477, -1.0782578237498215]]
Test Values: {Rule[k, 2], Rule[x, 1.5]}

... skip entries to safe data
22.15.E5 arcsn ⁑ ( x , k ) ≀ K inverse-Jacobi-elliptic-sn π‘₯ π‘˜ 𝐾 {\displaystyle{\displaystyle\operatorname{arcsn}\left(x,k\right)\leq K}}
\aJacobiellsnk@{x}{k} \leq K		 

InverseJacobiSN(x, k) <= EllipticK(k)

InverseJacobiSN[x, (k)^2] <= EllipticK[(k)^2]
Failure Failure Error
Failed [9 / 9]
Result: LessEqual[Complex[0.8047189562170503, -1.5707963267948966], DirectedInfinity[]]
Test Values: {Rule[k, 1], Rule[x, 1.5]}


Result: LessEqual[Complex[0.372543189356477, -1.0782578237498215], Complex[0.8428751774062981, -1.0782578237498217]]
Test Values: {Rule[k, 2], Rule[x, 1.5]}

... skip entries to safe data
22.15.E6 0 ≀ arccn ⁑ ( x , k ) 0 inverse-Jacobi-elliptic-cn π‘₯ π‘˜ {\displaystyle{\displaystyle 0\leq\operatorname{arccn}\left(x,k\right)}}
0 \leq \aJacobiellcnk@{x}{k}		 

0 <= InverseJacobiCN(x, k)

0 <= InverseJacobiCN[x, (k)^2]
Failure Failure Successful [Tested: 9]
Failed [8 / 9]
Result: LessEqual[0.0, Complex[0.0, 0.8410686705679303]]
Test Values: {Rule[k, 1], Rule[x, 1.5]}


Result: LessEqual[0.0, Complex[5.551115123125783*^-16, 0.6872864564092609]]
Test Values: {Rule[k, 2], Rule[x, 1.5]}

... skip entries to safe data
22.15.E6 arccn ⁑ ( x , k ) ≀ 2 ⁒ K inverse-Jacobi-elliptic-cn π‘₯ π‘˜ 2 𝐾 {\displaystyle{\displaystyle\operatorname{arccn}\left(x,k\right)\leq 2K}}
\aJacobiellcnk@{x}{k} \leq 2K		 

InverseJacobiCN(x, k) <= 2*EllipticK(k)

InverseJacobiCN[x, (k)^2] <= 2*EllipticK[(k)^2]
Failure Failure Error
Failed [9 / 9]
Result: LessEqual[Complex[0.0, 0.8410686705679303], DirectedInfinity[]]
Test Values: {Rule[k, 1], Rule[x, 1.5]}


Result: LessEqual[Complex[5.551115123125783*^-16, 0.6872864564092609], Complex[1.6857503548125963, -2.1565156474996434]]
Test Values: {Rule[k, 2], Rule[x, 1.5]}

... skip entries to safe data
22.15.E7 0 ≀ arcdn ⁑ ( x , k ) 0 inverse-Jacobi-elliptic-dn π‘₯ π‘˜ {\displaystyle{\displaystyle 0\leq\operatorname{arcdn}\left(x,k\right)}}
0 \leq \aJacobielldnk@{x}{k}		 

0 <= InverseJacobiDN(x, k)

0 <= InverseJacobiDN[x, (k)^2]
Failure Failure Successful [Tested: 9]
Failed [8 / 9]
Result: LessEqual[0.0, Complex[0.0, 0.8410686705679303]]
Test Values: {Rule[k, 1], Rule[x, 1.5]}


Result: LessEqual[0.0, Complex[1.6857503548125963, -1.6950867772240739]]
Test Values: {Rule[k, 2], Rule[x, 1.5]}

... skip entries to safe data
22.15.E7 arcdn ⁑ ( x , k ) ≀ K inverse-Jacobi-elliptic-dn π‘₯ π‘˜ 𝐾 {\displaystyle{\displaystyle\operatorname{arcdn}\left(x,k\right)\leq K}}
\aJacobielldnk@{x}{k} \leq K		 

InverseJacobiDN(x, k) <= EllipticK(k)

InverseJacobiDN[x, (k)^2] <= EllipticK[(k)^2]
Failure Failure Error
Failed [9 / 9]
Result: LessEqual[Complex[0.0, 0.8410686705679303], DirectedInfinity[]]
Test Values: {Rule[k, 1], Rule[x, 1.5]}


Result: LessEqual[Complex[1.6857503548125963, -1.6950867772240739], Complex[0.8428751774062981, -1.0782578237498217]]
Test Values: {Rule[k, 2], Rule[x, 1.5]}

... skip entries to safe data
22.15.E8 ΞΎ = ( - 1 ) m ⁒ arcsn ⁑ ( x , k ) + 2 ⁒ m ⁒ K πœ‰ superscript 1 π‘š inverse-Jacobi-elliptic-sn π‘₯ π‘˜ 2 π‘š 𝐾 {\displaystyle{\displaystyle\xi=(-1)^{m}\operatorname{arcsn}\left(x,k\right)+2% mK}}
\xi = (-1)^{m}\aJacobiellsnk@{x}{k}+2mK		 

xi = (- 1)^(m)* InverseJacobiSN(x, k)+ 2*m*K

\[Xi] == (- 1)^(m)* InverseJacobiSN[x, (k)^2]+ 2*m*K
Failure Failure
Failed [300 / 300]
Result: -.613064478e-1-2.070796327*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, x = 3/2, xi = 1/2*3^(1/2)+1/2*I, k = 1, m = 1}


Result: -3.402795168+.70796327e-1*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, x = 3/2, xi = 1/2*3^(1/2)+1/2*I, k = 1, m = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.061306447567388456, -2.0707963267948966]
Test Values: {Rule[k, 1], Rule[K, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[m, 1], Rule[x, 1.5], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-3.4027951675703663, 0.07079632679489672]
Test Values: {Rule[k, 1], Rule[K, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[m, 2], Rule[x, 1.5], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.15.E9 Ξ· = + arccn ⁑ ( x , k ) + 4 ⁒ m ⁒ K πœ‚ inverse-Jacobi-elliptic-cn π‘₯ π‘˜ 4 π‘š 𝐾 {\displaystyle{\displaystyle\eta=+\operatorname{arccn}\left(x,k\right)+4mK}}
\eta = +\aJacobiellcnk@{x}{k}+4mK		 

eta = + InverseJacobiCN(x, k)+ 4*m*K

\[Eta] == + InverseJacobiCN[x, (k)^2]+ 4*m*K
Failure Failure
Failed [300 / 300]
Result: -2.598076212-2.341068671*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, eta = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 1, m = 1}


Result: -6.062177828-4.341068671*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, eta = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 1, m = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-2.598076211353316, -2.34106867056793]
Test Values: {Rule[k, 1], Rule[K, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[m, 1], Rule[x, 1.5], Rule[Ξ·, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-6.062177826491071, -4.34106867056793]
Test Values: {Rule[k, 1], Rule[K, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[m, 2], Rule[x, 1.5], Rule[Ξ·, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.15.E9 Ξ· = - arccn ⁑ ( x , k ) + 4 ⁒ m ⁒ K πœ‚ inverse-Jacobi-elliptic-cn π‘₯ π‘˜ 4 π‘š 𝐾 {\displaystyle{\displaystyle\eta=-\operatorname{arccn}\left(x,k\right)+4mK}}
\eta = -\aJacobiellcnk@{x}{k}+4mK		 

eta = - InverseJacobiCN(x, k)+ 4*m*K

\[Eta] == - InverseJacobiCN[x, (k)^2]+ 4*m*K
Failure Failure
Failed [300 / 300]
Result: -2.598076212-.6589313294*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, eta = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 1, m = 1}


Result: -6.062177828-2.658931329*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, eta = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 1, m = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-2.598076211353316, -0.6589313294320696]
Test Values: {Rule[k, 1], Rule[K, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[m, 1], Rule[x, 1.5], Rule[Ξ·, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-6.062177826491071, -2.658931329432069]
Test Values: {Rule[k, 1], Rule[K, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[m, 2], Rule[x, 1.5], Rule[Ξ·, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.15.E10 ΞΆ = + arcdn ⁑ ( x , k ) + 2 ⁒ m ⁒ K 𝜁 inverse-Jacobi-elliptic-dn π‘₯ π‘˜ 2 π‘š 𝐾 {\displaystyle{\displaystyle\zeta=+\operatorname{arcdn}\left(x,k\right)+2mK}}
\zeta = +\aJacobielldnk@{x}{k}+2mK		 

(InverseJacobiDN(x, k)) = + InverseJacobiDN(x, k)+ 2*m*K

(InverseJacobiDN[x, (k)^2]) == + InverseJacobiDN[x, (k)^2]+ 2*m*K
Failure Failure
Failed [270 / 270]
Result: -1.732050808-1.000000000*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 1, m = 1}


Result: -3.464101616-2.*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 1, m = 2}

... skip entries to safe data
Failed [270 / 270]
Result: Complex[-1.7320508075688774, -0.9999999999999999]
Test Values: {Rule[k, 1], Rule[K, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[m, 1], Rule[x, 1.5]}


Result: Complex[-3.464101615137755, -1.9999999999999998]
Test Values: {Rule[k, 1], Rule[K, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[m, 2], Rule[x, 1.5]}

... skip entries to safe data
22.15.E10 ΞΆ = - arcdn ⁑ ( x , k ) + 2 ⁒ m ⁒ K 𝜁 inverse-Jacobi-elliptic-dn π‘₯ π‘˜ 2 π‘š 𝐾 {\displaystyle{\displaystyle\zeta=-\operatorname{arcdn}\left(x,k\right)+2mK}}
\zeta = -\aJacobielldnk@{x}{k}+2mK		 

(InverseJacobiDN(x, k)) = - InverseJacobiDN(x, k)+ 2*m*K

(InverseJacobiDN[x, (k)^2]) == - InverseJacobiDN[x, (k)^2]+ 2*m*K
Failure Failure
Failed [270 / 270]
Result: -1.732050808+.682137341*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 1, m = 1}


Result: -3.464101616-.317862659*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 1, m = 2}

... skip entries to safe data
Failed [270 / 270]
Result: Complex[-1.7320508075688774, 0.6821373411358608]
Test Values: {Rule[k, 1], Rule[K, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[m, 1], Rule[x, 1.5]}


Result: Complex[-3.464101615137755, -0.3178626588641391]
Test Values: {Rule[k, 1], Rule[K, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[m, 2], Rule[x, 1.5]}

... skip entries to safe data
22.15.E11 x = ∫ 0 sn ⁑ ( x , k ) d t ( 1 - t 2 ) ⁒ ( 1 - k 2 ⁒ t 2 ) π‘₯ superscript subscript 0 Jacobi-elliptic-sn π‘₯ π‘˜ 𝑑 1 superscript 𝑑 2 1 superscript π‘˜ 2 superscript 𝑑 2 {\displaystyle{\displaystyle x=\int_{0}^{\operatorname{sn}\left(x,k\right)}% \frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-k^{2}t^{2})}}}}
x = \int_{0}^{\Jacobiellsnk@{x}{k}}\frac{\diff{t}}{\sqrt{(1-t^{2})(1-k^{2}t^{2})}}		 - 1 ≀ x , x ≀ 1 , 0 ≀ k , k ≀ 1 formulae-sequence 1 π‘₯ formulae-sequence π‘₯ 1 formulae-sequence 0 π‘˜ π‘˜ 1 {\displaystyle{\displaystyle-1\leq x,x\leq 1,0\leq k,k\leq 1}} 
x = int((1)/(sqrt((1 - (t)^(2))*(1 - (k)^(2)* (t)^(2)))), t = 0..JacobiSN(x, k))

x == Integrate[Divide[1,Sqrt[(1 - (t)^(2))*(1 - (k)^(2)* (t)^(2))]], {t, 0, JacobiSN[x, (k)^2]}, GenerateConditions->None]
Failure Aborted Successful [Tested: 1] Skipped - Because timed out
22.15.E12 arcsn ⁑ ( x , k ) = ∫ 0 x d t ( 1 - t 2 ) ⁒ ( 1 - k 2 ⁒ t 2 ) inverse-Jacobi-elliptic-sn π‘₯ π‘˜ superscript subscript 0 π‘₯ 𝑑 1 superscript 𝑑 2 1 superscript π‘˜ 2 superscript 𝑑 2 {\displaystyle{\displaystyle\operatorname{arcsn}\left(x,k\right)=\int_{0}^{x}% \frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-k^{2}t^{2})}}}}
\aJacobiellsnk@{x}{k} = \int_{0}^{x}\frac{\diff{t}}{\sqrt{(1-t^{2})(1-k^{2}t^{2})}}		 - 1 ≀ x , x ≀ 1 formulae-sequence 1 π‘₯ π‘₯ 1 {\displaystyle{\displaystyle-1\leq x,x\leq 1}} 
InverseJacobiSN(x, k) = int((1)/(sqrt((1 - (t)^(2))*(1 - (k)^(2)* (t)^(2)))), t = 0..x)

InverseJacobiSN[x, (k)^2] == Integrate[Divide[1,Sqrt[(1 - (t)^(2))*(1 - (k)^(2)* (t)^(2))]], {t, 0, x}, GenerateConditions->None]
Error Aborted - Skipped - Because timed out
22.15.E13 arccn ⁑ ( x , k ) = ∫ x 1 d t ( 1 - t 2 ) ⁒ ( k β€² 2 + k 2 ⁒ t 2 ) inverse-Jacobi-elliptic-cn π‘₯ π‘˜ superscript subscript π‘₯ 1 𝑑 1 superscript 𝑑 2 superscript superscript π‘˜ β€² 2 superscript π‘˜ 2 superscript 𝑑 2 {\displaystyle{\displaystyle\operatorname{arccn}\left(x,k\right)=\int_{x}^{1}% \frac{\mathrm{d}t}{\sqrt{(1-t^{2})({k^{\prime}}^{2}+k^{2}t^{2})}}}}
\aJacobiellcnk@{x}{k} = \int_{x}^{1}\frac{\diff{t}}{\sqrt{(1-t^{2})({k^{\prime}}^{2}+k^{2}t^{2})}}		 - 1 ≀ x , x ≀ 1 formulae-sequence 1 π‘₯ π‘₯ 1 {\displaystyle{\displaystyle-1\leq x,x\leq 1}} 
InverseJacobiCN(x, k) = int((1)/(sqrt((1 - (t)^(2))*(1 - (k)^(2)+ (k)^(2)* (t)^(2)))), t = x..1)

InverseJacobiCN[x, (k)^2] == Integrate[Divide[1,Sqrt[(1 - (t)^(2))*(1 - (k)^(2)+ (k)^(2)* (t)^(2))]], {t, x, 1}, GenerateConditions->None]
Error Aborted - Skipped - Because timed out
22.15.E14 arcdn ⁑ ( x , k ) = ∫ x 1 d t ( 1 - t 2 ) ⁒ ( t 2 - k β€² 2 ) inverse-Jacobi-elliptic-dn π‘₯ π‘˜ superscript subscript π‘₯ 1 𝑑 1 superscript 𝑑 2 superscript 𝑑 2 superscript superscript π‘˜ β€² 2 {\displaystyle{\displaystyle\operatorname{arcdn}\left(x,k\right)=\int_{x}^{1}% \frac{\mathrm{d}t}{\sqrt{(1-t^{2})(t^{2}-{k^{\prime}}^{2})}}}}
\aJacobielldnk@{x}{k} = \int_{x}^{1}\frac{\diff{t}}{\sqrt{(1-t^{2})(t^{2}-{k^{\prime}}^{2})}}		 k β€² ≀ x , x ≀ 1 formulae-sequence superscript π‘˜ β€² π‘₯ π‘₯ 1 {\displaystyle{\displaystyle k^{\prime}\leq x,x\leq 1}} 
InverseJacobiDN(x, k) = int((1)/(sqrt((1 - (t)^(2))*((t)^(2)-1 - (k)^(2)))), t = x..1)

InverseJacobiDN[x, (k)^2] == Integrate[Divide[1,Sqrt[(1 - (t)^(2))*((t)^(2)-1 - (k)^(2))]], {t, x, 1}, GenerateConditions->None]
Error Aborted - Skipped - Because timed out
22.15.E15 arccd ⁑ ( x , k ) = ∫ x 1 d t ( 1 - t 2 ) ⁒ ( 1 - k 2 ⁒ t 2 ) inverse-Jacobi-elliptic-cd π‘₯ π‘˜ superscript subscript π‘₯ 1 𝑑 1 superscript 𝑑 2 1 superscript π‘˜ 2 superscript 𝑑 2 {\displaystyle{\displaystyle\operatorname{arccd}\left(x,k\right)=\int_{x}^{1}% \frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-k^{2}t^{2})}}}}
\aJacobiellcdk@{x}{k} = \int_{x}^{1}\frac{\diff{t}}{\sqrt{(1-t^{2})(1-k^{2}t^{2})}}		 - 1 ≀ x , x ≀ 1 formulae-sequence 1 π‘₯ π‘₯ 1 {\displaystyle{\displaystyle-1\leq x,x\leq 1}} 
InverseJacobiCD(x, k) = int((1)/(sqrt((1 - (t)^(2))*(1 - (k)^(2)* (t)^(2)))), t = x..1)

InverseJacobiCD[x, (k)^2] == Integrate[Divide[1,Sqrt[(1 - (t)^(2))*(1 - (k)^(2)* (t)^(2))]], {t, x, 1}, GenerateConditions->None]
Failure Aborted Error Skipped - Because timed out
22.15.E16 arcsd ⁑ ( x , k ) = ∫ 0 x d t ( 1 - k β€² 2 ⁒ t 2 ) ⁒ ( 1 + k 2 ⁒ t 2 ) inverse-Jacobi-elliptic-sd π‘₯ π‘˜ superscript subscript 0 π‘₯ 𝑑 1 superscript superscript π‘˜ β€² 2 superscript 𝑑 2 1 superscript π‘˜ 2 superscript 𝑑 2 {\displaystyle{\displaystyle\operatorname{arcsd}\left(x,k\right)=\int_{0}^{x}% \frac{\mathrm{d}t}{\sqrt{(1-{k^{\prime}}^{2}t^{2})(1+k^{2}t^{2})}}}}
\aJacobiellsdk@{x}{k} = \int_{0}^{x}\frac{\diff{t}}{\sqrt{(1-{k^{\prime}}^{2}t^{2})(1+k^{2}t^{2})}}		 - 1 / k β€² ≀ x , x ≀ 1 / k β€² formulae-sequence 1 superscript π‘˜ β€² π‘₯ π‘₯ 1 superscript π‘˜ β€² {\displaystyle{\displaystyle-1/k^{\prime}\leq x,x\leq 1/k^{\prime}}} 
InverseJacobiSD(x, k) = int((1)/(sqrt((1 -1 - (k)^(2)*(t)^(2))*(1 + (k)^(2)* (t)^(2)))), t = 0..x)

InverseJacobiSD[x, (k)^2] == Integrate[Divide[1,Sqrt[(1 -1 - (k)^(2)*(t)^(2))*(1 + (k)^(2)* (t)^(2))]], {t, 0, x}, GenerateConditions->None]
Error Failure - Skip - No test values generated
22.15.E17 arcnd ⁑ ( x , k ) = ∫ 1 x d t ( t 2 - 1 ) ⁒ ( 1 - k β€² 2 ⁒ t 2 ) inverse-Jacobi-elliptic-nd π‘₯ π‘˜ superscript subscript 1 π‘₯ 𝑑 superscript 𝑑 2 1 1 superscript superscript π‘˜ β€² 2 superscript 𝑑 2 {\displaystyle{\displaystyle\operatorname{arcnd}\left(x,k\right)=\int_{1}^{x}% \frac{\mathrm{d}t}{\sqrt{(t^{2}-1)(1-{k^{\prime}}^{2}t^{2})}}}}
\aJacobiellndk@{x}{k} = \int_{1}^{x}\frac{\diff{t}}{\sqrt{(t^{2}-1)(1-{k^{\prime}}^{2}t^{2})}}		 1 ≀ x , x ≀ 1 / k β€² formulae-sequence 1 π‘₯ π‘₯ 1 superscript π‘˜ β€² {\displaystyle{\displaystyle 1\leq x,x\leq 1/k^{\prime}}} 
InverseJacobiND(x, k) = int((1)/(sqrt(((t)^(2)- 1)*(1 -1 - (k)^(2)*(t)^(2)))), t = 1..x)
 
InverseJacobiND[x, (k)^2] == Integrate[Divide[1,Sqrt[((t)^(2)- 1)*(1 -1 - (k)^(2)*(t)^(2))]], {t, 1, x}, GenerateConditions->None]
 Error Failure - Skip - No test values generated
22.15.E18 arcdc ⁑ ( x , k ) = ∫ 1 x d t ( t 2 - 1 ) ⁒ ( t 2 - k 2 ) inverse-Jacobi-elliptic-dc π‘₯ π‘˜ superscript subscript 1 π‘₯ 𝑑 superscript 𝑑 2 1 superscript 𝑑 2 superscript π‘˜ 2 {\displaystyle{\displaystyle\operatorname{arcdc}\left(x,k\right)=\int_{1}^{x}% \frac{\mathrm{d}t}{\sqrt{(t^{2}-1)(t^{2}-k^{2})}}}}
\aJacobielldck@{x}{k} = \int_{1}^{x}\frac{\diff{t}}{\sqrt{(t^{2}-1)(t^{2}-k^{2})}}		 1 ≀ x , x < ∞ formulae-sequence 1 π‘₯ π‘₯ {\displaystyle{\displaystyle 1\leq x,x<\infty}} 
InverseJacobiDC(x, k) = int((1)/(sqrt(((t)^(2)- 1)*((t)^(2)- (k)^(2)))), t = 1..x)
 
InverseJacobiDC[x, (k)^2] == Integrate[Divide[1,Sqrt[((t)^(2)- 1)*((t)^(2)- (k)^(2))]], {t, 1, x}, GenerateConditions->None]
 Failure Aborted Error Skipped - Because timed out
22.15.E19 arcnc ⁑ ( x , k ) = ∫ 1 x d t ( t 2 - 1 ) ⁒ ( k 2 + k β€² 2 ⁒ t 2 ) inverse-Jacobi-elliptic-nc π‘₯ π‘˜ superscript subscript 1 π‘₯ 𝑑 superscript 𝑑 2 1 superscript π‘˜ 2 superscript superscript π‘˜ β€² 2 superscript 𝑑 2 {\displaystyle{\displaystyle\operatorname{arcnc}\left(x,k\right)=\int_{1}^{x}% \frac{\mathrm{d}t}{\sqrt{(t^{2}-1)(k^{2}+{k^{\prime}}^{2}t^{2})}}}}
\aJacobiellnck@{x}{k} = \int_{1}^{x}\frac{\diff{t}}{\sqrt{(t^{2}-1)(k^{2}+{k^{\prime}}^{2}t^{2})}}		 1 ≀ x , x < ∞ formulae-sequence 1 π‘₯ π‘₯ {\displaystyle{\displaystyle 1\leq x,x<\infty}} 
InverseJacobiNC(x, k) = int((1)/(sqrt(((t)^(2)- 1)*((k)^(2)+1 - (k)^(2)*(t)^(2)))), t = 1..x)
 
InverseJacobiNC[x, (k)^2] == Integrate[Divide[1,Sqrt[((t)^(2)- 1)*((k)^(2)+1 - (k)^(2)*(t)^(2))]], {t, 1, x}, GenerateConditions->None]
 Failure Aborted Skipped - Because timed out Skipped - Because timed out
22.15.E20 arcsc ⁑ ( x , k ) = ∫ 0 x d t ( 1 + t 2 ) ⁒ ( 1 + k β€² 2 ⁒ t 2 ) inverse-Jacobi-elliptic-sc π‘₯ π‘˜ superscript subscript 0 π‘₯ 𝑑 1 superscript 𝑑 2 1 superscript superscript π‘˜ β€² 2 superscript 𝑑 2 {\displaystyle{\displaystyle\operatorname{arcsc}\left(x,k\right)=\int_{0}^{x}% \frac{\mathrm{d}t}{\sqrt{(1+t^{2})(1+{k^{\prime}}^{2}t^{2})}}}}
\aJacobiellsck@{x}{k} = \int_{0}^{x}\frac{\diff{t}}{\sqrt{(1+t^{2})(1+{k^{\prime}}^{2}t^{2})}}		 - ∞ < x , x < ∞ formulae-sequence π‘₯ π‘₯ {\displaystyle{\displaystyle-\infty<x,x<\infty}} 
InverseJacobiSC(x, k) = int((1)/(sqrt((1 + (t)^(2))*(1 +1 - (k)^(2)*(t)^(2)))), t = 0..x)
 
InverseJacobiSC[x, (k)^2] == Integrate[Divide[1,Sqrt[(1 + (t)^(2))*(1 +1 - (k)^(2)*(t)^(2))]], {t, 0, x}, GenerateConditions->None]
 Error Aborted - Skipped - Because timed out
22.15.E21 arcns ⁑ ( x , k ) = ∫ x ∞ d t ( t 2 - 1 ) ⁒ ( t 2 - k 2 ) inverse-Jacobi-elliptic-ns π‘₯ π‘˜ superscript subscript π‘₯ 𝑑 superscript 𝑑 2 1 superscript 𝑑 2 superscript π‘˜ 2 {\displaystyle{\displaystyle\operatorname{arcns}\left(x,k\right)=\int_{x}^{% \infty}\frac{\mathrm{d}t}{\sqrt{(t^{2}-1)(t^{2}-k^{2})}}}}
\aJacobiellnsk@{x}{k} = \int_{x}^{\infty}\frac{\diff{t}}{\sqrt{(t^{2}-1)(t^{2}-k^{2})}}		 1 ≀ x , x < ∞ formulae-sequence 1 π‘₯ π‘₯ {\displaystyle{\displaystyle 1\leq x,x<\infty}} 
InverseJacobiNS(x, k) = int((1)/(sqrt(((t)^(2)- 1)*((t)^(2)- (k)^(2)))), t = x..infinity)
 
InverseJacobiNS[x, (k)^2] == Integrate[Divide[1,Sqrt[((t)^(2)- 1)*((t)^(2)- (k)^(2))]], {t, x, Infinity}, GenerateConditions->None]
 Failure Aborted Skipped - Because timed out Skipped - Because timed out
22.15.E22 arcds ⁑ ( x , k ) = ∫ x ∞ d t ( t 2 + k 2 ) ⁒ ( t 2 - k β€² 2 ) inverse-Jacobi-elliptic-ds π‘₯ π‘˜ superscript subscript π‘₯ 𝑑 superscript 𝑑 2 superscript π‘˜ 2 superscript 𝑑 2 superscript superscript π‘˜ β€² 2 {\displaystyle{\displaystyle\operatorname{arcds}\left(x,k\right)=\int_{x}^{% \infty}\frac{\mathrm{d}t}{\sqrt{(t^{2}+k^{2})(t^{2}-{k^{\prime}}^{2})}}}}
\aJacobielldsk@{x}{k} = \int_{x}^{\infty}\frac{\diff{t}}{\sqrt{(t^{2}+k^{2})(t^{2}-{k^{\prime}}^{2})}}		 k β€² ≀ x , x < ∞ formulae-sequence superscript π‘˜ β€² π‘₯ π‘₯ {\displaystyle{\displaystyle k^{\prime}\leq x,x<\infty}} 
InverseJacobiDS(x, k) = int((1)/(sqrt(((t)^(2)+ (k)^(2))*((t)^(2)-1 - (k)^(2)))), t = x..infinity)
 
InverseJacobiDS[x, (k)^2] == Integrate[Divide[1,Sqrt[((t)^(2)+ (k)^(2))*((t)^(2)-1 - (k)^(2))]], {t, x, Infinity}, GenerateConditions->None]
 Error Aborted - Skipped - Because timed out
22.15.E23 arccs ⁑ ( x , k ) = ∫ x ∞ d t ( 1 + t 2 ) ⁒ ( t 2 + k β€² 2 ) inverse-Jacobi-elliptic-cs π‘₯ π‘˜ superscript subscript π‘₯ 𝑑 1 superscript 𝑑 2 superscript 𝑑 2 superscript superscript π‘˜ β€² 2 {\displaystyle{\displaystyle\operatorname{arccs}\left(x,k\right)=\int_{x}^{% \infty}\frac{\mathrm{d}t}{\sqrt{(1+t^{2})(t^{2}+{k^{\prime}}^{2})}}}}
\aJacobiellcsk@{x}{k} = \int_{x}^{\infty}\frac{\diff{t}}{\sqrt{(1+t^{2})(t^{2}+{k^{\prime}}^{2})}}		 - ∞ < x , x < ∞ formulae-sequence π‘₯ π‘₯ {\displaystyle{\displaystyle-\infty<x,x<\infty}} 
InverseJacobiCS(x, k) = int((1)/(sqrt((1 + (t)^(2))*((t)^(2)+1 - (k)^(2)))), t = x..infinity)
 
InverseJacobiCS[x, (k)^2] == Integrate[Divide[1,Sqrt[(1 + (t)^(2))*((t)^(2)+1 - (k)^(2))]], {t, x, Infinity}, GenerateConditions->None]
 Error Aborted - Skipped - Because timed out
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