Elliptic Integrals - 19.14 Reduction of General Elliptic Integrals

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
19.14.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{1}^{x}\frac{\diff{t}}{\sqrt{t^{3}-1}} = 3^{-1/4}\incellintFk@{\phi}{k}}
\int_{1}^{x}\frac{\diff{t}}{\sqrt{t^{3}-1}} = 3^{-1/4}\incellintFk@{\phi}{k}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \cos@@{\phi} = \dfrac{\sqrt{3}+1-x}{\sqrt{3}-1+x}, k^{2} = \dfrac{2-\sqrt{3}}{4}} 
int((1)/(sqrt((t)^(3)- 1)), t = 1..x) = (3)^(- 1/4)* EllipticF(sin(phi), k)

Integrate[Divide[1,Sqrt[(t)^(3)- 1]], {t, 1, x}, GenerateConditions->None] == (3)^(- 1/4)* EllipticF[\[Phi], (k)^2]
Failure Aborted Error Skipped - Because timed out
19.14.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{x}^{1}\frac{\diff{t}}{\sqrt{1-t^{3}}} = 3^{-1/4}\incellintFk@{\phi}{k}}
\int_{x}^{1}\frac{\diff{t}}{\sqrt{1-t^{3}}} = 3^{-1/4}\incellintFk@{\phi}{k}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \cos@@{\phi} = \dfrac{\sqrt{3}-1+x}{\sqrt{3}+1-x}, k^{2} = \dfrac{2+\sqrt{3}}{4}} 
int((1)/(sqrt(1 - (t)^(3))), t = x..1) = (3)^(- 1/4)* EllipticF(sin(phi), k)

Integrate[Divide[1,Sqrt[1 - (t)^(3)]], {t, x, 1}, GenerateConditions->None] == (3)^(- 1/4)* EllipticF[\[Phi], (k)^2]
Failure Aborted Error Skip - No test values generated
19.14.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{x}\frac{\diff{t}}{\sqrt{1+t^{4}}} = \frac{\sign@{x}}{2}\incellintFk@{\phi}{k}}
\int_{0}^{x}\frac{\diff{t}}{\sqrt{1+t^{4}}} = \frac{\sign@{x}}{2}\incellintFk@{\phi}{k}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \cos@@{\phi} = \dfrac{1-x^{2}}{1+x^{2}}, k^{2} = \dfrac{1}{2}} 
int((1)/(sqrt(1 + (t)^(4))), t = 0..x) = (signum(x))/(2)*EllipticF(sin(phi), k)

Integrate[Divide[1,Sqrt[1 + (t)^(4)]], {t, 0, x}, GenerateConditions->None] == Divide[Sign[x],2]*EllipticF[\[Phi], (k)^2]
Failure Failure Error Skip - No test values generated
19.14.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{y}^{x}\frac{\diff{t}}{\sqrt{(a_{1}+b_{1}t^{2})(a_{2}+b_{2}t^{2})}} = \frac{1}{\sqrt{\gamma-\alpha}}\incellintFk@{\phi}{k}}
\int_{y}^{x}\frac{\diff{t}}{\sqrt{(a_{1}+b_{1}t^{2})(a_{2}+b_{2}t^{2})}} = \frac{1}{\sqrt{\gamma-\alpha}}\incellintFk@{\phi}{k}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle k^{2} = \ifrac{(\gamma-\beta)}{(\gamma-\alpha)}} 
int((1)/(sqrt((a[1]+ b[1]*(t)^(2))*(a[2]+ b[2]*(t)^(2)))), t = y..x) = (1)/(sqrt(gamma - alpha))*EllipticF(sin(phi), k)

Integrate[Divide[1,Sqrt[(Subscript[a, 1]+ Subscript[b, 1]*(t)^(2))*(Subscript[a, 2]+ Subscript[b, 2]*(t)^(2))]], {t, y, x}, GenerateConditions->None] == Divide[1,Sqrt[\[Gamma]- \[Alpha]]]*EllipticF[\[Phi], (k)^2]
Skipped - Unable to analyze test case: Null Skipped - Unable to analyze test case: Null - -
19.14.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sin^{2}@@{\phi} = \frac{\gamma-\alpha}{U^{2}+\gamma}}
\sin^{2}@@{\phi} = \frac{\gamma-\alpha}{U^{2}+\gamma}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(sin(phi))^(2) = (gamma - alpha)/((U)^(2)+ gamma)

(Sin[\[Phi]])^(2) == Divide[\[Gamma]- \[Alpha],(U)^(2)+ \[Gamma]]
Failure Failure
Failed [300 / 300]
Result: 1.144207228+.1616580578*I
Test Values: {U = 1/2*3^(1/2)+1/2*I, alpha = 3/2, gamma = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I}


Result: .2329549284-1.570148532*I
Test Values: {U = 1/2*3^(1/2)+1/2*I, alpha = 3/2, gamma = 1/2*3^(1/2)+1/2*I, phi = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[1.0397570908067482, -1.0061601508735134]
Test Values: {Rule[U, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[α, 1.5], Rule[γ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[0.7911458419033055, -1.4391726141222814]
Test Values: {Rule[U, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[α, 1.5], Rule[γ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.14.E6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (x^{2}-y^{2})U = x\sqrt{(a_{1}+b_{1}y^{2})(a_{2}+b_{2}y^{2})}+y\sqrt{(a_{1}+b_{1}x^{2})(a_{2}+b_{2}x^{2})}}
(x^{2}-y^{2})U = x\sqrt{(a_{1}+b_{1}y^{2})(a_{2}+b_{2}y^{2})}+y\sqrt{(a_{1}+b_{1}x^{2})(a_{2}+b_{2}x^{2})}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
((x)^(2)- (y)^(2))*U = x*sqrt((a[1]+ b[1]*(y)^(2))*(a[2]+ b[2]*(y)^(2)))+ y*sqrt((a[1]+ b[1]*(x)^(2))*(a[2]+ b[2]*(x)^(2)))
 
((x)^(2)- (y)^(2))*U == x*Sqrt[(Subscript[a, 1]+ Subscript[b, 1]*(y)^(2))*(Subscript[a, 2]+ Subscript[b, 2]*(y)^(2))]+ y*Sqrt[(Subscript[a, 1]+ Subscript[b, 1]*(x)^(2))*(Subscript[a, 2]+ Subscript[b, 2]*(x)^(2))]
 Skipped - no semantic math Skipped - no semantic math - -
19.14.E7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sin^{2}@@{\phi} = \frac{(\gamma-\alpha)x^{2}}{a_{1}a_{2}+\gamma x^{2}}}
\sin^{2}@@{\phi} = \frac{(\gamma-\alpha)x^{2}}{a_{1}a_{2}+\gamma x^{2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(sin(phi))^(2) = ((gamma - alpha)*(x)^(2))/(a[1]*a[2]+ gamma*(x)^(2))

(Sin[\[Phi]])^(2) == Divide[(\[Gamma]- \[Alpha])*(x)^(2),Subscript[a, 1]*Subscript[a, 2]+ \[Gamma]*(x)^(2)]
Failure Failure
Failed [300 / 300]
Result: 1.560947444+.1288116535*I
Test Values: {alpha = 3/2, gamma = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, x = 3/2, a[1] = 1/2*3^(1/2)+1/2*I, a[2] = 1/2*3^(1/2)+1/2*I}


Result: 2.678639127-1.794319469*I
Test Values: {alpha = 3/2, gamma = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, x = 3/2, a[1] = 1/2*3^(1/2)+1/2*I, a[2] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[1.3471528039744003, -1.172411794219179]
Test Values: {Rule[x, 1.5], Rule[α, 1.5], Rule[γ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[a, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[a, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[1.5030688086095803, -1.7852795940180226]
Test Values: {Rule[x, 1.5], Rule[α, 1.5], Rule[γ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[a, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[a, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.14.E8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sin^{2}@@{\phi} = \frac{\gamma-\alpha}{b_{1}b_{2}y^{2}+\gamma}}
\sin^{2}@@{\phi} = \frac{\gamma-\alpha}{b_{1}b_{2}y^{2}+\gamma}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(sin(phi))^(2) = (gamma - alpha)/(b[1]*b[2]*(y)^(2)+ gamma)

(Sin[\[Phi]])^(2) == Divide[\[Gamma]- \[Alpha],Subscript[b, 1]*Subscript[b, 2]*(y)^(2)+ \[Gamma]]
Failure Failure
Failed [300 / 300]
Result: .8585159693+.3113806358*I
Test Values: {alpha = 3/2, gamma = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, y = -3/2, b[1] = 1/2*3^(1/2)+1/2*I, b[2] = 1/2*3^(1/2)+1/2*I}


Result: .2216600130+.2500138214*I
Test Values: {alpha = 3/2, gamma = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, y = -3/2, b[1] = 1/2*3^(1/2)+1/2*I, b[2] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.683185473382228, -0.7175596041712626]
Test Values: {Rule[y, -1.5], Rule[α, 1.5], Rule[γ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[b, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[b, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-0.5335538340604822, -1.7418837307419275]
Test Values: {Rule[y, -1.5], Rule[α, 1.5], Rule[γ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[b, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[b, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.14.E9 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sin^{2}@@{\phi} = \frac{(\gamma-\alpha)(x^{2}-y^{2})}{\gamma(x^{2}-y^{2})-a_{1}(a_{2}+b_{2}x^{2})}}
\sin^{2}@@{\phi} = \frac{(\gamma-\alpha)(x^{2}-y^{2})}{\gamma(x^{2}-y^{2})-a_{1}(a_{2}+b_{2}x^{2})}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(sin(phi))^(2) = ((gamma - alpha)*((x)^(2)- (y)^(2)))/(gamma*((x)^(2)- (y)^(2))- a[1]*(a[2]+ b[2]*(x)^(2)))

(Sin[\[Phi]])^(2) == Divide[(\[Gamma]- \[Alpha])*((x)^(2)- (y)^(2)),\[Gamma]*((x)^(2)- (y)^(2))- Subscript[a, 1]*(Subscript[a, 2]+ Subscript[b, 2]*(x)^(2))]
Failure Failure Manual Skip! Skipped - Because timed out
19.14.E10 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sin^{2}@@{\phi} = \frac{(\gamma-\alpha)(y^{2}-x^{2})}{\gamma(y^{2}-x^{2})-a_{1}(a_{2}+b_{2}y^{2})}}
\sin^{2}@@{\phi} = \frac{(\gamma-\alpha)(y^{2}-x^{2})}{\gamma(y^{2}-x^{2})-a_{1}(a_{2}+b_{2}y^{2})}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(sin(phi))^(2) = ((gamma - alpha)*((y)^(2)- (x)^(2)))/(gamma*((y)^(2)- (x)^(2))- a[1]*(a[2]+ b[2]*(y)^(2)))

(Sin[\[Phi]])^(2) == Divide[(\[Gamma]- \[Alpha])*((y)^(2)- (x)^(2)),\[Gamma]*((y)^(2)- (x)^(2))- Subscript[a, 1]*(Subscript[a, 2]+ Subscript[b, 2]*(y)^(2))]
Failure Failure Manual Skip! Skipped - Because timed out
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