Elliptic Integrals - 19.30 Lengths of Plane Curves

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
19.30#Ex1 x = a ⁢ sin ⁡ ϕ 𝑥 𝑎 italic-ϕ {\displaystyle{\displaystyle x=a\sin\phi}}
x = a\sin@@{\phi}		 

x = a*sin(phi)

x == a*Sin[\[Phi]]
Failure Failure
Failed [180 / 180]
Result: 2.788470502+.5063946946*I
Test Values: {a = -3/2, phi = 1/2*3^(1/2)+1/2*I, x = 3/2}


Result: 1.788470502+.5063946946*I
Test Values: {a = -3/2, phi = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [180 / 180]
Result: Complex[2.1491827752870476, 0.34394646701016035]
Test Values: {Rule[a, -1.5], Rule[x, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[1.093555858156998, 0.6491787480429551]
Test Values: {Rule[a, -1.5], Rule[x, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.30#Ex2 y = b ⁢ cos ⁡ ϕ 𝑦 𝑏 italic-ϕ {\displaystyle{\displaystyle y=b\cos\phi}}
y = b\cos@@{\phi}		 0 ≤ ϕ , ϕ ≤ 2 ⁢ π formulae-sequence 0 italic-ϕ italic-ϕ 2 𝜋 {\displaystyle{\displaystyle 0\leq\phi,\phi\leq 2\pi}} 
y = b*cos(phi)

y == b*Cos[\[Phi]]
Failure Failure
Failed [108 / 108]
Result: -1.393894198
Test Values: {b = -3/2, phi = 3/2, y = -3/2}


Result: 1.606105802
Test Values: {b = -3/2, phi = 3/2, y = 3/2}

... skip entries to safe data
Failed [108 / 108]
Result: -1.3938941974984456
Test Values: {Rule[b, -1.5], Rule[y, -1.5], Rule[ϕ, 1.5]}


Result: -0.18362615716444086
Test Values: {Rule[b, -1.5], Rule[y, -1.5], Rule[ϕ, 0.5]}

... skip entries to safe data
19.30.E2 s = a ⁢ ∫ 0 ϕ 1 - k 2 ⁢ sin 2 ⁡ θ ⁢ d θ 𝑠 𝑎 superscript subscript 0 italic-ϕ 1 superscript 𝑘 2 2 𝜃 𝜃 {\displaystyle{\displaystyle s=a\int_{0}^{\phi}\sqrt{1-k^{2}{\sin^{2}}\theta}% \mathrm{d}\theta}}
s = a\int_{0}^{\phi}\sqrt{1-k^{2}\sin^{2}@@{\theta}}\diff{\theta}		 

s = a*int(sqrt(1 - (k)^(2)* (sin(theta))^(2)), theta = 0..phi)

s == a*Integrate[Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)], {\[Theta], 0, \[Phi]}, GenerateConditions->None]
Aborted Aborted Skipped - Because timed out Skipped - Because timed out
19.30.E3 s / a = E ⁡ ( ϕ , k ) 𝑠 𝑎 elliptic-integral-second-kind-E italic-ϕ 𝑘 {\displaystyle{\displaystyle s/a=E\left(\phi,k\right)}}
s/a = \incellintEk@{\phi}{k}		 

s/a = EllipticE(sin(phi), k)

s/a == EllipticE[\[Phi], (k)^2]
Failure Failure
Failed [300 / 300]
Result: .1410196655-.3375964631*I
Test Values: {a = -3/2, phi = 1/2*3^(1/2)+1/2*I, s = -3/2, k = 1}


Result: -.36391978e-1+.5433649104e-1*I
Test Values: {a = -3/2, phi = 1/2*3^(1/2)+1/2*I, s = -3/2, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.5672114831419685, -0.22929764467344024]
Test Values: {Rule[a, -1.5], Rule[k, 1], Rule[s, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[0.5579190406370536, -0.16535187593702125]
Test Values: {Rule[a, -1.5], Rule[k, 2], Rule[s, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.30.E3 E ⁡ ( ϕ , k ) = R F ⁡ ( c - 1 , c - k 2 , c ) - 1 3 ⁢ k 2 ⁢ R D ⁡ ( c - 1 , c - k 2 , c ) elliptic-integral-second-kind-E italic-ϕ 𝑘 Carlson-integral-RF 𝑐 1 𝑐 superscript 𝑘 2 𝑐 1 3 superscript 𝑘 2 Carlson-integral-RD 𝑐 1 𝑐 superscript 𝑘 2 𝑐 {\displaystyle{\displaystyle E\left(\phi,k\right)={R_{F}\left(c-1,c-k^{2},c% \right)-\tfrac{1}{3}k^{2}R_{D}\left(c-1,c-k^{2},c\right)}}}
\incellintEk@{\phi}{k} = {\CarlsonsymellintRF@{c-1}{c-k^{2}}{c}-\tfrac{1}{3}k^{2}\CarlsonsymellintRD@{c-1}{c-k^{2}}{c}}		 

Error

EllipticE[\[Phi], (k)^2] == EllipticF[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)]/Sqrt[c-c - 1]-Divide[1,3]*(k)^(2)* 3*(EllipticF[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)]-EllipticE[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)])/((c-c - (k)^(2))*(c-c - 1)^(1/2))
Missing Macro Error Failure Skip - symbolical successful subtest
Failed [180 / 180]
Result: Complex[3.5743811704478246, 0.7698502565730785]
Test Values: {Rule[c, -1.5], Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[3.9424508382496875, -1.017653751864599]
Test Values: {Rule[c, -1.5], Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.30#Ex3 k 2 = 1 - ( b 2 / a 2 ) superscript 𝑘 2 1 superscript 𝑏 2 superscript 𝑎 2 {\displaystyle{\displaystyle k^{2}=1-(b^{2}/a^{2})}}
k^{2} = 1-(b^{2}/a^{2})		 

(k)^(2) = 1 -((b)^(2)/(a)^(2))
 
(k)^(2) == 1 -((b)^(2)/(a)^(2))
 Skipped - no semantic math Skipped - no semantic math - -
19.30#Ex4 c = csc 2 ⁡ ϕ 𝑐 2 italic-ϕ {\displaystyle{\displaystyle c={\csc^{2}}\phi}}
c = \csc^{2}@@{\phi}		 

c = (csc(phi))^(2)

c == (Csc[\[Phi]])^(2)
Failure Failure
Failed [60 / 60]
Result: -2.359812877+.7993130071*I
Test Values: {c = -3/2, phi = 1/2*3^(1/2)+1/2*I}


Result: -1.296085040-.8173084059*I
Test Values: {c = -3/2, phi = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [60 / 60]
Result: Complex[-3.841312467237177, 3.4490957612740374]
Test Values: {Rule[c, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[0.17530792640393877, -3.4502399957777015]
Test Values: {Rule[c, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.30.E5 L ⁢ ( a , b ) = 4 ⁢ a ⁢ E ⁡ ( k ) 𝐿 𝑎 𝑏 4 𝑎 complete-elliptic-integral-second-kind-E 𝑘 {\displaystyle{\displaystyle L(a,b)=4aE\left(k\right)}}
L(a,b) = 4a\compellintEk@{k}		 

L(a , b) = 4*a*EllipticE(k)

L[a , b] == 4*a*EllipticE[(k)^2]
Failure Failure
Failed [300 / 300]
Result: (.8660254040+.5000000000*I)*(-1.500000000, -1.500000000)+6.000000000
Test Values: {L = 1/2*3^(1/2)+1/2*I, a = -3/2, b = -3/2, k = 1}


Result: (.8660254040+.5000000000*I)*(-1.500000000, -1.500000000)+2.437793319+8.063125386*I
Test Values: {L = 1/2*3^(1/2)+1/2*I, a = -3/2, b = -3/2, k = 2}

... skip entries to safe data
 Error
19.30.E5 4 ⁢ a ⁢ E ⁡ ( k ) = 8 ⁢ a ⁢ R G ⁡ ( 0 , b 2 / a 2 , 1 ) 4 𝑎 complete-elliptic-integral-second-kind-E 𝑘 8 𝑎 Carlson-integral-RG 0 superscript 𝑏 2 superscript 𝑎 2 1 {\displaystyle{\displaystyle 4aE\left(k\right)=8aR_{G}\left(0,b^{2}/a^{2},1% \right)}}
4a\compellintEk@{k} = 8a\CarlsonsymellintRG@{0}{b^{2}/a^{2}}{1}		 

Error

4*a*EllipticE[(k)^2] == 8*a*Sqrt[1-0]*(EllipticE[ArcCos[Sqrt[0/1]],(1-(b)^(2)/(a)^(2))/(1-0)]+(Cot[ArcCos[Sqrt[0/1]]])^2*EllipticF[ArcCos[Sqrt[0/1]],(1-(b)^(2)/(a)^(2))/(1-0)]+Cot[ArcCos[Sqrt[0/1]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/1]]]^2])
Missing Macro Error Failure Skip - symbolical successful subtest
Failed [108 / 108]
Result: 12.849555921538759
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[k, 1]}


Result: Complex[16.411762602778996, -8.063125388322588]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[k, 2]}

... skip entries to safe data
19.30.E5 8 ⁢ a ⁢ R G ⁡ ( 0 , b 2 / a 2 , 1 ) = 8 ⁢ R G ⁡ ( 0 , a 2 , b 2 ) 8 𝑎 Carlson-integral-RG 0 superscript 𝑏 2 superscript 𝑎 2 1 8 Carlson-integral-RG 0 superscript 𝑎 2 superscript 𝑏 2 {\displaystyle{\displaystyle 8aR_{G}\left(0,b^{2}/a^{2},1\right)=8R_{G}\left(0% ,a^{2},b^{2}\right)}}
8a\CarlsonsymellintRG@{0}{b^{2}/a^{2}}{1} = 8\CarlsonsymellintRG@{0}{a^{2}}{b^{2}}		 

Error

8*a*Sqrt[1-0]*(EllipticE[ArcCos[Sqrt[0/1]],(1-(b)^(2)/(a)^(2))/(1-0)]+(Cot[ArcCos[Sqrt[0/1]]])^2*EllipticF[ArcCos[Sqrt[0/1]],(1-(b)^(2)/(a)^(2))/(1-0)]+Cot[ArcCos[Sqrt[0/1]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/1]]]^2]) == 8*Sqrt[(b)^(2)-0]*(EllipticE[ArcCos[Sqrt[0/(b)^(2)]],((b)^(2)-(a)^(2))/((b)^(2)-0)]+(Cot[ArcCos[Sqrt[0/(b)^(2)]]])^2*EllipticF[ArcCos[Sqrt[0/(b)^(2)]],((b)^(2)-(a)^(2))/((b)^(2)-0)]+Cot[ArcCos[Sqrt[0/(b)^(2)]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/(b)^(2)]]]^2])
Missing Macro Error Failure Skip - symbolical successful subtest
Failed [18 / 36]
Result: -37.69911184307752
Test Values: {Rule[a, -1.5], Rule[b, -1.5]}


Result: -37.69911184307752
Test Values: {Rule[a, -1.5], Rule[b, 1.5]}

... skip entries to safe data
19.30.E5 8 ⁢ R G ⁡ ( 0 , a 2 , b 2 ) = 8 ⁢ a ⁢ b ⁢ R G ⁡ ( 0 , a - 2 , b - 2 ) 8 Carlson-integral-RG 0 superscript 𝑎 2 superscript 𝑏 2 8 𝑎 𝑏 Carlson-integral-RG 0 superscript 𝑎 2 superscript 𝑏 2 {\displaystyle{\displaystyle 8R_{G}\left(0,a^{2},b^{2}\right)=8abR_{G}\left(0,% a^{-2},b^{-2}\right)}}
8\CarlsonsymellintRG@{0}{a^{2}}{b^{2}} = 8ab\CarlsonsymellintRG@{0}{a^{-2}}{b^{-2}}		 

Error

8*Sqrt[(b)^(2)-0]*(EllipticE[ArcCos[Sqrt[0/(b)^(2)]],((b)^(2)-(a)^(2))/((b)^(2)-0)]+(Cot[ArcCos[Sqrt[0/(b)^(2)]]])^2*EllipticF[ArcCos[Sqrt[0/(b)^(2)]],((b)^(2)-(a)^(2))/((b)^(2)-0)]+Cot[ArcCos[Sqrt[0/(b)^(2)]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/(b)^(2)]]]^2]) == 8*a*b*Sqrt[(b)^(- 2)-0]*(EllipticE[ArcCos[Sqrt[0/(b)^(- 2)]],((b)^(- 2)-(a)^(- 2))/((b)^(- 2)-0)]+(Cot[ArcCos[Sqrt[0/(b)^(- 2)]]])^2*EllipticF[ArcCos[Sqrt[0/(b)^(- 2)]],((b)^(- 2)-(a)^(- 2))/((b)^(- 2)-0)]+Cot[ArcCos[Sqrt[0/(b)^(- 2)]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/(b)^(- 2)]]]^2])
Missing Macro Error Failure Skip - symbolical successful subtest
Failed [18 / 36]
Result: 37.69911184307752
Test Values: {Rule[a, -1.5], Rule[b, 1.5]}


Result: 26.729786441110512
Test Values: {Rule[a, -1.5], Rule[b, 0.5]}

... skip entries to safe data
19.30.E6 ∂ ⁡ s ∂ ⁡ ( 1 / k ) = a 2 - b 2 ⁢ F ⁡ ( ϕ , k ) partial-derivative 𝑠 1 𝑘 superscript 𝑎 2 superscript 𝑏 2 elliptic-integral-first-kind-F italic-ϕ 𝑘 {\displaystyle{\displaystyle\frac{\partial s}{\partial(1/k)}=\sqrt{a^{2}-b^{2}% }F\left(\phi,k\right)}}
\pderiv{s}{(1/k)} = \sqrt{a^{2}-b^{2}}\incellintFk@{\phi}{k}		 k 2 = ( a 2 - b 2 ) / ( a 2 + λ ) , c = csc 2 ⁡ ϕ formulae-sequence superscript 𝑘 2 superscript 𝑎 2 superscript 𝑏 2 superscript 𝑎 2 𝜆 𝑐 2 italic-ϕ {\displaystyle{\displaystyle k^{2}=(a^{2}-b^{2})/(a^{2}+\lambda),c={\csc^{2}}% \phi}} 
subs( temp=(1/k), diff( s, temp$(1) ) ) = sqrt((a)^(2)- (b)^(2))*EllipticF(sin(phi), k)

(D[s, {temp, 1}]/.temp-> (1/k)) == Sqrt[(a)^(2)- (b)^(2)]*EllipticF[\[Phi], (k)^2]
Failure Failure Successful [Tested: 300]
Failed [20 / 300]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[k, 1], Rule[s, -1.5], Rule[ϕ, -2]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[k, 1], Rule[s, -1.5], Rule[ϕ, 2]}

... skip entries to safe data
19.30.E6 a 2 - b 2 ⁢ F ⁡ ( ϕ , k ) = a 2 - b 2 ⁢ R F ⁡ ( c - 1 , c - k 2 , c ) superscript 𝑎 2 superscript 𝑏 2 elliptic-integral-first-kind-F italic-ϕ 𝑘 superscript 𝑎 2 superscript 𝑏 2 Carlson-integral-RF 𝑐 1 𝑐 superscript 𝑘 2 𝑐 {\displaystyle{\displaystyle\sqrt{a^{2}-b^{2}}F\left(\phi,k\right)=\sqrt{a^{2}% -b^{2}}R_{F}\left(c-1,c-k^{2},c\right)}}
\sqrt{a^{2}-b^{2}}\incellintFk@{\phi}{k} = \sqrt{a^{2}-b^{2}}\CarlsonsymellintRF@{c-1}{c-k^{2}}{c}		 k 2 = ( a 2 - b 2 ) / ( a 2 + λ ) , c = csc 2 ⁡ ϕ formulae-sequence superscript 𝑘 2 superscript 𝑎 2 superscript 𝑏 2 superscript 𝑎 2 𝜆 𝑐 2 italic-ϕ {\displaystyle{\displaystyle k^{2}=(a^{2}-b^{2})/(a^{2}+\lambda),c={\csc^{2}}% \phi}} 
sqrt((a)^(2)- (b)^(2))*EllipticF(sin(phi), k) = sqrt((a)^(2)- (b)^(2))*0.5*int(1/(sqrt(t+c - 1)*sqrt(t+c - (k)^(2))*sqrt(t+c)), t = 0..infinity)

Sqrt[(a)^(2)- (b)^(2)]*EllipticF[\[Phi], (k)^2] == Sqrt[(a)^(2)- (b)^(2)]*EllipticF[ArcCos[Sqrt[c - 1/c]],(c-c - (k)^(2))/(c-c - 1)]/Sqrt[c-c - 1]
Error Failure Skip - symbolical successful subtest Skip - No test values generated
19.30#Ex5 x = a ⁢ t + 1 𝑥 𝑎 𝑡 1 {\displaystyle{\displaystyle x=a\sqrt{t+1}}}
x = a\sqrt{t+1}		 

x = a*sqrt(t + 1)
 
x == a*Sqrt[t + 1]
 Skipped - no semantic math Skipped - no semantic math - -
19.30#Ex6 y = b ⁢ t 𝑦 𝑏 𝑡 {\displaystyle{\displaystyle y=b\sqrt{t}}}
y = b\sqrt{t}		 

y = b*sqrt(t)
 
y == b*Sqrt[t]
 Skipped - no semantic math Skipped - no semantic math - -
19.30.E8 s = 1 2 ⁢ ∫ 0 y 2 / b 2 ( a 2 + b 2 ) ⁢ t + b 2 t ⁢ ( t + 1 ) ⁢ d t 𝑠 1 2 superscript subscript 0 superscript 𝑦 2 superscript 𝑏 2 superscript 𝑎 2 superscript 𝑏 2 𝑡 superscript 𝑏 2 𝑡 𝑡 1 𝑡 {\displaystyle{\displaystyle s=\frac{1}{2}\int_{0}^{y^{2}/b^{2}}\sqrt{\frac{(a% ^{2}+b^{2})t+b^{2}}{t(t+1)}}\mathrm{d}t}}
s = \frac{1}{2}\int_{0}^{y^{2}/b^{2}}\sqrt{\frac{(a^{2}+b^{2})t+b^{2}}{t(t+1)}}\diff{t}		 

s = (1)/(2)*int(sqrt((((a)^(2)+ (b)^(2))*t + (b)^(2))/(t*(t + 1))), t = 0..(y)^(2)/(b)^(2))

s == Divide[1,2]*Integrate[Sqrt[Divide[((a)^(2)+ (b)^(2))*t + (b)^(2),t*(t + 1)]], {t, 0, (y)^(2)/(b)^(2)}, GenerateConditions->None]
Failure Aborted
Failed [300 / 300]
Result: -3.149531120
Test Values: {a = -3/2, b = -3/2, s = -3/2, y = -3/2}


Result: -3.149531120
Test Values: {a = -3/2, b = -3/2, s = -3/2, y = 3/2}

... skip entries to safe data
 Skipped - Because timed out
19.30.E9 s = 1 2 ⁢ I ⁢ ( 𝐞 1 ) 𝑠 1 2 𝐼 subscript 𝐞 1 {\displaystyle{\displaystyle s=\tfrac{1}{2}I(\mathbf{e}_{1})}}
s = \tfrac{1}{2}I(\mathbf{e}_{1})		 r = b 4 / y 2 𝑟 superscript 𝑏 4 superscript 𝑦 2 {\displaystyle{\displaystyle r=b^{4}/y^{2}}} 
s = (1)/(2)*I(e[1])

s == Divide[1,2]*I[Subscript[e, 1]]
Failure Failure
Failed [298 / 300]
Result: -1.750000000-.4330127020*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, s = -3/2, e[1] = 1/2*3^(1/2)+1/2*I}


Result: -1.066987298-.2500000002*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, s = -3/2, e[1] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [180 / 180]
Result: Complex[-1.375, -0.21650635094610968]
Test Values: {Rule[Complex[0, 1], 1], Rule[s, -1.5], Rule[Subscript[e, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-1.375, -0.21650635094610968]
Test Values: {Rule[Complex[0, 1], 2], Rule[s, -1.5], Rule[Subscript[e, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.30.E9 1 2 ⁢ I ⁢ ( 𝐞 1 ) = - 1 3 ⁢ a 2 ⁢ b 2 ⁢ R D ⁡ ( r , r + b 2 + a 2 , r + b 2 ) + y ⁢ r + b 2 + a 2 r + b 2 1 2 𝐼 subscript 𝐞 1 1 3 superscript 𝑎 2 superscript 𝑏 2 Carlson-integral-RD 𝑟 𝑟 superscript 𝑏 2 superscript 𝑎 2 𝑟 superscript 𝑏 2 𝑦 𝑟 superscript 𝑏 2 superscript 𝑎 2 𝑟 superscript 𝑏 2 {\displaystyle{\displaystyle\tfrac{1}{2}I(\mathbf{e}_{1})=-\tfrac{1}{3}a^{2}b^% {2}R_{D}\left(r,r+b^{2}+a^{2},r+b^{2}\right)+y\sqrt{\frac{r+b^{2}+a^{2}}{r+b^{% 2}}}}}
\tfrac{1}{2}I(\mathbf{e}_{1}) = -\tfrac{1}{3}a^{2}b^{2}\CarlsonsymellintRD@{r}{r+b^{2}+a^{2}}{r+b^{2}}+y\sqrt{\frac{r+b^{2}+a^{2}}{r+b^{2}}}		 r = b 4 / y 2 𝑟 superscript 𝑏 4 superscript 𝑦 2 {\displaystyle{\displaystyle r=b^{4}/y^{2}}} 
Error

Divide[1,2]*I[Subscript[e, 1]] == -Divide[1,3]*(a)^(2)* (b)^(2)* 3*(EllipticF[ArcCos[Sqrt[r/r + (b)^(2)]],(r + (b)^(2)-r + (b)^(2)+ (a)^(2))/(r + (b)^(2)-r)]-EllipticE[ArcCos[Sqrt[r/r + (b)^(2)]],(r + (b)^(2)-r + (b)^(2)+ (a)^(2))/(r + (b)^(2)-r)])/((r + (b)^(2)-r + (b)^(2)+ (a)^(2))*(r + (b)^(2)-r)^(1/2))+ y*Sqrt[Divide[r + (b)^(2)+ (a)^(2),r + (b)^(2)]]
Missing Macro Error Failure Skip - symbolical successful subtest Skip - No test values generated
19.30.E10 r 2 = 2 ⁢ a 2 ⁢ cos ⁡ ( 2 ⁢ θ ) superscript 𝑟 2 2 superscript 𝑎 2 2 𝜃 {\displaystyle{\displaystyle r^{2}=2a^{2}\cos\left(2\theta\right)}}
r^{2} = 2a^{2}\cos@{2\theta}		 0 ≤ θ , θ ≤ 2 ⁢ π formulae-sequence 0 𝜃 𝜃 2 𝜋 {\displaystyle{\displaystyle 0\leq\theta,\theta\leq 2\pi}} 
(r)^(2) = 2*(a)^(2)* cos(2*theta)

(r)^(2) == 2*(a)^(2)* Cos[2*\[Theta]]
Failure Failure
Failed [108 / 108]
Result: 6.704966234
Test Values: {a = -3/2, r = -3/2, theta = 3/2}


Result: -.181360376
Test Values: {a = -3/2, r = -3/2, theta = 1/2}

... skip entries to safe data
Failed [108 / 108]
Result: 6.704966234702004
Test Values: {Rule[a, -1.5], Rule[r, -1.5], Rule[θ, 1.5]}


Result: -0.18136037640662916
Test Values: {Rule[a, -1.5], Rule[r, -1.5], Rule[θ, 0.5]}

... skip entries to safe data
19.30.E11 s = 2 ⁢ a 2 ⁢ ∫ 0 r d t 4 ⁢ a 4 - t 4 𝑠 2 superscript 𝑎 2 superscript subscript 0 𝑟 𝑡 4 superscript 𝑎 4 superscript 𝑡 4 {\displaystyle{\displaystyle s=2a^{2}\int_{0}^{r}\frac{\mathrm{d}t}{\sqrt{4a^{% 4}-t^{4}}}}}
s = 2a^{2}\int_{0}^{r}\frac{\diff{t}}{\sqrt{4a^{4}-t^{4}}}		 q = 2 ⁢ a 2 / r 2 , 2 ⁢ a 2 / r 2 = sec ⁡ ( 2 ⁢ θ ) formulae-sequence 𝑞 2 superscript 𝑎 2 superscript 𝑟 2 2 superscript 𝑎 2 superscript 𝑟 2 2 𝜃 {\displaystyle{\displaystyle q=2a^{2}/r^{2},2a^{2}/r^{2}=\sec\left(2\theta% \right)}} 
s = 2*(a)^(2)* int((1)/(sqrt(4*(a)^(4)- (t)^(4))), t = 0..r)

s == 2*(a)^(2)* Integrate[Divide[1,Sqrt[4*(a)^(4)- (t)^(4)]], {t, 0, r}, GenerateConditions->None]
Error Failure -
Failed [208 / 216]
Result: 0.042085201578189846
Test Values: {Rule[a, -1.5], Rule[r, -1.5], Rule[s, -1.5]}


Result: 3.04208520157819
Test Values: {Rule[a, -1.5], Rule[r, -1.5], Rule[s, 1.5]}

... skip entries to safe data
19.30.E11 2 ⁢ a 2 ⁢ ∫ 0 r d t 4 ⁢ a 4 - t 4 = 2 ⁢ a 2 ⁢ R F ⁡ ( q - 1 , q , q + 1 ) 2 superscript 𝑎 2 superscript subscript 0 𝑟 𝑡 4 superscript 𝑎 4 superscript 𝑡 4 2 superscript 𝑎 2 Carlson-integral-RF 𝑞 1 𝑞 𝑞 1 {\displaystyle{\displaystyle 2a^{2}\int_{0}^{r}\frac{\mathrm{d}t}{\sqrt{4a^{4}% -t^{4}}}=\sqrt{2a^{2}}R_{F}\left(q-1,q,q+1\right)}}
2a^{2}\int_{0}^{r}\frac{\diff{t}}{\sqrt{4a^{4}-t^{4}}} = \sqrt{2a^{2}}\CarlsonsymellintRF@{q-1}{q}{q+1}		 q = 2 ⁢ a 2 / r 2 , 2 ⁢ a 2 / r 2 = sec ⁡ ( 2 ⁢ θ ) formulae-sequence 𝑞 2 superscript 𝑎 2 superscript 𝑟 2 2 superscript 𝑎 2 superscript 𝑟 2 2 𝜃 {\displaystyle{\displaystyle q=2a^{2}/r^{2},2a^{2}/r^{2}=\sec\left(2\theta% \right)}} 
2*(a)^(2)* int((1)/(sqrt(4*(a)^(4)- (t)^(4))), t = 0..r) = sqrt(2*(a)^(2))*0.5*int(1/(sqrt(t+q - 1)*sqrt(t+q)*sqrt(t+q + 1)), t = 0..infinity)

2*(a)^(2)* Integrate[Divide[1,Sqrt[4*(a)^(4)- (t)^(4)]], {t, 0, r}, GenerateConditions->None] == Sqrt[2*(a)^(2)]*EllipticF[ArcCos[Sqrt[q - 1/q + 1]],(q + 1-q)/(q + 1-q - 1)]/Sqrt[q + 1-q - 1]
Error Failure -
Failed [12 / 12]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[q, 2], Rule[r, -1.5]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[q, 2], Rule[r, 1.5]}

... skip entries to safe data
19.30.E12 s = a ⁢ F ⁡ ( ϕ , 1 / 2 ) 𝑠 𝑎 elliptic-integral-first-kind-F italic-ϕ 1 2 {\displaystyle{\displaystyle s=aF\left(\phi,1/\sqrt{2}\right)}}
s = a\incellintFk@{\phi}{1/\sqrt{2}}		 ϕ = arcsin ⁡ 2 / ( q + 1 ) , arcsin ⁡ 2 / ( q + 1 ) = arccos ⁡ ( tan ⁡ θ ) formulae-sequence italic-ϕ 2 𝑞 1 2 𝑞 1 𝜃 {\displaystyle{\displaystyle\phi=\operatorname{arcsin}\sqrt{2/(q+1)},% \operatorname{arcsin}\sqrt{2/(q+1)}=\operatorname{arccos}\left(\tan\theta% \right)}} 
s = a*EllipticF(sin(phi), 1/(sqrt(2)))
 
s == a*EllipticF[\[Phi], (1/(Sqrt[2]))^2]
 Failure Failure
Failed [300 / 300]
Result: -.201379324+.8785912788*I
Test Values: {a = -3/2, phi = 1/2*3^(1/2)+1/2*I, s = -3/2}

Result: 2.798620676+.8785912788*I
Test Values: {a = -3/2, phi = 1/2*3^(1/2)+1/2*I, s = 3/2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.8505476575870029, 0.390685462269601]
Test Values: {Rule[a, -1.5], Rule[s, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

Result: Complex[-1.859414812385125, 0.6494166239344216]
Test Values: {Rule[a, -1.5], Rule[s, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.30.E13 P = 4 ⁢ 2 ⁢ a 2 ⁢ R F ⁡ ( 0 , 1 , 2 ) 𝑃 4 2 superscript 𝑎 2 Carlson-integral-RF 0 1 2 {\displaystyle{\displaystyle P=4\sqrt{2a^{2}}R_{F}\left(0,1,2\right)}}
P = 4\sqrt{2a^{2}}\CarlsonsymellintRF@{0}{1}{2}		 

P = 4*sqrt(2*(a)^(2))*0.5*int(1/(sqrt(t+0)*sqrt(t+1)*sqrt(t+2)), t = 0..infinity)
 
P == 4*Sqrt[2*(a)^(2)]*EllipticF[ArcCos[Sqrt[0/2]],(2-1)/(2-0)]/Sqrt[2-0]
 Failure Failure
Failed [60 / 60]
Result: -10.25842266+.5000000000*I
Test Values: {P = 1/2*3^(1/2)+1/2*I, a = -3/2}

Result: -10.25842266+.5000000000*I
Test Values: {P = 1/2*3^(1/2)+1/2*I, a = 3/2}

... skip entries to safe data
Failed [60 / 60]
Result: Complex[-10.691435361916012, 0.24999999999999997]
Test Values: {Rule[a, -1.5], Rule[P, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

Result: Complex[-11.37444806380823, 0.43301270189221935]
Test Values: {Rule[a, -1.5], Rule[P, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
19.30.E13 4 ⁢ 2 ⁢ a 2 ⁢ R F ⁡ ( 0 , 1 , 2 ) = 2 ⁢ a 2 × 5.24411 51 ⁢ … 4 2 superscript 𝑎 2 Carlson-integral-RF 0 1 2 2 superscript 𝑎 2 5.24411 51 … {\displaystyle{\displaystyle 4\sqrt{2a^{2}}R_{F}\left(0,1,2\right)=\sqrt{2a^{2% }}\times 5.24411\;51\ldots}}
4\sqrt{2a^{2}}\CarlsonsymellintRF@{0}{1}{2} = \sqrt{2a^{2}}\times 5.24411\;51\ldots		 

4*sqrt(2*(a)^(2))*0.5*int(1/(sqrt(t+0)*sqrt(t+1)*sqrt(t+2)), t = 0..infinity) = sqrt(2*(a)^(2)) * 5.2441151
 
4*Sqrt[2*(a)^(2)]*EllipticF[ArcCos[Sqrt[0/2]],(2-1)/(2-0)]/Sqrt[2-0] == Sqrt[2*(a)^(2)] * 5.2441151
 Translation Error Translation Error Skip - symbolical successful subtest Skip - symbolical successful subtest
19.30.E13 2 ⁢ a 2 × 5.24411 51 ⁢ … = 4 ⁢ a ⁢ K ⁡ ( 1 / 2 ) 2 superscript 𝑎 2 5.24411 51 … 4 𝑎 complete-elliptic-integral-first-kind-K 1 2 {\displaystyle{\displaystyle\sqrt{2a^{2}}\times 5.24411\;51\ldots=4aK\left(1/% \sqrt{2}\right)}}
\sqrt{2a^{2}}\times 5.24411\;51\ldots = 4a\compellintKk@{1/\sqrt{2}}		 

sqrt(2*(a)^(2)) * 5.2441151 = 4*a*EllipticK(1/(sqrt(2)))
 
Sqrt[2*(a)^(2)] * 5.2441151 == 4*a*EllipticK[(1/(Sqrt[2]))^2]
 Translation Error Translation Error Skip - symbolical successful subtest Skip - symbolical successful subtest
19.30.E13 4 ⁢ a ⁢ K ⁡ ( 1 / 2 ) = a × 7.41629 87 ⁢ … 4 𝑎 complete-elliptic-integral-first-kind-K 1 2 𝑎 7.41629 87 … {\displaystyle{\displaystyle 4aK\left(1/\sqrt{2}\right)=a\times 7.41629\;87% \dots}}
4a\compellintKk@{1/\sqrt{2}} = a\times 7.41629\;87\dots		 

4*a*EllipticK(1/(sqrt(2))) = a * 7.4162987
 
4*a*EllipticK[(1/(Sqrt[2]))^2] == a * 7.4162987
 Translation Error Translation Error Skip - symbolical successful subtest Skip - symbolical successful subtest
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