3.28.97 \(\int \frac {-b^{12}+a^{12} x^{12}}{\sqrt {-b^4+a^4 x^4} (b^{12}+a^{12} x^{12})} \, dx\)

Optimal. Leaf size=270 \[ -\frac {\tan ^{-1}\left (\frac {-\frac {a^3 x^4}{2 b}+\frac {b^3}{2 a}+a b x^2}{x \sqrt {a^4 x^4-b^4}}\right )}{12 a b}+\frac {\tanh ^{-1}\left (\frac {-\frac {a^3 x^4}{2 b}+\frac {b^3}{2 a}-a b x^2}{x \sqrt {a^4 x^4-b^4}}\right )}{12 a b}+\frac {\tanh ^{-1}\left (\frac {-\frac {a^3 x^4}{\sqrt {2} b}+\frac {b^3}{\sqrt {2} a}-\frac {a b x^2}{\sqrt {2}}}{x \sqrt {a^4 x^4-b^4}}\right )}{3 \sqrt {2} a b}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} a b x \sqrt {a^4 x^4-b^4}}{-a^4 x^4+a^2 b^2 x^2+b^4}\right )}{3 \sqrt {2} a b} \]

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Rubi [C]  time = 9.29, antiderivative size = 1043, normalized size of antiderivative = 3.86, number of steps used = 207, number of rules used = 18, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {1586, 6725, 1729, 1209, 1201, 224, 221, 1200, 1199, 424, 1219, 1218, 1248, 735, 844, 217, 206, 725} \begin {gather*} -\frac {(-1)^{2/3} \left (a^2-(-1)^{2/3} \sqrt [6]{-a^{12}}\right ) \left (\sqrt [3]{-1} a^8-(-1)^{2/3} \sqrt [3]{-a^{12}} a^4-\left (-a^{12}\right )^{2/3}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{6 a^{11} \sqrt {a^4 x^4-b^4}}-\frac {(-1)^{2/3} \left (a^2+(-1)^{2/3} \sqrt [6]{-a^{12}}\right ) \left (\sqrt [3]{-1} a^8-(-1)^{2/3} \sqrt [3]{-a^{12}} a^4-\left (-a^{12}\right )^{2/3}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{6 a^{11} \sqrt {a^4 x^4-b^4}}+\frac {\left (a^2-\sqrt [6]{-a^{12}}\right ) \left (a^8+\sqrt [3]{-a^{12}} a^4+\left (-a^{12}\right )^{2/3}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{6 a^{11} \sqrt {a^4 x^4-b^4}}+\frac {\left (a^2+\sqrt [6]{-a^{12}}\right ) \left (a^8+\sqrt [3]{-a^{12}} a^4+\left (-a^{12}\right )^{2/3}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{6 a^{11} \sqrt {a^4 x^4-b^4}}-\frac {\sqrt [3]{-1} \left (a^2-\sqrt [3]{-1} \sqrt [6]{-a^{12}}\right ) \left ((-1)^{2/3} a^8+\left (-a^{12}\right )^{2/3}+\frac {\sqrt [3]{-1} \left (-a^{12}\right )^{4/3}}{a^8}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{6 a^{11} \sqrt {a^4 x^4-b^4}}-\frac {\sqrt [3]{-1} \left (a^2+\sqrt [3]{-1} \sqrt [6]{-a^{12}}\right ) \left ((-1)^{2/3} a^8+\left (-a^{12}\right )^{2/3}+\frac {\sqrt [3]{-1} \left (-a^{12}\right )^{4/3}}{a^8}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{6 a^{11} \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {a^{10}}{\left (-a^{12}\right )^{5/6}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{3 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt [3]{-1} a^{10}}{\left (-a^{12}\right )^{5/6}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{3 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {(-1)^{2/3} a^{10}}{\left (-a^{12}\right )^{5/6}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{3 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt [6]{-a^{12}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{3 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt [3]{-1} \sqrt [6]{-a^{12}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{3 a \sqrt {a^4 x^4-b^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {(-1)^{2/3} \sqrt [6]{-a^{12}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{3 a \sqrt {a^4 x^4-b^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-b^12 + a^12*x^12)/(Sqrt[-b^4 + a^4*x^4]*(b^12 + a^12*x^12)),x]

[Out]

-1/6*((-1)^(2/3)*(a^2 - (-1)^(2/3)*(-a^12)^(1/6))*((-1)^(1/3)*a^8 - (-1)^(2/3)*a^4*(-a^12)^(1/3) - (-a^12)^(2/
3))*b*Sqrt[1 - (a^4*x^4)/b^4]*EllipticF[ArcSin[(a*x)/b], -1])/(a^11*Sqrt[-b^4 + a^4*x^4]) - ((-1)^(2/3)*(a^2 +
(-1)^(2/3)*(-a^12)^(1/6))*((-1)^(1/3)*a^8 - (-1)^(2/3)*a^4*(-a^12)^(1/3) - (-a^12)^(2/3))*b*Sqrt[1 - (a^4*x^4
)/b^4]*EllipticF[ArcSin[(a*x)/b], -1])/(6*a^11*Sqrt[-b^4 + a^4*x^4]) + ((a^2 - (-a^12)^(1/6))*(a^8 + a^4*(-a^1
2)^(1/3) + (-a^12)^(2/3))*b*Sqrt[1 - (a^4*x^4)/b^4]*EllipticF[ArcSin[(a*x)/b], -1])/(6*a^11*Sqrt[-b^4 + a^4*x^
4]) + ((a^2 + (-a^12)^(1/6))*(a^8 + a^4*(-a^12)^(1/3) + (-a^12)^(2/3))*b*Sqrt[1 - (a^4*x^4)/b^4]*EllipticF[Arc
Sin[(a*x)/b], -1])/(6*a^11*Sqrt[-b^4 + a^4*x^4]) - ((-1)^(1/3)*(a^2 - (-1)^(1/3)*(-a^12)^(1/6))*((-1)^(2/3)*a^
8 + (-a^12)^(2/3) + ((-1)^(1/3)*(-a^12)^(4/3))/a^8)*b*Sqrt[1 - (a^4*x^4)/b^4]*EllipticF[ArcSin[(a*x)/b], -1])/
(6*a^11*Sqrt[-b^4 + a^4*x^4]) - ((-1)^(1/3)*(a^2 + (-1)^(1/3)*(-a^12)^(1/6))*((-1)^(2/3)*a^8 + (-a^12)^(2/3) +
((-1)^(1/3)*(-a^12)^(4/3))/a^8)*b*Sqrt[1 - (a^4*x^4)/b^4]*EllipticF[ArcSin[(a*x)/b], -1])/(6*a^11*Sqrt[-b^4 +
a^4*x^4]) - (b*Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[a^10/(-a^12)^(5/6), ArcSin[(a*x)/b], -1])/(3*a*Sqrt[-b^4 +
a^4*x^4]) - (b*Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[((-1)^(1/3)*a^10)/(-a^12)^(5/6), ArcSin[(a*x)/b], -1])/(3*a*
Sqrt[-b^4 + a^4*x^4]) - (b*Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[((-1)^(2/3)*a^10)/(-a^12)^(5/6), ArcSin[(a*x)/b]
, -1])/(3*a*Sqrt[-b^4 + a^4*x^4]) - (b*Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[(-a^12)^(1/6)/a^2, ArcSin[(a*x)/b],
-1])/(3*a*Sqrt[-b^4 + a^4*x^4]) - (b*Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[((-1)^(1/3)*(-a^12)^(1/6))/a^2, ArcSin
[(a*x)/b], -1])/(3*a*Sqrt[-b^4 + a^4*x^4]) - (b*Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[((-1)^(2/3)*(-a^12)^(1/6))/
a^2, ArcSin[(a*x)/b], -1])/(3*a*Sqrt[-b^4 + a^4*x^4])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[(Rt[-b, 4]*x)/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + (b*x^4)/a]/Sqrt[a + b*x^4], Int[1/Sqrt[1 + (b*x^4)
/a], x], x] /; FreeQ[{a, b}, x] && NegQ[b/a] &&  !GtQ[a, 0]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 735

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(a + c*x^2)^p)/(
e*(m + 2*p + 1)), x] + Dist[(2*p)/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[a*e - c*d*x, x]*(a + c*x^2)^(p - 1),
x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !Ration
alQ[m] || LtQ[m, 1]) &&  !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 1199

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Dist[d/Sqrt[a], Int[Sqrt[1 + (e*x^2)/d]/Sqrt
[1 - (e*x^2)/d], x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] && GtQ[a, 0]

Rule 1200

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + (c*x^4)/a]/Sqrt[a + c*x^4], In
t[(d + e*x^2)/Sqrt[1 + (c*x^4)/a], x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] &&
!GtQ[a, 0]

Rule 1201

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-(c/a), 2]}, Dist[(d*q - e)/q,
Int[1/Sqrt[a + c*x^4], x], x] + Dist[e/q, Int[(1 + q*x^2)/Sqrt[a + c*x^4], x], x]] /; FreeQ[{a, c, d, e}, x] &
& NegQ[c/a] && NeQ[c*d^2 + a*e^2, 0]

Rule 1209

Int[((a_) + (c_.)*(x_)^4)^(p_)/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Dist[(e^2)^(-1), Int[(c*d - c*e*x^2)*(a +
c*x^4)^(p - 1), x], x] + Dist[(c*d^2 + a*e^2)/e^2, Int[(a + c*x^4)^(p - 1)/(d + e*x^2), x], x] /; FreeQ[{a, c,
d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p + 1/2, 0]

Rule 1218

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[-(c/a), 4]}, Simp[(1*Ellipt
icPi[-(e/(d*q^2)), ArcSin[q*x], -1])/(d*Sqrt[a]*q), x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]

Rule 1219

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[Sqrt[1 + (c*x^4)/a]/Sqrt[a + c*x^4]
, Int[1/((d + e*x^2)*Sqrt[1 + (c*x^4)/a]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] &&  !GtQ[a, 0]

Rule 1248

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 1729

Int[((a_) + (c_.)*(x_)^4)^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Dist[d, Int[(a + c*x^4)^p/(d^2 - e^2*x^2), x
], x] - Dist[e, Int[(x*(a + c*x^4)^p)/(d^2 - e^2*x^2), x], x] /; FreeQ[{a, c, d, e}, x] && IntegerQ[p + 1/2]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {-b^{12}+a^{12} x^{12}}{\sqrt {-b^4+a^4 x^4} \left (b^{12}+a^{12} x^{12}\right )} \, dx &=\int \frac {\sqrt {-b^4+a^4 x^4} \left (b^8+a^4 b^4 x^4+a^8 x^8\right )}{b^{12}+a^{12} x^{12}} \, dx\\ &=\int \left (\frac {\left (b^9+\frac {a^8 b^9}{\left (-a^{12}\right )^{2/3}}+\frac {a^4 b^9}{\sqrt [3]{-a^{12}}}\right ) \sqrt {-b^4+a^4 x^4}}{12 b^{12} \left (b-\sqrt [12]{-a^{12}} x\right )}+\frac {\left (b^9+\frac {a^8 b^9}{\left (-a^{12}\right )^{2/3}}+\frac {a^4 b^9}{\sqrt [3]{-a^{12}}}\right ) \sqrt {-b^4+a^4 x^4}}{12 b^{12} \left (b-i \sqrt [12]{-a^{12}} x\right )}+\frac {\left (b^9+\frac {a^8 b^9}{\left (-a^{12}\right )^{2/3}}+\frac {a^4 b^9}{\sqrt [3]{-a^{12}}}\right ) \sqrt {-b^4+a^4 x^4}}{12 b^{12} \left (b+i \sqrt [12]{-a^{12}} x\right )}+\frac {\left (b^9+\frac {a^8 b^9}{\left (-a^{12}\right )^{2/3}}+\frac {a^4 b^9}{\sqrt [3]{-a^{12}}}\right ) \sqrt {-b^4+a^4 x^4}}{12 b^{12} \left (b+\sqrt [12]{-a^{12}} x\right )}+\frac {\left (b^9+\frac {(-1)^{2/3} a^8 b^9}{\left (-a^{12}\right )^{2/3}}-\frac {\sqrt [3]{-1} a^4 b^9}{\sqrt [3]{-a^{12}}}\right ) \sqrt {-b^4+a^4 x^4}}{12 b^{12} \left (b-\sqrt [6]{-1} \sqrt [12]{-a^{12}} x\right )}+\frac {\left (b^9+\frac {(-1)^{2/3} a^8 b^9}{\left (-a^{12}\right )^{2/3}}-\frac {\sqrt [3]{-1} a^4 b^9}{\sqrt [3]{-a^{12}}}\right ) \sqrt {-b^4+a^4 x^4}}{12 b^{12} \left (b+\sqrt [6]{-1} \sqrt [12]{-a^{12}} x\right )}+\frac {\left (b^9-\frac {\sqrt [3]{-1} a^8 b^9}{\left (-a^{12}\right )^{2/3}}+\frac {(-1)^{2/3} a^4 b^9}{\sqrt [3]{-a^{12}}}\right ) \sqrt {-b^4+a^4 x^4}}{12 b^{12} \left (b-\sqrt [3]{-1} \sqrt [12]{-a^{12}} x\right )}+\frac {\left (b^9-\frac {\sqrt [3]{-1} a^8 b^9}{\left (-a^{12}\right )^{2/3}}+\frac {(-1)^{2/3} a^4 b^9}{\sqrt [3]{-a^{12}}}\right ) \sqrt {-b^4+a^4 x^4}}{12 b^{12} \left (b+\sqrt [3]{-1} \sqrt [12]{-a^{12}} x\right )}+\frac {\left (b^9+\frac {(-1)^{2/3} a^8 b^9}{\left (-a^{12}\right )^{2/3}}-\frac {\sqrt [3]{-1} a^4 b^9}{\sqrt [3]{-a^{12}}}\right ) \sqrt {-b^4+a^4 x^4}}{12 b^{12} \left (b-(-1)^{2/3} \sqrt [12]{-a^{12}} x\right )}+\frac {\left (b^9+\frac {(-1)^{2/3} a^8 b^9}{\left (-a^{12}\right )^{2/3}}-\frac {\sqrt [3]{-1} a^4 b^9}{\sqrt [3]{-a^{12}}}\right ) \sqrt {-b^4+a^4 x^4}}{12 b^{12} \left (b+(-1)^{2/3} \sqrt [12]{-a^{12}} x\right )}+\frac {\left (b^9-\frac {\sqrt [3]{-1} a^8 b^9}{\left (-a^{12}\right )^{2/3}}+\frac {(-1)^{2/3} a^4 b^9}{\sqrt [3]{-a^{12}}}\right ) \sqrt {-b^4+a^4 x^4}}{12 b^{12} \left (b-(-1)^{5/6} \sqrt [12]{-a^{12}} x\right )}+\frac {\left (b^9-\frac {\sqrt [3]{-1} a^8 b^9}{\left (-a^{12}\right )^{2/3}}+\frac {(-1)^{2/3} a^4 b^9}{\sqrt [3]{-a^{12}}}\right ) \sqrt {-b^4+a^4 x^4}}{12 b^{12} \left (b+(-1)^{5/6} \sqrt [12]{-a^{12}} x\right )}\right ) \, dx\\ &=\frac {\left (1+\frac {a^8}{\left (-a^{12}\right )^{2/3}}+\frac {a^4}{\sqrt [3]{-a^{12}}}\right ) \int \frac {\sqrt {-b^4+a^4 x^4}}{b-\sqrt [12]{-a^{12}} x} \, dx}{12 b^3}+\frac {\left (1+\frac {a^8}{\left (-a^{12}\right )^{2/3}}+\frac {a^4}{\sqrt [3]{-a^{12}}}\right ) \int \frac {\sqrt {-b^4+a^4 x^4}}{b-i \sqrt [12]{-a^{12}} x} \, dx}{12 b^3}+\frac {\left (1+\frac {a^8}{\left (-a^{12}\right )^{2/3}}+\frac {a^4}{\sqrt [3]{-a^{12}}}\right ) \int \frac {\sqrt {-b^4+a^4 x^4}}{b+i \sqrt [12]{-a^{12}} x} \, dx}{12 b^3}+\frac {\left (1+\frac {a^8}{\left (-a^{12}\right )^{2/3}}+\frac {a^4}{\sqrt [3]{-a^{12}}}\right ) \int \frac {\sqrt {-b^4+a^4 x^4}}{b+\sqrt [12]{-a^{12}} x} \, dx}{12 b^3}+\frac {\left (1+\frac {(-1)^{2/3} a^4}{\sqrt [3]{-a^{12}}}+\frac {\sqrt [3]{-1} \sqrt [3]{-a^{12}}}{a^4}\right ) \int \frac {\sqrt {-b^4+a^4 x^4}}{b-\sqrt [3]{-1} \sqrt [12]{-a^{12}} x} \, dx}{12 b^3}+\frac {\left (1+\frac {(-1)^{2/3} a^4}{\sqrt [3]{-a^{12}}}+\frac {\sqrt [3]{-1} \sqrt [3]{-a^{12}}}{a^4}\right ) \int \frac {\sqrt {-b^4+a^4 x^4}}{b+\sqrt [3]{-1} \sqrt [12]{-a^{12}} x} \, dx}{12 b^3}+\frac {\left (1+\frac {(-1)^{2/3} a^4}{\sqrt [3]{-a^{12}}}+\frac {\sqrt [3]{-1} \sqrt [3]{-a^{12}}}{a^4}\right ) \int \frac {\sqrt {-b^4+a^4 x^4}}{b-(-1)^{5/6} \sqrt [12]{-a^{12}} x} \, dx}{12 b^3}+\frac {\left (1+\frac {(-1)^{2/3} a^4}{\sqrt [3]{-a^{12}}}+\frac {\sqrt [3]{-1} \sqrt [3]{-a^{12}}}{a^4}\right ) \int \frac {\sqrt {-b^4+a^4 x^4}}{b+(-1)^{5/6} \sqrt [12]{-a^{12}} x} \, dx}{12 b^3}+\frac {\left (1+\frac {(-1)^{2/3} a^8}{\left (-a^{12}\right )^{2/3}}+\frac {\sqrt [3]{-1} \left (-a^{12}\right )^{2/3}}{a^8}\right ) \int \frac {\sqrt {-b^4+a^4 x^4}}{b-\sqrt [6]{-1} \sqrt [12]{-a^{12}} x} \, dx}{12 b^3}+\frac {\left (1+\frac {(-1)^{2/3} a^8}{\left (-a^{12}\right )^{2/3}}+\frac {\sqrt [3]{-1} \left (-a^{12}\right )^{2/3}}{a^8}\right ) \int \frac {\sqrt {-b^4+a^4 x^4}}{b+\sqrt [6]{-1} \sqrt [12]{-a^{12}} x} \, dx}{12 b^3}+\frac {\left (1+\frac {(-1)^{2/3} a^8}{\left (-a^{12}\right )^{2/3}}+\frac {\sqrt [3]{-1} \left (-a^{12}\right )^{2/3}}{a^8}\right ) \int \frac {\sqrt {-b^4+a^4 x^4}}{b-(-1)^{2/3} \sqrt [12]{-a^{12}} x} \, dx}{12 b^3}+\frac {\left (1+\frac {(-1)^{2/3} a^8}{\left (-a^{12}\right )^{2/3}}+\frac {\sqrt [3]{-1} \left (-a^{12}\right )^{2/3}}{a^8}\right ) \int \frac {\sqrt {-b^4+a^4 x^4}}{b+(-1)^{2/3} \sqrt [12]{-a^{12}} x} \, dx}{12 b^3}\\ &=2 \frac {\left (1+\frac {a^8}{\left (-a^{12}\right )^{2/3}}+\frac {a^4}{\sqrt [3]{-a^{12}}}\right ) \int \frac {\sqrt {-b^4+a^4 x^4}}{b^2-\sqrt [6]{-a^{12}} x^2} \, dx}{12 b^2}+2 \frac {\left (1+\frac {a^8}{\left (-a^{12}\right )^{2/3}}+\frac {a^4}{\sqrt [3]{-a^{12}}}\right ) \int \frac {\sqrt {-b^4+a^4 x^4}}{b^2+\sqrt [6]{-a^{12}} x^2} \, dx}{12 b^2}+2 \frac {\left (1+\frac {(-1)^{2/3} a^4}{\sqrt [3]{-a^{12}}}+\frac {\sqrt [3]{-1} \sqrt [3]{-a^{12}}}{a^4}\right ) \int \frac {\sqrt {-b^4+a^4 x^4}}{b^2-(-1)^{2/3} \sqrt [6]{-a^{12}} x^2} \, dx}{12 b^2}+2 \frac {\left (1+\frac {(-1)^{2/3} a^4}{\sqrt [3]{-a^{12}}}+\frac {\sqrt [3]{-1} \sqrt [3]{-a^{12}}}{a^4}\right ) \int \frac {\sqrt {-b^4+a^4 x^4}}{b^2+(-1)^{2/3} \sqrt [6]{-a^{12}} x^2} \, dx}{12 b^2}+2 \frac {\left (1+\frac {(-1)^{2/3} a^8}{\left (-a^{12}\right )^{2/3}}+\frac {\sqrt [3]{-1} \left (-a^{12}\right )^{2/3}}{a^8}\right ) \int \frac {\sqrt {-b^4+a^4 x^4}}{b^2-\sqrt [3]{-1} \sqrt [6]{-a^{12}} x^2} \, dx}{12 b^2}+2 \frac {\left (1+\frac {(-1)^{2/3} a^8}{\left (-a^{12}\right )^{2/3}}+\frac {\sqrt [3]{-1} \left (-a^{12}\right )^{2/3}}{a^8}\right ) \int \frac {\sqrt {-b^4+a^4 x^4}}{b^2+\sqrt [3]{-1} \sqrt [6]{-a^{12}} x^2} \, dx}{12 b^2}\\ &=2 \left (\frac {\left (a^8+a^4 \sqrt [3]{-a^{12}}+\left (-a^{12}\right )^{2/3}\right ) \int \frac {a^4 b^2+a^4 \sqrt [6]{-a^{12}} x^2}{\sqrt {-b^4+a^4 x^4}} \, dx}{12 a^{12} b^2}-\frac {1}{6} b^2 \int \frac {1}{\left (b^2-\sqrt [6]{-a^{12}} x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx\right )+2 \left (\frac {\left (a^8+a^4 \sqrt [3]{-a^{12}}+\left (-a^{12}\right )^{2/3}\right ) \int \frac {a^4 b^2-a^4 \sqrt [6]{-a^{12}} x^2}{\sqrt {-b^4+a^4 x^4}} \, dx}{12 a^{12} b^2}-\frac {1}{6} b^2 \int \frac {1}{\left (b^2+\sqrt [6]{-a^{12}} x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx\right )+2 \left (\frac {\left (a^8+(-1)^{2/3} a^4 \sqrt [3]{-a^{12}}-\sqrt [3]{-1} \left (-a^{12}\right )^{2/3}\right ) \int \frac {a^4 b^2+\sqrt [3]{-1} a^4 \sqrt [6]{-a^{12}} x^2}{\sqrt {-b^4+a^4 x^4}} \, dx}{12 a^{12} b^2}-\frac {1}{6} b^2 \int \frac {1}{\left (b^2-\sqrt [3]{-1} \sqrt [6]{-a^{12}} x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx\right )+2 \left (\frac {\left (a^8+(-1)^{2/3} a^4 \sqrt [3]{-a^{12}}-\sqrt [3]{-1} \left (-a^{12}\right )^{2/3}\right ) \int \frac {a^4 b^2-\sqrt [3]{-1} a^4 \sqrt [6]{-a^{12}} x^2}{\sqrt {-b^4+a^4 x^4}} \, dx}{12 a^{12} b^2}-\frac {1}{6} b^2 \int \frac {1}{\left (b^2+\sqrt [3]{-1} \sqrt [6]{-a^{12}} x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx\right )+2 \left (-\frac {\left ((-1)^{2/3} \left (1+\frac {(-1)^{2/3} a^4}{\sqrt [3]{-a^{12}}}+\frac {\sqrt [3]{-1} \sqrt [3]{-a^{12}}}{a^4}\right )\right ) \int \frac {a^4 b^2+(-1)^{2/3} a^4 \sqrt [6]{-a^{12}} x^2}{\sqrt {-b^4+a^4 x^4}} \, dx}{12 \sqrt [3]{-a^{12}} b^2}-\frac {1}{6} b^2 \int \frac {1}{\left (b^2-(-1)^{2/3} \sqrt [6]{-a^{12}} x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx\right )+2 \left (-\frac {\left ((-1)^{2/3} \left (1+\frac {(-1)^{2/3} a^4}{\sqrt [3]{-a^{12}}}+\frac {\sqrt [3]{-1} \sqrt [3]{-a^{12}}}{a^4}\right )\right ) \int \frac {a^4 b^2-(-1)^{2/3} a^4 \sqrt [6]{-a^{12}} x^2}{\sqrt {-b^4+a^4 x^4}} \, dx}{12 \sqrt [3]{-a^{12}} b^2}-\frac {1}{6} b^2 \int \frac {1}{\left (b^2+(-1)^{2/3} \sqrt [6]{-a^{12}} x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx\right )\\ &=2 \left (\frac {\left (\sqrt [6]{-a^{12}} \left (a^8+a^4 \sqrt [3]{-a^{12}}+\left (-a^{12}\right )^{2/3}\right )\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\sqrt {-b^4+a^4 x^4}} \, dx}{12 a^{10}}+\frac {\left (\left (a^2-\sqrt [6]{-a^{12}}\right ) \left (a^8+a^4 \sqrt [3]{-a^{12}}+\left (-a^{12}\right )^{2/3}\right )\right ) \int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx}{12 a^{10}}-\frac {\left (b^2 \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1}{\left (b^2-\sqrt [6]{-a^{12}} x^2\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{6 \sqrt {-b^4+a^4 x^4}}\right )+2 \left (-\frac {\left (\sqrt [6]{-a^{12}} \left (a^8+a^4 \sqrt [3]{-a^{12}}+\left (-a^{12}\right )^{2/3}\right )\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\sqrt {-b^4+a^4 x^4}} \, dx}{12 a^{10}}+\frac {\left (\left (a^2+\sqrt [6]{-a^{12}}\right ) \left (a^8+a^4 \sqrt [3]{-a^{12}}+\left (-a^{12}\right )^{2/3}\right )\right ) \int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx}{12 a^{10}}-\frac {\left (b^2 \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1}{\left (b^2+\sqrt [6]{-a^{12}} x^2\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{6 \sqrt {-b^4+a^4 x^4}}\right )+2 \left (\frac {\left (\sqrt [3]{-1} \sqrt [6]{-a^{12}} \left (a^8+(-1)^{2/3} a^4 \sqrt [3]{-a^{12}}-\sqrt [3]{-1} \left (-a^{12}\right )^{2/3}\right )\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\sqrt {-b^4+a^4 x^4}} \, dx}{12 a^{10}}+\frac {\left (\left (a^2-\sqrt [3]{-1} \sqrt [6]{-a^{12}}\right ) \left (a^8+(-1)^{2/3} a^4 \sqrt [3]{-a^{12}}-\sqrt [3]{-1} \left (-a^{12}\right )^{2/3}\right )\right ) \int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx}{12 a^{10}}-\frac {\left (b^2 \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1}{\left (b^2-\sqrt [3]{-1} \sqrt [6]{-a^{12}} x^2\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{6 \sqrt {-b^4+a^4 x^4}}\right )+2 \left (-\frac {\left (\sqrt [3]{-1} \sqrt [6]{-a^{12}} \left (a^8+(-1)^{2/3} a^4 \sqrt [3]{-a^{12}}-\sqrt [3]{-1} \left (-a^{12}\right )^{2/3}\right )\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\sqrt {-b^4+a^4 x^4}} \, dx}{12 a^{10}}+\frac {\left (\left (a^2+\sqrt [3]{-1} \sqrt [6]{-a^{12}}\right ) \left (a^8+(-1)^{2/3} a^4 \sqrt [3]{-a^{12}}-\sqrt [3]{-1} \left (-a^{12}\right )^{2/3}\right )\right ) \int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx}{12 a^{10}}-\frac {\left (b^2 \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1}{\left (b^2+\sqrt [3]{-1} \sqrt [6]{-a^{12}} x^2\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{6 \sqrt {-b^4+a^4 x^4}}\right )+2 \left (\frac {\left (\sqrt [3]{-1} a^2 \left (1+\frac {(-1)^{2/3} a^4}{\sqrt [3]{-a^{12}}}+\frac {\sqrt [3]{-1} \sqrt [3]{-a^{12}}}{a^4}\right )\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\sqrt {-b^4+a^4 x^4}} \, dx}{12 \sqrt [6]{-a^{12}}}-\frac {\left ((-1)^{2/3} a^2 \left (a^2-(-1)^{2/3} \sqrt [6]{-a^{12}}\right ) \left (1+\frac {(-1)^{2/3} a^4}{\sqrt [3]{-a^{12}}}+\frac {\sqrt [3]{-1} \sqrt [3]{-a^{12}}}{a^4}\right )\right ) \int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx}{12 \sqrt [3]{-a^{12}}}-\frac {\left (b^2 \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1}{\left (b^2-(-1)^{2/3} \sqrt [6]{-a^{12}} x^2\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{6 \sqrt {-b^4+a^4 x^4}}\right )+2 \left (-\frac {\left (\sqrt [3]{-1} a^2 \left (1+\frac {(-1)^{2/3} a^4}{\sqrt [3]{-a^{12}}}+\frac {\sqrt [3]{-1} \sqrt [3]{-a^{12}}}{a^4}\right )\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\sqrt {-b^4+a^4 x^4}} \, dx}{12 \sqrt [6]{-a^{12}}}-\frac {\left ((-1)^{2/3} \left (a^2+(-1)^{2/3} \sqrt [6]{-a^{12}}\right ) \left ((-1)^{2/3} a^8+a^4 \sqrt [3]{-a^{12}}+\sqrt [3]{-1} \left (-a^{12}\right )^{2/3}\right )\right ) \int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx}{12 a^2 \left (-a^{12}\right )^{2/3}}-\frac {\left (b^2 \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1}{\left (b^2+(-1)^{2/3} \sqrt [6]{-a^{12}} x^2\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{6 \sqrt {-b^4+a^4 x^4}}\right )\\ &=2 \left (-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {(-1)^{2/3} \sqrt [6]{-a^{12}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{6 a \sqrt {-b^4+a^4 x^4}}+\frac {\left (\sqrt [3]{-1} a^2 \left (1+\frac {(-1)^{2/3} a^4}{\sqrt [3]{-a^{12}}}+\frac {\sqrt [3]{-1} \sqrt [3]{-a^{12}}}{a^4}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{12 \sqrt [6]{-a^{12}} \sqrt {-b^4+a^4 x^4}}-\frac {\left ((-1)^{2/3} a^2 \left (a^2-(-1)^{2/3} \sqrt [6]{-a^{12}}\right ) \left (1+\frac {(-1)^{2/3} a^4}{\sqrt [3]{-a^{12}}}+\frac {\sqrt [3]{-1} \sqrt [3]{-a^{12}}}{a^4}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{12 \sqrt [3]{-a^{12}} \sqrt {-b^4+a^4 x^4}}\right )+2 \left (-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt [6]{-a^{12}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{6 a \sqrt {-b^4+a^4 x^4}}+\frac {\left (\sqrt [6]{-a^{12}} \left (a^8+a^4 \sqrt [3]{-a^{12}}+\left (-a^{12}\right )^{2/3}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{12 a^{10} \sqrt {-b^4+a^4 x^4}}+\frac {\left (\left (a^2-\sqrt [6]{-a^{12}}\right ) \left (a^8+a^4 \sqrt [3]{-a^{12}}+\left (-a^{12}\right )^{2/3}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{12 a^{10} \sqrt {-b^4+a^4 x^4}}\right )+2 \left (-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {a^{10}}{\left (-a^{12}\right )^{5/6}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{6 a \sqrt {-b^4+a^4 x^4}}-\frac {\left (\sqrt [6]{-a^{12}} \left (a^8+a^4 \sqrt [3]{-a^{12}}+\left (-a^{12}\right )^{2/3}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{12 a^{10} \sqrt {-b^4+a^4 x^4}}+\frac {\left (\left (a^2+\sqrt [6]{-a^{12}}\right ) \left (a^8+a^4 \sqrt [3]{-a^{12}}+\left (-a^{12}\right )^{2/3}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{12 a^{10} \sqrt {-b^4+a^4 x^4}}\right )+2 \left (-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt [3]{-1} \sqrt [6]{-a^{12}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{6 a \sqrt {-b^4+a^4 x^4}}+\frac {\left (\sqrt [3]{-1} \sqrt [6]{-a^{12}} \left (a^8+(-1)^{2/3} a^4 \sqrt [3]{-a^{12}}-\sqrt [3]{-1} \left (-a^{12}\right )^{2/3}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{12 a^{10} \sqrt {-b^4+a^4 x^4}}+\frac {\left (\left (a^2-\sqrt [3]{-1} \sqrt [6]{-a^{12}}\right ) \left (a^8+(-1)^{2/3} a^4 \sqrt [3]{-a^{12}}-\sqrt [3]{-1} \left (-a^{12}\right )^{2/3}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{12 a^{10} \sqrt {-b^4+a^4 x^4}}\right )+2 \left (-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt [3]{-1} a^{10}}{\left (-a^{12}\right )^{5/6}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{6 a \sqrt {-b^4+a^4 x^4}}-\frac {\left (\sqrt [3]{-1} \sqrt [6]{-a^{12}} \left (a^8+(-1)^{2/3} a^4 \sqrt [3]{-a^{12}}-\sqrt [3]{-1} \left (-a^{12}\right )^{2/3}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{12 a^{10} \sqrt {-b^4+a^4 x^4}}+\frac {\left (\left (a^2+\sqrt [3]{-1} \sqrt [6]{-a^{12}}\right ) \left (a^8+(-1)^{2/3} a^4 \sqrt [3]{-a^{12}}-\sqrt [3]{-1} \left (-a^{12}\right )^{2/3}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{12 a^{10} \sqrt {-b^4+a^4 x^4}}\right )+2 \left (-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {(-1)^{2/3} a^{10}}{\left (-a^{12}\right )^{5/6}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{6 a \sqrt {-b^4+a^4 x^4}}-\frac {\left (\sqrt [3]{-1} a^2 \left (1+\frac {(-1)^{2/3} a^4}{\sqrt [3]{-a^{12}}}+\frac {\sqrt [3]{-1} \sqrt [3]{-a^{12}}}{a^4}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{12 \sqrt [6]{-a^{12}} \sqrt {-b^4+a^4 x^4}}-\frac {\left ((-1)^{2/3} \left (a^2+(-1)^{2/3} \sqrt [6]{-a^{12}}\right ) \left ((-1)^{2/3} a^8+a^4 \sqrt [3]{-a^{12}}+\sqrt [3]{-1} \left (-a^{12}\right )^{2/3}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{12 a^2 \left (-a^{12}\right )^{2/3} \sqrt {-b^4+a^4 x^4}}\right )\\ &=2 \left (-\frac {(-1)^{2/3} \left (a^2+(-1)^{2/3} \sqrt [6]{-a^{12}}\right ) \left ((-1)^{2/3} a^8+a^4 \sqrt [3]{-a^{12}}+\sqrt [3]{-1} \left (-a^{12}\right )^{2/3}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{12 a^3 \left (-a^{12}\right )^{2/3} \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {(-1)^{2/3} a^{10}}{\left (-a^{12}\right )^{5/6}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{6 a \sqrt {-b^4+a^4 x^4}}-\frac {\left (\sqrt [3]{-1} a^2 \left (1+\frac {(-1)^{2/3} a^4}{\sqrt [3]{-a^{12}}}+\frac {\sqrt [3]{-1} \sqrt [3]{-a^{12}}}{a^4}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {\sqrt {1+\frac {a^2 x^2}{b^2}}}{\sqrt {1-\frac {a^2 x^2}{b^2}}} \, dx}{12 \sqrt [6]{-a^{12}} \sqrt {-b^4+a^4 x^4}}\right )+2 \left (-\frac {(-1)^{2/3} a \left (a^2-(-1)^{2/3} \sqrt [6]{-a^{12}}\right ) \left (1+\frac {(-1)^{2/3} a^4}{\sqrt [3]{-a^{12}}}+\frac {\sqrt [3]{-1} \sqrt [3]{-a^{12}}}{a^4}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{12 \sqrt [3]{-a^{12}} \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {(-1)^{2/3} \sqrt [6]{-a^{12}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{6 a \sqrt {-b^4+a^4 x^4}}+\frac {\left (\sqrt [3]{-1} a^2 \left (1+\frac {(-1)^{2/3} a^4}{\sqrt [3]{-a^{12}}}+\frac {\sqrt [3]{-1} \sqrt [3]{-a^{12}}}{a^4}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {\sqrt {1+\frac {a^2 x^2}{b^2}}}{\sqrt {1-\frac {a^2 x^2}{b^2}}} \, dx}{12 \sqrt [6]{-a^{12}} \sqrt {-b^4+a^4 x^4}}\right )+2 \left (\frac {\left (a^2+\sqrt [6]{-a^{12}}\right ) \left (a^8+a^4 \sqrt [3]{-a^{12}}+\left (-a^{12}\right )^{2/3}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{12 a^{11} \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {a^{10}}{\left (-a^{12}\right )^{5/6}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{6 a \sqrt {-b^4+a^4 x^4}}-\frac {\left (\sqrt [6]{-a^{12}} \left (a^8+a^4 \sqrt [3]{-a^{12}}+\left (-a^{12}\right )^{2/3}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {\sqrt {1+\frac {a^2 x^2}{b^2}}}{\sqrt {1-\frac {a^2 x^2}{b^2}}} \, dx}{12 a^{10} \sqrt {-b^4+a^4 x^4}}\right )+2 \left (\frac {\left (a^2-\sqrt [6]{-a^{12}}\right ) \left (a^8+a^4 \sqrt [3]{-a^{12}}+\left (-a^{12}\right )^{2/3}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{12 a^{11} \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt [6]{-a^{12}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{6 a \sqrt {-b^4+a^4 x^4}}+\frac {\left (\sqrt [6]{-a^{12}} \left (a^8+a^4 \sqrt [3]{-a^{12}}+\left (-a^{12}\right )^{2/3}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {\sqrt {1+\frac {a^2 x^2}{b^2}}}{\sqrt {1-\frac {a^2 x^2}{b^2}}} \, dx}{12 a^{10} \sqrt {-b^4+a^4 x^4}}\right )+2 \left (\frac {\left (a^2+\sqrt [3]{-1} \sqrt [6]{-a^{12}}\right ) \left (a^8+(-1)^{2/3} a^4 \sqrt [3]{-a^{12}}-\sqrt [3]{-1} \left (-a^{12}\right )^{2/3}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{12 a^{11} \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt [3]{-1} a^{10}}{\left (-a^{12}\right )^{5/6}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{6 a \sqrt {-b^4+a^4 x^4}}-\frac {\left (\sqrt [3]{-1} \sqrt [6]{-a^{12}} \left (a^8+(-1)^{2/3} a^4 \sqrt [3]{-a^{12}}-\sqrt [3]{-1} \left (-a^{12}\right )^{2/3}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {\sqrt {1+\frac {a^2 x^2}{b^2}}}{\sqrt {1-\frac {a^2 x^2}{b^2}}} \, dx}{12 a^{10} \sqrt {-b^4+a^4 x^4}}\right )+2 \left (\frac {\left (a^2-\sqrt [3]{-1} \sqrt [6]{-a^{12}}\right ) \left (a^8+(-1)^{2/3} a^4 \sqrt [3]{-a^{12}}-\sqrt [3]{-1} \left (-a^{12}\right )^{2/3}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{12 a^{11} \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt [3]{-1} \sqrt [6]{-a^{12}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{6 a \sqrt {-b^4+a^4 x^4}}+\frac {\left (\sqrt [3]{-1} \sqrt [6]{-a^{12}} \left (a^8+(-1)^{2/3} a^4 \sqrt [3]{-a^{12}}-\sqrt [3]{-1} \left (-a^{12}\right )^{2/3}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {\sqrt {1+\frac {a^2 x^2}{b^2}}}{\sqrt {1-\frac {a^2 x^2}{b^2}}} \, dx}{12 a^{10} \sqrt {-b^4+a^4 x^4}}\right )\\ &=2 \left (\frac {a \left (a^8+a^4 \sqrt [3]{-a^{12}}+\left (-a^{12}\right )^{2/3}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} E\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{12 \left (-a^{12}\right )^{5/6} \sqrt {-b^4+a^4 x^4}}+\frac {\left (a^2+\sqrt [6]{-a^{12}}\right ) \left (a^8+a^4 \sqrt [3]{-a^{12}}+\left (-a^{12}\right )^{2/3}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{12 a^{11} \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {a^{10}}{\left (-a^{12}\right )^{5/6}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{6 a \sqrt {-b^4+a^4 x^4}}\right )+2 \left (\frac {\sqrt [3]{-1} a \left (a^8+(-1)^{2/3} a^4 \sqrt [3]{-a^{12}}-\sqrt [3]{-1} \left (-a^{12}\right )^{2/3}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} E\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{12 \left (-a^{12}\right )^{5/6} \sqrt {-b^4+a^4 x^4}}+\frac {\left (a^2+\sqrt [3]{-1} \sqrt [6]{-a^{12}}\right ) \left (a^8+(-1)^{2/3} a^4 \sqrt [3]{-a^{12}}-\sqrt [3]{-1} \left (-a^{12}\right )^{2/3}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{12 a^{11} \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt [3]{-1} a^{10}}{\left (-a^{12}\right )^{5/6}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{6 a \sqrt {-b^4+a^4 x^4}}\right )+2 \left (-\frac {\sqrt [3]{-1} a \left (1+\frac {(-1)^{2/3} a^4}{\sqrt [3]{-a^{12}}}+\frac {\sqrt [3]{-1} \sqrt [3]{-a^{12}}}{a^4}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} E\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{12 \sqrt [6]{-a^{12}} \sqrt {-b^4+a^4 x^4}}-\frac {(-1)^{2/3} \left (a^2+(-1)^{2/3} \sqrt [6]{-a^{12}}\right ) \left ((-1)^{2/3} a^8+a^4 \sqrt [3]{-a^{12}}+\sqrt [3]{-1} \left (-a^{12}\right )^{2/3}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{12 a^3 \left (-a^{12}\right )^{2/3} \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {(-1)^{2/3} a^{10}}{\left (-a^{12}\right )^{5/6}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{6 a \sqrt {-b^4+a^4 x^4}}\right )+2 \left (-\frac {a \left (a^8+a^4 \sqrt [3]{-a^{12}}+\left (-a^{12}\right )^{2/3}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} E\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{12 \left (-a^{12}\right )^{5/6} \sqrt {-b^4+a^4 x^4}}+\frac {\left (a^2-\sqrt [6]{-a^{12}}\right ) \left (a^8+a^4 \sqrt [3]{-a^{12}}+\left (-a^{12}\right )^{2/3}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{12 a^{11} \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt [6]{-a^{12}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{6 a \sqrt {-b^4+a^4 x^4}}\right )+2 \left (-\frac {\sqrt [3]{-1} a \left (a^8+(-1)^{2/3} a^4 \sqrt [3]{-a^{12}}-\sqrt [3]{-1} \left (-a^{12}\right )^{2/3}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} E\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{12 \left (-a^{12}\right )^{5/6} \sqrt {-b^4+a^4 x^4}}+\frac {\left (a^2-\sqrt [3]{-1} \sqrt [6]{-a^{12}}\right ) \left (a^8+(-1)^{2/3} a^4 \sqrt [3]{-a^{12}}-\sqrt [3]{-1} \left (-a^{12}\right )^{2/3}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{12 a^{11} \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt [3]{-1} \sqrt [6]{-a^{12}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{6 a \sqrt {-b^4+a^4 x^4}}\right )+2 \left (\frac {\sqrt [3]{-1} a \left (1+\frac {(-1)^{2/3} a^4}{\sqrt [3]{-a^{12}}}+\frac {\sqrt [3]{-1} \sqrt [3]{-a^{12}}}{a^4}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} E\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{12 \sqrt [6]{-a^{12}} \sqrt {-b^4+a^4 x^4}}-\frac {(-1)^{2/3} a \left (a^2-(-1)^{2/3} \sqrt [6]{-a^{12}}\right ) \left (1+\frac {(-1)^{2/3} a^4}{\sqrt [3]{-a^{12}}}+\frac {\sqrt [3]{-1} \sqrt [3]{-a^{12}}}{a^4}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{12 \sqrt [3]{-a^{12}} \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {(-1)^{2/3} \sqrt [6]{-a^{12}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{6 a \sqrt {-b^4+a^4 x^4}}\right )\\ \end {align*}

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Mathematica [C]  time = 1.80, size = 276, normalized size = 1.02 \begin {gather*} -\frac {i \sqrt {1-\frac {a^4 x^4}{b^4}} \left (3 F\left (\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )-\Pi \left (-i;\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )-\Pi \left (i;\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )-\Pi \left (-\frac {i}{2}-\frac {\sqrt {3}}{2};\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )-\Pi \left (\frac {i}{2}-\frac {\sqrt {3}}{2};\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )-\Pi \left (\frac {1}{2} \left (-i+\sqrt {3}\right );\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )-\Pi \left (\frac {1}{2} \left (i+\sqrt {3}\right );\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )\right )}{3 \sqrt {-\frac {a^2}{b^2}} \sqrt {a^4 x^4-b^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-b^12 + a^12*x^12)/(Sqrt[-b^4 + a^4*x^4]*(b^12 + a^12*x^12)),x]

[Out]

((-1/3*I)*Sqrt[1 - (a^4*x^4)/b^4]*(3*EllipticF[I*ArcSinh[Sqrt[-(a^2/b^2)]*x], -1] - EllipticPi[-I, I*ArcSinh[S
qrt[-(a^2/b^2)]*x], -1] - EllipticPi[I, I*ArcSinh[Sqrt[-(a^2/b^2)]*x], -1] - EllipticPi[-1/2*I - Sqrt[3]/2, I*
ArcSinh[Sqrt[-(a^2/b^2)]*x], -1] - EllipticPi[I/2 - Sqrt[3]/2, I*ArcSinh[Sqrt[-(a^2/b^2)]*x], -1] - EllipticPi
[(-I + Sqrt[3])/2, I*ArcSinh[Sqrt[-(a^2/b^2)]*x], -1] - EllipticPi[(I + Sqrt[3])/2, I*ArcSinh[Sqrt[-(a^2/b^2)]
*x], -1]))/(Sqrt[-(a^2/b^2)]*Sqrt[-b^4 + a^4*x^4])

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IntegrateAlgebraic [C]  time = 23.74, size = 598, normalized size = 2.21 \begin {gather*} -\frac {\left (\frac {1}{6}-\frac {i}{6}\right ) \tan ^{-1}\left (\frac {(1+i) a b x}{i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}}\right )}{a b}+\frac {(-1)^{3/4} \tan ^{-1}\left (\frac {\sqrt {2} a b x}{i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}}\right )}{3 a b}-\frac {i \left (-\sqrt {2} \sqrt [4]{-7-4 \sqrt {3}}+\sqrt {6} \sqrt [4]{-7-4 \sqrt {3}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{-7-4 \sqrt {3}} \left (b^2-i a^2 x^2-i \sqrt {-b^4+a^4 x^4}\right )}{\sqrt {2} a b x}\right )}{6 a b}+\frac {i \left (\sqrt {2} \sqrt [4]{-7+4 \sqrt {3}}+\sqrt {6} \sqrt [4]{-7+4 \sqrt {3}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{-7+4 \sqrt {3}} \left (b^2-i a^2 x^2-i \sqrt {-b^4+a^4 x^4}\right )}{\sqrt {2} a b x}\right )}{6 a b}-\frac {\sqrt [4]{-1} \tanh ^{-1}\left (\frac {\sqrt {2} a b x}{i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}}\right )}{3 a b}+\frac {\left (\frac {1}{24}-\frac {i}{24}\right ) \log \left (i b^4-(1+i) a b^3 x-(1-i) a^3 b x^3-i a^4 x^4+\left (b^2-(1-i) a b x-i a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}\right )}{a b}-\frac {\left (\frac {1}{24}-\frac {i}{24}\right ) \log \left (-a b^5-(1-i) a^2 b^4 x+(1+i) a^4 b^2 x^3+a^5 b x^4+\left (i a b^3+(1+i) a^2 b^2 x+a^3 b x^2\right ) \sqrt {-b^4+a^4 x^4}\right )}{a b} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

IntegrateAlgebraic[(-b^12 + a^12*x^12)/(Sqrt[-b^4 + a^4*x^4]*(b^12 + a^12*x^12)),x]

[Out]

((-1/6 + I/6)*ArcTan[((1 + I)*a*b*x)/(I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4])])/(a*b) + ((-1)^(3/4)*ArcTan[(Sq
rt[2]*a*b*x)/(I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4])])/(3*a*b) - ((I/6)*(-(Sqrt[2]*(-7 - 4*Sqrt[3])^(1/4)) +
Sqrt[6]*(-7 - 4*Sqrt[3])^(1/4))*ArcTanh[((-7 - 4*Sqrt[3])^(1/4)*(b^2 - I*a^2*x^2 - I*Sqrt[-b^4 + a^4*x^4]))/(S
qrt[2]*a*b*x)])/(a*b) + ((I/6)*(Sqrt[2]*(-7 + 4*Sqrt[3])^(1/4) + Sqrt[6]*(-7 + 4*Sqrt[3])^(1/4))*ArcTanh[((-7
+ 4*Sqrt[3])^(1/4)*(b^2 - I*a^2*x^2 - I*Sqrt[-b^4 + a^4*x^4]))/(Sqrt[2]*a*b*x)])/(a*b) - ((-1)^(1/4)*ArcTanh[(
Sqrt[2]*a*b*x)/(I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4])])/(3*a*b) + ((1/24 - I/24)*Log[I*b^4 - (1 + I)*a*b^3*x
- (1 - I)*a^3*b*x^3 - I*a^4*x^4 + (b^2 - (1 - I)*a*b*x - I*a^2*x^2)*Sqrt[-b^4 + a^4*x^4]])/(a*b) - ((1/24 - I
/24)*Log[-(a*b^5) - (1 - I)*a^2*b^4*x + (1 + I)*a^4*b^2*x^3 + a^5*b*x^4 + (I*a*b^3 + (1 + I)*a^2*b^2*x + a^3*b
*x^2)*Sqrt[-b^4 + a^4*x^4]])/(a*b)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^12*x^12-b^12)/(a^4*x^4-b^4)^(1/2)/(a^12*x^12+b^12),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{12} x^{12} - b^{12}}{{\left (a^{12} x^{12} + b^{12}\right )} \sqrt {a^{4} x^{4} - b^{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^12*x^12-b^12)/(a^4*x^4-b^4)^(1/2)/(a^12*x^12+b^12),x, algorithm="giac")

[Out]

integrate((a^12*x^12 - b^12)/((a^12*x^12 + b^12)*sqrt(a^4*x^4 - b^4)), x)

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maple [A]  time = 0.27, size = 435, normalized size = 1.61

method	result	size
elliptic	\(\frac {\left (-\frac {\ln \left (a^{2} b^{2}+\frac {a b \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{x}+\frac {a^{4} x^{4}-b^{4}}{x^{2}}\right )}{6 a b}+\frac {\arctan \left (\frac {2 a b +\frac {2 \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{x}}{2 a b}\right )}{3 a b}+\frac {\sqrt {2}\, \ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}\right )}{24 \left (a^{4} b^{4}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}+1\right )}{12 \left (a^{4} b^{4}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}-1\right )}{12 \left (a^{4} b^{4}\right )^{\frac {1}{4}}}+\frac {\ln \left (a^{2} b^{2}-\frac {a b \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{x}+\frac {a^{4} x^{4}-b^{4}}{x^{2}}\right )}{6 a b}+\frac {\arctan \left (\frac {-2 a b +\frac {2 \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{x}}{2 a b}\right )}{3 a b}\right ) \sqrt {2}}{2}\)	\(435\)
default	\(\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )}{\sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}-\frac {b^{4} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{8} \textit {\_Z}^{8}-a^{4} b^{4} \textit {\_Z}^{4}+b^{8}\right )}{\sum }\frac {\left (-a^{4} \underline {\hspace {1.25 ex}}\alpha ^{4}+2 b^{4}\right ) \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{6} a^{4}-\underline {\hspace {1.25 ex}}\alpha ^{2} b^{4}+b^{4} x^{2}\right ) a^{4}}{b^{4} \sqrt {a^{4} \underline {\hspace {1.25 ex}}\alpha ^{4}-b^{4}}\, \sqrt {a^{4} x^{4}-b^{4}}}\right )}{\sqrt {a^{4} \underline {\hspace {1.25 ex}}\alpha ^{4}-b^{4}}}+\frac {2 a^{4} \underline {\hspace {1.25 ex}}\alpha ^{3} \left (a^{4} \underline {\hspace {1.25 ex}}\alpha ^{4}-b^{4}\right ) \sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (a^{4} \underline {\hspace {1.25 ex}}\alpha ^{4}-b^{4}\right ) a^{2}}{b^{6}}, \frac {\sqrt {\frac {a^{2}}{b^{2}}}}{\sqrt {-\frac {a^{2}}{b^{2}}}}\right )}{\sqrt {-\frac {a^{2}}{b^{2}}}\, b^{8} \sqrt {a^{4} x^{4}-b^{4}}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3} \left (2 a^{4} \underline {\hspace {1.25 ex}}\alpha ^{4}-b^{4}\right )}\right )}{12 a^{4}}-\frac {b^{4} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} a^{4}+b^{4}\right )}{\sum }\frac {-\frac {\sqrt {2}\, \arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}\right ) a^{4}}{\sqrt {-2 b^{4}}\, \sqrt {a^{4} x^{4}-b^{4}}}\right )}{\sqrt {-b^{4}}}+\frac {4 \underline {\hspace {1.25 ex}}\alpha ^{3} a^{4} \sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}}{b^{2}}, \frac {\sqrt {\frac {a^{2}}{b^{2}}}}{\sqrt {-\frac {a^{2}}{b^{2}}}}\right )}{\sqrt {-\frac {a^{2}}{b^{2}}}\, b^{4} \sqrt {a^{4} x^{4}-b^{4}}}}{\underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{24 a^{4}}\)	\(541\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^12*x^12-b^12)/(a^4*x^4-b^4)^(1/2)/(a^12*x^12+b^12),x,method=_RETURNVERBOSE)

[Out]

1/2*(-1/6/a/b*ln(a^2*b^2+a*b*(a^4*x^4-b^4)^(1/2)*2^(1/2)/x+(a^4*x^4-b^4)/x^2)+1/3/a/b*arctan(1/2*(2*a*b+2*(a^4
*x^4-b^4)^(1/2)*2^(1/2)/x)/a/b)+1/24/(a^4*b^4)^(1/4)*2^(1/2)*ln((1/2*(a^4*x^4-b^4)/x^2-(a^4*b^4)^(1/4)*(a^4*x^
4-b^4)^(1/2)/x+(a^4*b^4)^(1/2))/(1/2*(a^4*x^4-b^4)/x^2+(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1/2)/x+(a^4*b^4)^(1/2)))
+1/12/(a^4*b^4)^(1/4)*2^(1/2)*arctan(1/(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1/2)/x+1)+1/12/(a^4*b^4)^(1/4)*2^(1/2)*a
rctan(1/(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1/2)/x-1)+1/6/a/b*ln(a^2*b^2-a*b*(a^4*x^4-b^4)^(1/2)*2^(1/2)/x+(a^4*x^4
-b^4)/x^2)+1/3/a/b*arctan(1/2*(-2*a*b+2*(a^4*x^4-b^4)^(1/2)*2^(1/2)/x)/a/b))*2^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{12} x^{12} - b^{12}}{{\left (a^{12} x^{12} + b^{12}\right )} \sqrt {a^{4} x^{4} - b^{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^12*x^12-b^12)/(a^4*x^4-b^4)^(1/2)/(a^12*x^12+b^12),x, algorithm="maxima")

[Out]

integrate((a^12*x^12 - b^12)/((a^12*x^12 + b^12)*sqrt(a^4*x^4 - b^4)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {b^{12}-a^{12}\,x^{12}}{\sqrt {a^4\,x^4-b^4}\,\left (a^{12}\,x^{12}+b^{12}\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(b^12 - a^12*x^12)/((a^4*x^4 - b^4)^(1/2)*(b^12 + a^12*x^12)),x)

[Out]

int(-(b^12 - a^12*x^12)/((a^4*x^4 - b^4)^(1/2)*(b^12 + a^12*x^12)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**12*x**12-b**12)/(a**4*x**4-b**4)**(1/2)/(a**12*x**12+b**12),x)

[Out]

Timed out

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