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

Optimal. Leaf size=376 \[ -\frac {x}{3 \sqrt {a^4 x^4-b^4}}+\frac {1}{6} \text {RootSum}\left [\text {$\#$1}^8-8 i \text {$\#$1}^6 a^2 b^2+24 \text {$\#$1}^4 a^4 b^4+32 i \text {$\#$1}^2 a^6 b^6+16 a^8 b^8\& ,\frac {-i \text {$\#$1}^6 \log \left (-\text {$\#$1} x+\sqrt {a^4 x^4-b^4}+a^2 x^2+i b^2\right )+i \text {$\#$1}^6 \log (x)+2 \text {$\#$1}^4 a^2 b^2 \log (x)-2 \text {$\#$1}^4 a^2 b^2 \log \left (-\text {$\#$1} x+\sqrt {a^4 x^4-b^4}+a^2 x^2+i b^2\right )+4 i \text {$\#$1}^2 a^4 b^4 \log (x)-4 i \text {$\#$1}^2 a^4 b^4 \log \left (-\text {$\#$1} x+\sqrt {a^4 x^4-b^4}+a^2 x^2+i b^2\right )-8 a^6 b^6 \log \left (-\text {$\#$1} x+\sqrt {a^4 x^4-b^4}+a^2 x^2+i b^2\right )+8 a^6 b^6 \log (x)}{-i \text {$\#$1}^7-6 \text {$\#$1}^5 a^2 b^2-12 i \text {$\#$1}^3 a^4 b^4+8 \text {$\#$1} a^6 b^6}\& \right ] \]

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Rubi [A]  time = 3.74, antiderivative size = 428, normalized size of antiderivative = 1.14, number of steps used = 56, number of rules used = 19, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.432, Rules used = {6725, 224, 221, 2073, 1152, 414, 21, 423, 427, 426, 424, 253, 6728, 1725, 1219, 1218, 1248, 725, 204} \begin {gather*} \frac {2 b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\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 {4 a^2}{\left (a+\sqrt {3} \sqrt {-a^2}\right )^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 {2 a}{a+\sqrt {3} \sqrt {-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 {2 a^2}{a^2-\sqrt {3} \sqrt {-a^4}};\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 {2 a^2}{a^2+\sqrt {3} \sqrt {-a^4}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{3 a \sqrt {a^4 x^4-b^4}}-\frac {x \left (b^2-a^2 x^2\right )}{6 b^2 \sqrt {a^4 x^4-b^4}}-\frac {x \left (a^2 x^2+b^2\right )}{6 b^2 \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*(x*(b^2 - a^2*x^2))/(b^2*Sqrt[-b^4 + a^4*x^4]) - (x*(b^2 + a^2*x^2))/(6*b^2*Sqrt[-b^4 + a^4*x^4]) + (2*b*
Sqrt[1 - (a^4*x^4)/b^4]*EllipticF[ArcSin[(a*x)/b], -1])/(3*a*Sqrt[-b^4 + a^4*x^4]) - (b*Sqrt[1 - (a^4*x^4)/b^4
]*EllipticPi[(4*a^2)/(a + Sqrt[3]*Sqrt[-a^2])^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[(-2*a)/(a + Sqrt[3]*Sqrt[-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[(2*a^2)/(a^2 - Sqrt[3]*Sqrt[-a^4]), ArcSin[(a*x)/b], -1])/(3*a*Sqrt[-b
^4 + a^4*x^4]) - (b*Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[(2*a^2)/(a^2 + Sqrt[3]*Sqrt[-a^4]), ArcSin[(a*x)/b], -1
])/(3*a*Sqrt[-b^4 + a^4*x^4])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 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 253

Int[((a1_.) + (b1_.)*(x_)^(n_))^(p_)*((a2_.) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[((a1 + b1*x^n)^FracPa
rt[p]*(a2 + b2*x^n)^FracPart[p])/(a1*a2 + b1*b2*x^(2*n))^FracPart[p], Int[(a1*a2 + b1*b2*x^(2*n))^p, x], x] /;
FreeQ[{a1, b1, a2, b2, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] &&  !IntegerQ[p]

Rule 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 423

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[b/d, Int[Sqrt[c + d*x^2]/Sqrt[a + b
*x^2], x], x] - Dist[(b*c - a*d)/d, Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x]
&& PosQ[d/c] && NegQ[b/a]

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 426

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

Rule 427

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + (d*x^2)/c]/Sqrt[c + d*x^2]
, Int[Sqrt[a + b*x^2]/Sqrt[1 + (d*x^2)/c], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] &&  !GtQ[c, 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 1152

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a + c*x^4)^FracPart[p]/((d + e*x
^2)^FracPart[p]*(a/d + (c*x^2)/e)^FracPart[p]), Int[(d + e*x^2)^(p + q)*(a/d + (c*x^2)/e)^p, x], x] /; FreeQ[{
a, c, d, e, p, q}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p]

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 1725

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

Rule 2073

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(PP /. x ->
x^2)^p*Q^q, x], x] /;  !SumQ[NonfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x^2] && PolyQ[Q, x] && ILtQ[p,
0]

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]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && 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 \left (\frac {1}{\sqrt {-b^4+a^4 x^4}}+\frac {2 b^{12}}{\sqrt {-b^4+a^4 x^4} \left (-b^{12}+a^{12} x^{12}\right )}\right ) \, dx\\ &=\left (2 b^{12}\right ) \int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (-b^{12}+a^{12} x^{12}\right )} \, dx+\int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx\\ &=\left (2 b^{12}\right ) \int \left (-\frac {1}{6 b^{10} \left (b^2-a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}-\frac {1}{6 b^{10} \left (b^2+a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}+\frac {-2 b+a x}{12 b^{11} \left (b^2-a b x+a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}+\frac {-2 b-a x}{12 b^{11} \left (b^2+a b x+a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}+\frac {-2 b^2+a^2 x^2}{6 b^{10} \sqrt {-b^4+a^4 x^4} \left (b^4-a^2 b^2 x^2+a^4 x^4\right )}\right ) \, dx+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{\sqrt {-b^4+a^4 x^4}}\\ &=\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{a \sqrt {-b^4+a^4 x^4}}+\frac {1}{6} b \int \frac {-2 b+a x}{\left (b^2-a b x+a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx+\frac {1}{6} b \int \frac {-2 b-a x}{\left (b^2+a b x+a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{3} b^2 \int \frac {1}{\left (b^2-a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{3} b^2 \int \frac {1}{\left (b^2+a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx+\frac {1}{3} b^2 \int \frac {-2 b^2+a^2 x^2}{\sqrt {-b^4+a^4 x^4} \left (b^4-a^2 b^2 x^2+a^4 x^4\right )} \, dx\\ &=\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{a \sqrt {-b^4+a^4 x^4}}+\frac {1}{6} b \int \left (\frac {a+\sqrt {3} \sqrt {-a^2}}{\left (-a b-\sqrt {3} \sqrt {-a^2} b+2 a^2 x\right ) \sqrt {-b^4+a^4 x^4}}+\frac {a-\sqrt {3} \sqrt {-a^2}}{\left (-a b+\sqrt {3} \sqrt {-a^2} b+2 a^2 x\right ) \sqrt {-b^4+a^4 x^4}}\right ) \, dx+\frac {1}{6} b \int \left (\frac {-a+\sqrt {3} \sqrt {-a^2}}{\left (a b-\sqrt {3} \sqrt {-a^2} b+2 a^2 x\right ) \sqrt {-b^4+a^4 x^4}}+\frac {-a-\sqrt {3} \sqrt {-a^2}}{\left (a b+\sqrt {3} \sqrt {-a^2} b+2 a^2 x\right ) \sqrt {-b^4+a^4 x^4}}\right ) \, dx+\frac {1}{3} b^2 \int \left (\frac {a^2+\sqrt {3} \sqrt {-a^4}}{\left (-a^2 b^2-\sqrt {3} \sqrt {-a^4} b^2+2 a^4 x^2\right ) \sqrt {-b^4+a^4 x^4}}+\frac {a^2-\sqrt {3} \sqrt {-a^4}}{\left (-a^2 b^2+\sqrt {3} \sqrt {-a^4} b^2+2 a^4 x^2\right ) \sqrt {-b^4+a^4 x^4}}\right ) \, dx-\frac {\left (b^2 \sqrt {-b^2-a^2 x^2} \sqrt {b^2-a^2 x^2}\right ) \int \frac {1}{\sqrt {-b^2-a^2 x^2} \left (b^2-a^2 x^2\right )^{3/2}} \, dx}{3 \sqrt {-b^4+a^4 x^4}}-\frac {\left (b^2 \sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}\right ) \int \frac {1}{\sqrt {-b^2+a^2 x^2} \left (b^2+a^2 x^2\right )^{3/2}} \, dx}{3 \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x \left (b^2-a^2 x^2\right )}{6 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{6 b^2 \sqrt {-b^4+a^4 x^4}}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{a \sqrt {-b^4+a^4 x^4}}-\frac {1}{6} \left (\left (a-\sqrt {3} \sqrt {-a^2}\right ) b\right ) \int \frac {1}{\left (a b-\sqrt {3} \sqrt {-a^2} b+2 a^2 x\right ) \sqrt {-b^4+a^4 x^4}} \, dx+\frac {1}{6} \left (\left (a-\sqrt {3} \sqrt {-a^2}\right ) b\right ) \int \frac {1}{\left (-a b+\sqrt {3} \sqrt {-a^2} b+2 a^2 x\right ) \sqrt {-b^4+a^4 x^4}} \, dx+\frac {1}{6} \left (\left (a+\sqrt {3} \sqrt {-a^2}\right ) b\right ) \int \frac {1}{\left (-a b-\sqrt {3} \sqrt {-a^2} b+2 a^2 x\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{6} \left (\left (a+\sqrt {3} \sqrt {-a^2}\right ) b\right ) \int \frac {1}{\left (a b+\sqrt {3} \sqrt {-a^2} b+2 a^2 x\right ) \sqrt {-b^4+a^4 x^4}} \, dx+\frac {1}{3} \left (\left (a^2-\sqrt {3} \sqrt {-a^4}\right ) b^2\right ) \int \frac {1}{\left (-a^2 b^2+\sqrt {3} \sqrt {-a^4} b^2+2 a^4 x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx+\frac {1}{3} \left (\left (a^2+\sqrt {3} \sqrt {-a^4}\right ) b^2\right ) \int \frac {1}{\left (-a^2 b^2-\sqrt {3} \sqrt {-a^4} b^2+2 a^4 x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx+\frac {\left (\sqrt {-b^2-a^2 x^2} \sqrt {b^2-a^2 x^2}\right ) \int \frac {-a^2 b^2+a^4 x^2}{\sqrt {-b^2-a^2 x^2} \sqrt {b^2-a^2 x^2}} \, dx}{6 a^2 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {\left (\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}\right ) \int \frac {a^2 b^2+a^4 x^2}{\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}} \, dx}{6 a^2 b^2 \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x \left (b^2-a^2 x^2\right )}{6 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{6 b^2 \sqrt {-b^4+a^4 x^4}}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{a \sqrt {-b^4+a^4 x^4}}+\frac {1}{3} \left (a^2 \left (a-\sqrt {3} \sqrt {-a^2}\right ) b\right ) \int \frac {x}{\left (\left (a b-\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{3} \left (a^2 \left (a-\sqrt {3} \sqrt {-a^2}\right ) b\right ) \int \frac {x}{\left (\left (-a b+\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{3} \left (a^2 \left (a+\sqrt {3} \sqrt {-a^2}\right ) b\right ) \int \frac {x}{\left (\left (-a b-\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx+\frac {1}{3} \left (a^2 \left (a+\sqrt {3} \sqrt {-a^2}\right ) b\right ) \int \frac {x}{\left (\left (a b+\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{6} \left (\left (a-\sqrt {3} \sqrt {-a^2}\right )^2 b^2\right ) \int \frac {1}{\left (\left (a b-\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{6} \left (\left (a-\sqrt {3} \sqrt {-a^2}\right )^2 b^2\right ) \int \frac {1}{\left (\left (-a b+\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{6} \left (\left (a+\sqrt {3} \sqrt {-a^2}\right )^2 b^2\right ) \int \frac {1}{\left (\left (-a b-\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{6} \left (\left (a+\sqrt {3} \sqrt {-a^2}\right )^2 b^2\right ) \int \frac {1}{\left (\left (a b+\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx-\frac {\left (\sqrt {-b^2-a^2 x^2} \sqrt {b^2-a^2 x^2}\right ) \int \frac {\sqrt {b^2-a^2 x^2}}{\sqrt {-b^2-a^2 x^2}} \, dx}{6 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {\left (\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}\right ) \int \frac {\sqrt {b^2+a^2 x^2}}{\sqrt {-b^2+a^2 x^2}} \, dx}{6 b^2 \sqrt {-b^4+a^4 x^4}}+\frac {\left (\left (a^2-\sqrt {3} \sqrt {-a^4}\right ) b^2 \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1}{\left (-a^2 b^2+\sqrt {3} \sqrt {-a^4} b^2+2 a^4 x^2\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{3 \sqrt {-b^4+a^4 x^4}}+\frac {\left (\left (a^2+\sqrt {3} \sqrt {-a^4}\right ) b^2 \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1}{\left (-a^2 b^2-\sqrt {3} \sqrt {-a^4} b^2+2 a^4 x^2\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{3 \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x \left (b^2-a^2 x^2\right )}{6 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{6 b^2 \sqrt {-b^4+a^4 x^4}}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {2 a^2}{a^2-\sqrt {3} \sqrt {-a^4}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{3 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {2 a^2}{a^2+\sqrt {3} \sqrt {-a^4}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{3 a \sqrt {-b^4+a^4 x^4}}+\frac {1}{6} \left (a^2 \left (a-\sqrt {3} \sqrt {-a^2}\right ) b\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (a b-\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x\right ) \sqrt {-b^4+a^4 x^2}} \, dx,x,x^2\right )-\frac {1}{6} \left (a^2 \left (a-\sqrt {3} \sqrt {-a^2}\right ) b\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (-a b+\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x\right ) \sqrt {-b^4+a^4 x^2}} \, dx,x,x^2\right )-\frac {1}{6} \left (a^2 \left (a+\sqrt {3} \sqrt {-a^2}\right ) b\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (-a b-\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x\right ) \sqrt {-b^4+a^4 x^2}} \, dx,x,x^2\right )+\frac {1}{6} \left (a^2 \left (a+\sqrt {3} \sqrt {-a^2}\right ) b\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (a b+\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x\right ) \sqrt {-b^4+a^4 x^2}} \, dx,x,x^2\right )-\frac {\left (\sqrt {-b^2-a^2 x^2} \sqrt {b^2-a^2 x^2}\right ) \int \frac {1}{\sqrt {-b^2-a^2 x^2} \sqrt {b^2-a^2 x^2}} \, dx}{3 \sqrt {-b^4+a^4 x^4}}-\frac {\left (\sqrt {-b^2-a^2 x^2} \sqrt {b^2-a^2 x^2}\right ) \int \frac {\sqrt {-b^2-a^2 x^2}}{\sqrt {b^2-a^2 x^2}} \, dx}{6 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {\left (\sqrt {b^2+a^2 x^2} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \int \frac {\sqrt {b^2+a^2 x^2}}{\sqrt {1-\frac {a^2 x^2}{b^2}}} \, dx}{6 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {\left (\left (a-\sqrt {3} \sqrt {-a^2}\right )^2 b^2 \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1}{\left (\left (a b-\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{6 \sqrt {-b^4+a^4 x^4}}-\frac {\left (\left (a-\sqrt {3} \sqrt {-a^2}\right )^2 b^2 \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1}{\left (\left (-a b+\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{6 \sqrt {-b^4+a^4 x^4}}-\frac {\left (\left (a+\sqrt {3} \sqrt {-a^2}\right )^2 b^2 \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1}{\left (\left (-a b-\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{6 \sqrt {-b^4+a^4 x^4}}-\frac {\left (\left (a+\sqrt {3} \sqrt {-a^2}\right )^2 b^2 \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1}{\left (\left (a b+\sqrt {3} \sqrt {-a^2} b\right )^2-4 a^4 x^2\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{6 \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x \left (b^2-a^2 x^2\right )}{6 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{6 b^2 \sqrt {-b^4+a^4 x^4}}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {4 a^2}{\left (a+\sqrt {3} \sqrt {-a^2}\right )^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{3 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (-\frac {2 a}{a+\sqrt {3} \sqrt {-a^2}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{3 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {2 a^2}{a^2-\sqrt {3} \sqrt {-a^4}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{3 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {2 a^2}{a^2+\sqrt {3} \sqrt {-a^4}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{3 a \sqrt {-b^4+a^4 x^4}}-\frac {1}{3} \int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx-\frac {1}{6} \left (a^2 \left (a-\sqrt {3} \sqrt {-a^2}\right ) b\right ) \operatorname {Subst}\left (\int \frac {1}{-16 a^8 b^4+a^4 \left (a b-\sqrt {3} \sqrt {-a^2} b\right )^4-x^2} \, dx,x,\frac {4 a^4 b^4-a^4 \left (a b-\sqrt {3} \sqrt {-a^2} b\right )^2 x^2}{\sqrt {-b^4+a^4 x^4}}\right )+\frac {1}{6} \left (a^2 \left (a-\sqrt {3} \sqrt {-a^2}\right ) b\right ) \operatorname {Subst}\left (\int \frac {1}{-16 a^8 b^4+a^4 \left (-a b+\sqrt {3} \sqrt {-a^2} b\right )^4-x^2} \, dx,x,\frac {4 a^4 b^4-a^4 \left (-a b+\sqrt {3} \sqrt {-a^2} b\right )^2 x^2}{\sqrt {-b^4+a^4 x^4}}\right )+\frac {1}{6} \left (a^2 \left (a+\sqrt {3} \sqrt {-a^2}\right ) b\right ) \operatorname {Subst}\left (\int \frac {1}{-16 a^8 b^4+a^4 \left (-a b-\sqrt {3} \sqrt {-a^2} b\right )^4-x^2} \, dx,x,\frac {4 a^4 b^4-a^4 \left (-a b-\sqrt {3} \sqrt {-a^2} b\right )^2 x^2}{\sqrt {-b^4+a^4 x^4}}\right )-\frac {1}{6} \left (a^2 \left (a+\sqrt {3} \sqrt {-a^2}\right ) b\right ) \operatorname {Subst}\left (\int \frac {1}{-16 a^8 b^4+a^4 \left (a b+\sqrt {3} \sqrt {-a^2} b\right )^4-x^2} \, dx,x,\frac {4 a^4 b^4-a^4 \left (a b+\sqrt {3} \sqrt {-a^2} b\right )^2 x^2}{\sqrt {-b^4+a^4 x^4}}\right )-\frac {\left (\sqrt {-b^2-a^2 x^2} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \int \frac {\sqrt {-b^2-a^2 x^2}}{\sqrt {1-\frac {a^2 x^2}{b^2}}} \, dx}{6 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {\left (\left (b^2+a^2 x^2\right ) \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \int \frac {\sqrt {1+\frac {a^2 x^2}{b^2}}}{\sqrt {1-\frac {a^2 x^2}{b^2}}} \, dx}{6 b^2 \sqrt {1+\frac {a^2 x^2}{b^2}} \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x \left (b^2-a^2 x^2\right )}{6 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{6 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {\left (b^2+a^2 x^2\right ) \sqrt {1-\frac {a^2 x^2}{b^2}} E\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{6 a b \sqrt {1+\frac {a^2 x^2}{b^2}} \sqrt {-b^4+a^4 x^4}}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {4 a^2}{\left (a+\sqrt {3} \sqrt {-a^2}\right )^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{3 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (-\frac {2 a}{a+\sqrt {3} \sqrt {-a^2}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{3 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {2 a^2}{a^2-\sqrt {3} \sqrt {-a^4}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{3 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {2 a^2}{a^2+\sqrt {3} \sqrt {-a^4}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{3 a \sqrt {-b^4+a^4 x^4}}-\frac {\left (\left (-b^2-a^2 x^2\right ) \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \int \frac {\sqrt {1+\frac {a^2 x^2}{b^2}}}{\sqrt {1-\frac {a^2 x^2}{b^2}}} \, dx}{6 b^2 \sqrt {1+\frac {a^2 x^2}{b^2}} \sqrt {-b^4+a^4 x^4}}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{3 \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x \left (b^2-a^2 x^2\right )}{6 b^2 \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{6 b^2 \sqrt {-b^4+a^4 x^4}}+\frac {2 b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{3 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {4 a^2}{\left (a+\sqrt {3} \sqrt {-a^2}\right )^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{3 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (-\frac {2 a}{a+\sqrt {3} \sqrt {-a^2}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{3 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {2 a^2}{a^2-\sqrt {3} \sqrt {-a^4}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{3 a \sqrt {-b^4+a^4 x^4}}-\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {2 a^2}{a^2+\sqrt {3} \sqrt {-a^4}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{3 a \sqrt {-b^4+a^4 x^4}}\\ \end {align*}

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Mathematica [A]  time = 2.04, size = 319, normalized size = 0.85 \begin {gather*} \frac {x \left (-\sqrt {-\frac {a^2}{b^2}}\right )-2 i \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )+i \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (-\frac {2 i}{-i+\sqrt {3}};\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )+i \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {2 i}{-i+\sqrt {3}};\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )+i \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (-\frac {2 i}{i+\sqrt {3}};\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )+i \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {2 i}{i+\sqrt {3}};\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\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]

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

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IntegrateAlgebraic [A]  time = 2.08, size = 517, normalized size = 1.38 \begin {gather*} -\frac {x}{3 \sqrt {-b^4+a^4 x^4}}+\left (\frac {1}{6}-\frac {i}{6}\right ) \text {RootSum}\left [4 a^4 b^4+(4-4 i) a^3 b^3 \text {$\#$1}-(2+2 i) a b \text {$\#$1}^3-\text {$\#$1}^4\&,\frac {-2 a^2 b^2 \log (x)+2 a^2 b^2 \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right )-(1-i) a b \log (x) \text {$\#$1}+(1-i) a b \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}+i \log (x) \text {$\#$1}^2-i \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{2 a^3 b^3-3 i a b \text {$\#$1}^2-(1+i) \text {$\#$1}^3}\&\right ]-\left (\frac {1}{6}-\frac {i}{6}\right ) \text {RootSum}\left [4 a^4 b^4-(4-4 i) a^3 b^3 \text {$\#$1}+(2+2 i) a b \text {$\#$1}^3-\text {$\#$1}^4\&,\frac {-2 a^2 b^2 \log (x)+2 a^2 b^2 \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right )+(1-i) a b \log (x) \text {$\#$1}-(1-i) a b \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}+i \log (x) \text {$\#$1}^2-i \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{2 a^3 b^3-3 i a b \text {$\#$1}^2+(1+i) \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*x/Sqrt[-b^4 + a^4*x^4] + (1/6 - I/6)*RootSum[4*a^4*b^4 + (4 - 4*I)*a^3*b^3*#1 - (2 + 2*I)*a*b*#1^3 - #1^4
& , (-2*a^2*b^2*Log[x] + 2*a^2*b^2*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1] - (1 - I)*a*b*Log[x]*#1
+ (1 - I)*a*b*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1]*#1 + I*Log[x]*#1^2 - I*Log[I*b^2 + a^2*x^2 +
Sqrt[-b^4 + a^4*x^4] - x*#1]*#1^2)/(2*a^3*b^3 - (3*I)*a*b*#1^2 - (1 + I)*#1^3) & ] - (1/6 - I/6)*RootSum[4*a^
4*b^4 - (4 - 4*I)*a^3*b^3*#1 + (2 + 2*I)*a*b*#1^3 - #1^4 & , (-2*a^2*b^2*Log[x] + 2*a^2*b^2*Log[I*b^2 + a^2*x^
2 + Sqrt[-b^4 + a^4*x^4] - x*#1] + (1 - I)*a*b*Log[x]*#1 - (1 - I)*a*b*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x
^4] - x*#1]*#1 + I*Log[x]*#1^2 - I*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1]*#1^2)/(2*a^3*b^3 - (3*I)
*a*b*#1^2 + (1 + I)*#1^3) & ]

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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 [B]  time = 0.33, size = 305, normalized size = 0.81

method	result	size
elliptic	\(\frac {\left (-\frac {\sqrt {2}\, x}{3 \sqrt {a^{4} x^{4}-b^{4}}}+\frac {\ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\sqrt {\sqrt {3}\, \sqrt {a^{4} b^{4}}}\, \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {3}\, \sqrt {a^{4} b^{4}}}{2}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\sqrt {\sqrt {3}\, \sqrt {a^{4} b^{4}}}\, \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {3}\, \sqrt {a^{4} b^{4}}}{2}}\right )}{6 \sqrt {\sqrt {3}\, \sqrt {a^{4} b^{4}}}}+\frac {\arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\sqrt {\sqrt {3}\, \sqrt {a^{4} b^{4}}}\, x}+1\right )}{3 \sqrt {\sqrt {3}\, \sqrt {a^{4} b^{4}}}}+\frac {\arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\sqrt {\sqrt {3}\, \sqrt {a^{4} b^{4}}}\, x}-1\right )}{3 \sqrt {\sqrt {3}\, \sqrt {a^{4} b^{4}}}}\right ) \sqrt {2}}{2}\)	\(305\)
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 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{2} a^{2}-\textit {\_Z} a b +b^{2}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha a +2 b \right ) \left (-\frac {\arctanh \left (\frac {\left (\underline {\hspace {1.25 ex}}\alpha a -b \right ) a b \left (a \,x^{2}+b \underline {\hspace {1.25 ex}}\alpha \right )}{\sqrt {-b^{3} \left (\underline {\hspace {1.25 ex}}\alpha a +b \right )}\, \sqrt {a^{4} x^{4}-b^{4}}}\right )}{\sqrt {-b^{3} \left (\underline {\hspace {1.25 ex}}\alpha a +b \right )}}+\frac {2 a \left (\underline {\hspace {1.25 ex}}\alpha a -b \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 a}{b}, \frac {\sqrt {\frac {a^{2}}{b^{2}}}}{\sqrt {-\frac {a^{2}}{b^{2}}}}\right )}{\sqrt {-\frac {a^{2}}{b^{2}}}\, b^{2} \sqrt {a^{4} x^{4}-b^{4}}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha a -b}\right )}{12 a}-\frac {b \left (\frac {a^{4} x^{3}-a^{3} b \,x^{2}+a^{2} b^{2} x -a \,b^{3}}{2 a^{2} b^{3} \sqrt {\left (x +\frac {b}{a}\right ) \left (a^{4} x^{3}-a^{3} b \,x^{2}+a^{2} b^{2} x -a \,b^{3}\right )}}+\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 )}{2 b \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}-\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \left (\EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )-\EllipticE \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )\right )}{2 b \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}\right )}{6}+\frac {b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{2} a^{2}+\textit {\_Z} a b +b^{2}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha a -2 b \right ) \left (\frac {\arctanh \left (\frac {\left (\underline {\hspace {1.25 ex}}\alpha a +b \right ) a b \left (a \,x^{2}-b \underline {\hspace {1.25 ex}}\alpha \right )}{\sqrt {b^{3} \left (\underline {\hspace {1.25 ex}}\alpha a -b \right )}\, \sqrt {a^{4} x^{4}-b^{4}}}\right )}{\sqrt {b^{3} \left (\underline {\hspace {1.25 ex}}\alpha a -b \right )}}+\frac {2 a \left (\underline {\hspace {1.25 ex}}\alpha a +b \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 a}{b}, \frac {\sqrt {\frac {a^{2}}{b^{2}}}}{\sqrt {-\frac {a^{2}}{b^{2}}}}\right )}{\sqrt {-\frac {a^{2}}{b^{2}}}\, b^{2} \sqrt {a^{4} x^{4}-b^{4}}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha a +b}\right )}{12 a}+\frac {b \left (-\frac {a^{4} x^{3}+a^{3} b \,x^{2}+a^{2} b^{2} x +a \,b^{3}}{2 a^{2} b^{3} \sqrt {\left (x -\frac {b}{a}\right ) \left (a^{4} x^{3}+a^{3} b \,x^{2}+a^{2} b^{2} x +a \,b^{3}\right )}}-\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 )}{2 b \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}+\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \left (\EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )-\EllipticE \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )\right )}{2 b \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}\right )}{6}-\frac {b^{2} \left (-\frac {\left (a^{4} x^{2}-a^{2} b^{2}\right ) x}{2 b^{4} a^{2} \sqrt {\left (x^{2}+\frac {b^{2}}{a^{2}}\right ) \left (a^{4} x^{2}-a^{2} b^{2}\right )}}+\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 )}{2 b^{2} \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}+\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \left (\EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )-\EllipticE \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )\right )}{2 b^{2} \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}\right )}{3}-\frac {b^{2} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} a^{4}-\textit {\_Z}^{2} a^{2} b^{2}+b^{4}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+2 b^{2}\right ) \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+a^{2} x^{2}-b^{2}\right ) a^{2}}{\sqrt {b^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-2 b^{2}\right )}\, \sqrt {a^{4} x^{4}-b^{4}}}\right )}{\sqrt {b^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-2 b^{2}\right )}}+\frac {2 a^{2} \underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-b^{2}\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} a^{2}-b^{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}}}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-b^{2}\right )}\right )}{12 a^{2}}\)	\(1496\)

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*(-1/3/(a^4*x^4-b^4)^(1/2)*2^(1/2)*x+1/6/(3^(1/2)*(a^4*b^4)^(1/2))^(1/2)*ln((1/2*(a^4*x^4-b^4)/x^2-1/2*(3^(
1/2)*(a^4*b^4)^(1/2))^(1/2)*(a^4*x^4-b^4)^(1/2)*2^(1/2)/x+1/2*3^(1/2)*(a^4*b^4)^(1/2))/(1/2*(a^4*x^4-b^4)/x^2+
1/2*(3^(1/2)*(a^4*b^4)^(1/2))^(1/2)*(a^4*x^4-b^4)^(1/2)*2^(1/2)/x+1/2*3^(1/2)*(a^4*b^4)^(1/2)))+1/3/(3^(1/2)*(
a^4*b^4)^(1/2))^(1/2)*arctan(1/(3^(1/2)*(a^4*b^4)^(1/2))^(1/2)*(a^4*x^4-b^4)^(1/2)*2^(1/2)/x+1)+1/3/(3^(1/2)*(
a^4*b^4)^(1/2))^(1/2)*arctan(1/(3^(1/2)*(a^4*b^4)^(1/2))^(1/2)*(a^4*x^4-b^4)^(1/2)*2^(1/2)/x-1))*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 {a^{12}\,x^{12}+b^{12}}{\sqrt {a^4\,x^4-b^4}\,\left (b^{12}-a^{12}\,x^{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]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a^{4} x^{4} + b^{4}\right ) \left (a^{8} x^{8} - a^{4} b^{4} x^{4} + b^{8}\right )}{\sqrt {\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right )} \left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right ) \left (a^{2} x^{2} - a b x + b^{2}\right ) \left (a^{2} x^{2} + a b x + b^{2}\right ) \left (a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}\right )}\, dx \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]

Integral((a**4*x**4 + b**4)*(a**8*x**8 - a**4*b**4*x**4 + b**8)/(sqrt((a*x - b)*(a*x + b)*(a**2*x**2 + b**2))*
(a*x - b)*(a*x + b)*(a**2*x**2 + b**2)*(a**2*x**2 - a*b*x + b**2)*(a**2*x**2 + a*b*x + b**2)*(a**4*x**4 - a**2
*b**2*x**2 + b**4)), x)

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