3.22.13 \(\int \frac {b^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} (-b^6+a^6 x^6)} \, dx\)

Optimal. Leaf size=153 \[ -\frac {2 \tan ^{-1}\left (\frac {a b x}{\sqrt {a^4 x^4+b^4}}\right )}{3 a b}+\frac {\tanh ^{-1}\left (\frac {\sqrt {6-4 \sqrt {2}} a b x}{\sqrt {a^4 x^4+b^4}+a^2 x^2+b^2}\right )}{3 \sqrt {2} a b}-\frac {\tanh ^{-1}\left (\frac {\sqrt {6+4 \sqrt {2}} a b x}{\sqrt {a^4 x^4+b^4}+a^2 x^2+b^2}\right )}{3 \sqrt {2} a b} \]

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Rubi [C]  time = 3.93, antiderivative size = 340, normalized size of antiderivative = 2.22, number of steps used = 41, number of rules used = 13, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {6725, 220, 2073, 1211, 1699, 208, 6728, 1725, 1217, 1707, 1248, 725, 206} \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {a b x}{\sqrt {a^4 x^4+b^4}}\right )}{3 a b}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {a^4 x^4+b^4}}\right )}{3 \sqrt {2} a b}-\frac {\left (a-\sqrt {3} \sqrt {-a^2}\right ) \left (a^2 x^2+b^2\right ) \sqrt {\frac {a^4 x^4+b^4}{\left (a^2 x^2+b^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{6 a^2 b \sqrt {a^4 x^4+b^4}}-\frac {\left (\sqrt {3} \sqrt {-a^2}+a\right ) \left (a^2 x^2+b^2\right ) \sqrt {\frac {a^4 x^4+b^4}{\left (a^2 x^2+b^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{6 a^2 b \sqrt {a^4 x^4+b^4}}+\frac {\left (a^2 x^2+b^2\right ) \sqrt {\frac {a^4 x^4+b^4}{\left (a^2 x^2+b^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{3 a b \sqrt {a^4 x^4+b^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b^6 + a^6*x^6)/(Sqrt[b^4 + a^4*x^4]*(-b^6 + a^6*x^6)),x]

[Out]

(-2*ArcTan[(a*b*x)/Sqrt[b^4 + a^4*x^4]])/(3*a*b) - ArcTanh[(Sqrt[2]*a*b*x)/Sqrt[b^4 + a^4*x^4]]/(3*Sqrt[2]*a*b
) + ((b^2 + a^2*x^2)*Sqrt[(b^4 + a^4*x^4)/(b^2 + a^2*x^2)^2]*EllipticF[2*ArcTan[(a*x)/b], 1/2])/(3*a*b*Sqrt[b^
4 + a^4*x^4]) - ((a - Sqrt[3]*Sqrt[-a^2])*(b^2 + a^2*x^2)*Sqrt[(b^4 + a^4*x^4)/(b^2 + a^2*x^2)^2]*EllipticF[2*
ArcTan[(a*x)/b], 1/2])/(6*a^2*b*Sqrt[b^4 + a^4*x^4]) - ((a + Sqrt[3]*Sqrt[-a^2])*(b^2 + a^2*x^2)*Sqrt[(b^4 + a
^4*x^4)/(b^2 + a^2*x^2)^2]*EllipticF[2*ArcTan[(a*x)/b], 1/2])/(6*a^2*b*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 208

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

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

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 1211

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

Rule 1217

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

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 1699

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

Rule 1707

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]
}, -Simp[((B*d - A*e)*ArcTan[(Rt[(c*d)/e + (a*e)/d, 2]*x)/Sqrt[a + c*x^4]])/(2*d*e*Rt[(c*d)/e + (a*e)/d, 2]),
x] + Simp[((B*d + A*e)*(A + B*x^2)*Sqrt[(A^2*(a + c*x^4))/(a*(A + B*x^2)^2)]*EllipticPi[Cancel[-((B*d - A*e)^2
/(4*d*e*A*B))], 2*ArcTan[q*x], 1/2])/(4*d*e*A*q*Sqrt[a + c*x^4]), x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c
*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0]

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^6+a^6 x^6}{\sqrt {b^4+a^4 x^4} \left (-b^6+a^6 x^6\right )} \, dx &=\int \left (\frac {1}{\sqrt {b^4+a^4 x^4}}+\frac {2 b^6}{\sqrt {b^4+a^4 x^4} \left (-b^6+a^6 x^6\right )}\right ) \, dx\\ &=\left (2 b^6\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4} \left (-b^6+a^6 x^6\right )} \, dx+\int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx\\ &=\frac {\left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{2 a b \sqrt {b^4+a^4 x^4}}+\left (2 b^6\right ) \int \left (-\frac {1}{3 b^4 \left (b^2-a^2 x^2\right ) \sqrt {b^4+a^4 x^4}}+\frac {-2 b+a x}{6 b^5 \left (b^2-a b x+a^2 x^2\right ) \sqrt {b^4+a^4 x^4}}+\frac {-2 b-a x}{6 b^5 \left (b^2+a b x+a^2 x^2\right ) \sqrt {b^4+a^4 x^4}}\right ) \, dx\\ &=\frac {\left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{2 a b \sqrt {b^4+a^4 x^4}}+\frac {1}{3} 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 \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} \left (2 b^2\right ) \int \frac {1}{\left (b^2-a^2 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx\\ &=\frac {\left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{2 a b \sqrt {b^4+a^4 x^4}}-\frac {1}{3} \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{3} \int \frac {b^2+a^2 x^2}{\left (b^2-a^2 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx+\frac {1}{3} 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 \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 {\left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{3 a b \sqrt {b^4+a^4 x^4}}-\frac {1}{3} \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-\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+\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+\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} b^2 \operatorname {Subst}\left (\int \frac {1}{b^2-2 a^2 b^4 x^2} \, dx,x,\frac {x}{\sqrt {b^4+a^4 x^4}}\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a b}+\frac {\left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{3 a b \sqrt {b^4+a^4 x^4}}+\frac {1}{3} \left (2 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 (2 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 (2 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 (2 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 (\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}{3} \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}{3} \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}{3} \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 {\tanh ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a b}+\frac {\left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{3 a b \sqrt {b^4+a^4 x^4}}-2 \frac {\left (a-\sqrt {3} \sqrt {-a^2}\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{6 a}-2 \frac {\left (a+\sqrt {3} \sqrt {-a^2}\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx}{6 a}+\frac {1}{3} \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}{3} \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}{3} \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}{3} \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}{3} \left (2 a \left (a-\sqrt {3} \sqrt {-a^2}\right ) b^2\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\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 (2 a \left (a-\sqrt {3} \sqrt {-a^2}\right ) b^2\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\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 (2 a \left (a+\sqrt {3} \sqrt {-a^2}\right ) b^2\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\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 (2 a \left (a+\sqrt {3} \sqrt {-a^2}\right ) b^2\right ) \int \frac {1+\frac {a^2 x^2}{b^2}}{\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 {2 \tan ^{-1}\left (\frac {a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 a b}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a b}+\frac {\left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{3 a b \sqrt {b^4+a^4 x^4}}-\frac {\left (a-\sqrt {3} \sqrt {-a^2}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{6 a^2 b \sqrt {b^4+a^4 x^4}}-\frac {\left (a+\sqrt {3} \sqrt {-a^2}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{6 a^2 b \sqrt {b^4+a^4 x^4}}-\frac {1}{3} \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}{3} \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}{3} \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}{3} \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 {2 \tan ^{-1}\left (\frac {a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 a b}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a b}+\frac {\left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{3 a b \sqrt {b^4+a^4 x^4}}-\frac {\left (a-\sqrt {3} \sqrt {-a^2}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{6 a^2 b \sqrt {b^4+a^4 x^4}}-\frac {\left (a+\sqrt {3} \sqrt {-a^2}\right ) \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{6 a^2 b \sqrt {b^4+a^4 x^4}}\\ \end {align*}

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Mathematica [C]  time = 0.71, size = 268, normalized size = 1.75 \begin {gather*} -\frac {i \sqrt {\frac {a^4 x^4}{b^4}+1} \left (3 \sqrt [4]{a^4} F\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )-2 \sqrt [4]{a^4} \Pi \left (-\frac {i}{2}-\frac {\sqrt {3}}{2};\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )-2 \sqrt [4]{a^4} \Pi \left (\frac {1}{2} \left (-i+\sqrt {3}\right );\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )-(1-i) \sqrt {2} \sqrt [4]{b^4} \sqrt {\frac {i a^2}{b^2}} \Pi \left (\frac {i \sqrt {a^4} \sqrt {b^4}}{a^2 b^2};\left .i \sinh ^{-1}\left (\frac {(1+i) \sqrt [4]{a^4} x}{\sqrt {2} \sqrt [4]{b^4}}\right )\right |-1\right )\right )}{3 \sqrt [4]{a^4} \sqrt {\frac {i a^2}{b^2}} \sqrt {a^4 x^4+b^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b^6 + a^6*x^6)/(Sqrt[b^4 + a^4*x^4]*(-b^6 + a^6*x^6)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 1.42, size = 71, normalized size = 0.46 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 a b}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{3 \sqrt {2} a b} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(b^6 + a^6*x^6)/(Sqrt[b^4 + a^4*x^4]*(-b^6 + a^6*x^6)),x]

[Out]

(-2*ArcTan[(a*b*x)/Sqrt[b^4 + a^4*x^4]])/(3*a*b) - ArcTanh[(Sqrt[2]*a*b*x)/Sqrt[b^4 + a^4*x^4]]/(3*Sqrt[2]*a*b
)

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fricas [A]  time = 0.79, size = 127, normalized size = 0.83 \begin {gather*} \frac {\sqrt {2} \log \left (\frac {a^{4} x^{4} + 2 \, a^{2} b^{2} x^{2} + b^{4} - 2 \, \sqrt {2} \sqrt {a^{4} x^{4} + b^{4}} a b x}{a^{4} x^{4} - 2 \, a^{2} b^{2} x^{2} + b^{4}}\right ) - 4 \, \arctan \left (\frac {2 \, \sqrt {a^{4} x^{4} + b^{4}} a b x}{a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}}\right )}{12 \, a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^6*x^6+b^6)/(a^4*x^4+b^4)^(1/2)/(a^6*x^6-b^6),x, algorithm="fricas")

[Out]

1/12*(sqrt(2)*log((a^4*x^4 + 2*a^2*b^2*x^2 + b^4 - 2*sqrt(2)*sqrt(a^4*x^4 + b^4)*a*b*x)/(a^4*x^4 - 2*a^2*b^2*x
^2 + b^4)) - 4*arctan(2*sqrt(a^4*x^4 + b^4)*a*b*x/(a^4*x^4 - a^2*b^2*x^2 + b^4)))/(a*b)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{6} x^{6} + b^{6}}{{\left (a^{6} x^{6} - b^{6}\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^6*x^6+b^6)/(a^4*x^4+b^4)^(1/2)/(a^6*x^6-b^6),x, algorithm="giac")

[Out]

integrate((a^6*x^6 + b^6)/((a^6*x^6 - b^6)*sqrt(a^4*x^4 + b^4)), x)

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maple [A]  time = 0.17, size = 111, normalized size = 0.73

method	result	size
elliptic	\(\frac {\left (\frac {\ln \left (-a b +\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}\right )}{6 a b}+\frac {2 \sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}}{a b x}\right )}{3 a b}-\frac {\ln \left (a b +\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x}\right )}{6 a b}\right ) \sqrt {2}}{2}\)	\(111\)
default	\(\frac {\sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, i\right )}{\sqrt {\frac {i 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}-\underline {\hspace {1.25 ex}}\alpha b \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 {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, \frac {i \underline {\hspace {1.25 ex}}\alpha a}{b}, \frac {\sqrt {-\frac {i a^{2}}{b^{2}}}}{\sqrt {\frac {i a^{2}}{b^{2}}}}\right )}{\sqrt {\frac {i a^{2}}{b^{2}}}\, b^{2} \sqrt {a^{4} x^{4}+b^{4}}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha a -b}\right )}{6 a}-\frac {b \left (-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 a^{2} b^{2} x^{2}+2 b^{4}\right ) \sqrt {2}}{4 \sqrt {b^{4}}\, \sqrt {a^{4} x^{4}+b^{4}}}\right )}{4 \sqrt {b^{4}}}+\frac {a \sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, -i, \frac {\sqrt {-\frac {i a^{2}}{b^{2}}}}{\sqrt {\frac {i a^{2}}{b^{2}}}}\right )}{\sqrt {\frac {i a^{2}}{b^{2}}}\, b \sqrt {a^{4} x^{4}+b^{4}}}\right )}{3 a}+\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}+\underline {\hspace {1.25 ex}}\alpha b \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 {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, -\frac {i \underline {\hspace {1.25 ex}}\alpha a}{b}, \frac {\sqrt {-\frac {i a^{2}}{b^{2}}}}{\sqrt {\frac {i a^{2}}{b^{2}}}}\right )}{\sqrt {\frac {i a^{2}}{b^{2}}}\, b^{2} \sqrt {a^{4} x^{4}+b^{4}}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha a +b}\right )}{6 a}+\frac {b \left (-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 a^{2} b^{2} x^{2}+2 b^{4}\right ) \sqrt {2}}{4 \sqrt {b^{4}}\, \sqrt {a^{4} x^{4}+b^{4}}}\right )}{4 \sqrt {b^{4}}}-\frac {a \sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, -i, \frac {\sqrt {-\frac {i a^{2}}{b^{2}}}}{\sqrt {\frac {i a^{2}}{b^{2}}}}\right )}{\sqrt {\frac {i a^{2}}{b^{2}}}\, b \sqrt {a^{4} x^{4}+b^{4}}}\right )}{3 a}\)	\(825\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^6*x^6+b^6)/(a^4*x^4+b^4)^(1/2)/(a^6*x^6-b^6),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{6} x^{6} + b^{6}}{{\left (a^{6} x^{6} - b^{6}\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^6*x^6+b^6)/(a^4*x^4+b^4)^(1/2)/(a^6*x^6-b^6),x, algorithm="maxima")

[Out]

integrate((a^6*x^6 + b^6)/((a^6*x^6 - b^6)*sqrt(a^4*x^4 + b^4)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {a^6\,x^6+b^6}{\sqrt {a^4\,x^4+b^4}\,\left (b^6-a^6\,x^6\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(b^6 + a^6*x^6)/((b^4 + a^4*x^4)^(1/2)*(b^6 - a^6*x^6)),x)

[Out]

int(-(b^6 + a^6*x^6)/((b^4 + a^4*x^4)^(1/2)*(b^6 - a^6*x^6)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a^{2} x^{2} + b^{2}\right ) \left (a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}\right )}{\left (a x - b\right ) \left (a x + b\right ) \sqrt {a^{4} x^{4} + b^{4}} \left (a^{2} x^{2} - a b x + b^{2}\right ) \left (a^{2} x^{2} + a b x + b^{2}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**6*x**6+b**6)/(a**4*x**4+b**4)**(1/2)/(a**6*x**6-b**6),x)

[Out]

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

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