3.12.52 \(\int \frac {1+x^{12}}{\sqrt {1+x^4} (-1+x^{12})} \, dx\)

Optimal. Leaf size=85 \[ -\frac {1}{3} \tan ^{-1}\left (\frac {x}{\sqrt {x^4+1}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{6 \sqrt {2}}-\frac {1}{3} \tanh ^{-1}\left (\frac {x}{\sqrt {x^4+1}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{6 \sqrt {2}} \]

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Rubi [C]  time = 3.07, antiderivative size = 420, normalized size of antiderivative = 4.94, number of steps used = 171, number of rules used = 18, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {1586, 6725, 1729, 1209, 1198, 220, 1196, 1211, 1699, 206, 1248, 735, 844, 215, 725, 203, 1217, 1707} \begin {gather*} -\frac {1}{3} \tan ^{-1}\left (\frac {x}{\sqrt {x^4+1}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{6 \sqrt {2}}-\frac {1}{3} \tanh ^{-1}\left (\frac {x}{\sqrt {x^4+1}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{6 \sqrt {2}}+\frac {\left (3+i \sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {x^4+1}}-\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {x^4+1}}+\frac {\left (3-i \sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {x^4+1}}-\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {x^4+1}}-\frac {\left (-\sqrt {3}+i\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \left (-\sqrt {3}+3 i\right ) \sqrt {x^4+1}}-\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \left (1+\sqrt [3]{-1}\right ) \sqrt {x^4+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x^12)/(Sqrt[1 + x^4]*(-1 + x^12)),x]

[Out]

-1/3*ArcTan[x/Sqrt[1 + x^4]] - ArcTan[(Sqrt[2]*x)/Sqrt[1 + x^4]]/(6*Sqrt[2]) - ArcTanh[x/Sqrt[1 + x^4]]/3 - Ar
cTanh[(Sqrt[2]*x)/Sqrt[1 + x^4]]/(6*Sqrt[2]) - ((1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1
/2])/(6*(1 + (-1)^(1/3))*Sqrt[1 + x^4]) - ((I - Sqrt[3])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*Arc
Tan[x], 1/2])/(6*(3*I - Sqrt[3])*Sqrt[1 + x^4]) - ((1 - I*Sqrt[3])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*Ellip
ticF[2*ArcTan[x], 1/2])/(6*Sqrt[1 + x^4]) + ((3 - I*Sqrt[3])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2
*ArcTan[x], 1/2])/(12*Sqrt[1 + x^4]) - ((1 + I*Sqrt[3])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcT
an[x], 1/2])/(6*Sqrt[1 + x^4]) + ((3 + I*Sqrt[3])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x],
1/2])/(12*Sqrt[1 + x^4])

Rule 203

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

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 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[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 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 1196

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

Rule 1198

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

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 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 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 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 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 {1+x^{12}}{\sqrt {1+x^4} \left (-1+x^{12}\right )} \, dx &=\int \frac {\sqrt {1+x^4} \left (1-x^4+x^8\right )}{-1+x^{12}} \, dx\\ &=\int \left (-\frac {\sqrt {1+x^4}}{12 (1-x)}-\frac {\sqrt {1+x^4}}{12 (1-i x)}-\frac {\sqrt {1+x^4}}{12 (1+i x)}-\frac {\sqrt {1+x^4}}{12 (1+x)}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \sqrt {1+x^4}}{12 \left (1-\sqrt [6]{-1} x\right )}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \sqrt {1+x^4}}{12 \left (1+\sqrt [6]{-1} x\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \sqrt {1+x^4}}{12 \left (1-\sqrt [3]{-1} x\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \sqrt {1+x^4}}{12 \left (1+\sqrt [3]{-1} x\right )}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \sqrt {1+x^4}}{12 \left (1-(-1)^{2/3} x\right )}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \sqrt {1+x^4}}{12 \left (1+(-1)^{2/3} x\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \sqrt {1+x^4}}{12 \left (1-(-1)^{5/6} x\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \sqrt {1+x^4}}{12 \left (1+(-1)^{5/6} x\right )}\right ) \, dx\\ &=-\left (\frac {1}{12} \int \frac {\sqrt {1+x^4}}{1-x} \, dx\right )-\frac {1}{12} \int \frac {\sqrt {1+x^4}}{1-i x} \, dx-\frac {1}{12} \int \frac {\sqrt {1+x^4}}{1+i x} \, dx-\frac {1}{12} \int \frac {\sqrt {1+x^4}}{1+x} \, dx+\frac {1}{12} \left (-1-i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{1-\sqrt [6]{-1} x} \, dx+\frac {1}{12} \left (-1-i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{1+\sqrt [6]{-1} x} \, dx+\frac {1}{12} \left (-1-i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{1-(-1)^{2/3} x} \, dx+\frac {1}{12} \left (-1-i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{1+(-1)^{2/3} x} \, dx+\frac {1}{12} \left (-1+i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{1-\sqrt [3]{-1} x} \, dx+\frac {1}{12} \left (-1+i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{1+\sqrt [3]{-1} x} \, dx+\frac {1}{12} \left (-1+i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{1-(-1)^{5/6} x} \, dx+\frac {1}{12} \left (-1+i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{1+(-1)^{5/6} x} \, dx\\ &=-2 \left (\frac {1}{12} \int \frac {\sqrt {1+x^4}}{1-x^2} \, dx\right )-2 \left (\frac {1}{12} \int \frac {\sqrt {1+x^4}}{1+x^2} \, dx\right )+2 \left (\frac {1}{12} \left (-1-i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{1-\sqrt [3]{-1} x^2} \, dx\right )+2 \left (\frac {1}{12} \left (-1-i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{1+\sqrt [3]{-1} x^2} \, dx\right )+2 \left (\frac {1}{12} \left (-1+i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{1-(-1)^{2/3} x^2} \, dx\right )+2 \left (\frac {1}{12} \left (-1+i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{1+(-1)^{2/3} x^2} \, dx\right )\\ &=-2 \left (-\left (\frac {1}{12} \int \frac {1+x^2}{\sqrt {1+x^4}} \, dx\right )+\frac {1}{6} \int \frac {1}{\left (1-x^2\right ) \sqrt {1+x^4}} \, dx\right )-2 \left (-\left (\frac {1}{12} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx\right )+\frac {1}{6} \int \frac {1}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx\right )+2 \left (-\left (\frac {1}{6} \int \frac {1}{\left (1+\sqrt [3]{-1} x^2\right ) \sqrt {1+x^4}} \, dx\right )+\frac {1}{12} \left (1-i \sqrt {3}\right ) \int \frac {1-\sqrt [3]{-1} x^2}{\sqrt {1+x^4}} \, dx\right )+2 \left (-\left (\frac {1}{6} \int \frac {1}{\left (1-\sqrt [3]{-1} x^2\right ) \sqrt {1+x^4}} \, dx\right )+\frac {1}{12} \left (1-i \sqrt {3}\right ) \int \frac {1+\sqrt [3]{-1} x^2}{\sqrt {1+x^4}} \, dx\right )+2 \left (-\left (\frac {1}{6} \int \frac {1}{\left (1+(-1)^{2/3} x^2\right ) \sqrt {1+x^4}} \, dx\right )+\frac {1}{12} \left (1+i \sqrt {3}\right ) \int \frac {1-(-1)^{2/3} x^2}{\sqrt {1+x^4}} \, dx\right )+2 \left (-\left (\frac {1}{6} \int \frac {1}{\left (1-(-1)^{2/3} x^2\right ) \sqrt {1+x^4}} \, dx\right )+\frac {1}{12} \left (1+i \sqrt {3}\right ) \int \frac {1+(-1)^{2/3} x^2}{\sqrt {1+x^4}} \, dx\right )\\ &=-2 \left (\frac {x \sqrt {1+x^4}}{12 \left (1+x^2\right )}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}+\frac {1}{12} \int \frac {1}{\sqrt {1+x^4}} \, dx+\frac {1}{12} \int \frac {1-x^2}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx\right )-2 \left (\frac {1}{12} \int \frac {1}{\sqrt {1+x^4}} \, dx+\frac {1}{12} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx+\frac {1}{12} \int \frac {1+x^2}{\left (1-x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{6} \int \frac {1}{\sqrt {1+x^4}} \, dx\right )+2 \left (\frac {1}{6} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx-\frac {\int \frac {1}{\sqrt {1+x^4}} \, dx}{6 \left (1-\sqrt [3]{-1}\right )}+\frac {\left (\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right )\right ) \int \frac {1+x^2}{\left (1+\sqrt [3]{-1} x^2\right ) \sqrt {1+x^4}} \, dx}{6 \left (1-(-1)^{2/3}\right )}+\frac {1}{12} \left (-1-i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx\right )+2 \left (-\left (\frac {1}{6} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx\right )-\frac {\int \frac {1}{\sqrt {1+x^4}} \, dx}{6 \left (1+\sqrt [3]{-1}\right )}-\frac {\sqrt [3]{-1} \int \frac {1+x^2}{\left (1-\sqrt [3]{-1} x^2\right ) \sqrt {1+x^4}} \, dx}{6 \left (1+\sqrt [3]{-1}\right )}+\frac {1}{12} \left (3-i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx\right )+2 \left (\frac {1}{6} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx-\frac {1}{6} \left (1-\sqrt [3]{-1}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {1}{6} \left (1+(-1)^{2/3}\right ) \int \frac {1+x^2}{\left (1-(-1)^{2/3} x^2\right ) \sqrt {1+x^4}} \, dx+\frac {1}{12} \left (-1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx\right )+2 \left (-\left (\frac {1}{6} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx\right )-\frac {\int \frac {1+x^2}{\left (1+(-1)^{2/3} x^2\right ) \sqrt {1+x^4}} \, dx}{6 \left (1+\sqrt [3]{-1}\right )}-\frac {\left (1+(-1)^{2/3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{6 \left (1+\sqrt [3]{-1}\right )}+\frac {1}{12} \left (3+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx\right )\\ &=2 \left (-\frac {x \sqrt {1+x^4}}{6 \left (1+x^2\right )}-\frac {1}{12} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {1+x^4}}-\frac {\left (1-\sqrt [3]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}-\frac {\left (1-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \sqrt {1+x^4}}-\frac {i \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \sqrt {3} \sqrt {1+x^4}}\right )+2 \left (-\frac {x \sqrt {1+x^4}}{6 \left (1+x^2\right )}-\frac {1}{12} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {1+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \left (1-\sqrt [3]{-1}\right ) \sqrt {1+x^4}}-\frac {\left (1+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \sqrt {1+x^4}}+\frac {i \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \sqrt {3} \sqrt {1+x^4}}\right )+2 \left (\frac {x \sqrt {1+x^4}}{6 \left (1+x^2\right )}-\frac {1}{12} \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {1+x^4}}-\frac {\left (1+(-1)^{2/3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \left (1+\sqrt [3]{-1}\right ) \sqrt {1+x^4}}+\frac {\left (3+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \sqrt {1+x^4}}-\frac {\left (1-\sqrt [3]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {3}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (1+\sqrt [3]{-1}\right ) \sqrt {1+x^4}}\right )+2 \left (\frac {x \sqrt {1+x^4}}{6 \left (1+x^2\right )}-\frac {1}{12} \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {1+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \left (1+\sqrt [3]{-1}\right ) \sqrt {1+x^4}}+\frac {\left (3-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \sqrt {1+x^4}}+\frac {\left (1-\sqrt [3]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {3}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (1+\sqrt [3]{-1}\right ) \sqrt {1+x^4}}\right )-2 \left (-\frac {x \sqrt {1+x^4}}{12 \left (1+x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \sqrt {1+x^4}}+\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt {1+x^4}}\right )\right )-2 \left (\frac {x \sqrt {1+x^4}}{12 \left (1+x^2\right )}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \sqrt {1+x^4}}+\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {x}{\sqrt {1+x^4}}\right )\right )\\ &=-2 \left (-\frac {x \sqrt {1+x^4}}{12 \left (1+x^2\right )}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{12 \sqrt {2}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \sqrt {1+x^4}}\right )-2 \left (\frac {x \sqrt {1+x^4}}{12 \left (1+x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{12 \sqrt {2}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \sqrt {1+x^4}}\right )+2 \left (-\frac {x \sqrt {1+x^4}}{6 \left (1+x^2\right )}-\frac {1}{12} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {1+x^4}}-\frac {\left (1-\sqrt [3]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}-\frac {\left (1-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \sqrt {1+x^4}}-\frac {i \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \sqrt {3} \sqrt {1+x^4}}\right )+2 \left (-\frac {x \sqrt {1+x^4}}{6 \left (1+x^2\right )}-\frac {1}{12} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {1+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \left (1-\sqrt [3]{-1}\right ) \sqrt {1+x^4}}-\frac {\left (1+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \sqrt {1+x^4}}+\frac {i \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \sqrt {3} \sqrt {1+x^4}}\right )+2 \left (\frac {x \sqrt {1+x^4}}{6 \left (1+x^2\right )}-\frac {1}{12} \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {1+x^4}}-\frac {\left (1+(-1)^{2/3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \left (1+\sqrt [3]{-1}\right ) \sqrt {1+x^4}}+\frac {\left (3+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \sqrt {1+x^4}}-\frac {\left (1-\sqrt [3]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {3}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (1+\sqrt [3]{-1}\right ) \sqrt {1+x^4}}\right )+2 \left (\frac {x \sqrt {1+x^4}}{6 \left (1+x^2\right )}-\frac {1}{12} \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {1+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \left (1+\sqrt [3]{-1}\right ) \sqrt {1+x^4}}+\frac {\left (3-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \sqrt {1+x^4}}+\frac {\left (1-\sqrt [3]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {3}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (1+\sqrt [3]{-1}\right ) \sqrt {1+x^4}}\right )\\ \end {align*}

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Mathematica [C]  time = 0.92, size = 204, normalized size = 2.40 \begin {gather*} \frac {(-1)^{5/12} \left (-3 F\left (\left .i \sinh ^{-1}\left (\frac {(1+i) x}{\sqrt {2}}\right )\right |-1\right )+\Pi \left (-i;\left .i \sinh ^{-1}\left (\frac {(1+i) x}{\sqrt {2}}\right )\right |-1\right )+\Pi \left (i;\left .i \sinh ^{-1}\left (\frac {(1+i) x}{\sqrt {2}}\right )\right |-1\right )+\Pi \left (-\frac {i}{2}-\frac {\sqrt {3}}{2};\left .i \sinh ^{-1}\left (\frac {(1+i) x}{\sqrt {2}}\right )\right |-1\right )+\Pi \left (\frac {i}{2}-\frac {\sqrt {3}}{2};\left .i \sinh ^{-1}\left (\frac {(1+i) x}{\sqrt {2}}\right )\right |-1\right )+\Pi \left (\frac {1}{2} \left (-i+\sqrt {3}\right );\left .i \sinh ^{-1}\left (\frac {(1+i) x}{\sqrt {2}}\right )\right |-1\right )+\Pi \left (\frac {1}{2} \left (i+\sqrt {3}\right );\left .i \sinh ^{-1}\left (\frac {(1+i) x}{\sqrt {2}}\right )\right |-1\right )\right )}{\sqrt {3} \left (1+\sqrt [3]{-1}\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(1 + x^12)/(Sqrt[1 + x^4]*(-1 + x^12)),x]

[Out]

((-1)^(5/12)*(-3*EllipticF[I*ArcSinh[((1 + I)*x)/Sqrt[2]], -1] + EllipticPi[-I, I*ArcSinh[((1 + I)*x)/Sqrt[2]]
, -1] + EllipticPi[I, I*ArcSinh[((1 + I)*x)/Sqrt[2]], -1] + EllipticPi[-1/2*I - Sqrt[3]/2, I*ArcSinh[((1 + I)*
x)/Sqrt[2]], -1] + EllipticPi[I/2 - Sqrt[3]/2, I*ArcSinh[((1 + I)*x)/Sqrt[2]], -1] + EllipticPi[(-I + Sqrt[3])
/2, I*ArcSinh[((1 + I)*x)/Sqrt[2]], -1] + EllipticPi[(I + Sqrt[3])/2, I*ArcSinh[((1 + I)*x)/Sqrt[2]], -1]))/(S
qrt[3]*(1 + (-1)^(1/3)))

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IntegrateAlgebraic [A]  time = 0.42, size = 85, normalized size = 1.00 \begin {gather*} -\frac {1}{3} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{6 \sqrt {2}}-\frac {1}{3} \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{6 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x^12)/(Sqrt[1 + x^4]*(-1 + x^12)),x]

[Out]

-1/3*ArcTan[x/Sqrt[1 + x^4]] - ArcTan[(Sqrt[2]*x)/Sqrt[1 + x^4]]/(6*Sqrt[2]) - ArcTanh[x/Sqrt[1 + x^4]]/3 - Ar
cTanh[(Sqrt[2]*x)/Sqrt[1 + x^4]]/(6*Sqrt[2])

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fricas [A]  time = 0.53, size = 120, normalized size = 1.41 \begin {gather*} -\frac {1}{12} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} x}{\sqrt {x^{4} + 1}}\right ) + \frac {1}{24} \, \sqrt {2} \log \left (\frac {x^{4} - 2 \, \sqrt {2} \sqrt {x^{4} + 1} x + 2 \, x^{2} + 1}{x^{4} - 2 \, x^{2} + 1}\right ) - \frac {1}{6} \, \arctan \left (\frac {2 \, \sqrt {x^{4} + 1} x}{x^{4} - x^{2} + 1}\right ) + \frac {1}{6} \, \log \left (\frac {x^{4} + x^{2} - 2 \, \sqrt {x^{4} + 1} x + 1}{x^{4} - x^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*sqrt(2)*arctan(sqrt(2)*x/sqrt(x^4 + 1)) + 1/24*sqrt(2)*log((x^4 - 2*sqrt(2)*sqrt(x^4 + 1)*x + 2*x^2 + 1)
/(x^4 - 2*x^2 + 1)) - 1/6*arctan(2*sqrt(x^4 + 1)*x/(x^4 - x^2 + 1)) + 1/6*log((x^4 + x^2 - 2*sqrt(x^4 + 1)*x +
1)/(x^4 - x^2 + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{12} + 1}{{\left (x^{12} - 1\right )} \sqrt {x^{4} + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^12 + 1)/((x^12 - 1)*sqrt(x^4 + 1)), x)

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maple [A]  time = 0.94, size = 99, normalized size = 1.16

method	result	size
elliptic	\(\frac {\left (\frac {\ln \left (-1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{12}+\frac {\arctan \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{6}-\frac {\ln \left (1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{12}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}+1}}{x}\right )}{3}-\frac {\sqrt {2}\, \arctanh \left (\frac {\sqrt {x^{4}+1}}{x}\right )}{3}\right ) \sqrt {2}}{2}\)	\(99\)
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) x +\sqrt {x^{4}+1}}{x^{2}+1}\right )}{12}+\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}-2\right ) x +\sqrt {x^{4}+1}}{\left (-1+x \right ) \left (1+x \right )}\right )}{12}-\frac {\ln \left (-\frac {x^{4}+2 \sqrt {x^{4}+1}\, x +x^{2}+1}{x^{4}-x^{2}+1}\right )}{6}+\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{4}-\RootOf \left (\textit {\_Z}^{2}+2\right ) \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}-2\right ) \RootOf \left (\textit {\_Z}^{2}+2\right )-4 \sqrt {x^{4}+1}\, x}{\left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}\right )}{12}\)	\(199\)
default	\(\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}+\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i, -\sqrt {-i}\, \left (-1\right )^{\frac {3}{4}}\right )}{3 \sqrt {x^{4}+1}}+\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , -i, -\sqrt {-i}\, \left (-1\right )^{\frac {3}{4}}\right )}{3 \sqrt {x^{4}+1}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}-\textit {\_Z}^{2}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}+1\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {x^{4}+1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {2 \left (-1\right )^{\frac {3}{4}} \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i \underline {\hspace {1.25 ex}}\alpha ^{2}-i, i\right )}{\sqrt {x^{4}+1}}\right )\right )}{12}+\frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\arctanh \left (\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x^{2}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+1}}\right )}{2 \sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}}+\frac {\left (-1\right )^{\frac {3}{4}} \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ), i\right )}{\sqrt {x^{4}+1}}\right )}{6}+\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\arctanh \left (\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x^{2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+1}}\right )}{2 \sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}}+\frac {\left (-1\right )^{\frac {3}{4}} \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ), i\right )}{\sqrt {x^{4}+1}}\right )}{6}+\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {\arctanh \left (\frac {\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \left (x^{2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\sqrt {x^{4}+1}}\right )}{2 \sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}}+\frac {\left (-1\right )^{\frac {3}{4}} \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , -i \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ), i\right )}{\sqrt {x^{4}+1}}\right )}{6}+\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (\frac {\arctanh \left (\frac {\sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}\, \left (x^{2}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\sqrt {x^{4}+1}}\right )}{2 \sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}}+\frac {\left (-1\right )^{\frac {3}{4}} \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , -i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ), i\right )}{\sqrt {x^{4}+1}}\right )}{6}\)	\(730\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(1/12*ln(-1+1/2*2^(1/2)/x*(x^4+1)^(1/2))+1/6*arctan(1/2*2^(1/2)/x*(x^4+1)^(1/2))-1/12*ln(1+1/2*2^(1/2)/x*(
x^4+1)^(1/2))+1/3*2^(1/2)*arctan((x^4+1)^(1/2)/x)-1/3*2^(1/2)*arctanh((x^4+1)^(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 {x^{12} + 1}{{\left (x^{12} - 1\right )} \sqrt {x^{4} + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^12 + 1)/((x^12 - 1)*sqrt(x^4 + 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^{12}+1}{\sqrt {x^4+1}\,\left (x^{12}-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} + 1} \left (x^{8} - x^{4} + 1\right )}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x**4 + 1)*(x**8 - x**4 + 1)/((x - 1)*(x + 1)*(x**2 + 1)*(x**2 - x + 1)*(x**2 + x + 1)*(x**4 - x*
*2 + 1)), x)

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