3.12.52 \(\int \frac {1+x^{12}}{\sqrt {1+x^4} (-1+x^{12})} \, dx\) [1152]

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

[Out]

-1/3*arctan(x/(x^4+1)^(1/2))-1/12*arctan(2^(1/2)*x/(x^4+1)^(1/2))*2^(1/2)-1/3*arctanh(x/(x^4+1)^(1/2))-1/12*ar
ctanh(2^(1/2)*x/(x^4+1)^(1/2))*2^(1/2)

________________________________________________________________________________________

Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 2.52, 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 = {1600, 6857, 1743, 1223, 1212, 226, 1210, 1225, 1713, 212, 1262, 749, 858, 221, 739, 209, 1231, 1721} \begin {gather*} -\frac {1}{3} \text {ArcTan}\left (\frac {x}{\sqrt {x^4+1}}\right )-\frac {\text {ArcTan}\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 \text {ArcTan}(x)\left |\frac {1}{2}\right .\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 \text {ArcTan}(x)\left |\frac {1}{2}\right .\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 \text {ArcTan}(x)\left |\frac {1}{2}\right .\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 \text {ArcTan}(x)\left |\frac {1}{2}\right .\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 \text {ArcTan}(x)\left |\frac {1}{2}\right .\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 \text {ArcTan}(x)\left |\frac {1}{2}\right .\right )}{6 \left (1+\sqrt [3]{-1}\right ) \sqrt {x^4+1}}-\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}} \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 209

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

Rule 212

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

Rule 221

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 226

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)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 739

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 749

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 858

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 1210

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)]/(q*Sqrt[a + c*x^4
]))*EllipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rule 1212

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 1223

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 1225

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 1231

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 1262

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 1600

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 1713

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 1721

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))]/(4*d*e*A*q*Sqrt[a + c*x^4]))*El
lipticPi[Cancel[-(B*d - A*e)^2/(4*d*e*A*B)], 2*ArcTan[q*x], 1/2], 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 1743

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 6857

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} \text {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} \text {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 [A]
time = 0.31, size = 81, normalized size = 0.95 \begin {gather*} \frac {1}{12} \left (-4 \text {ArcTan}\left (\frac {x}{\sqrt {1+x^4}}\right )-\sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )-4 \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*ArcTan[x/Sqrt[1 + x^4]] - Sqrt[2]*ArcTan[(Sqrt[2]*x)/Sqrt[1 + x^4]] - 4*ArcTanh[x/Sqrt[1 + x^4]] - Sqrt[2]
*ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^4]])/12

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.36, size = 730, normalized size = 8.59

method	result	size
elliptic	\(\frac {\left (\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}+1}}{x}\right )}{3}+\frac {\arctan \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{6}-\frac {\sqrt {2}\, \arctanh \left (\frac {\sqrt {x^{4}+1}}{x}\right )}{3}-\frac {\ln \left (1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{12}+\frac {\ln \left (-1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{12}\right ) \sqrt {2}}{2}\)	\(99\)
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) x +\sqrt {x^{4}+1}}{\left (-1+x \right ) \left (1+x \right )}\right )}{12}-\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+2\right ) x +\sqrt {x^{4}+1}}{x^{2}+1}\right )}{12}-\frac {\ln \left (-\frac {x^{4}+2 \sqrt {x^{4}+1}\, x +x^{2}+1}{x^{4}-x^{2}+1}\right )}{6}-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{4}-\RootOf \left (\textit {\_Z}^{2}+2\right ) \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}+4 \sqrt {x^{4}+1}\, x +\RootOf \left (\textit {\_Z}^{2}-2\right ) \RootOf \left (\textit {\_Z}^{2}+2\right )}{\left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}\right )}{12}\)	\(200\)
default	\(\text {Expression too large to display}\)	\(730\)

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/(1/2*2^(1/2)+1/2*I*2^(1/2))*(1-I*x^2)^(1/2)*(1+I*x^2)^(1/2)/(x^4+1)^(1/2)*EllipticF(x*(1/2*2^(1/2)+1/2*I*2^(
1/2)),I)+1/12*sum(_alpha*(-1/(_alpha^2)^(1/2)*arctanh(_alpha^2*(-_alpha^2+x^2+1)/(_alpha^2)^(1/2)/(x^4+1)^(1/2
))+2*(-1)^(3/4)*(-_alpha^3+_alpha)*(1-I*x^2)^(1/2)*(1+I*x^2)^(1/2)/(x^4+1)^(1/2)*EllipticPi((-1)^(1/4)*x,I*_al
pha^2-I,I)),_alpha=RootOf(_Z^4-_Z^2+1))+1/3*(-1)^(3/4)*(1-I*x^2)^(1/2)*(1+I*x^2)^(1/2)/(x^4+1)^(1/2)*EllipticP
i((-1)^(1/4)*x,-I,(-I)^(1/2)/(-1)^(1/4))+1/6*(-1/2+1/2*I*3^(1/2))*(1/2/(1/2+1/2*I*3^(1/2))^(1/2)*arctanh((1/2+
1/2*I*3^(1/2))^(1/2)*(x^2-1/2+1/2*I*3^(1/2))/(x^4+1)^(1/2))+(-1)^(3/4)*(-1/2-1/2*I*3^(1/2))*(1-I*x^2)^(1/2)*(1
+I*x^2)^(1/2)/(x^4+1)^(1/2)*EllipticPi((-1)^(1/4)*x,-I*(-1/2+1/2*I*3^(1/2)),I))+1/6*(-1/2-1/2*I*3^(1/2))*(1/2/
(1/2-1/2*I*3^(1/2))^(1/2)*arctanh((1/2-1/2*I*3^(1/2))^(1/2)*(x^2-1/2-1/2*I*3^(1/2))/(x^4+1)^(1/2))+(-1)^(3/4)*
(-1/2+1/2*I*3^(1/2))*(1-I*x^2)^(1/2)*(1+I*x^2)^(1/2)/(x^4+1)^(1/2)*EllipticPi((-1)^(1/4)*x,-I*(-1/2-1/2*I*3^(1
/2)),I))+1/3*(-1)^(3/4)*(1-I*x^2)^(1/2)*(1+I*x^2)^(1/2)/(x^4+1)^(1/2)*EllipticPi((-1)^(1/4)*x,I,(-I)^(1/2)/(-1
)^(1/4))+1/6*(1/2+1/2*I*3^(1/2))*(-1/2/(1/2-1/2*I*3^(1/2))^(1/2)*arctanh((-1/2+1/2*I*3^(1/2))*(x^2-1/2-1/2*I*3
^(1/2))/(1/2-1/2*I*3^(1/2))^(1/2)/(x^4+1)^(1/2))+(-1)^(3/4)*(1/2-1/2*I*3^(1/2))*(1-I*x^2)^(1/2)*(1+I*x^2)^(1/2
)/(x^4+1)^(1/2)*EllipticPi((-1)^(1/4)*x,I*(1/2+1/2*I*3^(1/2)),I))+1/6*(1/2-1/2*I*3^(1/2))*(-1/2/(1/2+1/2*I*3^(
1/2))^(1/2)*arctanh((-1/2-1/2*I*3^(1/2))*(x^2-1/2+1/2*I*3^(1/2))/(1/2+1/2*I*3^(1/2))^(1/2)/(x^4+1)^(1/2))+(-1)
^(3/4)*(1/2+1/2*I*3^(1/2))*(1-I*x^2)^(1/2)*(1+I*x^2)^(1/2)/(x^4+1)^(1/2)*EllipticPi((-1)^(1/4)*x,I*(1/2-1/2*I*
3^(1/2)),I))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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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Fricas [A]
time = 0.46, 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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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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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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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