3.7.78 \(\int \frac {1+x^6}{\sqrt {1-x^2+x^4} (1-x^6)} \, dx\)

Optimal. Leaf size=53 \[ \frac {1}{3} \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4-x^2+1}}\right )+\frac {1}{3} \tanh ^{-1}\left (\frac {x}{\sqrt {x^4-x^2+1}}\right ) \]

________________________________________________________________________________________

Rubi [C]  time = 2.15, antiderivative size = 228, normalized size of antiderivative = 4.30, number of steps used = 95, number of rules used = 19, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.655, Rules used = {1586, 6725, 1728, 1208, 1139, 1103, 1195, 1210, 1698, 206, 1247, 734, 843, 619, 215, 724, 1197, 1216, 1706} \begin {gather*} \frac {1}{3} \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4-x^2+1}}\right )+\frac {1}{3} \tanh ^{-1}\left (\frac {x}{\sqrt {x^4-x^2+1}}\right )+\frac {\left (1+(-1)^{2/3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4-x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{6 \sqrt {x^4-x^2+1}}+\frac {\left (1-\sqrt [3]{-1}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4-x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{6 \sqrt {x^4-x^2+1}}-\frac {\left (x^2+1\right ) \sqrt {\frac {x^4-x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{6 \sqrt {x^4-x^2+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x^6)/(Sqrt[1 - x^2 + x^4]*(1 - x^6)),x]

[Out]

(Sqrt[2]*ArcTan[(Sqrt[2]*x)/Sqrt[1 - x^2 + x^4]])/3 + ArcTanh[x/Sqrt[1 - x^2 + x^4]]/3 - ((1 + x^2)*Sqrt[(1 -
x^2 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 3/4])/(6*Sqrt[1 - x^2 + x^4]) + ((1 - (-1)^(1/3))*(1 + x^2)*Sqr
t[(1 - x^2 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 3/4])/(6*Sqrt[1 - x^2 + x^4]) + ((1 + (-1)^(2/3))*(1 + x
^2)*Sqrt[(1 - x^2 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 3/4])/(6*Sqrt[1 - x^2 + 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 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 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 724

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

Rule 734

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + 2*p + 1)), x] - Dist[p/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*
d - b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ
[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || Lt
Q[m, 1]) &&  !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 1103

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(
a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(2*q*Sqrt[a + b*x^2 + c
*x^4]), x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1139

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

Rule 1195

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

Rule 1197

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

Rule 1208

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

Rule 1210

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

Rule 1216

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

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, 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 1698

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

Rule 1706

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

Rule 1728

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Dist[d, Int[(a + b*x^2 + c*x^
4)^p/(d^2 - e^2*x^2), x], x] - Dist[e, Int[(x*(a + b*x^2 + c*x^4)^p)/(d^2 - e^2*x^2), x], x] /; FreeQ[{a, b, 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^6}{\sqrt {1-x^2+x^4} \left (1-x^6\right )} \, dx &=\int \frac {\left (1+x^2\right ) \sqrt {1-x^2+x^4}}{1-x^6} \, dx\\ &=\int \left (\frac {\sqrt {1-x^2+x^4}}{3 (1-x)}+\frac {\sqrt {1-x^2+x^4}}{3 (1+x)}+\frac {\left (1-\sqrt [3]{-1}\right ) \sqrt {1-x^2+x^4}}{6 \left (1-\sqrt [3]{-1} x\right )}+\frac {\left (1-\sqrt [3]{-1}\right ) \sqrt {1-x^2+x^4}}{6 \left (1+\sqrt [3]{-1} x\right )}+\frac {\left (1+(-1)^{2/3}\right ) \sqrt {1-x^2+x^4}}{6 \left (1-(-1)^{2/3} x\right )}+\frac {\left (1+(-1)^{2/3}\right ) \sqrt {1-x^2+x^4}}{6 \left (1+(-1)^{2/3} x\right )}\right ) \, dx\\ &=\frac {1}{3} \int \frac {\sqrt {1-x^2+x^4}}{1-x} \, dx+\frac {1}{3} \int \frac {\sqrt {1-x^2+x^4}}{1+x} \, dx+\frac {1}{6} \left (1-\sqrt [3]{-1}\right ) \int \frac {\sqrt {1-x^2+x^4}}{1-\sqrt [3]{-1} x} \, dx+\frac {1}{6} \left (1-\sqrt [3]{-1}\right ) \int \frac {\sqrt {1-x^2+x^4}}{1+\sqrt [3]{-1} x} \, dx+\frac {1}{6} \left (1+(-1)^{2/3}\right ) \int \frac {\sqrt {1-x^2+x^4}}{1-(-1)^{2/3} x} \, dx+\frac {1}{6} \left (1+(-1)^{2/3}\right ) \int \frac {\sqrt {1-x^2+x^4}}{1+(-1)^{2/3} x} \, dx\\ &=2 \left (\frac {1}{3} \int \frac {\sqrt {1-x^2+x^4}}{1-x^2} \, dx\right )+2 \left (\frac {1}{6} \left (1-\sqrt [3]{-1}\right ) \int \frac {\sqrt {1-x^2+x^4}}{1-(-1)^{2/3} x^2} \, dx\right )+2 \left (\frac {1}{6} \left (1+(-1)^{2/3}\right ) \int \frac {\sqrt {1-x^2+x^4}}{1+\sqrt [3]{-1} x^2} \, dx\right )\\ &=2 \left (-\left (\frac {1}{3} \int \frac {x^2}{\sqrt {1-x^2+x^4}} \, dx\right )+\frac {1}{3} \int \frac {1}{\left (1-x^2\right ) \sqrt {1-x^2+x^4}} \, dx\right )+2 \left (\frac {1}{3} \int \frac {1}{\left (1+\sqrt [3]{-1} x^2\right ) \sqrt {1-x^2+x^4}} \, dx+\frac {1}{6} \left (-1+\sqrt [3]{-1}\right ) \int \frac {1+\sqrt [3]{-1}-\sqrt [3]{-1} x^2}{\sqrt {1-x^2+x^4}} \, dx\right )+2 \left (\frac {1}{3} \int \frac {1}{\left (1-(-1)^{2/3} x^2\right ) \sqrt {1-x^2+x^4}} \, dx+\frac {1}{6} \left ((-1)^{2/3} \left (-1+\sqrt [3]{-1}\right )\right ) \int \frac {1-(-1)^{2/3}+(-1)^{2/3} x^2}{\sqrt {1-x^2+x^4}} \, dx\right )\\ &=2 \left (\frac {1}{6} \int \frac {1}{\sqrt {1-x^2+x^4}} \, dx+\frac {1}{6} \int \frac {1+x^2}{\left (1-x^2\right ) \sqrt {1-x^2+x^4}} \, dx-\frac {1}{3} \int \frac {1}{\sqrt {1-x^2+x^4}} \, dx+\frac {1}{3} \int \frac {1-x^2}{\sqrt {1-x^2+x^4}} \, dx\right )+2 \left (-\left (\frac {1}{6} \int \frac {1-x^2}{\sqrt {1-x^2+x^4}} \, dx\right )+\frac {\int \frac {1}{\sqrt {1-x^2+x^4}} \, dx}{3 \left (1-\sqrt [3]{-1}\right )}+\frac {1}{6} \left (-1+\sqrt [3]{-1}\right ) \int \frac {1}{\sqrt {1-x^2+x^4}} \, dx-\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^2+x^4}} \, dx}{3 \left (1-(-1)^{2/3}\right )}\right )+2 \left (-\left (\frac {1}{6} \int \frac {1-x^2}{\sqrt {1-x^2+x^4}} \, dx\right )+\frac {1}{3} \left (1-\sqrt [3]{-1}\right ) \int \frac {1}{\sqrt {1-x^2+x^4}} \, dx+\frac {1}{6} \left ((-1)^{2/3} \left (-1+\sqrt [3]{-1}\right )\right ) \int \frac {1}{\sqrt {1-x^2+x^4}} \, dx+\frac {1}{3} \left (1+(-1)^{2/3}\right ) \int \frac {1+x^2}{\left (1-(-1)^{2/3} x^2\right ) \sqrt {1-x^2+x^4}} \, dx\right )\\ &=2 \left (\frac {x \sqrt {1-x^2+x^4}}{6 \left (1+x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1-x^2+x^4}}\right )}{6 \sqrt {2}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{6 \sqrt {1-x^2+x^4}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{6 \left (1-\sqrt [3]{-1}\right ) \sqrt {1-x^2+x^4}}-\frac {\left (1-\sqrt [3]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{12 \sqrt {1-x^2+x^4}}-\frac {i \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {3}{4}\right )}{4 \sqrt {3} \sqrt {1-x^2+x^4}}\right )+2 \left (\frac {x \sqrt {1-x^2+x^4}}{6 \left (1+x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1-x^2+x^4}}\right )}{6 \sqrt {2}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{6 \sqrt {1-x^2+x^4}}+\frac {\left (1-\sqrt [3]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{6 \sqrt {1-x^2+x^4}}-\frac {(-1)^{2/3} \left (1-\sqrt [3]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{12 \sqrt {1-x^2+x^4}}+\frac {i \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {3}{4}\right )}{4 \sqrt {3} \sqrt {1-x^2+x^4}}\right )+2 \left (-\frac {x \sqrt {1-x^2+x^4}}{3 \left (1+x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{3 \sqrt {1-x^2+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{12 \sqrt {1-x^2+x^4}}+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {1-x^2+x^4}}\right )\right )\\ &=2 \left (-\frac {x \sqrt {1-x^2+x^4}}{3 \left (1+x^2\right )}+\frac {1}{6} \tanh ^{-1}\left (\frac {x}{\sqrt {1-x^2+x^4}}\right )+\frac {\left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{3 \sqrt {1-x^2+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{12 \sqrt {1-x^2+x^4}}\right )+2 \left (\frac {x \sqrt {1-x^2+x^4}}{6 \left (1+x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1-x^2+x^4}}\right )}{6 \sqrt {2}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{6 \sqrt {1-x^2+x^4}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{6 \left (1-\sqrt [3]{-1}\right ) \sqrt {1-x^2+x^4}}-\frac {\left (1-\sqrt [3]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{12 \sqrt {1-x^2+x^4}}-\frac {i \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {3}{4}\right )}{4 \sqrt {3} \sqrt {1-x^2+x^4}}\right )+2 \left (\frac {x \sqrt {1-x^2+x^4}}{6 \left (1+x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1-x^2+x^4}}\right )}{6 \sqrt {2}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{6 \sqrt {1-x^2+x^4}}+\frac {\left (1-\sqrt [3]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{6 \sqrt {1-x^2+x^4}}-\frac {(-1)^{2/3} \left (1-\sqrt [3]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{12 \sqrt {1-x^2+x^4}}+\frac {i \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {3}{4}\right )}{4 \sqrt {3} \sqrt {1-x^2+x^4}}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 1.80, size = 542, normalized size = 10.23 \begin {gather*} \frac {9 \sqrt [6]{-1} \sqrt {1-\sqrt [3]{-1} x^2} \sqrt {(-1)^{2/3} x^2+1} F\left (i \sinh ^{-1}\left (\sqrt [3]{-1} x\right )|(-1)^{2/3}\right )-\frac {18 i \sqrt {1-\sqrt [3]{-1} x^2} \sqrt {(-1)^{2/3} x^2+1} \Pi \left (-1;i \sinh ^{-1}\left (\sqrt [3]{-1} x\right )|(-1)^{2/3}\right )}{\left (1+\sqrt [3]{-1}\right )^2}-6 \sqrt [6]{-1} \sqrt {1-\sqrt [3]{-1} x^2} \sqrt {(-1)^{2/3} x^2+1} \Pi \left (-(-1)^{2/3};i \sinh ^{-1}\left (\sqrt [3]{-1} x\right )|(-1)^{2/3}\right )-4 \sqrt [3]{-1} \sqrt {3} \left (1+(-1)^{2/3}\right ) \sqrt {\frac {x+(-1)^{5/6}}{\left ((-1)^{2/3}-1\right ) \left (\sqrt [6]{-1}-x\right )}} \sqrt {\frac {2 i x+i \sqrt {3}+1}{-\sqrt {3} x+i x+2}} \sqrt {\frac {\left (\sqrt {3}+3 i\right ) x+2 i \sqrt {3}}{-2+\left (\sqrt {3}-i\right ) x}} \left (\sqrt [6]{-1}-x\right )^2 \left (\sqrt [6]{-1} \left (\Pi \left (-\frac {2 \left (1+(-1)^{2/3}\right )}{(3+6 i)+(2+3 i) \sqrt {3}};\sin ^{-1}\left (\sqrt {\frac {\left (3 i+\sqrt {3}\right ) x+2 i \sqrt {3}}{\left (-i+\sqrt {3}\right ) x-2}}\right )|-\frac {1}{3}\right )-\Pi \left (\frac {2 i \left (1+(-1)^{2/3}\right )}{(-6-3 i)+(3+2 i) \sqrt {3}};\sin ^{-1}\left (\sqrt {\frac {\left (3 i+\sqrt {3}\right ) x+2 i \sqrt {3}}{\left (-i+\sqrt {3}\right ) x-2}}\right )|-\frac {1}{3}\right )\right )-F\left (\sin ^{-1}\left (\sqrt {\frac {\left (3 i+\sqrt {3}\right ) x+2 i \sqrt {3}}{\left (-i+\sqrt {3}\right ) x-2}}\right )|-\frac {1}{3}\right )\right )}{9 \sqrt {x^4-x^2+1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(1 + x^6)/(Sqrt[1 - x^2 + x^4]*(1 - x^6)),x]

[Out]

(9*(-1)^(1/6)*Sqrt[1 - (-1)^(1/3)*x^2]*Sqrt[1 + (-1)^(2/3)*x^2]*EllipticF[I*ArcSinh[(-1)^(1/3)*x], (-1)^(2/3)]
- ((18*I)*Sqrt[1 - (-1)^(1/3)*x^2]*Sqrt[1 + (-1)^(2/3)*x^2]*EllipticPi[-1, I*ArcSinh[(-1)^(1/3)*x], (-1)^(2/3
)])/(1 + (-1)^(1/3))^2 - 6*(-1)^(1/6)*Sqrt[1 - (-1)^(1/3)*x^2]*Sqrt[1 + (-1)^(2/3)*x^2]*EllipticPi[-(-1)^(2/3)
, I*ArcSinh[(-1)^(1/3)*x], (-1)^(2/3)] - 4*(-1)^(1/3)*Sqrt[3]*(1 + (-1)^(2/3))*((-1)^(1/6) - x)^2*Sqrt[((-1)^(
5/6) + x)/((-1 + (-1)^(2/3))*((-1)^(1/6) - x))]*Sqrt[(1 + I*Sqrt[3] + (2*I)*x)/(2 + I*x - Sqrt[3]*x)]*Sqrt[((2
*I)*Sqrt[3] + (3*I + Sqrt[3])*x)/(-2 + (-I + Sqrt[3])*x)]*(-EllipticF[ArcSin[Sqrt[((2*I)*Sqrt[3] + (3*I + Sqrt
[3])*x)/(-2 + (-I + Sqrt[3])*x)]], -1/3] + (-1)^(1/6)*(EllipticPi[(-2*(1 + (-1)^(2/3)))/((3 + 6*I) + (2 + 3*I)
*Sqrt[3]), ArcSin[Sqrt[((2*I)*Sqrt[3] + (3*I + Sqrt[3])*x)/(-2 + (-I + Sqrt[3])*x)]], -1/3] - EllipticPi[((2*I
)*(1 + (-1)^(2/3)))/((-6 - 3*I) + (3 + 2*I)*Sqrt[3]), ArcSin[Sqrt[((2*I)*Sqrt[3] + (3*I + Sqrt[3])*x)/(-2 + (-
I + Sqrt[3])*x)]], -1/3])))/(9*Sqrt[1 - x^2 + x^4])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.28, size = 53, normalized size = 1.00 \begin {gather*} \frac {1}{3} \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1-x^2+x^4}}\right )+\frac {1}{3} \tanh ^{-1}\left (\frac {x}{\sqrt {1-x^2+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x^6)/(Sqrt[1 - x^2 + x^4]*(1 - x^6)),x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.70, size = 62, normalized size = 1.17 \begin {gather*} \frac {1}{6} \, \sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {x^{4} - x^{2} + 1} x}{x^{4} - 3 \, x^{2} + 1}\right ) + \frac {1}{3} \, \log \left (\frac {x + \sqrt {x^{4} - x^{2} + 1}}{x^{2} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(2)*arctan(2*sqrt(2)*sqrt(x^4 - x^2 + 1)*x/(x^4 - 3*x^2 + 1)) + 1/3*log((x + sqrt(x^4 - x^2 + 1))/(x^2
- 1))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {x^{6} + 1}{{\left (x^{6} - 1\right )} \sqrt {x^{4} - x^{2} + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)/(x^4-x^2+1)^(1/2)/(-x^6+1),x, algorithm="giac")

[Out]

integrate(-(x^6 + 1)/((x^6 - 1)*sqrt(x^4 - x^2 + 1)), x)

________________________________________________________________________________________

maple [A]  time = 0.67, size = 52, normalized size = 0.98

method	result	size
elliptic	\(\frac {\left (\frac {\sqrt {2}\, \arctanh \left (\frac {\sqrt {x^{4}-x^{2}+1}}{x}\right )}{3}-\frac {2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x^{4}-x^{2}+1}}{2 x}\right )}{3}\right ) \sqrt {2}}{2}\)	\(52\)
trager	\(-\frac {\ln \left (-\frac {\sqrt {x^{4}-x^{2}+1}-x}{\left (-1+x \right ) \left (1+x \right )}\right )}{3}+\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) x^{4}-3 \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{2}+4 \sqrt {x^{4}-x^{2}+1}\, x +\RootOf \left (\textit {\_Z}^{2}+2\right )}{\left (x^{2}-x +1\right ) \left (x^{2}+x +1\right )}\right )}{6}\)	\(105\)
default	\(-\frac {2 \sqrt {1-\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2-2 i \sqrt {3}}}{2}\right )}{\sqrt {2+2 i \sqrt {3}}\, \sqrt {x^{4}-x^{2}+1}}+\frac {2 \sqrt {1-\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , \frac {1}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}-x^{2}+1}}-\frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\arctanh \left (\frac {\left (-2+i \sqrt {3}\right ) \left (7 x^{2}-\frac {13}{2}-\frac {3 i \sqrt {3}}{2}\right )}{14 \sqrt {1-i \sqrt {3}}\, \sqrt {x^{4}-x^{2}+1}}\right )}{2 \sqrt {1-i \sqrt {3}}}-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {1-\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , \frac {i \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{2}-\frac {1}{4}-\frac {i \sqrt {3}}{4}, \frac {\sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}-x^{2}+1}}\right )}{3}-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\arctanh \left (\frac {\left (-2-i \sqrt {3}\right ) \left (7 x^{2}-\frac {13}{2}+\frac {3 i \sqrt {3}}{2}\right )}{14 \sqrt {1+i \sqrt {3}}\, \sqrt {x^{4}-x^{2}+1}}\right )}{2 \sqrt {1+i \sqrt {3}}}-\frac {\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {1-\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , \frac {i \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{2}-\frac {1}{4}+\frac {i \sqrt {3}}{4}, \frac {\sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{4}-x^{2}+1}}\right )}{3}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {\arctanh \left (\frac {\left (2+i \sqrt {3}\right ) \left (7 x^{2}-\frac {13}{2}+\frac {3 i \sqrt {3}}{2}\right )}{14 \sqrt {1+i \sqrt {3}}\, \sqrt {x^{4}-x^{2}+1}}\right )}{2 \sqrt {1+i \sqrt {3}}}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {1-\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {i \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{2}-\frac {1}{4}+\frac {i \sqrt {3}}{4}, \frac {\sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}-x^{2}+1}}\right )}{3}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (\frac {\arctanh \left (\frac {\left (2-i \sqrt {3}\right ) \left (7 x^{2}-\frac {13}{2}-\frac {3 i \sqrt {3}}{2}\right )}{14 \sqrt {1-i \sqrt {3}}\, \sqrt {x^{4}-x^{2}+1}}\right )}{2 \sqrt {1-i \sqrt {3}}}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {1-\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{2}-\frac {1}{4}-\frac {i \sqrt {3}}{4}, \frac {\sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}-x^{2}+1}}\right )}{3}\)	\(944\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {x^{6} + 1}{{\left (x^{6} - 1\right )} \sqrt {x^{4} - x^{2} + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)/(x^4-x^2+1)^(1/2)/(-x^6+1),x, algorithm="maxima")

[Out]

-integrate((x^6 + 1)/((x^6 - 1)*sqrt(x^4 - x^2 + 1)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int -\frac {x^6+1}{\left (x^6-1\right )\,\sqrt {x^4-x^2+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {\sqrt {x^{4} - x^{2} + 1}}{x^{6} - 1}\, dx - \int \frac {x^{2} \sqrt {x^{4} - x^{2} + 1}}{x^{6} - 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(sqrt(x**4 - x**2 + 1)/(x**6 - 1), x) - Integral(x**2*sqrt(x**4 - x**2 + 1)/(x**6 - 1), x)

________________________________________________________________________________________