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

Optimal. Leaf size=26 \[ \frac {\sqrt {x^4-1}}{x}+\tan ^{-1}\left (\frac {x}{\sqrt {x^4-1}}\right ) \]

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Rubi [C]  time = 0.98, antiderivative size = 291, normalized size of antiderivative = 11.19, number of steps used = 32, number of rules used = 13, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {6728, 277, 306, 222, 1185, 1209, 1188, 1215, 1457, 540, 253, 538, 537} \begin {gather*} \frac {\sqrt {x^4-1}}{x}+\frac {\left (1+\sqrt {5}\right ) \sqrt {x^2-1} \sqrt {x^2+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {x^4-1}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {x^2-1} \sqrt {x^2+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {x^4-1}}-\frac {\sqrt {2} \sqrt {x^2-1} \sqrt {x^2+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{\sqrt {x^4-1}}+\frac {\sqrt {1-x^2} \sqrt {x^2+1} \Pi \left (\frac {1}{2} \left (1-\sqrt {5}\right );\left .\sin ^{-1}(x)\right |-1\right )}{\sqrt {x^4-1}}+\frac {\sqrt {1-x^2} \sqrt {x^2+1} \Pi \left (\frac {1}{2} \left (1+\sqrt {5}\right );\left .\sin ^{-1}(x)\right |-1\right )}{\sqrt {x^4-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[-1 + x^4]/x - (Sqrt[2]*Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[(Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/2])/S
qrt[-1 + x^4] + ((1 - Sqrt[5])*Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[(Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/2]
)/(2*Sqrt[2]*Sqrt[-1 + x^4]) + ((1 + Sqrt[5])*Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[(Sqrt[2]*x)/Sqrt[-
1 + x^2]], 1/2])/(2*Sqrt[2]*Sqrt[-1 + x^4]) + (Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[(1 - Sqrt[5])/2, ArcSin[
x], -1])/Sqrt[-1 + x^4] + (Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[(1 + Sqrt[5])/2, ArcSin[x], -1])/Sqrt[-1 + x
^4]

Rule 222

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

Rule 253

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

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 306

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

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rule 538

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

Rule 540

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

Rule 1185

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

Rule 1188

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

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 1215

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

Rule 1457

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((f_) + (g_.)*(x_)^(n_))^(r_.)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :>
Dist[(a + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPart[p]), Int[(d + e*x^n)^(p
+ q)*(f + g*x^n)^r*(a/d + (c*x^n)/e)^p, x], x] /; FreeQ[{a, c, d, e, f, g, n, p, q, r}, x] && EqQ[n2, 2*n] &&
EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p]

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 {\sqrt {-1+x^4} \left (1+x^4\right )}{x^2 \left (-1+x^2+x^4\right )} \, dx &=\int \left (-\frac {\sqrt {-1+x^4}}{x^2}+\frac {\left (1+2 x^2\right ) \sqrt {-1+x^4}}{-1+x^2+x^4}\right ) \, dx\\ &=-\int \frac {\sqrt {-1+x^4}}{x^2} \, dx+\int \frac {\left (1+2 x^2\right ) \sqrt {-1+x^4}}{-1+x^2+x^4} \, dx\\ &=\frac {\sqrt {-1+x^4}}{x}-2 \int \frac {x^2}{\sqrt {-1+x^4}} \, dx+\int \left (\frac {2 \sqrt {-1+x^4}}{1-\sqrt {5}+2 x^2}+\frac {2 \sqrt {-1+x^4}}{1+\sqrt {5}+2 x^2}\right ) \, dx\\ &=\frac {\sqrt {-1+x^4}}{x}-2 \int \frac {1}{\sqrt {-1+x^4}} \, dx+2 \int \frac {1-x^2}{\sqrt {-1+x^4}} \, dx+2 \int \frac {\sqrt {-1+x^4}}{1-\sqrt {5}+2 x^2} \, dx+2 \int \frac {\sqrt {-1+x^4}}{1+\sqrt {5}+2 x^2} \, dx\\ &=-\frac {2 x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {-1+x^4}}{x}+\frac {2 \sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}-\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}-\frac {1}{2} \int \frac {1-\sqrt {5}-2 x^2}{\sqrt {-1+x^4}} \, dx-\frac {1}{2} \int \frac {1+\sqrt {5}-2 x^2}{\sqrt {-1+x^4}} \, dx+\left (1-\sqrt {5}\right ) \int \frac {1}{\left (1-\sqrt {5}+2 x^2\right ) \sqrt {-1+x^4}} \, dx+\left (1+\sqrt {5}\right ) \int \frac {1}{\left (1+\sqrt {5}+2 x^2\right ) \sqrt {-1+x^4}} \, dx\\ &=-\frac {2 x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {-1+x^4}}{x}+\frac {2 \sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}-\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}-\frac {1}{2} \left (-1-\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx+\frac {\left (1-\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx}{3-\sqrt {5}}+\frac {\left (2 \left (1-\sqrt {5}\right )\right ) \int \frac {1-x^2}{\left (1-\sqrt {5}+2 x^2\right ) \sqrt {-1+x^4}} \, dx}{3-\sqrt {5}}-\frac {1}{2} \left (-1+\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx+\frac {\left (1+\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx}{3+\sqrt {5}}+\frac {\left (2 \left (1+\sqrt {5}\right )\right ) \int \frac {1-x^2}{\left (1+\sqrt {5}+2 x^2\right ) \sqrt {-1+x^4}} \, dx}{3+\sqrt {5}}-2 \int \frac {1-x^2}{\sqrt {-1+x^4}} \, dx\\ &=-\frac {2 x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {-1+x^4}}{x}+\frac {2 \sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}-2 \left (-\frac {x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}\right )-\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {-1+x^4}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (3-\sqrt {5}\right ) \sqrt {-1+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {-1+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (3+\sqrt {5}\right ) \sqrt {-1+x^4}}+\frac {\left (2 \left (1-\sqrt {5}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2} \left (1-\sqrt {5}+2 x^2\right )} \, dx}{\left (3-\sqrt {5}\right ) \sqrt {-1+x^4}}+\frac {\left (2 \left (1+\sqrt {5}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2} \left (1+\sqrt {5}+2 x^2\right )} \, dx}{\left (3+\sqrt {5}\right ) \sqrt {-1+x^4}}\\ &=-\frac {2 x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {-1+x^4}}{x}+\frac {2 \sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}-2 \left (-\frac {x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}\right )-\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {-1+x^4}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (3-\sqrt {5}\right ) \sqrt {-1+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {-1+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (3+\sqrt {5}\right ) \sqrt {-1+x^4}}+\frac {\left (\left (1-\sqrt {5}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2} \left (1-\sqrt {5}+2 x^2\right )} \, dx}{\sqrt {-1+x^4}}-\frac {\left (\left (1-\sqrt {5}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2}} \, dx}{\left (3-\sqrt {5}\right ) \sqrt {-1+x^4}}+\frac {\left (\left (1+\sqrt {5}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2} \left (1+\sqrt {5}+2 x^2\right )} \, dx}{\sqrt {-1+x^4}}-\frac {\left (\left (1+\sqrt {5}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2}} \, dx}{\left (3+\sqrt {5}\right ) \sqrt {-1+x^4}}\\ &=-\frac {2 x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {-1+x^4}}{x}+\frac {2 \sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}-2 \left (-\frac {x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}\right )-\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {-1+x^4}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (3-\sqrt {5}\right ) \sqrt {-1+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {-1+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (3+\sqrt {5}\right ) \sqrt {-1+x^4}}-\frac {\left (1-\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx}{3-\sqrt {5}}-\frac {\left (1+\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx}{3+\sqrt {5}}+\frac {\left (\left (1-\sqrt {5}\right ) \sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1-\sqrt {5}+2 x^2\right )} \, dx}{\sqrt {-1+x^4}}+\frac {\left (\left (1+\sqrt {5}\right ) \sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1+\sqrt {5}+2 x^2\right )} \, dx}{\sqrt {-1+x^4}}\\ &=-\frac {2 x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {-1+x^4}}{x}+\frac {2 \sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}-2 \left (-\frac {x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}\right )-\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {-1+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {-1+x^4}}+\frac {\sqrt {1-x^2} \sqrt {1+x^2} \Pi \left (\frac {1}{2} \left (1-\sqrt {5}\right );\left .\sin ^{-1}(x)\right |-1\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {1-x^2} \sqrt {1+x^2} \Pi \left (\frac {1}{2} \left (1+\sqrt {5}\right );\left .\sin ^{-1}(x)\right |-1\right )}{\sqrt {-1+x^4}}\\ \end {align*}

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Mathematica [C]  time = 0.42, size = 92, normalized size = 3.54 \begin {gather*} \frac {x^4-\sqrt {1-x^4} x F\left (\left .\sin ^{-1}(x)\right |-1\right )+\sqrt {1-x^4} x \Pi \left (-\frac {2}{1+\sqrt {5}};\left .\sin ^{-1}(x)\right |-1\right )+\sqrt {1-x^4} x \Pi \left (\frac {1}{2} \left (1+\sqrt {5}\right );\left .\sin ^{-1}(x)\right |-1\right )-1}{x \sqrt {x^4-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1 + x^4 - x*Sqrt[1 - x^4]*EllipticF[ArcSin[x], -1] + x*Sqrt[1 - x^4]*EllipticPi[-2/(1 + Sqrt[5]), ArcSin[x],
-1] + x*Sqrt[1 - x^4]*EllipticPi[(1 + Sqrt[5])/2, ArcSin[x], -1])/(x*Sqrt[-1 + x^4])

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IntegrateAlgebraic [A]  time = 0.35, size = 26, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^4}}{x}+\tan ^{-1}\left (\frac {x}{\sqrt {-1+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[-1 + x^4]/x + ArcTan[x/Sqrt[-1 + x^4]]

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fricas [A]  time = 0.61, size = 40, normalized size = 1.54 \begin {gather*} \frac {x \arctan \left (\frac {2 \, \sqrt {x^{4} - 1} x}{x^{4} - x^{2} - 1}\right ) + 2 \, \sqrt {x^{4} - 1}}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.54, size = 38, normalized size = 1.46

method	result	size
elliptic	\(\frac {\left (\frac {\sqrt {x^{4}-1}\, \sqrt {2}}{x}-\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}-1}}{x}\right )\right ) \sqrt {2}}{2}\)	\(38\)
trager	\(\frac {\sqrt {x^{4}-1}}{x}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-2 x \sqrt {x^{4}-1}-\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{4}+x^{2}-1}\right )}{2}\)	\(73\)
default	\(\frac {\sqrt {x^{4}-1}}{x}+\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticF \left (i x , i\right )}{\sqrt {x^{4}-1}}-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z}^{2}-1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}-1\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {x^{4}-1}}\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {2 i \left (\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticPi \left (i x , -\underline {\hspace {1.25 ex}}\alpha ^{2}-1, i\right )}{\sqrt {x^{4}-1}}\right )\right )}{4}\)	\(146\)
risch	\(\frac {\sqrt {x^{4}-1}}{x}+\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticF \left (i x , i\right )}{\sqrt {x^{4}-1}}-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z}^{2}-1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}-1\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {x^{4}-1}}\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {2 i \left (\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticPi \left (i x , -\underline {\hspace {1.25 ex}}\alpha ^{2}-1, i\right )}{\sqrt {x^{4}-1}}\right )\right )}{4}\)	\(146\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\sqrt {x^4-1}\,\left (x^4+1\right )}{x^2\,\left (x^4+x^2-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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