3.7.71 \(\int \frac {(-1+x^4) \sqrt {1+x^4}}{(1+3 x^2+x^4)^2} \, dx\)

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

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Rubi [C]  time = 2.88, antiderivative size = 1165, normalized size of antiderivative = 21.98, number of steps used = 80, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6742, 1227, 1198, 220, 1196, 1217, 1707, 1209, 6728}

result too large to display

Antiderivative was successfully verified.

[In]

Int[((-1 + x^4)*Sqrt[1 + x^4])/(1 + 3*x^2 + x^4)^2,x]

[Out]

(-3*x*Sqrt[1 + x^4])/(5*(1 + x^2)) + (2*x*Sqrt[1 + x^4])/(5*(3 - Sqrt[5])*(1 + x^2)) + (2*x*Sqrt[1 + x^4])/(5*
(3 + Sqrt[5])*(1 + x^2)) + (3*x*Sqrt[1 + x^4])/(5*(3 - Sqrt[5] + 2*x^2)) - (4*x*Sqrt[1 + x^4])/(5*(3 - Sqrt[5]
)*(3 - Sqrt[5] + 2*x^2)) + (3*x*Sqrt[1 + x^4])/(5*(3 + Sqrt[5] + 2*x^2)) - (4*x*Sqrt[1 + x^4])/(5*(3 + Sqrt[5]
)*(3 + Sqrt[5] + 2*x^2)) - (Sqrt[3]*(11 - 5*Sqrt[5])*ArcTan[(Sqrt[3]*x)/Sqrt[1 + x^4]])/(4*(25 - 11*Sqrt[5]))
+ ((2 - Sqrt[5])*ArcTan[(Sqrt[3]*x)/Sqrt[1 + x^4]])/(Sqrt[3]*(25 - 11*Sqrt[5])) + ((2 + Sqrt[5])*ArcTan[(Sqrt[
3]*x)/Sqrt[1 + x^4]])/(Sqrt[3]*(25 + 11*Sqrt[5])) - (Sqrt[3]*(2 + Sqrt[5])*(3 + Sqrt[5])*ArcTan[(Sqrt[3]*x)/Sq
rt[1 + x^4]])/(4*(25 + 11*Sqrt[5])) + (3*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticE[2*ArcTan[x], 1/2])/(5
*Sqrt[1 + x^4]) - (2*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticE[2*ArcTan[x], 1/2])/(5*(3 - Sqrt[5])*Sqrt[
1 + x^4]) - (2*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticE[2*ArcTan[x], 1/2])/(5*(3 + Sqrt[5])*Sqrt[1 + x^
4]) + (3*(1 - Sqrt[5])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(40*Sqrt[1 + x^4]) -
((1 - Sqrt[5])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(10*(3 - Sqrt[5])*Sqrt[1 +
x^4]) - (3*(5 - Sqrt[5])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(40*Sqrt[1 + x^4])
+ ((5 - Sqrt[5])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(10*(3 - Sqrt[5])*Sqrt[1
+ x^4]) + (3*(1 + Sqrt[5])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(40*Sqrt[1 + x^4
]) - ((1 + Sqrt[5])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(10*(3 + Sqrt[5])*Sqrt[
1 + x^4]) - (3*(5 + Sqrt[5])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(40*Sqrt[1 + x
^4]) + ((5 + Sqrt[5])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(10*(3 + Sqrt[5])*Sqr
t[1 + x^4]) + (3*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticPi[-1/4, 2*ArcTan[x], 1/2])/(4*Sqrt[1 + x^4]) -
((5 - 2*Sqrt[5])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticPi[-1/4, 2*ArcTan[x], 1/2])/(2*(25 - 11*Sqrt[5
])*Sqrt[1 + x^4]) - ((1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticPi[-1/4, 2*ArcTan[x], 1/2])/(2*(3 + Sqrt[5]
)*Sqrt[1 + x^4])

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 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 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 1227

Int[Sqrt[(a_) + (c_.)*(x_)^4]/((d_) + (e_.)*(x_)^2)^2, x_Symbol] :> Simp[(x*Sqrt[a + c*x^4])/(2*d*(d + e*x^2))
, x] + (Dist[c/(2*d*e^2), Int[(d - e*x^2)/Sqrt[a + c*x^4], x], x] - Dist[(c*d^2 - a*e^2)/(2*d*e^2), Int[1/((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]

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 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]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^4\right ) \sqrt {1+x^4}}{\left (1+3 x^2+x^4\right )^2} \, dx &=\int \left (\frac {\left (-2-3 x^2\right ) \sqrt {1+x^4}}{\left (1+3 x^2+x^4\right )^2}+\frac {\sqrt {1+x^4}}{1+3 x^2+x^4}\right ) \, dx\\ &=\int \frac {\left (-2-3 x^2\right ) \sqrt {1+x^4}}{\left (1+3 x^2+x^4\right )^2} \, dx+\int \frac {\sqrt {1+x^4}}{1+3 x^2+x^4} \, dx\\ &=\int \left (-\frac {2 \sqrt {1+x^4}}{\sqrt {5} \left (-3+\sqrt {5}-2 x^2\right )}-\frac {2 \sqrt {1+x^4}}{\sqrt {5} \left (3+\sqrt {5}+2 x^2\right )}\right ) \, dx+\int \left (-\frac {2 \sqrt {1+x^4}}{\left (1+3 x^2+x^4\right )^2}-\frac {3 x^2 \sqrt {1+x^4}}{\left (1+3 x^2+x^4\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {1+x^4}}{\left (1+3 x^2+x^4\right )^2} \, dx\right )-3 \int \frac {x^2 \sqrt {1+x^4}}{\left (1+3 x^2+x^4\right )^2} \, dx-\frac {2 \int \frac {\sqrt {1+x^4}}{-3+\sqrt {5}-2 x^2} \, dx}{\sqrt {5}}-\frac {2 \int \frac {\sqrt {1+x^4}}{3+\sqrt {5}+2 x^2} \, dx}{\sqrt {5}}\\ &=-\left (2 \int \left (\frac {4 \sqrt {1+x^4}}{5 \left (-3+\sqrt {5}-2 x^2\right )^2}+\frac {4 \sqrt {1+x^4}}{5 \sqrt {5} \left (-3+\sqrt {5}-2 x^2\right )}+\frac {4 \sqrt {1+x^4}}{5 \left (3+\sqrt {5}+2 x^2\right )^2}+\frac {4 \sqrt {1+x^4}}{5 \sqrt {5} \left (3+\sqrt {5}+2 x^2\right )}\right ) \, dx\right )-3 \int \left (\frac {2 \left (-3+\sqrt {5}\right ) \sqrt {1+x^4}}{5 \left (-3+\sqrt {5}-2 x^2\right )^2}-\frac {6 \sqrt {1+x^4}}{5 \sqrt {5} \left (-3+\sqrt {5}-2 x^2\right )}+\frac {2 \left (-3-\sqrt {5}\right ) \sqrt {1+x^4}}{5 \left (3+\sqrt {5}+2 x^2\right )^2}-\frac {6 \sqrt {1+x^4}}{5 \sqrt {5} \left (3+\sqrt {5}+2 x^2\right )}\right ) \, dx+\frac {\int \frac {3+\sqrt {5}-2 x^2}{\sqrt {1+x^4}} \, dx}{2 \sqrt {5}}+\frac {\int \frac {-3+\sqrt {5}+2 x^2}{\sqrt {1+x^4}} \, dx}{2 \sqrt {5}}+\frac {1}{5} \left (3 \left (5-3 \sqrt {5}\right )\right ) \int \frac {1}{\left (-3+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{5} \left (3 \left (5+3 \sqrt {5}\right )\right ) \int \frac {1}{\left (3+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx\\ &=-\left (\frac {8}{5} \int \frac {\sqrt {1+x^4}}{\left (-3+\sqrt {5}-2 x^2\right )^2} \, dx\right )-\frac {8}{5} \int \frac {\sqrt {1+x^4}}{\left (3+\sqrt {5}+2 x^2\right )^2} \, dx-\frac {8 \int \frac {\sqrt {1+x^4}}{-3+\sqrt {5}-2 x^2} \, dx}{5 \sqrt {5}}-\frac {8 \int \frac {\sqrt {1+x^4}}{3+\sqrt {5}+2 x^2} \, dx}{5 \sqrt {5}}+\frac {18 \int \frac {\sqrt {1+x^4}}{-3+\sqrt {5}-2 x^2} \, dx}{5 \sqrt {5}}+\frac {18 \int \frac {\sqrt {1+x^4}}{3+\sqrt {5}+2 x^2} \, dx}{5 \sqrt {5}}+\frac {1}{5} \left (6 \left (3-\sqrt {5}\right )\right ) \int \frac {\sqrt {1+x^4}}{\left (-3+\sqrt {5}-2 x^2\right )^2} \, dx+\frac {1}{10} \left (5-\sqrt {5}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx+\frac {1}{5} \left (6 \left (3+\sqrt {5}\right )\right ) \int \frac {\sqrt {1+x^4}}{\left (3+\sqrt {5}+2 x^2\right )^2} \, dx+\frac {1}{10} \left (5+\sqrt {5}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx+\frac {\left (3 \left (5-3 \sqrt {5}\right ) \left (-5+\sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{5 \left (-4+\left (-3+\sqrt {5}\right )^2\right )}+\frac {\left (6 \left (5-3 \sqrt {5}\right ) \left (-5+\sqrt {5}\right )\right ) \int \frac {1+x^2}{\left (-3+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}} \, dx}{5 \left (-4+\left (-3+\sqrt {5}\right )^2\right )}-\frac {\left (3 \left (5+\sqrt {5}\right ) \left (5+3 \sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{5 \left (-4+\left (3+\sqrt {5}\right )^2\right )}+\frac {\left (6 \left (5+\sqrt {5}\right ) \left (5+3 \sqrt {5}\right )\right ) \int \frac {1+x^2}{\left (3+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx}{5 \left (-4+\left (3+\sqrt {5}\right )^2\right )}\\ &=\frac {3 x \sqrt {1+x^4}}{5 \left (3-\sqrt {5}+2 x^2\right )}-\frac {4 x \sqrt {1+x^4}}{5 \left (3-\sqrt {5}\right ) \left (3-\sqrt {5}+2 x^2\right )}+\frac {3 x \sqrt {1+x^4}}{5 \left (3+\sqrt {5}+2 x^2\right )}-\frac {4 x \sqrt {1+x^4}}{5 \left (3+\sqrt {5}\right ) \left (3+\sqrt {5}+2 x^2\right )}-\frac {\sqrt {3} \left (1-\sqrt {5}\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+x^4}}\right )}{2 \left (5-\sqrt {5}\right )}-\frac {\sqrt {3} \left (1+\sqrt {5}\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+x^4}}\right )}{2 \left (5+\sqrt {5}\right )}-\frac {3 \left (3-\sqrt {5}\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 )}{2 \left (5-\sqrt {5}\right ) \sqrt {1+x^4}}+\frac {\left (5-\sqrt {5}\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 )}{20 \sqrt {1+x^4}}-\frac {3 \left (3+\sqrt {5}\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 )}{2 \left (5+\sqrt {5}\right ) \sqrt {1+x^4}}+\frac {\left (5+\sqrt {5}\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 )}{20 \sqrt {1+x^4}}+\frac {3 \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 )}{4 \sqrt {1+x^4}}+\frac {3 \left (5+5 \sqrt {5}\right ) \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 )}{20 \left (1+\sqrt {5}\right ) \sqrt {1+x^4}}+\frac {3}{20} \int \frac {3+\sqrt {5}-2 x^2}{\sqrt {1+x^4}} \, dx-\frac {3}{20} \int \frac {-3+\sqrt {5}+2 x^2}{\sqrt {1+x^4}} \, dx+\frac {2 \int \frac {3+\sqrt {5}-2 x^2}{\sqrt {1+x^4}} \, dx}{5 \sqrt {5}}+\frac {2 \int \frac {-3+\sqrt {5}+2 x^2}{\sqrt {1+x^4}} \, dx}{5 \sqrt {5}}-\frac {9 \int \frac {3+\sqrt {5}-2 x^2}{\sqrt {1+x^4}} \, dx}{10 \sqrt {5}}-\frac {9 \int \frac {-3+\sqrt {5}+2 x^2}{\sqrt {1+x^4}} \, dx}{10 \sqrt {5}}+\frac {1}{10} \left (3 \left (5-3 \sqrt {5}\right )\right ) \int \frac {1}{\left (-3+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}} \, dx+\frac {1}{25} \left (12 \left (5-3 \sqrt {5}\right )\right ) \int \frac {1}{\left (-3+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{25} \left (27 \left (5-3 \sqrt {5}\right )\right ) \int \frac {1}{\left (-3+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}} \, dx-\frac {\int \frac {-3+\sqrt {5}+2 x^2}{\sqrt {1+x^4}} \, dx}{5 \left (-3+\sqrt {5}\right )}-\frac {\int \frac {3+\sqrt {5}-2 x^2}{\sqrt {1+x^4}} \, dx}{5 \left (3+\sqrt {5}\right )}-\frac {1}{10} \left (3 \left (5+3 \sqrt {5}\right )\right ) \int \frac {1}{\left (3+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{25} \left (12 \left (5+3 \sqrt {5}\right )\right ) \int \frac {1}{\left (3+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx+\frac {1}{25} \left (27 \left (5+3 \sqrt {5}\right )\right ) \int \frac {1}{\left (3+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx+\frac {\left (-4+\left (-3+\sqrt {5}\right )^2\right ) \int \frac {1}{\left (-3+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}} \, dx}{5 \left (-3+\sqrt {5}\right )}+\frac {\left (-4+\left (3+\sqrt {5}\right )^2\right ) \int \frac {1}{\left (3+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx}{5 \left (3+\sqrt {5}\right )}\\ &=\frac {3 x \sqrt {1+x^4}}{5 \left (3-\sqrt {5}+2 x^2\right )}-\frac {4 x \sqrt {1+x^4}}{5 \left (3-\sqrt {5}\right ) \left (3-\sqrt {5}+2 x^2\right )}+\frac {3 x \sqrt {1+x^4}}{5 \left (3+\sqrt {5}+2 x^2\right )}-\frac {4 x \sqrt {1+x^4}}{5 \left (3+\sqrt {5}\right ) \left (3+\sqrt {5}+2 x^2\right )}-\frac {\sqrt {3} \left (1-\sqrt {5}\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+x^4}}\right )}{2 \left (5-\sqrt {5}\right )}-\frac {\sqrt {3} \left (1+\sqrt {5}\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+x^4}}\right )}{2 \left (5+\sqrt {5}\right )}-\frac {3 \left (3-\sqrt {5}\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 )}{2 \left (5-\sqrt {5}\right ) \sqrt {1+x^4}}+\frac {\left (5-\sqrt {5}\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 )}{20 \sqrt {1+x^4}}-\frac {3 \left (3+\sqrt {5}\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 )}{2 \left (5+\sqrt {5}\right ) \sqrt {1+x^4}}+\frac {\left (5+\sqrt {5}\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 )}{20 \sqrt {1+x^4}}+\frac {3 \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 )}{4 \sqrt {1+x^4}}+\frac {3 \left (5+5 \sqrt {5}\right ) \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 )}{20 \left (1+\sqrt {5}\right ) \sqrt {1+x^4}}+2 \left (\frac {3}{10} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx\right )+\frac {1}{20} \left (3 \left (1-\sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx+\frac {1}{25} \left (2 \left (5-\sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {1}{50} \left (9 \left (5-\sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx+\frac {2 \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx}{5 \left (-3+\sqrt {5}\right )}+\frac {\left (-5+\sqrt {5}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{5 \left (-3+\sqrt {5}\right )}+\frac {\left (2 \left (-5+\sqrt {5}\right )\right ) \int \frac {1+x^2}{\left (-3+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}} \, dx}{5 \left (-3+\sqrt {5}\right )}-\frac {\left (-1+\sqrt {5}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{5 \left (-3+\sqrt {5}\right )}+\frac {1}{20} \left (3 \left (1+\sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {2 \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx}{5 \left (3+\sqrt {5}\right )}-\frac {\left (1+\sqrt {5}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{5 \left (3+\sqrt {5}\right )}+\frac {1}{25} \left (2 \left (5+\sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {1}{50} \left (9 \left (5+\sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx+\frac {\left (5+\sqrt {5}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{5 \left (3+\sqrt {5}\right )}-\frac {\left (2 \left (5+\sqrt {5}\right )\right ) \int \frac {1+x^2}{\left (3+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx}{5 \left (3+\sqrt {5}\right )}+\frac {\left (3 \left (5-3 \sqrt {5}\right ) \left (-5+\sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{10 \left (-4+\left (-3+\sqrt {5}\right )^2\right )}+\frac {\left (12 \left (5-3 \sqrt {5}\right ) \left (-5+\sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{25 \left (-4+\left (-3+\sqrt {5}\right )^2\right )}+\frac {\left (3 \left (5-3 \sqrt {5}\right ) \left (-5+\sqrt {5}\right )\right ) \int \frac {1+x^2}{\left (-3+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}} \, dx}{5 \left (-4+\left (-3+\sqrt {5}\right )^2\right )}+\frac {\left (24 \left (5-3 \sqrt {5}\right ) \left (-5+\sqrt {5}\right )\right ) \int \frac {1+x^2}{\left (-3+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}} \, dx}{25 \left (-4+\left (-3+\sqrt {5}\right )^2\right )}-\frac {\left (27 \left (5-3 \sqrt {5}\right ) \left (-5+\sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{25 \left (-4+\left (-3+\sqrt {5}\right )^2\right )}-\frac {\left (54 \left (5-3 \sqrt {5}\right ) \left (-5+\sqrt {5}\right )\right ) \int \frac {1+x^2}{\left (-3+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}} \, dx}{25 \left (-4+\left (-3+\sqrt {5}\right )^2\right )}-\frac {\left (3 \left (5+\sqrt {5}\right ) \left (5+3 \sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{10 \left (-4+\left (3+\sqrt {5}\right )^2\right )}-\frac {\left (12 \left (5+\sqrt {5}\right ) \left (5+3 \sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{25 \left (-4+\left (3+\sqrt {5}\right )^2\right )}+\frac {\left (3 \left (5+\sqrt {5}\right ) \left (5+3 \sqrt {5}\right )\right ) \int \frac {1+x^2}{\left (3+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx}{5 \left (-4+\left (3+\sqrt {5}\right )^2\right )}+\frac {\left (24 \left (5+\sqrt {5}\right ) \left (5+3 \sqrt {5}\right )\right ) \int \frac {1+x^2}{\left (3+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx}{25 \left (-4+\left (3+\sqrt {5}\right )^2\right )}+\frac {\left (27 \left (5+\sqrt {5}\right ) \left (5+3 \sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{25 \left (-4+\left (3+\sqrt {5}\right )^2\right )}-\frac {\left (54 \left (5+\sqrt {5}\right ) \left (5+3 \sqrt {5}\right )\right ) \int \frac {1+x^2}{\left (3+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx}{25 \left (-4+\left (3+\sqrt {5}\right )^2\right )}\\ &=\frac {2 x \sqrt {1+x^4}}{5 \left (3-\sqrt {5}\right ) \left (1+x^2\right )}+\frac {2 x \sqrt {1+x^4}}{5 \left (3+\sqrt {5}\right ) \left (1+x^2\right )}+\frac {3 x \sqrt {1+x^4}}{5 \left (3-\sqrt {5}+2 x^2\right )}-\frac {4 x \sqrt {1+x^4}}{5 \left (3-\sqrt {5}\right ) \left (3-\sqrt {5}+2 x^2\right )}+\frac {3 x \sqrt {1+x^4}}{5 \left (3+\sqrt {5}+2 x^2\right )}-\frac {4 x \sqrt {1+x^4}}{5 \left (3+\sqrt {5}\right ) \left (3+\sqrt {5}+2 x^2\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+x^4}}\right )}{\sqrt {15} \left (3-\sqrt {5}\right )}-\frac {\sqrt {3} \left (1-\sqrt {5}\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+x^4}}\right )}{4 \left (5-\sqrt {5}\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+x^4}}\right )}{\sqrt {15} \left (3+\sqrt {5}\right )}-\frac {\sqrt {3} \left (1+\sqrt {5}\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+x^4}}\right )}{4 \left (5+\sqrt {5}\right )}-\frac {2 \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 )}{5 \left (3-\sqrt {5}\right ) \sqrt {1+x^4}}-\frac {2 \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 )}{5 \left (3+\sqrt {5}\right ) \sqrt {1+x^4}}+2 \left (-\frac {3 x \sqrt {1+x^4}}{10 \left (1+x^2\right )}+\frac {3 \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 )}{10 \sqrt {1+x^4}}\right )+\frac {3 \left (1-\sqrt {5}\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 )}{40 \sqrt {1+x^4}}-\frac {\left (1-\sqrt {5}\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 )}{10 \left (3-\sqrt {5}\right ) \sqrt {1+x^4}}-\frac {3 \left (3-\sqrt {5}\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 )}{4 \left (5-\sqrt {5}\right ) \sqrt {1+x^4}}+\frac {\left (5-\sqrt {5}\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 )}{10 \left (3-\sqrt {5}\right ) \sqrt {1+x^4}}+\frac {3 \left (1+\sqrt {5}\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 )}{40 \sqrt {1+x^4}}-\frac {\left (1+\sqrt {5}\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 )}{10 \left (3+\sqrt {5}\right ) \sqrt {1+x^4}}-\frac {3 \left (3+\sqrt {5}\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 )}{4 \left (5+\sqrt {5}\right ) \sqrt {1+x^4}}+\frac {\left (5+\sqrt {5}\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 )}{10 \left (3+\sqrt {5}\right ) \sqrt {1+x^4}}+\frac {3 \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 {1+x^4}}-\frac {\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 )}{2 \left (3-\sqrt {5}\right ) \sqrt {1+x^4}}-\frac {\left (5+\sqrt {5}\right )^2 \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 )}{20 \left (3+\sqrt {5}\right )^2 \sqrt {1+x^4}}+\frac {3 \left (5+5 \sqrt {5}\right ) \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 )}{40 \left (1+\sqrt {5}\right ) \sqrt {1+x^4}}\\ \end {align*}

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Mathematica [C]  time = 1.37, size = 257, normalized size = 4.85 \begin {gather*} -\frac {x^5-\sqrt [4]{-1} \sqrt {x^4+1} x^4 \Pi \left (-\frac {2 i}{3+\sqrt {5}};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )-\sqrt [4]{-1} \sqrt {x^4+1} \Pi \left (-\frac {2 i}{3+\sqrt {5}};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\sqrt [4]{-1} \sqrt {x^4+1} \left (x^4+3 x^2+1\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )-3 \sqrt [4]{-1} \sqrt {x^4+1} x^2 \Pi \left (-\frac {2 i}{3+\sqrt {5}};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )-\sqrt [4]{-1} \sqrt {x^4+1} \left (x^4+3 x^2+1\right ) \Pi \left (\frac {2 i}{-3+\sqrt {5}};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+x}{2 \sqrt {x^4+1} \left (x^4+3 x^2+1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-1 + x^4)*Sqrt[1 + x^4])/(1 + 3*x^2 + x^4)^2,x]

[Out]

-1/2*(x + x^5 + (-1)^(1/4)*Sqrt[1 + x^4]*(1 + 3*x^2 + x^4)*EllipticF[I*ArcSinh[(-1)^(1/4)*x], -1] - (-1)^(1/4)
*Sqrt[1 + x^4]*(1 + 3*x^2 + x^4)*EllipticPi[(2*I)/(-3 + Sqrt[5]), I*ArcSinh[(-1)^(1/4)*x], -1] - (-1)^(1/4)*Sq
rt[1 + x^4]*EllipticPi[(-2*I)/(3 + Sqrt[5]), I*ArcSinh[(-1)^(1/4)*x], -1] - 3*(-1)^(1/4)*x^2*Sqrt[1 + x^4]*Ell
ipticPi[(-2*I)/(3 + Sqrt[5]), I*ArcSinh[(-1)^(1/4)*x], -1] - (-1)^(1/4)*x^4*Sqrt[1 + x^4]*EllipticPi[(-2*I)/(3
+ Sqrt[5]), I*ArcSinh[(-1)^(1/4)*x], -1])/(Sqrt[1 + x^4]*(1 + 3*x^2 + x^4))

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IntegrateAlgebraic [A]  time = 0.63, size = 53, normalized size = 1.00 \begin {gather*} -\frac {x \sqrt {1+x^4}}{2 \left (1+3 x^2+x^4\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+x^4}}\right )}{2 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^4)*Sqrt[1 + x^4])/(1 + 3*x^2 + x^4)^2,x]

[Out]

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

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fricas [A]  time = 0.51, size = 65, normalized size = 1.23 \begin {gather*} -\frac {\sqrt {3} {\left (x^{4} + 3 \, x^{2} + 1\right )} \arctan \left (\frac {2 \, \sqrt {3} \sqrt {x^{4} + 1} x}{x^{4} - 3 \, x^{2} + 1}\right ) + 6 \, \sqrt {x^{4} + 1} x}{12 \, {\left (x^{4} + 3 \, x^{2} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)*(x^4+1)^(1/2)/(x^4+3*x^2+1)^2,x, algorithm="fricas")

[Out]

-1/12*(sqrt(3)*(x^4 + 3*x^2 + 1)*arctan(2*sqrt(3)*sqrt(x^4 + 1)*x/(x^4 - 3*x^2 + 1)) + 6*sqrt(x^4 + 1)*x)/(x^4
+ 3*x^2 + 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} + 1} {\left (x^{4} - 1\right )}}{{\left (x^{4} + 3 \, x^{2} + 1\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)*(x^4+1)^(1/2)/(x^4+3*x^2+1)^2,x, algorithm="giac")

[Out]

integrate(sqrt(x^4 + 1)*(x^4 - 1)/(x^4 + 3*x^2 + 1)^2, x)

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maple [A]  time = 0.38, size = 60, normalized size = 1.13

method	result	size
elliptic	\(\frac {\left (-\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{4 x \left (\frac {x^{4}+1}{2 x^{2}}+\frac {3}{2}\right )}+\frac {\sqrt {6}\, \arctan \left (\frac {\sqrt {6}\, \sqrt {2}\, \sqrt {x^{4}+1}}{6 x}\right )}{6}\right ) \sqrt {2}}{2}\)	\(60\)
trager	\(-\frac {x \sqrt {x^{4}+1}}{2 \left (x^{4}+3 x^{2}+1\right )}-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) x^{4}-3 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{2}+6 \sqrt {x^{4}+1}\, x +\RootOf \left (\textit {\_Z}^{2}+3\right )}{x^{4}+3 x^{2}+1}\right )}{12}\)	\(85\)
risch	\(-\frac {x \sqrt {x^{4}+1}}{2 \left (x^{4}+3 x^{2}+1\right )}+\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{2 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\sqrt {3}\, \arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}-3\right )}{\sqrt {-3 \underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {x^{4}+1}}\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {6 \left (-1\right )^{\frac {3}{4}} \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}-3 \underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i \underline {\hspace {1.25 ex}}\alpha ^{2}+3 i, i\right )}{\sqrt {x^{4}+1}}\right )\right )}{24}\)	\(202\)
default	\(-\frac {x \sqrt {x^{4}+1}}{2 \left (x^{4}+3 x^{2}+1\right )}+\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 )}{2 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (12 \underline {\hspace {1.25 ex}}\alpha ^{2}+23\right ) \left (-\frac {\sqrt {3}\, \arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}-3\right )}{\sqrt {-3 \underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {x^{4}+1}}\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {6 \left (-1\right )^{\frac {3}{4}} \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}-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}+3 i, i\right )}{\sqrt {x^{4}+1}}\right )\right )}{120}-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{2}+3\right ) \left (-\frac {\sqrt {3}\, \arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}-3\right )}{\sqrt {-3 \underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {x^{4}+1}}\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {6 \left (-1\right )^{\frac {3}{4}} \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}-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}+3 i, i\right )}{\sqrt {x^{4}+1}}\right )\right )}{20}\)	\(334\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-1)*(x^4+1)^(1/2)/(x^4+3*x^2+1)^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} + 1} {\left (x^{4} - 1\right )}}{{\left (x^{4} + 3 \, x^{2} + 1\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)*(x^4+1)^(1/2)/(x^4+3*x^2+1)^2,x, algorithm="maxima")

[Out]

integrate(sqrt(x^4 + 1)*(x^4 - 1)/(x^4 + 3*x^2 + 1)^2, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\left (x^4-1\right )\,\sqrt {x^4+1}}{{\left (x^4+3\,x^2+1\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4 - 1)*(x^4 + 1)^(1/2))/(3*x^2 + x^4 + 1)^2,x)

[Out]

int(((x^4 - 1)*(x^4 + 1)^(1/2))/(3*x^2 + x^4 + 1)^2, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-1)*(x**4+1)**(1/2)/(x**4+3*x**2+1)**2,x)

[Out]

Timed out

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