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

Optimal. Leaf size=158 \[ \tan ^{-1}\left (\frac {x \sqrt {-2 x^4-3 x^2+1}}{2 x^4+3 x^2-1}\right )-i \sqrt {2} \tanh ^{-1}\left (\frac {2 \sqrt {2} x^3-2 i \sqrt {-2 x^4-3 x^2+1} x-2 i x}{2 \sqrt {2} x^4+3 \sqrt {2} x^2-2 i \sqrt {-2 x^4-3 x^2+1} x^2-\sqrt {2} \sqrt {-2 x^4-3 x^2+1}-\sqrt {2}}\right ) \]

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Rubi [C]  time = 1.47, antiderivative size = 467, normalized size of antiderivative = 2.96, number of steps used = 32, number of rules used = 8, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6728, 1208, 1180, 524, 424, 419, 1212, 537} \begin {gather*} -\sqrt {\frac {1}{2} \left (9 \sqrt {17}-37\right )} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+\frac {1}{2} \sqrt {3 \sqrt {17}-5} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+\frac {\left (1+2 \sqrt {3}-\sqrt {17}\right ) F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )}{\sqrt {2 \left (3+\sqrt {17}\right )}}+\frac {\left (1-2 \sqrt {3}-\sqrt {17}\right ) F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )}{\sqrt {2 \left (3+\sqrt {17}\right )}}+2 \sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {1}{4} \left (3-\sqrt {17}\right );\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {3-\sqrt {17}}{2 \left (1-\sqrt {3}\right )};\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {3-\sqrt {17}}{2 \left (1+\sqrt {3}\right )};\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+2 \sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {1}{2} \left (-3+\sqrt {17}\right );\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((1 - 2*Sqrt[3] - Sqrt[17])*EllipticF[ArcSin[(2*x)/Sqrt[-3 + Sqrt[17]]], (-13 + 3*Sqrt[17])/4])/Sqrt[2*(3 + Sq
rt[17])] + ((1 + 2*Sqrt[3] - Sqrt[17])*EllipticF[ArcSin[(2*x)/Sqrt[-3 + Sqrt[17]]], (-13 + 3*Sqrt[17])/4])/Sqr
t[2*(3 + Sqrt[17])] + (Sqrt[-5 + 3*Sqrt[17]]*EllipticF[ArcSin[(2*x)/Sqrt[-3 + Sqrt[17]]], (-13 + 3*Sqrt[17])/4
])/2 - Sqrt[(-37 + 9*Sqrt[17])/2]*EllipticF[ArcSin[(2*x)/Sqrt[-3 + Sqrt[17]]], (-13 + 3*Sqrt[17])/4] + 2*Sqrt[
2/(3 + Sqrt[17])]*EllipticPi[(3 - Sqrt[17])/4, ArcSin[(2*x)/Sqrt[-3 + Sqrt[17]]], (-13 + 3*Sqrt[17])/4] - Sqrt
[2/(3 + Sqrt[17])]*EllipticPi[(3 - Sqrt[17])/(2*(1 - Sqrt[3])), ArcSin[(2*x)/Sqrt[-3 + Sqrt[17]]], (-13 + 3*Sq
rt[17])/4] - Sqrt[2/(3 + Sqrt[17])]*EllipticPi[(3 - Sqrt[17])/(2*(1 + Sqrt[3])), ArcSin[(2*x)/Sqrt[-3 + Sqrt[1
7]]], (-13 + 3*Sqrt[17])/4] + 2*Sqrt[2/(3 + Sqrt[17])]*EllipticPi[(-3 + Sqrt[17])/2, ArcSin[(2*x)/Sqrt[-3 + Sq
rt[17]]], (-13 + 3*Sqrt[17])/4]

Rule 419

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

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 524

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

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 1180

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

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 1212

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

Rubi steps

\begin {align*} \int \frac {\sqrt {1-3 x^2-2 x^4} \left (1+2 x^4\right )}{\left (-1+x^2+2 x^4\right ) \left (-1+2 x^2+2 x^4\right )} \, dx &=\int \left (\frac {\sqrt {1-3 x^2-2 x^4}}{1+x^2}+\frac {2 \sqrt {1-3 x^2-2 x^4}}{-1+2 x^2}-\frac {2 \left (1+2 x^2\right ) \sqrt {1-3 x^2-2 x^4}}{-1+2 x^2+2 x^4}\right ) \, dx\\ &=2 \int \frac {\sqrt {1-3 x^2-2 x^4}}{-1+2 x^2} \, dx-2 \int \frac {\left (1+2 x^2\right ) \sqrt {1-3 x^2-2 x^4}}{-1+2 x^2+2 x^4} \, dx+\int \frac {\sqrt {1-3 x^2-2 x^4}}{1+x^2} \, dx\\ &=-\left (\frac {1}{2} \int \frac {8+4 x^2}{\sqrt {1-3 x^2-2 x^4}} \, dx\right )+2 \int \frac {1}{\left (1+x^2\right ) \sqrt {1-3 x^2-2 x^4}} \, dx-2 \int \frac {1}{\left (-1+2 x^2\right ) \sqrt {1-3 x^2-2 x^4}} \, dx-2 \int \left (\frac {2 \sqrt {1-3 x^2-2 x^4}}{2-2 \sqrt {3}+4 x^2}+\frac {2 \sqrt {1-3 x^2-2 x^4}}{2+2 \sqrt {3}+4 x^2}\right ) \, dx-\int \frac {1+2 x^2}{\sqrt {1-3 x^2-2 x^4}} \, dx\\ &=-\left (4 \int \frac {\sqrt {1-3 x^2-2 x^4}}{2-2 \sqrt {3}+4 x^2} \, dx\right )-4 \int \frac {\sqrt {1-3 x^2-2 x^4}}{2+2 \sqrt {3}+4 x^2} \, dx-\sqrt {2} \int \frac {8+4 x^2}{\sqrt {-3+\sqrt {17}-4 x^2} \sqrt {3+\sqrt {17}+4 x^2}} \, dx-\left (2 \sqrt {2}\right ) \int \frac {1+2 x^2}{\sqrt {-3+\sqrt {17}-4 x^2} \sqrt {3+\sqrt {17}+4 x^2}} \, dx+\left (4 \sqrt {2}\right ) \int \frac {1}{\sqrt {-3+\sqrt {17}-4 x^2} \left (1+x^2\right ) \sqrt {3+\sqrt {17}+4 x^2}} \, dx-\left (4 \sqrt {2}\right ) \int \frac {1}{\sqrt {-3+\sqrt {17}-4 x^2} \left (-1+2 x^2\right ) \sqrt {3+\sqrt {17}+4 x^2}} \, dx\\ &=2 \sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {1}{4} \left (3-\sqrt {17}\right );\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+2 \sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {1}{2} \left (-3+\sqrt {17}\right );\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+\frac {1}{4} \int \frac {12-2 \left (2-2 \sqrt {3}\right )+8 x^2}{\sqrt {1-3 x^2-2 x^4}} \, dx+\frac {1}{4} \int \frac {12-2 \left (2+2 \sqrt {3}\right )+8 x^2}{\sqrt {1-3 x^2-2 x^4}} \, dx-2 \left (\sqrt {2} \int \frac {\sqrt {3+\sqrt {17}+4 x^2}}{\sqrt {-3+\sqrt {17}-4 x^2}} \, dx\right )-\left (2 \left (1-\sqrt {3}\right )\right ) \int \frac {1}{\left (2-2 \sqrt {3}+4 x^2\right ) \sqrt {1-3 x^2-2 x^4}} \, dx-\left (2 \left (1+\sqrt {3}\right )\right ) \int \frac {1}{\left (2+2 \sqrt {3}+4 x^2\right ) \sqrt {1-3 x^2-2 x^4}} \, dx-\left (\sqrt {2} \left (5-\sqrt {17}\right )\right ) \int \frac {1}{\sqrt {-3+\sqrt {17}-4 x^2} \sqrt {3+\sqrt {17}+4 x^2}} \, dx+\left (\sqrt {2} \left (1+\sqrt {17}\right )\right ) \int \frac {1}{\sqrt {-3+\sqrt {17}-4 x^2} \sqrt {3+\sqrt {17}+4 x^2}} \, dx\\ &=-\sqrt {2 \left (3+\sqrt {17}\right )} E\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+\frac {1}{2} \sqrt {-5+3 \sqrt {17}} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {1}{2} \left (-37+9 \sqrt {17}\right )} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+2 \sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {1}{4} \left (3-\sqrt {17}\right );\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+2 \sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {1}{2} \left (-3+\sqrt {17}\right );\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+\frac {\int \frac {12-2 \left (2-2 \sqrt {3}\right )+8 x^2}{\sqrt {-3+\sqrt {17}-4 x^2} \sqrt {3+\sqrt {17}+4 x^2}} \, dx}{\sqrt {2}}+\frac {\int \frac {12-2 \left (2+2 \sqrt {3}\right )+8 x^2}{\sqrt {-3+\sqrt {17}-4 x^2} \sqrt {3+\sqrt {17}+4 x^2}} \, dx}{\sqrt {2}}-\left (4 \sqrt {2} \left (1-\sqrt {3}\right )\right ) \int \frac {1}{\sqrt {-3+\sqrt {17}-4 x^2} \left (2-2 \sqrt {3}+4 x^2\right ) \sqrt {3+\sqrt {17}+4 x^2}} \, dx-\left (4 \sqrt {2} \left (1+\sqrt {3}\right )\right ) \int \frac {1}{\sqrt {-3+\sqrt {17}-4 x^2} \left (2+2 \sqrt {3}+4 x^2\right ) \sqrt {3+\sqrt {17}+4 x^2}} \, dx\\ &=-\sqrt {2 \left (3+\sqrt {17}\right )} E\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+\frac {1}{2} \sqrt {-5+3 \sqrt {17}} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {1}{2} \left (-37+9 \sqrt {17}\right )} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+2 \sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {1}{4} \left (3-\sqrt {17}\right );\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {3-\sqrt {17}}{2 \left (1-\sqrt {3}\right )};\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {3-\sqrt {17}}{2 \left (1+\sqrt {3}\right )};\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+2 \sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {1}{2} \left (-3+\sqrt {17}\right );\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+2 \left (\sqrt {2} \int \frac {\sqrt {3+\sqrt {17}+4 x^2}}{\sqrt {-3+\sqrt {17}-4 x^2}} \, dx\right )+\left (\sqrt {2} \left (1-2 \sqrt {3}-\sqrt {17}\right )\right ) \int \frac {1}{\sqrt {-3+\sqrt {17}-4 x^2} \sqrt {3+\sqrt {17}+4 x^2}} \, dx+\left (\sqrt {2} \left (1+2 \sqrt {3}-\sqrt {17}\right )\right ) \int \frac {1}{\sqrt {-3+\sqrt {17}-4 x^2} \sqrt {3+\sqrt {17}+4 x^2}} \, dx\\ &=\frac {\left (1-2 \sqrt {3}-\sqrt {17}\right ) F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )}{\sqrt {2 \left (3+\sqrt {17}\right )}}+\frac {\left (1+2 \sqrt {3}-\sqrt {17}\right ) F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )}{\sqrt {2 \left (3+\sqrt {17}\right )}}+\frac {1}{2} \sqrt {-5+3 \sqrt {17}} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {1}{2} \left (-37+9 \sqrt {17}\right )} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+2 \sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {1}{4} \left (3-\sqrt {17}\right );\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {3-\sqrt {17}}{2 \left (1-\sqrt {3}\right )};\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {3-\sqrt {17}}{2 \left (1+\sqrt {3}\right )};\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+2 \sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {1}{2} \left (-3+\sqrt {17}\right );\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )\\ \end {align*}

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Mathematica [C]  time = 0.94, size = 251, normalized size = 1.59 \begin {gather*} i \sqrt {\frac {2}{\sqrt {17}-3}} \left (F\left (i \sinh ^{-1}\left (\frac {2 x}{\sqrt {3+\sqrt {17}}}\right )|-\frac {13}{4}-\frac {3 \sqrt {17}}{4}\right )-2 \Pi \left (-\frac {3}{2}-\frac {\sqrt {17}}{2};i \sinh ^{-1}\left (\frac {2 x}{\sqrt {3+\sqrt {17}}}\right )|-\frac {13}{4}-\frac {3 \sqrt {17}}{4}\right )-2 \Pi \left (\frac {1}{4} \left (3+\sqrt {17}\right );i \sinh ^{-1}\left (\frac {2 x}{\sqrt {3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13-3 \sqrt {17}\right )\right )+\Pi \left (\frac {3+\sqrt {17}}{2-2 \sqrt {3}};i \sinh ^{-1}\left (\frac {2 x}{\sqrt {3+\sqrt {17}}}\right )|-\frac {13}{4}-\frac {3 \sqrt {17}}{4}\right )+\Pi \left (\frac {3+\sqrt {17}}{2+2 \sqrt {3}};i \sinh ^{-1}\left (\frac {2 x}{\sqrt {3+\sqrt {17}}}\right )|-\frac {13}{4}-\frac {3 \sqrt {17}}{4}\right )\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

I*Sqrt[2/(-3 + Sqrt[17])]*(EllipticF[I*ArcSinh[(2*x)/Sqrt[3 + Sqrt[17]]], -13/4 - (3*Sqrt[17])/4] - 2*Elliptic
Pi[-3/2 - Sqrt[17]/2, I*ArcSinh[(2*x)/Sqrt[3 + Sqrt[17]]], -13/4 - (3*Sqrt[17])/4] - 2*EllipticPi[(3 + Sqrt[17
])/4, I*ArcSinh[(2*x)/Sqrt[3 + Sqrt[17]]], (-13 - 3*Sqrt[17])/4] + EllipticPi[(3 + Sqrt[17])/(2 - 2*Sqrt[3]),
I*ArcSinh[(2*x)/Sqrt[3 + Sqrt[17]]], -13/4 - (3*Sqrt[17])/4] + EllipticPi[(3 + Sqrt[17])/(2 + 2*Sqrt[3]), I*Ar
cSinh[(2*x)/Sqrt[3 + Sqrt[17]]], -13/4 - (3*Sqrt[17])/4])

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IntegrateAlgebraic [A]  time = 0.49, size = 79, normalized size = 0.50 \begin {gather*} \tan ^{-1}\left (\frac {x \sqrt {1-3 x^2-2 x^4}}{-1+3 x^2+2 x^4}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {1-3 x^2-2 x^4}}{-1+3 x^2+2 x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.58, size = 75, normalized size = 0.47 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-2 \, x^{4} - 3 \, x^{2} + 1} x}{2 \, x^{4} + 5 \, x^{2} - 1}\right ) + \frac {1}{2} \, \arctan \left (\frac {2 \, \sqrt {-2 \, x^{4} - 3 \, x^{2} + 1} x}{2 \, x^{4} + 4 \, x^{2} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(2)*arctan(2*sqrt(2)*sqrt(-2*x^4 - 3*x^2 + 1)*x/(2*x^4 + 5*x^2 - 1)) + 1/2*arctan(2*sqrt(-2*x^4 - 3*x
^2 + 1)*x/(2*x^4 + 4*x^2 - 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.78, size = 55, normalized size = 0.35

method	result	size
elliptic	\(\frac {\left (-2 \arctan \left (\frac {\sqrt {-2 x^{4}-3 x^{2}+1}\, \sqrt {2}}{2 x}\right )+\sqrt {2}\, \arctan \left (\frac {\sqrt {-2 x^{4}-3 x^{2}+1}}{x}\right )\right ) \sqrt {2}}{2}\)	\(55\)
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {-2 x^{4}-3 x^{2}+1}\, x -\RootOf \left (\textit {\_Z}^{2}+1\right )}{2 x^{4}+2 x^{2}-1}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {-2 \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{4}-5 \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{2}+4 \sqrt {-2 x^{4}-3 x^{2}+1}\, x +\RootOf \left (\textit {\_Z}^{2}+2\right )}{\left (x^{2}+1\right ) \left (2 x^{2}-1\right )}\right )}{2}\)	\(148\)
default	\(\frac {8 \sqrt {1-\left (\frac {3}{2}+\frac {\sqrt {17}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{2}-\frac {\sqrt {17}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {6+2 \sqrt {17}}}{2}, \frac {i \sqrt {34}}{4}-\frac {3 i \sqrt {2}}{4}\right )}{\sqrt {6+2 \sqrt {17}}\, \sqrt {-2 x^{4}-3 x^{2}+1}}-\frac {16 \sqrt {1-\left (\frac {3}{2}+\frac {\sqrt {17}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{2}-\frac {\sqrt {17}}{2}\right ) x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {6+2 \sqrt {17}}}{2}, \frac {i \sqrt {34}}{4}-\frac {3 i \sqrt {2}}{4}\right )-\EllipticE \left (\frac {x \sqrt {6+2 \sqrt {17}}}{2}, \frac {i \sqrt {34}}{4}-\frac {3 i \sqrt {2}}{4}\right )\right )}{\sqrt {6+2 \sqrt {17}}\, \sqrt {-2 x^{4}-3 x^{2}+1}\, \left (-3+\sqrt {17}\right )}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (2 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}-1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (\frac {\arctanh \left (\frac {\left (4 \underline {\hspace {1.25 ex}}\alpha ^{2}+3\right ) \left (-17 \underline {\hspace {1.25 ex}}\alpha ^{2}+11 x^{2}+4\right )}{22 \sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {-2 x^{4}-3 x^{2}+1}}\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{2}}}-\frac {2 \sqrt {2}\, \left (\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {-3 x^{2}+2-x^{2} \sqrt {17}}\, \sqrt {-3 x^{2}+2+x^{2} \sqrt {17}}\, \EllipticPi \left (\sqrt {\frac {3}{2}+\frac {\sqrt {17}}{2}}\, x , \frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {17}}{2}-\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {\sqrt {17}}{2}-\frac {3}{2}, \frac {\sqrt {\frac {3}{2}-\frac {\sqrt {17}}{2}}}{\sqrt {\frac {3}{2}+\frac {\sqrt {17}}{2}}}\right )}{\sqrt {3+\sqrt {17}}\, \sqrt {-2 x^{4}-3 x^{2}+1}}\right )\right )}{4}-\frac {10 \sqrt {1-\frac {3 x^{2}}{2}-\frac {x^{2} \sqrt {17}}{2}}\, \sqrt {1-\frac {3 x^{2}}{2}+\frac {x^{2} \sqrt {17}}{2}}\, \EllipticF \left (\frac {x \sqrt {6+2 \sqrt {17}}}{2}, \frac {i \sqrt {34}}{4}-\frac {3 i \sqrt {2}}{4}\right )}{\sqrt {6+2 \sqrt {17}}\, \sqrt {-2 x^{4}-3 x^{2}+1}}+\frac {16 \sqrt {1-\frac {3 x^{2}}{2}-\frac {x^{2} \sqrt {17}}{2}}\, \sqrt {1-\frac {3 x^{2}}{2}+\frac {x^{2} \sqrt {17}}{2}}\, \EllipticF \left (\frac {x \sqrt {6+2 \sqrt {17}}}{2}, \frac {i \sqrt {34}}{4}-\frac {3 i \sqrt {2}}{4}\right )}{\sqrt {6+2 \sqrt {17}}\, \sqrt {-2 x^{4}-3 x^{2}+1}\, \left (-3+\sqrt {17}\right )}-\frac {16 \sqrt {1-\frac {3 x^{2}}{2}-\frac {x^{2} \sqrt {17}}{2}}\, \sqrt {1-\frac {3 x^{2}}{2}+\frac {x^{2} \sqrt {17}}{2}}\, \EllipticE \left (\frac {x \sqrt {6+2 \sqrt {17}}}{2}, \frac {i \sqrt {34}}{4}-\frac {3 i \sqrt {2}}{4}\right )}{\sqrt {6+2 \sqrt {17}}\, \sqrt {-2 x^{4}-3 x^{2}+1}\, \left (-3+\sqrt {17}\right )}+\frac {2 \sqrt {1-\frac {3 x^{2}}{2}-\frac {x^{2} \sqrt {17}}{2}}\, \sqrt {1-\frac {3 x^{2}}{2}+\frac {x^{2} \sqrt {17}}{2}}\, \EllipticPi \left (\sqrt {\frac {3}{2}+\frac {\sqrt {17}}{2}}\, x , -\frac {1}{\frac {3}{2}+\frac {\sqrt {17}}{2}}, \frac {\sqrt {\frac {3}{2}-\frac {\sqrt {17}}{2}}}{\sqrt {\frac {3}{2}+\frac {\sqrt {17}}{2}}}\right )}{\sqrt {\frac {3}{2}+\frac {\sqrt {17}}{2}}\, \sqrt {-2 x^{4}-3 x^{2}+1}}+\frac {2 \sqrt {1-\frac {3 x^{2}}{2}-\frac {x^{2} \sqrt {17}}{2}}\, \sqrt {1-\frac {3 x^{2}}{2}+\frac {x^{2} \sqrt {17}}{2}}\, \EllipticPi \left (\sqrt {\frac {3}{2}+\frac {\sqrt {17}}{2}}\, x , \frac {2}{\frac {3}{2}+\frac {\sqrt {17}}{2}}, \frac {\sqrt {\frac {3}{2}-\frac {\sqrt {17}}{2}}}{\sqrt {\frac {3}{2}+\frac {\sqrt {17}}{2}}}\right )}{\sqrt {\frac {3}{2}+\frac {\sqrt {17}}{2}}\, \sqrt {-2 x^{4}-3 x^{2}+1}}\)	\(862\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (2\,x^4+1\right )\,\sqrt {-2\,x^4-3\,x^2+1}}{\left (2\,x^4+x^2-1\right )\,\left (2\,x^4+2\,x^2-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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