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

Optimal. Leaf size=248 \[ 2 \sqrt [4]{2} \text {RootSum}\left [\text {$\#$1}^8+12 \sqrt {2} \text {$\#$1}^6-4 \text {$\#$1}^6-24 \sqrt {2} \text {$\#$1}^4+70 \text {$\#$1}^4+12 \sqrt {2} \text {$\#$1}^2-4 \text {$\#$1}^2+1\& ,\frac {\text {$\#$1}^3 \left (-\log \left (-x^2+2^{3/4} x-\sqrt {2}\right )\right )+\text {$\#$1}^3 \log \left (\text {$\#$1} x^2-2^{3/4} \text {$\#$1} x+\sqrt {2} \text {$\#$1}+\sqrt {x^4+2}\right )-\text {$\#$1} \log \left (-x^2+2^{3/4} x-\sqrt {2}\right )+\text {$\#$1} \log \left (\text {$\#$1} x^2-2^{3/4} \text {$\#$1} x+\sqrt {2} \text {$\#$1}+\sqrt {x^4+2}\right )}{\text {$\#$1}^6+9 \sqrt {2} \text {$\#$1}^4-3 \text {$\#$1}^4-12 \sqrt {2} \text {$\#$1}^2+35 \text {$\#$1}^2+3 \sqrt {2}-1}\& \right ] \]

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Rubi [C]  time = 1.84, antiderivative size = 905, normalized size of antiderivative = 3.65, number of steps used = 34, number of rules used = 7, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {6728, 1209, 1198, 220, 1196, 1217, 1707} \begin {gather*} \tan ^{-1}\left (\frac {x}{\sqrt {x^4+2}}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+2}}\right )-\frac {\left (i-\sqrt {7}\right ) \left (1+2 \sqrt {2}+i \sqrt {7}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (7 i+\sqrt {7}\right ) \sqrt {x^4+2}}-\frac {\left (1+2 \sqrt {2}-i \sqrt {7}\right ) \left (i+\sqrt {7}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (7 i-\sqrt {7}\right ) \sqrt {x^4+2}}-\frac {\left (1-2 \sqrt {2}+i \sqrt {7}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {x^4+2}}-\frac {\left (1-2 \sqrt {2}-i \sqrt {7}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {x^4+2}}+\frac {i \left ((1+i)+\sqrt {2}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{2} \sqrt {x^4+2}}-\frac {\left ((-1+i)+\sqrt {2}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{2} \sqrt {x^4+2}}-\frac {\left ((-1-i)+\sqrt {2}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{2} \sqrt {x^4+2}}-\frac {\left ((1+i)+i \sqrt {2}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{2} \sqrt {x^4+2}}-\frac {\left (1+2 \sqrt {2}\right ) \left (3 i-\sqrt {7}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \Pi \left (\frac {1}{8} \left (4-\sqrt {2}\right );2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (7 i+3 \sqrt {7}\right ) \sqrt {x^4+2}}-\frac {\left (1+2 \sqrt {2}\right ) \left (3 i+\sqrt {7}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \Pi \left (\frac {1}{8} \left (4-\sqrt {2}\right );2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (7 i-3 \sqrt {7}\right ) \sqrt {x^4+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[x/Sqrt[2 + x^4]] - Sqrt[2]*ArcTan[(Sqrt[2]*x)/Sqrt[2 + x^4]] - (((1 + I) + I*Sqrt[2])*(Sqrt[2] + x^2)*S
qrt[(2 + x^4)/(Sqrt[2] + x^2)^2]*EllipticF[2*ArcTan[x/2^(1/4)], 1/2])/(2*2^(1/4)*Sqrt[2 + x^4]) - (((-1 - I) +
Sqrt[2])*(Sqrt[2] + x^2)*Sqrt[(2 + x^4)/(Sqrt[2] + x^2)^2]*EllipticF[2*ArcTan[x/2^(1/4)], 1/2])/(2*2^(1/4)*Sq
rt[2 + x^4]) - (((-1 + I) + Sqrt[2])*(Sqrt[2] + x^2)*Sqrt[(2 + x^4)/(Sqrt[2] + x^2)^2]*EllipticF[2*ArcTan[x/2^
(1/4)], 1/2])/(2*2^(1/4)*Sqrt[2 + x^4]) + ((I/2)*((1 + I) + Sqrt[2])*(Sqrt[2] + x^2)*Sqrt[(2 + x^4)/(Sqrt[2] +
x^2)^2]*EllipticF[2*ArcTan[x/2^(1/4)], 1/2])/(2^(1/4)*Sqrt[2 + x^4]) - ((1 - 2*Sqrt[2] - I*Sqrt[7])*(Sqrt[2]
+ x^2)*Sqrt[(2 + x^4)/(Sqrt[2] + x^2)^2]*EllipticF[2*ArcTan[x/2^(1/4)], 1/2])/(4*2^(1/4)*Sqrt[2 + x^4]) - ((1
- 2*Sqrt[2] + I*Sqrt[7])*(Sqrt[2] + x^2)*Sqrt[(2 + x^4)/(Sqrt[2] + x^2)^2]*EllipticF[2*ArcTan[x/2^(1/4)], 1/2]
)/(4*2^(1/4)*Sqrt[2 + x^4]) - ((1 + 2*Sqrt[2] - I*Sqrt[7])*(I + Sqrt[7])*(Sqrt[2] + x^2)*Sqrt[(2 + x^4)/(Sqrt[
2] + x^2)^2]*EllipticF[2*ArcTan[x/2^(1/4)], 1/2])/(4*2^(1/4)*(7*I - Sqrt[7])*Sqrt[2 + x^4]) - ((I - Sqrt[7])*(
1 + 2*Sqrt[2] + I*Sqrt[7])*(Sqrt[2] + x^2)*Sqrt[(2 + x^4)/(Sqrt[2] + x^2)^2]*EllipticF[2*ArcTan[x/2^(1/4)], 1/
2])/(4*2^(1/4)*(7*I + Sqrt[7])*Sqrt[2 + x^4]) - ((1 + 2*Sqrt[2])*(3*I + Sqrt[7])*(Sqrt[2] + x^2)*Sqrt[(2 + x^4
)/(Sqrt[2] + x^2)^2]*EllipticPi[(4 - Sqrt[2])/8, 2*ArcTan[x/2^(1/4)], 1/2])/(4*2^(1/4)*(7*I - 3*Sqrt[7])*Sqrt[
2 + x^4]) - ((1 + 2*Sqrt[2])*(3*I - Sqrt[7])*(Sqrt[2] + x^2)*Sqrt[(2 + x^4)/(Sqrt[2] + x^2)^2]*EllipticPi[(4 -
Sqrt[2])/8, 2*ArcTan[x/2^(1/4)], 1/2])/(4*2^(1/4)*(7*I + 3*Sqrt[7])*Sqrt[2 + 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 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]

Rubi steps

\begin {align*} \int \frac {\left (-2+x^4\right ) \sqrt {2+x^4}}{\left (2+x^2+x^4\right ) \left (2+2 x^2+x^4\right )} \, dx &=\int \left (\frac {\left (1+2 x^2\right ) \sqrt {2+x^4}}{2+x^2+x^4}-\frac {2 \left (1+x^2\right ) \sqrt {2+x^4}}{2+2 x^2+x^4}\right ) \, dx\\ &=-\left (2 \int \frac {\left (1+x^2\right ) \sqrt {2+x^4}}{2+2 x^2+x^4} \, dx\right )+\int \frac {\left (1+2 x^2\right ) \sqrt {2+x^4}}{2+x^2+x^4} \, dx\\ &=-\left (2 \int \left (\frac {\sqrt {2+x^4}}{(2-2 i)+2 x^2}+\frac {\sqrt {2+x^4}}{(2+2 i)+2 x^2}\right ) \, dx\right )+\int \left (\frac {2 \sqrt {2+x^4}}{1-i \sqrt {7}+2 x^2}+\frac {2 \sqrt {2+x^4}}{1+i \sqrt {7}+2 x^2}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {2+x^4}}{(2-2 i)+2 x^2} \, dx\right )-2 \int \frac {\sqrt {2+x^4}}{(2+2 i)+2 x^2} \, dx+2 \int \frac {\sqrt {2+x^4}}{1-i \sqrt {7}+2 x^2} \, dx+2 \int \frac {\sqrt {2+x^4}}{1+i \sqrt {7}+2 x^2} \, dx\\ &=\frac {1}{2} \int \frac {(2-2 i)-2 x^2}{\sqrt {2+x^4}} \, dx+\frac {1}{2} \int \frac {(2+2 i)-2 x^2}{\sqrt {2+x^4}} \, dx-\frac {1}{2} \int \frac {1-i \sqrt {7}-2 x^2}{\sqrt {2+x^4}} \, dx-\frac {1}{2} \int \frac {1+i \sqrt {7}-2 x^2}{\sqrt {2+x^4}} \, dx-(4-4 i) \int \frac {1}{\left ((2-2 i)+2 x^2\right ) \sqrt {2+x^4}} \, dx-(4+4 i) \int \frac {1}{\left ((2+2 i)+2 x^2\right ) \sqrt {2+x^4}} \, dx+\left (1-i \sqrt {7}\right ) \int \frac {1}{\left (1-i \sqrt {7}+2 x^2\right ) \sqrt {2+x^4}} \, dx+\left (1+i \sqrt {7}\right ) \int \frac {1}{\left (1+i \sqrt {7}+2 x^2\right ) \sqrt {2+x^4}} \, dx\\ &=\left ((1-i)-\sqrt {2}\right ) \int \frac {1}{\sqrt {2+x^4}} \, dx+\left ((1+i)-\sqrt {2}\right ) \int \frac {1}{\sqrt {2+x^4}} \, dx-\left ((1+i)+i \sqrt {2}\right ) \int \frac {1}{\sqrt {2+x^4}} \, dx-\left ((-2-2 i) \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {1+\frac {x^2}{\sqrt {2}}}{\left ((2-2 i)+2 x^2\right ) \sqrt {2+x^4}} \, dx+\left (i \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {1}{\sqrt {2+x^4}} \, dx-\left (2 i \left (2+(1+i) \sqrt {2}\right )\right ) \int \frac {1+\frac {x^2}{\sqrt {2}}}{\left ((2+2 i)+2 x^2\right ) \sqrt {2+x^4}} \, dx-\frac {1}{2} \left (1-2 \sqrt {2}-i \sqrt {7}\right ) \int \frac {1}{\sqrt {2+x^4}} \, dx-\frac {1}{2} \left (1-2 \sqrt {2}+i \sqrt {7}\right ) \int \frac {1}{\sqrt {2+x^4}} \, dx+\frac {\left (\left (-1+i \sqrt {7}\right ) \left (i+2 i \sqrt {2}+\sqrt {7}\right )\right ) \int \frac {1}{\sqrt {2+x^4}} \, dx}{2 \left (7 i-\sqrt {7}\right )}+\frac {\left (4 \left (-1+i \sqrt {7}\right ) \left (2+\frac {1-i \sqrt {7}}{\sqrt {2}}\right )\right ) \int \frac {1+\frac {x^2}{\sqrt {2}}}{\left (1-i \sqrt {7}+2 x^2\right ) \sqrt {2+x^4}} \, dx}{-8+\left (1-i \sqrt {7}\right )^2}+\frac {\left (\left (1+i \sqrt {7}\right ) \left (1+2 \sqrt {2}+i \sqrt {7}\right )\right ) \int \frac {1}{\sqrt {2+x^4}} \, dx}{-8+\left (1+i \sqrt {7}\right )^2}+\frac {\left (4 \left (-1-i \sqrt {7}\right ) \left (2+\frac {1+i \sqrt {7}}{\sqrt {2}}\right )\right ) \int \frac {1+\frac {x^2}{\sqrt {2}}}{\left (1+i \sqrt {7}+2 x^2\right ) \sqrt {2+x^4}} \, dx}{-8+\left (1+i \sqrt {7}\right )^2}\\ &=\tan ^{-1}\left (\frac {x}{\sqrt {2+x^4}}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {2+x^4}}\right )-\frac {\left ((1+i)+i \sqrt {2}\right ) \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{2} \sqrt {2+x^4}}-\frac {\left ((-1-i)+\sqrt {2}\right ) \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{2} \sqrt {2+x^4}}-\frac {\left ((-1+i)+\sqrt {2}\right ) \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{2} \sqrt {2+x^4}}+\frac {i \left ((1+i)+\sqrt {2}\right ) \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{2} \sqrt {2+x^4}}-\frac {\left (1-2 \sqrt {2}-i \sqrt {7}\right ) \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {2+x^4}}-\frac {\left (1-2 \sqrt {2}+i \sqrt {7}\right ) \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {2+x^4}}-\frac {\left (1+2 \sqrt {2}-i \sqrt {7}\right ) \left (i+\sqrt {7}\right ) \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (7 i-\sqrt {7}\right ) \sqrt {2+x^4}}-\frac {\left (i-\sqrt {7}\right ) \left (1+2 \sqrt {2}+i \sqrt {7}\right ) \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (7 i+\sqrt {7}\right ) \sqrt {2+x^4}}-\frac {i \left (1+\sqrt {2}\right ) \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} \Pi \left (\frac {1}{4} \left (2-\sqrt {2}\right );2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{2} \sqrt {2+x^4}}+\frac {i \left (2+\sqrt {2}\right ) \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} \Pi \left (\frac {1}{4} \left (2-\sqrt {2}\right );2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{2\ 2^{3/4} \sqrt {2+x^4}}+\frac {\left (1+2 \sqrt {2}-i \sqrt {7}\right )^2 \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} \Pi \left (\frac {1}{8} \left (4-\sqrt {2}\right );2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{8 \sqrt [4]{2} \left (7+i \sqrt {7}\right ) \sqrt {2+x^4}}+\frac {\left (1+2 \sqrt {2}+i \sqrt {7}\right )^2 \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} \Pi \left (\frac {1}{8} \left (4-\sqrt {2}\right );2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{8 \sqrt [4]{2} \left (7-i \sqrt {7}\right ) \sqrt {2+x^4}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.89, size = 148, normalized size = 0.60 \begin {gather*} -\sqrt [4]{-\frac {1}{2}} \left (F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac {1}{2}} x\right )\right |-1\right )-2 \Pi \left (-\frac {1+i}{\sqrt {2}};\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac {1}{2}} x\right )\right |-1\right )-2 \Pi \left (\frac {1-i}{\sqrt {2}};\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac {1}{2}} x\right )\right |-1\right )+\Pi \left (-\frac {2 \sqrt {2}}{-i+\sqrt {7}};\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac {1}{2}} x\right )\right |-1\right )+\Pi \left (\frac {2 \sqrt {2}}{i+\sqrt {7}};\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac {1}{2}} x\right )\right |-1\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((-1/2)^(1/4)*(EllipticF[I*ArcSinh[(-1/2)^(1/4)*x], -1] - 2*EllipticPi[(-1 - I)/Sqrt[2], I*ArcSinh[(-1/2)^(1/
4)*x], -1] - 2*EllipticPi[(1 - I)/Sqrt[2], I*ArcSinh[(-1/2)^(1/4)*x], -1] + EllipticPi[(-2*Sqrt[2])/(-I + Sqrt
[7]), I*ArcSinh[(-1/2)^(1/4)*x], -1] + EllipticPi[(2*Sqrt[2])/(I + Sqrt[7]), I*ArcSinh[(-1/2)^(1/4)*x], -1]))

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IntegrateAlgebraic [A]  time = 0.98, size = 37, normalized size = 0.15 \begin {gather*} \tan ^{-1}\left (\frac {x}{\sqrt {2+x^4}}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {2+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.56, size = 57, normalized size = 0.23 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {x^{4} + 2} x}{x^{4} - 2 \, x^{2} + 2}\right ) + \frac {1}{2} \, \arctan \left (\frac {2 \, \sqrt {x^{4} + 2} x}{x^{4} - x^{2} + 2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.74, size = 42, normalized size = 0.17

method	result	size
elliptic	\(\frac {\left (2 \arctan \left (\frac {\sqrt {x^{4}+2}\, \sqrt {2}}{2 x}\right )-\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}+2}}{x}\right )\right ) \sqrt {2}}{2}\)	\(42\)
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{2}+4 \sqrt {x^{4}+2}\, x +2 \RootOf \left (\textit {\_Z}^{2}+2\right )}{x^{4}+2 x^{2}+2}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-2 \sqrt {x^{4}+2}\, x +2 \RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{4}+x^{2}+2}\right )}{2}\)	\(125\)
default	\(\frac {\sqrt {2}\, \sqrt {4-2 i \sqrt {2}\, x^{2}}\, \sqrt {4+2 i \sqrt {2}\, x^{2}}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {i \sqrt {2}}}{2}, i\right )}{4 \sqrt {i \sqrt {2}}\, \sqrt {x^{4}+2}}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+2 \textit {\_Z}^{2}+2\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {2 \arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}-2\right )}{\sqrt {-2 \underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {x^{4}+2}}\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{2}}}-\frac {2^{\frac {3}{4}} \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}-2 \underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {2-i \sqrt {2}\, x^{2}}\, \sqrt {2+i \sqrt {2}\, x^{2}}\, \EllipticPi \left (\frac {x \sqrt {2}\, \sqrt {i \sqrt {2}}}{2}, \frac {i \sqrt {2}\, \underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+i \sqrt {2}, \frac {\sqrt {-\frac {i \sqrt {2}}{2}}\, \sqrt {2}}{\sqrt {i \sqrt {2}}}\right )}{\sqrt {i}\, \sqrt {x^{4}+2}}\right )\right )}{8}-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z}^{2}+2\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {4 \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}+2}}\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{2}}}-\frac {2 \,2^{\frac {1}{4}} \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {2-i \sqrt {2}\, x^{2}}\, \sqrt {2+i \sqrt {2}\, x^{2}}\, \EllipticPi \left (\frac {x \sqrt {2}\, \sqrt {i \sqrt {2}}}{2}, \frac {i \sqrt {2}\, \underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {i \sqrt {2}}{2}, \frac {\sqrt {-\frac {i \sqrt {2}}{2}}\, \sqrt {2}}{\sqrt {i \sqrt {2}}}\right )}{\sqrt {i}\, \sqrt {x^{4}+2}}\right )\right )}{16}\)	\(370\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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