3.123 \(\int \frac {4+x^2+3 x^4+5 x^6}{x^2 (3+2 x^2+x^4)^3} \, dx\)

Optimal. Leaf size=253 \[ -\frac {\sqrt {\frac {1}{3} \left (55161 \sqrt {3}-59711\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )}{4608}+\frac {\sqrt {\frac {1}{3} \left (55161 \sqrt {3}-59711\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )}{4608}-\frac {25 x \left (x^2+5\right )}{144 \left (x^4+2 x^2+3\right )^2}-\frac {x \left (242 x^2+325\right )}{1728 \left (x^4+2 x^2+3\right )}-\frac {4}{27 x}+\frac {\sqrt {\frac {1}{3} \left (59711+55161 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )}{2304}-\frac {\sqrt {\frac {1}{3} \left (59711+55161 \sqrt {3}\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )}{2304} \]

[Out]

-4/27/x-25/144*x*(x^2+5)/(x^4+2*x^2+3)^2-1/1728*x*(242*x^2+325)/(x^4+2*x^2+3)-1/13824*ln(x^2+3^(1/2)-x*(-2+2*3
^(1/2))^(1/2))*(-179133+165483*3^(1/2))^(1/2)+1/13824*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(-179133+165483*3
^(1/2))^(1/2)+1/6912*arctan((-2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(179133+165483*3^(1/2))^(1/2)-1/6
912*arctan((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(179133+165483*3^(1/2))^(1/2)

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Rubi [A]  time = 0.34, antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {1669, 1664, 1169, 634, 618, 204, 628} \[ -\frac {25 x \left (x^2+5\right )}{144 \left (x^4+2 x^2+3\right )^2}-\frac {x \left (242 x^2+325\right )}{1728 \left (x^4+2 x^2+3\right )}-\frac {\sqrt {\frac {1}{3} \left (55161 \sqrt {3}-59711\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )}{4608}+\frac {\sqrt {\frac {1}{3} \left (55161 \sqrt {3}-59711\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )}{4608}-\frac {4}{27 x}+\frac {\sqrt {\frac {1}{3} \left (59711+55161 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )}{2304}-\frac {\sqrt {\frac {1}{3} \left (59711+55161 \sqrt {3}\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )}{2304} \]

Antiderivative was successfully verified.

[In]

Int[(4 + x^2 + 3*x^4 + 5*x^6)/(x^2*(3 + 2*x^2 + x^4)^3),x]

[Out]

-4/(27*x) - (25*x*(5 + x^2))/(144*(3 + 2*x^2 + x^4)^2) - (x*(325 + 242*x^2))/(1728*(3 + 2*x^2 + x^4)) + (Sqrt[
(59711 + 55161*Sqrt[3])/3]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/2304 - (Sqrt[(59711 +
55161*Sqrt[3])/3]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/2304 - (Sqrt[(-59711 + 55161*
Sqrt[3])/3]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/4608 + (Sqrt[(-59711 + 55161*Sqrt[3])/3]*Log[Sqrt[3
] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/4608

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 618

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

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

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

Rule 1169

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

Rule 1664

Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x
)^m*Pq*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && PolyQ[Pq, x^2] && IGtQ[p, -2]

Rule 1669

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainde
r[x^m*Pq, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 2]}, S
imp[(x*(a + b*x^2 + c*x^4)^(p + 1)*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2))/(2*a*(p + 1)*(b^2 - 4*a*c)
), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[x^m*(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[(2*a*(p + 1)*(b^2
- 4*a*c)*PolynomialQuotient[x^m*Pq, a + b*x^2 + c*x^4, x])/x^m + (b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e)
/x^m + c*(4*p + 7)*(b*d - 2*a*e)*x^(2 - m), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && GtQ[Expon[
Pq, x^2], 1] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && ILtQ[m/2, 0]

Rubi steps

\begin {align*} \int \frac {4+x^2+3 x^4+5 x^6}{x^2 \left (3+2 x^2+x^4\right )^3} \, dx &=-\frac {25 x \left (5+x^2\right )}{144 \left (3+2 x^2+x^4\right )^2}+\frac {1}{96} \int \frac {128+30 x^2-\frac {250 x^4}{3}}{x^2 \left (3+2 x^2+x^4\right )^2} \, dx\\ &=-\frac {25 x \left (5+x^2\right )}{144 \left (3+2 x^2+x^4\right )^2}-\frac {x \left (325+242 x^2\right )}{1728 \left (3+2 x^2+x^4\right )}+\frac {\int \frac {2048-\frac {56 x^2}{3}-\frac {1936 x^4}{3}}{x^2 \left (3+2 x^2+x^4\right )} \, dx}{4608}\\ &=-\frac {25 x \left (5+x^2\right )}{144 \left (3+2 x^2+x^4\right )^2}-\frac {x \left (325+242 x^2\right )}{1728 \left (3+2 x^2+x^4\right )}+\frac {\int \left (\frac {2048}{3 x^2}-\frac {8 \left (173+166 x^2\right )}{3+2 x^2+x^4}\right ) \, dx}{4608}\\ &=-\frac {4}{27 x}-\frac {25 x \left (5+x^2\right )}{144 \left (3+2 x^2+x^4\right )^2}-\frac {x \left (325+242 x^2\right )}{1728 \left (3+2 x^2+x^4\right )}-\frac {1}{576} \int \frac {173+166 x^2}{3+2 x^2+x^4} \, dx\\ &=-\frac {4}{27 x}-\frac {25 x \left (5+x^2\right )}{144 \left (3+2 x^2+x^4\right )^2}-\frac {x \left (325+242 x^2\right )}{1728 \left (3+2 x^2+x^4\right )}-\frac {\int \frac {173 \sqrt {2 \left (-1+\sqrt {3}\right )}-\left (173-166 \sqrt {3}\right ) x}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{1152 \sqrt {6 \left (-1+\sqrt {3}\right )}}-\frac {\int \frac {173 \sqrt {2 \left (-1+\sqrt {3}\right )}+\left (173-166 \sqrt {3}\right ) x}{\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{1152 \sqrt {6 \left (-1+\sqrt {3}\right )}}\\ &=-\frac {4}{27 x}-\frac {25 x \left (5+x^2\right )}{144 \left (3+2 x^2+x^4\right )^2}-\frac {x \left (325+242 x^2\right )}{1728 \left (3+2 x^2+x^4\right )}-\frac {\sqrt {\frac {1}{3} \left (-59711+55161 \sqrt {3}\right )} \int \frac {-\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{4608}+\frac {\sqrt {\frac {1}{3} \left (-59711+55161 \sqrt {3}\right )} \int \frac {\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x}{\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{4608}-\frac {\sqrt {\frac {1}{3} \left (112597+57436 \sqrt {3}\right )} \int \frac {1}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{2304}-\frac {\sqrt {\frac {1}{3} \left (112597+57436 \sqrt {3}\right )} \int \frac {1}{\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{2304}\\ &=-\frac {4}{27 x}-\frac {25 x \left (5+x^2\right )}{144 \left (3+2 x^2+x^4\right )^2}-\frac {x \left (325+242 x^2\right )}{1728 \left (3+2 x^2+x^4\right )}-\frac {\sqrt {\frac {1}{3} \left (-59711+55161 \sqrt {3}\right )} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )}{4608}+\frac {\sqrt {\frac {1}{3} \left (-59711+55161 \sqrt {3}\right )} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )}{4608}+\frac {\sqrt {\frac {1}{3} \left (112597+57436 \sqrt {3}\right )} \operatorname {Subst}\left (\int \frac {1}{-2 \left (1+\sqrt {3}\right )-x^2} \, dx,x,-\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x\right )}{1152}+\frac {\sqrt {\frac {1}{3} \left (112597+57436 \sqrt {3}\right )} \operatorname {Subst}\left (\int \frac {1}{-2 \left (1+\sqrt {3}\right )-x^2} \, dx,x,\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x\right )}{1152}\\ &=-\frac {4}{27 x}-\frac {25 x \left (5+x^2\right )}{144 \left (3+2 x^2+x^4\right )^2}-\frac {x \left (325+242 x^2\right )}{1728 \left (3+2 x^2+x^4\right )}+\frac {\sqrt {\frac {1}{3} \left (59711+55161 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )}{2304}-\frac {\sqrt {\frac {1}{3} \left (59711+55161 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )}{2304}-\frac {\sqrt {\frac {1}{3} \left (-59711+55161 \sqrt {3}\right )} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )}{4608}+\frac {\sqrt {\frac {1}{3} \left (-59711+55161 \sqrt {3}\right )} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )}{4608}\\ \end {align*}

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Mathematica [C]  time = 0.37, size = 140, normalized size = 0.55 \[ \frac {-\frac {12 \left (166 x^8+611 x^6+1412 x^4+1849 x^2+768\right )}{x \left (x^4+2 x^2+3\right )^2}+\frac {3 i \left (7 \sqrt {2}+332 i\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )}{\sqrt {1-i \sqrt {2}}}-\frac {3 i \left (7 \sqrt {2}-332 i\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+i \sqrt {2}}}\right )}{\sqrt {1+i \sqrt {2}}}}{6912} \]

Antiderivative was successfully verified.

[In]

Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x^2*(3 + 2*x^2 + x^4)^3),x]

[Out]

((-12*(768 + 1849*x^2 + 1412*x^4 + 611*x^6 + 166*x^8))/(x*(3 + 2*x^2 + x^4)^2) + ((3*I)*(332*I + 7*Sqrt[2])*Ar
cTan[x/Sqrt[1 - I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] - ((3*I)*(-332*I + 7*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/
Sqrt[1 + I*Sqrt[2]])/6912

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fricas [B]  time = 0.91, size = 630, normalized size = 2.49 \[ -\frac {858518351136 \, x^{8} + 3159968147856 \, x^{6} + 210956 \cdot 1391283^{\frac {1}{4}} \sqrt {681} \sqrt {6} \sqrt {3} \sqrt {2} {\left (x^{9} + 4 \, x^{7} + 10 \, x^{5} + 12 \, x^{3} + 9 \, x\right )} \sqrt {59711 \, \sqrt {3} + 165483} \arctan \left (\frac {1}{15811665652336538898} \, \sqrt {11971753} 1391283^{\frac {3}{4}} \sqrt {681} \sqrt {6} \sqrt {1391283^{\frac {1}{4}} \sqrt {681} \sqrt {6} {\left (166 \, \sqrt {3} x - 173 \, x\right )} \sqrt {59711 \, \sqrt {3} + 165483} + 107745777 \, x^{2} + 107745777 \, \sqrt {3}} {\left (173 \, \sqrt {3} \sqrt {2} - 498 \, \sqrt {2}\right )} \sqrt {59711 \, \sqrt {3} + 165483} - \frac {1}{440249244822} \cdot 1391283^{\frac {3}{4}} \sqrt {681} \sqrt {6} {\left (173 \, \sqrt {3} \sqrt {2} x - 498 \, \sqrt {2} x\right )} \sqrt {59711 \, \sqrt {3} + 165483} + \frac {1}{2} \, \sqrt {3} \sqrt {2} - \frac {1}{2} \, \sqrt {2}\right ) + 210956 \cdot 1391283^{\frac {1}{4}} \sqrt {681} \sqrt {6} \sqrt {3} \sqrt {2} {\left (x^{9} + 4 \, x^{7} + 10 \, x^{5} + 12 \, x^{3} + 9 \, x\right )} \sqrt {59711 \, \sqrt {3} + 165483} \arctan \left (\frac {1}{47434996957009616694} \, \sqrt {11971753} 1391283^{\frac {3}{4}} \sqrt {681} \sqrt {6} \sqrt {-9 \cdot 1391283^{\frac {1}{4}} \sqrt {681} \sqrt {6} {\left (166 \, \sqrt {3} x - 173 \, x\right )} \sqrt {59711 \, \sqrt {3} + 165483} + 969711993 \, x^{2} + 969711993 \, \sqrt {3}} {\left (173 \, \sqrt {3} \sqrt {2} - 498 \, \sqrt {2}\right )} \sqrt {59711 \, \sqrt {3} + 165483} - \frac {1}{440249244822} \cdot 1391283^{\frac {3}{4}} \sqrt {681} \sqrt {6} {\left (173 \, \sqrt {3} \sqrt {2} x - 498 \, \sqrt {2} x\right )} \sqrt {59711 \, \sqrt {3} + 165483} - \frac {1}{2} \, \sqrt {3} \sqrt {2} + \frac {1}{2} \, \sqrt {2}\right ) + 7302577781952 \, x^{4} - 1391283^{\frac {1}{4}} \sqrt {681} \sqrt {6} {\left (165483 \, x^{9} + 661932 \, x^{7} + 1654830 \, x^{5} + 1985796 \, x^{3} - 59711 \, \sqrt {3} {\left (x^{9} + 4 \, x^{7} + 10 \, x^{5} + 12 \, x^{3} + 9 \, x\right )} + 1489347 \, x\right )} \sqrt {59711 \, \sqrt {3} + 165483} \log \left (9 \cdot 1391283^{\frac {1}{4}} \sqrt {681} \sqrt {6} {\left (166 \, \sqrt {3} x - 173 \, x\right )} \sqrt {59711 \, \sqrt {3} + 165483} + 969711993 \, x^{2} + 969711993 \, \sqrt {3}\right ) + 1391283^{\frac {1}{4}} \sqrt {681} \sqrt {6} {\left (165483 \, x^{9} + 661932 \, x^{7} + 1654830 \, x^{5} + 1985796 \, x^{3} - 59711 \, \sqrt {3} {\left (x^{9} + 4 \, x^{7} + 10 \, x^{5} + 12 \, x^{3} + 9 \, x\right )} + 1489347 \, x\right )} \sqrt {59711 \, \sqrt {3} + 165483} \log \left (-9 \cdot 1391283^{\frac {1}{4}} \sqrt {681} \sqrt {6} {\left (166 \, \sqrt {3} x - 173 \, x\right )} \sqrt {59711 \, \sqrt {3} + 165483} + 969711993 \, x^{2} + 969711993 \, \sqrt {3}\right ) + 9562653200304 \, x^{2} + 3971940323328}{2978955242496 \, {\left (x^{9} + 4 \, x^{7} + 10 \, x^{5} + 12 \, x^{3} + 9 \, x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^6+3*x^4+x^2+4)/x^2/(x^4+2*x^2+3)^3,x, algorithm="fricas")

[Out]

-1/2978955242496*(858518351136*x^8 + 3159968147856*x^6 + 210956*1391283^(1/4)*sqrt(681)*sqrt(6)*sqrt(3)*sqrt(2
)*(x^9 + 4*x^7 + 10*x^5 + 12*x^3 + 9*x)*sqrt(59711*sqrt(3) + 165483)*arctan(1/15811665652336538898*sqrt(119717
53)*1391283^(3/4)*sqrt(681)*sqrt(6)*sqrt(1391283^(1/4)*sqrt(681)*sqrt(6)*(166*sqrt(3)*x - 173*x)*sqrt(59711*sq
rt(3) + 165483) + 107745777*x^2 + 107745777*sqrt(3))*(173*sqrt(3)*sqrt(2) - 498*sqrt(2))*sqrt(59711*sqrt(3) +
165483) - 1/440249244822*1391283^(3/4)*sqrt(681)*sqrt(6)*(173*sqrt(3)*sqrt(2)*x - 498*sqrt(2)*x)*sqrt(59711*sq
rt(3) + 165483) + 1/2*sqrt(3)*sqrt(2) - 1/2*sqrt(2)) + 210956*1391283^(1/4)*sqrt(681)*sqrt(6)*sqrt(3)*sqrt(2)*
(x^9 + 4*x^7 + 10*x^5 + 12*x^3 + 9*x)*sqrt(59711*sqrt(3) + 165483)*arctan(1/47434996957009616694*sqrt(11971753
)*1391283^(3/4)*sqrt(681)*sqrt(6)*sqrt(-9*1391283^(1/4)*sqrt(681)*sqrt(6)*(166*sqrt(3)*x - 173*x)*sqrt(59711*s
qrt(3) + 165483) + 969711993*x^2 + 969711993*sqrt(3))*(173*sqrt(3)*sqrt(2) - 498*sqrt(2))*sqrt(59711*sqrt(3) +
165483) - 1/440249244822*1391283^(3/4)*sqrt(681)*sqrt(6)*(173*sqrt(3)*sqrt(2)*x - 498*sqrt(2)*x)*sqrt(59711*s
qrt(3) + 165483) - 1/2*sqrt(3)*sqrt(2) + 1/2*sqrt(2)) + 7302577781952*x^4 - 1391283^(1/4)*sqrt(681)*sqrt(6)*(1
65483*x^9 + 661932*x^7 + 1654830*x^5 + 1985796*x^3 - 59711*sqrt(3)*(x^9 + 4*x^7 + 10*x^5 + 12*x^3 + 9*x) + 148
9347*x)*sqrt(59711*sqrt(3) + 165483)*log(9*1391283^(1/4)*sqrt(681)*sqrt(6)*(166*sqrt(3)*x - 173*x)*sqrt(59711*
sqrt(3) + 165483) + 969711993*x^2 + 969711993*sqrt(3)) + 1391283^(1/4)*sqrt(681)*sqrt(6)*(165483*x^9 + 661932*
x^7 + 1654830*x^5 + 1985796*x^3 - 59711*sqrt(3)*(x^9 + 4*x^7 + 10*x^5 + 12*x^3 + 9*x) + 1489347*x)*sqrt(59711*
sqrt(3) + 165483)*log(-9*1391283^(1/4)*sqrt(681)*sqrt(6)*(166*sqrt(3)*x - 173*x)*sqrt(59711*sqrt(3) + 165483)
+ 969711993*x^2 + 969711993*sqrt(3)) + 9562653200304*x^2 + 3971940323328)/(x^9 + 4*x^7 + 10*x^5 + 12*x^3 + 9*x
)

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giac [B]  time = 3.27, size = 582, normalized size = 2.30 \[ \frac {1}{746496} \, \sqrt {2} {\left (83 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 1494 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 1494 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 83 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 3114 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 3114 \cdot 3^{\frac {1}{4}} \sqrt {-6 \, \sqrt {3} + 18}\right )} \arctan \left (\frac {3^{\frac {3}{4}} {\left (x + 3^{\frac {1}{4}} \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}\right )}}{3 \, \sqrt {\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}}\right ) + \frac {1}{746496} \, \sqrt {2} {\left (83 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 1494 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 1494 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 83 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 3114 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 3114 \cdot 3^{\frac {1}{4}} \sqrt {-6 \, \sqrt {3} + 18}\right )} \arctan \left (\frac {3^{\frac {3}{4}} {\left (x - 3^{\frac {1}{4}} \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}\right )}}{3 \, \sqrt {\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}}\right ) + \frac {1}{1492992} \, \sqrt {2} {\left (1494 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 83 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 83 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 1494 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 3114 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 3114 \cdot 3^{\frac {1}{4}} \sqrt {6 \, \sqrt {3} + 18}\right )} \log \left (x^{2} + 2 \cdot 3^{\frac {1}{4}} x \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}} + \sqrt {3}\right ) - \frac {1}{1492992} \, \sqrt {2} {\left (1494 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 83 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 83 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 1494 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 3114 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 3114 \cdot 3^{\frac {1}{4}} \sqrt {6 \, \sqrt {3} + 18}\right )} \log \left (x^{2} - 2 \cdot 3^{\frac {1}{4}} x \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}} + \sqrt {3}\right ) - \frac {242 \, x^{7} + 809 \, x^{5} + 1676 \, x^{3} + 2475 \, x}{1728 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}^{2}} - \frac {4}{27 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^6+3*x^4+x^2+4)/x^2/(x^4+2*x^2+3)^3,x, algorithm="giac")

[Out]

1/746496*sqrt(2)*(83*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 1494*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(
3) - 3) - 1494*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) + 83*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) - 3114*3^(1/4)
*sqrt(2)*sqrt(6*sqrt(3) + 18) + 3114*3^(1/4)*sqrt(-6*sqrt(3) + 18))*arctan(1/3*3^(3/4)*(x + 3^(1/4)*sqrt(-1/6*
sqrt(3) + 1/2))/sqrt(1/6*sqrt(3) + 1/2)) + 1/746496*sqrt(2)*(83*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 1494*
3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 1494*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) + 83*3^(
3/4)*(-6*sqrt(3) + 18)^(3/2) - 3114*3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) + 3114*3^(1/4)*sqrt(-6*sqrt(3) + 18))
*arctan(1/3*3^(3/4)*(x - 3^(1/4)*sqrt(-1/6*sqrt(3) + 1/2))/sqrt(1/6*sqrt(3) + 1/2)) + 1/1492992*sqrt(2)*(1494*
3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 83*3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2) + 83*3^(3/4)*
(6*sqrt(3) + 18)^(3/2) + 1494*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 3114*3^(1/4)*sqrt(2)*sqrt(-6*sqrt(3
) + 18) - 3114*3^(1/4)*sqrt(6*sqrt(3) + 18))*log(x^2 + 2*3^(1/4)*x*sqrt(-1/6*sqrt(3) + 1/2) + sqrt(3)) - 1/149
2992*sqrt(2)*(1494*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 83*3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^
(3/2) + 83*3^(3/4)*(6*sqrt(3) + 18)^(3/2) + 1494*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 3114*3^(1/4)*sqr
t(2)*sqrt(-6*sqrt(3) + 18) - 3114*3^(1/4)*sqrt(6*sqrt(3) + 18))*log(x^2 - 2*3^(1/4)*x*sqrt(-1/6*sqrt(3) + 1/2)
+ sqrt(3)) - 1/1728*(242*x^7 + 809*x^5 + 1676*x^3 + 2475*x)/(x^4 + 2*x^2 + 3)^2 - 4/27/x

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maple [B]  time = 0.03, size = 424, normalized size = 1.68 \[ -\frac {325 \left (-2+2 \sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{13824 \sqrt {2+2 \sqrt {3}}}+\frac {7 \left (-2+2 \sqrt {3}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{4608 \sqrt {2+2 \sqrt {3}}}-\frac {173 \sqrt {3}\, \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{1728 \sqrt {2+2 \sqrt {3}}}-\frac {325 \left (-2+2 \sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{13824 \sqrt {2+2 \sqrt {3}}}+\frac {7 \left (-2+2 \sqrt {3}\right ) \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{4608 \sqrt {2+2 \sqrt {3}}}-\frac {173 \sqrt {3}\, \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{1728 \sqrt {2+2 \sqrt {3}}}-\frac {325 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}\, \ln \left (x^{2}-\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{27648}+\frac {7 \sqrt {-2+2 \sqrt {3}}\, \ln \left (x^{2}-\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{9216}+\frac {325 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}\, \ln \left (x^{2}+\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{27648}-\frac {7 \sqrt {-2+2 \sqrt {3}}\, \ln \left (x^{2}+\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{9216}-\frac {4}{27 x}-\frac {\frac {121}{32} x^{7}+\frac {809}{64} x^{5}+\frac {419}{16} x^{3}+\frac {2475}{64} x}{27 \left (x^{4}+2 x^{2}+3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^6+3*x^4+x^2+4)/x^2/(x^4+2*x^2+3)^3,x)

[Out]

-4/27/x-1/27*(121/32*x^7+809/64*x^5+419/16*x^3+2475/64*x)/(x^4+2*x^2+3)^2-325/27648*(-2+2*3^(1/2))^(1/2)*3^(1/
2)*ln(x^2-(-2+2*3^(1/2))^(1/2)*x+3^(1/2))+7/9216*(-2+2*3^(1/2))^(1/2)*ln(x^2-(-2+2*3^(1/2))^(1/2)*x+3^(1/2))-3
25/13824/(2+2*3^(1/2))^(1/2)*(-2+2*3^(1/2))*3^(1/2)*arctan((2*x-(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))+7/4
608/(2+2*3^(1/2))^(1/2)*(-2+2*3^(1/2))*arctan((2*x-(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))-173/1728/(2+2*3^
(1/2))^(1/2)*3^(1/2)*arctan((2*x-(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))+325/27648*(-2+2*3^(1/2))^(1/2)*3^(
1/2)*ln(x^2+(-2+2*3^(1/2))^(1/2)*x+3^(1/2))-7/9216*(-2+2*3^(1/2))^(1/2)*ln(x^2+(-2+2*3^(1/2))^(1/2)*x+3^(1/2))
-325/13824/(2+2*3^(1/2))^(1/2)*(-2+2*3^(1/2))*3^(1/2)*arctan((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))+7
/4608/(2+2*3^(1/2))^(1/2)*(-2+2*3^(1/2))*arctan((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))-173/1728/(2+2*
3^(1/2))^(1/2)*3^(1/2)*arctan((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {166 \, x^{8} + 611 \, x^{6} + 1412 \, x^{4} + 1849 \, x^{2} + 768}{576 \, {\left (x^{9} + 4 \, x^{7} + 10 \, x^{5} + 12 \, x^{3} + 9 \, x\right )}} - \frac {1}{576} \, \int \frac {166 \, x^{2} + 173}{x^{4} + 2 \, x^{2} + 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^6+3*x^4+x^2+4)/x^2/(x^4+2*x^2+3)^3,x, algorithm="maxima")

[Out]

-1/576*(166*x^8 + 611*x^6 + 1412*x^4 + 1849*x^2 + 768)/(x^9 + 4*x^7 + 10*x^5 + 12*x^3 + 9*x) - 1/576*integrate
((166*x^2 + 173)/(x^4 + 2*x^2 + 3), x)

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mupad [B]  time = 0.99, size = 179, normalized size = 0.71 \[ -\frac {\frac {83\,x^8}{288}+\frac {611\,x^6}{576}+\frac {353\,x^4}{144}+\frac {1849\,x^2}{576}+\frac {4}{3}}{x^9+4\,x^7+10\,x^5+12\,x^3+9\,x}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-358266-\sqrt {2}\,316434{}\mathrm {i}}\,52739{}\mathrm {i}}{859963392\,\left (-\frac {17140175}{286654464}+\frac {\sqrt {2}\,9123847{}\mathrm {i}}{286654464}\right )}+\frac {52739\,\sqrt {2}\,x\,\sqrt {-358266-\sqrt {2}\,316434{}\mathrm {i}}}{1719926784\,\left (-\frac {17140175}{286654464}+\frac {\sqrt {2}\,9123847{}\mathrm {i}}{286654464}\right )}\right )\,\sqrt {-358266-\sqrt {2}\,316434{}\mathrm {i}}\,1{}\mathrm {i}}{6912}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-358266+\sqrt {2}\,316434{}\mathrm {i}}\,52739{}\mathrm {i}}{859963392\,\left (\frac {17140175}{286654464}+\frac {\sqrt {2}\,9123847{}\mathrm {i}}{286654464}\right )}-\frac {52739\,\sqrt {2}\,x\,\sqrt {-358266+\sqrt {2}\,316434{}\mathrm {i}}}{1719926784\,\left (\frac {17140175}{286654464}+\frac {\sqrt {2}\,9123847{}\mathrm {i}}{286654464}\right )}\right )\,\sqrt {-358266+\sqrt {2}\,316434{}\mathrm {i}}\,1{}\mathrm {i}}{6912} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 + 3*x^4 + 5*x^6 + 4)/(x^2*(2*x^2 + x^4 + 3)^3),x)

[Out]

(atan((x*(- 2^(1/2)*316434i - 358266)^(1/2)*52739i)/(859963392*((2^(1/2)*9123847i)/286654464 - 17140175/286654
464)) + (52739*2^(1/2)*x*(- 2^(1/2)*316434i - 358266)^(1/2))/(1719926784*((2^(1/2)*9123847i)/286654464 - 17140
175/286654464)))*(- 2^(1/2)*316434i - 358266)^(1/2)*1i)/6912 - (atan((x*(2^(1/2)*316434i - 358266)^(1/2)*52739
i)/(859963392*((2^(1/2)*9123847i)/286654464 + 17140175/286654464)) - (52739*2^(1/2)*x*(2^(1/2)*316434i - 35826
6)^(1/2))/(1719926784*((2^(1/2)*9123847i)/286654464 + 17140175/286654464)))*(2^(1/2)*316434i - 358266)^(1/2)*1
i)/6912 - ((1849*x^2)/576 + (353*x^4)/144 + (611*x^6)/576 + (83*x^8)/288 + 4/3)/(9*x + 12*x^3 + 10*x^5 + 4*x^7
+ x^9)

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sympy [A]  time = 0.67, size = 75, normalized size = 0.30 \[ \frac {- 166 x^{8} - 611 x^{6} - 1412 x^{4} - 1849 x^{2} - 768}{576 x^{9} + 2304 x^{7} + 5760 x^{5} + 6912 x^{3} + 5184 x} + \operatorname {RootSum} {\left (4174708211712 t^{4} + 15652880384 t^{2} + 37564641, \left (t \mapsto t \log {\left (- \frac {98146713600 t^{3}}{11971753} - \frac {9639364864 t}{323237331} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x**6+3*x**4+x**2+4)/x**2/(x**4+2*x**2+3)**3,x)

[Out]

(-166*x**8 - 611*x**6 - 1412*x**4 - 1849*x**2 - 768)/(576*x**9 + 2304*x**7 + 5760*x**5 + 6912*x**3 + 5184*x) +
RootSum(4174708211712*_t**4 + 15652880384*_t**2 + 37564641, Lambda(_t, _t*log(-98146713600*_t**3/11971753 - 9
639364864*_t/323237331 + x)))

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