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

Optimal. Leaf size=262 \[ -\frac {4}{81 x^3}+\frac {\sqrt {\frac {1}{3} \left (11240451 \sqrt {3}-10004741\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )}{41472}-\frac {\sqrt {\frac {1}{3} \left (11240451 \sqrt {3}-10004741\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )}{41472}+\frac {25 x \left (5 x^2+7\right )}{432 \left (x^4+2 x^2+3\right )^2}+\frac {x \left (1025 x^2+1474\right )}{5184 \left (x^4+2 x^2+3\right )}+\frac {7}{27 x}-\frac {\sqrt {\frac {1}{3} \left (10004741+11240451 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )}{20736}+\frac {\sqrt {\frac {1}{3} \left (10004741+11240451 \sqrt {3}\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )}{20736} \]

[Out]

-4/81/x^3+7/27/x+25/432*x*(5*x^2+7)/(x^4+2*x^2+3)^2+1/5184*x*(1025*x^2+1474)/(x^4+2*x^2+3)+1/124416*ln(x^2+3^(
1/2)-x*(-2+2*3^(1/2))^(1/2))*(-30014223+33721353*3^(1/2))^(1/2)-1/124416*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2)
)*(-30014223+33721353*3^(1/2))^(1/2)-1/62208*arctan((-2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(30014223
[phone]*3^(1/2))^(1/2)+1/62208*arctan((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(30014223+33721353*3^(
1/2))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.37, antiderivative size = 262, 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} \[ \frac {25 x \left (5 x^2+7\right )}{432 \left (x^4+2 x^2+3\right )^2}+\frac {x \left (1025 x^2+1474\right )}{5184 \left (x^4+2 x^2+3\right )}-\frac {4}{81 x^3}+\frac {\sqrt {\frac {1}{3} \left (11240451 \sqrt {3}-10004741\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )}{41472}-\frac {\sqrt {\frac {1}{3} \left (11240451 \sqrt {3}-10004741\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )}{41472}+\frac {7}{27 x}-\frac {\sqrt {\frac {1}{3} \left (10004741+11240451 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )}{20736}+\frac {\sqrt {\frac {1}{3} \left (10004741+11240451 \sqrt {3}\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )}{20736} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-4/(81*x^3) + 7/(27*x) + (25*x*(7 + 5*x^2))/(432*(3 + 2*x^2 + x^4)^2) + (x*(1474 + 1025*x^2))/(5184*(3 + 2*x^2
+ x^4)) - (Sqrt[(10004741 + 11240451*Sqrt[3])/3]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]]
)/20736 + (Sqrt[(10004741 + 11240451*Sqrt[3])/3]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])
/20736 + (Sqrt[(-10004741 + 11240451*Sqrt[3])/3]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/41472 - (Sqrt[
(-10004741 + 11240451*Sqrt[3])/3]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/41472

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^4 \left (3+2 x^2+x^4\right )^3} \, dx &=\frac {25 x \left (7+5 x^2\right )}{432 \left (3+2 x^2+x^4\right )^2}+\frac {1}{96} \int \frac {128-\frac {160 x^2}{3}+50 x^4+\frac {1250 x^6}{9}}{x^4 \left (3+2 x^2+x^4\right )^2} \, dx\\ &=\frac {25 x \left (7+5 x^2\right )}{432 \left (3+2 x^2+x^4\right )^2}+\frac {x \left (1474+1025 x^2\right )}{5184 \left (3+2 x^2+x^4\right )}+\frac {\int \frac {2048-\frac {6656 x^2}{3}+\frac {2576 x^4}{9}+\frac {8200 x^6}{9}}{x^4 \left (3+2 x^2+x^4\right )} \, dx}{4608}\\ &=\frac {25 x \left (7+5 x^2\right )}{432 \left (3+2 x^2+x^4\right )^2}+\frac {x \left (1474+1025 x^2\right )}{5184 \left (3+2 x^2+x^4\right )}+\frac {\int \left (\frac {2048}{3 x^4}-\frac {3584}{3 x^2}+\frac {8 \left (2242+2369 x^2\right )}{9 \left (3+2 x^2+x^4\right )}\right ) \, dx}{4608}\\ &=-\frac {4}{81 x^3}+\frac {7}{27 x}+\frac {25 x \left (7+5 x^2\right )}{432 \left (3+2 x^2+x^4\right )^2}+\frac {x \left (1474+1025 x^2\right )}{5184 \left (3+2 x^2+x^4\right )}+\frac {\int \frac {2242+2369 x^2}{3+2 x^2+x^4} \, dx}{5184}\\ &=-\frac {4}{81 x^3}+\frac {7}{27 x}+\frac {25 x \left (7+5 x^2\right )}{432 \left (3+2 x^2+x^4\right )^2}+\frac {x \left (1474+1025 x^2\right )}{5184 \left (3+2 x^2+x^4\right )}+\frac {\int \frac {2242 \sqrt {2 \left (-1+\sqrt {3}\right )}-\left (2242-2369 \sqrt {3}\right ) x}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{10368 \sqrt {6 \left (-1+\sqrt {3}\right )}}+\frac {\int \frac {2242 \sqrt {2 \left (-1+\sqrt {3}\right )}+\left (2242-2369 \sqrt {3}\right ) x}{\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{10368 \sqrt {6 \left (-1+\sqrt {3}\right )}}\\ &=-\frac {4}{81 x^3}+\frac {7}{27 x}+\frac {25 x \left (7+5 x^2\right )}{432 \left (3+2 x^2+x^4\right )^2}+\frac {x \left (1474+1025 x^2\right )}{5184 \left (3+2 x^2+x^4\right )}+\frac {\left (2242-2369 \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}{20736 \sqrt {6 \left (-1+\sqrt {3}\right )}}+\frac {\left (7107+2242 \sqrt {3}\right ) \int \frac {1}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{62208}+\frac {\left (7107+2242 \sqrt {3}\right ) \int \frac {1}{\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{62208}+\frac {\left (-2242+2369 \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}{20736 \sqrt {6 \left (-1+\sqrt {3}\right )}}\\ &=-\frac {4}{81 x^3}+\frac {7}{27 x}+\frac {25 x \left (7+5 x^2\right )}{432 \left (3+2 x^2+x^4\right )^2}+\frac {x \left (1474+1025 x^2\right )}{5184 \left (3+2 x^2+x^4\right )}+\frac {\sqrt {-\frac {10004741}{12}+\frac {3746817 \sqrt {3}}{4}} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )}{20736}-\frac {\sqrt {-\frac {10004741}{12}+\frac {3746817 \sqrt {3}}{4}} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )}{20736}-\frac {\left (7107+2242 \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 )}{31104}-\frac {\left (7107+2242 \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 )}{31104}\\ &=-\frac {4}{81 x^3}+\frac {7}{27 x}+\frac {25 x \left (7+5 x^2\right )}{432 \left (3+2 x^2+x^4\right )^2}+\frac {x \left (1474+1025 x^2\right )}{5184 \left (3+2 x^2+x^4\right )}-\frac {\sqrt {\frac {1}{3} \left (10004741+11240451 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )}{20736}+\frac {\sqrt {\frac {1}{3} \left (10004741+11240451 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )}{20736}+\frac {\sqrt {-\frac {10004741}{12}+\frac {3746817 \sqrt {3}}{4}} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )}{20736}-\frac {\sqrt {-\frac {10004741}{12}+\frac {3746817 \sqrt {3}}{4}} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )}{20736}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.32, size = 139, normalized size = 0.53 \[ \frac {\frac {4 \left (2369 x^{10}+8644 x^8+19939 x^6+20090 x^4+9024 x^2-2304\right )}{x^3 \left (x^4+2 x^2+3\right )^2}+\frac {\left (4738+127 i \sqrt {2}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )}{\sqrt {1-i \sqrt {2}}}+\frac {\left (4738-127 i \sqrt {2}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+i \sqrt {2}}}\right )}{\sqrt {1+i \sqrt {2}}}}{20736} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*(-2304 + 9024*x^2 + 20090*x^4 + 19939*x^6 + 8644*x^8 + 2369*x^10))/(x^3*(3 + 2*x^2 + x^4)^2) + ((4738 + (1
27*I)*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] + ((4738 - (127*I)*Sqrt[2])*ArcTan[x/Sqrt[1
+ I*Sqrt[2]]])/Sqrt[1 + I*Sqrt[2]])/20736

________________________________________________________________________________________

fricas [B]  time = 0.87, size = 652, normalized size = 2.49 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/135934787413472256*(62119890312985296*x^10 + 226662866975704896*x^8 + 522840224968600176*x^6 + 47239676*7132
36683^(1/4)*sqrt(15419)*sqrt(6)*sqrt(3)*sqrt(2)*(x^11 + 4*x^9 + 10*x^7 + 12*x^5 + 9*x^3)*sqrt(10004741*sqrt(3)
+ 33721353)*arctan(1/27609352591972558367520653346*sqrt(182097141061)*713236683^(3/4)*sqrt(15419)*sqrt(6)*sqr
t(3)*sqrt(713236683^(1/4)*sqrt(15419)*sqrt(6)*(2369*sqrt(3)*x - 2242*x)*sqrt(10004741*sqrt(3) + 33721353) + 54
6291423183*x^2 + 546291423183*sqrt(3))*(2242*sqrt(3)*sqrt(2) - 7107*sqrt(2))*sqrt(10004741*sqrt(3) + 33721353)
- 1/50539604724352062*713236683^(3/4)*sqrt(15419)*sqrt(6)*(2242*sqrt(3)*sqrt(2)*x - 7107*sqrt(2)*x)*sqrt(1000
4741*sqrt(3) + 33721353) + 1/2*sqrt(3)*sqrt(2) - 1/2*sqrt(2)) + 47239676*713236683^(1/4)*sqrt(15419)*sqrt(6)*s
qrt(3)*sqrt(2)*(x^11 + 4*x^9 + 10*x^7 + 12*x^5 + 9*x^3)*sqrt(10004741*sqrt(3) + 33721353)*arctan(1/82828057775
917675102561960038*sqrt(182097141061)*713236683^(3/4)*sqrt(15419)*sqrt(6)*sqrt(-27*713236683^(1/4)*sqrt(15419)
*sqrt(6)*(2369*sqrt(3)*x - 2242*x)*sqrt(10004741*sqrt(3) + 33721353) + 14749868425941*x^2 + 14749868425941*sqr
t(3))*(2242*sqrt(3)*sqrt(2) - 7107*sqrt(2))*sqrt(10004741*sqrt(3) + 33721353) - 1/50539604724352062*713236683^
(3/4)*sqrt(15419)*sqrt(6)*(2242*sqrt(3)*sqrt(2)*x - 7107*sqrt(2)*x)*sqrt(10004741*sqrt(3) + 33721353) - 1/2*sq
rt(3)*sqrt(2) + 1/2*sqrt(2)) + 526799745203830560*x^4 - 713236683^(1/4)*sqrt(15419)*sqrt(6)*(33721353*x^11 + 1
34885412*x^9 + 337213530*x^7 + 404656236*x^5 + 303492177*x^3 - 10004741*sqrt(3)*(x^11 + 4*x^9 + 10*x^7 + 12*x^
5 + 9*x^3))*sqrt(10004741*sqrt(3) + 33721353)*log(27*713236683^(1/4)*sqrt(15419)*sqrt(6)*(2369*sqrt(3)*x - 224
2*x)*sqrt(10004741*sqrt(3) + 33721353) + 14749868425941*x^2 + 14749868425941*sqrt(3)) + 713236683^(1/4)*sqrt(1
5419)*sqrt(6)*(33721353*x^11 + 134885412*x^9 + 337213530*x^7 + 404656236*x^5 + 303492177*x^3 - 10004741*sqrt(3
)*(x^11 + 4*x^9 + 10*x^7 + 12*x^5 + 9*x^3))*sqrt(10004741*sqrt(3) + 33721353)*log(-27*713236683^(1/4)*sqrt(154
19)*sqrt(6)*(2369*sqrt(3)*x - 2242*x)*sqrt(10004741*sqrt(3) + 33721353) + 14749868425941*x^2 + 14749868425941*
sqrt(3)) + 236627222534562816*x^2 - 60415461072654336)/(x^11 + 4*x^9 + 10*x^7 + 12*x^5 + 9*x^3)

________________________________________________________________________________________

giac [B]  time = 2.99, size = 589, normalized size = 2.25 \[ -\frac {1}{13436928} \, \sqrt {2} {\left (2369 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 42642 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 42642 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 2369 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 80712 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 80712 \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}{13436928} \, \sqrt {2} {\left (2369 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 42642 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 42642 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 2369 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 80712 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 80712 \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}{26873856} \, \sqrt {2} {\left (42642 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 2369 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 2369 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 42642 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 80712 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 80712 \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}{26873856} \, \sqrt {2} {\left (42642 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 2369 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 2369 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 42642 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 80712 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 80712 \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 {1025 \, x^{7} + 3524 \, x^{5} + 7523 \, x^{3} + 6522 \, x}{5184 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}^{2}} + \frac {21 \, x^{2} - 4}{81 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/13436928*sqrt(2)*(2369*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 42642*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) + 18)*
(sqrt(3) - 3) - 42642*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) + 2369*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) - 807
12*3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) + 80712*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)) - 1/13436928*sqrt(2)*(2369*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18
)^(3/2) + 42642*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 42642*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(
3) + 18) + 2369*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) - 80712*3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) + 80712*3^(1/4)*s
qrt(-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)) - 1/2
6873856*sqrt(2)*(42642*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 2369*3^(3/4)*sqrt(2)*(-6*sqrt(3)
+ 18)^(3/2) + 2369*3^(3/4)*(6*sqrt(3) + 18)^(3/2) + 42642*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 80712*3
^(1/4)*sqrt(2)*sqrt(-6*sqrt(3) + 18) - 80712*3^(1/4)*sqrt(6*sqrt(3) + 18))*log(x^2 + 2*3^(1/4)*x*sqrt(-1/6*sqr
t(3) + 1/2) + sqrt(3)) + 1/26873856*sqrt(2)*(42642*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 2369*
3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2) + 2369*3^(3/4)*(6*sqrt(3) + 18)^(3/2) + 42642*3^(3/4)*sqrt(6*sqrt(3) +
18)*(sqrt(3) - 3) - 80712*3^(1/4)*sqrt(2)*sqrt(-6*sqrt(3) + 18) - 80712*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)) + 1/5184*(1025*x^7 + 3524*x^5 + 7523*x^3 + 6522*x)/(x^4 + 2
*x^2 + 3)^2 + 1/81*(21*x^2 - 4)/x^3

________________________________________________________________________________________

maple [B]  time = 0.04, size = 429, normalized size = 1.64 \[ \frac {4865 \left (-2+2 \sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{124416 \sqrt {2+2 \sqrt {3}}}+\frac {127 \left (-2+2 \sqrt {3}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{41472 \sqrt {2+2 \sqrt {3}}}+\frac {1121 \sqrt {3}\, \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{7776 \sqrt {2+2 \sqrt {3}}}+\frac {4865 \left (-2+2 \sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{124416 \sqrt {2+2 \sqrt {3}}}+\frac {127 \left (-2+2 \sqrt {3}\right ) \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{41472 \sqrt {2+2 \sqrt {3}}}+\frac {1121 \sqrt {3}\, \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{7776 \sqrt {2+2 \sqrt {3}}}+\frac {4865 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}\, \ln \left (x^{2}-\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{248832}+\frac {127 \sqrt {-2+2 \sqrt {3}}\, \ln \left (x^{2}-\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{82944}-\frac {4865 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}\, \ln \left (x^{2}+\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{248832}-\frac {127 \sqrt {-2+2 \sqrt {3}}\, \ln \left (x^{2}+\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{82944}+\frac {7}{27 x}-\frac {4}{81 x^{3}}+\frac {\frac {1025}{192} x^{7}+\frac {881}{48} x^{5}+\frac {7523}{192} x^{3}+\frac {1087}{32} x}{27 \left (x^{4}+2 x^{2}+3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/81/x^3+7/27/x+1/27*(1025/192*x^7+881/48*x^5+7523/192*x^3+1087/32*x)/(x^4+2*x^2+3)^2+4865/248832*(-2+2*3^(1/
2))^(1/2)*3^(1/2)*ln(x^2-(-2+2*3^(1/2))^(1/2)*x+3^(1/2))+127/82944*(-2+2*3^(1/2))^(1/2)*ln(x^2-(-2+2*3^(1/2))^
(1/2)*x+3^(1/2))+4865/124416/(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))+127/41472/(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))+1121/7776/(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))-4865/24883
2*(-2+2*3^(1/2))^(1/2)*3^(1/2)*ln(x^2+(-2+2*3^(1/2))^(1/2)*x+3^(1/2))-127/82944*(-2+2*3^(1/2))^(1/2)*ln(x^2+(-
2+2*3^(1/2))^(1/2)*x+3^(1/2))+4865/124416/(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))+127/41472/(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))+1121/7776/(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
))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2369 \, x^{10} + 8644 \, x^{8} + 19939 \, x^{6} + 20090 \, x^{4} + 9024 \, x^{2} - 2304}{5184 \, {\left (x^{11} + 4 \, x^{9} + 10 \, x^{7} + 12 \, x^{5} + 9 \, x^{3}\right )}} + \frac {1}{5184} \, \int \frac {2369 \, x^{2} + 2242}{x^{4} + 2 \, x^{2} + 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5184*(2369*x^10 + 8644*x^8 + 19939*x^6 + 20090*x^4 + 9024*x^2 - 2304)/(x^11 + 4*x^9 + 10*x^7 + 12*x^5 + 9*x^
3) + 1/5184*integrate((2369*x^2 + 2242)/(x^4 + 2*x^2 + 3), x)

________________________________________________________________________________________

mupad [B]  time = 1.02, size = 185, normalized size = 0.71 \[ \frac {\frac {2369\,x^{10}}{5184}+\frac {2161\,x^8}{1296}+\frac {19939\,x^6}{5184}+\frac {10045\,x^4}{2592}+\frac {47\,x^2}{27}-\frac {4}{9}}{x^{11}+4\,x^9+10\,x^7+12\,x^5+9\,x^3}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-60028446-\sqrt {2}\,70859514{}\mathrm {i}}\,11809919{}\mathrm {i}}{626913312768\,\left (-\frac {57455255935}{208971104256}+\frac {\sqrt {2}\,13238919199{}\mathrm {i}}{104485552128}\right )}+\frac {11809919\,\sqrt {2}\,x\,\sqrt {-60028446-\sqrt {2}\,70859514{}\mathrm {i}}}{1253826625536\,\left (-\frac {57455255935}{208971104256}+\frac {\sqrt {2}\,13238919199{}\mathrm {i}}{104485552128}\right )}\right )\,\sqrt {-60028446-\sqrt {2}\,70859514{}\mathrm {i}}\,1{}\mathrm {i}}{62208}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-60028446+\sqrt {2}\,70859514{}\mathrm {i}}\,11809919{}\mathrm {i}}{626913312768\,\left (\frac {57455255935}{208971104256}+\frac {\sqrt {2}\,13238919199{}\mathrm {i}}{104485552128}\right )}-\frac {11809919\,\sqrt {2}\,x\,\sqrt {-60028446+\sqrt {2}\,70859514{}\mathrm {i}}}{1253826625536\,\left (\frac {57455255935}{208971104256}+\frac {\sqrt {2}\,13238919199{}\mathrm {i}}{104485552128}\right )}\right )\,\sqrt {-60028446+\sqrt {2}\,70859514{}\mathrm {i}}\,1{}\mathrm {i}}{62208} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((47*x^2)/27 + (10045*x^4)/2592 + (19939*x^6)/5184 + (2161*x^8)/1296 + (2369*x^10)/5184 - 4/9)/(9*x^3 + 12*x^5
+ 10*x^7 + 4*x^9 + x^11) - (atan((x*(- 2^(1/2)*70859514i - 60028446)^(1/2)*11809919i)/(626913312768*((2^(1/2)
*13238919199i)/104485552128 - 57455255935/208971104256)) + (11809919*2^(1/2)*x*(- 2^(1/2)*70859514i - 60028446
)^(1/2))/(1253826625536*((2^(1/2)*13238919199i)/104485552128 - 57455255935/208971104256)))*(- 2^(1/2)*70859514
i - 60028446)^(1/2)*1i)/62208 + (atan((x*(2^(1/2)*70859514i - 60028446)^(1/2)*11809919i)/(626913312768*((2^(1/
2)*13238919199i)/104485552128 + 57455255935/208971104256)) - (11809919*2^(1/2)*x*(2^(1/2)*70859514i - 60028446
)^(1/2))/(1253826625536*((2^(1/2)*13238919199i)/104485552128 + 57455255935/208971104256)))*(2^(1/2)*70859514i
- 60028446)^(1/2)*1i)/62208

________________________________________________________________________________________

sympy [A]  time = 0.69, size = 80, normalized size = 0.31 \[ \operatorname {RootSum} {\left (338151365148672 t^{4} + 2622682824704 t^{2} + 19257390441, \left (t \mapsto t \log {\left (\frac {357010935644160 t^{3}}{182097141061} + \frac {26016957890816 t}{1638874269549} + x \right )} \right )\right )} + \frac {2369 x^{10} + 8644 x^{8} + 19939 x^{6} + 20090 x^{4} + 9024 x^{2} - 2304}{5184 x^{11} + 20736 x^{9} + 51840 x^{7} + 62208 x^{5} + 46656 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(338151365148672*_t**4 + 2622682824704*_t**2 + 19257390441, Lambda(_t, _t*log(357010935644160*_t**3/182
097141061 + 26016957890816*_t/1638874269549 + x))) + (2369*x**10 + 8644*x**8 + 19939*x**6 + 20090*x**4 + 9024*
x**2 - 2304)/(5184*x**11 + 20736*x**9 + 51840*x**7 + 62208*x**5 + 46656*x**3)

________________________________________________________________________________________