\(\int \frac {4+x^2+3 x^4+5 x^6}{x^6 (3+2 x^2+x^4)^2} \, dx\) [116]

Optimal result

Integrand size = 31, antiderivative size = 245 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x^6 \left (3+2 x^2+x^4\right )^2} \, dx=-\frac {4}{45 x^5}+\frac {13}{81 x^3}-\frac {13}{27 x}+\frac {25 x \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}+\frac {\sqrt {\frac {1}{6} \left (-1139381+688419 \sqrt {3}\right )} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )}{1296}-\frac {\sqrt {\frac {1}{6} \left (-1139381+688419 \sqrt {3}\right )} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )}{1296}-\frac {\sqrt {\frac {1}{6} \left (1139381+688419 \sqrt {3}\right )} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )}{2592}+\frac {\sqrt {\frac {1}{6} \left (1139381+688419 \sqrt {3}\right )} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )}{2592} \]

[Out]

-4/45/x^5+13/81/x^3-13/27/x+25/648*x*(-7*x^2+1)/(x^4+2*x^2+3)+1/7776*arctan((-2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3
^(1/2))^(1/2))*(-6836286+4130514*3^(1/2))^(1/2)-1/7776*arctan((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*
(-6836286+4130514*3^(1/2))^(1/2)-1/15552*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(6836286+4130514*3^(1/2))^(1/2
)+1/15552*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(6836286+4130514*3^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {1683, 1678, 1183, 648, 632, 210, 642} \[ \int \frac {4+x^2+3 x^4+5 x^6}{x^6 \left (3+2 x^2+x^4\right )^2} \, dx=\frac {\sqrt {\frac {1}{6} \left (688419 \sqrt {3}-1139381\right )} \arctan \left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )}{1296}-\frac {\sqrt {\frac {1}{6} \left (688419 \sqrt {3}-1139381\right )} \arctan \left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )}{1296}-\frac {4}{45 x^5}+\frac {13}{81 x^3}-\frac {\sqrt {\frac {1}{6} \left (1139381+688419 \sqrt {3}\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )}{2592}+\frac {\sqrt {\frac {1}{6} \left (1139381+688419 \sqrt {3}\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )}{2592}+\frac {25 x \left (1-7 x^2\right )}{648 \left (x^4+2 x^2+3\right )}-\frac {13}{27 x} \]

[In]

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

[Out]

-4/(45*x^5) + 13/(81*x^3) - 13/(27*x) + (25*x*(1 - 7*x^2))/(648*(3 + 2*x^2 + x^4)) + (Sqrt[(-1139381 + 688419*
Sqrt[3])/6]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/1296 - (Sqrt[(-1139381 + 688419*Sqrt
[3])/6]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/1296 - (Sqrt[(1139381 + 688419*Sqrt[3])/
6]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/2592 + (Sqrt[(1139381 + 688419*Sqrt[3])/6]*Log[Sqrt[3] + Sqr
t[2*(-1 + Sqrt[3])]*x + x^2])/2592

Rule 210

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

Rule 632

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 642

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 648

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 1183

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 1678

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 1683

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*} \text {integral}& = \frac {25 x \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}+\frac {1}{48} \int \frac {64-\frac {80 x^2}{3}+\frac {400 x^4}{9}+\frac {1550 x^6}{27}-\frac {350 x^8}{27}}{x^6 \left (3+2 x^2+x^4\right )} \, dx \\ & = \frac {25 x \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}+\frac {1}{48} \int \left (\frac {64}{3 x^6}-\frac {208}{9 x^4}+\frac {208}{9 x^2}-\frac {2 \left (-463+487 x^2\right )}{27 \left (3+2 x^2+x^4\right )}\right ) \, dx \\ & = -\frac {4}{45 x^5}+\frac {13}{81 x^3}-\frac {13}{27 x}+\frac {25 x \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}-\frac {1}{648} \int \frac {-463+487 x^2}{3+2 x^2+x^4} \, dx \\ & = -\frac {4}{45 x^5}+\frac {13}{81 x^3}-\frac {13}{27 x}+\frac {25 x \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}-\frac {\int \frac {-463 \sqrt {2 \left (-1+\sqrt {3}\right )}-\left (-463-487 \sqrt {3}\right ) x}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{1296 \sqrt {6 \left (-1+\sqrt {3}\right )}}-\frac {\int \frac {-463 \sqrt {2 \left (-1+\sqrt {3}\right )}+\left (-463-487 \sqrt {3}\right ) x}{\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{1296 \sqrt {6 \left (-1+\sqrt {3}\right )}} \\ & = -\frac {4}{45 x^5}+\frac {13}{81 x^3}-\frac {13}{27 x}+\frac {25 x \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}-\frac {\left (1461-463 \sqrt {3}\right ) \int \frac {1}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{7776}+\frac {\left (-1461+463 \sqrt {3}\right ) \int \frac {1}{\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{7776}-\frac {\sqrt {\frac {1}{6} \left (1139381+688419 \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}{2592}+\frac {\sqrt {\frac {1}{6} \left (1139381+688419 \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}{2592} \\ & = -\frac {4}{45 x^5}+\frac {13}{81 x^3}-\frac {13}{27 x}+\frac {25 x \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}-\frac {\sqrt {\frac {1}{6} \left (1139381+688419 \sqrt {3}\right )} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )}{2592}+\frac {\sqrt {\frac {1}{6} \left (1139381+688419 \sqrt {3}\right )} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )}{2592}+\frac {\left (1461-463 \sqrt {3}\right ) \text {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 )}{3888}-\frac {\left (-1461+463 \sqrt {3}\right ) \text {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 )}{3888} \\ & = -\frac {4}{45 x^5}+\frac {13}{81 x^3}-\frac {13}{27 x}+\frac {25 x \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}+\frac {\sqrt {\frac {1}{6} \left (-1139381+688419 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )}{1296}-\frac {\sqrt {\frac {1}{6} \left (-1139381+688419 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )}{1296}-\frac {\sqrt {\frac {1}{6} \left (1139381+688419 \sqrt {3}\right )} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )}{2592}+\frac {\sqrt {\frac {1}{6} \left (1139381+688419 \sqrt {3}\right )} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )}{2592} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.21 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.57 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x^6 \left (3+2 x^2+x^4\right )^2} \, dx=\frac {-\frac {4 \left (864-984 x^2+3928 x^4+2475 x^6+2435 x^8\right )}{x^5 \left (3+2 x^2+x^4\right )}-\frac {10 i \left (-487 i+475 \sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )}{\sqrt {1-i \sqrt {2}}}+\frac {10 i \left (487 i+475 \sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt {1+i \sqrt {2}}}\right )}{\sqrt {1+i \sqrt {2}}}}{12960} \]

[In]

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

[Out]

((-4*(864 - 984*x^2 + 3928*x^4 + 2475*x^6 + 2435*x^8))/(x^5*(3 + 2*x^2 + x^4)) - ((10*I)*(-487*I + 475*Sqrt[2]
)*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] + ((10*I)*(487*I + 475*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[
2]]])/Sqrt[1 + I*Sqrt[2]])/12960

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.30

method	result	size
risch	\(\frac {-\frac {487}{648} x^{8}-\frac {55}{72} x^{6}-\frac {491}{405} x^{4}+\frac {41}{135} x^{2}-\frac {4}{15}}{x^{5} \left (x^{4}+2 x^{2}+3\right )}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{4}-2278762 \textit {\_Z}^{2}[phone]\right )}{\sum }\textit {\_R} \ln \left (-2886 \textit {\_R}^{3}[phone] \textit {\_R} [phone] x \right )\right )}{2592}\)	\(73\)
default	\(-\frac {4}{45 x^{5}}+\frac {13}{81 x^{3}}-\frac {13}{27 x}-\frac {\frac {175}{24} x^{3}-\frac {25}{24} x}{27 \left (x^{4}+2 x^{2}+3\right )}-\frac {\left (962 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+1425 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+\sqrt {3}-x \sqrt {-2+2 \sqrt {3}}\right )}{15552}-\frac {\left (-926 \sqrt {3}+\frac {\left (962 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+1425 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{3888 \sqrt {2+2 \sqrt {3}}}-\frac {\left (-962 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-1425 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+\sqrt {3}+x \sqrt {-2+2 \sqrt {3}}\right )}{15552}-\frac {\left (-926 \sqrt {3}-\frac {\left (-962 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-1425 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{3888 \sqrt {2+2 \sqrt {3}}}\)	\(293\)

[In]

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

[Out]

(-487/648*x^8-55/72*x^6-491/405*x^4+41/135*x^2-4/15)/x^5/(x^4+2*x^2+3)+1/2592*sum(_R*ln(-2886*_R^3+1211171969*
_R+171119622411*x),_R=RootOf(3*_Z^4-2278762*_Z^2+473920719561))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.03 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x^6 \left (3+2 x^2+x^4\right )^2} \, dx=-\frac {29220 \, x^{8} + 29700 \, x^{6} + 47136 \, x^{4} + 5 \, \sqrt {3} {\left (x^{9} + 2 \, x^{7} + 3 \, x^{5}\right )} \sqrt {248569 i \, \sqrt {2} + 1139381} \log \left (\sqrt {3} \sqrt {248569 i \, \sqrt {2} + 1139381} {\left (962 i \, \sqrt {2} - 463\right )} + 2065257 \, x\right ) - 5 \, \sqrt {3} {\left (x^{9} + 2 \, x^{7} + 3 \, x^{5}\right )} \sqrt {248569 i \, \sqrt {2} + 1139381} \log \left (\sqrt {3} \sqrt {248569 i \, \sqrt {2} + 1139381} {\left (-962 i \, \sqrt {2} + 463\right )} + 2065257 \, x\right ) - 5 \, \sqrt {3} {\left (x^{9} + 2 \, x^{7} + 3 \, x^{5}\right )} \sqrt {-248569 i \, \sqrt {2} + 1139381} \log \left (\sqrt {3} {\left (962 i \, \sqrt {2} + 463\right )} \sqrt {-248569 i \, \sqrt {2} + 1139381} + 2065257 \, x\right ) + 5 \, \sqrt {3} {\left (x^{9} + 2 \, x^{7} + 3 \, x^{5}\right )} \sqrt {-248569 i \, \sqrt {2} + 1139381} \log \left (\sqrt {3} {\left (-962 i \, \sqrt {2} - 463\right )} \sqrt {-248569 i \, \sqrt {2} + 1139381} + 2065257 \, x\right ) - 11808 \, x^{2} + 10368}{38880 \, {\left (x^{9} + 2 \, x^{7} + 3 \, x^{5}\right )}} \]

[In]

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

[Out]

-1/38880*(29220*x^8 + 29700*x^6 + 47136*x^4 + 5*sqrt(3)*(x^9 + 2*x^7 + 3*x^5)*sqrt(248569*I*sqrt(2) + 1139381)
*log(sqrt(3)*sqrt(248569*I*sqrt(2) + 1139381)*(962*I*sqrt(2) - 463) + 2065257*x) - 5*sqrt(3)*(x^9 + 2*x^7 + 3*
x^5)*sqrt(248569*I*sqrt(2) + 1139381)*log(sqrt(3)*sqrt(248569*I*sqrt(2) + 1139381)*(-962*I*sqrt(2) + 463) + 20
65257*x) - 5*sqrt(3)*(x^9 + 2*x^7 + 3*x^5)*sqrt(-248569*I*sqrt(2) + 1139381)*log(sqrt(3)*(962*I*sqrt(2) + 463)
*sqrt(-248569*I*sqrt(2) + 1139381) + 2065257*x) + 5*sqrt(3)*(x^9 + 2*x^7 + 3*x^5)*sqrt(-248569*I*sqrt(2) + 113
9381)*log(sqrt(3)*(-962*I*sqrt(2) - 463)*sqrt(-248569*I*sqrt(2) + 1139381) + 2065257*x) - 11808*x^2 + 10368)/(
x^9 + 2*x^7 + 3*x^5)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1202 vs. \(2 (199) = 398\).

Time = 0.77 (sec) , antiderivative size = 1202, normalized size of antiderivative = 4.91 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x^6 \left (3+2 x^2+x^4\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-sqrt(1139381/40310784 + 2833*sqrt(3)/165888)*log(x**2 + x*(-3848*sqrt(2)*sqrt(1139381 + 688419*sqrt(3))/24856
9 - 769085497*sqrt(6)*sqrt(1139381 + 688419*sqrt(3))/171119622411 + 1924*sqrt(3)*sqrt(1139381 + 688419*sqrt(3)
)*sqrt(784371528639*sqrt(3) + 1359975610922)/171119622411) - 8677510907569510603*sqrt(2)*sqrt(784371528639*sqr
t(3) + 1359975610922)/29281925174083213452921 - 21752950947364*sqrt(6)*sqrt(784371528639*sqrt(3) + 13599756109
22)/127605100269239577 + 20196165220927340076543947/29281925174083213452921 + 50945036826336313070*sqrt(3)/127
605100269239577) + sqrt(1139381/40310784 + 2833*sqrt(3)/165888)*log(x**2 + x*(-1924*sqrt(3)*sqrt(1139381 + 688
419*sqrt(3))*sqrt(784371528639*sqrt(3) + 1359975610922)/171119622411 + 769085497*sqrt(6)*sqrt(1139381 + 688419
*sqrt(3))/171119622411 + 3848*sqrt(2)*sqrt(1139381 + 688419*sqrt(3))/248569) - 8677510907569510603*sqrt(2)*sqr
t(784371528639*sqrt(3) + 1359975610922)/29281925174083213452921 - 21752950947364*sqrt(6)*sqrt(784371528639*sqr
t(3) + 1359975610922)/127605100269239577 + 20196165220927340076543947/29281925174083213452921 + 50945036826336
313070*sqrt(3)/127605100269239577) + 2*sqrt(-sqrt(2)*sqrt(784371528639*sqrt(3) + 1359975610922)/20155392 + 113
9381/40310784 + 2833*sqrt(3)/55296)*atan(342239244822*sqrt(3)*x/(-1924*sqrt(784371528639*sqrt(3) + 13599756109
22)*sqrt(-2*sqrt(2)*sqrt(784371528639*sqrt(3) + 1359975610922) + 1139381 + 2065257*sqrt(3)) + 115087447*sqrt(2
)*sqrt(-2*sqrt(2)*sqrt(784371528639*sqrt(3) + 1359975610922) + 1139381 + 2065257*sqrt(3))) + 2649036312*sqrt(6
)*sqrt(1139381 + 688419*sqrt(3))/(-1924*sqrt(784371528639*sqrt(3) + 1359975610922)*sqrt(-2*sqrt(2)*sqrt(784371
528639*sqrt(3) + 1359975610922) + 1139381 + 2065257*sqrt(3)) + 115087447*sqrt(2)*sqrt(-2*sqrt(2)*sqrt(78437152
8639*sqrt(3) + 1359975610922) + 1139381 + 2065257*sqrt(3))) + 2307256491*sqrt(2)*sqrt(1139381 + 688419*sqrt(3)
)/(-1924*sqrt(784371528639*sqrt(3) + 1359975610922)*sqrt(-2*sqrt(2)*sqrt(784371528639*sqrt(3) + 1359975610922)
+ 1139381 + 2065257*sqrt(3)) + 115087447*sqrt(2)*sqrt(-2*sqrt(2)*sqrt(784371528639*sqrt(3) + 1359975610922) +
1139381 + 2065257*sqrt(3))) - 5772*sqrt(1139381 + 688419*sqrt(3))*sqrt(784371528639*sqrt(3) + 1359975610922)/
(-1924*sqrt(784371528639*sqrt(3) + 1359975610922)*sqrt(-2*sqrt(2)*sqrt(784371528639*sqrt(3) + 1359975610922) +
1139381 + 2065257*sqrt(3)) + 115087447*sqrt(2)*sqrt(-2*sqrt(2)*sqrt(784371528639*sqrt(3) + 1359975610922) + 1
139381 + 2065257*sqrt(3)))) + 2*sqrt(-sqrt(2)*sqrt(784371528639*sqrt(3) + 1359975610922)/20155392 + 1139381/40
310784 + 2833*sqrt(3)/55296)*atan(342239244822*sqrt(3)*x/(-1924*sqrt(784371528639*sqrt(3) + 1359975610922)*sqr
t(-2*sqrt(2)*sqrt(784371528639*sqrt(3) + 1359975610922) + 1139381 + 2065257*sqrt(3)) + 115087447*sqrt(2)*sqrt(
-2*sqrt(2)*sqrt(784371528639*sqrt(3) + 1359975610922) + 1139381 + 2065257*sqrt(3))) + 5772*sqrt(1139381 + 6884
19*sqrt(3))*sqrt(784371528639*sqrt(3) + 1359975610922)/(-1924*sqrt(784371528639*sqrt(3) + 1359975610922)*sqrt(
-2*sqrt(2)*sqrt(784371528639*sqrt(3) + 1359975610922) + 1139381 + 2065257*sqrt(3)) + 115087447*sqrt(2)*sqrt(-2
*sqrt(2)*sqrt(784371528639*sqrt(3) + 1359975610922) + 1139381 + 2065257*sqrt(3))) - 2307256491*sqrt(2)*sqrt(11
39381 + 688419*sqrt(3))/(-1924*sqrt(784371528639*sqrt(3) + 1359975610922)*sqrt(-2*sqrt(2)*sqrt(784371528639*sq
rt(3) + 1359975610922) + 1139381 + 2065257*sqrt(3)) + 115087447*sqrt(2)*sqrt(-2*sqrt(2)*sqrt(784371528639*sqrt
(3) + 1359975610922) + 1139381 + 2065257*sqrt(3))) - 2649036312*sqrt(6)*sqrt(1139381 + 688419*sqrt(3))/(-1924*
sqrt(784371528639*sqrt(3) + 1359975610922)*sqrt(-2*sqrt(2)*sqrt(784371528639*sqrt(3) + 1359975610922) + 113938
1 + 2065257*sqrt(3)) + 115087447*sqrt(2)*sqrt(-2*sqrt(2)*sqrt(784371528639*sqrt(3) + 1359975610922) + 1139381
+ 2065257*sqrt(3)))) + (-2435*x**8 - 2475*x**6 - 3928*x**4 + 984*x**2 - 864)/(3240*x**9 + 6480*x**7 + 9720*x**
5)

Maxima [F]

\[ \int \frac {4+x^2+3 x^4+5 x^6}{x^6 \left (3+2 x^2+x^4\right )^2} \, dx=\int { \frac {5 \, x^{6} + 3 \, x^{4} + x^{2} + 4}{{\left (x^{4} + 2 \, x^{2} + 3\right )}^{2} x^{6}} \,d x } \]

[In]

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

[Out]

-1/3240*(2435*x^8 + 2475*x^6 + 3928*x^4 - 984*x^2 + 864)/(x^9 + 2*x^7 + 3*x^5) - 1/648*integrate((487*x^2 - 46
3)/(x^4 + 2*x^2 + 3), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 584 vs. \(2 (172) = 344\).

Time = 0.63 (sec) , antiderivative size = 584, normalized size of antiderivative = 2.38 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x^6 \left (3+2 x^2+x^4\right )^2} \, dx=\frac {1}{1679616} \, \sqrt {2} {\left (487 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 8766 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 8766 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 487 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 16668 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} - 16668 \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}{1679616} \, \sqrt {2} {\left (487 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 8766 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 8766 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 487 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 16668 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} - 16668 \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}{3359232} \, \sqrt {2} {\left (8766 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 487 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 487 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 8766 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} + 16668 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} + 16668 \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}{3359232} \, \sqrt {2} {\left (8766 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 487 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 487 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 8766 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} + 16668 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} + 16668 \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 {25 \, {\left (7 \, x^{3} - x\right )}}{648 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac {195 \, x^{4} - 65 \, x^{2} + 36}{405 \, x^{5}} \]

[In]

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

[Out]

1/1679616*sqrt(2)*(487*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 8766*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqr
t(3) - 3) - 8766*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) + 487*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) + 16668*3^(
1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) - 16668*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)) + 1/1679616*sqrt(2)*(487*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2)
+ 8766*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 8766*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) +
487*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) + 16668*3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) - 16668*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)) + 1/3359232*sqrt
(2)*(8766*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 487*3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2) +
487*3^(3/4)*(6*sqrt(3) + 18)^(3/2) + 8766*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) + 16668*3^(1/4)*sqrt(2)*s
qrt(-6*sqrt(3) + 18) + 16668*3^(1/4)*sqrt(6*sqrt(3) + 18))*log(x^2 + 2*3^(1/4)*x*sqrt(-1/6*sqrt(3) + 1/2) + sq
rt(3)) - 1/3359232*sqrt(2)*(8766*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 487*3^(3/4)*sqrt(2)*(-6
*sqrt(3) + 18)^(3/2) + 487*3^(3/4)*(6*sqrt(3) + 18)^(3/2) + 8766*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) +
16668*3^(1/4)*sqrt(2)*sqrt(-6*sqrt(3) + 18) + 16668*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)) - 25/648*(7*x^3 - x)/(x^4 + 2*x^2 + 3) - 1/405*(195*x^4 - 65*x^2 + 36)/x^5

Mupad [B] (verification not implemented)

Time = 8.77 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.70 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x^6 \left (3+2 x^2+x^4\right )^2} \, dx=-\frac {\frac {487\,x^8}{648}+\frac {55\,x^6}{72}+\frac {491\,x^4}{405}-\frac {41\,x^2}{135}+\frac {4}{15}}{x^9+2\,x^7+3\,x^5}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {3418143-\sqrt {2}\,745707{}\mathrm {i}}\,248569{}\mathrm {i}}{306110016\,\left (\frac {119561689}{51018336}+\frac {\sqrt {2}\,115087447{}\mathrm {i}}{204073344}\right )}+\frac {248569\,\sqrt {2}\,x\,\sqrt {3418143-\sqrt {2}\,745707{}\mathrm {i}}}{612220032\,\left (\frac {119561689}{51018336}+\frac {\sqrt {2}\,115087447{}\mathrm {i}}{204073344}\right )}\right )\,\sqrt {3418143-\sqrt {2}\,745707{}\mathrm {i}}\,1{}\mathrm {i}}{3888}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {3418143+\sqrt {2}\,745707{}\mathrm {i}}\,248569{}\mathrm {i}}{306110016\,\left (-\frac {119561689}{51018336}+\frac {\sqrt {2}\,115087447{}\mathrm {i}}{204073344}\right )}-\frac {248569\,\sqrt {2}\,x\,\sqrt {3418143+\sqrt {2}\,745707{}\mathrm {i}}}{612220032\,\left (-\frac {119561689}{51018336}+\frac {\sqrt {2}\,115087447{}\mathrm {i}}{204073344}\right )}\right )\,\sqrt {3418143+\sqrt {2}\,745707{}\mathrm {i}}\,1{}\mathrm {i}}{3888} \]

[In]

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

[Out]

(atan((x*(2^(1/2)*745707i + 3418143)^(1/2)*248569i)/(306110016*((2^(1/2)*115087447i)/204073344 - 119561689/510
18336)) - (248569*2^(1/2)*x*(2^(1/2)*745707i + 3418143)^(1/2))/(612220032*((2^(1/2)*115087447i)/204073344 - 11
9561689/51018336)))*(2^(1/2)*745707i + 3418143)^(1/2)*1i)/3888 - (atan((x*(3418143 - 2^(1/2)*745707i)^(1/2)*24
8569i)/(306110016*((2^(1/2)*115087447i)/204073344 + 119561689/51018336)) + (248569*2^(1/2)*x*(3418143 - 2^(1/2
)*745707i)^(1/2))/(612220032*((2^(1/2)*115087447i)/204073344 + 119561689/51018336)))*(3418143 - 2^(1/2)*745707
i)^(1/2)*1i)/3888 - ((491*x^4)/405 - (41*x^2)/135 + (55*x^6)/72 + (487*x^8)/648 + 4/15)/(3*x^5 + 2*x^7 + x^9)