\(\int \frac {4+x^2+3 x^4+5 x^6}{x^2 (3+2 x^2+x^4)^2} \, dx\) [114]

Optimal result

Integrand size = 31, antiderivative size = 229 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x^2 \left (3+2 x^2+x^4\right )^2} \, dx=-\frac {4}{9 x}-\frac {25 x \left (5+x^2\right )}{72 \left (3+2 x^2+x^4\right )}+\frac {1}{48} \sqrt {\frac {1}{6} \left (-965+699 \sqrt {3}\right )} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {1}{48} \sqrt {\frac {1}{6} \left (-965+699 \sqrt {3}\right )} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {1}{96} \sqrt {\frac {1}{6} \left (965+699 \sqrt {3}\right )} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {1}{96} \sqrt {\frac {1}{6} \left (965+699 \sqrt {3}\right )} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right ) \]

[Out]

-4/9/x-25/72*x*(x^2+5)/(x^4+2*x^2+3)+1/288*arctan((-2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(-5790+4194
*3^(1/2))^(1/2)-1/288*arctan((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(-5790+4194*3^(1/2))^(1/2)-1/576*
ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(5790+4194*3^(1/2))^(1/2)+1/576*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*
(5790+4194*3^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 229, 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} \[ \int \frac {4+x^2+3 x^4+5 x^6}{x^2 \left (3+2 x^2+x^4\right )^2} \, dx=\frac {1}{48} \sqrt {\frac {1}{6} \left (699 \sqrt {3}-965\right )} \arctan \left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {1}{48} \sqrt {\frac {1}{6} \left (699 \sqrt {3}-965\right )} \arctan \left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {1}{96} \sqrt {\frac {1}{6} \left (965+699 \sqrt {3}\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+\frac {1}{96} \sqrt {\frac {1}{6} \left (965+699 \sqrt {3}\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )-\frac {25 x \left (x^2+5\right )}{72 \left (x^4+2 x^2+3\right )}-\frac {4}{9 x} \]

[In]

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

[Out]

-4/(9*x) - (25*x*(5 + x^2))/(72*(3 + 2*x^2 + x^4)) + (Sqrt[(-965 + 699*Sqrt[3])/6]*ArcTan[(Sqrt[2*(-1 + Sqrt[3
])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/48 - (Sqrt[(-965 + 699*Sqrt[3])/6]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sq
rt[2*(1 + Sqrt[3])]])/48 - (Sqrt[(965 + 699*Sqrt[3])/6]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/96 + (S
qrt[(965 + 699*Sqrt[3])/6]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/96

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 (5+x^2\right )}{72 \left (3+2 x^2+x^4\right )}+\frac {1}{48} \int \frac {64+\frac {170 x^2}{3}-\frac {50 x^4}{3}}{x^2 \left (3+2 x^2+x^4\right )} \, dx \\ & = -\frac {25 x \left (5+x^2\right )}{72 \left (3+2 x^2+x^4\right )}+\frac {1}{48} \int \left (\frac {64}{3 x^2}-\frac {2 \left (-7+19 x^2\right )}{3+2 x^2+x^4}\right ) \, dx \\ & = -\frac {4}{9 x}-\frac {25 x \left (5+x^2\right )}{72 \left (3+2 x^2+x^4\right )}-\frac {1}{24} \int \frac {-7+19 x^2}{3+2 x^2+x^4} \, dx \\ & = -\frac {4}{9 x}-\frac {25 x \left (5+x^2\right )}{72 \left (3+2 x^2+x^4\right )}-\frac {\int \frac {-7 \sqrt {2 \left (-1+\sqrt {3}\right )}-\left (-7-19 \sqrt {3}\right ) x}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{48 \sqrt {6 \left (-1+\sqrt {3}\right )}}-\frac {\int \frac {-7 \sqrt {2 \left (-1+\sqrt {3}\right )}+\left (-7-19 \sqrt {3}\right ) x}{\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{48 \sqrt {6 \left (-1+\sqrt {3}\right )}} \\ & = -\frac {4}{9 x}-\frac {25 x \left (5+x^2\right )}{72 \left (3+2 x^2+x^4\right )}-\frac {1}{48} \sqrt {\frac {1}{6} \left (566-133 \sqrt {3}\right )} \int \frac {1}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx-\frac {1}{48} \sqrt {\frac {1}{6} \left (566-133 \sqrt {3}\right )} \int \frac {1}{\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx-\frac {1}{96} \sqrt {\frac {1}{6} \left (965+699 \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+\frac {1}{96} \sqrt {\frac {1}{6} \left (965+699 \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 \\ & = -\frac {4}{9 x}-\frac {25 x \left (5+x^2\right )}{72 \left (3+2 x^2+x^4\right )}-\frac {1}{96} \sqrt {\frac {1}{6} \left (965+699 \sqrt {3}\right )} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {1}{96} \sqrt {\frac {1}{6} \left (965+699 \sqrt {3}\right )} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {1}{24} \sqrt {\frac {1}{6} \left (566-133 \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 )+\frac {1}{24} \sqrt {\frac {1}{6} \left (566-133 \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 ) \\ & = -\frac {4}{9 x}-\frac {25 x \left (5+x^2\right )}{72 \left (3+2 x^2+x^4\right )}+\frac {1}{48} \sqrt {\frac {1}{6} \left (-965+699 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {1}{48} \sqrt {\frac {1}{6} \left (-965+699 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {1}{96} \sqrt {\frac {1}{6} \left (965+699 \sqrt {3}\right )} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {1}{96} \sqrt {\frac {1}{6} \left (965+699 \sqrt {3}\right )} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right ) \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.55 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x^2 \left (3+2 x^2+x^4\right )^2} \, dx=-\frac {4}{9 x}-\frac {25 x \left (5+x^2\right )}{72 \left (3+2 x^2+x^4\right )}-\frac {\left (26 i+19 \sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )}{48 \sqrt {2-2 i \sqrt {2}}}-\frac {\left (-26 i+19 \sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt {1+i \sqrt {2}}}\right )}{48 \sqrt {2+2 i \sqrt {2}}} \]

[In]

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

[Out]

-4/(9*x) - (25*x*(5 + x^2))/(72*(3 + 2*x^2 + x^4)) - ((26*I + 19*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/(48*S
qrt[2 - (2*I)*Sqrt[2]]) - ((-26*I + 19*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/(48*Sqrt[2 + (2*I)*Sqrt[2]])

Maple [C] (verified)

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

Time = 0.10 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.28

method	result	size
risch	\(\frac {-\frac {19}{24} x^{4}-\frac {21}{8} x^{2}-\frac {4}{3}}{x \left (x^{4}+2 x^{2}+3\right )}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{4}-1930 \textit {\_Z}^{2}+488601\right )}{\sum }\textit {\_R} \ln \left (-96 \textit {\_R}^{3}+34499 \textit {\_R} +361383 x \right )\right )}{96}\)	\(63\)
default	\(-\frac {4}{9 x}-\frac {\frac {25}{8} x^{3}+\frac {125}{8} x}{9 \left (x^{4}+2 x^{2}+3\right )}-\frac {\left (32 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+39 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+\sqrt {3}-x \sqrt {-2+2 \sqrt {3}}\right )}{576}-\frac {\left (-14 \sqrt {3}+\frac {\left (32 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+39 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{144 \sqrt {2+2 \sqrt {3}}}-\frac {\left (-32 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-39 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+\sqrt {3}+x \sqrt {-2+2 \sqrt {3}}\right )}{576}-\frac {\left (-14 \sqrt {3}-\frac {\left (-32 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-39 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{144 \sqrt {2+2 \sqrt {3}}}\)	\(283\)

[In]

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

[Out]

(-19/24*x^4-21/8*x^2-4/3)/x/(x^4+2*x^2+3)+1/96*sum(_R*ln(-96*_R^3+34499*_R+361383*x),_R=RootOf(3*_Z^4-1930*_Z^
2+488601))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.00 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x^2 \left (3+2 x^2+x^4\right )^2} \, dx=-\frac {228 \, x^{4} + \sqrt {3} {\left (x^{5} + 2 \, x^{3} + 3 \, x\right )} \sqrt {517 i \, \sqrt {2} + 965} \log \left (\sqrt {3} \sqrt {517 i \, \sqrt {2} + 965} {\left (32 i \, \sqrt {2} - 7\right )} + 2097 \, x\right ) - \sqrt {3} {\left (x^{5} + 2 \, x^{3} + 3 \, x\right )} \sqrt {517 i \, \sqrt {2} + 965} \log \left (\sqrt {3} \sqrt {517 i \, \sqrt {2} + 965} {\left (-32 i \, \sqrt {2} + 7\right )} + 2097 \, x\right ) - \sqrt {3} {\left (x^{5} + 2 \, x^{3} + 3 \, x\right )} \sqrt {-517 i \, \sqrt {2} + 965} \log \left (\sqrt {3} {\left (32 i \, \sqrt {2} + 7\right )} \sqrt {-517 i \, \sqrt {2} + 965} + 2097 \, x\right ) + \sqrt {3} {\left (x^{5} + 2 \, x^{3} + 3 \, x\right )} \sqrt {-517 i \, \sqrt {2} + 965} \log \left (\sqrt {3} {\left (-32 i \, \sqrt {2} - 7\right )} \sqrt {-517 i \, \sqrt {2} + 965} + 2097 \, x\right ) + 756 \, x^{2} + 384}{288 \, {\left (x^{5} + 2 \, x^{3} + 3 \, x\right )}} \]

[In]

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

[Out]

-1/288*(228*x^4 + sqrt(3)*(x^5 + 2*x^3 + 3*x)*sqrt(517*I*sqrt(2) + 965)*log(sqrt(3)*sqrt(517*I*sqrt(2) + 965)*
(32*I*sqrt(2) - 7) + 2097*x) - sqrt(3)*(x^5 + 2*x^3 + 3*x)*sqrt(517*I*sqrt(2) + 965)*log(sqrt(3)*sqrt(517*I*sq
rt(2) + 965)*(-32*I*sqrt(2) + 7) + 2097*x) - sqrt(3)*(x^5 + 2*x^3 + 3*x)*sqrt(-517*I*sqrt(2) + 965)*log(sqrt(3
)*(32*I*sqrt(2) + 7)*sqrt(-517*I*sqrt(2) + 965) + 2097*x) + sqrt(3)*(x^5 + 2*x^3 + 3*x)*sqrt(-517*I*sqrt(2) +
965)*log(sqrt(3)*(-32*I*sqrt(2) - 7)*sqrt(-517*I*sqrt(2) + 965) + 2097*x) + 756*x^2 + 384)/(x^5 + 2*x^3 + 3*x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1192 vs. \(2 (184) = 368\).

Time = 0.76 (sec) , antiderivative size = 1192, normalized size of antiderivative = 5.21 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x^2 \left (3+2 x^2+x^4\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(-19*x**4 - 63*x**2 - 32)/(24*x**5 + 48*x**3 + 72*x) - sqrt(965/55296 + 233*sqrt(3)/18432)*log(x**2 + x*(-128*
sqrt(2)*sqrt(965 + 699*sqrt(3))/517 - 21793*sqrt(6)*sqrt(965 + 699*sqrt(3))/361383 + 64*sqrt(3)*sqrt(965 + 699
*sqrt(3))*sqrt(674535*sqrt(3) + 1198514)/361383) - 8882635459*sqrt(2)*sqrt(674535*sqrt(3) + 1198514)/130597672
689 - 20458048*sqrt(6)*sqrt(674535*sqrt(3) + 1198514)/560505033 + 18567565928783/130597672689 + 46950427730*sq
rt(3)/560505033) + sqrt(965/55296 + 233*sqrt(3)/18432)*log(x**2 + x*(-64*sqrt(3)*sqrt(965 + 699*sqrt(3))*sqrt(
674535*sqrt(3) + 1198514)/361383 + 21793*sqrt(6)*sqrt(965 + 699*sqrt(3))/361383 + 128*sqrt(2)*sqrt(965 + 699*s
qrt(3))/517) - 8882635459*sqrt(2)*sqrt(674535*sqrt(3) + 1198514)/130597672689 - 20458048*sqrt(6)*sqrt(674535*s
qrt(3) + 1198514)/560505033 + 18567565928783/130597672689 + 46950427730*sqrt(3)/560505033) + 2*sqrt(-sqrt(2)*s
qrt(674535*sqrt(3) + 1198514)/27648 + 965/55296 + 233*sqrt(3)/6144)*atan(722766*sqrt(3)*x/(-64*sqrt(674535*sqr
t(3) + 1198514)*sqrt(-2*sqrt(2)*sqrt(674535*sqrt(3) + 1198514) + 965 + 2097*sqrt(3)) + 3619*sqrt(2)*sqrt(-2*sq
rt(2)*sqrt(674535*sqrt(3) + 1198514) + 965 + 2097*sqrt(3))) + 89472*sqrt(6)*sqrt(965 + 699*sqrt(3))/(-64*sqrt(
674535*sqrt(3) + 1198514)*sqrt(-2*sqrt(2)*sqrt(674535*sqrt(3) + 1198514) + 965 + 2097*sqrt(3)) + 3619*sqrt(2)*
sqrt(-2*sqrt(2)*sqrt(674535*sqrt(3) + 1198514) + 965 + 2097*sqrt(3))) + 65379*sqrt(2)*sqrt(965 + 699*sqrt(3))/
(-64*sqrt(674535*sqrt(3) + 1198514)*sqrt(-2*sqrt(2)*sqrt(674535*sqrt(3) + 1198514) + 965 + 2097*sqrt(3)) + 361
9*sqrt(2)*sqrt(-2*sqrt(2)*sqrt(674535*sqrt(3) + 1198514) + 965 + 2097*sqrt(3))) - 192*sqrt(965 + 699*sqrt(3))*
sqrt(674535*sqrt(3) + 1198514)/(-64*sqrt(674535*sqrt(3) + 1198514)*sqrt(-2*sqrt(2)*sqrt(674535*sqrt(3) + 11985
14) + 965 + 2097*sqrt(3)) + 3619*sqrt(2)*sqrt(-2*sqrt(2)*sqrt(674535*sqrt(3) + 1198514) + 965 + 2097*sqrt(3)))
) + 2*sqrt(-sqrt(2)*sqrt(674535*sqrt(3) + 1198514)/27648 + 965/55296 + 233*sqrt(3)/6144)*atan(722766*sqrt(3)*x
/(-64*sqrt(674535*sqrt(3) + 1198514)*sqrt(-2*sqrt(2)*sqrt(674535*sqrt(3) + 1198514) + 965 + 2097*sqrt(3)) + 36
19*sqrt(2)*sqrt(-2*sqrt(2)*sqrt(674535*sqrt(3) + 1198514) + 965 + 2097*sqrt(3))) + 192*sqrt(965 + 699*sqrt(3))
*sqrt(674535*sqrt(3) + 1198514)/(-64*sqrt(674535*sqrt(3) + 1198514)*sqrt(-2*sqrt(2)*sqrt(674535*sqrt(3) + 1198
514) + 965 + 2097*sqrt(3)) + 3619*sqrt(2)*sqrt(-2*sqrt(2)*sqrt(674535*sqrt(3) + 1198514) + 965 + 2097*sqrt(3))
) - 65379*sqrt(2)*sqrt(965 + 699*sqrt(3))/(-64*sqrt(674535*sqrt(3) + 1198514)*sqrt(-2*sqrt(2)*sqrt(674535*sqrt
(3) + 1198514) + 965 + 2097*sqrt(3)) + 3619*sqrt(2)*sqrt(-2*sqrt(2)*sqrt(674535*sqrt(3) + 1198514) + 965 + 209
7*sqrt(3))) - 89472*sqrt(6)*sqrt(965 + 699*sqrt(3))/(-64*sqrt(674535*sqrt(3) + 1198514)*sqrt(-2*sqrt(2)*sqrt(6
74535*sqrt(3) + 1198514) + 965 + 2097*sqrt(3)) + 3619*sqrt(2)*sqrt(-2*sqrt(2)*sqrt(674535*sqrt(3) + 1198514) +
965 + 2097*sqrt(3))))

Maxima [F]

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

[In]

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

[Out]

-1/24*(19*x^4 + 63*x^2 + 32)/(x^5 + 2*x^3 + 3*x) - 1/24*integrate((19*x^2 - 7)/(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. 572 vs. \(2 (160) = 320\).

Time = 0.58 (sec) , antiderivative size = 572, normalized size of antiderivative = 2.50 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x^2 \left (3+2 x^2+x^4\right )^2} \, dx=\frac {1}{62208} \, \sqrt {2} {\left (19 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 342 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 342 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 19 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 252 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} - 252 \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}{62208} \, \sqrt {2} {\left (19 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 342 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 342 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 19 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 252 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} - 252 \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}{124416} \, \sqrt {2} {\left (342 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 19 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 19 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 342 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} + 252 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} + 252 \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}{124416} \, \sqrt {2} {\left (342 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 19 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 19 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 342 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} + 252 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} + 252 \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 {19 \, x^{4} + 63 \, x^{2} + 32}{24 \, {\left (x^{5} + 2 \, x^{3} + 3 \, x\right )}} \]

[In]

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

[Out]

1/62208*sqrt(2)*(19*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 342*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3)
- 3) - 342*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) + 19*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) + 252*3^(1/4)*sqr
t(2)*sqrt(6*sqrt(3) + 18) - 252*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)) + 1/62208*sqrt(2)*(19*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 342*3^(3/4)
*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 342*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) + 19*3^(3/4)*(-6
*sqrt(3) + 18)^(3/2) + 252*3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) - 252*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)) + 1/124416*sqrt(2)*(342*3^(3/4)*sqrt
(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 19*3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2) + 19*3^(3/4)*(6*sqrt(3) +
18)^(3/2) + 342*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) + 252*3^(1/4)*sqrt(2)*sqrt(-6*sqrt(3) + 18) + 252*
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)) - 1/124416*sqrt(2)*(34
2*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 19*3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2) + 19*3^(3/4
)*(6*sqrt(3) + 18)^(3/2) + 342*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) + 252*3^(1/4)*sqrt(2)*sqrt(-6*sqrt(3
) + 18) + 252*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)) - 1/24*(
19*x^4 + 63*x^2 + 32)/(x^5 + 2*x^3 + 3*x)

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.69 \[ \int \frac {4+x^2+3 x^4+5 x^6}{x^2 \left (3+2 x^2+x^4\right )^2} \, dx=-\frac {\frac {19\,x^4}{24}+\frac {21\,x^2}{8}+\frac {4}{3}}{x^5+2\,x^3+3\,x}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {2895-\sqrt {2}\,1551{}\mathrm {i}}\,517{}\mathrm {i}}{15552\,\left (\frac {517}{162}+\frac {\sqrt {2}\,3619{}\mathrm {i}}{10368}\right )}+\frac {517\,\sqrt {2}\,x\,\sqrt {2895-\sqrt {2}\,1551{}\mathrm {i}}}{31104\,\left (\frac {517}{162}+\frac {\sqrt {2}\,3619{}\mathrm {i}}{10368}\right )}\right )\,\sqrt {2895-\sqrt {2}\,1551{}\mathrm {i}}\,1{}\mathrm {i}}{144}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {2895+\sqrt {2}\,1551{}\mathrm {i}}\,517{}\mathrm {i}}{15552\,\left (-\frac {517}{162}+\frac {\sqrt {2}\,3619{}\mathrm {i}}{10368}\right )}-\frac {517\,\sqrt {2}\,x\,\sqrt {2895+\sqrt {2}\,1551{}\mathrm {i}}}{31104\,\left (-\frac {517}{162}+\frac {\sqrt {2}\,3619{}\mathrm {i}}{10368}\right )}\right )\,\sqrt {2895+\sqrt {2}\,1551{}\mathrm {i}}\,1{}\mathrm {i}}{144} \]

[In]

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

[Out]

(atan((x*(2^(1/2)*1551i + 2895)^(1/2)*517i)/(15552*((2^(1/2)*3619i)/10368 - 517/162)) - (517*2^(1/2)*x*(2^(1/2
)*1551i + 2895)^(1/2))/(31104*((2^(1/2)*3619i)/10368 - 517/162)))*(2^(1/2)*1551i + 2895)^(1/2)*1i)/144 - (atan
((x*(2895 - 2^(1/2)*1551i)^(1/2)*517i)/(15552*((2^(1/2)*3619i)/10368 + 517/162)) + (517*2^(1/2)*x*(2895 - 2^(1
/2)*1551i)^(1/2))/(31104*((2^(1/2)*3619i)/10368 + 517/162)))*(2895 - 2^(1/2)*1551i)^(1/2)*1i)/144 - ((21*x^2)/
8 + (19*x^4)/24 + 4/3)/(3*x + 2*x^3 + x^5)