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

Optimal. Leaf size=229 \[ -\frac {1}{96} \sqrt {\frac {1}{6} \left (965+699 \sqrt {3}\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+\frac {1}{96} \sqrt {\frac {1}{6} \left (965+699 \sqrt {3}\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )-\frac {25 x \left (x^2+5\right )}{72 \left (x^4+2 x^2+3\right )}-\frac {4}{9 x}+\frac {1}{48} \sqrt {\frac {1}{6} \left (699 \sqrt {3}-965\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {1}{48} \sqrt {\frac {1}{6} \left (699 \sqrt {3}-965\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right ) \]

[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)

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Rubi [A]  time = 0.31, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 12, 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} \[ -\frac {25 x \left (x^2+5\right )}{72 \left (x^4+2 x^2+3\right )}-\frac {1}{96} \sqrt {\frac {1}{6} \left (965+699 \sqrt {3}\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+\frac {1}{96} \sqrt {\frac {1}{6} \left (965+699 \sqrt {3}\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )-\frac {4}{9 x}+\frac {1}{48} \sqrt {\frac {1}{6} \left (699 \sqrt {3}-965\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {1}{48} \sqrt {\frac {1}{6} \left (699 \sqrt {3}-965\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right ) \]

Antiderivative was successfully verified.

[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 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 )^2} \, dx &=-\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 )} \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 )+\frac {1}{24} \sqrt {\frac {1}{6} \left (566-133 \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 )\\ &=-\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*}

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Mathematica [C]  time = 0.18, size = 126, normalized size = 0.55 \[ -\frac {25 x \left (x^2+5\right )}{72 \left (x^4+2 x^2+3\right )}-\frac {4}{9 x}-\frac {\left (19 \sqrt {2}+26 i\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )}{48 \sqrt {2-2 i \sqrt {2}}}-\frac {\left (19 \sqrt {2}-26 i\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+i \sqrt {2}}}\right )}{48 \sqrt {2+2 i \sqrt {2}}} \]

Antiderivative was successfully verified.

[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]])

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fricas [B]  time = 0.83, size = 471, normalized size = 2.06 \[ -\frac {164790648 \, x^{4} - 2068 \cdot 1465803^{\frac {1}{4}} \sqrt {2} {\left (x^{5} + 2 \, x^{3} + 3 \, x\right )} \sqrt {-674535 \, \sqrt {3} + 1465803} \arctan \left (\frac {1}{547726639257666} \cdot 1465803^{\frac {3}{4}} \sqrt {120461} \sqrt {1084149 \, x^{2} + 1465803^{\frac {1}{4}} {\left (7 \, \sqrt {3} x + 57 \, x\right )} \sqrt {-674535 \, \sqrt {3} + 1465803} + 1084149 \, \sqrt {3}} {\left (19 \, \sqrt {3} \sqrt {2} + 7 \, \sqrt {2}\right )} \sqrt {-674535 \, \sqrt {3} + 1465803} - \frac {1}{1515640302} \cdot 1465803^{\frac {3}{4}} {\left (19 \, \sqrt {3} \sqrt {2} x + 7 \, \sqrt {2} x\right )} \sqrt {-674535 \, \sqrt {3} + 1465803} - \frac {1}{2} \, \sqrt {3} \sqrt {2} + \frac {1}{2} \, \sqrt {2}\right ) - 2068 \cdot 1465803^{\frac {1}{4}} \sqrt {2} {\left (x^{5} + 2 \, x^{3} + 3 \, x\right )} \sqrt {-674535 \, \sqrt {3} + 1465803} \arctan \left (\frac {1}{547726639257666} \cdot 1465803^{\frac {3}{4}} \sqrt {120461} \sqrt {1084149 \, x^{2} - 1465803^{\frac {1}{4}} {\left (7 \, \sqrt {3} x + 57 \, x\right )} \sqrt {-674535 \, \sqrt {3} + 1465803} + 1084149 \, \sqrt {3}} {\left (19 \, \sqrt {3} \sqrt {2} + 7 \, \sqrt {2}\right )} \sqrt {-674535 \, \sqrt {3} + 1465803} - \frac {1}{1515640302} \cdot 1465803^{\frac {3}{4}} {\left (19 \, \sqrt {3} \sqrt {2} x + 7 \, \sqrt {2} x\right )} \sqrt {-674535 \, \sqrt {3} + 1465803} + \frac {1}{2} \, \sqrt {3} \sqrt {2} - \frac {1}{2} \, \sqrt {2}\right ) - 1465803^{\frac {1}{4}} {\left (965 \, x^{5} + 1930 \, x^{3} + 699 \, \sqrt {3} {\left (x^{5} + 2 \, x^{3} + 3 \, x\right )} + 2895 \, x\right )} \sqrt {-674535 \, \sqrt {3} + 1465803} \log \left (1084149 \, x^{2} + 1465803^{\frac {1}{4}} {\left (7 \, \sqrt {3} x + 57 \, x\right )} \sqrt {-674535 \, \sqrt {3} + 1465803} + 1084149 \, \sqrt {3}\right ) + 1465803^{\frac {1}{4}} {\left (965 \, x^{5} + 1930 \, x^{3} + 699 \, \sqrt {3} {\left (x^{5} + 2 \, x^{3} + 3 \, x\right )} + 2895 \, x\right )} \sqrt {-674535 \, \sqrt {3} + 1465803} \log \left (1084149 \, x^{2} - 1465803^{\frac {1}{4}} {\left (7 \, \sqrt {3} x + 57 \, x\right )} \sqrt {-674535 \, \sqrt {3} + 1465803} + 1084149 \, \sqrt {3}\right ) + 546411096 \, x^{2} + 277542144}{208156608 \, {\left (x^{5} + 2 \, x^{3} + 3 \, 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)^2,x, algorithm="fricas")

[Out]

-1/208156608*(164790648*x^4 - 2068*1465803^(1/4)*sqrt(2)*(x^5 + 2*x^3 + 3*x)*sqrt(-674535*sqrt(3) + 1465803)*a
rctan(1/547726639257666*1465803^(3/4)*sqrt(120461)*sqrt(1084149*x^2 + 1465803^(1/4)*(7*sqrt(3)*x + 57*x)*sqrt(
-674535*sqrt(3) + 1465803) + 1084149*sqrt(3))*(19*sqrt(3)*sqrt(2) + 7*sqrt(2))*sqrt(-674535*sqrt(3) + 1465803)
- 1/1515640302*1465803^(3/4)*(19*sqrt(3)*sqrt(2)*x + 7*sqrt(2)*x)*sqrt(-674535*sqrt(3) + 1465803) - 1/2*sqrt(
3)*sqrt(2) + 1/2*sqrt(2)) - 2068*1465803^(1/4)*sqrt(2)*(x^5 + 2*x^3 + 3*x)*sqrt(-674535*sqrt(3) + 1465803)*arc
tan(1/547726639257666*1465803^(3/4)*sqrt(120461)*sqrt(1084149*x^2 - 1465803^(1/4)*(7*sqrt(3)*x + 57*x)*sqrt(-6
74535*sqrt(3) + 1465803) + 1084149*sqrt(3))*(19*sqrt(3)*sqrt(2) + 7*sqrt(2))*sqrt(-674535*sqrt(3) + 1465803) -
1/1515640302*1465803^(3/4)*(19*sqrt(3)*sqrt(2)*x + 7*sqrt(2)*x)*sqrt(-674535*sqrt(3) + 1465803) + 1/2*sqrt(3)
*sqrt(2) - 1/2*sqrt(2)) - 1465803^(1/4)*(965*x^5 + 1930*x^3 + 699*sqrt(3)*(x^5 + 2*x^3 + 3*x) + 2895*x)*sqrt(-
674535*sqrt(3) + 1465803)*log(1084149*x^2 + 1465803^(1/4)*(7*sqrt(3)*x + 57*x)*sqrt(-674535*sqrt(3) + 1465803)
+ 1084149*sqrt(3)) + 1465803^(1/4)*(965*x^5 + 1930*x^3 + 699*sqrt(3)*(x^5 + 2*x^3 + 3*x) + 2895*x)*sqrt(-6745
35*sqrt(3) + 1465803)*log(1084149*x^2 - 1465803^(1/4)*(7*sqrt(3)*x + 57*x)*sqrt(-674535*sqrt(3) + 1465803) + 1
084149*sqrt(3)) + 546411096*x^2 + 277542144)/(x^5 + 2*x^3 + 3*x)

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giac [B]  time = 1.94, size = 572, normalized size = 2.50 \[ \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 )}} \]

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)^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)

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maple [B]  time = 0.03, size = 414, normalized size = 1.81 \[ -\frac {\left (-2+2 \sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{9 \sqrt {2+2 \sqrt {3}}}-\frac {13 \left (-2+2 \sqrt {3}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{96 \sqrt {2+2 \sqrt {3}}}+\frac {7 \sqrt {3}\, \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{72 \sqrt {2+2 \sqrt {3}}}-\frac {\left (-2+2 \sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{9 \sqrt {2+2 \sqrt {3}}}-\frac {13 \left (-2+2 \sqrt {3}\right ) \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{96 \sqrt {2+2 \sqrt {3}}}+\frac {7 \sqrt {3}\, \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{72 \sqrt {2+2 \sqrt {3}}}-\frac {\sqrt {-2+2 \sqrt {3}}\, \sqrt {3}\, \ln \left (x^{2}-\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{18}-\frac {13 \sqrt {-2+2 \sqrt {3}}\, \ln \left (x^{2}-\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{192}+\frac {\sqrt {-2+2 \sqrt {3}}\, \sqrt {3}\, \ln \left (x^{2}+\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{18}+\frac {13 \sqrt {-2+2 \sqrt {3}}\, \ln \left (x^{2}+\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{192}-\frac {4}{9 x}-\frac {\frac {25}{8} x^{3}+\frac {125}{8} x}{9 \left (x^{4}+2 x^{2}+3\right )} \]

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)^2,x)

[Out]

-4/9/x-1/9*(25/8*x^3+125/8*x)/(x^4+2*x^2+3)-1/18*(-2+2*3^(1/2))^(1/2)*3^(1/2)*ln(x^2-(-2+2*3^(1/2))^(1/2)*x+3^
(1/2))-13/192*(-2+2*3^(1/2))^(1/2)*ln(x^2-(-2+2*3^(1/2))^(1/2)*x+3^(1/2))-1/9/(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))-13/96/(2+2*3^(1/2))^(1/2)*(-2+2*3^(1/2))*arc
tan((2*x-(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))+7/72/(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))+1/18*(-2+2*3^(1/2))^(1/2)*3^(1/2)*ln(x^2+(-2+2*3^(1/2))^(1/2)*x+3^(1/2))+13/192*
(-2+2*3^(1/2))^(1/2)*ln(x^2+(-2+2*3^(1/2))^(1/2)*x+3^(1/2))-1/9/(2+2*3^(1/2))^(1/2)*(-2+2*3^(1/2))*3^(1/2)*arc
tan((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))-13/96/(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))+7/72/(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 \[ -\frac {19 \, x^{4} + 63 \, x^{2} + 32}{24 \, {\left (x^{5} + 2 \, x^{3} + 3 \, x\right )}} - \frac {1}{24} \, \int \frac {19 \, x^{2} - 7}{x^{4} + 2 \, x^{2} + 3}\,{d x} \]

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)^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)

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mupad [B]  time = 0.14, size = 159, normalized size = 0.69 \[ -\frac {\frac {19\,x^4}{24}+\frac {21\,x^2}{8}+\frac {4}{3}}{x^5+2\,x^3+3\,x}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {2895-\sqrt {2}\,1551{}\mathrm {i}}\,517{}\mathrm {i}}{15552\,\left (\frac {517}{162}+\frac {\sqrt {2}\,3619{}\mathrm {i}}{10368}\right )}+\frac {517\,\sqrt {2}\,x\,\sqrt {2895-\sqrt {2}\,1551{}\mathrm {i}}}{31104\,\left (\frac {517}{162}+\frac {\sqrt {2}\,3619{}\mathrm {i}}{10368}\right )}\right )\,\sqrt {2895-\sqrt {2}\,1551{}\mathrm {i}}\,1{}\mathrm {i}}{144}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {2895+\sqrt {2}\,1551{}\mathrm {i}}\,517{}\mathrm {i}}{15552\,\left (-\frac {517}{162}+\frac {\sqrt {2}\,3619{}\mathrm {i}}{10368}\right )}-\frac {517\,\sqrt {2}\,x\,\sqrt {2895+\sqrt {2}\,1551{}\mathrm {i}}}{31104\,\left (-\frac {517}{162}+\frac {\sqrt {2}\,3619{}\mathrm {i}}{10368}\right )}\right )\,\sqrt {2895+\sqrt {2}\,1551{}\mathrm {i}}\,1{}\mathrm {i}}{144} \]

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)^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)

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sympy [B]  time = 1.32, size = 1192, normalized size = 5.21 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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