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

Optimal. Leaf size=245 \[ -\frac {4}{45 x^5}+\frac {13}{81 x^3}-\frac {\sqrt {\frac {1}{6} \left (1139381+688419 \sqrt {3}\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )}{2592}+\frac {\sqrt {\frac {1}{6} \left (1139381+688419 \sqrt {3}\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )}{2592}+\frac {25 x \left (1-7 x^2\right )}{648 \left (x^4+2 x^2+3\right )}-\frac {13}{27 x}+\frac {\sqrt {\frac {1}{6} \left (688419 \sqrt {3}-1139381\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )}{1296}-\frac {\sqrt {\frac {1}{6} \left (688419 \sqrt {3}-1139381\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )}{1296} \]

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Rubi [A]  time = 0.33, antiderivative size = 245, 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} \begin {gather*} \frac {25 x \left (1-7 x^2\right )}{648 \left (x^4+2 x^2+3\right )}+\frac {13}{81 x^3}-\frac {4}{45 x^5}-\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 {13}{27 x}+\frac {\sqrt {\frac {1}{6} \left (688419 \sqrt {3}-1139381\right )} \tan ^{-1}\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 )} \tan ^{-1}\left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )}{1296} \end {gather*}

Antiderivative was successfully verified.

[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 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^6 \left (3+2 x^2+x^4\right )^2} \, dx &=\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 ) \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 )}{3888}-\frac {\left (-1461+463 \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 )}{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*}

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Mathematica [C]  time = 0.29, size = 140, normalized size = 0.57 \begin {gather*} \frac {-\frac {4 \left (2435 x^8+2475 x^6+3928 x^4-984 x^2+864\right )}{x^5 \left (x^4+2 x^2+3\right )}-\frac {10 i \left (475 \sqrt {2}-487 i\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )}{\sqrt {1-i \sqrt {2}}}+\frac {10 i \left (475 \sqrt {2}+487 i\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+i \sqrt {2}}}\right )}{\sqrt {1+i \sqrt {2}}}}{12960} \end {gather*}

Antiderivative was successfully verified.

[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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4+x^2+3 x^4+5 x^6}{x^6 \left (3+2 x^2+x^4\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.44, size = 496, normalized size = 2.02 \begin {gather*} -\frac {1111136748188760 \, x^{8} + 1129389507912600 \, x^{6} + 1792421004881088 \, x^{4} - 4971380 \cdot 216699003^{\frac {1}{4}} \sqrt {2} {\left (x^{9} + 2 \, x^{7} + 3 \, x^{5}\right )} \sqrt {-784371528639 \, \sqrt {3} + 1421762158683} \arctan \left (\frac {1}{6144866223568721756453718} \, \sqrt {704195977} 216699003^{\frac {3}{4}} \sqrt {57039874137 \, x^{2} + 216699003^{\frac {1}{4}} {\left (463 \, \sqrt {3} x + 1461 \, x\right )} \sqrt {-784371528639 \, \sqrt {3} + 1421762158683} + 57039874137 \, \sqrt {3}} {\left (487 \, \sqrt {3} \sqrt {2} + 463 \, \sqrt {2}\right )} \sqrt {-784371528639 \, \sqrt {3} + 1421762158683} - \frac {1}{969563780580726} \cdot 216699003^{\frac {3}{4}} {\left (487 \, \sqrt {3} \sqrt {2} x + 463 \, \sqrt {2} x\right )} \sqrt {-784371528639 \, \sqrt {3} + 1421762158683} - \frac {1}{2} \, \sqrt {3} \sqrt {2} + \frac {1}{2} \, \sqrt {2}\right ) - 4971380 \cdot 216699003^{\frac {1}{4}} \sqrt {2} {\left (x^{9} + 2 \, x^{7} + 3 \, x^{5}\right )} \sqrt {-784371528639 \, \sqrt {3} + 1421762158683} \arctan \left (\frac {1}{6144866223568721756453718} \, \sqrt {704195977} 216699003^{\frac {3}{4}} \sqrt {57039874137 \, x^{2} - 216699003^{\frac {1}{4}} {\left (463 \, \sqrt {3} x + 1461 \, x\right )} \sqrt {-784371528639 \, \sqrt {3} + 1421762158683} + 57039874137 \, \sqrt {3}} {\left (487 \, \sqrt {3} \sqrt {2} + 463 \, \sqrt {2}\right )} \sqrt {-784371528639 \, \sqrt {3} + 1421762158683} - \frac {1}{969563780580726} \cdot 216699003^{\frac {3}{4}} {\left (487 \, \sqrt {3} \sqrt {2} x + 463 \, \sqrt {2} x\right )} \sqrt {-784371528639 \, \sqrt {3} + 1421762158683} + \frac {1}{2} \, \sqrt {3} \sqrt {2} - \frac {1}{2} \, \sqrt {2}\right ) - 5 \cdot 216699003^{\frac {1}{4}} {\left (1139381 \, x^{9} + 2278762 \, x^{7} + 3418143 \, x^{5} + 688419 \, \sqrt {3} {\left (x^{9} + 2 \, x^{7} + 3 \, x^{5}\right )}\right )} \sqrt {-784371528639 \, \sqrt {3} + 1421762158683} \log \left (57039874137 \, x^{2} + 216699003^{\frac {1}{4}} {\left (463 \, \sqrt {3} x + 1461 \, x\right )} \sqrt {-784371528639 \, \sqrt {3} + 1421762158683} + 57039874137 \, \sqrt {3}\right ) + 5 \cdot 216699003^{\frac {1}{4}} {\left (1139381 \, x^{9} + 2278762 \, x^{7} + 3418143 \, x^{5} + 688419 \, \sqrt {3} {\left (x^{9} + 2 \, x^{7} + 3 \, x^{5}\right )}\right )} \sqrt {-784371528639 \, \sqrt {3} + 1421762158683} \log \left (57039874137 \, x^{2} - 216699003^{\frac {1}{4}} {\left (463 \, \sqrt {3} x + 1461 \, x\right )} \sqrt {-784371528639 \, \sqrt {3} + 1421762158683} + 57039874137 \, \sqrt {3}\right ) - 449017889206464 \, x^{2} + 394259610034944}{1478473537631040 \, {\left (x^{9} + 2 \, x^{7} + 3 \, x^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1478473537631040*(1111136748188760*x^8 + 1129389507912600*x^6 + 1792421004881088*x^4 - 4971380*216699003^(1
/4)*sqrt(2)*(x^9 + 2*x^7 + 3*x^5)*sqrt(-784371528639*sqrt(3) + 1421762158683)*arctan(1/61448662235687217564537
18*sqrt(704195977)*216699003^(3/4)*sqrt(57039874137*x^2 + 216699003^(1/4)*(463*sqrt(3)*x + 1461*x)*sqrt(-78437
1528639*sqrt(3) + 1421762158683) + 57039874137*sqrt(3))*(487*sqrt(3)*sqrt(2) + 463*sqrt(2))*sqrt(-784371528639
*sqrt(3) + 1421762158683) - 1/969563780580726*216699003^(3/4)*(487*sqrt(3)*sqrt(2)*x + 463*sqrt(2)*x)*sqrt(-78
4371528639*sqrt(3) + 1421762158683) - 1/2*sqrt(3)*sqrt(2) + 1/2*sqrt(2)) - 4971380*216699003^(1/4)*sqrt(2)*(x^
9 + 2*x^7 + 3*x^5)*sqrt(-784371528639*sqrt(3) + 1421762158683)*arctan(1/6144866223568721756453718*sqrt(7041959
77)*216699003^(3/4)*sqrt(57039874137*x^2 - 216699003^(1/4)*(463*sqrt(3)*x + 1461*x)*sqrt(-784371528639*sqrt(3)
+ 1421762158683) + 57039874137*sqrt(3))*(487*sqrt(3)*sqrt(2) + 463*sqrt(2))*sqrt(-784371528639*sqrt(3) + 1421
762158683) - 1/969563780580726*216699003^(3/4)*(487*sqrt(3)*sqrt(2)*x + 463*sqrt(2)*x)*sqrt(-784371528639*sqrt
(3) + 1421762158683) + 1/2*sqrt(3)*sqrt(2) - 1/2*sqrt(2)) - 5*216699003^(1/4)*(1139381*x^9 + 2278762*x^7 + 341
8143*x^5 + 688419*sqrt(3)*(x^9 + 2*x^7 + 3*x^5))*sqrt(-784371528639*sqrt(3) + 1421762158683)*log(57039874137*x
^2 + 216699003^(1/4)*(463*sqrt(3)*x + 1461*x)*sqrt(-784371528639*sqrt(3) + 1421762158683) + 57039874137*sqrt(3
)) + 5*216699003^(1/4)*(1139381*x^9 + 2278762*x^7 + 3418143*x^5 + 688419*sqrt(3)*(x^9 + 2*x^7 + 3*x^5))*sqrt(-
784371528639*sqrt(3) + 1421762158683)*log(57039874137*x^2 - 216699003^(1/4)*(463*sqrt(3)*x + 1461*x)*sqrt(-784
371528639*sqrt(3) + 1421762158683) + 57039874137*sqrt(3)) - 449017889206464*x^2 + 394259610034944)/(x^9 + 2*x^
7 + 3*x^5)

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giac [B]  time = 1.78, size = 584, normalized size = 2.38 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [B]  time = 0.03, size = 424, normalized size = 1.73 \begin {gather*} -\frac {481 \left (-2+2 \sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{3888 \sqrt {2+2 \sqrt {3}}}-\frac {475 \left (-2+2 \sqrt {3}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{2592 \sqrt {2+2 \sqrt {3}}}+\frac {463 \sqrt {3}\, \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{1944 \sqrt {2+2 \sqrt {3}}}-\frac {481 \left (-2+2 \sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{3888 \sqrt {2+2 \sqrt {3}}}-\frac {475 \left (-2+2 \sqrt {3}\right ) \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{2592 \sqrt {2+2 \sqrt {3}}}+\frac {463 \sqrt {3}\, \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{1944 \sqrt {2+2 \sqrt {3}}}-\frac {481 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}\, \ln \left (x^{2}-\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{7776}-\frac {475 \sqrt {-2+2 \sqrt {3}}\, \ln \left (x^{2}-\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{5184}+\frac {481 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}\, \ln \left (x^{2}+\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{7776}+\frac {475 \sqrt {-2+2 \sqrt {3}}\, \ln \left (x^{2}+\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{5184}-\frac {13}{27 x}+\frac {13}{81 x^{3}}-\frac {4}{45 x^{5}}-\frac {\frac {175}{24} x^{3}-\frac {25}{24} x}{27 \left (x^{4}+2 x^{2}+3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/45/x^5+13/81/x^3-13/27/x-1/27*(175/24*x^3-25/24*x)/(x^4+2*x^2+3)-481/7776*(-2+2*3^(1/2))^(1/2)*3^(1/2)*ln(x
^2-(-2+2*3^(1/2))^(1/2)*x+3^(1/2))-475/5184*(-2+2*3^(1/2))^(1/2)*ln(x^2-(-2+2*3^(1/2))^(1/2)*x+3^(1/2))-481/38
88/(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))-475/2592/
(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))+463/1944/(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))+481/7776*(-2+2*3^(1/2))^(1/2)*3^(1/2)*
ln(x^2+(-2+2*3^(1/2))^(1/2)*x+3^(1/2))+475/5184*(-2+2*3^(1/2))^(1/2)*ln(x^2+(-2+2*3^(1/2))^(1/2)*x+3^(1/2))-48
1/3888/(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))-475/2
592/(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))+463/1944/(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 \begin {gather*} -\frac {2435 \, x^{8} + 2475 \, x^{6} + 3928 \, x^{4} - 984 \, x^{2} + 864}{3240 \, {\left (x^{9} + 2 \, x^{7} + 3 \, x^{5}\right )}} - \frac {1}{648} \, \int \frac {487 \, x^{2} - 463}{x^{4} + 2 \, x^{2} + 3}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.14, size = 171, normalized size = 0.70 \begin {gather*} -\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [B]  time = 1.33, size = 1202, normalized size = 4.91

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**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)

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