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

Optimal. Leaf size=238 \[ -\frac {4}{27 x^3}+\frac {1}{864} \sqrt {\frac {1}{6} \left (56673 \sqrt {3}-6073\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )-\frac {1}{864} \sqrt {\frac {1}{6} \left (56673 \sqrt {3}-6073\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+\frac {25 x \left (5 x^2+7\right )}{216 \left (x^4+2 x^2+3\right )}+\frac {13}{27 x}-\frac {1}{432} \sqrt {\frac {1}{6} \left (6073+56673 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {1}{432} \sqrt {\frac {1}{6} \left (6073+56673 \sqrt {3}\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right ) \]

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Rubi [A]  time = 0.34, antiderivative size = 238, 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 (5 x^2+7\right )}{216 \left (x^4+2 x^2+3\right )}-\frac {4}{27 x^3}+\frac {1}{864} \sqrt {\frac {1}{6} \left (56673 \sqrt {3}-6073\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )-\frac {1}{864} \sqrt {\frac {1}{6} \left (56673 \sqrt {3}-6073\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+\frac {13}{27 x}-\frac {1}{432} \sqrt {\frac {1}{6} \left (6073+56673 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {1}{432} \sqrt {\frac {1}{6} \left (6073+56673 \sqrt {3}\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-4/(27*x^3) + 13/(27*x) + (25*x*(7 + 5*x^2))/(216*(3 + 2*x^2 + x^4)) - (Sqrt[(6073 + 56673*Sqrt[3])/6]*ArcTan[
(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/432 + (Sqrt[(6073 + 56673*Sqrt[3])/6]*ArcTan[(Sqrt[2*(-
1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/432 + (Sqrt[(-6073 + 56673*Sqrt[3])/6]*Log[Sqrt[3] - Sqrt[2*(-1 +
Sqrt[3])]*x + x^2])/864 - (Sqrt[(-6073 + 56673*Sqrt[3])/6]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/864

Rule 204

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

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1169

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rule 1664

Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x
)^m*Pq*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && PolyQ[Pq, x^2] && IGtQ[p, -2]

Rule 1669

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainde
r[x^m*Pq, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 2]}, S
imp[(x*(a + b*x^2 + c*x^4)^(p + 1)*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2))/(2*a*(p + 1)*(b^2 - 4*a*c)
), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[x^m*(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[(2*a*(p + 1)*(b^2
- 4*a*c)*PolynomialQuotient[x^m*Pq, a + b*x^2 + c*x^4, x])/x^m + (b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e)
/x^m + c*(4*p + 7)*(b*d - 2*a*e)*x^(2 - m), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && GtQ[Expon[
Pq, x^2], 1] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && ILtQ[m/2, 0]

Rubi steps

\begin {align*} \int \frac {4+x^2+3 x^4+5 x^6}{x^4 \left (3+2 x^2+x^4\right )^2} \, dx &=\frac {25 x \left (7+5 x^2\right )}{216 \left (3+2 x^2+x^4\right )}+\frac {1}{48} \int \frac {64-\frac {80 x^2}{3}+\frac {50 x^4}{9}+\frac {250 x^6}{9}}{x^4 \left (3+2 x^2+x^4\right )} \, dx\\ &=\frac {25 x \left (7+5 x^2\right )}{216 \left (3+2 x^2+x^4\right )}+\frac {1}{48} \int \left (\frac {64}{3 x^4}-\frac {208}{9 x^2}+\frac {2 \left (137+229 x^2\right )}{9 \left (3+2 x^2+x^4\right )}\right ) \, dx\\ &=-\frac {4}{27 x^3}+\frac {13}{27 x}+\frac {25 x \left (7+5 x^2\right )}{216 \left (3+2 x^2+x^4\right )}+\frac {1}{216} \int \frac {137+229 x^2}{3+2 x^2+x^4} \, dx\\ &=-\frac {4}{27 x^3}+\frac {13}{27 x}+\frac {25 x \left (7+5 x^2\right )}{216 \left (3+2 x^2+x^4\right )}+\frac {\int \frac {137 \sqrt {2 \left (-1+\sqrt {3}\right )}-\left (137-229 \sqrt {3}\right ) x}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{432 \sqrt {6 \left (-1+\sqrt {3}\right )}}+\frac {\int \frac {137 \sqrt {2 \left (-1+\sqrt {3}\right )}+\left (137-229 \sqrt {3}\right ) x}{\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{432 \sqrt {6 \left (-1+\sqrt {3}\right )}}\\ &=-\frac {4}{27 x^3}+\frac {13}{27 x}+\frac {25 x \left (7+5 x^2\right )}{216 \left (3+2 x^2+x^4\right )}+\frac {1}{432} \sqrt {\frac {1}{6} \left (88046+31373 \sqrt {3}\right )} \int \frac {1}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx+\frac {1}{432} \sqrt {\frac {1}{6} \left (88046+31373 \sqrt {3}\right )} \int \frac {1}{\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx+\frac {1}{864} \sqrt {\frac {1}{6} \left (-6073+56673 \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}{864} \sqrt {\frac {1}{6} \left (-6073+56673 \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}{27 x^3}+\frac {13}{27 x}+\frac {25 x \left (7+5 x^2\right )}{216 \left (3+2 x^2+x^4\right )}+\frac {1}{864} \sqrt {\frac {1}{6} \left (-6073+56673 \sqrt {3}\right )} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )-\frac {1}{864} \sqrt {\frac {1}{6} \left (-6073+56673 \sqrt {3}\right )} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )-\frac {1}{216} \sqrt {\frac {1}{6} \left (88046+31373 \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}{216} \sqrt {\frac {1}{6} \left (88046+31373 \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}{27 x^3}+\frac {13}{27 x}+\frac {25 x \left (7+5 x^2\right )}{216 \left (3+2 x^2+x^4\right )}-\frac {1}{432} \sqrt {\frac {1}{6} \left (6073+56673 \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}{432} \sqrt {\frac {1}{6} \left (6073+56673 \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}{864} \sqrt {\frac {1}{6} \left (-6073+56673 \sqrt {3}\right )} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )-\frac {1}{864} \sqrt {\frac {1}{6} \left (-6073+56673 \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 = 0.29, size = 131, normalized size = 0.55 \begin {gather*} \frac {1}{864} \left (\frac {4 \left (229 x^6+351 x^4+248 x^2-96\right )}{x^3 \left (x^4+2 x^2+3\right )}+\frac {2 \left (229+46 i \sqrt {2}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )}{\sqrt {1-i \sqrt {2}}}+\frac {2 \left (229-46 i \sqrt {2}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+i \sqrt {2}}}\right )}{\sqrt {1+i \sqrt {2}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((4*(-96 + 248*x^2 + 351*x^4 + 229*x^6))/(x^3*(3 + 2*x^2 + x^4)) + (2*(229 + (46*I)*Sqrt[2])*ArcTan[x/Sqrt[1 -
I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] + (2*(229 - (46*I)*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/Sqrt[1 + I*Sqrt[2
]])/864

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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^4 \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^4*(3 + 2*x^2 + x^4)^2),x]

[Out]

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

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fricas [B]  time = 1.36, size = 528, normalized size = 2.22 \begin {gather*} \frac {2397560030424 \, x^{6} + 3674862754056 \, x^{4} - 277108 \cdot 118956627^{\frac {1}{4}} \sqrt {6297} \sqrt {2} {\left (x^{7} + 2 \, x^{5} + 3 \, x^{3}\right )} \sqrt {6073 \, \sqrt {3} + 170019} \arctan \left (\frac {1}{295480530439458889122} \cdot 118956627^{\frac {3}{4}} \sqrt {81861} \sqrt {6297} \sqrt {3 \cdot 118956627^{\frac {1}{4}} \sqrt {6297} {\left (137 \, \sqrt {3} x - 687 \, x\right )} \sqrt {6073 \, \sqrt {3} + 170019} + 3926135421 \, x^{2} + 3926135421 \, \sqrt {3}} {\left (229 \, \sqrt {3} \sqrt {2} - 137 \, \sqrt {2}\right )} \sqrt {6073 \, \sqrt {3} + 170019} - \frac {1}{16481916497358} \cdot 118956627^{\frac {3}{4}} \sqrt {6297} {\left (229 \, \sqrt {3} \sqrt {2} x - 137 \, \sqrt {2} x\right )} \sqrt {6073 \, \sqrt {3} + 170019} + \frac {1}{2} \, \sqrt {3} \sqrt {2} - \frac {1}{2} \, \sqrt {2}\right ) - 277108 \cdot 118956627^{\frac {1}{4}} \sqrt {6297} \sqrt {2} {\left (x^{7} + 2 \, x^{5} + 3 \, x^{3}\right )} \sqrt {6073 \, \sqrt {3} + 170019} \arctan \left (\frac {1}{295480530439458889122} \cdot 118956627^{\frac {3}{4}} \sqrt {81861} \sqrt {6297} \sqrt {-3 \cdot 118956627^{\frac {1}{4}} \sqrt {6297} {\left (137 \, \sqrt {3} x - 687 \, x\right )} \sqrt {6073 \, \sqrt {3} + 170019} + 3926135421 \, x^{2} + 3926135421 \, \sqrt {3}} {\left (229 \, \sqrt {3} \sqrt {2} - 137 \, \sqrt {2}\right )} \sqrt {6073 \, \sqrt {3} + 170019} - \frac {1}{16481916497358} \cdot 118956627^{\frac {3}{4}} \sqrt {6297} {\left (229 \, \sqrt {3} \sqrt {2} x - 137 \, \sqrt {2} x\right )} \sqrt {6073 \, \sqrt {3} + 170019} - \frac {1}{2} \, \sqrt {3} \sqrt {2} + \frac {1}{2} \, \sqrt {2}\right ) - 118956627^{\frac {1}{4}} \sqrt {6297} {\left (6073 \, x^{7} + 12146 \, x^{5} + 18219 \, x^{3} - 56673 \, \sqrt {3} {\left (x^{7} + 2 \, x^{5} + 3 \, x^{3}\right )}\right )} \sqrt {6073 \, \sqrt {3} + 170019} \log \left (3 \cdot 118956627^{\frac {1}{4}} \sqrt {6297} {\left (137 \, \sqrt {3} x - 687 \, x\right )} \sqrt {6073 \, \sqrt {3} + 170019} + 3926135421 \, x^{2} + 3926135421 \, \sqrt {3}\right ) + 118956627^{\frac {1}{4}} \sqrt {6297} {\left (6073 \, x^{7} + 12146 \, x^{5} + 18219 \, x^{3} - 56673 \, \sqrt {3} {\left (x^{7} + 2 \, x^{5} + 3 \, x^{3}\right )}\right )} \sqrt {6073 \, \sqrt {3} + 170019} \log \left (-3 \cdot 118956627^{\frac {1}{4}} \sqrt {6297} {\left (137 \, \sqrt {3} x - 687 \, x\right )} \sqrt {6073 \, \sqrt {3} + 170019} + 3926135421 \, x^{2} + 3926135421 \, \sqrt {3}\right ) + 2596484225088 \, x^{2} - 1005090667776}{2261454002496 \, {\left (x^{7} + 2 \, x^{5} + 3 \, x^{3}\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^4/(x^4+2*x^2+3)^2,x, algorithm="fricas")

[Out]

1/2261454002496*(2397560030424*x^6 + 3674862754056*x^4 - 277108*118956627^(1/4)*sqrt(6297)*sqrt(2)*(x^7 + 2*x^
5 + 3*x^3)*sqrt(6073*sqrt(3) + 170019)*arctan(1/295480530439458889122*118956627^(3/4)*sqrt(81861)*sqrt(6297)*s
qrt(3*118956627^(1/4)*sqrt(6297)*(137*sqrt(3)*x - 687*x)*sqrt(6073*sqrt(3) + 170019) + 3926135421*x^2 + 392613
5421*sqrt(3))*(229*sqrt(3)*sqrt(2) - 137*sqrt(2))*sqrt(6073*sqrt(3) + 170019) - 1/16481916497358*118956627^(3/
4)*sqrt(6297)*(229*sqrt(3)*sqrt(2)*x - 137*sqrt(2)*x)*sqrt(6073*sqrt(3) + 170019) + 1/2*sqrt(3)*sqrt(2) - 1/2*
sqrt(2)) - 277108*118956627^(1/4)*sqrt(6297)*sqrt(2)*(x^7 + 2*x^5 + 3*x^3)*sqrt(6073*sqrt(3) + 170019)*arctan(
1/295480530439458889122*118956627^(3/4)*sqrt(81861)*sqrt(6297)*sqrt(-3*118956627^(1/4)*sqrt(6297)*(137*sqrt(3)
*x - 687*x)*sqrt(6073*sqrt(3) + 170019) + 3926135421*x^2 + 3926135421*sqrt(3))*(229*sqrt(3)*sqrt(2) - 137*sqrt
(2))*sqrt(6073*sqrt(3) + 170019) - 1/16481916497358*118956627^(3/4)*sqrt(6297)*(229*sqrt(3)*sqrt(2)*x - 137*sq
rt(2)*x)*sqrt(6073*sqrt(3) + 170019) - 1/2*sqrt(3)*sqrt(2) + 1/2*sqrt(2)) - 118956627^(1/4)*sqrt(6297)*(6073*x
^7 + 12146*x^5 + 18219*x^3 - 56673*sqrt(3)*(x^7 + 2*x^5 + 3*x^3))*sqrt(6073*sqrt(3) + 170019)*log(3*118956627^
(1/4)*sqrt(6297)*(137*sqrt(3)*x - 687*x)*sqrt(6073*sqrt(3) + 170019) + 3926135421*x^2 + 3926135421*sqrt(3)) +
118956627^(1/4)*sqrt(6297)*(6073*x^7 + 12146*x^5 + 18219*x^3 - 56673*sqrt(3)*(x^7 + 2*x^5 + 3*x^3))*sqrt(6073*
sqrt(3) + 170019)*log(-3*118956627^(1/4)*sqrt(6297)*(137*sqrt(3)*x - 687*x)*sqrt(6073*sqrt(3) + 170019) + 3926
135421*x^2 + 3926135421*sqrt(3)) + 2596484225088*x^2 - 1005090667776)/(x^7 + 2*x^5 + 3*x^3)

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giac [B]  time = 1.85, size = 579, normalized size = 2.43 \begin {gather*} -\frac {1}{559872} \, \sqrt {2} {\left (229 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 4122 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 4122 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 229 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 4932 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 4932 \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}{559872} \, \sqrt {2} {\left (229 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 4122 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 4122 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 229 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 4932 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 4932 \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}{1119744} \, \sqrt {2} {\left (4122 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 229 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 229 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 4122 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 4932 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 4932 \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}{1119744} \, \sqrt {2} {\left (4122 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 229 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 229 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 4122 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 4932 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 4932 \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 (5 \, x^{3} + 7 \, x\right )}}{216 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} + \frac {13 \, x^{2} - 4}{27 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/559872*sqrt(2)*(229*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 4122*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqr
t(3) - 3) - 4122*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) + 229*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) - 4932*3^(1
/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) + 4932*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)) - 1/559872*sqrt(2)*(229*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 4
122*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 4122*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) + 22
9*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) - 4932*3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) + 4932*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)) - 1/1119744*sqrt(2)*(
4122*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 229*3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2) + 229*3
^(3/4)*(6*sqrt(3) + 18)^(3/2) + 4122*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 4932*3^(1/4)*sqrt(2)*sqrt(-6
*sqrt(3) + 18) - 4932*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))
+ 1/1119744*sqrt(2)*(4122*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 229*3^(3/4)*sqrt(2)*(-6*sqrt(3
) + 18)^(3/2) + 229*3^(3/4)*(6*sqrt(3) + 18)^(3/2) + 4122*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 4932*3^
(1/4)*sqrt(2)*sqrt(-6*sqrt(3) + 18) - 4932*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)) + 25/216*(5*x^3 + 7*x)/(x^4 + 2*x^2 + 3) + 1/27*(13*x^2 - 4)/x^3

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maple [B]  time = 0.04, size = 419, normalized size = 1.76 \begin {gather*} \frac {275 \left (-2+2 \sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{2592 \sqrt {2+2 \sqrt {3}}}+\frac {23 \left (-2+2 \sqrt {3}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{432 \sqrt {2+2 \sqrt {3}}}+\frac {137 \sqrt {3}\, \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{648 \sqrt {2+2 \sqrt {3}}}+\frac {275 \left (-2+2 \sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{2592 \sqrt {2+2 \sqrt {3}}}+\frac {23 \left (-2+2 \sqrt {3}\right ) \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{432 \sqrt {2+2 \sqrt {3}}}+\frac {137 \sqrt {3}\, \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{648 \sqrt {2+2 \sqrt {3}}}+\frac {275 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}\, \ln \left (x^{2}-\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{5184}+\frac {23 \sqrt {-2+2 \sqrt {3}}\, \ln \left (x^{2}-\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{864}-\frac {275 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}\, \ln \left (x^{2}+\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{5184}-\frac {23 \sqrt {-2+2 \sqrt {3}}\, \ln \left (x^{2}+\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{864}+\frac {13}{27 x}-\frac {4}{27 x^{3}}+\frac {\frac {125}{8} x^{3}+\frac {175}{8} x}{27 x^{4}+54 x^{2}+81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/27/x^3+13/27/x+1/27*(125/8*x^3+175/8*x)/(x^4+2*x^2+3)+275/5184*(-2+2*3^(1/2))^(1/2)*3^(1/2)*ln(x^2-(-2+2*3^
(1/2))^(1/2)*x+3^(1/2))+23/864*(-2+2*3^(1/2))^(1/2)*ln(x^2-(-2+2*3^(1/2))^(1/2)*x+3^(1/2))+275/2592/(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))+23/432/(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))+137/648/(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))-275/5184*(-2+2*3^(1/2))^(1/2)*3^(1/2)*ln(x^2+(-2+2*3^(
1/2))^(1/2)*x+3^(1/2))-23/864*(-2+2*3^(1/2))^(1/2)*ln(x^2+(-2+2*3^(1/2))^(1/2)*x+3^(1/2))+275/2592/(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))+23/432/(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))+137/648/(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 {229 \, x^{6} + 351 \, x^{4} + 248 \, x^{2} - 96}{216 \, {\left (x^{7} + 2 \, x^{5} + 3 \, x^{3}\right )}} + \frac {1}{216} \, \int \frac {229 \, x^{2} + 137}{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^4/(x^4+2*x^2+3)^2,x, algorithm="maxima")

[Out]

1/216*(229*x^6 + 351*x^4 + 248*x^2 - 96)/(x^7 + 2*x^5 + 3*x^3) + 1/216*integrate((229*x^2 + 137)/(x^4 + 2*x^2
+ 3), x)

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mupad [B]  time = 0.14, size = 165, normalized size = 0.69 \begin {gather*} \frac {\frac {229\,x^6}{216}+\frac {13\,x^4}{8}+\frac {31\,x^2}{27}-\frac {4}{9}}{x^7+2\,x^5+3\,x^3}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-18219-\sqrt {2}\,207831{}\mathrm {i}}\,69277{}\mathrm {i}}{11337408\,\left (-\frac {19051175}{3779136}+\frac {\sqrt {2}\,9490949{}\mathrm {i}}{7558272}\right )}+\frac {69277\,\sqrt {2}\,x\,\sqrt {-18219-\sqrt {2}\,207831{}\mathrm {i}}}{22674816\,\left (-\frac {19051175}{3779136}+\frac {\sqrt {2}\,9490949{}\mathrm {i}}{7558272}\right )}\right )\,\sqrt {-18219-\sqrt {2}\,207831{}\mathrm {i}}\,1{}\mathrm {i}}{1296}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-18219+\sqrt {2}\,207831{}\mathrm {i}}\,69277{}\mathrm {i}}{11337408\,\left (\frac {19051175}{3779136}+\frac {\sqrt {2}\,9490949{}\mathrm {i}}{7558272}\right )}-\frac {69277\,\sqrt {2}\,x\,\sqrt {-18219+\sqrt {2}\,207831{}\mathrm {i}}}{22674816\,\left (\frac {19051175}{3779136}+\frac {\sqrt {2}\,9490949{}\mathrm {i}}{7558272}\right )}\right )\,\sqrt {-18219+\sqrt {2}\,207831{}\mathrm {i}}\,1{}\mathrm {i}}{1296} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((31*x^2)/27 + (13*x^4)/8 + (229*x^6)/216 - 4/9)/(3*x^3 + 2*x^5 + x^7) - (atan((x*(- 2^(1/2)*207831i - 18219)^
(1/2)*69277i)/(11337408*((2^(1/2)*9490949i)/7558272 - 19051175/3779136)) + (69277*2^(1/2)*x*(- 2^(1/2)*207831i
- 18219)^(1/2))/(22674816*((2^(1/2)*9490949i)/7558272 - 19051175/3779136)))*(- 2^(1/2)*207831i - 18219)^(1/2)
*1i)/1296 + (atan((x*(2^(1/2)*207831i - 18219)^(1/2)*69277i)/(11337408*((2^(1/2)*9490949i)/7558272 + 19051175/
3779136)) - (69277*2^(1/2)*x*(2^(1/2)*207831i - 18219)^(1/2))/(22674816*((2^(1/2)*9490949i)/7558272 + 19051175
/3779136)))*(2^(1/2)*207831i - 18219)^(1/2)*1i)/1296

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sympy [A]  time = 0.65, size = 60, normalized size = 0.25 \begin {gather*} \operatorname {RootSum} {\left (2293235712 t^{4} + 12437504 t^{2} + 4405801, \left (t \mapsto t \log {\left (\frac {19707494400 t^{3}}{145412423} + \frac {357152768 t}{145412423} + x \right )} \right )\right )} + \frac {229 x^{6} + 351 x^{4} + 248 x^{2} - 96}{216 x^{7} + 432 x^{5} + 648 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(2293235712*_t**4 + 12437504*_t**2 + 4405801, Lambda(_t, _t*log(19707494400*_t**3/145412423 + 357152768
*_t/145412423 + x))) + (229*x**6 + 351*x**4 + 248*x**2 - 96)/(216*x**7 + 432*x**5 + 648*x**3)

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