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

Optimal. Leaf size=238 \[ \frac {1}{512} \sqrt {3 \left (48835+32827 \sqrt {3}\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )-\frac {1}{512} \sqrt {3 \left (48835+32827 \sqrt {3}\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+\frac {x \left (238-59 x^2\right )}{64 \left (x^4+2 x^2+3\right )}-\frac {25 x \left (x^2+3\right )}{16 \left (x^4+2 x^2+3\right )^2}-\frac {1}{256} \sqrt {3 \left (32827 \sqrt {3}-48835\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {1}{256} \sqrt {3 \left (32827 \sqrt {3}-48835\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right ) \]

[Out]

-25/16*x*(x^2+3)/(x^4+2*x^2+3)^2+1/64*x*(-59*x^2+238)/(x^4+2*x^2+3)-1/256*arctan((-2*x+(-2+2*3^(1/2))^(1/2))/(
2+2*3^(1/2))^(1/2))*(-146505+98481*3^(1/2))^(1/2)+1/256*arctan((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))
*(-146505+98481*3^(1/2))^(1/2)+1/512*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(146505+98481*3^(1/2))^(1/2)-1/512
*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(146505+98481*3^(1/2))^(1/2)

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Rubi [A]  time = 0.29, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {1668, 1678, 1169, 634, 618, 204, 628} \[ \frac {x \left (238-59 x^2\right )}{64 \left (x^4+2 x^2+3\right )}-\frac {25 x \left (x^2+3\right )}{16 \left (x^4+2 x^2+3\right )^2}+\frac {1}{512} \sqrt {3 \left (48835+32827 \sqrt {3}\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )-\frac {1}{512} \sqrt {3 \left (48835+32827 \sqrt {3}\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )-\frac {1}{256} \sqrt {3 \left (32827 \sqrt {3}-48835\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {1}{256} \sqrt {3 \left (32827 \sqrt {3}-48835\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-25*x*(3 + x^2))/(16*(3 + 2*x^2 + x^4)^2) + (x*(238 - 59*x^2))/(64*(3 + 2*x^2 + x^4)) - (Sqrt[3*(-48835 + 328
27*Sqrt[3])]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 + (Sqrt[3*(-48835 + 32827*Sqrt[
3])]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/256 + (Sqrt[3*(48835 + 32827*Sqrt[3])]*Log[
Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/512 - (Sqrt[3*(48835 + 32827*Sqrt[3])]*Log[Sqrt[3] + Sqrt[2*(-1 + S
qrt[3])]*x + x^2])/512

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 1668

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[(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] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4*p +
7)*(b*d - 2*a*e)*x^2, 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] && IGtQ[m/2, 0]

Rule 1678

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Pq, a +
b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[Pq, a + b*x^2 + c*x^4, x], x, 2]}, Simp[(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[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuot
ient[Pq, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4*p + 7)*(b*d - 2*a*e)*x^2,
x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rubi steps

\begin {align*} \int \frac {x^4 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^3} \, dx &=-\frac {25 x \left (3+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac {1}{96} \int \frac {450-750 x^2-672 x^4+480 x^6}{\left (3+2 x^2+x^4\right )^2} \, dx\\ &=-\frac {25 x \left (3+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac {x \left (238-59 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac {\int \frac {-9936+18792 x^2}{3+2 x^2+x^4} \, dx}{4608}\\ &=-\frac {25 x \left (3+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac {x \left (238-59 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac {\int \frac {-9936 \sqrt {2 \left (-1+\sqrt {3}\right )}-\left (-9936-18792 \sqrt {3}\right ) x}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{9216 \sqrt {6 \left (-1+\sqrt {3}\right )}}+\frac {\int \frac {-9936 \sqrt {2 \left (-1+\sqrt {3}\right )}+\left (-9936-18792 \sqrt {3}\right ) x}{\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{9216 \sqrt {6 \left (-1+\sqrt {3}\right )}}\\ &=-\frac {25 x \left (3+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac {x \left (238-59 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac {1}{256} \left (261-46 \sqrt {3}\right ) \int \frac {1}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx+\frac {1}{256} \left (261-46 \sqrt {3}\right ) \int \frac {1}{\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx+\frac {1}{256} \left (\sqrt {\frac {3}{2 \left (-1+\sqrt {3}\right )}} \left (46+87 \sqrt {3}\right )\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}{256} \left (\sqrt {\frac {3}{2 \left (-1+\sqrt {3}\right )}} \left (46+87 \sqrt {3}\right )\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 {25 x \left (3+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac {x \left (238-59 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac {1}{512} \sqrt {146505+98481 \sqrt {3}} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )-\frac {1}{512} \sqrt {146505+98481 \sqrt {3}} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {1}{128} \left (-261+46 \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}{128} \left (-261+46 \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 {25 x \left (3+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac {x \left (238-59 x^2\right )}{64 \left (3+2 x^2+x^4\right )}-\frac {1}{256} \sqrt {3 \left (-48835+32827 \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}{256} \sqrt {3 \left (-48835+32827 \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}{512} \sqrt {146505+98481 \sqrt {3}} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )-\frac {1}{512} \sqrt {146505+98481 \sqrt {3}} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )\\ \end {align*}

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Mathematica [C]  time = 0.30, size = 129, normalized size = 0.54 \[ \frac {1}{256} \left (\frac {4 x \left (-59 x^6+120 x^4+199 x^2+414\right )}{\left (x^4+2 x^2+3\right )^2}+\frac {3 \left (174+133 i \sqrt {2}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )}{\sqrt {1-i \sqrt {2}}}+\frac {3 \left (174-133 i \sqrt {2}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+i \sqrt {2}}}\right )}{\sqrt {1+i \sqrt {2}}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*x*(414 + 199*x^2 + 120*x^4 - 59*x^6))/(3 + 2*x^2 + x^4)^2 + (3*(174 + (133*I)*Sqrt[2])*ArcTan[x/Sqrt[1 - I
*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] + (3*(174 - (133*I)*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/Sqrt[1 + I*Sqrt[2]
])/256

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fricas [B]  time = 0.88, size = 546, normalized size = 2.29 \[ -\frac {1914264223824 \, x^{7} - 3893418760320 \, x^{5} + 164728 \cdot 29095522083^{\frac {1}{4}} \sqrt {3} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} \sqrt {-1603106545 \, \sqrt {3} + 3232835787} \arctan \left (\frac {1}{1214880276996365518761363} \cdot 29095522083^{\frac {3}{4}} \sqrt {2027822271} \sqrt {2027822271 \, x^{2} + 29095522083^{\frac {1}{4}} {\left (87 \, \sqrt {3} \sqrt {2} x + 46 \, \sqrt {2} x\right )} \sqrt {-1603106545 \, \sqrt {3} + 3232835787} + 2027822271 \, \sqrt {3}} {\left (46 \, \sqrt {3} + 261\right )} \sqrt {-1603106545 \, \sqrt {3} + 3232835787} - \frac {1}{599105895211053} \cdot 29095522083^{\frac {3}{4}} {\left (46 \, \sqrt {3} x + 261 \, x\right )} \sqrt {-1603106545 \, \sqrt {3} + 3232835787} - \frac {1}{2} \, \sqrt {3} \sqrt {2} + \frac {1}{2} \, \sqrt {2}\right ) + 164728 \cdot 29095522083^{\frac {1}{4}} \sqrt {3} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} \sqrt {-1603106545 \, \sqrt {3} + 3232835787} \arctan \left (\frac {1}{1214880276996365518761363} \cdot 29095522083^{\frac {3}{4}} \sqrt {2027822271} \sqrt {2027822271 \, x^{2} - 29095522083^{\frac {1}{4}} {\left (87 \, \sqrt {3} \sqrt {2} x + 46 \, \sqrt {2} x\right )} \sqrt {-1603106545 \, \sqrt {3} + 3232835787} + 2027822271 \, \sqrt {3}} {\left (46 \, \sqrt {3} + 261\right )} \sqrt {-1603106545 \, \sqrt {3} + 3232835787} - \frac {1}{599105895211053} \cdot 29095522083^{\frac {3}{4}} {\left (46 \, \sqrt {3} x + 261 \, x\right )} \sqrt {-1603106545 \, \sqrt {3} + 3232835787} + \frac {1}{2} \, \sqrt {3} \sqrt {2} - \frac {1}{2} \, \sqrt {2}\right ) - 6456586110864 \, x^{3} + 29095522083^{\frac {1}{4}} {\left (48835 \, \sqrt {3} \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} + 98481 \, \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}\right )} \sqrt {-1603106545 \, \sqrt {3} + 3232835787} \log \left (2027822271 \, x^{2} + 29095522083^{\frac {1}{4}} {\left (87 \, \sqrt {3} \sqrt {2} x + 46 \, \sqrt {2} x\right )} \sqrt {-1603106545 \, \sqrt {3} + 3232835787} + 2027822271 \, \sqrt {3}\right ) - 29095522083^{\frac {1}{4}} {\left (48835 \, \sqrt {3} \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} + 98481 \, \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}\right )} \sqrt {-1603106545 \, \sqrt {3} + 3232835787} \log \left (2027822271 \, x^{2} - 29095522083^{\frac {1}{4}} {\left (87 \, \sqrt {3} \sqrt {2} x + 46 \, \sqrt {2} x\right )} \sqrt {-1603106545 \, \sqrt {3} + 3232835787} + 2027822271 \, \sqrt {3}\right ) - 13432294723104 \, x}{2076490005504 \, {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2076490005504*(1914264223824*x^7 - 3893418760320*x^5 + 164728*29095522083^(1/4)*sqrt(3)*(x^8 + 4*x^6 + 10*x
^4 + 12*x^2 + 9)*sqrt(-1603106545*sqrt(3) + 3232835787)*arctan(1/1214880276996365518761363*29095522083^(3/4)*s
qrt(2027822271)*sqrt(2027822271*x^2 + 29095522083^(1/4)*(87*sqrt(3)*sqrt(2)*x + 46*sqrt(2)*x)*sqrt(-1603106545
*sqrt(3) + 3232835787) + 2027822271*sqrt(3))*(46*sqrt(3) + 261)*sqrt(-1603106545*sqrt(3) + 3232835787) - 1/599
105895211053*29095522083^(3/4)*(46*sqrt(3)*x + 261*x)*sqrt(-1603106545*sqrt(3) + 3232835787) - 1/2*sqrt(3)*sqr
t(2) + 1/2*sqrt(2)) + 164728*29095522083^(1/4)*sqrt(3)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)*sqrt(-1603106545*sq
rt(3) + 3232835787)*arctan(1/1214880276996365518761363*29095522083^(3/4)*sqrt(2027822271)*sqrt(2027822271*x^2
- 29095522083^(1/4)*(87*sqrt(3)*sqrt(2)*x + 46*sqrt(2)*x)*sqrt(-1603106545*sqrt(3) + 3232835787) + 2027822271*
sqrt(3))*(46*sqrt(3) + 261)*sqrt(-1603106545*sqrt(3) + 3232835787) - 1/599105895211053*29095522083^(3/4)*(46*s
qrt(3)*x + 261*x)*sqrt(-1603106545*sqrt(3) + 3232835787) + 1/2*sqrt(3)*sqrt(2) - 1/2*sqrt(2)) - 6456586110864*
x^3 + 29095522083^(1/4)*(48835*sqrt(3)*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9) + 98481*sqrt(2)*(x^8 + 4*x^
6 + 10*x^4 + 12*x^2 + 9))*sqrt(-1603106545*sqrt(3) + 3232835787)*log(2027822271*x^2 + 29095522083^(1/4)*(87*sq
rt(3)*sqrt(2)*x + 46*sqrt(2)*x)*sqrt(-1603106545*sqrt(3) + 3232835787) + 2027822271*sqrt(3)) - 29095522083^(1/
4)*(48835*sqrt(3)*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9) + 98481*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 +
9))*sqrt(-1603106545*sqrt(3) + 3232835787)*log(2027822271*x^2 - 29095522083^(1/4)*(87*sqrt(3)*sqrt(2)*x + 46*
sqrt(2)*x)*sqrt(-1603106545*sqrt(3) + 3232835787) + 2027822271*sqrt(3)) - 13432294723104*x)/(x^8 + 4*x^6 + 10*
x^4 + 12*x^2 + 9)

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giac [B]  time = 2.60, size = 577, normalized size = 2.42 \[ -\frac {1}{18432} \, \sqrt {2} {\left (29 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 522 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 522 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 29 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 552 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} - 552 \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}{18432} \, \sqrt {2} {\left (29 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 522 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 522 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 29 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 552 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} - 552 \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}{36864} \, \sqrt {2} {\left (522 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 29 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 29 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 522 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} + 552 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} + 552 \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}{36864} \, \sqrt {2} {\left (522 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 29 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 29 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 522 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} + 552 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} + 552 \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 {59 \, x^{7} - 120 \, x^{5} - 199 \, x^{3} - 414 \, x}{64 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18432*sqrt(2)*(29*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 522*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3
) - 3) - 522*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) + 29*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) + 552*3^(1/4)*sq
rt(2)*sqrt(6*sqrt(3) + 18) - 552*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)) - 1/18432*sqrt(2)*(29*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 522*3^(3/4
)*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 522*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) + 29*3^(3/4)*(-
6*sqrt(3) + 18)^(3/2) + 552*3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) - 552*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)) - 1/36864*sqrt(2)*(522*3^(3/4)*sqrt
(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 29*3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2) + 29*3^(3/4)*(6*sqrt(3) +
18)^(3/2) + 522*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) + 552*3^(1/4)*sqrt(2)*sqrt(-6*sqrt(3) + 18) + 552*
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)) + 1/36864*sqrt(2)*(522
*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 29*3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2) + 29*3^(3/4)
*(6*sqrt(3) + 18)^(3/2) + 522*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) + 552*3^(1/4)*sqrt(2)*sqrt(-6*sqrt(3)
+ 18) + 552*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)) - 1/64*(5
9*x^7 - 120*x^5 - 199*x^3 - 414*x)/(x^4 + 2*x^2 + 3)^2

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maple [B]  time = 0.03, size = 418, normalized size = 1.76 \[ \frac {307 \left (-2+2 \sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{512 \sqrt {2+2 \sqrt {3}}}+\frac {399 \left (-2+2 \sqrt {3}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{512 \sqrt {2+2 \sqrt {3}}}-\frac {23 \sqrt {3}\, \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{32 \sqrt {2+2 \sqrt {3}}}+\frac {307 \left (-2+2 \sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{512 \sqrt {2+2 \sqrt {3}}}+\frac {399 \left (-2+2 \sqrt {3}\right ) \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{512 \sqrt {2+2 \sqrt {3}}}-\frac {23 \sqrt {3}\, \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{32 \sqrt {2+2 \sqrt {3}}}+\frac {307 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}\, \ln \left (x^{2}-\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{1024}+\frac {399 \sqrt {-2+2 \sqrt {3}}\, \ln \left (x^{2}-\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{1024}-\frac {307 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}\, \ln \left (x^{2}+\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{1024}-\frac {399 \sqrt {-2+2 \sqrt {3}}\, \ln \left (x^{2}+\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{1024}+\frac {-\frac {59}{64} x^{7}+\frac {15}{8} x^{5}+\frac {199}{64} x^{3}+\frac {207}{32} x}{\left (x^{4}+2 x^{2}+3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-59/64*x^7+15/8*x^5+199/64*x^3+207/32*x)/(x^4+2*x^2+3)^2+307/1024*(-2+2*3^(1/2))^(1/2)*3^(1/2)*ln(x^2-(-2+2*3
^(1/2))^(1/2)*x+3^(1/2))+399/1024*(-2+2*3^(1/2))^(1/2)*ln(x^2-(-2+2*3^(1/2))^(1/2)*x+3^(1/2))+307/512/(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))+399/512/(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))-23/32/(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))-307/1024*(-2+2*3^(1/2))^(1/2)*3^(1/2)*ln(x^2+(-2+2*3^
(1/2))^(1/2)*x+3^(1/2))-399/1024*(-2+2*3^(1/2))^(1/2)*ln(x^2+(-2+2*3^(1/2))^(1/2)*x+3^(1/2))+307/512/(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))+399/512/(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))-23/32/(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 \[ -\frac {59 \, x^{7} - 120 \, x^{5} - 199 \, x^{3} - 414 \, x}{64 \, {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}} + \frac {3}{64} \, \int \frac {87 \, x^{2} - 46}{x^{4} + 2 \, x^{2} + 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64*(59*x^7 - 120*x^5 - 199*x^3 - 414*x)/(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9) + 3/64*integrate((87*x^2 - 46)/
(x^4 + 2*x^2 + 3), x)

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mupad [B]  time = 0.15, size = 173, normalized size = 0.73 \[ \frac {-\frac {59\,x^7}{64}+\frac {15\,x^5}{8}+\frac {199\,x^3}{64}+\frac {207\,x}{32}}{x^8+4\,x^6+10\,x^4+12\,x^2+9}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {293010-\sqrt {2}\,123546{}\mathrm {i}}\,61773{}\mathrm {i}}{131072\,\left (\frac {56892933}{131072}+\frac {\sqrt {2}\,4262337{}\mathrm {i}}{65536}\right )}+\frac {61773\,\sqrt {2}\,x\,\sqrt {293010-\sqrt {2}\,123546{}\mathrm {i}}}{262144\,\left (\frac {56892933}{131072}+\frac {\sqrt {2}\,4262337{}\mathrm {i}}{65536}\right )}\right )\,\sqrt {293010-\sqrt {2}\,123546{}\mathrm {i}}\,1{}\mathrm {i}}{256}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {293010+\sqrt {2}\,123546{}\mathrm {i}}\,61773{}\mathrm {i}}{131072\,\left (-\frac {56892933}{131072}+\frac {\sqrt {2}\,4262337{}\mathrm {i}}{65536}\right )}-\frac {61773\,\sqrt {2}\,x\,\sqrt {293010+\sqrt {2}\,123546{}\mathrm {i}}}{262144\,\left (-\frac {56892933}{131072}+\frac {\sqrt {2}\,4262337{}\mathrm {i}}{65536}\right )}\right )\,\sqrt {293010+\sqrt {2}\,123546{}\mathrm {i}}\,1{}\mathrm {i}}{256} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((207*x)/32 + (199*x^3)/64 + (15*x^5)/8 - (59*x^7)/64)/(12*x^2 + 10*x^4 + 4*x^6 + x^8 + 9) + (atan((x*(293010
- 2^(1/2)*123546i)^(1/2)*61773i)/(131072*((2^(1/2)*4262337i)/65536 + 56892933/131072)) + (61773*2^(1/2)*x*(293
010 - 2^(1/2)*123546i)^(1/2))/(262144*((2^(1/2)*4262337i)/65536 + 56892933/131072)))*(293010 - 2^(1/2)*123546i
)^(1/2)*1i)/256 - (atan((x*(2^(1/2)*123546i + 293010)^(1/2)*61773i)/(131072*((2^(1/2)*4262337i)/65536 - 568929
33/131072)) - (61773*2^(1/2)*x*(2^(1/2)*123546i + 293010)^(1/2))/(262144*((2^(1/2)*4262337i)/65536 - 56892933/
131072)))*(2^(1/2)*123546i + 293010)^(1/2)*1i)/256

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sympy [B]  time = 1.31, size = 1198, normalized size = 5.03 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-59*x**7 + 120*x**5 + 199*x**3 + 414*x)/(64*x**8 + 256*x**6 + 640*x**4 + 768*x**2 + 576) - sqrt(146505/262144
+ 98481*sqrt(3)/262144)*log(x**2 + x*(-307*sqrt(6)*sqrt(48835 + 32827*sqrt(3))*sqrt(1603106545*sqrt(3) + 2808
846506)/675940757 + 10626354*sqrt(3)*sqrt(48835 + 32827*sqrt(3))/675940757 + 1228*sqrt(48835 + 32827*sqrt(3))/
20591) - 941929306825573*sqrt(2)*sqrt(1603106545*sqrt(3) + 2808846506)/456895906973733049 - 47771215762*sqrt(6
)*sqrt(1603106545*sqrt(3) + 2808846506)/41754888382161 + 97477949666790882353/456895906973733049 + 52004501305
96150*sqrt(3)/41754888382161) + sqrt(146505/262144 + 98481*sqrt(3)/262144)*log(x**2 + x*(-1228*sqrt(48835 + 32
827*sqrt(3))/20591 - 10626354*sqrt(3)*sqrt(48835 + 32827*sqrt(3))/675940757 + 307*sqrt(6)*sqrt(48835 + 32827*s
qrt(3))*sqrt(1603106545*sqrt(3) + 2808846506)/675940757) - 941929306825573*sqrt(2)*sqrt(1603106545*sqrt(3) + 2
808846506)/456895906973733049 - 47771215762*sqrt(6)*sqrt(1603106545*sqrt(3) + 2808846506)/41754888382161 + 974
77949666790882353/456895906973733049 + 5200450130596150*sqrt(3)/41754888382161) + 2*sqrt(-3*sqrt(2)*sqrt(16031
06545*sqrt(3) + 2808846506)/131072 + 146505/262144 + 295443*sqrt(3)/262144)*atan(1351881514*sqrt(3)*x/(-189437
2*sqrt(-2*sqrt(2)*sqrt(1603106545*sqrt(3) + 2808846506) + 48835 + 98481*sqrt(3)) + 307*sqrt(2)*sqrt(1603106545
*sqrt(3) + 2808846506)*sqrt(-2*sqrt(2)*sqrt(1603106545*sqrt(3) + 2808846506) + 48835 + 98481*sqrt(3))) - 40311
556*sqrt(3)*sqrt(48835 + 32827*sqrt(3))/(-1894372*sqrt(-2*sqrt(2)*sqrt(1603106545*sqrt(3) + 2808846506) + 4883
5 + 98481*sqrt(3)) + 307*sqrt(2)*sqrt(1603106545*sqrt(3) + 2808846506)*sqrt(-2*sqrt(2)*sqrt(1603106545*sqrt(3)
+ 2808846506) + 48835 + 98481*sqrt(3))) - 31879062*sqrt(48835 + 32827*sqrt(3))/(-1894372*sqrt(-2*sqrt(2)*sqrt
(1603106545*sqrt(3) + 2808846506) + 48835 + 98481*sqrt(3)) + 307*sqrt(2)*sqrt(1603106545*sqrt(3) + 2808846506)
*sqrt(-2*sqrt(2)*sqrt(1603106545*sqrt(3) + 2808846506) + 48835 + 98481*sqrt(3))) + 921*sqrt(2)*sqrt(48835 + 32
827*sqrt(3))*sqrt(1603106545*sqrt(3) + 2808846506)/(-1894372*sqrt(-2*sqrt(2)*sqrt(1603106545*sqrt(3) + 2808846
506) + 48835 + 98481*sqrt(3)) + 307*sqrt(2)*sqrt(1603106545*sqrt(3) + 2808846506)*sqrt(-2*sqrt(2)*sqrt(1603106
545*sqrt(3) + 2808846506) + 48835 + 98481*sqrt(3)))) + 2*sqrt(-3*sqrt(2)*sqrt(1603106545*sqrt(3) + 2808846506)
/131072 + 146505/262144 + 295443*sqrt(3)/262144)*atan(1351881514*sqrt(3)*x/(-1894372*sqrt(-2*sqrt(2)*sqrt(1603
106545*sqrt(3) + 2808846506) + 48835 + 98481*sqrt(3)) + 307*sqrt(2)*sqrt(1603106545*sqrt(3) + 2808846506)*sqrt
(-2*sqrt(2)*sqrt(1603106545*sqrt(3) + 2808846506) + 48835 + 98481*sqrt(3))) - 921*sqrt(2)*sqrt(48835 + 32827*s
qrt(3))*sqrt(1603106545*sqrt(3) + 2808846506)/(-1894372*sqrt(-2*sqrt(2)*sqrt(1603106545*sqrt(3) + 2808846506)
+ 48835 + 98481*sqrt(3)) + 307*sqrt(2)*sqrt(1603106545*sqrt(3) + 2808846506)*sqrt(-2*sqrt(2)*sqrt(1603106545*s
qrt(3) + 2808846506) + 48835 + 98481*sqrt(3))) + 31879062*sqrt(48835 + 32827*sqrt(3))/(-1894372*sqrt(-2*sqrt(2
)*sqrt(1603106545*sqrt(3) + 2808846506) + 48835 + 98481*sqrt(3)) + 307*sqrt(2)*sqrt(1603106545*sqrt(3) + 28088
46506)*sqrt(-2*sqrt(2)*sqrt(1603106545*sqrt(3) + 2808846506) + 48835 + 98481*sqrt(3))) + 40311556*sqrt(3)*sqrt
(48835 + 32827*sqrt(3))/(-1894372*sqrt(-2*sqrt(2)*sqrt(1603106545*sqrt(3) + 2808846506) + 48835 + 98481*sqrt(3
)) + 307*sqrt(2)*sqrt(1603106545*sqrt(3) + 2808846506)*sqrt(-2*sqrt(2)*sqrt(1603106545*sqrt(3) + 2808846506) +
48835 + 98481*sqrt(3))))

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