3.2.21 \(\int \frac {x^2 (4+x^2+3 x^4+5 x^6)}{(3+2 x^2+x^4)^3} \, dx\) [121]

Optimal. Leaf size=246 \[ \frac {25 x \left (1+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac {x \left (353+88 x^2\right )}{192 \left (3+2 x^2+x^4\right )}-\frac {11}{768} \sqrt {\frac {1}{3} \left (-1825+1089 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {11}{768} \sqrt {\frac {1}{3} \left (-1825+1089 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {11 \sqrt {\frac {1}{3} \left (1825+1089 \sqrt {3}\right )} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )}{1536}+\frac {11 \sqrt {\frac {1}{3} \left (1825+1089 \sqrt {3}\right )} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )}{1536} \]

[Out]

25/16*x*(x^2+1)/(x^4+2*x^2+3)^2-1/192*x*(88*x^2+353)/(x^4+2*x^2+3)-11/2304*arctan((-2*x+(-2+2*3^(1/2))^(1/2))/
(2+2*3^(1/2))^(1/2))*(-5475+3267*3^(1/2))^(1/2)+11/2304*arctan((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))
*(-5475+3267*3^(1/2))^(1/2)-11/4608*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(5475+3267*3^(1/2))^(1/2)+11/4608*l
n(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*(5475+3267*3^(1/2))^(1/2)

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Rubi [A]
time = 0.19, antiderivative size = 246, 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 = {1682, 1692, 1183, 648, 632, 210, 642} \begin {gather*} -\frac {11}{768} \sqrt {\frac {1}{3} \left (1089 \sqrt {3}-1825\right )} \text {ArcTan}\left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {11}{768} \sqrt {\frac {1}{3} \left (1089 \sqrt {3}-1825\right )} \text {ArcTan}\left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {11 \sqrt {\frac {1}{3} \left (1825+1089 \sqrt {3}\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )}{1536}+\frac {11 \sqrt {\frac {1}{3} \left (1825+1089 \sqrt {3}\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )}{1536}+\frac {25 x \left (x^2+1\right )}{16 \left (x^4+2 x^2+3\right )^2}-\frac {x \left (88 x^2+353\right )}{192 \left (x^4+2 x^2+3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(25*x*(1 + x^2))/(16*(3 + 2*x^2 + x^4)^2) - (x*(353 + 88*x^2))/(192*(3 + 2*x^2 + x^4)) - (11*Sqrt[(-1825 + 108
9*Sqrt[3])/3]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*(1 + Sqrt[3])]])/768 + (11*Sqrt[(-1825 + 1089*Sqrt[
3])/3]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/768 - (11*Sqrt[(1825 + 1089*Sqrt[3])/3]*L
og[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/1536 + (11*Sqrt[(1825 + 1089*Sqrt[3])/3]*Log[Sqrt[3] + Sqrt[2*(-
1 + Sqrt[3])]*x + x^2])/1536

Rule 210

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

Rule 632

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

Rule 642

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

Rule 648

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

Rule 1183

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

Rule 1682

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 1692

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^2 \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 (1+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac {1}{96} \int \frac {-150+78 x^2+480 x^4}{\left (3+2 x^2+x^4\right )^2} \, dx\\ &=\frac {25 x \left (1+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac {x \left (353+88 x^2\right )}{192 \left (3+2 x^2+x^4\right )}+\frac {\int \frac {6072-2112 x^2}{3+2 x^2+x^4} \, dx}{4608}\\ &=\frac {25 x \left (1+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac {x \left (353+88 x^2\right )}{192 \left (3+2 x^2+x^4\right )}+\frac {\int \frac {6072 \sqrt {2 \left (-1+\sqrt {3}\right )}-\left (6072+2112 \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 {6072 \sqrt {2 \left (-1+\sqrt {3}\right )}+\left (6072+2112 \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 (1+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac {x \left (353+88 x^2\right )}{192 \left (3+2 x^2+x^4\right )}-\frac {\left (11 \left (24-23 \sqrt {3}\right )\right ) \int \frac {1}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{2304}-\frac {\left (11 \left (24-23 \sqrt {3}\right )\right ) \int \frac {1}{\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{2304}-\frac {\left (11 \left (23+8 \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}{768 \sqrt {6 \left (-1+\sqrt {3}\right )}}+\frac {\left (11 \left (23+8 \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}{768 \sqrt {6 \left (-1+\sqrt {3}\right )}}\\ &=\frac {25 x \left (1+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac {x \left (353+88 x^2\right )}{192 \left (3+2 x^2+x^4\right )}-\frac {11}{768} \sqrt {\frac {1825}{12}+\frac {363 \sqrt {3}}{4}} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {11}{768} \sqrt {\frac {1825}{12}+\frac {363 \sqrt {3}}{4}} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {\left (11 \left (24-23 \sqrt {3}\right )\right ) \text {Subst}\left (\int \frac {1}{-2 \left (1+\sqrt {3}\right )-x^2} \, dx,x,-\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x\right )}{1152}+\frac {\left (11 \left (24-23 \sqrt {3}\right )\right ) \text {Subst}\left (\int \frac {1}{-2 \left (1+\sqrt {3}\right )-x^2} \, dx,x,\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x\right )}{1152}\\ &=\frac {25 x \left (1+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac {x \left (353+88 x^2\right )}{192 \left (3+2 x^2+x^4\right )}-\frac {11}{768} \sqrt {\frac {1}{3} \left (-1825+1089 \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 {11}{768} \sqrt {\frac {1}{3} \left (-1825+1089 \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 {11}{768} \sqrt {\frac {1825}{12}+\frac {363 \sqrt {3}}{4}} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {11}{768} \sqrt {\frac {1825}{12}+\frac {363 \sqrt {3}}{4}} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.20, size = 133, normalized size = 0.54 \begin {gather*} \frac {1}{768} \left (-\frac {4 x \left (759+670 x^2+529 x^4+88 x^6\right )}{\left (3+2 x^2+x^4\right )^2}-\frac {11 i \left (-16 i+31 \sqrt {2}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )}{\sqrt {1-i \sqrt {2}}}+\frac {11 i \left (16 i+31 \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[(x^2*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^3,x]

[Out]

((-4*x*(759 + 670*x^2 + 529*x^4 + 88*x^6))/(3 + 2*x^2 + x^4)^2 - ((11*I)*(-16*I + 31*Sqrt[2])*ArcTan[x/Sqrt[1
- I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] + ((11*I)*(16*I + 31*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/Sqrt[1 + I*Sqr
t[2]])/768

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Maple [A]
time = 0.04, size = 287, normalized size = 1.17

method	result	size
risch	\(\frac {-\frac {11}{24} x^{7}-\frac {529}{192} x^{5}-\frac {335}{96} x^{3}-\frac {253}{64} x}{\left (x^{4}+2 x^{2}+3\right )^{2}}+\frac {11 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+2 \textit {\_Z}^{2}+3\right )}{\sum }\frac {\left (-8 \textit {\_R}^{2}+23\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+\textit {\_R}}\right )}{768}\)	\(71\)
default	\(\frac {-\frac {11}{24} x^{7}-\frac {529}{192} x^{5}-\frac {335}{96} x^{3}-\frac {253}{64} x}{\left (x^{4}+2 x^{2}+3\right )^{2}}-\frac {11 \left (47 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+93 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+\sqrt {3}-x \sqrt {-2+2 \sqrt {3}}\right )}{9216}-\frac {11 \left (-92 \sqrt {3}+\frac {\left (47 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+93 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{2304 \sqrt {2+2 \sqrt {3}}}+\frac {11 \left (47 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+93 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+\sqrt {3}+x \sqrt {-2+2 \sqrt {3}}\right )}{9216}+\frac {11 \left (92 \sqrt {3}-\frac {\left (47 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+93 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{2304 \sqrt {2+2 \sqrt {3}}}\)	\(287\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-11/24*x^7-529/192*x^5-335/96*x^3-253/64*x)/(x^4+2*x^2+3)^2-11/9216*(47*(-2+2*3^(1/2))^(1/2)*3^(1/2)+93*(-2+2
*3^(1/2))^(1/2))*ln(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))-11/2304*(-92*3^(1/2)+1/2*(47*(-2+2*3^(1/2))^(1/2)*3^(1
/2)+93*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2)*arctan((2*x-(-2+2*3^(1/2))^(1/2))/(2+2*
3^(1/2))^(1/2))+11/9216*(47*(-2+2*3^(1/2))^(1/2)*3^(1/2)+93*(-2+2*3^(1/2))^(1/2))*ln(x^2+3^(1/2)+x*(-2+2*3^(1/
2))^(1/2))+11/2304*(92*3^(1/2)-1/2*(47*(-2+2*3^(1/2))^(1/2)*3^(1/2)+93*(-2+2*3^(1/2))^(1/2))*(-2+2*3^(1/2))^(1
/2))/(2+2*3^(1/2))^(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*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192*(88*x^7 + 529*x^5 + 670*x^3 + 759*x)/(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9) - 11/192*integrate((8*x^2 - 23
)/(x^4 + 2*x^2 + 3), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 574 vs. \(2 (177) = 354\).
time = 0.39, size = 574, normalized size = 2.33 \begin {gather*} -\frac {12811392 \, x^{7} + 77013936 \, x^{5} + 1348 \, \sqrt {6} 3^{\frac {3}{4}} \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} \sqrt {-1987425 \, \sqrt {3} + 3557763} \arctan \left (\frac {1}{2226179538} \, \sqrt {337} \sqrt {11} \sqrt {6} 3^{\frac {3}{4}} \sqrt {\sqrt {6} 3^{\frac {1}{4}} {\left (8 \, \sqrt {3} x + 23 \, x\right )} \sqrt {-1987425 \, \sqrt {3} + 3557763} + 33363 \, x^{2} + 33363 \, \sqrt {3}} {\left (23 \, \sqrt {3} \sqrt {2} + 24 \, \sqrt {2}\right )} \sqrt {-1987425 \, \sqrt {3} + 3557763} - \frac {1}{200178} \, \sqrt {6} 3^{\frac {3}{4}} {\left (23 \, \sqrt {3} \sqrt {2} x + 24 \, \sqrt {2} x\right )} \sqrt {-1987425 \, \sqrt {3} + 3557763} - \frac {1}{2} \, \sqrt {3} \sqrt {2} + \frac {1}{2} \, \sqrt {2}\right ) + 1348 \, \sqrt {6} 3^{\frac {3}{4}} \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} \sqrt {-1987425 \, \sqrt {3} + 3557763} \arctan \left (\frac {1}{4848619033764} \, \sqrt {337} \sqrt {6} 3^{\frac {3}{4}} \sqrt {-52180524 \, \sqrt {6} 3^{\frac {1}{4}} {\left (8 \, \sqrt {3} x + 23 \, x\right )} \sqrt {-1987425 \, \sqrt {3} + 3557763} + 1740898822212 \, x^{2} + 1740898822212 \, \sqrt {3}} {\left (23 \, \sqrt {3} \sqrt {2} + 24 \, \sqrt {2}\right )} \sqrt {-1987425 \, \sqrt {3} + 3557763} - \frac {1}{200178} \, \sqrt {6} 3^{\frac {3}{4}} {\left (23 \, \sqrt {3} \sqrt {2} x + 24 \, \sqrt {2} x\right )} \sqrt {-1987425 \, \sqrt {3} + 3557763} + \frac {1}{2} \, \sqrt {3} \sqrt {2} - \frac {1}{2} \, \sqrt {2}\right ) - \sqrt {6} 3^{\frac {1}{4}} {\left (3267 \, x^{8} + 13068 \, x^{6} + 32670 \, x^{4} + 39204 \, x^{2} + 1825 \, \sqrt {3} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} + 29403\right )} \sqrt {-1987425 \, \sqrt {3} + 3557763} \log \left (\frac {52180524}{337} \, \sqrt {6} 3^{\frac {1}{4}} {\left (8 \, \sqrt {3} x + 23 \, x\right )} \sqrt {-1987425 \, \sqrt {3} + 3557763} + 5165871876 \, x^{2} + 5165871876 \, \sqrt {3}\right ) + \sqrt {6} 3^{\frac {1}{4}} {\left (3267 \, x^{8} + 13068 \, x^{6} + 32670 \, x^{4} + 39204 \, x^{2} + 1825 \, \sqrt {3} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} + 29403\right )} \sqrt {-1987425 \, \sqrt {3} + 3557763} \log \left (-\frac {52180524}{337} \, \sqrt {6} 3^{\frac {1}{4}} {\left (8 \, \sqrt {3} x + 23 \, x\right )} \sqrt {-1987425 \, \sqrt {3} + 3557763} + 5165871876 \, x^{2} + 5165871876 \, \sqrt {3}\right ) + 97541280 \, x^{3} + 110498256 \, x}{27952128 \, {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27952128*(12811392*x^7 + 77013936*x^5 + 1348*sqrt(6)*3^(3/4)*sqrt(2)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)*sq
rt(-1987425*sqrt(3) + 3557763)*arctan(1/2226179538*sqrt(337)*sqrt(11)*sqrt(6)*3^(3/4)*sqrt(sqrt(6)*3^(1/4)*(8*
sqrt(3)*x + 23*x)*sqrt(-1987425*sqrt(3) + 3557763) + 33363*x^2 + 33363*sqrt(3))*(23*sqrt(3)*sqrt(2) + 24*sqrt(
2))*sqrt(-1987425*sqrt(3) + 3557763) - 1/200178*sqrt(6)*3^(3/4)*(23*sqrt(3)*sqrt(2)*x + 24*sqrt(2)*x)*sqrt(-19
87425*sqrt(3) + 3557763) - 1/2*sqrt(3)*sqrt(2) + 1/2*sqrt(2)) + 1348*sqrt(6)*3^(3/4)*sqrt(2)*(x^8 + 4*x^6 + 10
*x^4 + 12*x^2 + 9)*sqrt(-1987425*sqrt(3) + 3557763)*arctan(1/4848619033764*sqrt(337)*sqrt(6)*3^(3/4)*sqrt(-521
80524*sqrt(6)*3^(1/4)*(8*sqrt(3)*x + 23*x)*sqrt(-1987425*sqrt(3) + 3557763) + 1740898822212*x^2 + 174089882221
2*sqrt(3))*(23*sqrt(3)*sqrt(2) + 24*sqrt(2))*sqrt(-1987425*sqrt(3) + 3557763) - 1/200178*sqrt(6)*3^(3/4)*(23*s
qrt(3)*sqrt(2)*x + 24*sqrt(2)*x)*sqrt(-1987425*sqrt(3) + 3557763) + 1/2*sqrt(3)*sqrt(2) - 1/2*sqrt(2)) - sqrt(
6)*3^(1/4)*(3267*x^8 + 13068*x^6 + 32670*x^4 + 39204*x^2 + 1825*sqrt(3)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9) +
29403)*sqrt(-1987425*sqrt(3) + 3557763)*log(52180524/337*sqrt(6)*3^(1/4)*(8*sqrt(3)*x + 23*x)*sqrt(-1987425*sq
rt(3) + 3557763) + 5165871876*x^2 + 5165871876*sqrt(3)) + sqrt(6)*3^(1/4)*(3267*x^8 + 13068*x^6 + 32670*x^4 +
39204*x^2 + 1825*sqrt(3)*(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9) + 29403)*sqrt(-1987425*sqrt(3) + 3557763)*log(-52
180524/337*sqrt(6)*3^(1/4)*(8*sqrt(3)*x + 23*x)*sqrt(-1987425*sqrt(3) + 3557763) + 5165871876*x^2 + 5165871876
*sqrt(3)) + 97541280*x^3 + 110498256*x)/(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1200 vs. \(2 (207) = 414\).
time = 0.72, size = 1200, normalized size = 4.88 \begin {gather*} \frac {- 88 x^{7} - 529 x^{5} - 670 x^{3} - 759 x}{192 x^{8} + 768 x^{6} + 1920 x^{4} + 2304 x^{2} + 1728} - \sqrt {\frac {220825}{7077888} + \frac {14641 \sqrt {3}}{786432}} \log {\left (x^{2} + x \left (- \frac {47 \sqrt {6} \sqrt {1825 + 1089 \sqrt {3}} \sqrt {1987425 \sqrt {3} + 3444194}}{366993} + \frac {52016 \sqrt {3} \sqrt {1825 + 1089 \sqrt {3}}}{366993} + \frac {188 \sqrt {1825 + 1089 \sqrt {3}}}{337}\right ) - \frac {24765218375 \sqrt {2} \sqrt {1987425 \sqrt {3} + 3444194}}{134683862049} - \frac {38128468 \sqrt {6} \sqrt {1987425 \sqrt {3} + 3444194}}{371029923} + \frac {90413874433403}{134683862049} + \frac {144251139148 \sqrt {3}}{371029923} \right )} + \sqrt {\frac {220825}{7077888} + \frac {14641 \sqrt {3}}{786432}} \log {\left (x^{2} + x \left (- \frac {188 \sqrt {1825 + 1089 \sqrt {3}}}{337} - \frac {52016 \sqrt {3} \sqrt {1825 + 1089 \sqrt {3}}}{366993} + \frac {47 \sqrt {6} \sqrt {1825 + 1089 \sqrt {3}} \sqrt {1987425 \sqrt {3} + 3444194}}{366993}\right ) - \frac {24765218375 \sqrt {2} \sqrt {1987425 \sqrt {3} + 3444194}}{134683862049} - \frac {38128468 \sqrt {6} \sqrt {1987425 \sqrt {3} + 3444194}}{371029923} + \frac {90413874433403}{134683862049} + \frac {144251139148 \sqrt {3}}{371029923} \right )} + 2 \sqrt {- \frac {121 \sqrt {2} \sqrt {1987425 \sqrt {3} + 3444194}}{3538944} + \frac {220825}{7077888} + \frac {14641 \sqrt {3}}{262144}} \operatorname {atan}{\left (\frac {733986 \sqrt {3} x}{15502 \sqrt {- 2 \sqrt {2} \sqrt {1987425 \sqrt {3} + 3444194} + 1825 + 3267 \sqrt {3}} + 47 \sqrt {2} \sqrt {1987425 \sqrt {3} + 3444194} \sqrt {- 2 \sqrt {2} \sqrt {1987425 \sqrt {3} + 3444194} + 1825 + 3267 \sqrt {3}}} - \frac {204732 \sqrt {3} \sqrt {1825 + 1089 \sqrt {3}}}{15502 \sqrt {- 2 \sqrt {2} \sqrt {1987425 \sqrt {3} + 3444194} + 1825 + 3267 \sqrt {3}} + 47 \sqrt {2} \sqrt {1987425 \sqrt {3} + 3444194} \sqrt {- 2 \sqrt {2} \sqrt {1987425 \sqrt {3} + 3444194} + 1825 + 3267 \sqrt {3}}} - \frac {156048 \sqrt {1825 + 1089 \sqrt {3}}}{15502 \sqrt {- 2 \sqrt {2} \sqrt {1987425 \sqrt {3} + 3444194} + 1825 + 3267 \sqrt {3}} + 47 \sqrt {2} \sqrt {1987425 \sqrt {3} + 3444194} \sqrt {- 2 \sqrt {2} \sqrt {1987425 \sqrt {3} + 3444194} + 1825 + 3267 \sqrt {3}}} + \frac {141 \sqrt {2} \sqrt {1825 + 1089 \sqrt {3}} \sqrt {1987425 \sqrt {3} + 3444194}}{15502 \sqrt {- 2 \sqrt {2} \sqrt {1987425 \sqrt {3} + 3444194} + 1825 + 3267 \sqrt {3}} + 47 \sqrt {2} \sqrt {1987425 \sqrt {3} + 3444194} \sqrt {- 2 \sqrt {2} \sqrt {1987425 \sqrt {3} + 3444194} + 1825 + 3267 \sqrt {3}}} \right )} + 2 \sqrt {- \frac {121 \sqrt {2} \sqrt {1987425 \sqrt {3} + 3444194}}{3538944} + \frac {220825}{7077888} + \frac {14641 \sqrt {3}}{262144}} \operatorname {atan}{\left (\frac {733986 \sqrt {3} x}{15502 \sqrt {- 2 \sqrt {2} \sqrt {1987425 \sqrt {3} + 3444194} + 1825 + 3267 \sqrt {3}} + 47 \sqrt {2} \sqrt {1987425 \sqrt {3} + 3444194} \sqrt {- 2 \sqrt {2} \sqrt {1987425 \sqrt {3} + 3444194} + 1825 + 3267 \sqrt {3}}} - \frac {141 \sqrt {2} \sqrt {1825 + 1089 \sqrt {3}} \sqrt {1987425 \sqrt {3} + 3444194}}{15502 \sqrt {- 2 \sqrt {2} \sqrt {1987425 \sqrt {3} + 3444194} + 1825 + 3267 \sqrt {3}} + 47 \sqrt {2} \sqrt {1987425 \sqrt {3} + 3444194} \sqrt {- 2 \sqrt {2} \sqrt {1987425 \sqrt {3} + 3444194} + 1825 + 3267 \sqrt {3}}} + \frac {156048 \sqrt {1825 + 1089 \sqrt {3}}}{15502 \sqrt {- 2 \sqrt {2} \sqrt {1987425 \sqrt {3} + 3444194} + 1825 + 3267 \sqrt {3}} + 47 \sqrt {2} \sqrt {1987425 \sqrt {3} + 3444194} \sqrt {- 2 \sqrt {2} \sqrt {1987425 \sqrt {3} + 3444194} + 1825 + 3267 \sqrt {3}}} + \frac {204732 \sqrt {3} \sqrt {1825 + 1089 \sqrt {3}}}{15502 \sqrt {- 2 \sqrt {2} \sqrt {1987425 \sqrt {3} + 3444194} + 1825 + 3267 \sqrt {3}} + 47 \sqrt {2} \sqrt {1987425 \sqrt {3} + 3444194} \sqrt {- 2 \sqrt {2} \sqrt {1987425 \sqrt {3} + 3444194} + 1825 + 3267 \sqrt {3}}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-88*x**7 - 529*x**5 - 670*x**3 - 759*x)/(192*x**8 + 768*x**6 + 1920*x**4 + 2304*x**2 + 1728) - sqrt(220825/70
77888 + 14641*sqrt(3)/786432)*log(x**2 + x*(-47*sqrt(6)*sqrt(1825 + 1089*sqrt(3))*sqrt(1987425*sqrt(3) + 34441
94)/366993 + 52016*sqrt(3)*sqrt(1825 + 1089*sqrt(3))/366993 + 188*sqrt(1825 + 1089*sqrt(3))/337) - 24765218375
*sqrt(2)*sqrt(1987425*sqrt(3) + 3444194)/134683862049 - 38128468*sqrt(6)*sqrt(1987425*sqrt(3) + 3444194)/37102
9923 + 90413874433403/134683862049 + 144251139148*sqrt(3)/371029923) + sqrt(220825/7077888 + 14641*sqrt(3)/786
432)*log(x**2 + x*(-188*sqrt(1825 + 1089*sqrt(3))/337 - 52016*sqrt(3)*sqrt(1825 + 1089*sqrt(3))/366993 + 47*sq
rt(6)*sqrt(1825 + 1089*sqrt(3))*sqrt(1987425*sqrt(3) + 3444194)/366993) - 24765218375*sqrt(2)*sqrt(1987425*sqr
t(3) + 3444194)/134683862049 - 38128468*sqrt(6)*sqrt(1987425*sqrt(3) + 3444194)/371029923 + 90413874433403/134
683862049 + 144251139148*sqrt(3)/371029923) + 2*sqrt(-121*sqrt(2)*sqrt(1987425*sqrt(3) + 3444194)/3538944 + 22
0825/7077888 + 14641*sqrt(3)/262144)*atan(733986*sqrt(3)*x/(15502*sqrt(-2*sqrt(2)*sqrt(1987425*sqrt(3) + 34441
94) + 1825 + 3267*sqrt(3)) + 47*sqrt(2)*sqrt(1987425*sqrt(3) + 3444194)*sqrt(-2*sqrt(2)*sqrt(1987425*sqrt(3) +
3444194) + 1825 + 3267*sqrt(3))) - 204732*sqrt(3)*sqrt(1825 + 1089*sqrt(3))/(15502*sqrt(-2*sqrt(2)*sqrt(19874
25*sqrt(3) + 3444194) + 1825 + 3267*sqrt(3)) + 47*sqrt(2)*sqrt(1987425*sqrt(3) + 3444194)*sqrt(-2*sqrt(2)*sqrt
(1987425*sqrt(3) + 3444194) + 1825 + 3267*sqrt(3))) - 156048*sqrt(1825 + 1089*sqrt(3))/(15502*sqrt(-2*sqrt(2)*
sqrt(1987425*sqrt(3) + 3444194) + 1825 + 3267*sqrt(3)) + 47*sqrt(2)*sqrt(1987425*sqrt(3) + 3444194)*sqrt(-2*sq
rt(2)*sqrt(1987425*sqrt(3) + 3444194) + 1825 + 3267*sqrt(3))) + 141*sqrt(2)*sqrt(1825 + 1089*sqrt(3))*sqrt(198
7425*sqrt(3) + 3444194)/(15502*sqrt(-2*sqrt(2)*sqrt(1987425*sqrt(3) + 3444194) + 1825 + 3267*sqrt(3)) + 47*sqr
t(2)*sqrt(1987425*sqrt(3) + 3444194)*sqrt(-2*sqrt(2)*sqrt(1987425*sqrt(3) + 3444194) + 1825 + 3267*sqrt(3))))
+ 2*sqrt(-121*sqrt(2)*sqrt(1987425*sqrt(3) + 3444194)/3538944 + 220825/7077888 + 14641*sqrt(3)/262144)*atan(73
3986*sqrt(3)*x/(15502*sqrt(-2*sqrt(2)*sqrt(1987425*sqrt(3) + 3444194) + 1825 + 3267*sqrt(3)) + 47*sqrt(2)*sqrt
(1987425*sqrt(3) + 3444194)*sqrt(-2*sqrt(2)*sqrt(1987425*sqrt(3) + 3444194) + 1825 + 3267*sqrt(3))) - 141*sqrt
(2)*sqrt(1825 + 1089*sqrt(3))*sqrt(1987425*sqrt(3) + 3444194)/(15502*sqrt(-2*sqrt(2)*sqrt(1987425*sqrt(3) + 34
44194) + 1825 + 3267*sqrt(3)) + 47*sqrt(2)*sqrt(1987425*sqrt(3) + 3444194)*sqrt(-2*sqrt(2)*sqrt(1987425*sqrt(3
) + 3444194) + 1825 + 3267*sqrt(3))) + 156048*sqrt(1825 + 1089*sqrt(3))/(15502*sqrt(-2*sqrt(2)*sqrt(1987425*sq
rt(3) + 3444194) + 1825 + 3267*sqrt(3)) + 47*sqrt(2)*sqrt(1987425*sqrt(3) + 3444194)*sqrt(-2*sqrt(2)*sqrt(1987
425*sqrt(3) + 3444194) + 1825 + 3267*sqrt(3))) + 204732*sqrt(3)*sqrt(1825 + 1089*sqrt(3))/(15502*sqrt(-2*sqrt(
2)*sqrt(1987425*sqrt(3) + 3444194) + 1825 + 3267*sqrt(3)) + 47*sqrt(2)*sqrt(1987425*sqrt(3) + 3444194)*sqrt(-2
*sqrt(2)*sqrt(1987425*sqrt(3) + 3444194) + 1825 + 3267*sqrt(3))))

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 577 vs. \(2 (177) = 354\).
time = 4.57, size = 577, normalized size = 2.35 \begin {gather*} \frac {11}{124416} \, \sqrt {2} {\left (2 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 36 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 36 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 2 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 207 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} - 207 \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 {11}{124416} \, \sqrt {2} {\left (2 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 36 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 36 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 2 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 207 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} - 207 \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 {11}{248832} \, \sqrt {2} {\left (36 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 2 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 2 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 36 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} + 207 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} + 207 \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 {11}{248832} \, \sqrt {2} {\left (36 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 2 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 2 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 36 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} + 207 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} + 207 \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 {88 \, x^{7} + 529 \, x^{5} + 670 \, x^{3} + 759 \, x}{192 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11/124416*sqrt(2)*(2*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 36*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3)
- 3) - 36*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) + 2*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) + 207*3^(1/4)*sqrt(
2)*sqrt(6*sqrt(3) + 18) - 207*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)) + 11/124416*sqrt(2)*(2*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 36*3^(3/4)*s
qrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 36*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) + 2*3^(3/4)*(-6*sqr
t(3) + 18)^(3/2) + 207*3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) - 207*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)) + 11/248832*sqrt(2)*(36*3^(3/4)*sqrt(2)*
(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 2*3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2) + 2*3^(3/4)*(6*sqrt(3) + 18)^(
3/2) + 36*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) + 207*3^(1/4)*sqrt(2)*sqrt(-6*sqrt(3) + 18) + 207*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)) - 11/248832*sqrt(2)*(36*3^(3/
4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 2*3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2) + 2*3^(3/4)*(6*sqrt
(3) + 18)^(3/2) + 36*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) + 207*3^(1/4)*sqrt(2)*sqrt(-6*sqrt(3) + 18) +
207*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)) - 1/192*(88*x^7 +
529*x^5 + 670*x^3 + 759*x)/(x^4 + 2*x^2 + 3)^2

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Mupad [B]
time = 1.01, size = 174, normalized size = 0.71 \begin {gather*} -\frac {\frac {11\,x^7}{24}+\frac {529\,x^5}{192}+\frac {335\,x^3}{96}+\frac {253\,x}{64}}{x^8+4\,x^6+10\,x^4+12\,x^2+9}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {10950-\sqrt {2}\,2022{}\mathrm {i}}\,448547{}\mathrm {i}}{31850496\,\left (-\frac {21081709}{10616832}+\frac {\sqrt {2}\,10316581{}\mathrm {i}}{10616832}\right )}-\frac {448547\,\sqrt {2}\,x\,\sqrt {10950-\sqrt {2}\,2022{}\mathrm {i}}}{63700992\,\left (-\frac {21081709}{10616832}+\frac {\sqrt {2}\,10316581{}\mathrm {i}}{10616832}\right )}\right )\,\sqrt {10950-\sqrt {2}\,2022{}\mathrm {i}}\,11{}\mathrm {i}}{2304}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {10950+\sqrt {2}\,2022{}\mathrm {i}}\,448547{}\mathrm {i}}{31850496\,\left (\frac {21081709}{10616832}+\frac {\sqrt {2}\,10316581{}\mathrm {i}}{10616832}\right )}+\frac {448547\,\sqrt {2}\,x\,\sqrt {10950+\sqrt {2}\,2022{}\mathrm {i}}}{63700992\,\left (\frac {21081709}{10616832}+\frac {\sqrt {2}\,10316581{}\mathrm {i}}{10616832}\right )}\right )\,\sqrt {10950+\sqrt {2}\,2022{}\mathrm {i}}\,11{}\mathrm {i}}{2304} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan((x*(10950 - 2^(1/2)*2022i)^(1/2)*448547i)/(31850496*((2^(1/2)*10316581i)/10616832 - 21081709/10616832))
- (448547*2^(1/2)*x*(10950 - 2^(1/2)*2022i)^(1/2))/(63700992*((2^(1/2)*10316581i)/10616832 - 21081709/10616832
)))*(10950 - 2^(1/2)*2022i)^(1/2)*11i)/2304 - ((253*x)/64 + (335*x^3)/96 + (529*x^5)/192 + (11*x^7)/24)/(12*x^
2 + 10*x^4 + 4*x^6 + x^8 + 9) - (atan((x*(2^(1/2)*2022i + 10950)^(1/2)*448547i)/(31850496*((2^(1/2)*10316581i)
/10616832 + 21081709/10616832)) + (448547*2^(1/2)*x*(2^(1/2)*2022i + 10950)^(1/2))/(63700992*((2^(1/2)*1031658
1i)/10616832 + 21081709/10616832)))*(2^(1/2)*2022i + 10950)^(1/2)*11i)/2304

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