3.2.13 \(\int \frac {4+x^2+3 x^4+5 x^6}{(3+2 x^2+x^4)^2} \, dx\) [113]

Optimal. Leaf size=224 \[ \frac {25 x \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}-\frac {1}{48} \sqrt {\frac {1}{6} \left (-11567+12897 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {1}{48} \sqrt {\frac {1}{6} \left (-11567+12897 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {1}{96} \sqrt {\frac {1}{6} \left (11567+12897 \sqrt {3}\right )} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )-\frac {1}{96} \sqrt {\frac {1}{6} \left (11567+12897 \sqrt {3}\right )} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right ) \]

[Out]

25/24*x*(-x^2+1)/(x^4+2*x^2+3)-1/288*arctan((-2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(-69402+77382*3^(
1/2))^(1/2)+1/288*arctan((2*x+(-2+2*3^(1/2))^(1/2))/(2+2*3^(1/2))^(1/2))*(-69402+77382*3^(1/2))^(1/2)+1/576*ln
(x^2+3^(1/2)-x*(-2+2*3^(1/2))^(1/2))*(69402+77382*3^(1/2))^(1/2)-1/576*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))*
(69402+77382*3^(1/2))^(1/2)

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Rubi [A]
time = 0.14, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1692, 1183, 648, 632, 210, 642} \begin {gather*} -\frac {1}{48} \sqrt {\frac {1}{6} \left (12897 \sqrt {3}-11567\right )} \text {ArcTan}\left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {1}{48} \sqrt {\frac {1}{6} \left (12897 \sqrt {3}-11567\right )} \text {ArcTan}\left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {1}{96} \sqrt {\frac {1}{6} \left (11567+12897 \sqrt {3}\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )-\frac {1}{96} \sqrt {\frac {1}{6} \left (11567+12897 \sqrt {3}\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+\frac {25 x \left (1-x^2\right )}{24 \left (x^4+2 x^2+3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(25*x*(1 - x^2))/(24*(3 + 2*x^2 + x^4)) - (Sqrt[(-11567 + 12897*Sqrt[3])/6]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2
*x)/Sqrt[2*(1 + Sqrt[3])]])/48 + (Sqrt[(-11567 + 12897*Sqrt[3])/6]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[
2*(1 + Sqrt[3])]])/48 + (Sqrt[(11567 + 12897*Sqrt[3])/6]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/96 - (
Sqrt[(11567 + 12897*Sqrt[3])/6]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2])/96

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 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 {4+x^2+3 x^4+5 x^6}{\left (3+2 x^2+x^4\right )^2} \, dx &=\frac {25 x \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac {1}{48} \int \frac {14+190 x^2}{3+2 x^2+x^4} \, dx\\ &=\frac {25 x \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac {\int \frac {14 \sqrt {2 \left (-1+\sqrt {3}\right )}-\left (14-190 \sqrt {3}\right ) x}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{96 \sqrt {6 \left (-1+\sqrt {3}\right )}}+\frac {\int \frac {14 \sqrt {2 \left (-1+\sqrt {3}\right )}+\left (14-190 \sqrt {3}\right ) x}{\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{96 \sqrt {6 \left (-1+\sqrt {3}\right )}}\\ &=\frac {25 x \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac {\left (7-95 \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}{96 \sqrt {6 \left (-1+\sqrt {3}\right )}}+\frac {1}{288} \left (285+7 \sqrt {3}\right ) \int \frac {1}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx+\frac {1}{288} \left (285+7 \sqrt {3}\right ) \int \frac {1}{\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx+\frac {\left (-7+95 \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}{96 \sqrt {6 \left (-1+\sqrt {3}\right )}}\\ &=\frac {25 x \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac {1}{96} \sqrt {\frac {11567}{6}+\frac {4299 \sqrt {3}}{2}} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )-\frac {1}{96} \sqrt {\frac {11567}{6}+\frac {4299 \sqrt {3}}{2}} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )-\frac {1}{144} \left (285+7 \sqrt {3}\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 )-\frac {1}{144} \left (285+7 \sqrt {3}\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 )\\ &=\frac {25 x \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}-\frac {1}{48} \sqrt {\frac {1}{6} \left (-11567+12897 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {1}{48} \sqrt {\frac {1}{6} \left (-11567+12897 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {1}{96} \sqrt {\frac {11567}{6}+\frac {4299 \sqrt {3}}{2}} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )-\frac {1}{96} \sqrt {\frac {11567}{6}+\frac {4299 \sqrt {3}}{2}} \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 = 0.17, size = 115, normalized size = 0.51 \begin {gather*} \frac {1}{48} \left (-\frac {50 x \left (-1+x^2\right )}{3+2 x^2+x^4}+\frac {\left (95+44 i \sqrt {2}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )}{\sqrt {1-i \sqrt {2}}}+\frac {\left (95-44 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)/(3 + 2*x^2 + x^4)^2,x]

[Out]

((-50*x*(-1 + x^2))/(3 + 2*x^2 + x^4) + ((95 + (44*I)*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[
2]] + ((95 - (44*I)*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/Sqrt[1 + I*Sqrt[2]])/48

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Maple [A]
time = 0.04, size = 277, normalized size = 1.24

method	result	size
risch	\(\frac {-\frac {25}{24} x^{3}+\frac {25}{24} x}{x^{4}+2 x^{2}+3}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+2 \textit {\_Z}^{2}+3\right )}{\sum }\frac {\left (95 \textit {\_R}^{2}+7\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+\textit {\_R}}\right )}{96}\)	\(61\)
default	\(\frac {-\frac {25}{24} x^{3}+\frac {25}{24} x}{x^{4}+2 x^{2}+3}+\frac {\left (139 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+132 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+\sqrt {3}-x \sqrt {-2+2 \sqrt {3}}\right )}{576}+\frac {\left (14 \sqrt {3}+\frac {\left (139 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+132 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{144 \sqrt {2+2 \sqrt {3}}}+\frac {\left (-139 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-132 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+\sqrt {3}+x \sqrt {-2+2 \sqrt {3}}\right )}{576}+\frac {\left (14 \sqrt {3}-\frac {\left (-139 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-132 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{144 \sqrt {2+2 \sqrt {3}}}\)	\(277\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-25/24*x^3+25/24*x)/(x^4+2*x^2+3)+1/576*(139*(-2+2*3^(1/2))^(1/2)*3^(1/2)+132*(-2+2*3^(1/2))^(1/2))*ln(x^2+3^
(1/2)-x*(-2+2*3^(1/2))^(1/2))+1/144*(14*3^(1/2)+1/2*(139*(-2+2*3^(1/2))^(1/2)*3^(1/2)+132*(-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))+1/576*(-139
*(-2+2*3^(1/2))^(1/2)*3^(1/2)-132*(-2+2*3^(1/2))^(1/2))*ln(x^2+3^(1/2)+x*(-2+2*3^(1/2))^(1/2))+1/144*(14*3^(1/
2)-1/2*(-139*(-2+2*3^(1/2))^(1/2)*3^(1/2)-132*(-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((5*x^6+3*x^4+x^2+4)/(x^4+2*x^2+3)^2,x, algorithm="maxima")

[Out]

-25/24*(x^3 - x)/(x^4 + 2*x^2 + 3) + 1/24*integrate((95*x^2 + 7)/(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. 461 vs. \(2 (155) = 310\).
time = 0.37, size = 461, normalized size = 2.06 \begin {gather*} -\frac {54052 \cdot 6160467^{\frac {1}{4}} \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {-149179599 \, \sqrt {3} + 498997827} \arctan \left (\frac {1}{29015889224422097862} \cdot 6160467^{\frac {3}{4}} \sqrt {13513} \sqrt {1433} \sqrt {174277161 \, x^{2} + 6160467^{\frac {1}{4}} {\left (7 \, \sqrt {3} x - 285 \, x\right )} \sqrt {-149179599 \, \sqrt {3} + 498997827} + 174277161 \, \sqrt {3}} {\left (95 \, \sqrt {3} \sqrt {2} - 7 \, \sqrt {2}\right )} \sqrt {-149179599 \, \sqrt {3} + 498997827} - \frac {1}{499478343426} \cdot 6160467^{\frac {3}{4}} {\left (95 \, \sqrt {3} \sqrt {2} x - 7 \, \sqrt {2} x\right )} \sqrt {-149179599 \, \sqrt {3} + 498997827} + \frac {1}{2} \, \sqrt {3} \sqrt {2} - \frac {1}{2} \, \sqrt {2}\right ) + 54052 \cdot 6160467^{\frac {1}{4}} \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {-149179599 \, \sqrt {3} + 498997827} \arctan \left (\frac {1}{29015889224422097862} \cdot 6160467^{\frac {3}{4}} \sqrt {13513} \sqrt {1433} \sqrt {174277161 \, x^{2} - 6160467^{\frac {1}{4}} {\left (7 \, \sqrt {3} x - 285 \, x\right )} \sqrt {-149179599 \, \sqrt {3} + 498997827} + 174277161 \, \sqrt {3}} {\left (95 \, \sqrt {3} \sqrt {2} - 7 \, \sqrt {2}\right )} \sqrt {-149179599 \, \sqrt {3} + 498997827} - \frac {1}{499478343426} \cdot 6160467^{\frac {3}{4}} {\left (95 \, \sqrt {3} \sqrt {2} x - 7 \, \sqrt {2} x\right )} \sqrt {-149179599 \, \sqrt {3} + 498997827} - \frac {1}{2} \, \sqrt {3} \sqrt {2} + \frac {1}{2} \, \sqrt {2}\right ) + 34855432200 \, x^{3} - 6160467^{\frac {1}{4}} {\left (11567 \, x^{4} + 23134 \, x^{2} + 12897 \, \sqrt {3} {\left (x^{4} + 2 \, x^{2} + 3\right )} + 34701\right )} \sqrt {-149179599 \, \sqrt {3} + 498997827} \log \left (1496993481 \, x^{2} + \frac {116073}{13513} \cdot 6160467^{\frac {1}{4}} {\left (7 \, \sqrt {3} x - 285 \, x\right )} \sqrt {-149179599 \, \sqrt {3} + 498997827} + 1496993481 \, \sqrt {3}\right ) + 6160467^{\frac {1}{4}} {\left (11567 \, x^{4} + 23134 \, x^{2} + 12897 \, \sqrt {3} {\left (x^{4} + 2 \, x^{2} + 3\right )} + 34701\right )} \sqrt {-149179599 \, \sqrt {3} + 498997827} \log \left (1496993481 \, x^{2} - \frac {116073}{13513} \cdot 6160467^{\frac {1}{4}} {\left (7 \, \sqrt {3} x - 285 \, x\right )} \sqrt {-149179599 \, \sqrt {3} + 498997827} + 1496993481 \, \sqrt {3}\right ) - 34855432200 \, x}{33461214912 \, {\left (x^{4} + 2 \, x^{2} + 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+2*x^2+3)^2,x, algorithm="fricas")

[Out]

-1/33461214912*(54052*6160467^(1/4)*sqrt(2)*(x^4 + 2*x^2 + 3)*sqrt(-149179599*sqrt(3) + 498997827)*arctan(1/29
015889224422097862*6160467^(3/4)*sqrt(13513)*sqrt(1433)*sqrt(174277161*x^2 + 6160467^(1/4)*(7*sqrt(3)*x - 285*
x)*sqrt(-149179599*sqrt(3) + 498997827) + 174277161*sqrt(3))*(95*sqrt(3)*sqrt(2) - 7*sqrt(2))*sqrt(-149179599*
sqrt(3) + 498997827) - 1/499478343426*6160467^(3/4)*(95*sqrt(3)*sqrt(2)*x - 7*sqrt(2)*x)*sqrt(-149179599*sqrt(
3) + 498997827) + 1/2*sqrt(3)*sqrt(2) - 1/2*sqrt(2)) + 54052*6160467^(1/4)*sqrt(2)*(x^4 + 2*x^2 + 3)*sqrt(-149
179599*sqrt(3) + 498997827)*arctan(1/29015889224422097862*6160467^(3/4)*sqrt(13513)*sqrt(1433)*sqrt(174277161*
x^2 - 6160467^(1/4)*(7*sqrt(3)*x - 285*x)*sqrt(-149179599*sqrt(3) + 498997827) + 174277161*sqrt(3))*(95*sqrt(3
)*sqrt(2) - 7*sqrt(2))*sqrt(-149179599*sqrt(3) + 498997827) - 1/499478343426*6160467^(3/4)*(95*sqrt(3)*sqrt(2)
*x - 7*sqrt(2)*x)*sqrt(-149179599*sqrt(3) + 498997827) - 1/2*sqrt(3)*sqrt(2) + 1/2*sqrt(2)) + 34855432200*x^3
- 6160467^(1/4)*(11567*x^4 + 23134*x^2 + 12897*sqrt(3)*(x^4 + 2*x^2 + 3) + 34701)*sqrt(-149179599*sqrt(3) + 49
8997827)*log(1496993481*x^2 + 116073/13513*6160467^(1/4)*(7*sqrt(3)*x - 285*x)*sqrt(-149179599*sqrt(3) + 49899
7827) + 1496993481*sqrt(3)) + 6160467^(1/4)*(11567*x^4 + 23134*x^2 + 12897*sqrt(3)*(x^4 + 2*x^2 + 3) + 34701)*
sqrt(-149179599*sqrt(3) + 498997827)*log(1496993481*x^2 - 116073/13513*6160467^(1/4)*(7*sqrt(3)*x - 285*x)*sqr
t(-149179599*sqrt(3) + 498997827) + 1496993481*sqrt(3)) - 34855432200*x)/(x^4 + 2*x^2 + 3)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1185 vs. \(2 (178) = 356\).
time = 0.72, size = 1185, normalized size = 5.29 \begin {gather*} \frac {- 25 x^{3} + 25 x}{24 x^{4} + 48 x^{2} + 72} + \sqrt {\frac {11567}{55296} + \frac {1433 \sqrt {3}}{6144}} \log {\left (x^{2} + x \left (- \frac {556 \sqrt {2} \sqrt {11567 + 12897 \sqrt {3}}}{13513} - \frac {1040345 \sqrt {6} \sqrt {11567 + 12897 \sqrt {3}}}{174277161} + \frac {278 \sqrt {3} \sqrt {11567 + 12897 \sqrt {3}} \sqrt {149179599 \sqrt {3} + 316396658}}{174277161}\right ) - \frac {47610276200401 \sqrt {2} \sqrt {149179599 \sqrt {3} + 316396658}}{30372528846219921} - \frac {4390831246 \sqrt {6} \sqrt {149179599 \sqrt {3} + 316396658}}{7065021829779} + \frac {1281046481635939181}{30372528846219921} + \frac {200684595453464 \sqrt {3}}{7065021829779} \right )} - \sqrt {\frac {11567}{55296} + \frac {1433 \sqrt {3}}{6144}} \log {\left (x^{2} + x \left (- \frac {278 \sqrt {3} \sqrt {11567 + 12897 \sqrt {3}} \sqrt {149179599 \sqrt {3} + 316396658}}{174277161} + \frac {1040345 \sqrt {6} \sqrt {11567 + 12897 \sqrt {3}}}{174277161} + \frac {556 \sqrt {2} \sqrt {11567 + 12897 \sqrt {3}}}{13513}\right ) - \frac {47610276200401 \sqrt {2} \sqrt {149179599 \sqrt {3} + 316396658}}{30372528846219921} - \frac {4390831246 \sqrt {6} \sqrt {149179599 \sqrt {3} + 316396658}}{7065021829779} + \frac {1281046481635939181}{30372528846219921} + \frac {200684595453464 \sqrt {3}}{7065021829779} \right )} + 2 \sqrt {- \frac {\sqrt {2} \sqrt {149179599 \sqrt {3} + 316396658}}{27648} + \frac {11567}{55296} + \frac {1433 \sqrt {3}}{2048}} \operatorname {atan}{\left (\frac {348554322 \sqrt {3} x}{94591 \sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {149179599 \sqrt {3} + 316396658} + 11567 + 38691 \sqrt {3}} + 278 \sqrt {149179599 \sqrt {3} + 316396658} \sqrt {- 2 \sqrt {2} \sqrt {149179599 \sqrt {3} + 316396658} + 11567 + 38691 \sqrt {3}}} - \frac {7170732 \sqrt {6} \sqrt {11567 + 12897 \sqrt {3}}}{94591 \sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {149179599 \sqrt {3} + 316396658} + 11567 + 38691 \sqrt {3}} + 278 \sqrt {149179599 \sqrt {3} + 316396658} \sqrt {- 2 \sqrt {2} \sqrt {149179599 \sqrt {3} + 316396658} + 11567 + 38691 \sqrt {3}}} - \frac {3121035 \sqrt {2} \sqrt {11567 + 12897 \sqrt {3}}}{94591 \sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {149179599 \sqrt {3} + 316396658} + 11567 + 38691 \sqrt {3}} + 278 \sqrt {149179599 \sqrt {3} + 316396658} \sqrt {- 2 \sqrt {2} \sqrt {149179599 \sqrt {3} + 316396658} + 11567 + 38691 \sqrt {3}}} + \frac {834 \sqrt {11567 + 12897 \sqrt {3}} \sqrt {149179599 \sqrt {3} + 316396658}}{94591 \sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {149179599 \sqrt {3} + 316396658} + 11567 + 38691 \sqrt {3}} + 278 \sqrt {149179599 \sqrt {3} + 316396658} \sqrt {- 2 \sqrt {2} \sqrt {149179599 \sqrt {3} + 316396658} + 11567 + 38691 \sqrt {3}}} \right )} + 2 \sqrt {- \frac {\sqrt {2} \sqrt {149179599 \sqrt {3} + 316396658}}{27648} + \frac {11567}{55296} + \frac {1433 \sqrt {3}}{2048}} \operatorname {atan}{\left (\frac {348554322 \sqrt {3} x}{94591 \sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {149179599 \sqrt {3} + 316396658} + 11567 + 38691 \sqrt {3}} + 278 \sqrt {149179599 \sqrt {3} + 316396658} \sqrt {- 2 \sqrt {2} \sqrt {149179599 \sqrt {3} + 316396658} + 11567 + 38691 \sqrt {3}}} - \frac {834 \sqrt {11567 + 12897 \sqrt {3}} \sqrt {149179599 \sqrt {3} + 316396658}}{94591 \sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {149179599 \sqrt {3} + 316396658} + 11567 + 38691 \sqrt {3}} + 278 \sqrt {149179599 \sqrt {3} + 316396658} \sqrt {- 2 \sqrt {2} \sqrt {149179599 \sqrt {3} + 316396658} + 11567 + 38691 \sqrt {3}}} + \frac {3121035 \sqrt {2} \sqrt {11567 + 12897 \sqrt {3}}}{94591 \sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {149179599 \sqrt {3} + 316396658} + 11567 + 38691 \sqrt {3}} + 278 \sqrt {149179599 \sqrt {3} + 316396658} \sqrt {- 2 \sqrt {2} \sqrt {149179599 \sqrt {3} + 316396658} + 11567 + 38691 \sqrt {3}}} + \frac {7170732 \sqrt {6} \sqrt {11567 + 12897 \sqrt {3}}}{94591 \sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {149179599 \sqrt {3} + 316396658} + 11567 + 38691 \sqrt {3}} + 278 \sqrt {149179599 \sqrt {3} + 316396658} \sqrt {- 2 \sqrt {2} \sqrt {149179599 \sqrt {3} + 316396658} + 11567 + 38691 \sqrt {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+2*x**2+3)**2,x)

[Out]

(-25*x**3 + 25*x)/(24*x**4 + 48*x**2 + 72) + sqrt(11567/55296 + 1433*sqrt(3)/6144)*log(x**2 + x*(-556*sqrt(2)*
sqrt(11567 + 12897*sqrt(3))/13513 - 1040345*sqrt(6)*sqrt(11567 + 12897*sqrt(3))/174277161 + 278*sqrt(3)*sqrt(1
1567 + 12897*sqrt(3))*sqrt(149179599*sqrt(3) + 316396658)/174277161) - 47610276200401*sqrt(2)*sqrt(149179599*s
qrt(3) + 316396658)/30372528846219921 - 4390831246*sqrt(6)*sqrt(149179599*sqrt(3) + 316396658)/7065021829779 +
1281046481635939181/30372528846219921 + 200684595453464*sqrt(3)/7065021829779) - sqrt(11567/55296 + 1433*sqrt
(3)/6144)*log(x**2 + x*(-278*sqrt(3)*sqrt(11567 + 12897*sqrt(3))*sqrt(149179599*sqrt(3) + 316396658)/174277161
+ 1040345*sqrt(6)*sqrt(11567 + 12897*sqrt(3))/174277161 + 556*sqrt(2)*sqrt(11567 + 12897*sqrt(3))/13513) - 47
610276200401*sqrt(2)*sqrt(149179599*sqrt(3) + 316396658)/30372528846219921 - 4390831246*sqrt(6)*sqrt(149179599
*sqrt(3) + 316396658)/7065021829779 + 1281046481635939181/30372528846219921 + 200684595453464*sqrt(3)/70650218
29779) + 2*sqrt(-sqrt(2)*sqrt(149179599*sqrt(3) + 316396658)/27648 + 11567/55296 + 1433*sqrt(3)/2048)*atan(348
554322*sqrt(3)*x/(94591*sqrt(2)*sqrt(-2*sqrt(2)*sqrt(149179599*sqrt(3) + 316396658) + 11567 + 38691*sqrt(3)) +
278*sqrt(149179599*sqrt(3) + 316396658)*sqrt(-2*sqrt(2)*sqrt(149179599*sqrt(3) + 316396658) + 11567 + 38691*s
qrt(3))) - 7170732*sqrt(6)*sqrt(11567 + 12897*sqrt(3))/(94591*sqrt(2)*sqrt(-2*sqrt(2)*sqrt(149179599*sqrt(3) +
316396658) + 11567 + 38691*sqrt(3)) + 278*sqrt(149179599*sqrt(3) + 316396658)*sqrt(-2*sqrt(2)*sqrt(149179599*
sqrt(3) + 316396658) + 11567 + 38691*sqrt(3))) - 3121035*sqrt(2)*sqrt(11567 + 12897*sqrt(3))/(94591*sqrt(2)*sq
rt(-2*sqrt(2)*sqrt(149179599*sqrt(3) + 316396658) + 11567 + 38691*sqrt(3)) + 278*sqrt(149179599*sqrt(3) + 3163
96658)*sqrt(-2*sqrt(2)*sqrt(149179599*sqrt(3) + 316396658) + 11567 + 38691*sqrt(3))) + 834*sqrt(11567 + 12897*
sqrt(3))*sqrt(149179599*sqrt(3) + 316396658)/(94591*sqrt(2)*sqrt(-2*sqrt(2)*sqrt(149179599*sqrt(3) + 316396658
) + 11567 + 38691*sqrt(3)) + 278*sqrt(149179599*sqrt(3) + 316396658)*sqrt(-2*sqrt(2)*sqrt(149179599*sqrt(3) +
316396658) + 11567 + 38691*sqrt(3)))) + 2*sqrt(-sqrt(2)*sqrt(149179599*sqrt(3) + 316396658)/27648 + 11567/5529
6 + 1433*sqrt(3)/2048)*atan(348554322*sqrt(3)*x/(94591*sqrt(2)*sqrt(-2*sqrt(2)*sqrt(149179599*sqrt(3) + 316396
658) + 11567 + 38691*sqrt(3)) + 278*sqrt(149179599*sqrt(3) + 316396658)*sqrt(-2*sqrt(2)*sqrt(149179599*sqrt(3)
+ 316396658) + 11567 + 38691*sqrt(3))) - 834*sqrt(11567 + 12897*sqrt(3))*sqrt(149179599*sqrt(3) + 316396658)/
(94591*sqrt(2)*sqrt(-2*sqrt(2)*sqrt(149179599*sqrt(3) + 316396658) + 11567 + 38691*sqrt(3)) + 278*sqrt(1491795
99*sqrt(3) + 316396658)*sqrt(-2*sqrt(2)*sqrt(149179599*sqrt(3) + 316396658) + 11567 + 38691*sqrt(3))) + 312103
5*sqrt(2)*sqrt(11567 + 12897*sqrt(3))/(94591*sqrt(2)*sqrt(-2*sqrt(2)*sqrt(149179599*sqrt(3) + 316396658) + 115
67 + 38691*sqrt(3)) + 278*sqrt(149179599*sqrt(3) + 316396658)*sqrt(-2*sqrt(2)*sqrt(149179599*sqrt(3) + 3163966
58) + 11567 + 38691*sqrt(3))) + 7170732*sqrt(6)*sqrt(11567 + 12897*sqrt(3))/(94591*sqrt(2)*sqrt(-2*sqrt(2)*sqr
t(149179599*sqrt(3) + 316396658) + 11567 + 38691*sqrt(3)) + 278*sqrt(149179599*sqrt(3) + 316396658)*sqrt(-2*sq
rt(2)*sqrt(149179599*sqrt(3) + 316396658) + 11567 + 38691*sqrt(3))))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 565 vs. \(2 (155) = 310\).
time = 4.03, size = 565, normalized size = 2.52 \begin {gather*} -\frac {1}{62208} \, \sqrt {2} {\left (95 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 1710 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 1710 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 95 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 252 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 252 \cdot 3^{\frac {1}{4}} \sqrt {-6 \, \sqrt {3} + 18}\right )} \arctan \left (\frac {3^{\frac {3}{4}} {\left (x + 3^{\frac {1}{4}} \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}\right )}}{3 \, \sqrt {\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}}\right ) - \frac {1}{62208} \, \sqrt {2} {\left (95 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 1710 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 1710 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 95 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 252 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 252 \cdot 3^{\frac {1}{4}} \sqrt {-6 \, \sqrt {3} + 18}\right )} \arctan \left (\frac {3^{\frac {3}{4}} {\left (x - 3^{\frac {1}{4}} \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}\right )}}{3 \, \sqrt {\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}}\right ) - \frac {1}{124416} \, \sqrt {2} {\left (1710 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 95 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 95 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 1710 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 252 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 252 \cdot 3^{\frac {1}{4}} \sqrt {6 \, \sqrt {3} + 18}\right )} \log \left (x^{2} + 2 \cdot 3^{\frac {1}{4}} x \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}} + \sqrt {3}\right ) + \frac {1}{124416} \, \sqrt {2} {\left (1710 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 95 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 95 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 1710 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 252 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 252 \cdot 3^{\frac {1}{4}} \sqrt {6 \, \sqrt {3} + 18}\right )} \log \left (x^{2} - 2 \cdot 3^{\frac {1}{4}} x \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}} + \sqrt {3}\right ) - \frac {25 \, {\left (x^{3} - x\right )}}{24 \, {\left (x^{4} + 2 \, x^{2} + 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+2*x^2+3)^2,x, algorithm="giac")

[Out]

-1/62208*sqrt(2)*(95*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 1710*3^(3/4)*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(
3) - 3) - 1710*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) + 95*3^(3/4)*(-6*sqrt(3) + 18)^(3/2) - 252*3^(1/4)*
sqrt(2)*sqrt(6*sqrt(3) + 18) + 252*3^(1/4)*sqrt(-6*sqrt(3) + 18))*arctan(1/3*3^(3/4)*(x + 3^(1/4)*sqrt(-1/6*sq
rt(3) + 1/2))/sqrt(1/6*sqrt(3) + 1/2)) - 1/62208*sqrt(2)*(95*3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 1710*3^(
3/4)*sqrt(2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 1710*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) + 95*3^(3/4
)*(-6*sqrt(3) + 18)^(3/2) - 252*3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) + 252*3^(1/4)*sqrt(-6*sqrt(3) + 18))*arct
an(1/3*3^(3/4)*(x - 3^(1/4)*sqrt(-1/6*sqrt(3) + 1/2))/sqrt(1/6*sqrt(3) + 1/2)) - 1/124416*sqrt(2)*(1710*3^(3/4
)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 95*3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2) + 95*3^(3/4)*(6*sqr
t(3) + 18)^(3/2) + 1710*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 252*3^(1/4)*sqrt(2)*sqrt(-6*sqrt(3) + 18)
- 252*3^(1/4)*sqrt(6*sqrt(3) + 18))*log(x^2 + 2*3^(1/4)*x*sqrt(-1/6*sqrt(3) + 1/2) + sqrt(3)) + 1/124416*sqrt
(2)*(1710*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 95*3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2) + 9
5*3^(3/4)*(6*sqrt(3) + 18)^(3/2) + 1710*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 252*3^(1/4)*sqrt(2)*sqrt(
-6*sqrt(3) + 18) - 252*3^(1/4)*sqrt(6*sqrt(3) + 18))*log(x^2 - 2*3^(1/4)*x*sqrt(-1/6*sqrt(3) + 1/2) + sqrt(3))
- 25/24*(x^3 - x)/(x^4 + 2*x^2 + 3)

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Mupad [B]
time = 0.13, size = 153, normalized size = 0.68 \begin {gather*} \frac {\frac {25\,x}{24}-\frac {25\,x^3}{24}}{x^4+2\,x^2+3}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {34701-\sqrt {2}\,40539{}\mathrm {i}}\,13513{}\mathrm {i}}{15552\,\left (-\frac {1878307}{5184}+\frac {\sqrt {2}\,94591{}\mathrm {i}}{10368}\right )}+\frac {13513\,\sqrt {2}\,x\,\sqrt {34701-\sqrt {2}\,40539{}\mathrm {i}}}{31104\,\left (-\frac {1878307}{5184}+\frac {\sqrt {2}\,94591{}\mathrm {i}}{10368}\right )}\right )\,\sqrt {34701-\sqrt {2}\,40539{}\mathrm {i}}\,1{}\mathrm {i}}{144}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {34701+\sqrt {2}\,40539{}\mathrm {i}}\,13513{}\mathrm {i}}{15552\,\left (\frac {1878307}{5184}+\frac {\sqrt {2}\,94591{}\mathrm {i}}{10368}\right )}-\frac {13513\,\sqrt {2}\,x\,\sqrt {34701+\sqrt {2}\,40539{}\mathrm {i}}}{31104\,\left (\frac {1878307}{5184}+\frac {\sqrt {2}\,94591{}\mathrm {i}}{10368}\right )}\right )\,\sqrt {34701+\sqrt {2}\,40539{}\mathrm {i}}\,1{}\mathrm {i}}{144} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((25*x)/24 - (25*x^3)/24)/(2*x^2 + x^4 + 3) - (atan((x*(34701 - 2^(1/2)*40539i)^(1/2)*13513i)/(15552*((2^(1/2)
*94591i)/10368 - 1878307/5184)) + (13513*2^(1/2)*x*(34701 - 2^(1/2)*40539i)^(1/2))/(31104*((2^(1/2)*94591i)/10
368 - 1878307/5184)))*(34701 - 2^(1/2)*40539i)^(1/2)*1i)/144 + (atan((x*(2^(1/2)*40539i + 34701)^(1/2)*13513i)
/(15552*((2^(1/2)*94591i)/10368 + 1878307/5184)) - (13513*2^(1/2)*x*(2^(1/2)*40539i + 34701)^(1/2))/(31104*((2
^(1/2)*94591i)/10368 + 1878307/5184)))*(2^(1/2)*40539i + 34701)^(1/2)*1i)/144

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