### 3.108 $$\int \frac{4+x^2+3 x^4+5 x^6}{x^7 (3+2 x^2+x^4)^2} \, dx$$

Optimal. Leaf size=87 $\frac{25 \left (1-7 x^2\right )}{648 \left (x^4+2 x^2+3\right )}-\frac{13}{54 x^2}+\frac{13}{108 x^4}-\frac{2}{27 x^6}-\frac{61}{972} \log \left (x^4+2 x^2+3\right )-\frac{1237 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{1944 \sqrt{2}}+\frac{61 \log (x)}{243}$

[Out]

-2/(27*x^6) + 13/(108*x^4) - 13/(54*x^2) + (25*(1 - 7*x^2))/(648*(3 + 2*x^2 + x^4)) - (1237*ArcTan[(1 + x^2)/S
qrt[2]])/(1944*Sqrt[2]) + (61*Log[x])/243 - (61*Log[3 + 2*x^2 + x^4])/972

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Rubi [A]  time = 0.148986, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 31, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.226, Rules used = {1663, 1646, 1628, 634, 618, 204, 628} $\frac{25 \left (1-7 x^2\right )}{648 \left (x^4+2 x^2+3\right )}-\frac{13}{54 x^2}+\frac{13}{108 x^4}-\frac{2}{27 x^6}-\frac{61}{972} \log \left (x^4+2 x^2+3\right )-\frac{1237 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{1944 \sqrt{2}}+\frac{61 \log (x)}{243}$

Antiderivative was successfully veriﬁed.

[In]

Int[(4 + x^2 + 3*x^4 + 5*x^6)/(x^7*(3 + 2*x^2 + x^4)^2),x]

[Out]

-2/(27*x^6) + 13/(108*x^4) - 13/(54*x^2) + (25*(1 - 7*x^2))/(648*(3 + 2*x^2 + x^4)) - (1237*ArcTan[(1 + x^2)/S
qrt[2]])/(1944*Sqrt[2]) + (61*Log[x])/243 - (61*Log[3 + 2*x^2 + x^4])/972

Rule 1663

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)
*SubstFor[x^2, Pq, x]*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && Inte
gerQ[(m - 1)/2]

Rule 1646

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[(d + e*x)^m*Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2,
x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2
*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*f - b*
g))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1628

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

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 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 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 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]

Rubi steps

\begin{align*} \int \frac{4+x^2+3 x^4+5 x^6}{x^7 \left (3+2 x^2+x^4\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{4+x+3 x^2+5 x^3}{x^4 \left (3+2 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{25 \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}+\frac{1}{16} \operatorname{Subst}\left (\int \frac{\frac{32}{3}-\frac{40 x}{9}+\frac{200 x^2}{27}+\frac{800 x^3}{81}-\frac{350 x^4}{81}}{x^4 \left (3+2 x+x^2\right )} \, dx,x,x^2\right )\\ &=\frac{25 \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}+\frac{1}{16} \operatorname{Subst}\left (\int \left (\frac{32}{9 x^4}-\frac{104}{27 x^3}+\frac{104}{27 x^2}+\frac{488}{243 x}-\frac{2 (1481+244 x)}{243 \left (3+2 x+x^2\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{2}{27 x^6}+\frac{13}{108 x^4}-\frac{13}{54 x^2}+\frac{25 \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}+\frac{61 \log (x)}{243}-\frac{\operatorname{Subst}\left (\int \frac{1481+244 x}{3+2 x+x^2} \, dx,x,x^2\right )}{1944}\\ &=-\frac{2}{27 x^6}+\frac{13}{108 x^4}-\frac{13}{54 x^2}+\frac{25 \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}+\frac{61 \log (x)}{243}-\frac{61}{972} \operatorname{Subst}\left (\int \frac{2+2 x}{3+2 x+x^2} \, dx,x,x^2\right )-\frac{1237 \operatorname{Subst}\left (\int \frac{1}{3+2 x+x^2} \, dx,x,x^2\right )}{1944}\\ &=-\frac{2}{27 x^6}+\frac{13}{108 x^4}-\frac{13}{54 x^2}+\frac{25 \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}+\frac{61 \log (x)}{243}-\frac{61}{972} \log \left (3+2 x^2+x^4\right )+\frac{1237}{972} \operatorname{Subst}\left (\int \frac{1}{-8-x^2} \, dx,x,2 \left (1+x^2\right )\right )\\ &=-\frac{2}{27 x^6}+\frac{13}{108 x^4}-\frac{13}{54 x^2}+\frac{25 \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}-\frac{1237 \tan ^{-1}\left (\frac{1+x^2}{\sqrt{2}}\right )}{1944 \sqrt{2}}+\frac{61 \log (x)}{243}-\frac{61}{972} \log \left (3+2 x^2+x^4\right )\\ \end{align*}

Mathematica [C]  time = 0.0689521, size = 110, normalized size = 1.26 $\frac{-\frac{300 \left (7 x^2-1\right )}{x^4+2 x^2+3}-\frac{1872}{x^2}+\frac{936}{x^4}-\frac{576}{x^6}+\sqrt{2} \left (-244 \sqrt{2}+1237 i\right ) \log \left (x^2-i \sqrt{2}+1\right )-\sqrt{2} \left (244 \sqrt{2}+1237 i\right ) \log \left (x^2+i \sqrt{2}+1\right )+1952 \log (x)}{7776}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x^7*(3 + 2*x^2 + x^4)^2),x]

[Out]

(-576/x^6 + 936/x^4 - 1872/x^2 - (300*(-1 + 7*x^2))/(3 + 2*x^2 + x^4) + 1952*Log[x] + Sqrt[2]*(1237*I - 244*Sq
rt[2])*Log[1 - I*Sqrt[2] + x^2] - Sqrt[2]*(1237*I + 244*Sqrt[2])*Log[1 + I*Sqrt[2] + x^2])/7776

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Maple [A]  time = 0.013, size = 73, normalized size = 0.8 \begin{align*} -{\frac{1}{486\,{x}^{4}+972\,{x}^{2}+1458} \left ({\frac{525\,{x}^{2}}{4}}-{\frac{75}{4}} \right ) }-{\frac{61\,\ln \left ({x}^{4}+2\,{x}^{2}+3 \right ) }{972}}-{\frac{1237\,\sqrt{2}}{3888}\arctan \left ({\frac{ \left ( 2\,{x}^{2}+2 \right ) \sqrt{2}}{4}} \right ) }-{\frac{2}{27\,{x}^{6}}}+{\frac{13}{108\,{x}^{4}}}-{\frac{13}{54\,{x}^{2}}}+{\frac{61\,\ln \left ( x \right ) }{243}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^6+3*x^4+x^2+4)/x^7/(x^4+2*x^2+3)^2,x)

[Out]

-1/486*(525/4*x^2-75/4)/(x^4+2*x^2+3)-61/972*ln(x^4+2*x^2+3)-1237/3888*2^(1/2)*arctan(1/4*(2*x^2+2)*2^(1/2))-2
/27/x^6+13/108/x^4-13/54/x^2+61/243*ln(x)

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Maxima [A]  time = 1.49926, size = 103, normalized size = 1.18 \begin{align*} -\frac{1237}{3888} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - \frac{331 \, x^{8} + 209 \, x^{6} + 360 \, x^{4} - 138 \, x^{2} + 144}{648 \,{\left (x^{10} + 2 \, x^{8} + 3 \, x^{6}\right )}} - \frac{61}{972} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) + \frac{61}{486} \, \log \left (x^{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^6+3*x^4+x^2+4)/x^7/(x^4+2*x^2+3)^2,x, algorithm="maxima")

[Out]

-1237/3888*sqrt(2)*arctan(1/2*sqrt(2)*(x^2 + 1)) - 1/648*(331*x^8 + 209*x^6 + 360*x^4 - 138*x^2 + 144)/(x^10 +
2*x^8 + 3*x^6) - 61/972*log(x^4 + 2*x^2 + 3) + 61/486*log(x^2)

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Fricas [A]  time = 1.52743, size = 317, normalized size = 3.64 \begin{align*} -\frac{1986 \, x^{8} + 1254 \, x^{6} + 2160 \, x^{4} + 1237 \, \sqrt{2}{\left (x^{10} + 2 \, x^{8} + 3 \, x^{6}\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - 828 \, x^{2} + 244 \,{\left (x^{10} + 2 \, x^{8} + 3 \, x^{6}\right )} \log \left (x^{4} + 2 \, x^{2} + 3\right ) - 976 \,{\left (x^{10} + 2 \, x^{8} + 3 \, x^{6}\right )} \log \left (x\right ) + 864}{3888 \,{\left (x^{10} + 2 \, x^{8} + 3 \, x^{6}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^6+3*x^4+x^2+4)/x^7/(x^4+2*x^2+3)^2,x, algorithm="fricas")

[Out]

-1/3888*(1986*x^8 + 1254*x^6 + 2160*x^4 + 1237*sqrt(2)*(x^10 + 2*x^8 + 3*x^6)*arctan(1/2*sqrt(2)*(x^2 + 1)) -
828*x^2 + 244*(x^10 + 2*x^8 + 3*x^6)*log(x^4 + 2*x^2 + 3) - 976*(x^10 + 2*x^8 + 3*x^6)*log(x) + 864)/(x^10 + 2
*x^8 + 3*x^6)

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Sympy [A]  time = 0.236714, size = 85, normalized size = 0.98 \begin{align*} \frac{61 \log{\left (x \right )}}{243} - \frac{61 \log{\left (x^{4} + 2 x^{2} + 3 \right )}}{972} - \frac{1237 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x^{2}}{2} + \frac{\sqrt{2}}{2} \right )}}{3888} - \frac{331 x^{8} + 209 x^{6} + 360 x^{4} - 138 x^{2} + 144}{648 x^{10} + 1296 x^{8} + 1944 x^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x**6+3*x**4+x**2+4)/x**7/(x**4+2*x**2+3)**2,x)

[Out]

61*log(x)/243 - 61*log(x**4 + 2*x**2 + 3)/972 - 1237*sqrt(2)*atan(sqrt(2)*x**2/2 + sqrt(2)/2)/3888 - (331*x**8
+ 209*x**6 + 360*x**4 - 138*x**2 + 144)/(648*x**10 + 1296*x**8 + 1944*x**6)

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Giac [A]  time = 1.09403, size = 113, normalized size = 1.3 \begin{align*} -\frac{1237}{3888} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + \frac{122 \, x^{4} - 281 \, x^{2} + 441}{1944 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac{671 \, x^{6} + 702 \, x^{4} - 351 \, x^{2} + 216}{2916 \, x^{6}} - \frac{61}{972} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) + \frac{61}{486} \, \log \left (x^{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^6+3*x^4+x^2+4)/x^7/(x^4+2*x^2+3)^2,x, algorithm="giac")

[Out]

-1237/3888*sqrt(2)*arctan(1/2*sqrt(2)*(x^2 + 1)) + 1/1944*(122*x^4 - 281*x^2 + 441)/(x^4 + 2*x^2 + 3) - 1/2916
*(671*x^6 + 702*x^4 - 351*x^2 + 216)/x^6 - 61/972*log(x^4 + 2*x^2 + 3) + 61/486*log(x^2)