### 3.99 $$\int \frac{4+x^2+3 x^4+5 x^6}{x^6 (2+3 x^2+x^4)^3} \, dx$$

Optimal. Leaf size=93 $-\frac{x \left (3-5 x^2\right )}{32 \left (x^4+3 x^2+2\right )^2}-\frac{x \left (999 x^2+1771\right )}{128 \left (x^4+3 x^2+2\right )}+\frac{17}{24 x^3}-\frac{1}{10 x^5}-\frac{93}{16 x}+\frac{29}{8} \tan ^{-1}(x)-\frac{2207 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{128 \sqrt{2}}$

[Out]

-1/(10*x^5) + 17/(24*x^3) - 93/(16*x) - (x*(3 - 5*x^2))/(32*(2 + 3*x^2 + x^4)^2) - (x*(1771 + 999*x^2))/(128*(
2 + 3*x^2 + x^4)) + (29*ArcTan[x])/8 - (2207*ArcTan[x/Sqrt[2]])/(128*Sqrt[2])

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Rubi [A]  time = 0.134206, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 31, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.097, Rules used = {1669, 1664, 203} $-\frac{x \left (3-5 x^2\right )}{32 \left (x^4+3 x^2+2\right )^2}-\frac{x \left (999 x^2+1771\right )}{128 \left (x^4+3 x^2+2\right )}+\frac{17}{24 x^3}-\frac{1}{10 x^5}-\frac{93}{16 x}+\frac{29}{8} \tan ^{-1}(x)-\frac{2207 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{128 \sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-1/(10*x^5) + 17/(24*x^3) - 93/(16*x) - (x*(3 - 5*x^2))/(32*(2 + 3*x^2 + x^4)^2) - (x*(1771 + 999*x^2))/(128*(
2 + 3*x^2 + x^4)) + (29*ArcTan[x])/8 - (2207*ArcTan[x/Sqrt[2]])/(128*Sqrt[2])

Rule 1669

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

Rule 1664

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

Rule 203

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

Rubi steps

\begin{align*} \int \frac{4+x^2+3 x^4+5 x^6}{x^6 \left (2+3 x^2+x^4\right )^3} \, dx &=-\frac{x \left (3-5 x^2\right )}{32 \left (2+3 x^2+x^4\right )^2}-\frac{1}{8} \int \frac{-16+20 x^2-34 x^4+\frac{81 x^6}{4}-\frac{25 x^8}{4}}{x^6 \left (2+3 x^2+x^4\right )^2} \, dx\\ &=-\frac{x \left (3-5 x^2\right )}{32 \left (2+3 x^2+x^4\right )^2}-\frac{x \left (1771+999 x^2\right )}{128 \left (2+3 x^2+x^4\right )}+\frac{1}{32} \int \frac{32-88 x^2+184 x^4+\frac{681 x^6}{4}-\frac{999 x^8}{4}}{x^6 \left (2+3 x^2+x^4\right )} \, dx\\ &=-\frac{x \left (3-5 x^2\right )}{32 \left (2+3 x^2+x^4\right )^2}-\frac{x \left (1771+999 x^2\right )}{128 \left (2+3 x^2+x^4\right )}+\frac{1}{32} \int \left (\frac{16}{x^6}-\frac{68}{x^4}+\frac{186}{x^2}+\frac{116}{1+x^2}-\frac{2207}{4 \left (2+x^2\right )}\right ) \, dx\\ &=-\frac{1}{10 x^5}+\frac{17}{24 x^3}-\frac{93}{16 x}-\frac{x \left (3-5 x^2\right )}{32 \left (2+3 x^2+x^4\right )^2}-\frac{x \left (1771+999 x^2\right )}{128 \left (2+3 x^2+x^4\right )}+\frac{29}{8} \int \frac{1}{1+x^2} \, dx-\frac{2207}{128} \int \frac{1}{2+x^2} \, dx\\ &=-\frac{1}{10 x^5}+\frac{17}{24 x^3}-\frac{93}{16 x}-\frac{x \left (3-5 x^2\right )}{32 \left (2+3 x^2+x^4\right )^2}-\frac{x \left (1771+999 x^2\right )}{128 \left (2+3 x^2+x^4\right )}+\frac{29}{8} \tan ^{-1}(x)-\frac{2207 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{128 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.0788035, size = 73, normalized size = 0.78 $\frac{-\frac{2 \left (26145 x^{12}+137120 x^{10}+246477 x^8+170702 x^6+30816 x^4-3136 x^2+768\right )}{x^5 \left (x^4+3 x^2+2\right )^2}+13920 \tan ^{-1}(x)-33105 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{3840}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((-2*(768 - 3136*x^2 + 30816*x^4 + 170702*x^6 + 246477*x^8 + 137120*x^10 + 26145*x^12))/(x^5*(2 + 3*x^2 + x^4)
^2) + 13920*ArcTan[x] - 33105*Sqrt[2]*ArcTan[x/Sqrt[2]])/3840

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Maple [A]  time = 0.015, size = 68, normalized size = 0.7 \begin{align*} -{\frac{1}{16\, \left ({x}^{2}+2 \right ) ^{2}} \left ({\frac{311\,{x}^{3}}{8}}+{\frac{337\,x}{4}} \right ) }-{\frac{2207\,\sqrt{2}}{256}\arctan \left ({\frac{x\sqrt{2}}{2}} \right ) }+{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ( -{\frac{43\,{x}^{3}}{8}}-{\frac{45\,x}{8}} \right ) }+{\frac{29\,\arctan \left ( x \right ) }{8}}-{\frac{1}{10\,{x}^{5}}}+{\frac{17}{24\,{x}^{3}}}-{\frac{93}{16\,x}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(311/8*x^3+337/4*x)/(x^2+2)^2-2207/256*arctan(1/2*x*2^(1/2))*2^(1/2)+(-43/8*x^3-45/8*x)/(x^2+1)^2+29/8*a
rctan(x)-1/10/x^5+17/24/x^3-93/16/x

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Maxima [A]  time = 1.47447, size = 104, normalized size = 1.12 \begin{align*} -\frac{2207}{256} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{26145 \, x^{12} + 137120 \, x^{10} + 246477 \, x^{8} + 170702 \, x^{6} + 30816 \, x^{4} - 3136 \, x^{2} + 768}{1920 \,{\left (x^{13} + 6 \, x^{11} + 13 \, x^{9} + 12 \, x^{7} + 4 \, x^{5}\right )}} + \frac{29}{8} \, \arctan \left (x\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^6/(x^4+3*x^2+2)^3,x, algorithm="maxima")

[Out]

-2207/256*sqrt(2)*arctan(1/2*sqrt(2)*x) - 1/1920*(26145*x^12 + 137120*x^10 + 246477*x^8 + 170702*x^6 + 30816*x
^4 - 3136*x^2 + 768)/(x^13 + 6*x^11 + 13*x^9 + 12*x^7 + 4*x^5) + 29/8*arctan(x)

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Fricas [A]  time = 1.56934, size = 370, normalized size = 3.98 \begin{align*} -\frac{52290 \, x^{12} + 274240 \, x^{10} + 492954 \, x^{8} + 341404 \, x^{6} + 61632 \, x^{4} + 33105 \, \sqrt{2}{\left (x^{13} + 6 \, x^{11} + 13 \, x^{9} + 12 \, x^{7} + 4 \, x^{5}\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - 6272 \, x^{2} - 13920 \,{\left (x^{13} + 6 \, x^{11} + 13 \, x^{9} + 12 \, x^{7} + 4 \, x^{5}\right )} \arctan \left (x\right ) + 1536}{3840 \,{\left (x^{13} + 6 \, x^{11} + 13 \, x^{9} + 12 \, x^{7} + 4 \, x^{5}\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^6/(x^4+3*x^2+2)^3,x, algorithm="fricas")

[Out]

-1/3840*(52290*x^12 + 274240*x^10 + 492954*x^8 + 341404*x^6 + 61632*x^4 + 33105*sqrt(2)*(x^13 + 6*x^11 + 13*x^
9 + 12*x^7 + 4*x^5)*arctan(1/2*sqrt(2)*x) - 6272*x^2 - 13920*(x^13 + 6*x^11 + 13*x^9 + 12*x^7 + 4*x^5)*arctan(
x) + 1536)/(x^13 + 6*x^11 + 13*x^9 + 12*x^7 + 4*x^5)

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Sympy [A]  time = 0.300799, size = 82, normalized size = 0.88 \begin{align*} \frac{29 \operatorname{atan}{\left (x \right )}}{8} - \frac{2207 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{256} - \frac{26145 x^{12} + 137120 x^{10} + 246477 x^{8} + 170702 x^{6} + 30816 x^{4} - 3136 x^{2} + 768}{1920 x^{13} + 11520 x^{11} + 24960 x^{9} + 23040 x^{7} + 7680 x^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

29*atan(x)/8 - 2207*sqrt(2)*atan(sqrt(2)*x/2)/256 - (26145*x**12 + 137120*x**10 + 246477*x**8 + 170702*x**6 +
30816*x**4 - 3136*x**2 + 768)/(1920*x**13 + 11520*x**11 + 24960*x**9 + 23040*x**7 + 7680*x**5)

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Giac [A]  time = 1.12132, size = 90, normalized size = 0.97 \begin{align*} -\frac{2207}{256} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{999 \, x^{7} + 4768 \, x^{5} + 7291 \, x^{3} + 3554 \, x}{128 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} - \frac{1395 \, x^{4} - 170 \, x^{2} + 24}{240 \, x^{5}} + \frac{29}{8} \, \arctan \left (x\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^6/(x^4+3*x^2+2)^3,x, algorithm="giac")

[Out]

-2207/256*sqrt(2)*arctan(1/2*sqrt(2)*x) - 1/128*(999*x^7 + 4768*x^5 + 7291*x^3 + 3554*x)/(x^4 + 3*x^2 + 2)^2 -
1/240*(1395*x^4 - 170*x^2 + 24)/x^5 + 29/8*arctan(x)