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

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

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Rubi [A]  time = 0.13, antiderivative size = 93, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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}} \end {gather*}

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 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])

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 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]

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*}

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Mathematica [A]  time = 0.08, size = 73, normalized size = 0.78 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4+x^2+3 x^4+5 x^6}{x^6 \left (2+3 x^2+x^4\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.25, size = 124, normalized size = 1.33 \begin {gather*} -\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 \relax (x) + 1536}{3840 \, {\left (x^{13} + 6 \, x^{11} + 13 \, x^{9} + 12 \, x^{7} + 4 \, x^{5}\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^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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giac [A]  time = 0.36, size = 67, normalized size = 0.72 \begin {gather*} -\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 \relax (x) \end {gather*}

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)

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maple [A]  time = 0.02, size = 68, normalized size = 0.73 \begin {gather*} \frac {29 \arctan \relax (x )}{8}-\frac {2207 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{2}\right )}{256}-\frac {93}{16 x}+\frac {17}{24 x^{3}}-\frac {1}{10 x^{5}}+\frac {-\frac {43}{8} x^{3}-\frac {45}{8} x}{\left (x^{2}+1\right )^{2}}-\frac {\frac {311}{8} x^{3}+\frac {337}{4} x}{16 \left (x^{2}+2\right )^{2}} \end {gather*}

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]

-1/10/x^5+17/24/x^3-93/16/x+(-43/8*x^3-45/8*x)/(x^2+1)^2+29/8*arctan(x)-1/16*(311/8*x^3+337/4*x)/(x^2+2)^2-220
7/256*2^(1/2)*arctan(1/2*2^(1/2)*x)

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maxima [A]  time = 1.74, size = 77, normalized size = 0.83 \begin {gather*} -\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 \relax (x) \end {gather*}

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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mupad [B]  time = 0.93, size = 77, normalized size = 0.83 \begin {gather*} \frac {29\,\mathrm {atan}\relax (x)}{8}-\frac {2207\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )}{256}-\frac {\frac {1743\,x^{12}}{128}+\frac {857\,x^{10}}{12}+\frac {82159\,x^8}{640}+\frac {85351\,x^6}{960}+\frac {321\,x^4}{20}-\frac {49\,x^2}{30}+\frac {2}{5}}{x^{13}+6\,x^{11}+13\,x^9+12\,x^7+4\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(29*atan(x))/8 - (2207*2^(1/2)*atan((2^(1/2)*x)/2))/256 - ((321*x^4)/20 - (49*x^2)/30 + (85351*x^6)/960 + (821
59*x^8)/640 + (857*x^10)/12 + (1743*x^12)/128 + 2/5)/(4*x^5 + 12*x^7 + 13*x^9 + 6*x^11 + x^13)

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sympy [A]  time = 0.31, size = 82, normalized size = 0.88 \begin {gather*} \frac {29 \operatorname {atan}{\relax (x )}}{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 {gather*}

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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