3.1.88 \(\int \frac {4+x^2+3 x^4+5 x^6}{x^8 (2+3 x^2+x^4)^2} \, dx\)

Optimal. Leaf size=76 \[ -\frac {1}{7 x^7}+\frac {11}{20 x^5}-\frac {23}{12 x^3}+\frac {x \left (3 x^2+19\right )}{32 \left (x^4+3 x^2+2\right )}+\frac {137}{16 x}+\frac {25}{2} \tan ^{-1}(x)-\frac {123 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{32 \sqrt {2}} \]

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Rubi [A]  time = 0.10, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 5, 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 x^2+19\right )}{32 \left (x^4+3 x^2+2\right )}-\frac {23}{12 x^3}+\frac {11}{20 x^5}-\frac {1}{7 x^7}+\frac {137}{16 x}+\frac {25}{2} \tan ^{-1}(x)-\frac {123 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{32 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 + x^2 + 3*x^4 + 5*x^6)/(x^8*(2 + 3*x^2 + x^4)^2),x]

[Out]

-1/(7*x^7) + 11/(20*x^5) - 23/(12*x^3) + 137/(16*x) + (x*(19 + 3*x^2))/(32*(2 + 3*x^2 + x^4)) + (25*ArcTan[x])
/2 - (123*ArcTan[x/Sqrt[2]])/(32*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^8 \left (2+3 x^2+x^4\right )^2} \, dx &=\frac {x \left (19+3 x^2\right )}{32 \left (2+3 x^2+x^4\right )}-\frac {1}{4} \int \frac {-8+10 x^2-17 x^4+\frac {21 x^6}{2}-\frac {39 x^8}{8}-\frac {3 x^{10}}{8}}{x^8 \left (2+3 x^2+x^4\right )} \, dx\\ &=\frac {x \left (19+3 x^2\right )}{32 \left (2+3 x^2+x^4\right )}-\frac {1}{4} \int \left (-\frac {4}{x^8}+\frac {11}{x^6}-\frac {23}{x^4}+\frac {137}{4 x^2}-\frac {50}{1+x^2}+\frac {123}{8 \left (2+x^2\right )}\right ) \, dx\\ &=-\frac {1}{7 x^7}+\frac {11}{20 x^5}-\frac {23}{12 x^3}+\frac {137}{16 x}+\frac {x \left (19+3 x^2\right )}{32 \left (2+3 x^2+x^4\right )}-\frac {123}{32} \int \frac {1}{2+x^2} \, dx+\frac {25}{2} \int \frac {1}{1+x^2} \, dx\\ &=-\frac {1}{7 x^7}+\frac {11}{20 x^5}-\frac {23}{12 x^3}+\frac {137}{16 x}+\frac {x \left (19+3 x^2\right )}{32 \left (2+3 x^2+x^4\right )}+\frac {25}{2} \tan ^{-1}(x)-\frac {123 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{32 \sqrt {2}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 77, normalized size = 1.01 \begin {gather*} -\frac {1}{7 x^7}+\frac {11}{20 x^5}-\frac {23}{12 x^3}+\frac {3 x^3+19 x}{32 \left (x^4+3 x^2+2\right )}+\frac {137}{16 x}+\frac {25}{2} \tan ^{-1}(x)-\frac {123 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{32 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 + x^2 + 3*x^4 + 5*x^6)/(x^8*(2 + 3*x^2 + x^4)^2),x]

[Out]

-1/7*1/x^7 + 11/(20*x^5) - 23/(12*x^3) + 137/(16*x) + (19*x + 3*x^3)/(32*(2 + 3*x^2 + x^4)) + (25*ArcTan[x])/2
- (123*ArcTan[x/Sqrt[2]])/(32*Sqrt[2])

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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^8 \left (2+3 x^2+x^4\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(4 + x^2 + 3*x^4 + 5*x^6)/(x^8*(2 + 3*x^2 + x^4)^2),x]

[Out]

IntegrateAlgebraic[(4 + x^2 + 3*x^4 + 5*x^6)/(x^8*(2 + 3*x^2 + x^4)^2), x]

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fricas [A]  time = 1.30, size = 89, normalized size = 1.17 \begin {gather*} \frac {58170 \, x^{10} + 163730 \, x^{8} + 80136 \, x^{6} - 15632 \, x^{4} - 12915 \, \sqrt {2} {\left (x^{11} + 3 \, x^{9} + 2 \, x^{7}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + 4512 \, x^{2} + 84000 \, {\left (x^{11} + 3 \, x^{9} + 2 \, x^{7}\right )} \arctan \relax (x) - 1920}{6720 \, {\left (x^{11} + 3 \, x^{9} + 2 \, x^{7}\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^8/(x^4+3*x^2+2)^2,x, algorithm="fricas")

[Out]

1/6720*(58170*x^10 + 163730*x^8 + 80136*x^6 - 15632*x^4 - 12915*sqrt(2)*(x^11 + 3*x^9 + 2*x^7)*arctan(1/2*sqrt
(2)*x) + 4512*x^2 + 84000*(x^11 + 3*x^9 + 2*x^7)*arctan(x) - 1920)/(x^11 + 3*x^9 + 2*x^7)

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giac [A]  time = 0.45, size = 62, normalized size = 0.82 \begin {gather*} -\frac {123}{64} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + \frac {3 \, x^{3} + 19 \, x}{32 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} + \frac {14385 \, x^{6} - 3220 \, x^{4} + 924 \, x^{2} - 240}{1680 \, x^{7}} + \frac {25}{2} \, \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^8/(x^4+3*x^2+2)^2,x, algorithm="giac")

[Out]

-123/64*sqrt(2)*arctan(1/2*sqrt(2)*x) + 1/32*(3*x^3 + 19*x)/(x^4 + 3*x^2 + 2) + 1/1680*(14385*x^6 - 3220*x^4 +
924*x^2 - 240)/x^7 + 25/2*arctan(x)

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maple [A]  time = 0.02, size = 58, normalized size = 0.76 \begin {gather*} \frac {x}{2 x^{2}+2}-\frac {13 x}{32 \left (x^{2}+2\right )}+\frac {25 \arctan \relax (x )}{2}-\frac {123 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{2}\right )}{64}+\frac {137}{16 x}-\frac {23}{12 x^{3}}+\frac {11}{20 x^{5}}-\frac {1}{7 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^6+3*x^4+x^2+4)/x^8/(x^4+3*x^2+2)^2,x)

[Out]

-1/7/x^7+11/20/x^5-23/12/x^3+137/16/x+1/2/(x^2+1)*x+25/2*arctan(x)-13/32/(x^2+2)*x-123/64*2^(1/2)*arctan(1/2*2
^(1/2)*x)

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maxima [A]  time = 1.53, size = 62, normalized size = 0.82 \begin {gather*} -\frac {123}{64} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + \frac {29085 \, x^{10} + 81865 \, x^{8} + 40068 \, x^{6} - 7816 \, x^{4} + 2256 \, x^{2} - 960}{3360 \, {\left (x^{11} + 3 \, x^{9} + 2 \, x^{7}\right )}} + \frac {25}{2} \, \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^8/(x^4+3*x^2+2)^2,x, algorithm="maxima")

[Out]

-123/64*sqrt(2)*arctan(1/2*sqrt(2)*x) + 1/3360*(29085*x^10 + 81865*x^8 + 40068*x^6 - 7816*x^4 + 2256*x^2 - 960
)/(x^11 + 3*x^9 + 2*x^7) + 25/2*arctan(x)

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mupad [B]  time = 0.07, size = 61, normalized size = 0.80 \begin {gather*} \frac {25\,\mathrm {atan}\relax (x)}{2}-\frac {123\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )}{64}+\frac {\frac {277\,x^{10}}{32}+\frac {2339\,x^8}{96}+\frac {477\,x^6}{40}-\frac {977\,x^4}{420}+\frac {47\,x^2}{70}-\frac {2}{7}}{x^{11}+3\,x^9+2\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 + 3*x^4 + 5*x^6 + 4)/(x^8*(3*x^2 + x^4 + 2)^2),x)

[Out]

(25*atan(x))/2 - (123*2^(1/2)*atan((2^(1/2)*x)/2))/64 + ((47*x^2)/70 - (977*x^4)/420 + (477*x^6)/40 + (2339*x^
8)/96 + (277*x^10)/32 - 2/7)/(2*x^7 + 3*x^9 + x^11)

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sympy [A]  time = 0.28, size = 66, normalized size = 0.87 \begin {gather*} \frac {25 \operatorname {atan}{\relax (x )}}{2} - \frac {123 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )}}{64} + \frac {29085 x^{10} + 81865 x^{8} + 40068 x^{6} - 7816 x^{4} + 2256 x^{2} - 960}{3360 x^{11} + 10080 x^{9} + 6720 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x**6+3*x**4+x**2+4)/x**8/(x**4+3*x**2+2)**2,x)

[Out]

25*atan(x)/2 - 123*sqrt(2)*atan(sqrt(2)*x/2)/64 + (29085*x**10 + 81865*x**8 + 40068*x**6 - 7816*x**4 + 2256*x*
*2 - 960)/(3360*x**11 + 10080*x**9 + 6720*x**7)

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