3.1.59 \(\int \frac {-48-40 x^2+96 x^4-40 x^6+14 x^8}{24 x+x^2-20 x^3+16 x^5-4 x^7+x^9} \, dx\)

Optimal. Leaf size=26 \[ \log \left (\frac {\left (-12+x+4 x^2+\left (5+\left (-1+x^2\right )^2\right )^2\right )^2}{x^2}\right ) \]

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Rubi [A]  time = 0.11, antiderivative size = 29, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 2, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2074, 1587} \begin {gather*} 2 \log \left (x^8-4 x^6+16 x^4-20 x^2+x+24\right )-2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-48 - 40*x^2 + 96*x^4 - 40*x^6 + 14*x^8)/(24*x + x^2 - 20*x^3 + 16*x^5 - 4*x^7 + x^9),x]

[Out]

-2*Log[x] + 2*Log[24 + x - 20*x^2 + 16*x^4 - 4*x^6 + x^8]

Rule 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2}{x}+\frac {2 \left (1-40 x+64 x^3-24 x^5+8 x^7\right )}{24+x-20 x^2+16 x^4-4 x^6+x^8}\right ) \, dx\\ &=-2 \log (x)+2 \int \frac {1-40 x+64 x^3-24 x^5+8 x^7}{24+x-20 x^2+16 x^4-4 x^6+x^8} \, dx\\ &=-2 \log (x)+2 \log \left (24+x-20 x^2+16 x^4-4 x^6+x^8\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 29, normalized size = 1.12 \begin {gather*} 2 \left (-\log (x)+\log \left (24+x-20 x^2+16 x^4-4 x^6+x^8\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-48 - 40*x^2 + 96*x^4 - 40*x^6 + 14*x^8)/(24*x + x^2 - 20*x^3 + 16*x^5 - 4*x^7 + x^9),x]

[Out]

2*(-Log[x] + Log[24 + x - 20*x^2 + 16*x^4 - 4*x^6 + x^8])

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fricas [A]  time = 0.54, size = 29, normalized size = 1.12 \begin {gather*} 2 \, \log \left (x^{8} - 4 \, x^{6} + 16 \, x^{4} - 20 \, x^{2} + x + 24\right ) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((14*x^8-40*x^6+96*x^4-40*x^2-48)/(x^9-4*x^7+16*x^5-20*x^3+x^2+24*x),x, algorithm="fricas")

[Out]

2*log(x^8 - 4*x^6 + 16*x^4 - 20*x^2 + x + 24) - 2*log(x)

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giac [A]  time = 0.45, size = 30, normalized size = 1.15 \begin {gather*} 2 \, \log \left (x^{8} - 4 \, x^{6} + 16 \, x^{4} - 20 \, x^{2} + x + 24\right ) - 2 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((14*x^8-40*x^6+96*x^4-40*x^2-48)/(x^9-4*x^7+16*x^5-20*x^3+x^2+24*x),x, algorithm="giac")

[Out]

2*log(x^8 - 4*x^6 + 16*x^4 - 20*x^2 + x + 24) - 2*log(abs(x))

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maple [A]  time = 0.03, size = 30, normalized size = 1.15

method	result	size
default	\(2 \ln \left (x^{8}-4 x^{6}+16 x^{4}-20 x^{2}+x +24\right )-2 \ln \relax (x )\)	\(30\)
norman	\(2 \ln \left (x^{8}-4 x^{6}+16 x^{4}-20 x^{2}+x +24\right )-2 \ln \relax (x )\)	\(30\)
risch	\(2 \ln \left (x^{8}-4 x^{6}+16 x^{4}-20 x^{2}+x +24\right )-2 \ln \relax (x )\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((14*x^8-40*x^6+96*x^4-40*x^2-48)/(x^9-4*x^7+16*x^5-20*x^3+x^2+24*x),x,method=_RETURNVERBOSE)

[Out]

2*ln(x^8-4*x^6+16*x^4-20*x^2+x+24)-2*ln(x)

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maxima [A]  time = 0.80, size = 29, normalized size = 1.12 \begin {gather*} 2 \, \log \left (x^{8} - 4 \, x^{6} + 16 \, x^{4} - 20 \, x^{2} + x + 24\right ) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((14*x^8-40*x^6+96*x^4-40*x^2-48)/(x^9-4*x^7+16*x^5-20*x^3+x^2+24*x),x, algorithm="maxima")

[Out]

2*log(x^8 - 4*x^6 + 16*x^4 - 20*x^2 + x + 24) - 2*log(x)

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mupad [B]  time = 0.31, size = 29, normalized size = 1.12 \begin {gather*} 2\,\ln \left (x^8-4\,x^6+16\,x^4-20\,x^2+x+24\right )-2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(40*x^2 - 96*x^4 + 40*x^6 - 14*x^8 + 48)/(24*x + x^2 - 20*x^3 + 16*x^5 - 4*x^7 + x^9),x)

[Out]

2*log(x - 20*x^2 + 16*x^4 - 4*x^6 + x^8 + 24) - 2*log(x)

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sympy [A]  time = 0.13, size = 29, normalized size = 1.12 \begin {gather*} - 2 \log {\relax (x )} + 2 \log {\left (x^{8} - 4 x^{6} + 16 x^{4} - 20 x^{2} + x + 24 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((14*x**8-40*x**6+96*x**4-40*x**2-48)/(x**9-4*x**7+16*x**5-20*x**3+x**2+24*x),x)

[Out]

-2*log(x) + 2*log(x**8 - 4*x**6 + 16*x**4 - 20*x**2 + x + 24)

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