3.96.2 \(\int \frac {5832-3420 x+668 x^2-58 x^3+2 x^4+(-738 x+206 x^2-27 x^3+x^4) \log (\frac {1}{x^2})}{(2916 x-1710 x^2+334 x^3-29 x^4+x^5) \log (\frac {1}{x^2})} \, dx\)

Optimal. Leaf size=26 \[ \log \left (\frac {-4+x+\frac {x (1+2 x)}{(-9+x)^2}}{4 \log \left (\frac {1}{x^2}\right )}\right ) \]

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Rubi [A]  time = 0.21, antiderivative size = 32, normalized size of antiderivative = 1.23, number of steps used = 7, number of rules used = 5, integrand size = 73, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {6688, 2074, 1587, 2302, 29} \begin {gather*} -\log \left (\log \left (\frac {1}{x^2}\right )\right )+\log \left (-x^3+20 x^2-154 x+324\right )-2 \log (9-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5832 - 3420*x + 668*x^2 - 58*x^3 + 2*x^4 + (-738*x + 206*x^2 - 27*x^3 + x^4)*Log[x^(-2)])/((2916*x - 1710
*x^2 + 334*x^3 - 29*x^4 + x^5)*Log[x^(-2)]),x]

[Out]

-2*Log[9 - x] + Log[324 - 154*x + 20*x^2 - x^3] - Log[Log[x^(-2)]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

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]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-738+206 x-27 x^2+x^3}{2916-1710 x+334 x^2-29 x^3+x^4}+\frac {2}{x \log \left (\frac {1}{x^2}\right )}\right ) \, dx\\ &=2 \int \frac {1}{x \log \left (\frac {1}{x^2}\right )} \, dx+\int \frac {-738+206 x-27 x^2+x^3}{2916-1710 x+334 x^2-29 x^3+x^4} \, dx\\ &=\int \left (-\frac {2}{-9+x}+\frac {154-40 x+3 x^2}{-324+154 x-20 x^2+x^3}\right ) \, dx-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (\frac {1}{x^2}\right )\right )\\ &=-2 \log (9-x)-\log \left (\log \left (\frac {1}{x^2}\right )\right )+\int \frac {154-40 x+3 x^2}{-324+154 x-20 x^2+x^3} \, dx\\ &=-2 \log (9-x)+\log \left (324-154 x+20 x^2-x^3\right )-\log \left (\log \left (\frac {1}{x^2}\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 32, normalized size = 1.23 \begin {gather*} -2 \log (9-x)+\log \left (324-154 x+20 x^2-x^3\right )-\log \left (\log \left (\frac {1}{x^2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5832 - 3420*x + 668*x^2 - 58*x^3 + 2*x^4 + (-738*x + 206*x^2 - 27*x^3 + x^4)*Log[x^(-2)])/((2916*x
- 1710*x^2 + 334*x^3 - 29*x^4 + x^5)*Log[x^(-2)]),x]

[Out]

-2*Log[9 - x] + Log[324 - 154*x + 20*x^2 - x^3] - Log[Log[x^(-2)]]

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fricas [A]  time = 0.61, size = 28, normalized size = 1.08 \begin {gather*} \log \left (x^{3} - 20 \, x^{2} + 154 \, x - 324\right ) - 2 \, \log \left (x - 9\right ) - \log \left (\log \left (\frac {1}{x^{2}}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^4-27*x^3+206*x^2-738*x)*log(1/x^2)+2*x^4-58*x^3+668*x^2-3420*x+5832)/(x^5-29*x^4+334*x^3-1710*x^
2+2916*x)/log(1/x^2),x, algorithm="fricas")

[Out]

log(x^3 - 20*x^2 + 154*x - 324) - 2*log(x - 9) - log(log(x^(-2)))

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giac [A]  time = 0.25, size = 28, normalized size = 1.08 \begin {gather*} \log \left (x^{3} - 20 \, x^{2} + 154 \, x - 324\right ) - 2 \, \log \left (x - 9\right ) - \log \left (\log \left (x^{2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^4-27*x^3+206*x^2-738*x)*log(1/x^2)+2*x^4-58*x^3+668*x^2-3420*x+5832)/(x^5-29*x^4+334*x^3-1710*x^
2+2916*x)/log(1/x^2),x, algorithm="giac")

[Out]

log(x^3 - 20*x^2 + 154*x - 324) - 2*log(x - 9) - log(log(x^2))

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maple [A]  time = 0.13, size = 29, normalized size = 1.12

method	result	size
norman	\(-\ln \left (\ln \left (\frac {1}{x^{2}}\right )\right )-2 \ln \left (x -9\right )+\ln \left (x^{3}-20 x^{2}+154 x -324\right )\)	\(29\)
risch	\(-\ln \left (\ln \left (\frac {1}{x^{2}}\right )\right )-2 \ln \left (x -9\right )+\ln \left (x^{3}-20 x^{2}+154 x -324\right )\)	\(29\)
derivativedivides	\(-\ln \left (\ln \left (\frac {1}{x^{2}}\right )\right )-\ln \left (\frac {1}{x}\right )-2 \ln \left (\frac {9}{x}-1\right )+\ln \left (\frac {324}{x^{3}}-\frac {154}{x^{2}}+\frac {20}{x}-1\right )\)	\(43\)
default	\(-\ln \left (\ln \left (\frac {1}{x^{2}}\right )\right )-\ln \left (\frac {1}{x}\right )-2 \ln \left (\frac {9}{x}-1\right )+\ln \left (\frac {324}{x^{3}}-\frac {154}{x^{2}}+\frac {20}{x}-1\right )\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4-27*x^3+206*x^2-738*x)*ln(1/x^2)+2*x^4-58*x^3+668*x^2-3420*x+5832)/(x^5-29*x^4+334*x^3-1710*x^2+2916*
x)/ln(1/x^2),x,method=_RETURNVERBOSE)

[Out]

-ln(ln(1/x^2))-2*ln(x-9)+ln(x^3-20*x^2+154*x-324)

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maxima [A]  time = 0.55, size = 26, normalized size = 1.00 \begin {gather*} \log \left (x^{3} - 20 \, x^{2} + 154 \, x - 324\right ) - 2 \, \log \left (x - 9\right ) - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^4-27*x^3+206*x^2-738*x)*log(1/x^2)+2*x^4-58*x^3+668*x^2-3420*x+5832)/(x^5-29*x^4+334*x^3-1710*x^
2+2916*x)/log(1/x^2),x, algorithm="maxima")

[Out]

log(x^3 - 20*x^2 + 154*x - 324) - 2*log(x - 9) - log(log(x))

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mupad [B]  time = 7.68, size = 28, normalized size = 1.08 \begin {gather*} \ln \left (x^3-20\,x^2+154\,x-324\right )-\ln \left (\ln \left (\frac {1}{x^2}\right )\right )-2\,\ln \left (x-9\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3420*x + log(1/x^2)*(738*x - 206*x^2 + 27*x^3 - x^4) - 668*x^2 + 58*x^3 - 2*x^4 - 5832)/(log(1/x^2)*(291
6*x - 1710*x^2 + 334*x^3 - 29*x^4 + x^5)),x)

[Out]

log(154*x - 20*x^2 + x^3 - 324) - log(log(1/x^2)) - 2*log(x - 9)

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sympy [A]  time = 0.17, size = 29, normalized size = 1.12 \begin {gather*} - 2 \log {\left (x - 9 \right )} + \log {\left (x^{3} - 20 x^{2} + 154 x - 324 \right )} - \log {\left (\log {\left (\frac {1}{x^{2}} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**4-27*x**3+206*x**2-738*x)*ln(1/x**2)+2*x**4-58*x**3+668*x**2-3420*x+5832)/(x**5-29*x**4+334*x**
3-1710*x**2+2916*x)/ln(1/x**2),x)

[Out]

-2*log(x - 9) + log(x**3 - 20*x**2 + 154*x - 324) - log(log(x**(-2)))

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