3.42.12 \(\int \frac {-90+84 x-27 x^2+4 x^3+(-144+72 x-12 x^2) \log (12-6 x+x^2)}{-36 x+18 x^2-3 x^3+(108-54 x+9 x^2) \log (12-6 x+x^2)} \, dx\)

Optimal. Leaf size=20 \[ -255-\frac {4 x}{3}+\log \left (x-3 \log \left (3+(-3+x)^2\right )\right ) \]

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Rubi [A]  time = 0.36, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 72, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {6741, 12, 6728, 6684} \begin {gather*} \log \left (x-3 \log \left (x^2-6 x+12\right )\right )-\frac {4 x}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-90 + 84*x - 27*x^2 + 4*x^3 + (-144 + 72*x - 12*x^2)*Log[12 - 6*x + x^2])/(-36*x + 18*x^2 - 3*x^3 + (108
- 54*x + 9*x^2)*Log[12 - 6*x + x^2]),x]

[Out]

(-4*x)/3 + Log[x - 3*Log[12 - 6*x + x^2]]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rule 6741

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {90-84 x+27 x^2-4 x^3-\left (-144+72 x-12 x^2\right ) \log \left (12-6 x+x^2\right )}{3 \left (12-6 x+x^2\right ) \left (x-3 \log \left (12-6 x+x^2\right )\right )} \, dx\\ &=\frac {1}{3} \int \frac {90-84 x+27 x^2-4 x^3-\left (-144+72 x-12 x^2\right ) \log \left (12-6 x+x^2\right )}{\left (12-6 x+x^2\right ) \left (x-3 \log \left (12-6 x+x^2\right )\right )} \, dx\\ &=\frac {1}{3} \int \left (-4+\frac {3 \left (30-12 x+x^2\right )}{\left (12-6 x+x^2\right ) \left (x-3 \log \left (12-6 x+x^2\right )\right )}\right ) \, dx\\ &=-\frac {4 x}{3}+\int \frac {30-12 x+x^2}{\left (12-6 x+x^2\right ) \left (x-3 \log \left (12-6 x+x^2\right )\right )} \, dx\\ &=-\frac {4 x}{3}+\log \left (x-3 \log \left (12-6 x+x^2\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.64, size = 24, normalized size = 1.20 \begin {gather*} \frac {1}{3} \left (-4 x+3 \log \left (x-3 \log \left (12-6 x+x^2\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-90 + 84*x - 27*x^2 + 4*x^3 + (-144 + 72*x - 12*x^2)*Log[12 - 6*x + x^2])/(-36*x + 18*x^2 - 3*x^3 +
(108 - 54*x + 9*x^2)*Log[12 - 6*x + x^2]),x]

[Out]

(-4*x + 3*Log[x - 3*Log[12 - 6*x + x^2]])/3

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fricas [A]  time = 0.75, size = 20, normalized size = 1.00 \begin {gather*} -\frac {4}{3} \, x + \log \left (-x + 3 \, \log \left (x^{2} - 6 \, x + 12\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x^2+72*x-144)*log(x^2-6*x+12)+4*x^3-27*x^2+84*x-90)/((9*x^2-54*x+108)*log(x^2-6*x+12)-3*x^3+18
*x^2-36*x),x, algorithm="fricas")

[Out]

-4/3*x + log(-x + 3*log(x^2 - 6*x + 12))

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giac [A]  time = 1.32, size = 18, normalized size = 0.90 \begin {gather*} -\frac {4}{3} \, x + \log \left (x - 3 \, \log \left (x^{2} - 6 \, x + 12\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x^2+72*x-144)*log(x^2-6*x+12)+4*x^3-27*x^2+84*x-90)/((9*x^2-54*x+108)*log(x^2-6*x+12)-3*x^3+18
*x^2-36*x),x, algorithm="giac")

[Out]

-4/3*x + log(x - 3*log(x^2 - 6*x + 12))

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maple [A]  time = 0.03, size = 19, normalized size = 0.95

method	result	size
norman	\(-\frac {4 x}{3}+\ln \left (x -3 \ln \left (x^{2}-6 x +12\right )\right )\)	\(19\)
risch	\(-\frac {4 x}{3}+\ln \left (\ln \left (x^{2}-6 x +12\right )-\frac {x}{3}\right )\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-12*x^2+72*x-144)*ln(x^2-6*x+12)+4*x^3-27*x^2+84*x-90)/((9*x^2-54*x+108)*ln(x^2-6*x+12)-3*x^3+18*x^2-36*
x),x,method=_RETURNVERBOSE)

[Out]

-4/3*x+ln(x-3*ln(x^2-6*x+12))

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maxima [A]  time = 0.40, size = 18, normalized size = 0.90 \begin {gather*} -\frac {4}{3} \, x + \log \left (-\frac {1}{3} \, x + \log \left (x^{2} - 6 \, x + 12\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x^2+72*x-144)*log(x^2-6*x+12)+4*x^3-27*x^2+84*x-90)/((9*x^2-54*x+108)*log(x^2-6*x+12)-3*x^3+18
*x^2-36*x),x, algorithm="maxima")

[Out]

-4/3*x + log(-1/3*x + log(x^2 - 6*x + 12))

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mupad [B]  time = 0.31, size = 18, normalized size = 0.90 \begin {gather*} \ln \left (\ln \left (x^2-6\,x+12\right )-\frac {x}{3}\right )-\frac {4\,x}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x^2 - 6*x + 12)*(12*x^2 - 72*x + 144) - 84*x + 27*x^2 - 4*x^3 + 90)/(36*x - log(x^2 - 6*x + 12)*(9*x^
2 - 54*x + 108) - 18*x^2 + 3*x^3),x)

[Out]

log(log(x^2 - 6*x + 12) - x/3) - (4*x)/3

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sympy [A]  time = 0.27, size = 19, normalized size = 0.95 \begin {gather*} - \frac {4 x}{3} + \log {\left (- \frac {x}{3} + \log {\left (x^{2} - 6 x + 12 \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x**2+72*x-144)*ln(x**2-6*x+12)+4*x**3-27*x**2+84*x-90)/((9*x**2-54*x+108)*ln(x**2-6*x+12)-3*x*
*3+18*x**2-36*x),x)

[Out]

-4*x/3 + log(-x/3 + log(x**2 - 6*x + 12))

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