3.49.68 $\int \frac{-648+474x-60x^2+2x^3+(288-72x+4x^2)\log (\frac{1}{3}(-12+x))+(-24+2x){\log }^2(\frac{1}{3}(-12+x))}{-972+297x-30x^2+x^3+(216-42x+2x^2)\log (\frac{1}{3}(-12+x))+(-12+x){\log }^2(\frac{1}{3}(-12+x))}dx$

Optimal. Leaf size=20 $$2x+\frac{12x}{-9+x+\log (-4+\frac{x}{3})}$$

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Rubi [F]  time = 0.61, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.000, Rules used = {} $$\begin{array}{c}\int \frac{-648+474x-60x^2+2x^3+(288-72x+4x^2)\log (\frac{1}{3}(-12+x))+(-24+2x){\log }^2(\frac{1}{3}(-12+x))}{-972+297x-30x^2+x^3+(216-42x+2x^2)\log (\frac{1}{3}(-12+x))+(-12+x){\log }^2(\frac{1}{3}(-12+x))}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(-648 + 474*x - 60*x^2 + 2*x^3 + (288 - 72*x + 4*x^2)*Log[(-12 + x)/3] + (-24 + 2*x)*Log[(-12 + x)/3]^2)/(
-972 + 297*x - 30*x^2 + x^3 + (216 - 42*x + 2*x^2)*Log[(-12 + x)/3] + (-12 + x)*Log[(-12 + x)/3]^2),x]

[Out]

2*x - 12*Defer[Int][(-9 + x + Log[-4 + x/3])^(-2), x] - 144*Defer[Int][1/((-12 + x)*(-9 + x + Log[-4 + x/3])^2
), x] - 12*Defer[Int][x/(-9 + x + Log[-4 + x/3])^2, x] + 12*Defer[Int][(-9 + x + Log[-4 + x/3])^(-1), x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{2(324-237x+30x^2-x^3-2(72-18x+x^2)\log (-4+\frac{x}{3})-(-12+x){\log }^2(-4+\frac{x}{3}))}{(12-x){(9-x-\log (-4+\frac{x}{3}))}^2}dx\\ & =2\int \frac{324-237x+30x^2-x^3-2(72-18x+x^2)\log (-4+\frac{x}{3})-(-12+x){\log }^2(-4+\frac{x}{3})}{(12-x){(9-x-\log (-4+\frac{x}{3}))}^2}dx\\ & =2\int (1-\frac{6(-11+x)x}{(-12+x){(-9+x+\log (-4+\frac{x}{3}))}^2}+\frac{6}{-9+x+\log (-4+\frac{x}{3})})dx\\ & =2x-12\int \frac{(-11+x)x}{(-12+x){(-9+x+\log (-4+\frac{x}{3}))}^2}dx+12\int \frac{1}{-9+x+\log (-4+\frac{x}{3})}dx\\ & =2x+12\int \frac{1}{-9+x+\log (-4+\frac{x}{3})}dx-12\int (\frac{1}{{(-9+x+\log (-4+\frac{x}{3}))}^2}+\frac{12}{(-12+x){(-9+x+\log (-4+\frac{x}{3}))}^2}+\frac{x}{{(-9+x+\log (-4+\frac{x}{3}))}^2})dx\\ & =2x-12\int \frac{1}{{(-9+x+\log (-4+\frac{x}{3}))}^2}dx-12\int \frac{x}{{(-9+x+\log (-4+\frac{x}{3}))}^2}dx+12\int \frac{1}{-9+x+\log (-4+\frac{x}{3})}dx-144\int \frac{1}{(-12+x){(-9+x+\log (-4+\frac{x}{3}))}^2}dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.21, size = 20, normalized size = 1.00 $$\begin{array}{c}2(x+\frac{6x}{-9+x+\log (-4+\frac{x}{3})})\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-648 + 474*x - 60*x^2 + 2*x^3 + (288 - 72*x + 4*x^2)*Log[(-12 + x)/3] + (-24 + 2*x)*Log[(-12 + x)/3
]^2)/(-972 + 297*x - 30*x^2 + x^3 + (216 - 42*x + 2*x^2)*Log[(-12 + x)/3] + (-12 + x)*Log[(-12 + x)/3]^2),x]

[Out]

2*(x + (6*x)/(-9 + x + Log[-4 + x/3]))

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fricas [A]  time = 0.92, size = 28, normalized size = 1.40 $$\begin{array}{c}\frac{2(x^2+x\log (\frac{1}{3}x-4)-3x)}{x+\log (\frac{1}{3}x-4)-9}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-24)*log(1/3*x-4)^2+(4*x^2-72*x+288)*log(1/3*x-4)+2*x^3-60*x^2+474*x-648)/((x-12)*log(1/3*x-4)^
2+(2*x^2-42*x+216)*log(1/3*x-4)+x^3-30*x^2+297*x-972),x, algorithm="fricas")

[Out]

2*(x^2 + x*log(1/3*x - 4) - 3*x)/(x + log(1/3*x - 4) - 9)

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giac [A]  time = 0.24, size = 18, normalized size = 0.90 $$\begin{array}{c}2x+\frac{12x}{x+\log (\frac{1}{3}x-4)-9}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-24)*log(1/3*x-4)^2+(4*x^2-72*x+288)*log(1/3*x-4)+2*x^3-60*x^2+474*x-648)/((x-12)*log(1/3*x-4)^
2+(2*x^2-42*x+216)*log(1/3*x-4)+x^3-30*x^2+297*x-972),x, algorithm="giac")

[Out]

2*x + 12*x/(x + log(1/3*x - 4) - 9)

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maple [A]  time = 0.09, size = 19, normalized size = 0.95

method	result	size
risch	$2x+\frac{12x}{\ln (\frac{x}{3}-4)-9+x}$	$19$
norman	$\frac{-2\ln {(\frac{x}{3}-4)}^2+24\ln (\frac{x}{3}-4)+2x^2-54}{\ln (\frac{x}{3}-4)-9+x}+2\ln (x-12)$	$45$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x-24)*ln(1/3*x-4)^2+(4*x^2-72*x+288)*ln(1/3*x-4)+2*x^3-60*x^2+474*x-648)/((x-12)*ln(1/3*x-4)^2+(2*x^2-
42*x+216)*ln(1/3*x-4)+x^3-30*x^2+297*x-972),x,method=_RETURNVERBOSE)

[Out]

2*x+12*x/(ln(1/3*x-4)-9+x)

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maxima [A]  time = 0.48, size = 32, normalized size = 1.60 $$\begin{array}{c}\frac{2(x^2-x(\log (3)+3)+x\log (x-12))}{x-\log (3)+\log (x-12)-9}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-24)*log(1/3*x-4)^2+(4*x^2-72*x+288)*log(1/3*x-4)+2*x^3-60*x^2+474*x-648)/((x-12)*log(1/3*x-4)^
2+(2*x^2-42*x+216)*log(1/3*x-4)+x^3-30*x^2+297*x-972),x, algorithm="maxima")

[Out]

2*(x^2 - x*(log(3) + 3) + x*log(x - 12))/(x - log(3) + log(x - 12) - 9)

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mupad [B]  time = 3.59, size = 37, normalized size = 1.85 $$\begin{array}{c}\frac{2(x\ln (\frac{x}{3}-4)-6\ln (\frac{x}{3}-4)-9x+x^2+54)}{x+\ln (\frac{x}{3}-4)-9}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((474*x + log(x/3 - 4)*(4*x^2 - 72*x + 288) + log(x/3 - 4)^2*(2*x - 24) - 60*x^2 + 2*x^3 - 648)/(297*x + lo
g(x/3 - 4)^2*(x - 12) + log(x/3 - 4)*(2*x^2 - 42*x + 216) - 30*x^2 + x^3 - 972),x)

[Out]

(2*(x*log(x/3 - 4) - 6*log(x/3 - 4) - 9*x + x^2 + 54))/(x + log(x/3 - 4) - 9)

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sympy [A]  time = 0.15, size = 15, normalized size = 0.75 $$\begin{array}{c}2x+\frac{12x}{x+\log (\frac{x}{3}-4)-9}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-24)*ln(1/3*x-4)**2+(4*x**2-72*x+288)*ln(1/3*x-4)+2*x**3-60*x**2+474*x-648)/((x-12)*ln(1/3*x-4)
**2+(2*x**2-42*x+216)*ln(1/3*x-4)+x**3-30*x**2+297*x-972),x)

[Out]

2*x + 12*x/(x + log(x/3 - 4) - 9)

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