3.49.68 \(\int \frac {-648+474 x-60 x^2+2 x^3+(288-72 x+4 x^2) \log (\frac {1}{3} (-12+x))+(-24+2 x) \log ^2(\frac {1}{3} (-12+x))}{-972+297 x-30 x^2+x^3+(216-42 x+2 x^2) \log (\frac {1}{3} (-12+x))+(-12+x) \log ^2(\frac {1}{3} (-12+x))} \, dx\) [4868]

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

[Out]

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

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Rubi [F]
time = 0.63, 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 {gather*} \int \frac {-648+474 x-60 x^2+2 x^3+\left (288-72 x+4 x^2\right ) \log \left (\frac {1}{3} (-12+x)\right )+(-24+2 x) \log ^2\left (\frac {1}{3} (-12+x)\right )}{-972+297 x-30 x^2+x^3+\left (216-42 x+2 x^2\right ) \log \left (\frac {1}{3} (-12+x)\right )+(-12+x) \log ^2\left (\frac {1}{3} (-12+x)\right )} \, dx \end {gather*}

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 {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (324-237 x+30 x^2-x^3-2 \left (72-18 x+x^2\right ) \log \left (-4+\frac {x}{3}\right )-(-12+x) \log ^2\left (-4+\frac {x}{3}\right )\right )}{(12-x) \left (9-x-\log \left (-4+\frac {x}{3}\right )\right )^2} \, dx\\ &=2 \int \frac {324-237 x+30 x^2-x^3-2 \left (72-18 x+x^2\right ) \log \left (-4+\frac {x}{3}\right )-(-12+x) \log ^2\left (-4+\frac {x}{3}\right )}{(12-x) \left (9-x-\log \left (-4+\frac {x}{3}\right )\right )^2} \, dx\\ &=2 \int \left (1-\frac {6 (-11+x) x}{(-12+x) \left (-9+x+\log \left (-4+\frac {x}{3}\right )\right )^2}+\frac {6}{-9+x+\log \left (-4+\frac {x}{3}\right )}\right ) \, dx\\ &=2 x-12 \int \frac {(-11+x) x}{(-12+x) \left (-9+x+\log \left (-4+\frac {x}{3}\right )\right )^2} \, dx+12 \int \frac {1}{-9+x+\log \left (-4+\frac {x}{3}\right )} \, dx\\ &=2 x+12 \int \frac {1}{-9+x+\log \left (-4+\frac {x}{3}\right )} \, dx-12 \int \left (\frac {1}{\left (-9+x+\log \left (-4+\frac {x}{3}\right )\right )^2}+\frac {12}{(-12+x) \left (-9+x+\log \left (-4+\frac {x}{3}\right )\right )^2}+\frac {x}{\left (-9+x+\log \left (-4+\frac {x}{3}\right )\right )^2}\right ) \, dx\\ &=2 x-12 \int \frac {1}{\left (-9+x+\log \left (-4+\frac {x}{3}\right )\right )^2} \, dx-12 \int \frac {x}{\left (-9+x+\log \left (-4+\frac {x}{3}\right )\right )^2} \, dx+12 \int \frac {1}{-9+x+\log \left (-4+\frac {x}{3}\right )} \, dx-144 \int \frac {1}{(-12+x) \left (-9+x+\log \left (-4+\frac {x}{3}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.10, size = 20, normalized size = 1.00 \begin {gather*} 2 \left (x+\frac {6 x}{-9+x+\log \left (-4+\frac {x}{3}\right )}\right ) \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(39\) vs. \(2(18)=36\).
time = 9.22, size = 40, normalized size = 2.00

method	result	size
risch	\(2 x +\frac {12 x}{\ln \left (\frac {x}{3}-4\right )-9+x}\)	\(19\)
derivativedivides	\(\frac {6 \ln \left (\frac {x}{3}-4\right ) \left (\frac {x}{3}-4\right )+18 x -72+18 \left (\frac {x}{3}-4\right )^{2}}{\ln \left (\frac {x}{3}-4\right )-9+x}\)	\(40\)
default	\(\frac {6 \ln \left (\frac {x}{3}-4\right ) \left (\frac {x}{3}-4\right )+18 x -72+18 \left (\frac {x}{3}-4\right )^{2}}{\ln \left (\frac {x}{3}-4\right )-9+x}\)	\(40\)
norman	\(\frac {-2 \ln \left (\frac {x}{3}-4\right )^{2}+24 \ln \left (\frac {x}{3}-4\right )+2 x^{2}-54}{\ln \left (\frac {x}{3}-4\right )-9+x}+2 \ln \left (x -12\right )\)	\(45\)

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]

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

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Maxima [A]
time = 0.54, size = 32, normalized size = 1.60 \begin {gather*} \frac {2 \, {\left (x^{2} - x {\left (\log \left (3\right ) + 3\right )} + x \log \left (x - 12\right )\right )}}{x - \log \left (3\right ) + \log \left (x - 12\right ) - 9} \end {gather*}

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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Fricas [A]
time = 0.37, size = 28, normalized size = 1.40 \begin {gather*} \frac {2 \, {\left (x^{2} + x \log \left (\frac {1}{3} \, x - 4\right ) - 3 \, x\right )}}{x + \log \left (\frac {1}{3} \, x - 4\right ) - 9} \end {gather*}

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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Sympy [A]
time = 0.05, size = 15, normalized size = 0.75 \begin {gather*} 2 x + \frac {12 x}{x + \log {\left (\frac {x}{3} - 4 \right )} - 9} \end {gather*}

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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Giac [A]
time = 0.41, size = 18, normalized size = 0.90 \begin {gather*} 2 \, x + \frac {12 \, x}{x + \log \left (\frac {1}{3} \, x - 4\right ) - 9} \end {gather*}

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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Mupad [B]
time = 3.59, size = 37, normalized size = 1.85 \begin {gather*} \frac {2\,\left (x\,\ln \left (\frac {x}{3}-4\right )-6\,\ln \left (\frac {x}{3}-4\right )-9\,x+x^2+54\right )}{x+\ln \left (\frac {x}{3}-4\right )-9} \end {gather*}

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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