3.3.95 \(\int \frac {768+384 x+(-768+64 x+64 x^2) \log (\frac {1}{3} (-3+x))+(768-64 x-64 x^2) \log (x)}{(3 x^3-x^4) \log ^3(\frac {1}{3} (-3+x))+(-9 x^3+3 x^4) \log ^2(\frac {1}{3} (-3+x)) \log (x)+(9 x^3-3 x^4) \log (\frac {1}{3} (-3+x)) \log ^2(x)+(-3 x^3+x^4) \log ^3(x)} \, dx\)

Optimal. Leaf size=23 \[ \frac {64 (2+x)}{x^2 \left (\log \left (\frac {1}{3} (-3+x)\right )-\log (x)\right )^2} \]

________________________________________________________________________________________

Rubi [F]  time = 1.37, 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 {768+384 x+\left (-768+64 x+64 x^2\right ) \log \left (\frac {1}{3} (-3+x)\right )+\left (768-64 x-64 x^2\right ) \log (x)}{\left (3 x^3-x^4\right ) \log ^3\left (\frac {1}{3} (-3+x)\right )+\left (-9 x^3+3 x^4\right ) \log ^2\left (\frac {1}{3} (-3+x)\right ) \log (x)+\left (9 x^3-3 x^4\right ) \log \left (\frac {1}{3} (-3+x)\right ) \log ^2(x)+\left (-3 x^3+x^4\right ) \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(768 + 384*x + (-768 + 64*x + 64*x^2)*Log[(-3 + x)/3] + (768 - 64*x - 64*x^2)*Log[x])/((3*x^3 - x^4)*Log[(
-3 + x)/3]^3 + (-9*x^3 + 3*x^4)*Log[(-3 + x)/3]^2*Log[x] + (9*x^3 - 3*x^4)*Log[(-3 + x)/3]*Log[x]^2 + (-3*x^3
+ x^4)*Log[x]^3),x]

[Out]

-256*Defer[Int][1/(x^3*(Log[(-3 + x)/3] - Log[x])^2), x] - 64*Defer[Int][1/(x^2*(Log[(-3 + x)/3] - Log[x])^2),
x] - (640*Defer[Int][1/((-3 + x)*(Log[-3 + x] - Log[3*x])^3), x])/9 + 256*Defer[Int][1/(x^3*(Log[-3 + x] - Lo
g[3*x])^3), x] + (640*Defer[Int][1/(x^2*(Log[-3 + x] - Log[3*x])^3), x])/3 + (640*Defer[Int][1/(x*(Log[-3 + x]
- Log[3*x])^3), x])/9

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {64 \left (6 (2+x)+\left (-12+x+x^2\right ) \log \left (\frac {1}{3} (-3+x)\right )-\left (-12+x+x^2\right ) \log (x)\right )}{(3-x) x^3 \left (\log \left (\frac {1}{3} (-3+x)\right )-\log (x)\right )^3} \, dx\\ &=64 \int \frac {6 (2+x)+\left (-12+x+x^2\right ) \log \left (\frac {1}{3} (-3+x)\right )-\left (-12+x+x^2\right ) \log (x)}{(3-x) x^3 \left (\log \left (\frac {1}{3} (-3+x)\right )-\log (x)\right )^3} \, dx\\ &=64 \int \left (\frac {-4-x}{x^3 \left (\log \left (\frac {1}{3} (-3+x)\right )-\log (x)\right )^2}-\frac {6 (2+x)}{(-3+x) x^3 (\log (-3+x)-\log (3 x))^3}\right ) \, dx\\ &=64 \int \frac {-4-x}{x^3 \left (\log \left (\frac {1}{3} (-3+x)\right )-\log (x)\right )^2} \, dx-384 \int \frac {2+x}{(-3+x) x^3 (\log (-3+x)-\log (3 x))^3} \, dx\\ &=64 \int \left (-\frac {4}{x^3 \left (\log \left (\frac {1}{3} (-3+x)\right )-\log (x)\right )^2}-\frac {1}{x^2 \left (\log \left (\frac {1}{3} (-3+x)\right )-\log (x)\right )^2}\right ) \, dx-384 \int \left (\frac {5}{27 (-3+x) (\log (-3+x)-\log (3 x))^3}-\frac {2}{3 x^3 (\log (-3+x)-\log (3 x))^3}-\frac {5}{9 x^2 (\log (-3+x)-\log (3 x))^3}-\frac {5}{27 x (\log (-3+x)-\log (3 x))^3}\right ) \, dx\\ &=-\left (64 \int \frac {1}{x^2 \left (\log \left (\frac {1}{3} (-3+x)\right )-\log (x)\right )^2} \, dx\right )-\frac {640}{9} \int \frac {1}{(-3+x) (\log (-3+x)-\log (3 x))^3} \, dx+\frac {640}{9} \int \frac {1}{x (\log (-3+x)-\log (3 x))^3} \, dx+\frac {640}{3} \int \frac {1}{x^2 (\log (-3+x)-\log (3 x))^3} \, dx-256 \int \frac {1}{x^3 \left (\log \left (\frac {1}{3} (-3+x)\right )-\log (x)\right )^2} \, dx+256 \int \frac {1}{x^3 (\log (-3+x)-\log (3 x))^3} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.32, size = 25, normalized size = 1.09 \begin {gather*} -\frac {64 (-2-x)}{x^2 \left (-\log \left (\frac {1}{3} (-3+x)\right )+\log (x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(768 + 384*x + (-768 + 64*x + 64*x^2)*Log[(-3 + x)/3] + (768 - 64*x - 64*x^2)*Log[x])/((3*x^3 - x^4)
*Log[(-3 + x)/3]^3 + (-9*x^3 + 3*x^4)*Log[(-3 + x)/3]^2*Log[x] + (9*x^3 - 3*x^4)*Log[(-3 + x)/3]*Log[x]^2 + (-
3*x^3 + x^4)*Log[x]^3),x]

[Out]

(-64*(-2 - x))/(x^2*(-Log[(-3 + x)/3] + Log[x])^2)

________________________________________________________________________________________

fricas [A]  time = 0.77, size = 41, normalized size = 1.78 \begin {gather*} \frac {64 \, {\left (x + 2\right )}}{x^{2} \log \relax (x)^{2} - 2 \, x^{2} \log \relax (x) \log \left (\frac {1}{3} \, x - 1\right ) + x^{2} \log \left (\frac {1}{3} \, x - 1\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-64*x^2-64*x+768)*log(x)+(64*x^2+64*x-768)*log(1/3*x-1)+384*x+768)/((x^4-3*x^3)*log(x)^3+(-3*x^4+9
*x^3)*log(1/3*x-1)*log(x)^2+(3*x^4-9*x^3)*log(1/3*x-1)^2*log(x)+(-x^4+3*x^3)*log(1/3*x-1)^3),x, algorithm="fri
cas")

[Out]

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

________________________________________________________________________________________

giac [B]  time = 0.45, size = 65, normalized size = 2.83 \begin {gather*} \frac {64 \, {\left (x + 2\right )}}{x^{2} \log \relax (3)^{2} - 2 \, x^{2} \log \relax (3) \log \left (x - 3\right ) + x^{2} \log \left (x - 3\right )^{2} + 2 \, x^{2} \log \relax (3) \log \relax (x) - 2 \, x^{2} \log \left (x - 3\right ) \log \relax (x) + x^{2} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-64*x^2-64*x+768)*log(x)+(64*x^2+64*x-768)*log(1/3*x-1)+384*x+768)/((x^4-3*x^3)*log(x)^3+(-3*x^4+9
*x^3)*log(1/3*x-1)*log(x)^2+(3*x^4-9*x^3)*log(1/3*x-1)^2*log(x)+(-x^4+3*x^3)*log(1/3*x-1)^3),x, algorithm="gia
c")

[Out]

64*(x + 2)/(x^2*log(3)^2 - 2*x^2*log(3)*log(x - 3) + x^2*log(x - 3)^2 + 2*x^2*log(3)*log(x) - 2*x^2*log(x - 3)
*log(x) + x^2*log(x)^2)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 22, normalized size = 0.96

method	result	size
risch	\(\frac {64 x +128}{x^{2} \left (\ln \relax (x )-\ln \left (\frac {x}{3}-1\right )\right )^{2}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-64*x^2-64*x+768)*ln(x)+(64*x^2+64*x-768)*ln(1/3*x-1)+384*x+768)/((x^4-3*x^3)*ln(x)^3+(-3*x^4+9*x^3)*ln(
1/3*x-1)*ln(x)^2+(3*x^4-9*x^3)*ln(1/3*x-1)^2*ln(x)+(-x^4+3*x^3)*ln(1/3*x-1)^3),x,method=_RETURNVERBOSE)

[Out]

64*(2+x)/x^2/(ln(x)-ln(1/3*x-1))^2

________________________________________________________________________________________

maxima [B]  time = 0.64, size = 62, normalized size = 2.70 \begin {gather*} \frac {64 \, {\left (x + 2\right )}}{x^{2} \log \relax (3)^{2} + x^{2} \log \left (x - 3\right )^{2} + 2 \, x^{2} \log \relax (3) \log \relax (x) + x^{2} \log \relax (x)^{2} - 2 \, {\left (x^{2} \log \relax (3) + x^{2} \log \relax (x)\right )} \log \left (x - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-64*x^2-64*x+768)*log(x)+(64*x^2+64*x-768)*log(1/3*x-1)+384*x+768)/((x^4-3*x^3)*log(x)^3+(-3*x^4+9
*x^3)*log(1/3*x-1)*log(x)^2+(3*x^4-9*x^3)*log(1/3*x-1)^2*log(x)+(-x^4+3*x^3)*log(1/3*x-1)^3),x, algorithm="max
ima")

[Out]

64*(x + 2)/(x^2*log(3)^2 + x^2*log(x - 3)^2 + 2*x^2*log(3)*log(x) + x^2*log(x)^2 - 2*(x^2*log(3) + x^2*log(x))
*log(x - 3))

________________________________________________________________________________________

mupad [B]  time = 0.80, size = 58, normalized size = 2.52 \begin {gather*} \frac {32\,\left (-x^2\,{\ln \left (\frac {x}{3}-1\right )}^2+2\,x^2\,\ln \left (\frac {x}{3}-1\right )\,\ln \relax (x)-x^2\,{\ln \relax (x)}^2+18\,x+36\right )}{9\,x^2\,{\left (\ln \left (\frac {x}{3}-1\right )-\ln \relax (x)\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((384*x + log(x/3 - 1)*(64*x + 64*x^2 - 768) - log(x)*(64*x + 64*x^2 - 768) + 768)/(log(x/3 - 1)^3*(3*x^3 -
x^4) - log(x)^3*(3*x^3 - x^4) + log(x/3 - 1)*log(x)^2*(9*x^3 - 3*x^4) - log(x/3 - 1)^2*log(x)*(9*x^3 - 3*x^4)
),x)

[Out]

(32*(18*x - x^2*log(x/3 - 1)^2 - x^2*log(x)^2 + 2*x^2*log(x/3 - 1)*log(x) + 36))/(9*x^2*(log(x/3 - 1) - log(x)
)^2)

________________________________________________________________________________________

sympy [B]  time = 0.42, size = 39, normalized size = 1.70 \begin {gather*} \frac {64 x + 128}{x^{2} \log {\relax (x )}^{2} - 2 x^{2} \log {\relax (x )} \log {\left (\frac {x}{3} - 1 \right )} + x^{2} \log {\left (\frac {x}{3} - 1 \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-64*x**2-64*x+768)*ln(x)+(64*x**2+64*x-768)*ln(1/3*x-1)+384*x+768)/((x**4-3*x**3)*ln(x)**3+(-3*x**
4+9*x**3)*ln(1/3*x-1)*ln(x)**2+(3*x**4-9*x**3)*ln(1/3*x-1)**2*ln(x)+(-x**4+3*x**3)*ln(1/3*x-1)**3),x)

[Out]

(64*x + 128)/(x**2*log(x)**2 - 2*x**2*log(x)*log(x/3 - 1) + x**2*log(x/3 - 1)**2)

________________________________________________________________________________________