3.87.45 \(\int \frac {-6+3 x+(-3 x+(12-6 x) \log (4)) \log (x)+(12-6 x) \log (x) \log (\frac {-2+x}{\log (x)})}{(-2 x^3+x^4) \log ^2(4) \log (x)+(-4 x^3+2 x^4) \log (4) \log (x) \log (\frac {-2+x}{\log (x)})+(-2 x^3+x^4) \log (x) \log ^2(\frac {-2+x}{\log (x)})} \, dx\)

Optimal. Leaf size=19 \[ \frac {3}{x^2 \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )} \]

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Rubi [F]  time = 3.04, 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 {-6+3 x+(-3 x+(12-6 x) \log (4)) \log (x)+(12-6 x) \log (x) \log \left (\frac {-2+x}{\log (x)}\right )}{\left (-2 x^3+x^4\right ) \log ^2(4) \log (x)+\left (-4 x^3+2 x^4\right ) \log (4) \log (x) \log \left (\frac {-2+x}{\log (x)}\right )+\left (-2 x^3+x^4\right ) \log (x) \log ^2\left (\frac {-2+x}{\log (x)}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-3*Defer[Int][1/((-2 + x)*(Log[4] + Log[(-2 + x)/Log[x]])^2), x])/4 + (3*Defer[Int][1/(x^2*(Log[4] + Log[(-2
+ x)/Log[x]])^2), x])/2 + (3*Defer[Int][1/(x*(Log[4] + Log[(-2 + x)/Log[x]])^2), x])/4 + 3*Defer[Int][1/(x^3*L
og[x]*(Log[4] + Log[(-2 + x)/Log[x]])^2), x] - 6*Defer[Int][1/(x^3*(Log[4] + Log[(-2 + x)/Log[x]])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {6-3 x+3 \log (x) \left (x-4 \log (4)+x \log (16)+2 (-2+x) \log \left (\frac {-2+x}{\log (x)}\right )\right )}{(2-x) x^3 \log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2} \, dx\\ &=\int \left (-\frac {3 (2-x+x \log (x))}{(-2+x) x^3 \log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2}-\frac {6}{x^3 \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )}\right ) \, dx\\ &=-\left (3 \int \frac {2-x+x \log (x)}{(-2+x) x^3 \log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2} \, dx\right )-6 \int \frac {1}{x^3 \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )} \, dx\\ &=-\left (3 \int \left (\frac {-2+x-x \log (x)}{2 x^3 \log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2}+\frac {-2+x-x \log (x)}{4 x^2 \log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2}+\frac {-2+x-x \log (x)}{8 x \log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2}+\frac {2-x+x \log (x)}{8 (-2+x) \log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2}\right ) \, dx\right )-6 \int \frac {1}{x^3 \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )} \, dx\\ &=-\left (\frac {3}{8} \int \frac {-2+x-x \log (x)}{x \log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2} \, dx\right )-\frac {3}{8} \int \frac {2-x+x \log (x)}{(-2+x) \log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2} \, dx-\frac {3}{4} \int \frac {-2+x-x \log (x)}{x^2 \log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2} \, dx-\frac {3}{2} \int \frac {-2+x-x \log (x)}{x^3 \log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2} \, dx-6 \int \frac {1}{x^3 \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )} \, dx\\ &=-\left (\frac {3}{8} \int \left (-\frac {1}{\left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2}+\frac {1}{\log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2}-\frac {2}{x \log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2}\right ) \, dx\right )-\frac {3}{8} \int \left (\frac {x}{(-2+x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2}+\frac {2}{(-2+x) \log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2}-\frac {x}{(-2+x) \log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2}\right ) \, dx-\frac {3}{4} \int \left (-\frac {1}{x \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2}-\frac {2}{x^2 \log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2}+\frac {1}{x \log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2}\right ) \, dx-\frac {3}{2} \int \left (-\frac {1}{x^2 \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2}-\frac {2}{x^3 \log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2}+\frac {1}{x^2 \log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2}\right ) \, dx-6 \int \frac {1}{x^3 \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )} \, dx\\ &=\frac {3}{8} \int \frac {1}{\left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2} \, dx-\frac {3}{8} \int \frac {x}{(-2+x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2} \, dx-\frac {3}{8} \int \frac {1}{\log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2} \, dx+\frac {3}{8} \int \frac {x}{(-2+x) \log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2} \, dx+\frac {3}{4} \int \frac {1}{x \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2} \, dx-\frac {3}{4} \int \frac {1}{(-2+x) \log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2} \, dx+\frac {3}{2} \int \frac {1}{x^2 \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2} \, dx+3 \int \frac {1}{x^3 \log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2} \, dx-6 \int \frac {1}{x^3 \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )} \, dx\\ &=\frac {3}{8} \int \frac {1}{\left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2} \, dx-\frac {3}{8} \int \frac {1}{\log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2} \, dx-\frac {3}{8} \int \left (\frac {1}{\left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2}+\frac {2}{(-2+x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2}\right ) \, dx+\frac {3}{8} \int \left (\frac {1}{\log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2}+\frac {2}{(-2+x) \log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2}\right ) \, dx+\frac {3}{4} \int \frac {1}{x \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2} \, dx-\frac {3}{4} \int \frac {1}{(-2+x) \log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2} \, dx+\frac {3}{2} \int \frac {1}{x^2 \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2} \, dx+3 \int \frac {1}{x^3 \log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2} \, dx-6 \int \frac {1}{x^3 \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )} \, dx\\ &=-\left (\frac {3}{4} \int \frac {1}{(-2+x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2} \, dx\right )+\frac {3}{4} \int \frac {1}{x \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2} \, dx+\frac {3}{2} \int \frac {1}{x^2 \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2} \, dx+3 \int \frac {1}{x^3 \log (x) \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )^2} \, dx-6 \int \frac {1}{x^3 \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.24, size = 19, normalized size = 1.00 \begin {gather*} \frac {3}{x^2 \left (\log (4)+\log \left (\frac {-2+x}{\log (x)}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.67, size = 25, normalized size = 1.32 \begin {gather*} \frac {3}{2 \, x^{2} \log \relax (2) + x^{2} \log \left (\frac {x - 2}{\log \relax (x)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12-6*x)*log(x)*log((x-2)/log(x))+(2*(12-6*x)*log(2)-3*x)*log(x)+3*x-6)/((x^4-2*x^3)*log(x)*log((x-
2)/log(x))^2+2*(2*x^4-4*x^3)*log(2)*log(x)*log((x-2)/log(x))+4*(x^4-2*x^3)*log(2)^2*log(x)),x, algorithm="fric
as")

[Out]

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

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giac [A]  time = 0.44, size = 28, normalized size = 1.47 \begin {gather*} \frac {3}{2 \, x^{2} \log \relax (2) + x^{2} \log \left (x - 2\right ) - x^{2} \log \left (\log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12-6*x)*log(x)*log((x-2)/log(x))+(2*(12-6*x)*log(2)-3*x)*log(x)+3*x-6)/((x^4-2*x^3)*log(x)*log((x-
2)/log(x))^2+2*(2*x^4-4*x^3)*log(2)*log(x)*log((x-2)/log(x))+4*(x^4-2*x^3)*log(2)^2*log(x)),x, algorithm="giac
")

[Out]

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

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maple [C]  time = 0.10, size = 118, normalized size = 6.21

method	result	size
risch	\(-\frac {6 i}{x^{2} \left (\pi \,\mathrm {csgn}\left (i \left (x -2\right )\right ) \mathrm {csgn}\left (\frac {i \left (x -2\right )}{\ln \relax (x )}\right )^{2}-\pi \,\mathrm {csgn}\left (i \left (x -2\right )\right ) \mathrm {csgn}\left (\frac {i \left (x -2\right )}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right )-\pi \mathrm {csgn}\left (\frac {i \left (x -2\right )}{\ln \relax (x )}\right )^{3}+\pi \mathrm {csgn}\left (\frac {i \left (x -2\right )}{\ln \relax (x )}\right )^{2} \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right )-4 i \ln \relax (2)+2 i \ln \left (\ln \relax (x )\right )-2 i \ln \left (x -2\right )\right )}\)	\(118\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((12-6*x)*ln(x)*ln((x-2)/ln(x))+(2*(12-6*x)*ln(2)-3*x)*ln(x)+3*x-6)/((x^4-2*x^3)*ln(x)*ln((x-2)/ln(x))^2+2
*(2*x^4-4*x^3)*ln(2)*ln(x)*ln((x-2)/ln(x))+4*(x^4-2*x^3)*ln(2)^2*ln(x)),x,method=_RETURNVERBOSE)

[Out]

-6*I/x^2/(Pi*csgn(I*(x-2))*csgn(I/ln(x)*(x-2))^2-Pi*csgn(I*(x-2))*csgn(I/ln(x)*(x-2))*csgn(I/ln(x))-Pi*csgn(I/
ln(x)*(x-2))^3+Pi*csgn(I/ln(x)*(x-2))^2*csgn(I/ln(x))-4*I*ln(2)+2*I*ln(ln(x))-2*I*ln(x-2))

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maxima [A]  time = 0.50, size = 28, normalized size = 1.47 \begin {gather*} \frac {3}{2 \, x^{2} \log \relax (2) + x^{2} \log \left (x - 2\right ) - x^{2} \log \left (\log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12-6*x)*log(x)*log((x-2)/log(x))+(2*(12-6*x)*log(2)-3*x)*log(x)+3*x-6)/((x^4-2*x^3)*log(x)*log((x-
2)/log(x))^2+2*(2*x^4-4*x^3)*log(2)*log(x)*log((x-2)/log(x))+4*(x^4-2*x^3)*log(2)^2*log(x)),x, algorithm="maxi
ma")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {\ln \relax (x)\,\left (3\,x+2\,\ln \relax (2)\,\left (6\,x-12\right )\right )-3\,x+\ln \relax (x)\,\ln \left (\frac {x-2}{\ln \relax (x)}\right )\,\left (6\,x-12\right )+6}{\ln \relax (x)\,\left (2\,x^3-x^4\right )\,{\ln \left (\frac {x-2}{\ln \relax (x)}\right )}^2+2\,\ln \relax (2)\,\ln \relax (x)\,\left (4\,x^3-2\,x^4\right )\,\ln \left (\frac {x-2}{\ln \relax (x)}\right )+4\,{\ln \relax (2)}^2\,\ln \relax (x)\,\left (2\,x^3-x^4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.36, size = 20, normalized size = 1.05 \begin {gather*} \frac {3}{x^{2} \log {\left (\frac {x - 2}{\log {\relax (x )}} \right )} + 2 x^{2} \log {\relax (2 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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