3.12.95 \(\int \frac {(20-22 x+6 x^2) \log (3)+(12 x-3 x^2) \log (3) \log (4 x-x^2) \log (\log (4 x-x^2))+(8 x^2-2 x^3) \log (3) \log (4 x-x^2) \log ^2(\log (4 x-x^2))}{(-4 x+x^2) \log (4 x-x^2) \log ^2(\log (4 x-x^2))} \, dx\)

Optimal. Leaf size=29 \[ \log (3) \left (-x^2+\frac {\left (-3+\frac {5}{x}\right ) x}{\log (\log ((4-x) x))}\right ) \]

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Rubi [F]  time = 2.01, 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 {\left (20-22 x+6 x^2\right ) \log (3)+\left (12 x-3 x^2\right ) \log (3) \log \left (4 x-x^2\right ) \log \left (\log \left (4 x-x^2\right )\right )+\left (8 x^2-2 x^3\right ) \log (3) \log \left (4 x-x^2\right ) \log ^2\left (\log \left (4 x-x^2\right )\right )}{\left (-4 x+x^2\right ) \log \left (4 x-x^2\right ) \log ^2\left (\log \left (4 x-x^2\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((20 - 22*x + 6*x^2)*Log[3] + (12*x - 3*x^2)*Log[3]*Log[4*x - x^2]*Log[Log[4*x - x^2]] + (8*x^2 - 2*x^3)*L
og[3]*Log[4*x - x^2]*Log[Log[4*x - x^2]]^2)/((-4*x + x^2)*Log[4*x - x^2]*Log[Log[4*x - x^2]]^2),x]

[Out]

-(x^2*Log[3]) + 6*Log[3]*Defer[Int][1/(Log[(4 - x)*x]*Log[Log[(4 - x)*x]]^2), x] + 7*Log[3]*Defer[Int][1/((-4
+ x)*Log[(4 - x)*x]*Log[Log[(4 - x)*x]]^2), x] - 5*Log[3]*Defer[Int][1/(x*Log[(4 - x)*x]*Log[Log[(4 - x)*x]]^2
), x] - 3*Log[3]*Defer[Int][Log[Log[(4 - x)*x]]^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (20-22 x+6 x^2\right ) \log (3)+\left (12 x-3 x^2\right ) \log (3) \log \left (4 x-x^2\right ) \log \left (\log \left (4 x-x^2\right )\right )+\left (8 x^2-2 x^3\right ) \log (3) \log \left (4 x-x^2\right ) \log ^2\left (\log \left (4 x-x^2\right )\right )}{(-4+x) x \log \left (4 x-x^2\right ) \log ^2\left (\log \left (4 x-x^2\right )\right )} \, dx\\ &=\int \frac {\log (3) \left (-20+22 x-6 x^2+(-4+x) x \log (-((-4+x) x)) \log (\log (-((-4+x) x))) (3+2 x \log (\log (-((-4+x) x))))\right )}{(4-x) x \log ((4-x) x) \log ^2(\log ((4-x) x))} \, dx\\ &=\log (3) \int \frac {-20+22 x-6 x^2+(-4+x) x \log (-((-4+x) x)) \log (\log (-((-4+x) x))) (3+2 x \log (\log (-((-4+x) x))))}{(4-x) x \log ((4-x) x) \log ^2(\log ((4-x) x))} \, dx\\ &=\log (3) \int \left (-2 x+\frac {2 \left (-10+11 x-3 x^2\right )}{(4-x) x \log ((4-x) x) \log ^2(\log ((4-x) x))}-\frac {3}{\log (\log ((4-x) x))}\right ) \, dx\\ &=-x^2 \log (3)+(2 \log (3)) \int \frac {-10+11 x-3 x^2}{(4-x) x \log ((4-x) x) \log ^2(\log ((4-x) x))} \, dx-(3 \log (3)) \int \frac {1}{\log (\log ((4-x) x))} \, dx\\ &=-x^2 \log (3)+(2 \log (3)) \int \left (\frac {3}{\log ((4-x) x) \log ^2(\log ((4-x) x))}+\frac {7}{2 (-4+x) \log ((4-x) x) \log ^2(\log ((4-x) x))}-\frac {5}{2 x \log ((4-x) x) \log ^2(\log ((4-x) x))}\right ) \, dx-(3 \log (3)) \int \frac {1}{\log (\log ((4-x) x))} \, dx\\ &=-x^2 \log (3)-(3 \log (3)) \int \frac {1}{\log (\log ((4-x) x))} \, dx-(5 \log (3)) \int \frac {1}{x \log ((4-x) x) \log ^2(\log ((4-x) x))} \, dx+(6 \log (3)) \int \frac {1}{\log ((4-x) x) \log ^2(\log ((4-x) x))} \, dx+(7 \log (3)) \int \frac {1}{(-4+x) \log ((4-x) x) \log ^2(\log ((4-x) x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.21, size = 25, normalized size = 0.86 \begin {gather*} \log (3) \left (-x^2+\frac {5-3 x}{\log (\log (-((-4+x) x)))}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((20 - 22*x + 6*x^2)*Log[3] + (12*x - 3*x^2)*Log[3]*Log[4*x - x^2]*Log[Log[4*x - x^2]] + (8*x^2 - 2*
x^3)*Log[3]*Log[4*x - x^2]*Log[Log[4*x - x^2]]^2)/((-4*x + x^2)*Log[4*x - x^2]*Log[Log[4*x - x^2]]^2),x]

[Out]

Log[3]*(-x^2 + (5 - 3*x)/Log[Log[-((-4 + x)*x)]])

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fricas [A]  time = 0.74, size = 41, normalized size = 1.41 \begin {gather*} -\frac {x^{2} \log \relax (3) \log \left (\log \left (-x^{2} + 4 \, x\right )\right ) + {\left (3 \, x - 5\right )} \log \relax (3)}{\log \left (\log \left (-x^{2} + 4 \, x\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+8*x^2)*log(3)*log(-x^2+4*x)*log(log(-x^2+4*x))^2+(-3*x^2+12*x)*log(3)*log(-x^2+4*x)*log(log
(-x^2+4*x))+(6*x^2-22*x+20)*log(3))/(x^2-4*x)/log(-x^2+4*x)/log(log(-x^2+4*x))^2,x, algorithm="fricas")

[Out]

-(x^2*log(3)*log(log(-x^2 + 4*x)) + (3*x - 5)*log(3))/log(log(-x^2 + 4*x))

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giac [A]  time = 0.86, size = 33, normalized size = 1.14 \begin {gather*} -x^{2} \log \relax (3) - \frac {3 \, x \log \relax (3) - 5 \, \log \relax (3)}{\log \left (\log \left (-x^{2} + 4 \, x\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+8*x^2)*log(3)*log(-x^2+4*x)*log(log(-x^2+4*x))^2+(-3*x^2+12*x)*log(3)*log(-x^2+4*x)*log(log
(-x^2+4*x))+(6*x^2-22*x+20)*log(3))/(x^2-4*x)/log(-x^2+4*x)/log(log(-x^2+4*x))^2,x, algorithm="giac")

[Out]

-x^2*log(3) - (3*x*log(3) - 5*log(3))/log(log(-x^2 + 4*x))

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (-2 x^{3}+8 x^{2}\right ) \ln \relax (3) \ln \left (-x^{2}+4 x \right ) \ln \left (\ln \left (-x^{2}+4 x \right )\right )^{2}+\left (-3 x^{2}+12 x \right ) \ln \relax (3) \ln \left (-x^{2}+4 x \right ) \ln \left (\ln \left (-x^{2}+4 x \right )\right )+\left (6 x^{2}-22 x +20\right ) \ln \relax (3)}{\left (x^{2}-4 x \right ) \ln \left (-x^{2}+4 x \right ) \ln \left (\ln \left (-x^{2}+4 x \right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^3+8*x^2)*ln(3)*ln(-x^2+4*x)*ln(ln(-x^2+4*x))^2+(-3*x^2+12*x)*ln(3)*ln(-x^2+4*x)*ln(ln(-x^2+4*x))+(6
*x^2-22*x+20)*ln(3))/(x^2-4*x)/ln(-x^2+4*x)/ln(ln(-x^2+4*x))^2,x)

[Out]

int(((-2*x^3+8*x^2)*ln(3)*ln(-x^2+4*x)*ln(ln(-x^2+4*x))^2+(-3*x^2+12*x)*ln(3)*ln(-x^2+4*x)*ln(ln(-x^2+4*x))+(6
*x^2-22*x+20)*ln(3))/(x^2-4*x)/ln(-x^2+4*x)/ln(ln(-x^2+4*x))^2,x)

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maxima [A]  time = 0.62, size = 40, normalized size = 1.38 \begin {gather*} -\frac {x^{2} \log \relax (3) \log \left (\log \relax (x) + \log \left (-x + 4\right )\right ) + 3 \, x \log \relax (3) - 5 \, \log \relax (3)}{\log \left (\log \relax (x) + \log \left (-x + 4\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+8*x^2)*log(3)*log(-x^2+4*x)*log(log(-x^2+4*x))^2+(-3*x^2+12*x)*log(3)*log(-x^2+4*x)*log(log
(-x^2+4*x))+(6*x^2-22*x+20)*log(3))/(x^2-4*x)/log(-x^2+4*x)/log(log(-x^2+4*x))^2,x, algorithm="maxima")

[Out]

-(x^2*log(3)*log(log(x) + log(-x + 4)) + 3*x*log(3) - 5*log(3))/log(log(x) + log(-x + 4))

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mupad [B]  time = 1.38, size = 106, normalized size = 3.66 \begin {gather*} 6\,\ln \left (x\,\left (x-4\right )\right )\,\ln \relax (3)-x^2\,\ln \relax (3)+\frac {5\,\ln \relax (3)-3\,x\,\ln \relax (3)+\frac {3\,x\,\ln \left (\ln \left (4\,x-x^2\right )\right )\,\ln \relax (3)\,\ln \left (4\,x-x^2\right )\,\left (x-4\right )}{2\,\left (x-2\right )}}{\ln \left (\ln \left (4\,x-x^2\right )\right )}+\frac {\ln \left (4\,x-x^2\right )\,\left (12\,\ln \relax (3)-\frac {3\,x^2\,\ln \relax (3)}{2}\right )}{x-2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(3)*(6*x^2 - 22*x + 20) + log(log(4*x - x^2))^2*log(3)*log(4*x - x^2)*(8*x^2 - 2*x^3) + log(log(4*x -
x^2))*log(3)*log(4*x - x^2)*(12*x - 3*x^2))/(log(log(4*x - x^2))^2*log(4*x - x^2)*(4*x - x^2)),x)

[Out]

6*log(x*(x - 4))*log(3) - x^2*log(3) + (5*log(3) - 3*x*log(3) + (3*x*log(log(4*x - x^2))*log(3)*log(4*x - x^2)
*(x - 4))/(2*(x - 2)))/log(log(4*x - x^2)) + (log(4*x - x^2)*(12*log(3) - (3*x^2*log(3))/2))/(x - 2)

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sympy [A]  time = 0.33, size = 27, normalized size = 0.93 \begin {gather*} - x^{2} \log {\relax (3 )} + \frac {- 3 x \log {\relax (3 )} + 5 \log {\relax (3 )}}{\log {\left (\log {\left (- x^{2} + 4 x \right )} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**3+8*x**2)*ln(3)*ln(-x**2+4*x)*ln(ln(-x**2+4*x))**2+(-3*x**2+12*x)*ln(3)*ln(-x**2+4*x)*ln(ln(
-x**2+4*x))+(6*x**2-22*x+20)*ln(3))/(x**2-4*x)/ln(-x**2+4*x)/ln(ln(-x**2+4*x))**2,x)

[Out]

-x**2*log(3) + (-3*x*log(3) + 5*log(3))/log(log(-x**2 + 4*x))

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