3.27.26 \(\int \frac {-4 x+2 x^2+(-134 x-2 x^2+2 x \log (4 x^2)) \log (67+x-\log (4 x^2))+(134+2 x-2 \log (4 x^2)) \log ^2(67+x-\log (4 x^2))}{(-67 x-x^2+x \log (4 x^2)) \log ^2(67+x-\log (4 x^2))} \, dx\)

Optimal. Leaf size=23 \[ 2 \left (-\log (x)+\frac {x}{\log \left (67+x-\log \left (4 x^2\right )\right )}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 0.47, 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 {-4 x+2 x^2+\left (-134 x-2 x^2+2 x \log \left (4 x^2\right )\right ) \log \left (67+x-\log \left (4 x^2\right )\right )+\left (134+2 x-2 \log \left (4 x^2\right )\right ) \log ^2\left (67+x-\log \left (4 x^2\right )\right )}{\left (-67 x-x^2+x \log \left (4 x^2\right )\right ) \log ^2\left (67+x-\log \left (4 x^2\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-4*x + 2*x^2 + (-134*x - 2*x^2 + 2*x*Log[4*x^2])*Log[67 + x - Log[4*x^2]] + (134 + 2*x - 2*Log[4*x^2])*Lo
g[67 + x - Log[4*x^2]]^2)/((-67*x - x^2 + x*Log[4*x^2])*Log[67 + x - Log[4*x^2]]^2),x]

[Out]

-2*Log[x] + 4*Defer[Int][1/((67 + x - Log[4*x^2])*Log[67 + x - Log[4*x^2]]^2), x] - 2*Defer[Int][x/((67 + x -
Log[4*x^2])*Log[67 + x - Log[4*x^2]]^2), x] + 2*Defer[Int][Log[67 + x - Log[4*x^2]]^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int 2 \left (-\frac {1}{x}+\frac {2-x}{\left (67+x-\log \left (4 x^2\right )\right ) \log ^2\left (67+x-\log \left (4 x^2\right )\right )}+\frac {1}{\log \left (67+x-\log \left (4 x^2\right )\right )}\right ) \, dx\\ &=2 \int \left (-\frac {1}{x}+\frac {2-x}{\left (67+x-\log \left (4 x^2\right )\right ) \log ^2\left (67+x-\log \left (4 x^2\right )\right )}+\frac {1}{\log \left (67+x-\log \left (4 x^2\right )\right )}\right ) \, dx\\ &=-2 \log (x)+2 \int \frac {2-x}{\left (67+x-\log \left (4 x^2\right )\right ) \log ^2\left (67+x-\log \left (4 x^2\right )\right )} \, dx+2 \int \frac {1}{\log \left (67+x-\log \left (4 x^2\right )\right )} \, dx\\ &=-2 \log (x)+2 \int \left (\frac {2}{\left (67+x-\log \left (4 x^2\right )\right ) \log ^2\left (67+x-\log \left (4 x^2\right )\right )}-\frac {x}{\left (67+x-\log \left (4 x^2\right )\right ) \log ^2\left (67+x-\log \left (4 x^2\right )\right )}\right ) \, dx+2 \int \frac {1}{\log \left (67+x-\log \left (4 x^2\right )\right )} \, dx\\ &=-2 \log (x)-2 \int \frac {x}{\left (67+x-\log \left (4 x^2\right )\right ) \log ^2\left (67+x-\log \left (4 x^2\right )\right )} \, dx+2 \int \frac {1}{\log \left (67+x-\log \left (4 x^2\right )\right )} \, dx+4 \int \frac {1}{\left (67+x-\log \left (4 x^2\right )\right ) \log ^2\left (67+x-\log \left (4 x^2\right )\right )} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.16, size = 23, normalized size = 1.00 \begin {gather*} 2 \left (-\log (x)+\frac {x}{\log \left (67+x-\log \left (4 x^2\right )\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*x + 2*x^2 + (-134*x - 2*x^2 + 2*x*Log[4*x^2])*Log[67 + x - Log[4*x^2]] + (134 + 2*x - 2*Log[4*x^
2])*Log[67 + x - Log[4*x^2]]^2)/((-67*x - x^2 + x*Log[4*x^2])*Log[67 + x - Log[4*x^2]]^2),x]

[Out]

2*(-Log[x] + x/Log[67 + x - Log[4*x^2]])

________________________________________________________________________________________

fricas [A]  time = 0.62, size = 39, normalized size = 1.70 \begin {gather*} -\frac {\log \left (4 \, x^{2}\right ) \log \left (x - \log \left (4 \, x^{2}\right ) + 67\right ) - 2 \, x}{\log \left (x - \log \left (4 \, x^{2}\right ) + 67\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*log(4*x^2)+2*x+134)*log(-log(4*x^2)+x+67)^2+(2*x*log(4*x^2)-2*x^2-134*x)*log(-log(4*x^2)+x+67)+
2*x^2-4*x)/(x*log(4*x^2)-x^2-67*x)/log(-log(4*x^2)+x+67)^2,x, algorithm="fricas")

[Out]

-(log(4*x^2)*log(x - log(4*x^2) + 67) - 2*x)/log(x - log(4*x^2) + 67)

________________________________________________________________________________________

giac [A]  time = 0.51, size = 22, normalized size = 0.96 \begin {gather*} \frac {2 \, x}{\log \left (x - \log \left (4 \, x^{2}\right ) + 67\right )} - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*log(4*x^2)+2*x+134)*log(-log(4*x^2)+x+67)^2+(2*x*log(4*x^2)-2*x^2-134*x)*log(-log(4*x^2)+x+67)+
2*x^2-4*x)/(x*log(4*x^2)-x^2-67*x)/log(-log(4*x^2)+x+67)^2,x, algorithm="giac")

[Out]

2*x/log(x - log(4*x^2) + 67) - 2*log(x)

________________________________________________________________________________________

maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (-2 \ln \left (4 x^{2}\right )+2 x +134\right ) \ln \left (-\ln \left (4 x^{2}\right )+x +67\right )^{2}+\left (2 x \ln \left (4 x^{2}\right )-2 x^{2}-134 x \right ) \ln \left (-\ln \left (4 x^{2}\right )+x +67\right )+2 x^{2}-4 x}{\left (x \ln \left (4 x^{2}\right )-x^{2}-67 x \right ) \ln \left (-\ln \left (4 x^{2}\right )+x +67\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*ln(4*x^2)+2*x+134)*ln(-ln(4*x^2)+x+67)^2+(2*x*ln(4*x^2)-2*x^2-134*x)*ln(-ln(4*x^2)+x+67)+2*x^2-4*x)/(
x*ln(4*x^2)-x^2-67*x)/ln(-ln(4*x^2)+x+67)^2,x)

[Out]

int(((-2*ln(4*x^2)+2*x+134)*ln(-ln(4*x^2)+x+67)^2+(2*x*ln(4*x^2)-2*x^2-134*x)*ln(-ln(4*x^2)+x+67)+2*x^2-4*x)/(
x*ln(4*x^2)-x^2-67*x)/ln(-ln(4*x^2)+x+67)^2,x)

________________________________________________________________________________________

maxima [A]  time = 0.90, size = 22, normalized size = 0.96 \begin {gather*} \frac {2 \, x}{\log \left (x - 2 \, \log \relax (2) - 2 \, \log \relax (x) + 67\right )} - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*log(4*x^2)+2*x+134)*log(-log(4*x^2)+x+67)^2+(2*x*log(4*x^2)-2*x^2-134*x)*log(-log(4*x^2)+x+67)+
2*x^2-4*x)/(x*log(4*x^2)-x^2-67*x)/log(-log(4*x^2)+x+67)^2,x, algorithm="maxima")

[Out]

2*x/log(x - 2*log(2) - 2*log(x) + 67) - 2*log(x)

________________________________________________________________________________________

mupad [B]  time = 1.83, size = 78, normalized size = 3.39 \begin {gather*} 2\,x-6\,\ln \relax (x)+\frac {276}{x-2}-\frac {4\,\ln \left (4\,x^2\right )}{x-2}+\frac {2\,x-\frac {2\,x\,\ln \left (x-\ln \left (4\,x^2\right )+67\right )\,\left (x-\ln \left (4\,x^2\right )+67\right )}{x-2}}{\ln \left (x-\ln \left (4\,x^2\right )+67\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x - log(x - log(4*x^2) + 67)^2*(2*x - 2*log(4*x^2) + 134) + log(x - log(4*x^2) + 67)*(134*x - 2*x*log(4
*x^2) + 2*x^2) - 2*x^2)/(log(x - log(4*x^2) + 67)^2*(67*x - x*log(4*x^2) + x^2)),x)

[Out]

2*x - 6*log(x) + 276/(x - 2) - (4*log(4*x^2))/(x - 2) + (2*x - (2*x*log(x - log(4*x^2) + 67)*(x - log(4*x^2) +
67))/(x - 2))/log(x - log(4*x^2) + 67)

________________________________________________________________________________________

sympy [A]  time = 0.33, size = 19, normalized size = 0.83 \begin {gather*} \frac {2 x}{\log {\left (x - \log {\left (4 x^{2} \right )} + 67 \right )}} - 2 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*ln(4*x**2)+2*x+134)*ln(-ln(4*x**2)+x+67)**2+(2*x*ln(4*x**2)-2*x**2-134*x)*ln(-ln(4*x**2)+x+67)+
2*x**2-4*x)/(x*ln(4*x**2)-x**2-67*x)/ln(-ln(4*x**2)+x+67)**2,x)

[Out]

2*x/log(x - log(4*x**2) + 67) - 2*log(x)

________________________________________________________________________________________