\(\int \frac {2 x^2 \log (12)+(2 x^2+(-4 x+x^2) \log (12)) \log (x)+(-2 x^2+(2-2 x) \log (12)) \log ^2(x)+\log (12) \log ^3(x)+(x^2 \log (12) \log (x)-2 x \log (12) \log ^2(x)+\log (12) \log ^3(x)) \log (\frac {5}{x \log ^2(x)})}{x^4 \log (x)-2 x^3 \log ^2(x)+x^2 \log ^3(x)} \, dx\) [2681]

Optimal result

Integrand size = 116, antiderivative size = 29 \[ \int \frac {2 x^2 \log (12)+\left (2 x^2+\left (-4 x+x^2\right ) \log (12)\right ) \log (x)+\left (-2 x^2+(2-2 x) \log (12)\right ) \log ^2(x)+\log (12) \log ^3(x)+\left (x^2 \log (12) \log (x)-2 x \log (12) \log ^2(x)+\log (12) \log ^3(x)\right ) \log \left (\frac {5}{x \log ^2(x)}\right )}{x^4 \log (x)-2 x^3 \log ^2(x)+x^2 \log ^3(x)} \, dx=\frac {2 x}{x-\log (x)}-\frac {\log (12) \log \left (\frac {5}{x \log ^2(x)}\right )}{x} \]

[Out]

2/(x-ln(x))*x-ln(5/x/ln(x)^2)*ln(12)/x

Rubi [F]

\[ \int \frac {2 x^2 \log (12)+\left (2 x^2+\left (-4 x+x^2\right ) \log (12)\right ) \log (x)+\left (-2 x^2+(2-2 x) \log (12)\right ) \log ^2(x)+\log (12) \log ^3(x)+\left (x^2 \log (12) \log (x)-2 x \log (12) \log ^2(x)+\log (12) \log ^3(x)\right ) \log \left (\frac {5}{x \log ^2(x)}\right )}{x^4 \log (x)-2 x^3 \log ^2(x)+x^2 \log ^3(x)} \, dx=\int \frac {2 x^2 \log (12)+\left (2 x^2+\left (-4 x+x^2\right ) \log (12)\right ) \log (x)+\left (-2 x^2+(2-2 x) \log (12)\right ) \log ^2(x)+\log (12) \log ^3(x)+\left (x^2 \log (12) \log (x)-2 x \log (12) \log ^2(x)+\log (12) \log ^3(x)\right ) \log \left (\frac {5}{x \log ^2(x)}\right )}{x^4 \log (x)-2 x^3 \log ^2(x)+x^2 \log ^3(x)} \, dx \]

[In]

Int[(2*x^2*Log[12] + (2*x^2 + (-4*x + x^2)*Log[12])*Log[x] + (-2*x^2 + (2 - 2*x)*Log[12])*Log[x]^2 + Log[12]*L
og[x]^3 + (x^2*Log[12]*Log[x] - 2*x*Log[12]*Log[x]^2 + Log[12]*Log[x]^3)*Log[5/(x*Log[x]^2)])/(x^4*Log[x] - 2*
x^3*Log[x]^2 + x^2*Log[x]^3),x]

[Out]

2*ExpIntegralEi[-Log[x]]*Log[12] + ExpIntegralEi[-Log[x]]*Log[12]*Log[x] - ExpIntegralEi[-Log[x]]*Log[12]*(2 +
Log[x]) - (Log[12]*Log[5/(x*Log[x]^2)])/x - Log[12]*Defer[Int][(x - Log[x])^(-2), x] + (2 + Log[12])*Defer[In
t][(x - Log[x])^(-2), x] - 2*Defer[Int][x/(x - Log[x])^2, x] - 2*Defer[Int][(-x + Log[x])^(-1), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 x^2 \log (12)+\left (2 x^2+\left (-4 x+x^2\right ) \log (12)\right ) \log (x)+\left (-2 x^2+(2-2 x) \log (12)\right ) \log ^2(x)+\log (12) \log ^3(x)+\left (x^2 \log (12) \log (x)-2 x \log (12) \log ^2(x)+\log (12) \log ^3(x)\right ) \log \left (\frac {5}{x \log ^2(x)}\right )}{x^2 (x-\log (x))^2 \log (x)} \, dx \\ & = \int \left (\frac {-4 \log (12)+x (2+\log (12))}{x (x-\log (x))^2}+\frac {2 \log (12)}{(x-\log (x))^2 \log (x)}-\frac {2 \left (x^2-\log (12)+x \log (12)\right ) \log (x)}{x^2 (x-\log (x))^2}+\frac {\log (12) \log ^2(x)}{x^2 (x-\log (x))^2}+\frac {\log (12) \log \left (\frac {5}{x \log ^2(x)}\right )}{x^2}\right ) \, dx \\ & = -\left (2 \int \frac {\left (x^2-\log (12)+x \log (12)\right ) \log (x)}{x^2 (x-\log (x))^2} \, dx\right )+\log (12) \int \frac {\log ^2(x)}{x^2 (x-\log (x))^2} \, dx+\log (12) \int \frac {\log \left (\frac {5}{x \log ^2(x)}\right )}{x^2} \, dx+(2 \log (12)) \int \frac {1}{(x-\log (x))^2 \log (x)} \, dx+\int \frac {-4 \log (12)+x (2+\log (12))}{x (x-\log (x))^2} \, dx \\ & = -\frac {\log (12) \log \left (\frac {5}{x \log ^2(x)}\right )}{x}-2 \int \left (\frac {x^2-\log (12)+x \log (12)}{x (x-\log (x))^2}+\frac {-x^2+\log (12)-x \log (12)}{x^2 (x-\log (x))}\right ) \, dx+\log (12) \int \left (\frac {1}{x^2}+\frac {1}{(x-\log (x))^2}-\frac {2}{x (x-\log (x))}\right ) \, dx-\log (12) \int \frac {2+\log (x)}{x^2 \log (x)} \, dx+(2 \log (12)) \int \left (\frac {1}{x (x-\log (x))^2}+\frac {1}{x^2 (x-\log (x))}+\frac {1}{x^2 \log (x)}\right ) \, dx+\int \left (\frac {2 \left (1+\frac {\log (12)}{2}\right )}{(x-\log (x))^2}-\frac {4 \log (12)}{x (x-\log (x))^2}\right ) \, dx \\ & = -\frac {\log (12)}{x}-\text {Ei}(-\log (x)) \log (12) (2+\log (x))-\frac {\log (12) \log \left (\frac {5}{x \log ^2(x)}\right )}{x}-2 \int \frac {x^2-\log (12)+x \log (12)}{x (x-\log (x))^2} \, dx-2 \int \frac {-x^2+\log (12)-x \log (12)}{x^2 (x-\log (x))} \, dx+\log (12) \int \frac {\text {Ei}(-\log (x))}{x} \, dx+\log (12) \int \frac {1}{(x-\log (x))^2} \, dx+(2 \log (12)) \int \frac {1}{x (x-\log (x))^2} \, dx+(2 \log (12)) \int \frac {1}{x^2 (x-\log (x))} \, dx-(2 \log (12)) \int \frac {1}{x (x-\log (x))} \, dx+(2 \log (12)) \int \frac {1}{x^2 \log (x)} \, dx-(4 \log (12)) \int \frac {1}{x (x-\log (x))^2} \, dx+(2+\log (12)) \int \frac {1}{(x-\log (x))^2} \, dx \\ & = -\frac {\log (12)}{x}-\text {Ei}(-\log (x)) \log (12) (2+\log (x))-\frac {\log (12) \log \left (\frac {5}{x \log ^2(x)}\right )}{x}-2 \int \left (\frac {x}{(x-\log (x))^2}+\frac {\log (12)}{(x-\log (x))^2}-\frac {\log (12)}{x (x-\log (x))^2}\right ) \, dx-2 \int \left (\frac {\log (12)}{x^2 (x-\log (x))}-\frac {\log (12)}{x (x-\log (x))}+\frac {1}{-x+\log (x)}\right ) \, dx+\log (12) \int \frac {1}{(x-\log (x))^2} \, dx+\log (12) \text {Subst}(\int \text {Ei}(-x) \, dx,x,\log (x))+(2 \log (12)) \int \frac {1}{x (x-\log (x))^2} \, dx+(2 \log (12)) \int \frac {1}{x^2 (x-\log (x))} \, dx-(2 \log (12)) \int \frac {1}{x (x-\log (x))} \, dx+(2 \log (12)) \text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right )-(4 \log (12)) \int \frac {1}{x (x-\log (x))^2} \, dx+(2+\log (12)) \int \frac {1}{(x-\log (x))^2} \, dx \\ & = 2 \text {Ei}(-\log (x)) \log (12)+\text {Ei}(-\log (x)) \log (12) \log (x)-\text {Ei}(-\log (x)) \log (12) (2+\log (x))-\frac {\log (12) \log \left (\frac {5}{x \log ^2(x)}\right )}{x}-2 \int \frac {x}{(x-\log (x))^2} \, dx-2 \int \frac {1}{-x+\log (x)} \, dx+\log (12) \int \frac {1}{(x-\log (x))^2} \, dx-(2 \log (12)) \int \frac {1}{(x-\log (x))^2} \, dx+2 \left ((2 \log (12)) \int \frac {1}{x (x-\log (x))^2} \, dx\right )-(4 \log (12)) \int \frac {1}{x (x-\log (x))^2} \, dx+(2+\log (12)) \int \frac {1}{(x-\log (x))^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {2 x^2 \log (12)+\left (2 x^2+\left (-4 x+x^2\right ) \log (12)\right ) \log (x)+\left (-2 x^2+(2-2 x) \log (12)\right ) \log ^2(x)+\log (12) \log ^3(x)+\left (x^2 \log (12) \log (x)-2 x \log (12) \log ^2(x)+\log (12) \log ^3(x)\right ) \log \left (\frac {5}{x \log ^2(x)}\right )}{x^4 \log (x)-2 x^3 \log ^2(x)+x^2 \log ^3(x)} \, dx=\frac {2 x}{x-\log (x)}-\frac {\log (12) \log \left (\frac {5}{x \log ^2(x)}\right )}{x} \]

[In]

Integrate[(2*x^2*Log[12] + (2*x^2 + (-4*x + x^2)*Log[12])*Log[x] + (-2*x^2 + (2 - 2*x)*Log[12])*Log[x]^2 + Log
[12]*Log[x]^3 + (x^2*Log[12]*Log[x] - 2*x*Log[12]*Log[x]^2 + Log[12]*Log[x]^3)*Log[5/(x*Log[x]^2)])/(x^4*Log[x
] - 2*x^3*Log[x]^2 + x^2*Log[x]^3),x]

[Out]

(2*x)/(x - Log[x]) - (Log[12]*Log[5/(x*Log[x]^2)])/x

Maple [A] (verified)

Time = 2.06 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31

method	result	size
default	\(-\frac {\ln \left (12\right ) \ln \left (\frac {1}{x \ln \left (x \right )^{2}}\right )}{x}-\frac {\ln \left (12\right ) \ln \left (5\right )}{x}-\frac {2 x}{\ln \left (x \right )-x}\)	\(38\)
parallelrisch	\(\frac {-2 \ln \left (12\right ) x \ln \left (\frac {5}{x \ln \left (x \right )^{2}}\right )+2 \ln \left (12\right ) \ln \left (\frac {5}{x \ln \left (x \right )^{2}}\right ) \ln \left (x \right )+4 x^{2}}{2 x \left (x -\ln \left (x \right )\right )}\)	\(51\)
risch	\(\text {Expression too large to display}\)	\(794\)

[In]

int(((ln(12)*ln(x)^3-2*x*ln(12)*ln(x)^2+x^2*ln(12)*ln(x))*ln(5/x/ln(x)^2)+ln(12)*ln(x)^3+((2-2*x)*ln(12)-2*x^2
)*ln(x)^2+((x^2-4*x)*ln(12)+2*x^2)*ln(x)+2*x^2*ln(12))/(x^2*ln(x)^3-2*x^3*ln(x)^2+x^4*ln(x)),x,method=_RETURNV
ERBOSE)

[Out]

-ln(12)*ln(1/x/ln(x)^2)/x-ln(12)*ln(5)/x-2*x/(ln(x)-x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41 \[ \int \frac {2 x^2 \log (12)+\left (2 x^2+\left (-4 x+x^2\right ) \log (12)\right ) \log (x)+\left (-2 x^2+(2-2 x) \log (12)\right ) \log ^2(x)+\log (12) \log ^3(x)+\left (x^2 \log (12) \log (x)-2 x \log (12) \log ^2(x)+\log (12) \log ^3(x)\right ) \log \left (\frac {5}{x \log ^2(x)}\right )}{x^4 \log (x)-2 x^3 \log ^2(x)+x^2 \log ^3(x)} \, dx=\frac {2 \, x^{2} - {\left (x \log \left (12\right ) - \log \left (12\right ) \log \left (x\right )\right )} \log \left (\frac {5}{x \log \left (x\right )^{2}}\right )}{x^{2} - x \log \left (x\right )} \]

[In]

integrate(((log(12)*log(x)^3-2*x*log(12)*log(x)^2+x^2*log(12)*log(x))*log(5/x/log(x)^2)+log(12)*log(x)^3+((2-2
*x)*log(12)-2*x^2)*log(x)^2+((x^2-4*x)*log(12)+2*x^2)*log(x)+2*x^2*log(12))/(x^2*log(x)^3-2*x^3*log(x)^2+x^4*l
og(x)),x, algorithm="fricas")

[Out]

(2*x^2 - (x*log(12) - log(12)*log(x))*log(5/(x*log(x)^2)))/(x^2 - x*log(x))

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {2 x^2 \log (12)+\left (2 x^2+\left (-4 x+x^2\right ) \log (12)\right ) \log (x)+\left (-2 x^2+(2-2 x) \log (12)\right ) \log ^2(x)+\log (12) \log ^3(x)+\left (x^2 \log (12) \log (x)-2 x \log (12) \log ^2(x)+\log (12) \log ^3(x)\right ) \log \left (\frac {5}{x \log ^2(x)}\right )}{x^4 \log (x)-2 x^3 \log ^2(x)+x^2 \log ^3(x)} \, dx=- \frac {2 x}{- x + \log {\left (x \right )}} - \frac {\log {\left (12 \right )} \log {\left (\frac {5}{x \log {\left (x \right )}^{2}} \right )}}{x} \]

[In]

integrate(((ln(12)*ln(x)**3-2*x*ln(12)*ln(x)**2+x**2*ln(12)*ln(x))*ln(5/x/ln(x)**2)+ln(12)*ln(x)**3+((2-2*x)*l
n(12)-2*x**2)*ln(x)**2+((x**2-4*x)*ln(12)+2*x**2)*ln(x)+2*x**2*ln(12))/(x**2*ln(x)**3-2*x**3*ln(x)**2+x**4*ln(
x)),x)

[Out]

-2*x/(-x + log(x)) - log(12)*log(5/(x*log(x)**2))/x

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (29) = 58\).

Time = 0.32 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.31 \[ \int \frac {2 x^2 \log (12)+\left (2 x^2+\left (-4 x+x^2\right ) \log (12)\right ) \log (x)+\left (-2 x^2+(2-2 x) \log (12)\right ) \log ^2(x)+\log (12) \log ^3(x)+\left (x^2 \log (12) \log (x)-2 x \log (12) \log ^2(x)+\log (12) \log ^3(x)\right ) \log \left (\frac {5}{x \log ^2(x)}\right )}{x^4 \log (x)-2 x^3 \log ^2(x)+x^2 \log ^3(x)} \, dx=-\frac {{\left (\log \left (3\right ) + 2 \, \log \left (2\right )\right )} \log \left (x\right )^{2} + {\left (\log \left (5\right ) \log \left (3\right ) + 2 \, \log \left (5\right ) \log \left (2\right )\right )} x - 2 \, x^{2} - {\left (x {\left (\log \left (3\right ) + 2 \, \log \left (2\right )\right )} + \log \left (5\right ) \log \left (3\right ) + 2 \, \log \left (5\right ) \log \left (2\right )\right )} \log \left (x\right ) - 2 \, {\left (x {\left (\log \left (3\right ) + 2 \, \log \left (2\right )\right )} - {\left (\log \left (3\right ) + 2 \, \log \left (2\right )\right )} \log \left (x\right )\right )} \log \left (\log \left (x\right )\right )}{x^{2} - x \log \left (x\right )} \]

[In]

integrate(((log(12)*log(x)^3-2*x*log(12)*log(x)^2+x^2*log(12)*log(x))*log(5/x/log(x)^2)+log(12)*log(x)^3+((2-2
*x)*log(12)-2*x^2)*log(x)^2+((x^2-4*x)*log(12)+2*x^2)*log(x)+2*x^2*log(12))/(x^2*log(x)^3-2*x^3*log(x)^2+x^4*l
og(x)),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.38 \[ \int \frac {2 x^2 \log (12)+\left (2 x^2+\left (-4 x+x^2\right ) \log (12)\right ) \log (x)+\left (-2 x^2+(2-2 x) \log (12)\right ) \log ^2(x)+\log (12) \log ^3(x)+\left (x^2 \log (12) \log (x)-2 x \log (12) \log ^2(x)+\log (12) \log ^3(x)\right ) \log \left (\frac {5}{x \log ^2(x)}\right )}{x^4 \log (x)-2 x^3 \log ^2(x)+x^2 \log ^3(x)} \, dx=-\frac {\log \left (12\right ) \log \left (5\right )}{x} + \frac {\log \left (12\right ) \log \left (\log \left (x\right )^{2}\right )}{x} + \frac {\log \left (12\right ) \log \left (x\right )}{x} + \frac {2 \, x}{x - \log \left (x\right )} \]

[In]

integrate(((log(12)*log(x)^3-2*x*log(12)*log(x)^2+x^2*log(12)*log(x))*log(5/x/log(x)^2)+log(12)*log(x)^3+((2-2
*x)*log(12)-2*x^2)*log(x)^2+((x^2-4*x)*log(12)+2*x^2)*log(x)+2*x^2*log(12))/(x^2*log(x)^3-2*x^3*log(x)^2+x^4*l
og(x)),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.84 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.86 \[ \int \frac {2 x^2 \log (12)+\left (2 x^2+\left (-4 x+x^2\right ) \log (12)\right ) \log (x)+\left (-2 x^2+(2-2 x) \log (12)\right ) \log ^2(x)+\log (12) \log ^3(x)+\left (x^2 \log (12) \log (x)-2 x \log (12) \log ^2(x)+\log (12) \log ^3(x)\right ) \log \left (\frac {5}{x \log ^2(x)}\right )}{x^4 \log (x)-2 x^3 \log ^2(x)+x^2 \log ^3(x)} \, dx=\frac {2}{x-1}-\frac {\frac {2\,x}{x-1}-\frac {2\,x\,\ln \left (x\right )}{x-1}}{x-\ln \left (x\right )}-\frac {\ln \left (12\right )\,\ln \left (\frac {5}{x\,{\ln \left (x\right )}^2}\right )}{x} \]

[In]

int((log(12)*log(x)^3 - log(x)*(log(12)*(4*x - x^2) - 2*x^2) - log(x)^2*(log(12)*(2*x - 2) + 2*x^2) + 2*x^2*lo
g(12) + log(5/(x*log(x)^2))*(log(12)*log(x)^3 - 2*x*log(12)*log(x)^2 + x^2*log(12)*log(x)))/(x^4*log(x) + x^2*
log(x)^3 - 2*x^3*log(x)^2),x)

[Out]

2/(x - 1) - ((2*x)/(x - 1) - (2*x*log(x))/(x - 1))/(x - log(x)) - (log(12)*log(5/(x*log(x)^2)))/x