\(\int \frac {3+3 x+(3 x+3 \log (x)) \log (x+\log (x)) \log (\frac {3}{\log (x+\log (x))})+(-x-2 x^2+e^x (-x-x^2)+(-1+e^x (-1-x)-2 x) \log (x)) \log (x+\log (x)) \log ^2(\frac {3}{\log (x+\log (x))})}{(x+\log (x)) \log (x+\log (x)) \log ^2(\frac {3}{\log (x+\log (x))})} \, dx\) [9900]

Optimal result

Integrand size = 112, antiderivative size = 30 \[ \int \frac {3+3 x+(3 x+3 \log (x)) \log (x+\log (x)) \log \left (\frac {3}{\log (x+\log (x))}\right )+\left (-x-2 x^2+e^x \left (-x-x^2\right )+\left (-1+e^x (-1-x)-2 x\right ) \log (x)\right ) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )}{(x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )} \, dx=5-x+x \left (-e^x-x+\frac {3}{\log \left (\frac {3}{\log (x+\log (x))}\right )}\right ) \]

[Out]

(3/ln(3/ln(x+ln(x)))-x-exp(x))*x+5-x

Rubi [F]

\[ \int \frac {3+3 x+(3 x+3 \log (x)) \log (x+\log (x)) \log \left (\frac {3}{\log (x+\log (x))}\right )+\left (-x-2 x^2+e^x \left (-x-x^2\right )+\left (-1+e^x (-1-x)-2 x\right ) \log (x)\right ) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )}{(x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )} \, dx=\int \frac {3+3 x+(3 x+3 \log (x)) \log (x+\log (x)) \log \left (\frac {3}{\log (x+\log (x))}\right )+\left (-x-2 x^2+e^x \left (-x-x^2\right )+\left (-1+e^x (-1-x)-2 x\right ) \log (x)\right ) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )}{(x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )} \, dx \]

[In]

Int[(3 + 3*x + (3*x + 3*Log[x])*Log[x + Log[x]]*Log[3/Log[x + Log[x]]] + (-x - 2*x^2 + E^x*(-x - x^2) + (-1 +
E^x*(-1 - x) - 2*x)*Log[x])*Log[x + Log[x]]*Log[3/Log[x + Log[x]]]^2)/((x + Log[x])*Log[x + Log[x]]*Log[3/Log[
x + Log[x]]]^2),x]

[Out]

E^x - x - x^2 - E^x*(1 + x) + 3*Defer[Int][1/((x + Log[x])*Log[x + Log[x]]*Log[3/Log[x + Log[x]]]^2), x] + 3*D
efer[Int][x/((x + Log[x])*Log[x + Log[x]]*Log[3/Log[x + Log[x]]]^2), x] + 3*Defer[Int][Log[3/Log[x + Log[x]]]^
(-1), x]

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {3+3 x+(3 x+3 \log (x)) \log (x+\log (x)) \log \left (\frac {3}{\log (x+\log (x))}\right )+\left (-x-2 x^2+e^x \left (-x-x^2\right )+\left (-1+e^x (-1-x)-2 x\right ) \log (x)\right ) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )}{(x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )} \, dx=x \left (-1-e^x-x+\frac {3}{\log \left (\frac {3}{\log (x+\log (x))}\right )}\right ) \]

[In]

Integrate[(3 + 3*x + (3*x + 3*Log[x])*Log[x + Log[x]]*Log[3/Log[x + Log[x]]] + (-x - 2*x^2 + E^x*(-x - x^2) +
(-1 + E^x*(-1 - x) - 2*x)*Log[x])*Log[x + Log[x]]*Log[3/Log[x + Log[x]]]^2)/((x + Log[x])*Log[x + Log[x]]*Log[
3/Log[x + Log[x]]]^2),x]

[Out]

x*(-1 - E^x - x + 3/Log[3/Log[x + Log[x]]])

Maple [A] (verified)

Time = 21.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03

method	result	size
default	\(-{\mathrm e}^{x} x -x^{2}-x +\frac {3 x}{\ln \left (3\right )-\ln \left (\ln \left (x +\ln \left (x \right )\right )\right )}\)	\(31\)
parts	\(-{\mathrm e}^{x} x -x^{2}-x +\frac {3 x}{\ln \left (3\right )-\ln \left (\ln \left (x +\ln \left (x \right )\right )\right )}\)	\(31\)
risch	\(-x^{2}-{\mathrm e}^{x} x -x -\frac {6 i x}{-2 i \ln \left (3\right )+2 i \ln \left (\ln \left (x +\ln \left (x \right )\right )\right )}\)	\(36\)
parallelrisch	\(\frac {-2 \ln \left (\frac {3}{\ln \left (x +\ln \left (x \right )\right )}\right ) x^{2}-2 \ln \left (\frac {3}{\ln \left (x +\ln \left (x \right )\right )}\right ) {\mathrm e}^{x} x -2 \ln \left (\frac {3}{\ln \left (x +\ln \left (x \right )\right )}\right ) x +6 x}{2 \ln \left (\frac {3}{\ln \left (x +\ln \left (x \right )\right )}\right )}\)	\(62\)

[In]

int(((((-1-x)*exp(x)-2*x-1)*ln(x)+(-x^2-x)*exp(x)-2*x^2-x)*ln(x+ln(x))*ln(3/ln(x+ln(x)))^2+(3*x+3*ln(x))*ln(x+
ln(x))*ln(3/ln(x+ln(x)))+3*x+3)/(x+ln(x))/ln(x+ln(x))/ln(3/ln(x+ln(x)))^2,x,method=_RETURNVERBOSE)

[Out]

-exp(x)*x-x^2-x+3*x/(ln(3)-ln(ln(x+ln(x))))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27 \[ \int \frac {3+3 x+(3 x+3 \log (x)) \log (x+\log (x)) \log \left (\frac {3}{\log (x+\log (x))}\right )+\left (-x-2 x^2+e^x \left (-x-x^2\right )+\left (-1+e^x (-1-x)-2 x\right ) \log (x)\right ) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )}{(x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )} \, dx=-\frac {{\left (x^{2} + x e^{x} + x\right )} \log \left (\frac {3}{\log \left (x + \log \left (x\right )\right )}\right ) - 3 \, x}{\log \left (\frac {3}{\log \left (x + \log \left (x\right )\right )}\right )} \]

[In]

integrate(((((-1-x)*exp(x)-2*x-1)*log(x)+(-x^2-x)*exp(x)-2*x^2-x)*log(x+log(x))*log(3/log(x+log(x)))^2+(3*x+3*
log(x))*log(x+log(x))*log(3/log(x+log(x)))+3*x+3)/(x+log(x))/log(x+log(x))/log(3/log(x+log(x)))^2,x, algorithm
="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 1.54 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73 \[ \int \frac {3+3 x+(3 x+3 \log (x)) \log (x+\log (x)) \log \left (\frac {3}{\log (x+\log (x))}\right )+\left (-x-2 x^2+e^x \left (-x-x^2\right )+\left (-1+e^x (-1-x)-2 x\right ) \log (x)\right ) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )}{(x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )} \, dx=- x^{2} - x e^{x} - x + \frac {3 x}{\log {\left (\frac {3}{\log {\left (x + \log {\left (x \right )} \right )}} \right )}} \]

[In]

integrate(((((-1-x)*exp(x)-2*x-1)*ln(x)+(-x**2-x)*exp(x)-2*x**2-x)*ln(x+ln(x))*ln(3/ln(x+ln(x)))**2+(3*x+3*ln(
x))*ln(x+ln(x))*ln(3/ln(x+ln(x)))+3*x+3)/(x+ln(x))/ln(x+ln(x))/ln(3/ln(x+ln(x)))**2,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.70 \[ \int \frac {3+3 x+(3 x+3 \log (x)) \log (x+\log (x)) \log \left (\frac {3}{\log (x+\log (x))}\right )+\left (-x-2 x^2+e^x \left (-x-x^2\right )+\left (-1+e^x (-1-x)-2 x\right ) \log (x)\right ) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )}{(x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )} \, dx=-\frac {x^{2} \log \left (3\right ) + x e^{x} \log \left (3\right ) + x {\left (\log \left (3\right ) - 3\right )} - {\left (x^{2} + x e^{x} + x\right )} \log \left (\log \left (x + \log \left (x\right )\right )\right )}{\log \left (3\right ) - \log \left (\log \left (x + \log \left (x\right )\right )\right )} \]

[In]

integrate(((((-1-x)*exp(x)-2*x-1)*log(x)+(-x^2-x)*exp(x)-2*x^2-x)*log(x+log(x))*log(3/log(x+log(x)))^2+(3*x+3*
log(x))*log(x+log(x))*log(3/log(x+log(x)))+3*x+3)/(x+log(x))/log(x+log(x))/log(3/log(x+log(x)))^2,x, algorithm
="maxima")

[Out]

-(x^2*log(3) + x*e^x*log(3) + x*(log(3) - 3) - (x^2 + x*e^x + x)*log(log(x + log(x))))/(log(3) - log(log(x + l
og(x))))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (26) = 52\).

Time = 0.46 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.20 \[ \int \frac {3+3 x+(3 x+3 \log (x)) \log (x+\log (x)) \log \left (\frac {3}{\log (x+\log (x))}\right )+\left (-x-2 x^2+e^x \left (-x-x^2\right )+\left (-1+e^x (-1-x)-2 x\right ) \log (x)\right ) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )}{(x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )} \, dx=-\frac {x^{2} \log \left (3\right ) + x e^{x} \log \left (3\right ) - x^{2} \log \left (\log \left (x + \log \left (x\right )\right )\right ) - x e^{x} \log \left (\log \left (x + \log \left (x\right )\right )\right ) + x \log \left (3\right ) - x \log \left (\log \left (x + \log \left (x\right )\right )\right ) - 3 \, x}{\log \left (3\right ) - \log \left (\log \left (x + \log \left (x\right )\right )\right )} \]

[In]

integrate(((((-1-x)*exp(x)-2*x-1)*log(x)+(-x^2-x)*exp(x)-2*x^2-x)*log(x+log(x))*log(3/log(x+log(x)))^2+(3*x+3*
log(x))*log(x+log(x))*log(3/log(x+log(x)))+3*x+3)/(x+log(x))/log(x+log(x))/log(3/log(x+log(x)))^2,x, algorithm
="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 16.90 (sec) , antiderivative size = 204, normalized size of antiderivative = 6.80 \[ \int \frac {3+3 x+(3 x+3 \log (x)) \log (x+\log (x)) \log \left (\frac {3}{\log (x+\log (x))}\right )+\left (-x-2 x^2+e^x \left (-x-x^2\right )+\left (-1+e^x (-1-x)-2 x\right ) \log (x)\right ) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )}{(x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )} \, dx=\frac {3\,x+\frac {3\,x\,\ln \left (x+\ln \left (x\right )\right )\,\ln \left (\frac {3}{\ln \left (x+\ln \left (x\right )\right )}\right )\,\left (x+\ln \left (x\right )\right )}{x+1}}{\ln \left (\frac {3}{\ln \left (x+\ln \left (x\right )\right )}\right )}-\ln \left (x+\ln \left (x\right )\right )\,\left (\ln \left (x\right )\,\left (\frac {3\,\left (x^3+2\,x^2+x\right )}{x\,{\left (x+1\right )}^2}-\frac {3\,x^2+3\,x}{x\,{\left (x+1\right )}^2}\right )-\frac {3\,x^4+12\,x^3+12\,x^2+3\,x}{x\,{\left (x+1\right )}^2}+\frac {6\,x^4+12\,x^3+6\,x^2}{x\,{\left (x+1\right )}^2}+\frac {3\,\left (x^3+3\,x^2+2\,x\right )}{x\,{\left (x+1\right )}^2}-\frac {3\,x^2+3\,x}{x\,{\left (x+1\right )}^2}\right )-x-x\,{\mathrm {e}}^x-x^2 \]

[In]

int((3*x + log(x + log(x))*log(3/log(x + log(x)))*(3*x + 3*log(x)) - log(x + log(x))*log(3/log(x + log(x)))^2*
(x + log(x)*(2*x + exp(x)*(x + 1) + 1) + 2*x^2 + exp(x)*(x + x^2)) + 3)/(log(x + log(x))*log(3/log(x + log(x))
)^2*(x + log(x))),x)

[Out]

(3*x + (3*x*log(x + log(x))*log(3/log(x + log(x)))*(x + log(x)))/(x + 1))/log(3/log(x + log(x))) - log(x + log
(x))*(log(x)*((3*(x + 2*x^2 + x^3))/(x*(x + 1)^2) - (3*x + 3*x^2)/(x*(x + 1)^2)) - (3*x + 12*x^2 + 12*x^3 + 3*
x^4)/(x*(x + 1)^2) + (6*x^2 + 12*x^3 + 6*x^4)/(x*(x + 1)^2) + (3*(2*x + 3*x^2 + x^3))/(x*(x + 1)^2) - (3*x + 3
*x^2)/(x*(x + 1)^2)) - x - x*exp(x) - x^2