\(\int \frac {(-x+6 x^2-3 x^3) \log (2)+(-x+2 x^2-x^3) \log (2) \log (4)+((1+3 x-3 x^2) \log (2)+(x-x^2) \log (2) \log (4)) \log (x)+((x-x^2) \log (2)-x \log (2) \log (x)) \log (-x+x^2+x \log (x))}{e^x (-2 x+2 x^2)+2 e^x x \log (x)} \, dx\) [6922]

Optimal result

Integrand size = 122, antiderivative size = 30 \[ \int \frac {\left (-x+6 x^2-3 x^3\right ) \log (2)+\left (-x+2 x^2-x^3\right ) \log (2) \log (4)+\left (\left (1+3 x-3 x^2\right ) \log (2)+\left (x-x^2\right ) \log (2) \log (4)\right ) \log (x)+\left (\left (x-x^2\right ) \log (2)-x \log (2) \log (x)\right ) \log \left (-x+x^2+x \log (x)\right )}{e^x \left (-2 x+2 x^2\right )+2 e^x x \log (x)} \, dx=\frac {1}{2} e^{-x} \log (2) \left (x (3+\log (4))+\log \left (-x+x^2+x \log (x)\right )\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {\left (-x+6 x^2-3 x^3\right ) \log (2)+\left (-x+2 x^2-x^3\right ) \log (2) \log (4)+\left (\left (1+3 x-3 x^2\right ) \log (2)+\left (x-x^2\right ) \log (2) \log (4)\right ) \log (x)+\left (\left (x-x^2\right ) \log (2)-x \log (2) \log (x)\right ) \log \left (-x+x^2+x \log (x)\right )}{e^x \left (-2 x+2 x^2\right )+2 e^x x \log (x)} \, dx=\int \frac {\left (-x+6 x^2-3 x^3\right ) \log (2)+\left (-x+2 x^2-x^3\right ) \log (2) \log (4)+\left (\left (1+3 x-3 x^2\right ) \log (2)+\left (x-x^2\right ) \log (2) \log (4)\right ) \log (x)+\left (\left (x-x^2\right ) \log (2)-x \log (2) \log (x)\right ) \log \left (-x+x^2+x \log (x)\right )}{e^x \left (-2 x+2 x^2\right )+2 e^x x \log (x)} \, dx \]

[In]

Int[((-x + 6*x^2 - 3*x^3)*Log[2] + (-x + 2*x^2 - x^3)*Log[2]*Log[4] + ((1 + 3*x - 3*x^2)*Log[2] + (x - x^2)*Lo
g[2]*Log[4])*Log[x] + ((x - x^2)*Log[2] - x*Log[2]*Log[x])*Log[-x + x^2 + x*Log[x]])/(E^x*(-2*x + 2*x^2) + 2*E
^x*x*Log[x]),x]

[Out]

(ExpIntegralEi[-x]*Log[2])/2 + (x*Log[2]*(3 + Log[4]))/(2*E^x) + (Log[2]*Defer[Int][1/(E^x*(1 - x - Log[x])),
x])/2 + (Log[2]*(1 + Log[4])*Defer[Int][1/(E^x*(1 - x - Log[x])), x])/2 + (Log[2]*(3 + Log[4])*Defer[Int][1/(E
^x*(-1 + x + Log[x])), x])/2 + (Log[2]*Defer[Int][1/(E^x*x*(-1 + x + Log[x])), x])/2 - (Log[2]*Defer[Int][Log[
-x + x^2 + x*Log[x]]/E^x, x])/2

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {\left (-x+6 x^2-3 x^3\right ) \log (2)+\left (-x+2 x^2-x^3\right ) \log (2) \log (4)+\left (\left (1+3 x-3 x^2\right ) \log (2)+\left (x-x^2\right ) \log (2) \log (4)\right ) \log (x)+\left (\left (x-x^2\right ) \log (2)-x \log (2) \log (x)\right ) \log \left (-x+x^2+x \log (x)\right )}{e^x \left (-2 x+2 x^2\right )+2 e^x x \log (x)} \, dx=\frac {1}{2} e^{-x} \log (2) (x (3+\log (4))+\log (x (-1+x+\log (x)))) \]

[In]

Integrate[((-x + 6*x^2 - 3*x^3)*Log[2] + (-x + 2*x^2 - x^3)*Log[2]*Log[4] + ((1 + 3*x - 3*x^2)*Log[2] + (x - x
^2)*Log[2]*Log[4])*Log[x] + ((x - x^2)*Log[2] - x*Log[2]*Log[x])*Log[-x + x^2 + x*Log[x]])/(E^x*(-2*x + 2*x^2)
+ 2*E^x*x*Log[x]),x]

[Out]

(Log[2]*(x*(3 + Log[4]) + Log[x*(-1 + x + Log[x])]))/(2*E^x)

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03

method	result	size
parallelrisch	\(\frac {\left (2 x \ln \left (2\right )^{2}+3 x \ln \left (2\right )+\ln \left (2\right ) \ln \left (x \left (-1+\ln \left (x \right )+x \right )\right )\right ) {\mathrm e}^{-x}}{2}\)	\(31\)
risch	\(\frac {\ln \left (2\right ) {\mathrm e}^{-x} \ln \left (-1+\ln \left (x \right )+x \right )}{2}+\frac {\ln \left (2\right ) \left (i \pi \,\operatorname {csgn}\left (i \left (-1+\ln \left (x \right )+x \right )\right ) \operatorname {csgn}\left (i x \left (-1+\ln \left (x \right )+x \right )\right )^{2}-i \pi \,\operatorname {csgn}\left (i \left (-1+\ln \left (x \right )+x \right )\right ) \operatorname {csgn}\left (i x \left (-1+\ln \left (x \right )+x \right )\right ) \operatorname {csgn}\left (i x \right )-i \pi \operatorname {csgn}\left (i x \left (-1+\ln \left (x \right )+x \right )\right )^{3}+i \pi \operatorname {csgn}\left (i x \left (-1+\ln \left (x \right )+x \right )\right )^{2} \operatorname {csgn}\left (i x \right )+4 x \ln \left (2\right )+6 x +2 \ln \left (x \right )\right ) {\mathrm e}^{-x}}{4}\)	\(127\)

[In]

int(((-x*ln(2)*ln(x)+(-x^2+x)*ln(2))*ln(x*ln(x)+x^2-x)+(2*(-x^2+x)*ln(2)^2+(-3*x^2+3*x+1)*ln(2))*ln(x)+2*(-x^3
+2*x^2-x)*ln(2)^2+(-3*x^3+6*x^2-x)*ln(2))/(2*x*exp(x)*ln(x)+(2*x^2-2*x)*exp(x)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {\left (-x+6 x^2-3 x^3\right ) \log (2)+\left (-x+2 x^2-x^3\right ) \log (2) \log (4)+\left (\left (1+3 x-3 x^2\right ) \log (2)+\left (x-x^2\right ) \log (2) \log (4)\right ) \log (x)+\left (\left (x-x^2\right ) \log (2)-x \log (2) \log (x)\right ) \log \left (-x+x^2+x \log (x)\right )}{e^x \left (-2 x+2 x^2\right )+2 e^x x \log (x)} \, dx=\frac {1}{2} \, {\left (2 \, x \log \left (2\right )^{2} + 3 \, x \log \left (2\right ) + \log \left (2\right ) \log \left (x^{2} + x \log \left (x\right ) - x\right )\right )} e^{\left (-x\right )} \]

[In]

integrate(((-x*log(2)*log(x)+(-x^2+x)*log(2))*log(x*log(x)+x^2-x)+(2*(-x^2+x)*log(2)^2+(-3*x^2+3*x+1)*log(2))*
log(x)+2*(-x^3+2*x^2-x)*log(2)^2+(-3*x^3+6*x^2-x)*log(2))/(2*x*exp(x)*log(x)+(2*x^2-2*x)*exp(x)),x, algorithm=
"fricas")

[Out]

1/2*(2*x*log(2)^2 + 3*x*log(2) + log(2)*log(x^2 + x*log(x) - x))*e^(-x)

Sympy [A] (verification not implemented)

Time = 12.78 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {\left (-x+6 x^2-3 x^3\right ) \log (2)+\left (-x+2 x^2-x^3\right ) \log (2) \log (4)+\left (\left (1+3 x-3 x^2\right ) \log (2)+\left (x-x^2\right ) \log (2) \log (4)\right ) \log (x)+\left (\left (x-x^2\right ) \log (2)-x \log (2) \log (x)\right ) \log \left (-x+x^2+x \log (x)\right )}{e^x \left (-2 x+2 x^2\right )+2 e^x x \log (x)} \, dx=\frac {\left (2 x \log {\left (2 \right )}^{2} + 3 x \log {\left (2 \right )} + \log {\left (2 \right )} \log {\left (x^{2} + x \log {\left (x \right )} - x \right )}\right ) e^{- x}}{2} \]

[In]

integrate(((-x*ln(2)*ln(x)+(-x**2+x)*ln(2))*ln(x*ln(x)+x**2-x)+(2*(-x**2+x)*ln(2)**2+(-3*x**2+3*x+1)*ln(2))*ln
(x)+2*(-x**3+2*x**2-x)*ln(2)**2+(-3*x**3+6*x**2-x)*ln(2))/(2*x*exp(x)*ln(x)+(2*x**2-2*x)*exp(x)),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {\left (-x+6 x^2-3 x^3\right ) \log (2)+\left (-x+2 x^2-x^3\right ) \log (2) \log (4)+\left (\left (1+3 x-3 x^2\right ) \log (2)+\left (x-x^2\right ) \log (2) \log (4)\right ) \log (x)+\left (\left (x-x^2\right ) \log (2)-x \log (2) \log (x)\right ) \log \left (-x+x^2+x \log (x)\right )}{e^x \left (-2 x+2 x^2\right )+2 e^x x \log (x)} \, dx=\frac {1}{2} \, {\left ({\left (2 \, \log \left (2\right )^{2} + 3 \, \log \left (2\right )\right )} x + \log \left (2\right ) \log \left (x + \log \left (x\right ) - 1\right ) + \log \left (2\right ) \log \left (x\right )\right )} e^{\left (-x\right )} \]

[In]

integrate(((-x*log(2)*log(x)+(-x^2+x)*log(2))*log(x*log(x)+x^2-x)+(2*(-x^2+x)*log(2)^2+(-3*x^2+3*x+1)*log(2))*
log(x)+2*(-x^3+2*x^2-x)*log(2)^2+(-3*x^3+6*x^2-x)*log(2))/(2*x*exp(x)*log(x)+(2*x^2-2*x)*exp(x)),x, algorithm=
"maxima")

[Out]

1/2*((2*log(2)^2 + 3*log(2))*x + log(2)*log(x + log(x) - 1) + log(2)*log(x))*e^(-x)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47 \[ \int \frac {\left (-x+6 x^2-3 x^3\right ) \log (2)+\left (-x+2 x^2-x^3\right ) \log (2) \log (4)+\left (\left (1+3 x-3 x^2\right ) \log (2)+\left (x-x^2\right ) \log (2) \log (4)\right ) \log (x)+\left (\left (x-x^2\right ) \log (2)-x \log (2) \log (x)\right ) \log \left (-x+x^2+x \log (x)\right )}{e^x \left (-2 x+2 x^2\right )+2 e^x x \log (x)} \, dx=x e^{\left (-x\right )} \log \left (2\right )^{2} + \frac {3}{2} \, x e^{\left (-x\right )} \log \left (2\right ) + \frac {1}{2} \, e^{\left (-x\right )} \log \left (2\right ) \log \left (x + \log \left (x\right ) - 1\right ) + \frac {1}{2} \, e^{\left (-x\right )} \log \left (2\right ) \log \left (x\right ) \]

[In]

integrate(((-x*log(2)*log(x)+(-x^2+x)*log(2))*log(x*log(x)+x^2-x)+(2*(-x^2+x)*log(2)^2+(-3*x^2+3*x+1)*log(2))*
log(x)+2*(-x^3+2*x^2-x)*log(2)^2+(-3*x^3+6*x^2-x)*log(2))/(2*x*exp(x)*log(x)+(2*x^2-2*x)*exp(x)),x, algorithm=
"giac")

[Out]

x*e^(-x)*log(2)^2 + 3/2*x*e^(-x)*log(2) + 1/2*e^(-x)*log(2)*log(x + log(x) - 1) + 1/2*e^(-x)*log(2)*log(x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-x+6 x^2-3 x^3\right ) \log (2)+\left (-x+2 x^2-x^3\right ) \log (2) \log (4)+\left (\left (1+3 x-3 x^2\right ) \log (2)+\left (x-x^2\right ) \log (2) \log (4)\right ) \log (x)+\left (\left (x-x^2\right ) \log (2)-x \log (2) \log (x)\right ) \log \left (-x+x^2+x \log (x)\right )}{e^x \left (-2 x+2 x^2\right )+2 e^x x \log (x)} \, dx=\int \frac {\ln \left (2\right )\,\left (3\,x^3-6\,x^2+x\right )+2\,{\ln \left (2\right )}^2\,\left (x^3-2\,x^2+x\right )-\ln \left (x\right )\,\left (\ln \left (2\right )\,\left (-3\,x^2+3\,x+1\right )+2\,{\ln \left (2\right )}^2\,\left (x-x^2\right )\right )-\ln \left (x\,\ln \left (x\right )-x+x^2\right )\,\left (\ln \left (2\right )\,\left (x-x^2\right )-x\,\ln \left (2\right )\,\ln \left (x\right )\right )}{{\mathrm {e}}^x\,\left (2\,x-2\,x^2\right )-2\,x\,{\mathrm {e}}^x\,\ln \left (x\right )} \,d x \]

[In]

int((log(2)*(x - 6*x^2 + 3*x^3) + 2*log(2)^2*(x - 2*x^2 + x^3) - log(x)*(log(2)*(3*x - 3*x^2 + 1) + 2*log(2)^2
*(x - x^2)) - log(x*log(x) - x + x^2)*(log(2)*(x - x^2) - x*log(2)*log(x)))/(exp(x)*(2*x - 2*x^2) - 2*x*exp(x)
*log(x)),x)

[Out]

int((log(2)*(x - 6*x^2 + 3*x^3) + 2*log(2)^2*(x - 2*x^2 + x^3) - log(x)*(log(2)*(3*x - 3*x^2 + 1) + 2*log(2)^2
*(x - x^2)) - log(x*log(x) - x + x^2)*(log(2)*(x - x^2) - x*log(2)*log(x)))/(exp(x)*(2*x - 2*x^2) - 2*x*exp(x)
*log(x)), x)