\(\int \frac {e^{-x} (-1+(1-x) \log (-\frac {x}{2})-\log ^2(-\frac {x}{2}))}{\log ^2(-\frac {x}{2})} \, dx\) [7683]

Optimal result

Integrand size = 38, antiderivative size = 23 \[ \int \frac {e^{-x} \left (-1+(1-x) \log \left (-\frac {x}{2}\right )-\log ^2\left (-\frac {x}{2}\right )\right )}{\log ^2\left (-\frac {x}{2}\right )} \, dx=\frac {e^{-x} \left (x+\frac {x^2}{\log \left (-\frac {x}{2}\right )}\right )}{x} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \frac {e^{-x} \left (-1+(1-x) \log \left (-\frac {x}{2}\right )-\log ^2\left (-\frac {x}{2}\right )\right )}{\log ^2\left (-\frac {x}{2}\right )} \, dx=e^{-x} \left (1+\frac {x}{\log \left (-\frac {x}{2}\right )}\right ) \]

[In]

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

[Out]

(1 + x/Log[-1/2*x])/E^x

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78

method	result	size
norman	\(\frac {{\mathrm e}^{-x} \left (x +\ln \left (-\frac {x}{2}\right )\right )}{\ln \left (-\frac {x}{2}\right )}\)	\(18\)
risch	\({\mathrm e}^{-x}+\frac {x \,{\mathrm e}^{-x}}{\ln \left (-\frac {x}{2}\right )}\)	\(18\)
parallelrisch	\(-\frac {\left (-x -\ln \left (-\frac {x}{2}\right )\right ) {\mathrm e}^{-x}}{\ln \left (-\frac {x}{2}\right )}\)	\(23\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-x} \left (-1+(1-x) \log \left (-\frac {x}{2}\right )-\log ^2\left (-\frac {x}{2}\right )\right )}{\log ^2\left (-\frac {x}{2}\right )} \, dx=\frac {x e^{\left (-x\right )} + e^{\left (-x\right )} \log \left (-\frac {1}{2} \, x\right )}{\log \left (-\frac {1}{2} \, x\right )} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {e^{-x} \left (-1+(1-x) \log \left (-\frac {x}{2}\right )-\log ^2\left (-\frac {x}{2}\right )\right )}{\log ^2\left (-\frac {x}{2}\right )} \, dx=\frac {\left (x + \log {\left (- \frac {x}{2} \right )}\right ) e^{- x}}{\log {\left (- \frac {x}{2} \right )}} \]

[In]

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

[Out]

(x + log(-x/2))*exp(-x)/log(-x/2)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {e^{-x} \left (-1+(1-x) \log \left (-\frac {x}{2}\right )-\log ^2\left (-\frac {x}{2}\right )\right )}{\log ^2\left (-\frac {x}{2}\right )} \, dx=-\frac {x}{e^{x} \log \left (2\right ) - e^{x} \log \left (-x\right )} + e^{\left (-x\right )} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 12.57 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {e^{-x} \left (-1+(1-x) \log \left (-\frac {x}{2}\right )-\log ^2\left (-\frac {x}{2}\right )\right )}{\log ^2\left (-\frac {x}{2}\right )} \, dx={\mathrm {e}}^{-x}+\frac {x\,{\mathrm {e}}^{-x}}{\ln \left (-\frac {x}{2}\right )} \]

[In]

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

[Out]

exp(-x) + (x*exp(-x))/log(-x/2)