\(\int \frac {e^x (-1+x)}{e^x x+x^2} \, dx\) [9312]

Optimal result

Integrand size = 18, antiderivative size = 10 \[ \int \frac {e^x (-1+x)}{e^x x+x^2} \, dx=\log \left (\frac {e^x+x}{x}\right ) \]

[Out]

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

Rubi [F]

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

[In]

Int[(E^x*(-1 + x))/(E^x*x + x^2),x]

[Out]

Defer[Int][E^x/(E^x + x), x] - Defer[Int][E^x/(x*(E^x + x)), x]

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10 \[ \int \frac {e^x (-1+x)}{e^x x+x^2} \, dx=-\log (x)+\log \left (e^x+x\right ) \]

[In]

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

[Out]

-Log[x] + Log[E^x + x]

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10

method	result	size
norman	\(-\ln \left (x \right )+\ln \left ({\mathrm e}^{x}+x \right )\)	\(11\)
risch	\(-\ln \left (x \right )+\ln \left ({\mathrm e}^{x}+x \right )\)	\(11\)
parallelrisch	\(-\ln \left (x \right )+\ln \left ({\mathrm e}^{x}+x \right )\)	\(11\)

[In]

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

[Out]

-ln(x)+ln(exp(x)+x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {e^x (-1+x)}{e^x x+x^2} \, dx=\log \left (x + e^{x}\right ) - \log \left (x\right ) \]

[In]

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

[Out]

log(x + e^x) - log(x)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {e^x (-1+x)}{e^x x+x^2} \, dx=- \log {\left (x \right )} + \log {\left (x + e^{x} \right )} \]

[In]

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

[Out]

-log(x) + log(x + exp(x))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {e^x (-1+x)}{e^x x+x^2} \, dx=\log \left (x + e^{x}\right ) - \log \left (x\right ) \]

[In]

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

[Out]

log(x + e^x) - log(x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {e^x (-1+x)}{e^x x+x^2} \, dx=\log \left (x + e^{x}\right ) - \log \left (x\right ) \]

[In]

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

[Out]

log(x + e^x) - log(x)

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {e^x (-1+x)}{e^x x+x^2} \, dx=\ln \left (x+{\mathrm {e}}^x\right )-\ln \left (x\right ) \]

[In]

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

[Out]

log(x + exp(x)) - log(x)