\(\int \frac {1+x}{-4 x+x^2+x \log (x)} \, dx\) [5872]

Optimal result

Integrand size = 17, antiderivative size = 11 \[ \int \frac {1+x}{-4 x+x^2+x \log (x)} \, dx=-4+e^2+\log (-4+x+\log (x)) \]

[Out]

-4+ln(x+ln(x)-4)+exp(2)

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91 \[ \int \frac {1+x}{-4 x+x^2+x \log (x)} \, dx=\log (4-x-\log (x)) \]

[In]

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

[Out]

Log[4 - x - Log[x]]

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.64

method	result	size
default	\(\ln \left (x +\ln \left (x \right )-4\right )\)	\(7\)
norman	\(\ln \left (x +\ln \left (x \right )-4\right )\)	\(7\)
risch	\(\ln \left (x +\ln \left (x \right )-4\right )\)	\(7\)
parallelrisch	\(\ln \left (x +\ln \left (x \right )-4\right )\)	\(7\)

[In]

int((1+x)/(x*ln(x)+x^2-4*x),x,method=_RETURNVERBOSE)

[Out]

ln(x+ln(x)-4)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.55 \[ \int \frac {1+x}{-4 x+x^2+x \log (x)} \, dx=\log \left (x + \log \left (x\right ) - 4\right ) \]

[In]

integrate((1+x)/(x*log(x)+x^2-4*x),x, algorithm="fricas")

[Out]

log(x + log(x) - 4)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.64 \[ \int \frac {1+x}{-4 x+x^2+x \log (x)} \, dx=\log {\left (x + \log {\left (x \right )} - 4 \right )} \]

[In]

integrate((1+x)/(x*ln(x)+x**2-4*x),x)

[Out]

log(x + log(x) - 4)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.55 \[ \int \frac {1+x}{-4 x+x^2+x \log (x)} \, dx=\log \left (x + \log \left (x\right ) - 4\right ) \]

[In]

integrate((1+x)/(x*log(x)+x^2-4*x),x, algorithm="maxima")

[Out]

log(x + log(x) - 4)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.55 \[ \int \frac {1+x}{-4 x+x^2+x \log (x)} \, dx=\log \left (x + \log \left (x\right ) - 4\right ) \]

[In]

integrate((1+x)/(x*log(x)+x^2-4*x),x, algorithm="giac")

[Out]

log(x + log(x) - 4)

Mupad [B] (verification not implemented)

Time = 12.50 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.55 \[ \int \frac {1+x}{-4 x+x^2+x \log (x)} \, dx=\ln \left (x+\ln \left (x\right )-4\right ) \]

[In]

int((x + 1)/(x*log(x) - 4*x + x^2),x)

[Out]

log(x + log(x) - 4)