\(\int \frac {-1-4 x+e^x x}{x} \, dx\) [5761]

Optimal result

Integrand size = 14, antiderivative size = 12 \[ \int \frac {-1-4 x+e^x x}{x} \, dx=-2+e^x-4 x-\log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {14, 2225, 45} \[ \int \frac {-1-4 x+e^x x}{x} \, dx=-4 x+e^x-\log (x) \]

[In]

Int[(-1 - 4*x + E^x*x)/x,x]

[Out]

E^x - 4*x - Log[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92 \[ \int \frac {-1-4 x+e^x x}{x} \, dx=e^x-4 x-\log (x) \]

[In]

Integrate[(-1 - 4*x + E^x*x)/x,x]

[Out]

E^x - 4*x - Log[x]

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92

method	result	size
default	\(-\ln \left (x \right )-4 x +{\mathrm e}^{x}\)	\(11\)
norman	\(-\ln \left (x \right )-4 x +{\mathrm e}^{x}\)	\(11\)
risch	\(-\ln \left (x \right )-4 x +{\mathrm e}^{x}\)	\(11\)
parallelrisch	\(-\ln \left (x \right )-4 x +{\mathrm e}^{x}\)	\(11\)
parts	\(-\ln \left (x \right )-4 x +{\mathrm e}^{x}\)	\(11\)

[In]

int((exp(x)*x-4*x-1)/x,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {-1-4 x+e^x x}{x} \, dx=-4 \, x + e^{x} - \log \left (x\right ) \]

[In]

integrate((exp(x)*x-4*x-1)/x,x, algorithm="fricas")

[Out]

-4*x + e^x - log(x)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {-1-4 x+e^x x}{x} \, dx=- 4 x + e^{x} - \log {\left (x \right )} \]

[In]

integrate((exp(x)*x-4*x-1)/x,x)

[Out]

-4*x + exp(x) - log(x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {-1-4 x+e^x x}{x} \, dx=-4 \, x + e^{x} - \log \left (x\right ) \]

[In]

integrate((exp(x)*x-4*x-1)/x,x, algorithm="maxima")

[Out]

-4*x + e^x - log(x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {-1-4 x+e^x x}{x} \, dx=-4 \, x + e^{x} - \log \left (x\right ) \]

[In]

integrate((exp(x)*x-4*x-1)/x,x, algorithm="giac")

[Out]

-4*x + e^x - log(x)

Mupad [B] (verification not implemented)

Time = 12.07 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {-1-4 x+e^x x}{x} \, dx={\mathrm {e}}^x-4\,x-\ln \left (x\right ) \]

[In]

int(-(4*x - x*exp(x) + 1)/x,x)

[Out]

exp(x) - 4*x - log(x)