\(\int \frac {6+e^x (-1+x)}{x^2} \, dx\) [6424]

Optimal result

Integrand size = 13, antiderivative size = 9 \[ \int \frac {6+e^x (-1+x)}{x^2} \, dx=\frac {-6+e^x}{x} \]

[Out]

(exp(x)-6)/x

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.44, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {14, 2228} \[ \int \frac {6+e^x (-1+x)}{x^2} \, dx=\frac {e^x}{x}-\frac {6}{x} \]

[In]

Int[(6 + E^x*(-1 + x))/x^2,x]

[Out]

-6/x + E^x/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 2228

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[g*u^(m + 1)*(F^(c*v)/(b*c*
e*Log[F])), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {6}{x^2}+\frac {e^x (-1+x)}{x^2}\right ) \, dx \\ & = -\frac {6}{x}+\int \frac {e^x (-1+x)}{x^2} \, dx \\ & = -\frac {6}{x}+\frac {e^x}{x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {6+e^x (-1+x)}{x^2} \, dx=\frac {-6+e^x}{x} \]

[In]

Integrate[(6 + E^x*(-1 + x))/x^2,x]

[Out]

(-6 + E^x)/x

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00

method	result	size
norman	\(\frac {{\mathrm e}^{x}-6}{x}\)	\(9\)
parallelrisch	\(\frac {{\mathrm e}^{x}-6}{x}\)	\(9\)
default	\(-\frac {6}{x}+\frac {{\mathrm e}^{x}}{x}\)	\(13\)
risch	\(-\frac {6}{x}+\frac {{\mathrm e}^{x}}{x}\)	\(13\)
parts	\(-\frac {6}{x}+\frac {{\mathrm e}^{x}}{x}\)	\(13\)

[In]

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

[Out]

(exp(x)-6)/x

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.89 \[ \int \frac {6+e^x (-1+x)}{x^2} \, dx=\frac {e^{x} - 6}{x} \]

[In]

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

[Out]

(e^x - 6)/x

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \frac {6+e^x (-1+x)}{x^2} \, dx=\frac {e^{x}}{x} - \frac {6}{x} \]

[In]

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

[Out]

exp(x)/x - 6/x

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.67 \[ \int \frac {6+e^x (-1+x)}{x^2} \, dx=-\frac {6}{x} + {\rm Ei}\left (x\right ) - \Gamma \left (-1, -x\right ) \]

[In]

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

[Out]

-6/x + Ei(x) - gamma(-1, -x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.89 \[ \int \frac {6+e^x (-1+x)}{x^2} \, dx=\frac {e^{x} - 6}{x} \]

[In]

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

[Out]

(e^x - 6)/x

Mupad [B] (verification not implemented)

Time = 13.24 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.89 \[ \int \frac {6+e^x (-1+x)}{x^2} \, dx=\frac {{\mathrm {e}}^x-6}{x} \]

[In]

int((exp(x)*(x - 1) + 6)/x^2,x)

[Out]

(exp(x) - 6)/x