\(\int \frac {-2 e^3+x}{x^2} \, dx\) [733]

Optimal result

Integrand size = 11, antiderivative size = 21 \[ \int \frac {-2 e^3+x}{x^2} \, dx=\frac {2 e^3}{x}-\log \left (-\frac {2}{x \log (2)}\right ) \]

[Out]

2*exp(3)/x-ln(-2/x/ln(2))

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.52, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {-2 e^3+x}{x^2} \, dx=\frac {2 e^3}{x}+\log (x) \]

[In]

Int[(-2*E^3 + x)/x^2,x]

[Out]

(2*E^3)/x + Log[x]

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])

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 e^3}{x^2}+\frac {1}{x}\right ) \, dx \\ & = \frac {2 e^3}{x}+\log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.52 \[ \int \frac {-2 e^3+x}{x^2} \, dx=\frac {2 e^3}{x}+\log (x) \]

[In]

Integrate[(-2*E^3 + x)/x^2,x]

[Out]

(2*E^3)/x + Log[x]

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.52

method	result	size
default	\(\ln \left (x \right )+\frac {2 \,{\mathrm e}^{3}}{x}\)	\(11\)
norman	\(\ln \left (x \right )+\frac {2 \,{\mathrm e}^{3}}{x}\)	\(11\)
risch	\(\ln \left (x \right )+\frac {2 \,{\mathrm e}^{3}}{x}\)	\(11\)
parallelrisch	\(\frac {x \ln \left (x \right )+2 \,{\mathrm e}^{3}}{x}\)	\(14\)

[In]

int((-2*exp(3)+x)/x^2,x,method=_RETURNVERBOSE)

[Out]

ln(x)+2*exp(3)/x

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int \frac {-2 e^3+x}{x^2} \, dx=\frac {x \log \left (x\right ) + 2 \, e^{3}}{x} \]

[In]

integrate((-2*exp(3)+x)/x^2,x, algorithm="fricas")

[Out]

(x*log(x) + 2*e^3)/x

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.38 \[ \int \frac {-2 e^3+x}{x^2} \, dx=\log {\left (x \right )} + \frac {2 e^{3}}{x} \]

[In]

integrate((-2*exp(3)+x)/x**2,x)

[Out]

log(x) + 2*exp(3)/x

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.48 \[ \int \frac {-2 e^3+x}{x^2} \, dx=\frac {2 \, e^{3}}{x} + \log \left (x\right ) \]

[In]

integrate((-2*exp(3)+x)/x^2,x, algorithm="maxima")

[Out]

2*e^3/x + log(x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.52 \[ \int \frac {-2 e^3+x}{x^2} \, dx=\frac {2 \, e^{3}}{x} + \log \left ({\left | x \right |}\right ) \]

[In]

integrate((-2*exp(3)+x)/x^2,x, algorithm="giac")

[Out]

2*e^3/x + log(abs(x))

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.48 \[ \int \frac {-2 e^3+x}{x^2} \, dx=\ln \left (x\right )+\frac {2\,{\mathrm {e}}^3}{x} \]

[In]

int((x - 2*exp(3))/x^2,x)

[Out]

log(x) + (2*exp(3))/x