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

Optimal. Leaf size=21 \[ \frac {2 e^3}{x}-\log \left (-\frac {2}{x \log (2)}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 11, normalized size of antiderivative = 0.52, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {43} \begin {gather*} \frac {2 e^3}{x}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2*E^3 + x)/x^2,x]

[Out]

(2*E^3)/x + Log[x]

Rule 43

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

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 11, normalized size = 0.52 \begin {gather*} \frac {2 e^3}{x}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*E^3 + x)/x^2,x]

[Out]

(2*E^3)/x + Log[x]

________________________________________________________________________________________

fricas [A]  time = 0.61, size = 13, normalized size = 0.62 \begin {gather*} \frac {x \log \relax (x) + 2 \, e^{3}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*log(x) + 2*e^3)/x

________________________________________________________________________________________

giac [A]  time = 0.27, size = 11, normalized size = 0.52 \begin {gather*} \frac {2 \, e^{3}}{x} + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e^3/x + log(abs(x))

________________________________________________________________________________________

maple [A]  time = 0.03, size = 11, normalized size = 0.52

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.44, size = 10, normalized size = 0.48 \begin {gather*} \frac {2 \, e^{3}}{x} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e^3/x + log(x)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 10, normalized size = 0.48 \begin {gather*} \ln \relax (x)+\frac {2\,{\mathrm {e}}^3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 2*exp(3))/x^2,x)

[Out]

log(x) + (2*exp(3))/x

________________________________________________________________________________________

sympy [A]  time = 0.08, size = 8, normalized size = 0.38 \begin {gather*} \log {\relax (x )} + \frac {2 e^{3}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) + 2*exp(3)/x

________________________________________________________________________________________