\(\int \frac {4 e}{x^3} \, dx\) [3562]

Optimal result

Integrand size = 6, antiderivative size = 6 \[ \int \frac {4 e}{x^3} \, dx=-\frac {2 e}{x^2} \]

[Out]

-1/x^2*exp(1+ln(2))

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {12, 30} \[ \int \frac {4 e}{x^3} \, dx=-\frac {2 e}{x^2} \]

[In]

Int[(4*E)/x^3,x]

[Out]

(-2*E)/x^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = (4 e) \int \frac {1}{x^3} \, dx \\ & = -\frac {2 e}{x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {4 e}{x^3} \, dx=-\frac {2 e}{x^2} \]

[In]

Integrate[(4*E)/x^3,x]

[Out]

(-2*E)/x^2

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.33

method	result	size
norman	\(-\frac {2 \,{\mathrm e}}{x^{2}}\)	\(8\)
risch	\(-\frac {2 \,{\mathrm e}}{x^{2}}\)	\(8\)
gosper	\(-\frac {{\mathrm e}^{1+\ln \left (2\right )}}{x^{2}}\)	\(11\)
default	\(-\frac {{\mathrm e}^{1+\ln \left (2\right )}}{x^{2}}\)	\(11\)
parallelrisch	\(-\frac {{\mathrm e}^{1+\ln \left (2\right )}}{x^{2}}\)	\(11\)

[In]

int(2*exp(1+ln(2))/x^3,x,method=_RETURNVERBOSE)

[Out]

-2*exp(1)/x^2

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.67 \[ \int \frac {4 e}{x^3} \, dx=-\frac {e^{\left (\log \left (2\right ) + 1\right )}}{x^{2}} \]

[In]

integrate(2*exp(1+log(2))/x^3,x, algorithm="fricas")

[Out]

-e^(log(2) + 1)/x^2

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.33 \[ \int \frac {4 e}{x^3} \, dx=- \frac {2 e}{x^{2}} \]

[In]

integrate(2*exp(1+ln(2))/x**3,x)

[Out]

-2*E/x**2

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.17 \[ \int \frac {4 e}{x^3} \, dx=-\frac {2 \, e}{x^{2}} \]

[In]

integrate(2*exp(1+log(2))/x^3,x, algorithm="maxima")

[Out]

-2*e/x^2

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.67 \[ \int \frac {4 e}{x^3} \, dx=-\frac {e^{\left (\log \left (2\right ) + 1\right )}}{x^{2}} \]

[In]

integrate(2*exp(1+log(2))/x^3,x, algorithm="giac")

[Out]

-e^(log(2) + 1)/x^2

Mupad [B] (verification not implemented)

Time = 9.09 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.17 \[ \int \frac {4 e}{x^3} \, dx=-\frac {2\,\mathrm {e}}{x^2} \]

[In]

int((2*exp(log(2) + 1))/x^3,x)

[Out]

-(2*exp(1))/x^2