3.96.35 \(\int \frac {e^{-1+x} (-32+32 x)}{e^{2 x}-2 e^x x+x^2} \, dx\)

Optimal. Leaf size=20 \[ 2 \left (11-\frac {16 e^{-1+x}}{e^x-x}\right ) \]

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Rubi [A]  time = 0.21, antiderivative size = 17, normalized size of antiderivative = 0.85, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6688, 12, 6711, 32} \begin {gather*} \frac {32}{e \left (1-\frac {e^x}{x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(-1 + x)*(-32 + 32*x))/(E^(2*x) - 2*E^x*x + x^2),x]

[Out]

32/(E*(1 - E^x/x))

Rule 12

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6711

Int[(u_)*((a_.)*(v_)^(p_.) + (b_.)*(w_)^(q_.))^(m_.), x_Symbol] :> With[{c = Simplify[u/(p*w*D[v, x] - q*v*D[w
, x])]}, Dist[c*p, Subst[Int[(b + a*x^p)^m, x], x, v*w^(m*q + 1)], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p, q}
, x] && EqQ[p + q*(m*p + 1), 0] && IntegerQ[p] && IntegerQ[m]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {32 e^{-1+x} (-1+x)}{\left (e^x-x\right )^2} \, dx\\ &=32 \int \frac {e^{-1+x} (-1+x)}{\left (e^x-x\right )^2} \, dx\\ &=\frac {32 \operatorname {Subst}\left (\int \frac {1}{(-1+x)^2} \, dx,x,\frac {e^x}{x}\right )}{e}\\ &=\frac {32}{e \left (1-\frac {e^x}{x}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 15, normalized size = 0.75 \begin {gather*} -\frac {32 x}{e \left (e^x-x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-1 + x)*(-32 + 32*x))/(E^(2*x) - 2*E^x*x + x^2),x]

[Out]

(-32*x)/(E*(E^x - x))

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fricas [A]  time = 0.59, size = 16, normalized size = 0.80 \begin {gather*} \frac {32 \, x}{x e - e^{\left (x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((32*x-32)*exp(x-1)/(exp(x)^2-2*exp(x)*x+x^2),x, algorithm="fricas")

[Out]

32*x/(x*e - e^(x + 1))

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giac [A]  time = 0.21, size = 16, normalized size = 0.80 \begin {gather*} \frac {32 \, x}{x e - e^{\left (x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((32*x-32)*exp(x-1)/(exp(x)^2-2*exp(x)*x+x^2),x, algorithm="giac")

[Out]

32*x/(x*e - e^(x + 1))

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maple [A]  time = 0.08, size = 14, normalized size = 0.70

method	result	size
risch	\(\frac {32 \,{\mathrm e}^{-1} x}{x -{\mathrm e}^{x}}\)	\(14\)
norman	\(\frac {32 \,{\mathrm e}^{-1} {\mathrm e}^{x}}{x -{\mathrm e}^{x}}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((32*x-32)*exp(x-1)/(exp(x)^2-2*exp(x)*x+x^2),x,method=_RETURNVERBOSE)

[Out]

32*exp(-1)*x/(x-exp(x))

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maxima [A]  time = 0.40, size = 16, normalized size = 0.80 \begin {gather*} \frac {32 \, x}{x e - e^{\left (x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((32*x-32)*exp(x-1)/(exp(x)^2-2*exp(x)*x+x^2),x, algorithm="maxima")

[Out]

32*x/(x*e - e^(x + 1))

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mupad [B]  time = 9.74, size = 15, normalized size = 0.75 \begin {gather*} -\frac {32\,x}{{\mathrm {e}}^{x+1}-x\,\mathrm {e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x - 1)*(32*x - 32))/(exp(2*x) - 2*x*exp(x) + x^2),x)

[Out]

-(32*x)/(exp(x + 1) - x*exp(1))

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sympy [A]  time = 0.09, size = 15, normalized size = 0.75 \begin {gather*} - \frac {32 x}{- e x + e e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((32*x-32)*exp(x-1)/(exp(x)**2-2*exp(x)*x+x**2),x)

[Out]

-32*x/(-E*x + E*exp(x))

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