\(\int \frac {e^{-x} (e^x (-8-4 x)+20 x+(-40 x+30 x^2-5 x^3+e^x (8+6 x-2 x^2)) \log (\frac {4-x}{x}))}{-4+x} \, dx\) [4948]

Optimal result

Integrand size = 63, antiderivative size = 25 \[ \int \frac {e^{-x} \left (e^x (-8-4 x)+20 x+\left (-40 x+30 x^2-5 x^3+e^x \left (8+6 x-2 x^2\right )\right ) \log \left (\frac {4-x}{x}\right )\right )}{-4+x} \, dx=x \left (-2-x+5 e^{-x} x\right ) \log \left (\frac {4-x}{x}\right ) \]

[Out]

(5*x/exp(x)-x-2)*ln((-x+4)/x)*x

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76, number of steps used = 24, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {6874, 6820, 2230, 2225, 2209, 2227, 2207, 2634, 12, 45, 2513, 528, 84} \[ \int \frac {e^{-x} \left (e^x (-8-4 x)+20 x+\left (-40 x+30 x^2-5 x^3+e^x \left (8+6 x-2 x^2\right )\right ) \log \left (\frac {4-x}{x}\right )\right )}{-4+x} \, dx=5 e^{-x} x^2 \log \left (\frac {4}{x}-1\right )-(x+1)^2 \log \left (\frac {4}{x}-1\right )+\log (4-x)-\log (x) \]

[In]

Int[(E^x*(-8 - 4*x) + 20*x + (-40*x + 30*x^2 - 5*x^3 + E^x*(8 + 6*x - 2*x^2))*Log[(4 - x)/x])/(E^x*(-4 + x)),x
]

[Out]

(5*x^2*Log[-1 + 4/x])/E^x - (1 + x)^2*Log[-1 + 4/x] + Log[4 - x] - Log[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 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])

Rule 84

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 528

Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[x^(m - n*q)*
(a + b*x^n)^p*(d + c*x^n)^q, x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] |
|  !IntegerQ[p])

Rule 2207

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*
((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Dist[d*(m/(f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !TrueQ[$UseGamma]

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2227

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !TrueQ[$UseGamma]

Rule 2230

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !TrueQ[$UseGamma]

Rule 2513

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.) + (g_.)*(x_))^(r_.), x_Symbol] :> Simp[(f
+ g*x)^(r + 1)*((a + b*Log[c*(d + e*x^n)^p])/(g*(r + 1))), x] - Dist[b*e*n*(p/(g*(r + 1))), Int[x^(n - 1)*((f
+ g*x)^(r + 1)/(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, r}, x] && (IGtQ[r, 0] || RationalQ[n
]) && NeQ[r, -1]

Rule 2634

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[w*(D[u, x]
/u), x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rule 6820

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {5 e^{-x} x \left (-4+8 \log \left (-1+\frac {4}{x}\right )-6 x \log \left (-1+\frac {4}{x}\right )+x^2 \log \left (-1+\frac {4}{x}\right )\right )}{-4+x}-\frac {2 \left (4+2 x-4 \log \left (-1+\frac {4}{x}\right )-3 x \log \left (-1+\frac {4}{x}\right )+x^2 \log \left (-1+\frac {4}{x}\right )\right )}{-4+x}\right ) \, dx \\ & = -\left (2 \int \frac {4+2 x-4 \log \left (-1+\frac {4}{x}\right )-3 x \log \left (-1+\frac {4}{x}\right )+x^2 \log \left (-1+\frac {4}{x}\right )}{-4+x} \, dx\right )-5 \int \frac {e^{-x} x \left (-4+8 \log \left (-1+\frac {4}{x}\right )-6 x \log \left (-1+\frac {4}{x}\right )+x^2 \log \left (-1+\frac {4}{x}\right )\right )}{-4+x} \, dx \\ & = -\left (2 \int \frac {-2 (2+x)-\left (-4-3 x+x^2\right ) \log \left (-1+\frac {4}{x}\right )}{4-x} \, dx\right )-5 \int \frac {e^{-x} x \left (4-\left (8-6 x+x^2\right ) \log \left (-1+\frac {4}{x}\right )\right )}{4-x} \, dx \\ & = -\left (2 \int \left (\frac {2 (2+x)}{-4+x}+(1+x) \log \left (-1+\frac {4}{x}\right )\right ) \, dx\right )-5 \int \left (-\frac {4 e^{-x} x}{-4+x}+e^{-x} (-2+x) x \log \left (-1+\frac {4}{x}\right )\right ) \, dx \\ & = -\left (2 \int (1+x) \log \left (-1+\frac {4}{x}\right ) \, dx\right )-4 \int \frac {2+x}{-4+x} \, dx-5 \int e^{-x} (-2+x) x \log \left (-1+\frac {4}{x}\right ) \, dx+20 \int \frac {e^{-x} x}{-4+x} \, dx \\ & = 5 e^{-x} x^2 \log \left (-1+\frac {4}{x}\right )-(1+x)^2 \log \left (-1+\frac {4}{x}\right )-4 \int \left (1+\frac {6}{-4+x}\right ) \, dx-4 \int \frac {(1+x)^2}{\left (-1+\frac {4}{x}\right ) x^2} \, dx+5 \int \frac {4 e^{-x} x}{4-x} \, dx+20 \int \left (e^{-x}+\frac {4 e^{-x}}{-4+x}\right ) \, dx \\ & = -4 x+5 e^{-x} x^2 \log \left (-1+\frac {4}{x}\right )-(1+x)^2 \log \left (-1+\frac {4}{x}\right )-24 \log (4-x)-4 \int \frac {(1+x)^2}{(4-x) x} \, dx+20 \int e^{-x} \, dx+20 \int \frac {e^{-x} x}{4-x} \, dx+80 \int \frac {e^{-x}}{-4+x} \, dx \\ & = -20 e^{-x}-4 x+\frac {80 \text {Ei}(4-x)}{e^4}+5 e^{-x} x^2 \log \left (-1+\frac {4}{x}\right )-(1+x)^2 \log \left (-1+\frac {4}{x}\right )-24 \log (4-x)-4 \int \left (-1-\frac {25}{4 (-4+x)}+\frac {1}{4 x}\right ) \, dx+20 \int \left (-e^{-x}-\frac {4 e^{-x}}{-4+x}\right ) \, dx \\ & = -20 e^{-x}+\frac {80 \text {Ei}(4-x)}{e^4}+5 e^{-x} x^2 \log \left (-1+\frac {4}{x}\right )-(1+x)^2 \log \left (-1+\frac {4}{x}\right )+\log (4-x)-\log (x)-20 \int e^{-x} \, dx-80 \int \frac {e^{-x}}{-4+x} \, dx \\ & = 5 e^{-x} x^2 \log \left (-1+\frac {4}{x}\right )-(1+x)^2 \log \left (-1+\frac {4}{x}\right )+\log (4-x)-\log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 5.06 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {e^{-x} \left (e^x (-8-4 x)+20 x+\left (-40 x+30 x^2-5 x^3+e^x \left (8+6 x-2 x^2\right )\right ) \log \left (\frac {4-x}{x}\right )\right )}{-4+x} \, dx=x \left (-2-x+5 e^{-x} x\right ) \log \left (-1+\frac {4}{x}\right ) \]

[In]

Integrate[(E^x*(-8 - 4*x) + 20*x + (-40*x + 30*x^2 - 5*x^3 + E^x*(8 + 6*x - 2*x^2))*Log[(4 - x)/x])/(E^x*(-4 +
x)),x]

[Out]

x*(-2 - x + (5*x)/E^x)*Log[-1 + 4/x]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(51\) vs. \(2(24)=48\).

Time = 0.41 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.08

method	result	size
parallelrisch	\(\frac {\left (-16 \ln \left (-\frac {x -4}{x}\right ) {\mathrm e}^{x} x^{2}+80 \ln \left (-\frac {x -4}{x}\right ) x^{2}-32 \ln \left (-\frac {x -4}{x}\right ) {\mathrm e}^{x} x \right ) {\mathrm e}^{-x}}{16}\)	\(52\)
norman	\(\left (5 x^{2} \ln \left (\frac {-x +4}{x}\right )-2 \,{\mathrm e}^{x} x \ln \left (\frac {-x +4}{x}\right )-{\mathrm e}^{x} x^{2} \ln \left (\frac {-x +4}{x}\right )\right ) {\mathrm e}^{-x}\)	\(54\)
default	\(-24 \ln \left (\frac {4}{x}\right )+\ln \left (-1+\frac {4}{x}\right ) \left (-1+\frac {4}{x}\right ) \left (\frac {4}{x}+1\right ) x^{2}+2 \ln \left (-1+\frac {4}{x}\right ) \left (-1+\frac {4}{x}\right ) x +5 x^{2} \ln \left (\frac {-x +4}{x}\right ) {\mathrm e}^{-x}-24 \ln \left (x -4\right )\)	\(79\)
parts	\(-24 \ln \left (\frac {4}{x}\right )+\ln \left (-1+\frac {4}{x}\right ) \left (-1+\frac {4}{x}\right ) \left (\frac {4}{x}+1\right ) x^{2}+2 \ln \left (-1+\frac {4}{x}\right ) \left (-1+\frac {4}{x}\right ) x +5 x^{2} \ln \left (\frac {-x +4}{x}\right ) {\mathrm e}^{-x}-24 \ln \left (x -4\right )\)	\(79\)
risch	\(-x^{2} \ln \left (x -4\right )-2 x \ln \left (x -4\right )+5 \,{\mathrm e}^{-x} \ln \left (x -4\right ) x^{2}-5 i {\mathrm e}^{-x} \operatorname {csgn}\left (\frac {i \left (x -4\right )}{x}\right )^{2} \pi \,x^{2}+i \operatorname {csgn}\left (\frac {i \left (x -4\right )}{x}\right ) \operatorname {csgn}\left (i \left (x -4\right )\right ) \operatorname {csgn}\left (\frac {i}{x}\right ) x \pi +\frac {5 i {\mathrm e}^{-x} \operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left (x -4\right )}{x}\right )^{2} \pi \,x^{2}}{2}-\frac {i x^{2} \pi \operatorname {csgn}\left (\frac {i \left (x -4\right )}{x}\right )^{3}}{2}+5 i {\mathrm e}^{-x} \pi \,x^{2}-i \operatorname {csgn}\left (\frac {i \left (x -4\right )}{x}\right )^{3} x \pi -\frac {5 i {\mathrm e}^{-x} \operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left (x -4\right )}{x}\right ) \operatorname {csgn}\left (i \left (x -4\right )\right ) \pi \,x^{2}}{2}-i \operatorname {csgn}\left (\frac {i \left (x -4\right )}{x}\right )^{2} \operatorname {csgn}\left (i \left (x -4\right )\right ) x \pi -i \pi \,x^{2}+\frac {i x^{2} \pi \,\operatorname {csgn}\left (i \left (x -4\right )\right ) \operatorname {csgn}\left (\frac {i \left (x -4\right )}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right )}{2}-2 i x \pi +i x^{2} \pi \operatorname {csgn}\left (\frac {i \left (x -4\right )}{x}\right )^{2}+\frac {5 i {\mathrm e}^{-x} \operatorname {csgn}\left (\frac {i \left (x -4\right )}{x}\right )^{3} \pi \,x^{2}}{2}-\frac {i x^{2} \pi \operatorname {csgn}\left (\frac {i \left (x -4\right )}{x}\right )^{2} \operatorname {csgn}\left (\frac {i}{x}\right )}{2}-\frac {i x^{2} \pi \,\operatorname {csgn}\left (i \left (x -4\right )\right ) \operatorname {csgn}\left (\frac {i \left (x -4\right )}{x}\right )^{2}}{2}+2 i \operatorname {csgn}\left (\frac {i \left (x -4\right )}{x}\right )^{2} x \pi -i \operatorname {csgn}\left (\frac {i \left (x -4\right )}{x}\right )^{2} \operatorname {csgn}\left (\frac {i}{x}\right ) x \pi +\frac {5 i {\mathrm e}^{-x} \operatorname {csgn}\left (\frac {i \left (x -4\right )}{x}\right )^{2} \operatorname {csgn}\left (i \left (x -4\right )\right ) \pi \,x^{2}}{2}+x^{2} \ln \left (x \right )+2 x \ln \left (x \right )-5 \,{\mathrm e}^{-x} \ln \left (x \right ) x^{2}\)	\(449\)

[In]

int((((-2*x^2+6*x+8)*exp(x)-5*x^3+30*x^2-40*x)*ln((-x+4)/x)+(-4*x-8)*exp(x)+20*x)/(x-4)/exp(x),x,method=_RETUR
NVERBOSE)

[Out]

1/16*(-16*ln(-(x-4)/x)*exp(x)*x^2+80*ln(-(x-4)/x)*x^2-32*ln(-(x-4)/x)*exp(x)*x)/exp(x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {e^{-x} \left (e^x (-8-4 x)+20 x+\left (-40 x+30 x^2-5 x^3+e^x \left (8+6 x-2 x^2\right )\right ) \log \left (\frac {4-x}{x}\right )\right )}{-4+x} \, dx={\left (5 \, x^{2} - {\left (x^{2} + 2 \, x\right )} e^{x}\right )} e^{\left (-x\right )} \log \left (-\frac {x - 4}{x}\right ) \]

[In]

integrate((((-2*x^2+6*x+8)*exp(x)-5*x^3+30*x^2-40*x)*log((-x+4)/x)+(-4*x-8)*exp(x)+20*x)/(x-4)/exp(x),x, algor
ithm="fricas")

[Out]

(5*x^2 - (x^2 + 2*x)*e^x)*e^(-x)*log(-(x - 4)/x)

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {e^{-x} \left (e^x (-8-4 x)+20 x+\left (-40 x+30 x^2-5 x^3+e^x \left (8+6 x-2 x^2\right )\right ) \log \left (\frac {4-x}{x}\right )\right )}{-4+x} \, dx=5 x^{2} e^{- x} \log {\left (\frac {4 - x}{x} \right )} + \left (- x^{2} - 2 x\right ) \log {\left (\frac {4 - x}{x} \right )} \]

[In]

integrate((((-2*x**2+6*x+8)*exp(x)-5*x**3+30*x**2-40*x)*ln((-x+4)/x)+(-4*x-8)*exp(x)+20*x)/(x-4)/exp(x),x)

[Out]

5*x**2*exp(-x)*log((4 - x)/x) + (-x**2 - 2*x)*log((4 - x)/x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (23) = 46\).

Time = 0.23 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.88 \[ \int \frac {e^{-x} \left (e^x (-8-4 x)+20 x+\left (-40 x+30 x^2-5 x^3+e^x \left (8+6 x-2 x^2\right )\right ) \log \left (\frac {4-x}{x}\right )\right )}{-4+x} \, dx=-5 \, x^{2} e^{\left (-x\right )} \log \left (x\right ) + {\left (x^{2} + 2 \, x\right )} \log \left (x\right ) + {\left (5 \, x^{2} e^{\left (-x\right )} - x^{2} - 2 \, x\right )} \log \left (-x + 4\right ) \]

[In]

integrate((((-2*x^2+6*x+8)*exp(x)-5*x^3+30*x^2-40*x)*log((-x+4)/x)+(-4*x-8)*exp(x)+20*x)/(x-4)/exp(x),x, algor
ithm="maxima")

[Out]

-5*x^2*e^(-x)*log(x) + (x^2 + 2*x)*log(x) + (5*x^2*e^(-x) - x^2 - 2*x)*log(-x + 4)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.80 \[ \int \frac {e^{-x} \left (e^x (-8-4 x)+20 x+\left (-40 x+30 x^2-5 x^3+e^x \left (8+6 x-2 x^2\right )\right ) \log \left (\frac {4-x}{x}\right )\right )}{-4+x} \, dx=5 \, x^{2} e^{\left (-x\right )} \log \left (-\frac {x - 4}{x}\right ) - x^{2} \log \left (-\frac {x - 4}{x}\right ) - 2 \, x \log \left (-\frac {x - 4}{x}\right ) \]

[In]

integrate((((-2*x^2+6*x+8)*exp(x)-5*x^3+30*x^2-40*x)*log((-x+4)/x)+(-4*x-8)*exp(x)+20*x)/(x-4)/exp(x),x, algor
ithm="giac")

[Out]

5*x^2*e^(-x)*log(-(x - 4)/x) - x^2*log(-(x - 4)/x) - 2*x*log(-(x - 4)/x)

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-x} \left (e^x (-8-4 x)+20 x+\left (-40 x+30 x^2-5 x^3+e^x \left (8+6 x-2 x^2\right )\right ) \log \left (\frac {4-x}{x}\right )\right )}{-4+x} \, dx=\int -\frac {{\mathrm {e}}^{-x}\,\left (\ln \left (-\frac {x-4}{x}\right )\,\left (40\,x-{\mathrm {e}}^x\,\left (-2\,x^2+6\,x+8\right )-30\,x^2+5\,x^3\right )-20\,x+{\mathrm {e}}^x\,\left (4\,x+8\right )\right )}{x-4} \,d x \]

[In]

int(-(exp(-x)*(log(-(x - 4)/x)*(40*x - exp(x)*(6*x - 2*x^2 + 8) - 30*x^2 + 5*x^3) - 20*x + exp(x)*(4*x + 8)))/
(x - 4),x)

[Out]

int(-(exp(-x)*(log(-(x - 4)/x)*(40*x - exp(x)*(6*x - 2*x^2 + 8) - 30*x^2 + 5*x^3) - 20*x + exp(x)*(4*x + 8)))/
(x - 4), x)