3.1.5 \(\int e^{-x} (36+e^x (-1-6 x)+72 x-36 x^2+(36-36 x) \log (x)) \, dx\)

Optimal. Leaf size=24 \[ -4-x+x \left (x-4 \left (x-9 e^{-x} (x+\log (x))\right )\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.26, antiderivative size = 42, normalized size of antiderivative = 1.75, number of steps used = 10, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {6742, 2194, 2176, 2554} \begin {gather*} 36 e^{-x} x^2-3 x^2-x+36 e^{-x} \log (x)-36 e^{-x} (1-x) \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(36 + E^x*(-1 - 6*x) + 72*x - 36*x^2 + (36 - 36*x)*Log[x])/E^x,x]

[Out]

-x - 3*x^2 + (36*x^2)/E^x + (36*Log[x])/E^x - (36*(1 - x)*Log[x])/E^x

Rule 2176

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] &&  !$UseGamma === True

Rule 2194

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 2554

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 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1+36 e^{-x}-6 x+72 e^{-x} x-36 e^{-x} x^2-36 e^{-x} (-1+x) \log (x)\right ) \, dx\\ &=-x-3 x^2+36 \int e^{-x} \, dx-36 \int e^{-x} x^2 \, dx-36 \int e^{-x} (-1+x) \log (x) \, dx+72 \int e^{-x} x \, dx\\ &=-36 e^{-x}-x-72 e^{-x} x-3 x^2+36 e^{-x} x^2+36 e^{-x} \log (x)-36 e^{-x} (1-x) \log (x)-36 \int e^{-x} \, dx+72 \int e^{-x} \, dx-72 \int e^{-x} x \, dx\\ &=-72 e^{-x}-x-3 x^2+36 e^{-x} x^2+36 e^{-x} \log (x)-36 e^{-x} (1-x) \log (x)-72 \int e^{-x} \, dx\\ &=-x-3 x^2+36 e^{-x} x^2+36 e^{-x} \log (x)-36 e^{-x} (1-x) \log (x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.14, size = 18, normalized size = 0.75 \begin {gather*} x \left (-1-3 x+36 e^{-x} (x+\log (x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(36 + E^x*(-1 - 6*x) + 72*x - 36*x^2 + (36 - 36*x)*Log[x])/E^x,x]

[Out]

x*(-1 - 3*x + (36*(x + Log[x]))/E^x)

________________________________________________________________________________________

fricas [A]  time = 0.49, size = 27, normalized size = 1.12 \begin {gather*} {\left (36 \, x^{2} - {\left (3 \, x^{2} + x\right )} e^{x} + 36 \, x \log \relax (x)\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-36*x+36)*log(x)+(-6*x-1)*exp(x)-36*x^2+72*x+36)/exp(x),x, algorithm="fricas")

[Out]

(36*x^2 - (3*x^2 + x)*e^x + 36*x*log(x))*e^(-x)

________________________________________________________________________________________

giac [A]  time = 0.12, size = 27, normalized size = 1.12 \begin {gather*} 36 \, x^{2} e^{\left (-x\right )} + 36 \, x e^{\left (-x\right )} \log \relax (x) - 3 \, x^{2} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-36*x+36)*log(x)+(-6*x-1)*exp(x)-36*x^2+72*x+36)/exp(x),x, algorithm="giac")

[Out]

36*x^2*e^(-x) + 36*x*e^(-x)*log(x) - 3*x^2 - x

________________________________________________________________________________________

maple [A]  time = 0.11, size = 26, normalized size = 1.08

method	result	size
default	\(-x +\left (36 x^{2}+36 x \ln \relax (x )\right ) {\mathrm e}^{-x}-3 x^{2}\)	\(26\)
norman	\(\left (36 x^{2}+36 x \ln \relax (x )-{\mathrm e}^{x} x -3 \,{\mathrm e}^{x} x^{2}\right ) {\mathrm e}^{-x}\)	\(29\)
risch	\(36 x \,{\mathrm e}^{-x} \ln \relax (x )-x \left (3 \,{\mathrm e}^{x} x -36 x +{\mathrm e}^{x}\right ) {\mathrm e}^{-x}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-36*x+36)*ln(x)+(-6*x-1)*exp(x)-36*x^2+72*x+36)/exp(x),x,method=_RETURNVERBOSE)

[Out]

-x+(36*x^2+36*x*ln(x))/exp(x)-3*x^2

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 36 \, {\left (x + 1\right )} e^{\left (-x\right )} \log \relax (x) - 3 \, x^{2} + 36 \, {\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} - 72 \, {\left (x + 1\right )} e^{\left (-x\right )} - 36 \, e^{\left (-x\right )} \log \relax (x) - x + 36 \, {\rm Ei}\left (-x\right ) - 36 \, e^{\left (-x\right )} - 36 \, \int \frac {{\left (x + 1\right )} e^{\left (-x\right )}}{x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-36*x+36)*log(x)+(-6*x-1)*exp(x)-36*x^2+72*x+36)/exp(x),x, algorithm="maxima")

[Out]

36*(x + 1)*e^(-x)*log(x) - 3*x^2 + 36*(x^2 + 2*x + 2)*e^(-x) - 72*(x + 1)*e^(-x) - 36*e^(-x)*log(x) - x + 36*E
i(-x) - 36*e^(-x) - 36*integrate((x + 1)*e^(-x)/x, x)

________________________________________________________________________________________

mupad [B]  time = 0.27, size = 23, normalized size = 0.96 \begin {gather*} -x\,\left (3\,x-36\,x\,{\mathrm {e}}^{-x}-36\,{\mathrm {e}}^{-x}\,\ln \relax (x)+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-x)*(exp(x)*(6*x + 1) - 72*x + log(x)*(36*x - 36) + 36*x^2 - 36),x)

[Out]

-x*(3*x - 36*x*exp(-x) - 36*exp(-x)*log(x) + 1)

________________________________________________________________________________________

sympy [A]  time = 0.28, size = 20, normalized size = 0.83 \begin {gather*} - 3 x^{2} - x + \left (36 x^{2} + 36 x \log {\relax (x )}\right ) e^{- x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-36*x+36)*ln(x)+(-6*x-1)*exp(x)-36*x**2+72*x+36)/exp(x),x)

[Out]

-3*x**2 - x + (36*x**2 + 36*x*log(x))*exp(-x)

________________________________________________________________________________________