3.34.20 \(\int \frac {e^{-4-x} (4+10 x+e^{4+x} x-10 x^2-4 x \log (x))}{x} \, dx\)

Optimal. Leaf size=22 \[ x-2 e^{-4-x} x \left (-5+\log \left (x^{-2/x}\right )\right ) \]

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Rubi [A]  time = 0.38, antiderivative size = 25, normalized size of antiderivative = 1.14, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {6742, 2288} \begin {gather*} \frac {2 e^{-x-4} \left (5 x^2+2 x \log (x)\right )}{x}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(-4 - x)*(4 + 10*x + E^(4 + x)*x - 10*x^2 - 4*x*Log[x]))/x,x]

[Out]

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

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, 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-\frac {2 e^{-4-x} \left (-2-5 x+5 x^2+2 x \log (x)\right )}{x}\right ) \, dx\\ &=x-2 \int \frac {e^{-4-x} \left (-2-5 x+5 x^2+2 x \log (x)\right )}{x} \, dx\\ &=x+\frac {2 e^{-4-x} \left (5 x^2+2 x \log (x)\right )}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 23, normalized size = 1.05 \begin {gather*} x+10 e^{-4-x} x+4 e^{-4-x} \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-4 - x)*(4 + 10*x + E^(4 + x)*x - 10*x^2 - 4*x*Log[x]))/x,x]

[Out]

x + 10*E^(-4 - x)*x + 4*E^(-4 - x)*Log[x]

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fricas [A]  time = 0.58, size = 21, normalized size = 0.95 \begin {gather*} {\left (x e^{\left (x + 4\right )} + 10 \, x + 4 \, \log \relax (x)\right )} e^{\left (-x - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*log(x)+x*exp(4+x)-10*x^2+10*x+4)/x/exp(4+x),x, algorithm="fricas")

[Out]

(x*e^(x + 4) + 10*x + 4*log(x))*e^(-x - 4)

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giac [A]  time = 0.27, size = 23, normalized size = 1.05 \begin {gather*} {\left (x e^{4} + 10 \, x e^{\left (-x\right )} + 4 \, e^{\left (-x\right )} \log \relax (x)\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*log(x)+x*exp(4+x)-10*x^2+10*x+4)/x/exp(4+x),x, algorithm="giac")

[Out]

(x*e^4 + 10*x*e^(-x) + 4*e^(-x)*log(x))*e^(-4)

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maple [A]  time = 0.04, size = 18, normalized size = 0.82

method	result	size
default	\(x +\left (10 x +4 \ln \relax (x )\right ) {\mathrm e}^{-x -4}\)	\(18\)
norman	\(\left (x \,{\mathrm e}^{4+x}+10 x +4 \ln \relax (x )\right ) {\mathrm e}^{-x -4}\)	\(22\)
risch	\(4 \,{\mathrm e}^{-x -4} \ln \relax (x )+x \left ({\mathrm e}^{4+x}+10\right ) {\mathrm e}^{-x -4}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x*ln(x)+x*exp(4+x)-10*x^2+10*x+4)/x/exp(4+x),x,method=_RETURNVERBOSE)

[Out]

x+(10*x+4*ln(x))/exp(4+x)

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maxima [A]  time = 0.70, size = 31, normalized size = 1.41 \begin {gather*} 10 \, {\left (x + 1\right )} e^{\left (-x - 4\right )} + 4 \, e^{\left (-x - 4\right )} \log \relax (x) + x - 10 \, e^{\left (-x - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*log(x)+x*exp(4+x)-10*x^2+10*x+4)/x/exp(4+x),x, algorithm="maxima")

[Out]

10*(x + 1)*e^(-x - 4) + 4*e^(-x - 4)*log(x) + x - 10*e^(-x - 4)

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mupad [B]  time = 2.01, size = 21, normalized size = 0.95 \begin {gather*} x+10\,x\,{\mathrm {e}}^{-x-4}+4\,{\mathrm {e}}^{-x-4}\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(- x - 4)*(10*x + x*exp(x + 4) - 4*x*log(x) - 10*x^2 + 4))/x,x)

[Out]

x + 10*x*exp(- x - 4) + 4*exp(- x - 4)*log(x)

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sympy [A]  time = 0.30, size = 15, normalized size = 0.68 \begin {gather*} x + \left (10 x + 4 \log {\relax (x )}\right ) e^{- x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*ln(x)+x*exp(4+x)-10*x**2+10*x+4)/x/exp(4+x),x)

[Out]

x + (10*x + 4*log(x))*exp(-x - 4)

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