3.85.33 \(\int \frac {e^{-3 x} (36+72 x-2 e^{3 x} x^2+(24+36 x) \log (x))}{x^3} \, dx\)

Optimal. Leaf size=22 \[ 4-\frac {12 e^{-3 x} (2+\log (x))}{x^2}-\log \left (x^2\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.68, antiderivative size = 23, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6741, 12, 6742, 2288} \begin {gather*} -\frac {12 e^{-3 x} (2 x+x \log (x))}{x^3}-2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(36 + 72*x - 2*E^(3*x)*x^2 + (24 + 36*x)*Log[x])/(E^(3*x)*x^3),x]

[Out]

-2*Log[x] - (12*(2*x + x*Log[x]))/(E^(3*x)*x^3)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 6741

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

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 \frac {2 e^{-3 x} \left (18+36 x-e^{3 x} x^2+12 \log (x)+18 x \log (x)\right )}{x^3} \, dx\\ &=2 \int \frac {e^{-3 x} \left (18+36 x-e^{3 x} x^2+12 \log (x)+18 x \log (x)\right )}{x^3} \, dx\\ &=2 \int \left (-\frac {1}{x}+\frac {6 e^{-3 x} (3+6 x+2 \log (x)+3 x \log (x))}{x^3}\right ) \, dx\\ &=-2 \log (x)+12 \int \frac {e^{-3 x} (3+6 x+2 \log (x)+3 x \log (x))}{x^3} \, dx\\ &=-2 \log (x)-\frac {12 e^{-3 x} (2 x+x \log (x))}{x^3}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.30, size = 19, normalized size = 0.86 \begin {gather*} -2 \log (x)-\frac {12 e^{-3 x} (2+\log (x))}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(36 + 72*x - 2*E^(3*x)*x^2 + (24 + 36*x)*Log[x])/(E^(3*x)*x^3),x]

[Out]

-2*Log[x] - (12*(2 + Log[x]))/(E^(3*x)*x^2)

________________________________________________________________________________________

fricas [A]  time = 0.80, size = 24, normalized size = 1.09 \begin {gather*} -\frac {2 \, {\left ({\left (x^{2} e^{\left (3 \, x\right )} + 6\right )} \log \relax (x) + 12\right )} e^{\left (-3 \, x\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((36*x+24)*log(x)-2*x^2*exp(x)^3+72*x+36)/x^3/exp(x)^3,x, algorithm="fricas")

[Out]

-2*((x^2*e^(3*x) + 6)*log(x) + 12)*e^(-3*x)/x^2

________________________________________________________________________________________

giac [A]  time = 0.14, size = 26, normalized size = 1.18 \begin {gather*} -\frac {2 \, {\left (x^{2} \log \relax (x) + 6 \, e^{\left (-3 \, x\right )} \log \relax (x) + 12 \, e^{\left (-3 \, x\right )}\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((36*x+24)*log(x)-2*x^2*exp(x)^3+72*x+36)/x^3/exp(x)^3,x, algorithm="giac")

[Out]

-2*(x^2*log(x) + 6*e^(-3*x)*log(x) + 12*e^(-3*x))/x^2

________________________________________________________________________________________

maple [A]  time = 0.08, size = 34, normalized size = 1.55

method	result	size
risch	\(-\frac {12 \,{\mathrm e}^{-3 x} \ln \relax (x )}{x^{2}}-\frac {2 \left (\ln \relax (x ) x^{2} {\mathrm e}^{3 x}+12\right ) {\mathrm e}^{-3 x}}{x^{2}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((36*x+24)*ln(x)-2*x^2*exp(x)^3+72*x+36)/x^3/exp(x)^3,x,method=_RETURNVERBOSE)

[Out]

-12/x^2*exp(-3*x)*ln(x)-2*(ln(x)*x^2*exp(3*x)+12)/x^2*exp(-3*x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {12 \, e^{\left (-3 \, x\right )} \log \relax (x)}{x^{2}} - 216 \, \Gamma \left (-1, 3 \, x\right ) - 324 \, \Gamma \left (-2, 3 \, x\right ) + 12 \, \int \frac {e^{\left (-3 \, x\right )}}{x^{3}}\,{d x} - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((36*x+24)*log(x)-2*x^2*exp(x)^3+72*x+36)/x^3/exp(x)^3,x, algorithm="maxima")

[Out]

-12*e^(-3*x)*log(x)/x^2 - 216*gamma(-1, 3*x) - 324*gamma(-2, 3*x) + 12*integrate(e^(-3*x)/x^3, x) - 2*log(x)

________________________________________________________________________________________

mupad [B]  time = 5.30, size = 20, normalized size = 0.91 \begin {gather*} -2\,\ln \relax (x)-\frac {2\,{\mathrm {e}}^{-3\,x}\,\left (6\,\ln \relax (x)+12\right )}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-3*x)*(72*x - 2*x^2*exp(3*x) + log(x)*(36*x + 24) + 36))/x^3,x)

[Out]

- 2*log(x) - (2*exp(-3*x)*(6*log(x) + 12))/x^2

________________________________________________________________________________________

sympy [A]  time = 0.43, size = 20, normalized size = 0.91 \begin {gather*} - 2 \log {\relax (x )} + \frac {\left (- 12 \log {\relax (x )} - 24\right ) e^{- 3 x}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((36*x+24)*ln(x)-2*x**2*exp(x)**3+72*x+36)/x**3/exp(x)**3,x)

[Out]

-2*log(x) + (-12*log(x) - 24)*exp(-3*x)/x**2

________________________________________________________________________________________