3.64.65 \(\int \frac {e^{-2 x} (2 e^{8 x}+(-8 e^{5 x}-6 e^{8 x} x) \log (\frac {x}{3})+24 e^{5 x} x \log ^2(\frac {x}{3})+e^{1+3 x} x \log ^3(\frac {x}{3}))}{x \log ^3(\frac {x}{3})} \, dx\)

Optimal. Leaf size=28 \[ -1+e^{1+x}-\left (4-\frac {e^{3 x}}{\log \left (\frac {x}{3}\right )}\right )^2 \]

________________________________________________________________________________________

Rubi [A]  time = 1.41, antiderivative size = 36, normalized size of antiderivative = 1.29, number of steps used = 5, number of rules used = 3, integrand size = 81, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {6742, 2194, 2202} \begin {gather*} e^{x+1}-\frac {e^{6 x}}{\log ^2\left (\frac {x}{3}\right )}+\frac {8 e^{3 x}}{\log \left (\frac {x}{3}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*E^(8*x) + (-8*E^(5*x) - 6*E^(8*x)*x)*Log[x/3] + 24*E^(5*x)*x*Log[x/3]^2 + E^(1 + 3*x)*x*Log[x/3]^3)/(E^
(2*x)*x*Log[x/3]^3),x]

[Out]

E^(1 + x) - E^(6*x)/Log[x/3]^2 + (8*E^(3*x))/Log[x/3]

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 2202

Int[Log[(d_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*(x_)^(m_.)*((e_) + Log[(d_.)*(x_)]*(h_.)*((f_.) +
(g_.)*(x_))), x_Symbol] :> Simp[(e*x^(m + 1)*F^(c*(a + b*x))*Log[d*x]^(n + 1))/(n + 1), x] /; FreeQ[{F, a, b,
c, d, e, f, g, h, m, n}, x] && EqQ[e*(m + 1) - f*h*(n + 1), 0] && EqQ[g*h*(n + 1) - b*c*e*Log[F], 0] && NeQ[n,
-1]

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 (e^{1+x}-\frac {2 e^{6 x} \left (-1+3 x \log \left (\frac {x}{3}\right )\right )}{x \log ^3\left (\frac {x}{3}\right )}+\frac {8 e^{3 x} \left (-1+3 x \log \left (\frac {x}{3}\right )\right )}{x \log ^2\left (\frac {x}{3}\right )}\right ) \, dx\\ &=-\left (2 \int \frac {e^{6 x} \left (-1+3 x \log \left (\frac {x}{3}\right )\right )}{x \log ^3\left (\frac {x}{3}\right )} \, dx\right )+8 \int \frac {e^{3 x} \left (-1+3 x \log \left (\frac {x}{3}\right )\right )}{x \log ^2\left (\frac {x}{3}\right )} \, dx+\int e^{1+x} \, dx\\ &=e^{1+x}-\frac {e^{6 x}}{\log ^2\left (\frac {x}{3}\right )}+\frac {8 e^{3 x}}{\log \left (\frac {x}{3}\right )}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 1.12, size = 36, normalized size = 1.29 \begin {gather*} e^{1+x}-\frac {e^{6 x}}{\log ^2\left (\frac {x}{3}\right )}+\frac {8 e^{3 x}}{\log \left (\frac {x}{3}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*E^(8*x) + (-8*E^(5*x) - 6*E^(8*x)*x)*Log[x/3] + 24*E^(5*x)*x*Log[x/3]^2 + E^(1 + 3*x)*x*Log[x/3]^
3)/(E^(2*x)*x*Log[x/3]^3),x]

[Out]

E^(1 + x) - E^(6*x)/Log[x/3]^2 + (8*E^(3*x))/Log[x/3]

________________________________________________________________________________________

fricas [A]  time = 0.71, size = 36, normalized size = 1.29 \begin {gather*} \frac {{\left (e^{\left (-5 \, x + 1\right )} \log \left (\frac {1}{3} \, x\right )^{2} + 8 \, e^{\left (-3 \, x\right )} \log \left (\frac {1}{3} \, x\right ) - 1\right )} e^{\left (6 \, x\right )}}{\log \left (\frac {1}{3} \, x\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*exp(x)^2*exp(x+1)*log(1/3*x)^3+24*x*exp(x)*exp(4*x)*log(1/3*x)^2+(-6*x*exp(4*x)^2-8*exp(x)*exp(4*
x))*log(1/3*x)+2*exp(4*x)^2)/x/exp(x)^2/log(1/3*x)^3,x, algorithm="fricas")

[Out]

(e^(-5*x + 1)*log(1/3*x)^2 + 8*e^(-3*x)*log(1/3*x) - 1)*e^(6*x)/log(1/3*x)^2

________________________________________________________________________________________

giac [B]  time = 0.19, size = 69, normalized size = 2.46 \begin {gather*} \frac {e^{\left (x + 1\right )} \log \relax (3)^{2} - 2 \, e^{\left (x + 1\right )} \log \relax (3) \log \relax (x) + e^{\left (x + 1\right )} \log \relax (x)^{2} - 8 \, e^{\left (3 \, x\right )} \log \relax (3) + 8 \, e^{\left (3 \, x\right )} \log \relax (x) - e^{\left (6 \, x\right )}}{\log \relax (3)^{2} - 2 \, \log \relax (3) \log \relax (x) + \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*exp(x)^2*exp(x+1)*log(1/3*x)^3+24*x*exp(x)*exp(4*x)*log(1/3*x)^2+(-6*x*exp(4*x)^2-8*exp(x)*exp(4*
x))*log(1/3*x)+2*exp(4*x)^2)/x/exp(x)^2/log(1/3*x)^3,x, algorithm="giac")

[Out]

(e^(x + 1)*log(3)^2 - 2*e^(x + 1)*log(3)*log(x) + e^(x + 1)*log(x)^2 - 8*e^(3*x)*log(3) + 8*e^(3*x)*log(x) - e
^(6*x))/(log(3)^2 - 2*log(3)*log(x) + log(x)^2)

________________________________________________________________________________________

maple [A]  time = 0.10, size = 29, normalized size = 1.04

method	result	size
risch	\({\mathrm e}^{x +1}-\frac {{\mathrm e}^{3 x} \left ({\mathrm e}^{3 x}-8 \ln \left (\frac {x}{3}\right )\right )}{\ln \left (\frac {x}{3}\right )^{2}}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*exp(x)^2*exp(x+1)*ln(1/3*x)^3+24*x*exp(x)*exp(4*x)*ln(1/3*x)^2+(-6*x*exp(4*x)^2-8*exp(x)*exp(4*x))*ln(1
/3*x)+2*exp(4*x)^2)/x/exp(x)^2/ln(1/3*x)^3,x,method=_RETURNVERBOSE)

[Out]

exp(x+1)-exp(3*x)*(exp(3*x)-8*ln(1/3*x))/ln(1/3*x)^2

________________________________________________________________________________________

maxima [B]  time = 0.50, size = 64, normalized size = 2.29 \begin {gather*} -\frac {8 \, {\left (\log \relax (3) - \log \relax (x)\right )} e^{\left (3 \, x\right )} - {\left (e \log \relax (3)^{2} - 2 \, e \log \relax (3) \log \relax (x) + e \log \relax (x)^{2}\right )} e^{x} + e^{\left (6 \, x\right )}}{\log \relax (3)^{2} - 2 \, \log \relax (3) \log \relax (x) + \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*exp(x)^2*exp(x+1)*log(1/3*x)^3+24*x*exp(x)*exp(4*x)*log(1/3*x)^2+(-6*x*exp(4*x)^2-8*exp(x)*exp(4*
x))*log(1/3*x)+2*exp(4*x)^2)/x/exp(x)^2/log(1/3*x)^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 4.47, size = 192, normalized size = 6.86 \begin {gather*} {\mathrm {e}}^{x+1}+24\,x\,{\mathrm {e}}^{3\,x}-\frac {12\,x\,{\mathrm {e}}^{3\,x}\,{\ln \left (\frac {x}{3}\right )}^2-{\mathrm {e}}^{-6}\,\left (4\,{\mathrm {e}}^{3\,x+6}+3\,x\,{\mathrm {e}}^{6\,x+6}\right )\,\ln \left (\frac {x}{3}\right )+{\mathrm {e}}^{6\,x}}{{\ln \left (\frac {x}{3}\right )}^2}+\frac {-12\,x\,{\mathrm {e}}^{-3}\,\left ({\mathrm {e}}^{3\,x+3}+3\,x\,{\mathrm {e}}^{3\,x+3}\right )\,{\ln \left (\frac {x}{3}\right )}^2+3\,x\,{\mathrm {e}}^{-6}\,\left ({\mathrm {e}}^{6\,x+6}-4\,{\mathrm {e}}^{3\,x+6}+6\,x\,{\mathrm {e}}^{6\,x+6}\right )\,\ln \left (\frac {x}{3}\right )+{\mathrm {e}}^{-6}\,\left (4\,{\mathrm {e}}^{3\,x+6}-3\,x\,{\mathrm {e}}^{6\,x+6}\right )}{\ln \left (\frac {x}{3}\right )}-{\mathrm {e}}^{6\,x+6}\,\left (18\,{\mathrm {e}}^{-6}\,x^2+3\,{\mathrm {e}}^{-6}\,x\right )+\ln \left (\frac {x}{3}\right )\,{\mathrm {e}}^{3\,x}\,\left (36\,x^2+12\,x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-2*x)*(2*exp(8*x) - log(x/3)*(8*exp(5*x) + 6*x*exp(8*x)) + 24*x*log(x/3)^2*exp(5*x) + x*log(x/3)^3*ex
p(2*x)*exp(x + 1)))/(x*log(x/3)^3),x)

[Out]

exp(x + 1) + 24*x*exp(3*x) - (exp(6*x) + 12*x*log(x/3)^2*exp(3*x) - log(x/3)*exp(-6)*(4*exp(3*x + 6) + 3*x*exp
(6*x + 6)))/log(x/3)^2 + (exp(-6)*(4*exp(3*x + 6) - 3*x*exp(6*x + 6)) - 12*x*log(x/3)^2*exp(-3)*(exp(3*x + 3)
+ 3*x*exp(3*x + 3)) + 3*x*log(x/3)*exp(-6)*(exp(6*x + 6) - 4*exp(3*x + 6) + 6*x*exp(6*x + 6)))/log(x/3) - exp(
6*x + 6)*(3*x*exp(-6) + 18*x^2*exp(-6)) + log(x/3)*exp(3*x)*(12*x + 36*x^2)

________________________________________________________________________________________

sympy [B]  time = 0.42, size = 42, normalized size = 1.50 \begin {gather*} \frac {- e^{6 x} \log {\left (\frac {x}{3} \right )} + 8 e^{3 x} \log {\left (\frac {x}{3} \right )}^{2} + e e^{x} \log {\left (\frac {x}{3} \right )}^{3}}{\log {\left (\frac {x}{3} \right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*exp(x)**2*exp(x+1)*ln(1/3*x)**3+24*x*exp(x)*exp(4*x)*ln(1/3*x)**2+(-6*x*exp(4*x)**2-8*exp(x)*exp(
4*x))*ln(1/3*x)+2*exp(4*x)**2)/x/exp(x)**2/ln(1/3*x)**3,x)

[Out]

(-exp(6*x)*log(x/3) + 8*exp(3*x)*log(x/3)**2 + E*exp(x)*log(x/3)**3)/log(x/3)**3

________________________________________________________________________________________