3.76.4 \(\int \frac {e^{-x} ((-36+24 e^2-4 e^4) \log (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2})+(-9 x+6 e^2 x-e^4 x-27 x^2) \log ^2(\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}))}{9 x-6 e^2 x+e^4 x+27 x^2} \, dx\)

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

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Rubi [B]  time = 0.26, antiderivative size = 88, normalized size of antiderivative = 3.14, number of steps used = 4, number of rules used = 3, integrand size = 154, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {6, 1593, 2288} \begin {gather*} \frac {e^{-x} \left (27 x^2+e^4 x-6 e^2 x+9 x\right ) \log ^2\left (\frac {729 x^2+486 x+54 e^4 (x+1)-108 e^2 (3 x+1)+e^8-12 e^6+81}{81 x^2}\right )}{x \left (27 x+\left (e^2-3\right )^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-36 + 24*E^2 - 4*E^4)*Log[(81 - 12*E^6 + E^8 + E^2*(-108 - 324*x) + 486*x + 729*x^2 + E^4*(54 + 54*x))/(
81*x^2)] + (-9*x + 6*E^2*x - E^4*x - 27*x^2)*Log[(81 - 12*E^6 + E^8 + E^2*(-108 - 324*x) + 486*x + 729*x^2 + E
^4*(54 + 54*x))/(81*x^2)]^2)/(E^x*(9*x - 6*E^2*x + E^4*x + 27*x^2)),x]

[Out]

((9*x - 6*E^2*x + E^4*x + 27*x^2)*Log[(81 - 12*E^6 + E^8 + 486*x + 729*x^2 + 54*E^4*(1 + x) - 108*E^2*(1 + 3*x
))/(81*x^2)]^2)/(E^x*x*((-3 + E^2)^2 + 27*x))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x} \left (\left (-36+24 e^2-4 e^4\right ) \log \left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )+\left (-9 x+6 e^2 x-e^4 x-27 x^2\right ) \log ^2\left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )\right )}{e^4 x+\left (9-6 e^2\right ) x+27 x^2} \, dx\\ &=\int \frac {e^{-x} \left (\left (-36+24 e^2-4 e^4\right ) \log \left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )+\left (-9 x+6 e^2 x-e^4 x-27 x^2\right ) \log ^2\left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )\right )}{\left (9-6 e^2+e^4\right ) x+27 x^2} \, dx\\ &=\int \frac {e^{-x} \left (\left (-36+24 e^2-4 e^4\right ) \log \left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )+\left (-9 x+6 e^2 x-e^4 x-27 x^2\right ) \log ^2\left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )\right )}{x \left (9-6 e^2+e^4+27 x\right )} \, dx\\ &=\frac {e^{-x} \left (9 x-6 e^2 x+e^4 x+27 x^2\right ) \log ^2\left (\frac {81-12 e^6+e^8+486 x+729 x^2+54 e^4 (1+x)-108 e^2 (1+3 x)}{81 x^2}\right )}{x \left (\left (-3+e^2\right )^2+27 x\right )}\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.15, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{-x} \left (\left (-36+24 e^2-4 e^4\right ) \log \left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )+\left (-9 x+6 e^2 x-e^4 x-27 x^2\right ) \log ^2\left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )\right )}{9 x-6 e^2 x+e^4 x+27 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-36 + 24*E^2 - 4*E^4)*Log[(81 - 12*E^6 + E^8 + E^2*(-108 - 324*x) + 486*x + 729*x^2 + E^4*(54 + 54
*x))/(81*x^2)] + (-9*x + 6*E^2*x - E^4*x - 27*x^2)*Log[(81 - 12*E^6 + E^8 + E^2*(-108 - 324*x) + 486*x + 729*x
^2 + E^4*(54 + 54*x))/(81*x^2)]^2)/(E^x*(9*x - 6*E^2*x + E^4*x + 27*x^2)),x]

[Out]

Integrate[((-36 + 24*E^2 - 4*E^4)*Log[(81 - 12*E^6 + E^8 + E^2*(-108 - 324*x) + 486*x + 729*x^2 + E^4*(54 + 54
*x))/(81*x^2)] + (-9*x + 6*E^2*x - E^4*x - 27*x^2)*Log[(81 - 12*E^6 + E^8 + E^2*(-108 - 324*x) + 486*x + 729*x
^2 + E^4*(54 + 54*x))/(81*x^2)]^2)/(E^x*(9*x - 6*E^2*x + E^4*x + 27*x^2)), x]

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fricas [A]  time = 1.03, size = 45, normalized size = 1.61 \begin {gather*} e^{\left (-x\right )} \log \left (\frac {729 \, x^{2} + 54 \, {\left (x + 1\right )} e^{4} - 108 \, {\left (3 \, x + 1\right )} e^{2} + 486 \, x + e^{8} - 12 \, e^{6} + 81}{81 \, x^{2}}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x*exp(2)^2+6*exp(2)*x-27*x^2-9*x)*log(1/81*(exp(2)^4-12*exp(2)^3+(54*x+54)*exp(2)^2+(-324*x-108)*
exp(2)+729*x^2+486*x+81)/x^2)^2+(-4*exp(2)^2+24*exp(2)-36)*log(1/81*(exp(2)^4-12*exp(2)^3+(54*x+54)*exp(2)^2+(
-324*x-108)*exp(2)+729*x^2+486*x+81)/x^2))/(x*exp(2)^2-6*exp(2)*x+27*x^2+9*x)/exp(x),x, algorithm="fricas")

[Out]

e^(-x)*log(1/81*(729*x^2 + 54*(x + 1)*e^4 - 108*(3*x + 1)*e^2 + 486*x + e^8 - 12*e^6 + 81)/x^2)^2

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giac [A]  time = 3.05, size = 47, normalized size = 1.68 \begin {gather*} e^{\left (-x\right )} \log \left (\frac {729 \, x^{2} + 54 \, x e^{4} - 324 \, x e^{2} + 486 \, x + e^{8} - 12 \, e^{6} + 54 \, e^{4} - 108 \, e^{2} + 81}{81 \, x^{2}}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x*exp(2)^2+6*exp(2)*x-27*x^2-9*x)*log(1/81*(exp(2)^4-12*exp(2)^3+(54*x+54)*exp(2)^2+(-324*x-108)*
exp(2)+729*x^2+486*x+81)/x^2)^2+(-4*exp(2)^2+24*exp(2)-36)*log(1/81*(exp(2)^4-12*exp(2)^3+(54*x+54)*exp(2)^2+(
-324*x-108)*exp(2)+729*x^2+486*x+81)/x^2))/(x*exp(2)^2-6*exp(2)*x+27*x^2+9*x)/exp(x),x, algorithm="giac")

[Out]

e^(-x)*log(1/81*(729*x^2 + 54*x*e^4 - 324*x*e^2 + 486*x + e^8 - 12*e^6 + 54*e^4 - 108*e^2 + 81)/x^2)^2

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maple [B]  time = 0.76, size = 52, normalized size = 1.86

method	result	size
norman	\(\ln \left (\frac {{\mathrm e}^{8}-12 \,{\mathrm e}^{6}+\left (54 x +54\right ) {\mathrm e}^{4}+\left (-324 x -108\right ) {\mathrm e}^{2}+729 x^{2}+486 x +81}{81 x^{2}}\right )^{2} {\mathrm e}^{-x}\)	\(52\)
risch	\(\text {Expression too large to display}\)	\(3672\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x*exp(2)^2+6*exp(2)*x-27*x^2-9*x)*ln(1/81*(exp(2)^4-12*exp(2)^3+(54*x+54)*exp(2)^2+(-324*x-108)*exp(2)+
729*x^2+486*x+81)/x^2)^2+(-4*exp(2)^2+24*exp(2)-36)*ln(1/81*(exp(2)^4-12*exp(2)^3+(54*x+54)*exp(2)^2+(-324*x-1
08)*exp(2)+729*x^2+486*x+81)/x^2))/(x*exp(2)^2-6*exp(2)*x+27*x^2+9*x)/exp(x),x,method=_RETURNVERBOSE)

[Out]

ln(1/81*(exp(2)^4-12*exp(2)^3+(54*x+54)*exp(2)^2+(-324*x-108)*exp(2)+729*x^2+486*x+81)/x^2)^2/exp(x)

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maxima [B]  time = 0.54, size = 58, normalized size = 2.07 \begin {gather*} 4 \, {\left (4 \, \log \relax (3)^{2} - 2 \, {\left (2 \, \log \relax (3) + \log \relax (x)\right )} \log \left (27 \, x + e^{4} - 6 \, e^{2} + 9\right ) + \log \left (27 \, x + e^{4} - 6 \, e^{2} + 9\right )^{2} + 4 \, \log \relax (3) \log \relax (x) + \log \relax (x)^{2}\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x*exp(2)^2+6*exp(2)*x-27*x^2-9*x)*log(1/81*(exp(2)^4-12*exp(2)^3+(54*x+54)*exp(2)^2+(-324*x-108)*
exp(2)+729*x^2+486*x+81)/x^2)^2+(-4*exp(2)^2+24*exp(2)-36)*log(1/81*(exp(2)^4-12*exp(2)^3+(54*x+54)*exp(2)^2+(
-324*x-108)*exp(2)+729*x^2+486*x+81)/x^2))/(x*exp(2)^2-6*exp(2)*x+27*x^2+9*x)/exp(x),x, algorithm="maxima")

[Out]

4*(4*log(3)^2 - 2*(2*log(3) + log(x))*log(27*x + e^4 - 6*e^2 + 9) + log(27*x + e^4 - 6*e^2 + 9)^2 + 4*log(3)*l
og(x) + log(x)^2)*e^(-x)

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mupad [B]  time = 5.80, size = 48, normalized size = 1.71 \begin {gather*} {\mathrm {e}}^{-x}\,{\ln \left (\frac {6\,x-\frac {4\,{\mathrm {e}}^6}{27}+\frac {{\mathrm {e}}^8}{81}+9\,x^2+\frac {{\mathrm {e}}^4\,\left (54\,x+54\right )}{81}-\frac {{\mathrm {e}}^2\,\left (324\,x+108\right )}{81}+1}{x^2}\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-x)*(log((6*x - (4*exp(6))/27 + exp(8)/81 + 9*x^2 + (exp(4)*(54*x + 54))/81 - (exp(2)*(324*x + 108))
/81 + 1)/x^2)^2*(9*x - 6*x*exp(2) + x*exp(4) + 27*x^2) + log((6*x - (4*exp(6))/27 + exp(8)/81 + 9*x^2 + (exp(4
)*(54*x + 54))/81 - (exp(2)*(324*x + 108))/81 + 1)/x^2)*(4*exp(4) - 24*exp(2) + 36)))/(9*x - 6*x*exp(2) + x*ex
p(4) + 27*x^2),x)

[Out]

exp(-x)*log((6*x - (4*exp(6))/27 + exp(8)/81 + 9*x^2 + (exp(4)*(54*x + 54))/81 - (exp(2)*(324*x + 108))/81 + 1
)/x^2)^2

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x*exp(2)**2+6*exp(2)*x-27*x**2-9*x)*ln(1/81*(exp(2)**4-12*exp(2)**3+(54*x+54)*exp(2)**2+(-324*x-1
08)*exp(2)+729*x**2+486*x+81)/x**2)**2+(-4*exp(2)**2+24*exp(2)-36)*ln(1/81*(exp(2)**4-12*exp(2)**3+(54*x+54)*e
xp(2)**2+(-324*x-108)*exp(2)+729*x**2+486*x+81)/x**2))/(x*exp(2)**2-6*exp(2)*x+27*x**2+9*x)/exp(x),x)

[Out]

Timed out

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