\(\int \frac {e^{-4-\log ^2((9 x+x \log (3)) \log (x))} (-4-4 \log (x)+(-2-2 \log (x)) \log ((9 x+x \log (3)) \log (x)))}{x (9 x+x \log (3))^4 \log ^5(x)} \, dx\) [7655]

Optimal result

Integrand size = 63, antiderivative size = 17 \[ \int \frac {e^{-4-\log ^2((9 x+x \log (3)) \log (x))} (-4-4 \log (x)+(-2-2 \log (x)) \log ((9 x+x \log (3)) \log (x)))}{x (9 x+x \log (3))^4 \log ^5(x)} \, dx=e^{-(2+\log (x (9+\log (3)) \log (x)))^2} \]

[Out]

1/exp((2+ln((9+ln(3))*ln(x)*x))^2)

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(48\) vs. \(2(17)=34\).

Time = 0.10 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.82, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {6, 12, 2326} \[ \int \frac {e^{-4-\log ^2((9 x+x \log (3)) \log (x))} (-4-4 \log (x)+(-2-2 \log (x)) \log ((9 x+x \log (3)) \log (x)))}{x (9 x+x \log (3))^4 \log ^5(x)} \, dx=\frac {e^{-\log ^2(x (9+\log (3)) \log (x))-4} (\log (x)+1)}{x^4 (9+\log (3))^3 \log ^4(x) ((9+\log (3)) \log (x)+9+\log (3))} \]

[In]

Int[(E^(-4 - Log[(9*x + x*Log[3])*Log[x]]^2)*(-4 - 4*Log[x] + (-2 - 2*Log[x])*Log[(9*x + x*Log[3])*Log[x]]))/(
x*(9*x + x*Log[3])^4*Log[x]^5),x]

[Out]

(E^(-4 - Log[x*(9 + Log[3])*Log[x]]^2)*(1 + Log[x]))/(x^4*(9 + Log[3])^3*Log[x]^4*(9 + Log[3] + (9 + Log[3])*L
og[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 12

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

Rule 2326

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{align*} \text {integral}& = \int \frac {e^{-4-\log ^2((9 x+x \log (3)) \log (x))} (-4-4 \log (x)+(-2-2 \log (x)) \log ((9 x+x \log (3)) \log (x)))}{x^5 (9+\log (3))^4 \log ^5(x)} \, dx \\ & = \frac {\int \frac {e^{-4-\log ^2((9 x+x \log (3)) \log (x))} (-4-4 \log (x)+(-2-2 \log (x)) \log ((9 x+x \log (3)) \log (x)))}{x^5 \log ^5(x)} \, dx}{(9+\log (3))^4} \\ & = \frac {e^{-4-\log ^2(x (9+\log (3)) \log (x))} (1+\log (x))}{x^4 (9+\log (3))^3 \log ^4(x) (9+\log (3)+(9+\log (3)) \log (x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.82 \[ \int \frac {e^{-4-\log ^2((9 x+x \log (3)) \log (x))} (-4-4 \log (x)+(-2-2 \log (x)) \log ((9 x+x \log (3)) \log (x)))}{x (9 x+x \log (3))^4 \log ^5(x)} \, dx=\frac {e^{-4-\log ^2(x (9+\log (3)) \log (x))}}{x^4 (9+\log (3))^4 \log ^4(x)} \]

[In]

Integrate[(E^(-4 - Log[(9*x + x*Log[3])*Log[x]]^2)*(-4 - 4*Log[x] + (-2 - 2*Log[x])*Log[(9*x + x*Log[3])*Log[x
]]))/(x*(9*x + x*Log[3])^4*Log[x]^5),x]

[Out]

E^(-4 - Log[x*(9 + Log[3])*Log[x]]^2)/(x^4*(9 + Log[3])^4*Log[x]^4)

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.65

method	result	size
parallelrisch	\(\frac {{\mathrm e}^{-\ln \left (\left (9+\ln \left (3\right )\right ) \ln \left (x \right ) x \right )^{2}-4}}{\left (9+\ln \left (3\right )\right )^{4} x^{4} \ln \left (x \right )^{4}}\)	\(28\)
risch	\(\text {Expression too large to display}\)	\(567\)

[In]

int(((-2*ln(x)-2)*ln((x*ln(3)+9*x)*ln(x))-4*ln(x)-4)/x/ln(x)/exp(ln((x*ln(3)+9*x)*ln(x))^2+4*ln((x*ln(3)+9*x)*
ln(x))+4),x,method=_RETURNVERBOSE)

[Out]

1/exp(ln((9+ln(3))*ln(x)*x)^2+4*ln((9+ln(3))*ln(x)*x)+4)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (16) = 32\).

Time = 0.30 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.94 \[ \int \frac {e^{-4-\log ^2((9 x+x \log (3)) \log (x))} (-4-4 \log (x)+(-2-2 \log (x)) \log ((9 x+x \log (3)) \log (x)))}{x (9 x+x \log (3))^4 \log ^5(x)} \, dx=e^{\left (-\log \left ({\left (x \log \left (3\right ) + 9 \, x\right )} \log \left (x\right )\right )^{2} - 4 \, \log \left ({\left (x \log \left (3\right ) + 9 \, x\right )} \log \left (x\right )\right ) - 4\right )} \]

[In]

integrate(((-2*log(x)-2)*log((x*log(3)+9*x)*log(x))-4*log(x)-4)/x/log(x)/exp(log((x*log(3)+9*x)*log(x))^2+4*lo
g((x*log(3)+9*x)*log(x))+4),x, algorithm="fricas")

[Out]

e^(-log((x*log(3) + 9*x)*log(x))^2 - 4*log((x*log(3) + 9*x)*log(x)) - 4)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (17) = 34\).

Time = 0.26 (sec) , antiderivative size = 87, normalized size of antiderivative = 5.12 \[ \int \frac {e^{-4-\log ^2((9 x+x \log (3)) \log (x))} (-4-4 \log (x)+(-2-2 \log (x)) \log ((9 x+x \log (3)) \log (x)))}{x (9 x+x \log (3))^4 \log ^5(x)} \, dx=\frac {e^{- \log {\left (\left (x \log {\left (3 \right )} + 9 x\right ) \log {\left (x \right )} \right )}^{2} - 4}}{x^{4} \log {\left (3 \right )}^{4} \log {\left (x \right )}^{4} + 36 x^{4} \log {\left (3 \right )}^{3} \log {\left (x \right )}^{4} + 486 x^{4} \log {\left (3 \right )}^{2} \log {\left (x \right )}^{4} + 2916 x^{4} \log {\left (3 \right )} \log {\left (x \right )}^{4} + 6561 x^{4} \log {\left (x \right )}^{4}} \]

[In]

integrate(((-2*ln(x)-2)*ln((x*ln(3)+9*x)*ln(x))-4*ln(x)-4)/x/ln(x)/exp(ln((x*ln(3)+9*x)*ln(x))**2+4*ln((x*ln(3
)+9*x)*ln(x))+4),x)

[Out]

exp(-log((x*log(3) + 9*x)*log(x))**2 - 4)/(x**4*log(3)**4*log(x)**4 + 36*x**4*log(3)**3*log(x)**4 + 486*x**4*l
og(3)**2*log(x)**4 + 2916*x**4*log(3)*log(x)**4 + 6561*x**4*log(x)**4)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (16) = 32\).

Time = 0.40 (sec) , antiderivative size = 83, normalized size of antiderivative = 4.88 \[ \int \frac {e^{-4-\log ^2((9 x+x \log (3)) \log (x))} (-4-4 \log (x)+(-2-2 \log (x)) \log ((9 x+x \log (3)) \log (x)))}{x (9 x+x \log (3))^4 \log ^5(x)} \, dx=\frac {e^{\left (-\log \left (x\right )^{2} - 2 \, \log \left (x\right ) \log \left (\log \left (3\right ) + 9\right ) - \log \left (\log \left (3\right ) + 9\right )^{2} - 2 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) - 2 \, \log \left (\log \left (3\right ) + 9\right ) \log \left (\log \left (x\right )\right ) - \log \left (\log \left (x\right )\right )^{2} - 4\right )}}{{\left (\log \left (3\right )^{4} + 36 \, \log \left (3\right )^{3} + 486 \, \log \left (3\right )^{2} + 2916 \, \log \left (3\right ) + 6561\right )} x^{4} \log \left (x\right )^{4}} \]

[In]

integrate(((-2*log(x)-2)*log((x*log(3)+9*x)*log(x))-4*log(x)-4)/x/log(x)/exp(log((x*log(3)+9*x)*log(x))^2+4*lo
g((x*log(3)+9*x)*log(x))+4),x, algorithm="maxima")

[Out]

e^(-log(x)^2 - 2*log(x)*log(log(3) + 9) - log(log(3) + 9)^2 - 2*log(x)*log(log(x)) - 2*log(log(3) + 9)*log(log
(x)) - log(log(x))^2 - 4)/((log(3)^4 + 36*log(3)^3 + 486*log(3)^2 + 2916*log(3) + 6561)*x^4*log(x)^4)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (16) = 32\).

Time = 0.34 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.06 \[ \int \frac {e^{-4-\log ^2((9 x+x \log (3)) \log (x))} (-4-4 \log (x)+(-2-2 \log (x)) \log ((9 x+x \log (3)) \log (x)))}{x (9 x+x \log (3))^4 \log ^5(x)} \, dx=e^{\left (-\log \left (x \log \left (3\right ) \log \left (x\right ) + 9 \, x \log \left (x\right )\right )^{2} - 4 \, \log \left (x \log \left (3\right ) \log \left (x\right ) + 9 \, x \log \left (x\right )\right ) - 4\right )} \]

[In]

integrate(((-2*log(x)-2)*log((x*log(3)+9*x)*log(x))-4*log(x)-4)/x/log(x)/exp(log((x*log(3)+9*x)*log(x))^2+4*lo
g((x*log(3)+9*x)*log(x))+4),x, algorithm="giac")

[Out]

e^(-log(x*log(3)*log(x) + 9*x*log(x))^2 - 4*log(x*log(3)*log(x) + 9*x*log(x)) - 4)

Mupad [B] (verification not implemented)

Time = 13.51 (sec) , antiderivative size = 82, normalized size of antiderivative = 4.82 \[ \int \frac {e^{-4-\log ^2((9 x+x \log (3)) \log (x))} (-4-4 \log (x)+(-2-2 \log (x)) \log ((9 x+x \log (3)) \log (x)))}{x (9 x+x \log (3))^4 \log ^5(x)} \, dx=\frac {{\mathrm {e}}^{-4}\,{\mathrm {e}}^{-{\ln \left (9\,x\,\ln \left (x\right )+x\,\ln \left (3\right )\,\ln \left (x\right )\right )}^2}}{6561\,x^4\,{\ln \left (x\right )}^4+486\,x^4\,{\ln \left (3\right )}^2\,{\ln \left (x\right )}^4+36\,x^4\,{\ln \left (3\right )}^3\,{\ln \left (x\right )}^4+x^4\,{\ln \left (3\right )}^4\,{\ln \left (x\right )}^4+2916\,x^4\,\ln \left (3\right )\,{\ln \left (x\right )}^4} \]

[In]

int(-(exp(- 4*log(log(x)*(9*x + x*log(3))) - log(log(x)*(9*x + x*log(3)))^2 - 4)*(4*log(x) + log(log(x)*(9*x +
x*log(3)))*(2*log(x) + 2) + 4))/(x*log(x)),x)

[Out]

(exp(-4)*exp(-log(9*x*log(x) + x*log(3)*log(x))^2))/(6561*x^4*log(x)^4 + 486*x^4*log(3)^2*log(x)^4 + 36*x^4*lo
g(3)^3*log(x)^4 + x^4*log(3)^4*log(x)^4 + 2916*x^4*log(3)*log(x)^4)