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

Optimal. Leaf size=17 \[ e^{-(2+\log (x (9+\log (3)) \log (x)))^2} \]

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Rubi [B]  time = 0.16, antiderivative size = 48, normalized size of antiderivative = 2.82, number of steps used = 3, number of rules used = 3, integrand size = 63, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {6, 12, 2288} \begin {gather*} \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))} \end {gather*}

Antiderivative was successfully verified.

[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 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^{-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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 31, normalized size = 1.82 \begin {gather*} \frac {e^{-4-\log ^2(x (9+\log (3)) \log (x))}}{x^4 (9+\log (3))^4 \log ^4(x)} \end {gather*}

Antiderivative was successfully verified.

[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)

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fricas [B]  time = 0.73, size = 33, normalized size = 1.94 \begin {gather*} e^{\left (-\log \left ({\left (x \log \relax (3) + 9 \, x\right )} \log \relax (x)\right )^{2} - 4 \, \log \left ({\left (x \log \relax (3) + 9 \, x\right )} \log \relax (x)\right ) - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [B]  time = 0.35, size = 35, normalized size = 2.06 \begin {gather*} e^{\left (-\log \left (x \log \relax (3) \log \relax (x) + 9 \, x \log \relax (x)\right )^{2} - 4 \, \log \left (x \log \relax (3) \log \relax (x) + 9 \, x \log \relax (x)\right ) - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [C]  time = 0.22, size = 567, normalized size = 33.35

method	result	size
risch	\(\frac {\left (9+\ln \relax (3)\right )^{-i \pi \,\mathrm {csgn}\left (i x \right )} \ln \relax (x )^{-i \pi \,\mathrm {csgn}\left (i x \right )} \ln \relax (x )^{-i \pi \,\mathrm {csgn}\left (i \ln \relax (x )\right )} \ln \relax (x )^{i \pi \,\mathrm {csgn}\left (i x \ln \relax (x )\right )} \ln \relax (x )^{i \pi \,\mathrm {csgn}\left (i x \ln \relax (x )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right )} x^{i \pi \,\mathrm {csgn}\left (i x \ln \relax (x )\right )} \ln \relax (x )^{-2 \ln \left (9+\ln \relax (3)\right )} x^{-2 \ln \left (9+\ln \relax (3)\right )} x^{i \pi \,\mathrm {csgn}\left (i x \ln \relax (x )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right )} x^{-i \pi \,\mathrm {csgn}\left (i \ln \relax (x )\right )} x^{-i \pi \,\mathrm {csgn}\left (i x \right )} \left (9+\ln \relax (3)\right )^{i \pi \,\mathrm {csgn}\left (i x \ln \relax (x )\right )} \left (9+\ln \relax (3)\right )^{i \pi \,\mathrm {csgn}\left (i x \ln \relax (x )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right )} \left (9+\ln \relax (3)\right )^{-i \pi \,\mathrm {csgn}\left (i \ln \relax (x )\right )} \ln \relax (x )^{-2 \ln \relax (x )} {\mathrm e}^{-4-\ln \left (\ln \relax (x )\right )^{2}-\ln \relax (x )^{2}+\pi ^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{4} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right )-\frac {\pi ^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )}{2}-\frac {\pi ^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3} \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i x \right )}{2}+\frac {\pi ^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )^{2}}{4}-2 i \pi \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i x \right )-2 i \pi \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )+\frac {\pi ^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{6}}{4}+\frac {\pi ^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{4} \mathrm {csgn}\left (i \ln \relax (x )\right )^{2}}{4}-\frac {\pi ^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{5} \mathrm {csgn}\left (i x \right )}{2}-\frac {\pi ^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{5} \mathrm {csgn}\left (i \ln \relax (x )\right )}{2}+\frac {\pi ^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{4} \mathrm {csgn}\left (i x \right )^{2}}{4}+2 i \pi \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}+2 i \pi \,\mathrm {csgn}\left (i x \ln \relax (x )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right )-\ln \left (9+\ln \relax (3)\right )^{2}}}{\ln \relax (x )^{4} x^{4} \left (9+\ln \relax (3)\right )^{4}}\)	\(567\)

Verification of antiderivative is not currently implemented for this CAS.

[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/((9+ln(3))^(I*Pi*csgn(I*x)))/(ln(x)^(I*Pi*csgn(I*x)))/(ln(x)^(I*Pi*csgn(I*ln(x))))/(ln(x)^(-I*Pi*csgn(I*x*ln
(x))))/(ln(x)^(-I*Pi*csgn(I*x*ln(x))*csgn(I*x)*csgn(I*ln(x))))/(x^(-I*Pi*csgn(I*x*ln(x))))/(ln(x)^(2*ln(9+ln(3
))))/(x^(2*ln(9+ln(3))))/(x^(-I*Pi*csgn(I*x*ln(x))*csgn(I*x)*csgn(I*ln(x))))/(x^(I*Pi*csgn(I*ln(x))))/(x^(I*Pi
*csgn(I*x)))/((9+ln(3))^(-I*Pi*csgn(I*x*ln(x))))/((9+ln(3))^(-I*Pi*csgn(I*x*ln(x))*csgn(I*x)*csgn(I*ln(x))))/(
(9+ln(3))^(I*Pi*csgn(I*ln(x))))/(ln(x)^(2*ln(x)))/ln(x)^4/x^4/(9+ln(3))^4*exp(-4-ln(ln(x))^2-ln(x)^2+Pi^2*csgn
(I*x*ln(x))^4*csgn(I*x)*csgn(I*ln(x))-1/2*Pi^2*csgn(I*x*ln(x))^3*csgn(I*x)^2*csgn(I*ln(x))-1/2*Pi^2*csgn(I*x*l
n(x))^3*csgn(I*ln(x))^2*csgn(I*x)+1/4*Pi^2*csgn(I*x*ln(x))^2*csgn(I*x)^2*csgn(I*ln(x))^2-2*I*Pi*csgn(I*x*ln(x)
)^2*csgn(I*x)-2*I*Pi*csgn(I*x*ln(x))^2*csgn(I*ln(x))+1/4*Pi^2*csgn(I*x*ln(x))^6+1/4*Pi^2*csgn(I*x*ln(x))^4*csg
n(I*ln(x))^2-1/2*Pi^2*csgn(I*x*ln(x))^5*csgn(I*x)-1/2*Pi^2*csgn(I*x*ln(x))^5*csgn(I*ln(x))+1/4*Pi^2*csgn(I*x*l
n(x))^4*csgn(I*x)^2+2*I*Pi*csgn(I*x*ln(x))^3+2*I*Pi*csgn(I*x*ln(x))*csgn(I*x)*csgn(I*ln(x))-ln(9+ln(3))^2)

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maxima [B]  time = 0.59, size = 83, normalized size = 4.88 \begin {gather*} \frac {e^{\left (-\log \relax (x)^{2} - 2 \, \log \relax (x) \log \left (\log \relax (3) + 9\right ) - \log \left (\log \relax (3) + 9\right )^{2} - 2 \, \log \relax (x) \log \left (\log \relax (x)\right ) - 2 \, \log \left (\log \relax (3) + 9\right ) \log \left (\log \relax (x)\right ) - \log \left (\log \relax (x)\right )^{2} - 4\right )}}{{\left (\log \relax (3)^{4} + 36 \, \log \relax (3)^{3} + 486 \, \log \relax (3)^{2} + 2916 \, \log \relax (3) + 6561\right )} x^{4} \log \relax (x)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 4.81, size = 82, normalized size = 4.82 \begin {gather*} \frac {{\mathrm {e}}^{-4}\,{\mathrm {e}}^{-{\ln \left (9\,x\,\ln \relax (x)+x\,\ln \relax (3)\,\ln \relax (x)\right )}^2}}{6561\,x^4\,{\ln \relax (x)}^4+486\,x^4\,{\ln \relax (3)}^2\,{\ln \relax (x)}^4+36\,x^4\,{\ln \relax (3)}^3\,{\ln \relax (x)}^4+x^4\,{\ln \relax (3)}^4\,{\ln \relax (x)}^4+2916\,x^4\,\ln \relax (3)\,{\ln \relax (x)}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [B]  time = 0.57, size = 87, normalized size = 5.12 \begin {gather*} \frac {e^{- \log {\left (\left (x \log {\relax (3 )} + 9 x\right ) \log {\relax (x )} \right )}^{2} - 4}}{x^{4} \log {\relax (3 )}^{4} \log {\relax (x )}^{4} + 36 x^{4} \log {\relax (3 )}^{3} \log {\relax (x )}^{4} + 486 x^{4} \log {\relax (3 )}^{2} \log {\relax (x )}^{4} + 2916 x^{4} \log {\relax (3 )} \log {\relax (x )}^{4} + 6561 x^{4} \log {\relax (x )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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