\(\int -\frac {e^{\frac {1}{3+\log ^2(3+\log (x))}} \log (3+\log (x))}{54 x+18 x \log (x)+(36 x+12 x \log (x)) \log ^2(3+\log (x))+(6 x+2 x \log (x)) \log ^4(3+\log (x))} \, dx\) [4126]

Optimal result

Integrand size = 65, antiderivative size = 25 \[ \int -\frac {e^{\frac {1}{3+\log ^2(3+\log (x))}} \log (3+\log (x))}{54 x+18 x \log (x)+(36 x+12 x \log (x)) \log ^2(3+\log (x))+(6 x+2 x \log (x)) \log ^4(3+\log (x))} \, dx=\frac {1}{4} \left (7+e^{\frac {x}{3 x+x \log ^2(3+\log (x))}}\right ) \]

[Out]

7/4+1/4*exp(x/(ln(3+ln(x))^2*x+3*x))

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68, number of steps used = 3, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {12, 6874, 6857, 6838} \[ \int -\frac {e^{\frac {1}{3+\log ^2(3+\log (x))}} \log (3+\log (x))}{54 x+18 x \log (x)+(36 x+12 x \log (x)) \log ^2(3+\log (x))+(6 x+2 x \log (x)) \log ^4(3+\log (x))} \, dx=\frac {1}{4} e^{\frac {1}{\log ^2(\log (x)+3)+3}} \]

[In]

Int[-((E^(3 + Log[3 + Log[x]]^2)^(-1)*Log[3 + Log[x]])/(54*x + 18*x*Log[x] + (36*x + 12*x*Log[x])*Log[3 + Log[
x]]^2 + (6*x + 2*x*Log[x])*Log[3 + Log[x]]^4)),x]

[Out]

E^(3 + Log[3 + Log[x]]^2)^(-1)/4

Rule 12

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

Rule 6838

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[q*(F^v/Log[F]), x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {e^{\frac {1}{3+\log ^2(3+x)}} \log (3+x)}{2 (3+x) \left (3+\log ^2(3+x)\right )^2} \, dx,x,\log (x)\right ) \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {e^{\frac {1}{3+\log ^2(3+x)}} \log (3+x)}{(3+x) \left (3+\log ^2(3+x)\right )^2} \, dx,x,\log (x)\right )\right ) \\ & = \frac {1}{4} e^{\frac {1}{3+\log ^2(3+\log (x))}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68 \[ \int -\frac {e^{\frac {1}{3+\log ^2(3+\log (x))}} \log (3+\log (x))}{54 x+18 x \log (x)+(36 x+12 x \log (x)) \log ^2(3+\log (x))+(6 x+2 x \log (x)) \log ^4(3+\log (x))} \, dx=\frac {1}{4} e^{\frac {1}{3+\log ^2(3+\log (x))}} \]

[In]

Integrate[-((E^(3 + Log[3 + Log[x]]^2)^(-1)*Log[3 + Log[x]])/(54*x + 18*x*Log[x] + (36*x + 12*x*Log[x])*Log[3
+ Log[x]]^2 + (6*x + 2*x*Log[x])*Log[3 + Log[x]]^4)),x]

[Out]

E^(3 + Log[3 + Log[x]]^2)^(-1)/4

Maple [A] (verified)

Time = 50.22 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.60

method	result	size
risch	\(\frac {{\mathrm e}^{\frac {1}{\ln \left (3+\ln \left (x \right )\right )^{2}+3}}}{4}\)	\(15\)
parallelrisch	\(\frac {{\mathrm e}^{\frac {1}{\ln \left (3+\ln \left (x \right )\right )^{2}+3}}}{4}\)	\(15\)

[In]

int(-ln(3+ln(x))*exp(1/(ln(3+ln(x))^2+3))/((2*x*ln(x)+6*x)*ln(3+ln(x))^4+(12*x*ln(x)+36*x)*ln(3+ln(x))^2+18*x*
ln(x)+54*x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.56 \[ \int -\frac {e^{\frac {1}{3+\log ^2(3+\log (x))}} \log (3+\log (x))}{54 x+18 x \log (x)+(36 x+12 x \log (x)) \log ^2(3+\log (x))+(6 x+2 x \log (x)) \log ^4(3+\log (x))} \, dx=\frac {1}{4} \, e^{\left (\frac {1}{\log \left (\log \left (x\right ) + 3\right )^{2} + 3}\right )} \]

[In]

integrate(-log(3+log(x))*exp(1/(log(3+log(x))^2+3))/((2*x*log(x)+6*x)*log(3+log(x))^4+(12*x*log(x)+36*x)*log(3
+log(x))^2+18*x*log(x)+54*x),x, algorithm="fricas")

[Out]

1/4*e^(1/(log(log(x) + 3)^2 + 3))

Sympy [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.56 \[ \int -\frac {e^{\frac {1}{3+\log ^2(3+\log (x))}} \log (3+\log (x))}{54 x+18 x \log (x)+(36 x+12 x \log (x)) \log ^2(3+\log (x))+(6 x+2 x \log (x)) \log ^4(3+\log (x))} \, dx=\frac {e^{\frac {1}{\log {\left (\log {\left (x \right )} + 3 \right )}^{2} + 3}}}{4} \]

[In]

integrate(-ln(3+ln(x))*exp(1/(ln(3+ln(x))**2+3))/((2*x*ln(x)+6*x)*ln(3+ln(x))**4+(12*x*ln(x)+36*x)*ln(3+ln(x))
**2+18*x*ln(x)+54*x),x)

[Out]

exp(1/(log(log(x) + 3)**2 + 3))/4

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.56 \[ \int -\frac {e^{\frac {1}{3+\log ^2(3+\log (x))}} \log (3+\log (x))}{54 x+18 x \log (x)+(36 x+12 x \log (x)) \log ^2(3+\log (x))+(6 x+2 x \log (x)) \log ^4(3+\log (x))} \, dx=\frac {1}{4} \, e^{\left (\frac {1}{\log \left (\log \left (x\right ) + 3\right )^{2} + 3}\right )} \]

[In]

integrate(-log(3+log(x))*exp(1/(log(3+log(x))^2+3))/((2*x*log(x)+6*x)*log(3+log(x))^4+(12*x*log(x)+36*x)*log(3
+log(x))^2+18*x*log(x)+54*x),x, algorithm="maxima")

[Out]

1/4*e^(1/(log(log(x) + 3)^2 + 3))

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.56 \[ \int -\frac {e^{\frac {1}{3+\log ^2(3+\log (x))}} \log (3+\log (x))}{54 x+18 x \log (x)+(36 x+12 x \log (x)) \log ^2(3+\log (x))+(6 x+2 x \log (x)) \log ^4(3+\log (x))} \, dx=\frac {1}{4} \, e^{\left (\frac {1}{\log \left (\log \left (x\right ) + 3\right )^{2} + 3}\right )} \]

[In]

integrate(-log(3+log(x))*exp(1/(log(3+log(x))^2+3))/((2*x*log(x)+6*x)*log(3+log(x))^4+(12*x*log(x)+36*x)*log(3
+log(x))^2+18*x*log(x)+54*x),x, algorithm="giac")

[Out]

1/4*e^(1/(log(log(x) + 3)^2 + 3))

Mupad [B] (verification not implemented)

Time = 10.12 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.56 \[ \int -\frac {e^{\frac {1}{3+\log ^2(3+\log (x))}} \log (3+\log (x))}{54 x+18 x \log (x)+(36 x+12 x \log (x)) \log ^2(3+\log (x))+(6 x+2 x \log (x)) \log ^4(3+\log (x))} \, dx=\frac {{\mathrm {e}}^{\frac {1}{{\ln \left (\ln \left (x\right )+3\right )}^2+3}}}{4} \]

[In]

int(-(exp(1/(log(log(x) + 3)^2 + 3))*log(log(x) + 3))/(54*x + log(log(x) + 3)^4*(6*x + 2*x*log(x)) + log(log(x
) + 3)^2*(36*x + 12*x*log(x)) + 18*x*log(x)),x)

[Out]

exp(1/(log(log(x) + 3)^2 + 3))/4