3.6.51 $\int \frac{e^{\frac{6}{-6561-11502x-4717x^2+1580x^3+970x^4-174x^5-60x^6+16x^7-x^8+\log (\frac{3}{x})}}(6+69012x+56604x^2-28440x^3-23280x^4+5220x^5+2160x^6-672x^7+48x^8)}{43046721x+150929244x^2+194192478x^3+87777108x^4-26824571x^5-34936372x^6-1864564x^7+5877004x^8+602158x^9-655100x^{10}-26130x^{11}+48760x^{12}-3908x^{13}-1572x^{14}+376x^{15}-32x^{16}+x^{17}+(-13122x-23004x^2-9434x^3+3160x^4+1940x^5-348x^6-120x^7+32x^8-2x^9)\log (\frac{3}{x})+x{\log }^2(\frac{3}{x})}dx$

Optimal. Leaf size=32 $$e^{\frac{6}{-{(-x+(9+(4-x)x{)}^2)}^2+\log \left(\frac{3}{x}\right)}}$$

________________________________________________________________________________________

Rubi [A]  time = 5.93, antiderivative size = 35, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, integrand size = 238, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.013, Rules used = {6688, 12, 6706} $$\begin{array}{c}\exp (-\frac{6}{{(x^4-8x^3-2x^2+71x+81)}^2-\log \left(\frac{3}{x}\right)})\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(E^(6/(-6561 - 11502*x - 4717*x^2 + 1580*x^3 + 970*x^4 - 174*x^5 - 60*x^6 + 16*x^7 - x^8 + Log[3/x]))*(6 +
69012*x + 56604*x^2 - 28440*x^3 - 23280*x^4 + 5220*x^5 + 2160*x^6 - 672*x^7 + 48*x^8))/(43046721*x + 15092924
4*x^2 + 194192478*x^3 + 87777108*x^4 - 26824571*x^5 - 34936372*x^6 - 1864564*x^7 + 5877004*x^8 + 602158*x^9 -
655100*x^10 - 26130*x^11 + 48760*x^12 - 3908*x^13 - 1572*x^14 + 376*x^15 - 32*x^16 + x^17 + (-13122*x - 23004*
x^2 - 9434*x^3 + 3160*x^4 + 1940*x^5 - 348*x^6 - 120*x^7 + 32*x^8 - 2*x^9)*Log[3/x] + x*Log[3/x]^2),x]

[Out]

E^(-6/((81 + 71*x - 2*x^2 - 8*x^3 + x^4)^2 - Log[3/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 6688

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

Rule 6706

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

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{6\exp (-\frac{6}{{(81+71x-2x^2-8x^3+x^4)}^2-\log \left(\frac{3}{x}\right)})(1+11502x+9434x^2-4740x^3-3880x^4+870x^5+360x^6-112x^7+8x^8)}{x{({(81+71x-2x^2-8x^3+x^4)}^2-\log \left(\frac{3}{x}\right))}^2}dx\\ & =6\int \frac{\exp (-\frac{6}{{(81+71x-2x^2-8x^3+x^4)}^2-\log \left(\frac{3}{x}\right)})(1+11502x+9434x^2-4740x^3-3880x^4+870x^5+360x^6-112x^7+8x^8)}{x{({(81+71x-2x^2-8x^3+x^4)}^2-\log \left(\frac{3}{x}\right))}^2}dx\\ & =\exp (-\frac{6}{{(81+71x-2x^2-8x^3+x^4)}^2-\log \left(\frac{3}{x}\right)})\end{array}\end{array}$$

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 35, normalized size = 1.09 $$\begin{array}{c}{e}^{-\frac{6}{{(81+71x-2x^2-8x^3+x^4)}^2-\log \left(\frac{3}{x}\right)}}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(E^(6/(-6561 - 11502*x - 4717*x^2 + 1580*x^3 + 970*x^4 - 174*x^5 - 60*x^6 + 16*x^7 - x^8 + Log[3/x])
)*(6 + 69012*x + 56604*x^2 - 28440*x^3 - 23280*x^4 + 5220*x^5 + 2160*x^6 - 672*x^7 + 48*x^8))/(43046721*x + 15
0929244*x^2 + 194192478*x^3 + 87777108*x^4 - 26824571*x^5 - 34936372*x^6 - 1864564*x^7 + 5877004*x^8 + 602158*
x^9 - 655100*x^10 - 26130*x^11 + 48760*x^12 - 3908*x^13 - 1572*x^14 + 376*x^15 - 32*x^16 + x^17 + (-13122*x -
23004*x^2 - 9434*x^3 + 3160*x^4 + 1940*x^5 - 348*x^6 - 120*x^7 + 32*x^8 - 2*x^9)*Log[3/x] + x*Log[3/x]^2),x]

[Out]

E^(-6/((81 + 71*x - 2*x^2 - 8*x^3 + x^4)^2 - Log[3/x]))

________________________________________________________________________________________

fricas [A]  time = 0.77, size = 51, normalized size = 1.59 $$\begin{array}{c}{e}^{(-\frac{6}{x^8-16x^7+60x^6+174x^5-970x^4-1580x^3+4717x^2+11502x-\log \left(\frac{3}{x}\right)+6561})}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((48*x^8-672*x^7+2160*x^6+5220*x^5-23280*x^4-28440*x^3+56604*x^2+69012*x+6)*exp(6/(log(3/x)-x^8+16*x^
7-60*x^6-174*x^5+970*x^4+1580*x^3-4717*x^2-11502*x-6561))/(x*log(3/x)^2+(-2*x^9+32*x^8-120*x^7-348*x^6+1940*x^
5+3160*x^4-9434*x^3-23004*x^2-13122*x)*log(3/x)+x^17-32*x^16+376*x^15-1572*x^14-3908*x^13+48760*x^12-26130*x^1
1-655100*x^10+602158*x^9+5877004*x^8-1864564*x^7-34936372*x^6-26824571*x^5+87777108*x^4+194192478*x^3+15092924
4*x^2+43046721*x),x, algorithm="fricas")

[Out]

e^(-6/(x^8 - 16*x^7 + 60*x^6 + 174*x^5 - 970*x^4 - 1580*x^3 + 4717*x^2 + 11502*x - log(3/x) + 6561))

________________________________________________________________________________________

giac [A]  time = 1.10, size = 51, normalized size = 1.59 $$\begin{array}{c}{e}^{(-\frac{6}{x^8-16x^7+60x^6+174x^5-970x^4-1580x^3+4717x^2+11502x-\log \left(\frac{3}{x}\right)+6561})}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((48*x^8-672*x^7+2160*x^6+5220*x^5-23280*x^4-28440*x^3+56604*x^2+69012*x+6)*exp(6/(log(3/x)-x^8+16*x^
7-60*x^6-174*x^5+970*x^4+1580*x^3-4717*x^2-11502*x-6561))/(x*log(3/x)^2+(-2*x^9+32*x^8-120*x^7-348*x^6+1940*x^
5+3160*x^4-9434*x^3-23004*x^2-13122*x)*log(3/x)+x^17-32*x^16+376*x^15-1572*x^14-3908*x^13+48760*x^12-26130*x^1
1-655100*x^10+602158*x^9+5877004*x^8-1864564*x^7-34936372*x^6-26824571*x^5+87777108*x^4+194192478*x^3+15092924
4*x^2+43046721*x),x, algorithm="giac")

[Out]

e^(-6/(x^8 - 16*x^7 + 60*x^6 + 174*x^5 - 970*x^4 - 1580*x^3 + 4717*x^2 + 11502*x - log(3/x) + 6561))

________________________________________________________________________________________

maple [A]  time = 0.15, size = 52, normalized size = 1.62

method	result	size
risch	${\mathrm{e}}^{\frac{6}{\ln \left(\frac{3}{x}\right)-x^8+16x^7-60x^6-174x^5+970x^4+1580x^3-4717x^2-11502x-6561}}$	$52$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((48*x^8-672*x^7+2160*x^6+5220*x^5-23280*x^4-28440*x^3+56604*x^2+69012*x+6)*exp(6/(ln(3/x)-x^8+16*x^7-60*x^
6-174*x^5+970*x^4+1580*x^3-4717*x^2-11502*x-6561))/(x*ln(3/x)^2+(-2*x^9+32*x^8-120*x^7-348*x^6+1940*x^5+3160*x
^4-9434*x^3-23004*x^2-13122*x)*ln(3/x)+x^17-32*x^16+376*x^15-1572*x^14-3908*x^13+48760*x^12-26130*x^11-655100*
x^10+602158*x^9+5877004*x^8-1864564*x^7-34936372*x^6-26824571*x^5+87777108*x^4+194192478*x^3+150929244*x^2+430
46721*x),x,method=_RETURNVERBOSE)

[Out]

exp(6/(ln(3/x)-x^8+16*x^7-60*x^6-174*x^5+970*x^4+1580*x^3-4717*x^2-11502*x-6561))

________________________________________________________________________________________

maxima [B]  time = 2.61, size = 859, normalized size = 26.84 $$\begin{array}{c}\frac{8x^8{e}^{(-\frac{6}{x^8-16x^7+60x^6+174x^5-970x^4-1580x^3+4717x^2+11502x-\log (3)+\log (x)+6561})}}{8x^8-112x^7+360x^6+870x^5-3880x^4-4740x^3+9434x^2+11502x+1}-\frac{112x^7{e}^{(-\frac{6}{x^8-16x^7+60x^6+174x^5-970x^4-1580x^3+4717x^2+11502x-\log (3)+\log (x)+6561})}}{8x^8-112x^7+360x^6+870x^5-3880x^4-4740x^3+9434x^2+11502x+1}+\frac{360x^6{e}^{(-\frac{6}{x^8-16x^7+60x^6+174x^5-970x^4-1580x^3+4717x^2+11502x-\log (3)+\log (x)+6561})}}{8x^8-112x^7+360x^6+870x^5-3880x^4-4740x^3+9434x^2+11502x+1}+\frac{870x^5{e}^{(-\frac{6}{x^8-16x^7+60x^6+174x^5-970x^4-1580x^3+4717x^2+11502x-\log (3)+\log (x)+6561})}}{8x^8-112x^7+360x^6+870x^5-3880x^4-4740x^3+9434x^2+11502x+1}-\frac{3880x^4{e}^{(-\frac{6}{x^8-16x^7+60x^6+174x^5-970x^4-1580x^3+4717x^2+11502x-\log (3)+\log (x)+6561})}}{8x^8-112x^7+360x^6+870x^5-3880x^4-4740x^3+9434x^2+11502x+1}-\frac{4740x^3{e}^{(-\frac{6}{x^8-16x^7+60x^6+174x^5-970x^4-1580x^3+4717x^2+11502x-\log (3)+\log (x)+6561})}}{8x^8-112x^7+360x^6+870x^5-3880x^4-4740x^3+9434x^2+11502x+1}+\frac{9434x^2{e}^{(-\frac{6}{x^8-16x^7+60x^6+174x^5-970x^4-1580x^3+4717x^2+11502x-\log (3)+\log (x)+6561})}}{8x^8-112x^7+360x^6+870x^5-3880x^4-4740x^3+9434x^2+11502x+1}+\frac{11502x{e}^{(-\frac{6}{x^8-16x^7+60x^6+174x^5-970x^4-1580x^3+4717x^2+11502x-\log (3)+\log (x)+6561})}}{8x^8-112x^7+360x^6+870x^5-3880x^4-4740x^3+9434x^2+11502x+1}+\frac{e^{(-\frac{6}{x^8-16x^7+60x^6+174x^5-970x^4-1580x^3+4717x^2+11502x-\log (3)+\log (x)+6561})}}{8x^8-112x^7+360x^6+870x^5-3880x^4-4740x^3+9434x^2+11502x+1}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((48*x^8-672*x^7+2160*x^6+5220*x^5-23280*x^4-28440*x^3+56604*x^2+69012*x+6)*exp(6/(log(3/x)-x^8+16*x^
7-60*x^6-174*x^5+970*x^4+1580*x^3-4717*x^2-11502*x-6561))/(x*log(3/x)^2+(-2*x^9+32*x^8-120*x^7-348*x^6+1940*x^
5+3160*x^4-9434*x^3-23004*x^2-13122*x)*log(3/x)+x^17-32*x^16+376*x^15-1572*x^14-3908*x^13+48760*x^12-26130*x^1
1-655100*x^10+602158*x^9+5877004*x^8-1864564*x^7-34936372*x^6-26824571*x^5+87777108*x^4+194192478*x^3+15092924
4*x^2+43046721*x),x, algorithm="maxima")

[Out]

8*x^8*e^(-6/(x^8 - 16*x^7 + 60*x^6 + 174*x^5 - 970*x^4 - 1580*x^3 + 4717*x^2 + 11502*x - log(3) + log(x) + 656
1))/(8*x^8 - 112*x^7 + 360*x^6 + 870*x^5 - 3880*x^4 - 4740*x^3 + 9434*x^2 + 11502*x + 1) - 112*x^7*e^(-6/(x^8
- 16*x^7 + 60*x^6 + 174*x^5 - 970*x^4 - 1580*x^3 + 4717*x^2 + 11502*x - log(3) + log(x) + 6561))/(8*x^8 - 112*
x^7 + 360*x^6 + 870*x^5 - 3880*x^4 - 4740*x^3 + 9434*x^2 + 11502*x + 1) + 360*x^6*e^(-6/(x^8 - 16*x^7 + 60*x^6
+ 174*x^5 - 970*x^4 - 1580*x^3 + 4717*x^2 + 11502*x - log(3) + log(x) + 6561))/(8*x^8 - 112*x^7 + 360*x^6 + 8
70*x^5 - 3880*x^4 - 4740*x^3 + 9434*x^2 + 11502*x + 1) + 870*x^5*e^(-6/(x^8 - 16*x^7 + 60*x^6 + 174*x^5 - 970*
x^4 - 1580*x^3 + 4717*x^2 + 11502*x - log(3) + log(x) + 6561))/(8*x^8 - 112*x^7 + 360*x^6 + 870*x^5 - 3880*x^4
- 4740*x^3 + 9434*x^2 + 11502*x + 1) - 3880*x^4*e^(-6/(x^8 - 16*x^7 + 60*x^6 + 174*x^5 - 970*x^4 - 1580*x^3 +
4717*x^2 + 11502*x - log(3) + log(x) + 6561))/(8*x^8 - 112*x^7 + 360*x^6 + 870*x^5 - 3880*x^4 - 4740*x^3 + 94
34*x^2 + 11502*x + 1) - 4740*x^3*e^(-6/(x^8 - 16*x^7 + 60*x^6 + 174*x^5 - 970*x^4 - 1580*x^3 + 4717*x^2 + 1150
2*x - log(3) + log(x) + 6561))/(8*x^8 - 112*x^7 + 360*x^6 + 870*x^5 - 3880*x^4 - 4740*x^3 + 9434*x^2 + 11502*x
+ 1) + 9434*x^2*e^(-6/(x^8 - 16*x^7 + 60*x^6 + 174*x^5 - 970*x^4 - 1580*x^3 + 4717*x^2 + 11502*x - log(3) + l
og(x) + 6561))/(8*x^8 - 112*x^7 + 360*x^6 + 870*x^5 - 3880*x^4 - 4740*x^3 + 9434*x^2 + 11502*x + 1) + 11502*x*
e^(-6/(x^8 - 16*x^7 + 60*x^6 + 174*x^5 - 970*x^4 - 1580*x^3 + 4717*x^2 + 11502*x - log(3) + log(x) + 6561))/(8
*x^8 - 112*x^7 + 360*x^6 + 870*x^5 - 3880*x^4 - 4740*x^3 + 9434*x^2 + 11502*x + 1) + e^(-6/(x^8 - 16*x^7 + 60*
x^6 + 174*x^5 - 970*x^4 - 1580*x^3 + 4717*x^2 + 11502*x - log(3) + log(x) + 6561))/(8*x^8 - 112*x^7 + 360*x^6
+ 870*x^5 - 3880*x^4 - 4740*x^3 + 9434*x^2 + 11502*x + 1)

________________________________________________________________________________________

mupad [B]  time = 1.02, size = 51, normalized size = 1.59 $$\begin{array}{c}{\mathrm{e}}^{-\frac{6}{11502x-\ln \left(\frac{3}{x}\right)+4717x^2-1580x^3-970x^4+174x^5+60x^6-16x^7+x^8+6561}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-6/(11502*x - log(3/x) + 4717*x^2 - 1580*x^3 - 970*x^4 + 174*x^5 + 60*x^6 - 16*x^7 + x^8 + 6561))*(69
012*x + 56604*x^2 - 28440*x^3 - 23280*x^4 + 5220*x^5 + 2160*x^6 - 672*x^7 + 48*x^8 + 6))/(43046721*x + 1509292
44*x^2 + 194192478*x^3 + 87777108*x^4 - 26824571*x^5 - 34936372*x^6 - 1864564*x^7 + 5877004*x^8 + 602158*x^9 -
655100*x^10 - 26130*x^11 + 48760*x^12 - 3908*x^13 - 1572*x^14 + 376*x^15 - 32*x^16 + x^17 + x*log(3/x)^2 - lo
g(3/x)*(13122*x + 23004*x^2 + 9434*x^3 - 3160*x^4 - 1940*x^5 + 348*x^6 + 120*x^7 - 32*x^8 + 2*x^9)),x)

[Out]

exp(-6/(11502*x - log(3/x) + 4717*x^2 - 1580*x^3 - 970*x^4 + 174*x^5 + 60*x^6 - 16*x^7 + x^8 + 6561))

________________________________________________________________________________________

sympy [B]  time = 2.11, size = 46, normalized size = 1.44 $$\begin{array}{c}{e}^{\frac{6}{-x^8+16x^7-60x^6-174x^5+970x^4+1580x^3-4717x^2-11502x+\log \left(\frac{3}{x}\right)-6561}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((48*x**8-672*x**7+2160*x**6+5220*x**5-23280*x**4-28440*x**3+56604*x**2+69012*x+6)*exp(6/(ln(3/x)-x**
8+16*x**7-60*x**6-174*x**5+970*x**4+1580*x**3-4717*x**2-11502*x-6561))/(x*ln(3/x)**2+(-2*x**9+32*x**8-120*x**7
-348*x**6+1940*x**5+3160*x**4-9434*x**3-23004*x**2-13122*x)*ln(3/x)+x**17-32*x**16+376*x**15-1572*x**14-3908*x
**13+48760*x**12-26130*x**11-655100*x**10+602158*x**9+5877004*x**8-1864564*x**7-34936372*x**6-26824571*x**5+87
777108*x**4+194192478*x**3+150929244*x**2+43046721*x),x)

[Out]

exp(6/(-x**8 + 16*x**7 - 60*x**6 - 174*x**5 + 970*x**4 + 1580*x**3 - 4717*x**2 - 11502*x + log(3/x) - 6561))

________________________________________________________________________________________