3.6.51 \(\int \frac {e^{\frac {6}{-6561-11502 x-4717 x^2+1580 x^3+970 x^4-174 x^5-60 x^6+16 x^7-x^8+\log (\frac {3}{x})}} (6+69012 x+56604 x^2-28440 x^3-23280 x^4+5220 x^5+2160 x^6-672 x^7+48 x^8)}{43046721 x+150929244 x^2+194192478 x^3+87777108 x^4-26824571 x^5-34936372 x^6-1864564 x^7+5877004 x^8+602158 x^9-655100 x^{10}-26130 x^{11}+48760 x^{12}-3908 x^{13}-1572 x^{14}+376 x^{15}-32 x^{16}+x^{17}+(-13122 x-23004 x^2-9434 x^3+3160 x^4+1940 x^5-348 x^6-120 x^7+32 x^8-2 x^9) \log (\frac {3}{x})+x \log ^2(\frac {3}{x})} \, dx\)
Optimal. Leaf size=32 \[ e^{\frac {6}{-\left (-x+(9+(4-x) x)^2\right )^2+\log \left (\frac {3}{x}\right )}} \]
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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 {gather*} \exp \left (-\frac {6}{\left (x^4-8 x^3-2 x^2+71 x+81\right )^2-\log \left (\frac {3}{x}\right )}\right ) \end {gather*}
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 {gather*} \begin {aligned} \text {integral} &=\int \frac {6 \exp \left (-\frac {6}{\left (81+71 x-2 x^2-8 x^3+x^4\right )^2-\log \left (\frac {3}{x}\right )}\right ) \left (1+11502 x+9434 x^2-4740 x^3-3880 x^4+870 x^5+360 x^6-112 x^7+8 x^8\right )}{x \left (\left (81+71 x-2 x^2-8 x^3+x^4\right )^2-\log \left (\frac {3}{x}\right )\right )^2} \, dx\\ &=6 \int \frac {\exp \left (-\frac {6}{\left (81+71 x-2 x^2-8 x^3+x^4\right )^2-\log \left (\frac {3}{x}\right )}\right ) \left (1+11502 x+9434 x^2-4740 x^3-3880 x^4+870 x^5+360 x^6-112 x^7+8 x^8\right )}{x \left (\left (81+71 x-2 x^2-8 x^3+x^4\right )^2-\log \left (\frac {3}{x}\right )\right )^2} \, dx\\ &=\exp \left (-\frac {6}{\left (81+71 x-2 x^2-8 x^3+x^4\right )^2-\log \left (\frac {3}{x}\right )}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.07, size = 35, normalized size = 1.09 \begin {gather*} e^{-\frac {6}{\left (81+71 x-2 x^2-8 x^3+x^4\right )^2-\log \left (\frac {3}{x}\right )}} \end {gather*}
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]))
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fricas [A] time = 0.77, size = 51, normalized size = 1.59 \begin {gather*} e^{\left (-\frac {6}{x^{8} - 16 \, x^{7} + 60 \, x^{6} + 174 \, x^{5} - 970 \, x^{4} - 1580 \, x^{3} + 4717 \, x^{2} + 11502 \, x - \log \left (\frac {3}{x}\right ) + 6561}\right )} \end {gather*}
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))
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giac [A] time = 1.10, size = 51, normalized size = 1.59 \begin {gather*} e^{\left (-\frac {6}{x^{8} - 16 \, x^{7} + 60 \, x^{6} + 174 \, x^{5} - 970 \, x^{4} - 1580 \, x^{3} + 4717 \, x^{2} + 11502 \, x - \log \left (\frac {3}{x}\right ) + 6561}\right )} \end {gather*}
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))
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maple [A] time = 0.15, size = 52, normalized size = 1.62
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\({\mathrm e}^{\frac {6}{\ln \left (\frac {3}{x}\right )-x^{8}+16 x^{7}-60 x^{6}-174 x^{5}+970 x^{4}+1580 x^{3}-4717 x^{2}-11502 x -6561}}\) |
\(52\) |
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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))
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maxima [B] time = 2.61, size = 859, normalized size = 26.84 \begin {gather*} \frac {8 \, x^{8} e^{\left (-\frac {6}{x^{8} - 16 \, x^{7} + 60 \, x^{6} + 174 \, x^{5} - 970 \, x^{4} - 1580 \, x^{3} + 4717 \, x^{2} + 11502 \, x - \log \relax (3) + \log \relax (x) + 6561}\right )}}{8 \, x^{8} - 112 \, x^{7} + 360 \, x^{6} + 870 \, x^{5} - 3880 \, x^{4} - 4740 \, x^{3} + 9434 \, x^{2} + 11502 \, x + 1} - \frac {112 \, x^{7} e^{\left (-\frac {6}{x^{8} - 16 \, x^{7} + 60 \, x^{6} + 174 \, x^{5} - 970 \, x^{4} - 1580 \, x^{3} + 4717 \, x^{2} + 11502 \, x - \log \relax (3) + \log \relax (x) + 6561}\right )}}{8 \, x^{8} - 112 \, x^{7} + 360 \, x^{6} + 870 \, x^{5} - 3880 \, x^{4} - 4740 \, x^{3} + 9434 \, x^{2} + 11502 \, x + 1} + \frac {360 \, x^{6} e^{\left (-\frac {6}{x^{8} - 16 \, x^{7} + 60 \, x^{6} + 174 \, x^{5} - 970 \, x^{4} - 1580 \, x^{3} + 4717 \, x^{2} + 11502 \, x - \log \relax (3) + \log \relax (x) + 6561}\right )}}{8 \, x^{8} - 112 \, x^{7} + 360 \, x^{6} + 870 \, x^{5} - 3880 \, x^{4} - 4740 \, x^{3} + 9434 \, x^{2} + 11502 \, x + 1} + \frac {870 \, x^{5} e^{\left (-\frac {6}{x^{8} - 16 \, x^{7} + 60 \, x^{6} + 174 \, x^{5} - 970 \, x^{4} - 1580 \, x^{3} + 4717 \, x^{2} + 11502 \, x - \log \relax (3) + \log \relax (x) + 6561}\right )}}{8 \, x^{8} - 112 \, x^{7} + 360 \, x^{6} + 870 \, x^{5} - 3880 \, x^{4} - 4740 \, x^{3} + 9434 \, x^{2} + 11502 \, x + 1} - \frac {3880 \, x^{4} e^{\left (-\frac {6}{x^{8} - 16 \, x^{7} + 60 \, x^{6} + 174 \, x^{5} - 970 \, x^{4} - 1580 \, x^{3} + 4717 \, x^{2} + 11502 \, x - \log \relax (3) + \log \relax (x) + 6561}\right )}}{8 \, x^{8} - 112 \, x^{7} + 360 \, x^{6} + 870 \, x^{5} - 3880 \, x^{4} - 4740 \, x^{3} + 9434 \, x^{2} + 11502 \, x + 1} - \frac {4740 \, x^{3} e^{\left (-\frac {6}{x^{8} - 16 \, x^{7} + 60 \, x^{6} + 174 \, x^{5} - 970 \, x^{4} - 1580 \, x^{3} + 4717 \, x^{2} + 11502 \, x - \log \relax (3) + \log \relax (x) + 6561}\right )}}{8 \, x^{8} - 112 \, x^{7} + 360 \, x^{6} + 870 \, x^{5} - 3880 \, x^{4} - 4740 \, x^{3} + 9434 \, x^{2} + 11502 \, x + 1} + \frac {9434 \, x^{2} e^{\left (-\frac {6}{x^{8} - 16 \, x^{7} + 60 \, x^{6} + 174 \, x^{5} - 970 \, x^{4} - 1580 \, x^{3} + 4717 \, x^{2} + 11502 \, x - \log \relax (3) + \log \relax (x) + 6561}\right )}}{8 \, x^{8} - 112 \, x^{7} + 360 \, x^{6} + 870 \, x^{5} - 3880 \, x^{4} - 4740 \, x^{3} + 9434 \, x^{2} + 11502 \, x + 1} + \frac {11502 \, x e^{\left (-\frac {6}{x^{8} - 16 \, x^{7} + 60 \, x^{6} + 174 \, x^{5} - 970 \, x^{4} - 1580 \, x^{3} + 4717 \, x^{2} + 11502 \, x - \log \relax (3) + \log \relax (x) + 6561}\right )}}{8 \, x^{8} - 112 \, x^{7} + 360 \, x^{6} + 870 \, x^{5} - 3880 \, x^{4} - 4740 \, x^{3} + 9434 \, x^{2} + 11502 \, x + 1} + \frac {e^{\left (-\frac {6}{x^{8} - 16 \, x^{7} + 60 \, x^{6} + 174 \, x^{5} - 970 \, x^{4} - 1580 \, x^{3} + 4717 \, x^{2} + 11502 \, x - \log \relax (3) + \log \relax (x) + 6561}\right )}}{8 \, x^{8} - 112 \, x^{7} + 360 \, x^{6} + 870 \, x^{5} - 3880 \, x^{4} - 4740 \, x^{3} + 9434 \, x^{2} + 11502 \, x + 1} \end {gather*}
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)
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mupad [B] time = 1.02, size = 51, normalized size = 1.59 \begin {gather*} {\mathrm {e}}^{-\frac {6}{11502\,x-\ln \left (\frac {3}{x}\right )+4717\,x^2-1580\,x^3-970\,x^4+174\,x^5+60\,x^6-16\,x^7+x^8+6561}} \end {gather*}
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))
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sympy [B] time = 2.11, size = 46, normalized size = 1.44 \begin {gather*} e^{\frac {6}{- x^{8} + 16 x^{7} - 60 x^{6} - 174 x^{5} + 970 x^{4} + 1580 x^{3} - 4717 x^{2} - 11502 x + \log {\left (\frac {3}{x} \right )} - 6561}} \end {gather*}
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))
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