3.7.97 \(\int -\frac {100 e^{\frac {-20711+6912 x-864 x^2+48 x^3-x^4}{20736-6912 x+864 x^2-48 x^3+x^4}}}{-248832+103680 x-17280 x^2+1440 x^3-60 x^4+x^5} \, dx\) [697]

3.7.97.1 Optimal result

Integrand size = 70, antiderivative size = 11 \[ \int -\frac {100 e^{\frac {-20711+6912 x-864 x^2+48 x^3-x^4}{20736-6912 x+864 x^2-48 x^3+x^4}}}{-248832+103680 x-17280 x^2+1440 x^3-60 x^4+x^5} \, dx=e^{-1+\frac {25}{(-12+x)^4}} \]

output

exp(25/(x-12)^4-1)
 

3.7.97.2 Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int -\frac {100 e^{\frac {-20711+6912 x-864 x^2+48 x^3-x^4}{20736-6912 x+864 x^2-48 x^3+x^4}}}{-248832+103680 x-17280 x^2+1440 x^3-60 x^4+x^5} \, dx=e^{-1+\frac {25}{(-12+x)^4}} \]

input

Integrate[(-100*E^((-20711 + 6912*x - 864*x^2 + 48*x^3 - x^4)/(20736 - 691 
2*x + 864*x^2 - 48*x^3 + x^4)))/(-248832 + 103680*x - 17280*x^2 + 1440*x^3 
 - 60*x^4 + x^5),x]
 

output

E^(-1 + 25/(-12 + x)^4)
 

3.7.97.3 Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int -\frac {100 \exp \left (\frac {-x^4+48 x^3-864 x^2+6912 x-20711}{x^4-48 x^3+864 x^2-6912 x+20736}\right )}{x^5-60 x^4+1440 x^3-17280 x^2+103680 x-248832} \, dx\)

\(\Big \downarrow \) 27

\(\displaystyle -100 \int -\frac {\exp \left (-\frac {x^4-48 x^3+864 x^2-6912 x+20711}{x^4-48 x^3+864 x^2-6912 x+20736}\right )}{-x^5+60 x^4-1440 x^3+17280 x^2-103680 x+248832}dx\)

\(\Big \downarrow \) 25

\(\displaystyle 100 \int \frac {\exp \left (-\frac {x^4-48 x^3+864 x^2-6912 x+20711}{x^4-48 x^3+864 x^2-6912 x+20736}\right )}{-x^5+60 x^4-1440 x^3+17280 x^2-103680 x+248832}dx\)

\(\Big \downarrow \) 2007

\(\displaystyle 100 \int \frac {\exp \left (-\frac {x^4-48 x^3+864 x^2-6912 x+20711}{x^4-48 x^3+864 x^2-6912 x+20736}\right )}{(12-x)^5}dx\)

\(\Big \downarrow \) 7299

\(\displaystyle 100 \int \frac {\exp \left (-\frac {x^4-48 x^3+864 x^2-6912 x+20711}{x^4-48 x^3+864 x^2-6912 x+20736}\right )}{(12-x)^5}dx\)

input

Int[(-100*E^((-20711 + 6912*x - 864*x^2 + 48*x^3 - x^4)/(20736 - 6912*x + 
864*x^2 - 48*x^3 + x^4)))/(-248832 + 103680*x - 17280*x^2 + 1440*x^3 - 60* 
x^4 + x^5),x]
 

output

$Aborted
 

3.7.97.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2007

Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, 
x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex 
pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol 
yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
 

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

3.7.97.4 Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\({\mathrm e}^{\frac {25}{\left (x -12\right )^{4}}} {\mathrm e}^{-1}\)	\(12\)
default	\({\mathrm e}^{\frac {25}{\left (x -12\right )^{4}}} {\mathrm e}^{-1}\)	\(12\)
risch	\({\mathrm e}^{-\frac {\left (x^{2}-24 x +149\right ) \left (x^{2}-24 x +139\right )}{\left (x -12\right )^{4}}}\)	\(25\)
gosper	\({\mathrm e}^{-\frac {x^{4}-48 x^{3}+864 x^{2}-6912 x +20711}{x^{4}-48 x^{3}+864 x^{2}-6912 x +20736}}\)	\(42\)
parallelrisch	\({\mathrm e}^{-\frac {x^{4}-48 x^{3}+864 x^{2}-6912 x +20711}{x^{4}-48 x^{3}+864 x^{2}-6912 x +20736}}\)	\(42\)
norman	\(\frac {x^{4} {\mathrm e}^{\frac {-x^{4}+48 x^{3}-864 x^{2}+6912 x -20711}{x^{4}-48 x^{3}+864 x^{2}-6912 x +20736}}-6912 x \,{\mathrm e}^{\frac {-x^{4}+48 x^{3}-864 x^{2}+6912 x -20711}{x^{4}-48 x^{3}+864 x^{2}-6912 x +20736}}+864 x^{2} {\mathrm e}^{\frac {-x^{4}+48 x^{3}-864 x^{2}+6912 x -20711}{x^{4}-48 x^{3}+864 x^{2}-6912 x +20736}}-48 x^{3} {\mathrm e}^{\frac {-x^{4}+48 x^{3}-864 x^{2}+6912 x -20711}{x^{4}-48 x^{3}+864 x^{2}-6912 x +20736}}+20736 \,{\mathrm e}^{\frac {-x^{4}+48 x^{3}-864 x^{2}+6912 x -20711}{x^{4}-48 x^{3}+864 x^{2}-6912 x +20736}}}{\left (x -12\right )^{4}}\)	\(237\)

input

int(-100*exp((-x^4+48*x^3-864*x^2+6912*x-20711)/(x^4-48*x^3+864*x^2-6912*x 
+20736))/(x^5-60*x^4+1440*x^3-17280*x^2+103680*x-248832),x,method=_RETURNV 
ERBOSE)
 

output

exp(1/(x-12)^4)^25*exp(-1)
 

3.7.97.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (10) = 20\).

Time = 0.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 3.73 \[ \int -\frac {100 e^{\frac {-20711+6912 x-864 x^2+48 x^3-x^4}{20736-6912 x+864 x^2-48 x^3+x^4}}}{-248832+103680 x-17280 x^2+1440 x^3-60 x^4+x^5} \, dx=e^{\left (-\frac {x^{4} - 48 \, x^{3} + 864 \, x^{2} - 6912 \, x + 20711}{x^{4} - 48 \, x^{3} + 864 \, x^{2} - 6912 \, x + 20736}\right )} \]

input

integrate(-100*exp((-x^4+48*x^3-864*x^2+6912*x-20711)/(x^4-48*x^3+864*x^2- 
6912*x+20736))/(x^5-60*x^4+1440*x^3-17280*x^2+103680*x-248832),x, algorith 
m=\
 

output

e^(-(x^4 - 48*x^3 + 864*x^2 - 6912*x + 20711)/(x^4 - 48*x^3 + 864*x^2 - 69 
12*x + 20736))
 

3.7.97.6 Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (8) = 16\).

Time = 0.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 3.36 \[ \int -\frac {100 e^{\frac {-20711+6912 x-864 x^2+48 x^3-x^4}{20736-6912 x+864 x^2-48 x^3+x^4}}}{-248832+103680 x-17280 x^2+1440 x^3-60 x^4+x^5} \, dx=e^{\frac {- x^{4} + 48 x^{3} - 864 x^{2} + 6912 x - 20711}{x^{4} - 48 x^{3} + 864 x^{2} - 6912 x + 20736}} \]

input

integrate(-100*exp((-x**4+48*x**3-864*x**2+6912*x-20711)/(x**4-48*x**3+864 
*x**2-6912*x+20736))/(x**5-60*x**4+1440*x**3-17280*x**2+103680*x-248832),x 
)
 

output

exp((-x**4 + 48*x**3 - 864*x**2 + 6912*x - 20711)/(x**4 - 48*x**3 + 864*x* 
*2 - 6912*x + 20736))
 

3.7.97.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (10) = 20\).

Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 2.27 \[ \int -\frac {100 e^{\frac {-20711+6912 x-864 x^2+48 x^3-x^4}{20736-6912 x+864 x^2-48 x^3+x^4}}}{-248832+103680 x-17280 x^2+1440 x^3-60 x^4+x^5} \, dx=e^{\left (\frac {25}{x^{4} - 48 \, x^{3} + 864 \, x^{2} - 6912 \, x + 20736} - 1\right )} \]

input

integrate(-100*exp((-x^4+48*x^3-864*x^2+6912*x-20711)/(x^4-48*x^3+864*x^2- 
6912*x+20736))/(x^5-60*x^4+1440*x^3-17280*x^2+103680*x-248832),x, algorith 
m=\
 

output

e^(25/(x^4 - 48*x^3 + 864*x^2 - 6912*x + 20736) - 1)
 

3.7.97.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (10) = 20\).

Time = 0.29 (sec) , antiderivative size = 122, normalized size of antiderivative = 11.09 \[ \int -\frac {100 e^{\frac {-20711+6912 x-864 x^2+48 x^3-x^4}{20736-6912 x+864 x^2-48 x^3+x^4}}}{-248832+103680 x-17280 x^2+1440 x^3-60 x^4+x^5} \, dx=e^{\left (-\frac {x^{4}}{x^{4} - 48 \, x^{3} + 864 \, x^{2} - 6912 \, x + 20736} + \frac {48 \, x^{3}}{x^{4} - 48 \, x^{3} + 864 \, x^{2} - 6912 \, x + 20736} - \frac {864 \, x^{2}}{x^{4} - 48 \, x^{3} + 864 \, x^{2} - 6912 \, x + 20736} + \frac {6912 \, x}{x^{4} - 48 \, x^{3} + 864 \, x^{2} - 6912 \, x + 20736} - \frac {20711}{x^{4} - 48 \, x^{3} + 864 \, x^{2} - 6912 \, x + 20736}\right )} \]

input

integrate(-100*exp((-x^4+48*x^3-864*x^2+6912*x-20711)/(x^4-48*x^3+864*x^2- 
6912*x+20736))/(x^5-60*x^4+1440*x^3-17280*x^2+103680*x-248832),x, algorith 
m=\
 

output

e^(-x^4/(x^4 - 48*x^3 + 864*x^2 - 6912*x + 20736) + 48*x^3/(x^4 - 48*x^3 + 
 864*x^2 - 6912*x + 20736) - 864*x^2/(x^4 - 48*x^3 + 864*x^2 - 6912*x + 20 
736) + 6912*x/(x^4 - 48*x^3 + 864*x^2 - 6912*x + 20736) - 20711/(x^4 - 48* 
x^3 + 864*x^2 - 6912*x + 20736))
 

3.7.97.9 Mupad [B] (verification not implemented)

Time = 11.70 (sec) , antiderivative size = 126, normalized size of antiderivative = 11.45 \[ \int -\frac {100 e^{\frac {-20711+6912 x-864 x^2+48 x^3-x^4}{20736-6912 x+864 x^2-48 x^3+x^4}}}{-248832+103680 x-17280 x^2+1440 x^3-60 x^4+x^5} \, dx={\mathrm {e}}^{\frac {6912\,x}{x^4-48\,x^3+864\,x^2-6912\,x+20736}}\,{\mathrm {e}}^{-\frac {x^4}{x^4-48\,x^3+864\,x^2-6912\,x+20736}}\,{\mathrm {e}}^{\frac {48\,x^3}{x^4-48\,x^3+864\,x^2-6912\,x+20736}}\,{\mathrm {e}}^{-\frac {864\,x^2}{x^4-48\,x^3+864\,x^2-6912\,x+20736}}\,{\mathrm {e}}^{-\frac {20711}{x^4-48\,x^3+864\,x^2-6912\,x+20736}} \]

input

int(-(100*exp(-(864*x^2 - 6912*x - 48*x^3 + x^4 + 20711)/(864*x^2 - 6912*x 
 - 48*x^3 + x^4 + 20736)))/(103680*x - 17280*x^2 + 1440*x^3 - 60*x^4 + x^5 
 - 248832),x)
 

output

exp((6912*x)/(864*x^2 - 6912*x - 48*x^3 + x^4 + 20736))*exp(-x^4/(864*x^2 
- 6912*x - 48*x^3 + x^4 + 20736))*exp((48*x^3)/(864*x^2 - 6912*x - 48*x^3 
+ x^4 + 20736))*exp(-(864*x^2)/(864*x^2 - 6912*x - 48*x^3 + x^4 + 20736))* 
exp(-20711/(864*x^2 - 6912*x - 48*x^3 + x^4 + 20736))