3.12.19 \(\int \frac {e (-1250+2000 x-1250 x^2+390 x^3-52 x^4)}{-5625 x+14625 x^2-14550 x^3+7110 x^4-1729 x^5+169 x^6+(3750 x-9750 x^2+9650 x^3-4650 x^4+1104 x^5-104 x^6) \log (\frac {1-2 x+x^2}{x^2})+(-625 x+1625 x^2-1600 x^3+760 x^4-176 x^5+16 x^6) \log ^2(\frac {1-2 x+x^2}{x^2})} \, dx\) [1119]

3.12.19.1 Optimal result

Integrand size = 141, antiderivative size = 32 \[ \int \frac {e \left (-1250+2000 x-1250 x^2+390 x^3-52 x^4\right )}{-5625 x+14625 x^2-14550 x^3+7110 x^4-1729 x^5+169 x^6+\left (3750 x-9750 x^2+9650 x^3-4650 x^4+1104 x^5-104 x^6\right ) \log \left (\frac {1-2 x+x^2}{x^2}\right )+\left (-625 x+1625 x^2-1600 x^3+760 x^4-176 x^5+16 x^6\right ) \log ^2\left (\frac {1-2 x+x^2}{x^2}\right )} \, dx=5+\frac {e}{-3-\frac {x^2}{(-5+2 x)^2}+\log \left (\frac {(1-x)^2}{x^2}\right )} \]

output

5+exp(1)/(ln((1-x)^2/x^2)-x^2/(-5+2*x)^2-3)
 

3.12.19.2 Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22 \[ \int \frac {e \left (-1250+2000 x-1250 x^2+390 x^3-52 x^4\right )}{-5625 x+14625 x^2-14550 x^3+7110 x^4-1729 x^5+169 x^6+\left (3750 x-9750 x^2+9650 x^3-4650 x^4+1104 x^5-104 x^6\right ) \log \left (\frac {1-2 x+x^2}{x^2}\right )+\left (-625 x+1625 x^2-1600 x^3+760 x^4-176 x^5+16 x^6\right ) \log ^2\left (\frac {1-2 x+x^2}{x^2}\right )} \, dx=\frac {e (5-2 x)^2}{-75+60 x-13 x^2+(5-2 x)^2 \log \left (\frac {(-1+x)^2}{x^2}\right )} \]

input

Integrate[(E*(-1250 + 2000*x - 1250*x^2 + 390*x^3 - 52*x^4))/(-5625*x + 14 
625*x^2 - 14550*x^3 + 7110*x^4 - 1729*x^5 + 169*x^6 + (3750*x - 9750*x^2 + 
 9650*x^3 - 4650*x^4 + 1104*x^5 - 104*x^6)*Log[(1 - 2*x + x^2)/x^2] + (-62 
5*x + 1625*x^2 - 1600*x^3 + 760*x^4 - 176*x^5 + 16*x^6)*Log[(1 - 2*x + x^2 
)/x^2]^2),x]
 

output

(E*(5 - 2*x)^2)/(-75 + 60*x - 13*x^2 + (5 - 2*x)^2*Log[(-1 + x)^2/x^2])
 

3.12.19.3 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.

\(\displaystyle \int \frac {e \left (-52 x^4+390 x^3-1250 x^2+2000 x-1250\right )}{169 x^6-1729 x^5+7110 x^4-14550 x^3+14625 x^2+\left (16 x^6-176 x^5+760 x^4-1600 x^3+1625 x^2-625 x\right ) \log ^2\left (\frac {x^2-2 x+1}{x^2}\right )+\left (-104 x^6+1104 x^5-4650 x^4+9650 x^3-9750 x^2+3750 x\right ) \log \left (\frac {x^2-2 x+1}{x^2}\right )-5625 x} \, dx\)

\(\Big \downarrow \) 27

\(\displaystyle e \int \frac {2 \left (26 x^4-195 x^3+625 x^2-1000 x+625\right )}{-169 x^6+1729 x^5-7110 x^4+14550 x^3-14625 x^2+5625 x+\left (-16 x^6+176 x^5-760 x^4+1600 x^3-1625 x^2+625 x\right ) \log ^2\left (\frac {x^2-2 x+1}{x^2}\right )-2 \left (-52 x^6+552 x^5-2325 x^4+4825 x^3-4875 x^2+1875 x\right ) \log \left (\frac {x^2-2 x+1}{x^2}\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e \int \frac {26 x^4-195 x^3+625 x^2-1000 x+625}{-169 x^6+1729 x^5-7110 x^4+14550 x^3-14625 x^2+5625 x+\left (-16 x^6+176 x^5-760 x^4+1600 x^3-1625 x^2+625 x\right ) \log ^2\left (\frac {x^2-2 x+1}{x^2}\right )-2 \left (-52 x^6+552 x^5-2325 x^4+4825 x^3-4875 x^2+1875 x\right ) \log \left (\frac {x^2-2 x+1}{x^2}\right )}dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 e \int \frac {26 x^4-195 x^3+625 x^2-1000 x+625}{(1-x) x \left (-\log \left (\frac {(x-1)^2}{x^2}\right ) (5-2 x)^2+13 x^2-60 x+75\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 e \int \left (-\frac {26 x^2}{\left (4 \log \left (\frac {(x-1)^2}{x^2}\right ) x^2-13 x^2-20 \log \left (\frac {(x-1)^2}{x^2}\right ) x+60 x+25 \log \left (\frac {(x-1)^2}{x^2}\right )-75\right )^2}+\frac {169 x}{\left (4 \log \left (\frac {(x-1)^2}{x^2}\right ) x^2-13 x^2-20 \log \left (\frac {(x-1)^2}{x^2}\right ) x+60 x+25 \log \left (\frac {(x-1)^2}{x^2}\right )-75\right )^2}-\frac {81}{(x-1) \left (4 \log \left (\frac {(x-1)^2}{x^2}\right ) x^2-13 x^2-20 \log \left (\frac {(x-1)^2}{x^2}\right ) x+60 x+25 \log \left (\frac {(x-1)^2}{x^2}\right )-75\right )^2}-\frac {456}{\left (4 \log \left (\frac {(x-1)^2}{x^2}\right ) x^2-13 x^2-20 \log \left (\frac {(x-1)^2}{x^2}\right ) x+60 x+25 \log \left (\frac {(x-1)^2}{x^2}\right )-75\right )^2}+\frac {625}{\left (4 \log \left (\frac {(x-1)^2}{x^2}\right ) x^2-13 x^2-20 \log \left (\frac {(x-1)^2}{x^2}\right ) x+60 x+25 \log \left (\frac {(x-1)^2}{x^2}\right )-75\right )^2 x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 e \left (-456 \int \frac {1}{\left (-\log \left (\frac {(x-1)^2}{x^2}\right ) (5-2 x)^2+13 x^2-60 x+75\right )^2}dx-81 \int \frac {1}{(x-1) \left (-\log \left (\frac {(x-1)^2}{x^2}\right ) (5-2 x)^2+13 x^2-60 x+75\right )^2}dx+625 \int \frac {1}{x \left (-\log \left (\frac {(x-1)^2}{x^2}\right ) (5-2 x)^2+13 x^2-60 x+75\right )^2}dx+169 \int \frac {x}{\left (-\log \left (\frac {(x-1)^2}{x^2}\right ) (5-2 x)^2+13 x^2-60 x+75\right )^2}dx-26 \int \frac {x^2}{\left (-\log \left (\frac {(x-1)^2}{x^2}\right ) (5-2 x)^2+13 x^2-60 x+75\right )^2}dx\right )\)

input

Int[(E*(-1250 + 2000*x - 1250*x^2 + 390*x^3 - 52*x^4))/(-5625*x + 14625*x^ 
2 - 14550*x^3 + 7110*x^4 - 1729*x^5 + 169*x^6 + (3750*x - 9750*x^2 + 9650* 
x^3 - 4650*x^4 + 1104*x^5 - 104*x^6)*Log[(1 - 2*x + x^2)/x^2] + (-625*x + 
1625*x^2 - 1600*x^3 + 760*x^4 - 176*x^5 + 16*x^6)*Log[(1 - 2*x + x^2)/x^2] 
^2),x]
 

output

$Aborted
 

3.12.19.3.1 Defintions of rubi rules used

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7239

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

rule 7293

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

3.12.19.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(71\) vs. \(2(33)=66\).

Time = 1.62 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.25

method	result	size
risch	\(\frac {{\mathrm e} \left (-5+2 x \right )^{2}}{4 \ln \left (\frac {x^{2}-2 x +1}{x^{2}}\right ) x^{2}-20 \ln \left (\frac {x^{2}-2 x +1}{x^{2}}\right ) x -13 x^{2}+25 \ln \left (\frac {x^{2}-2 x +1}{x^{2}}\right )+60 x -75}\)	\(72\)
parallelrisch	\(-\frac {{\mathrm e} \left (-32 x^{2}+160 x -200\right )}{8 \left (4 \ln \left (\frac {x^{2}-2 x +1}{x^{2}}\right ) x^{2}-20 \ln \left (\frac {x^{2}-2 x +1}{x^{2}}\right ) x -13 x^{2}+25 \ln \left (\frac {x^{2}-2 x +1}{x^{2}}\right )+60 x -75\right )}\)	\(76\)
norman	\(\frac {-20 x \,{\mathrm e}+4 x^{2} {\mathrm e}+25 \,{\mathrm e}}{4 \ln \left (\frac {x^{2}-2 x +1}{x^{2}}\right ) x^{2}-20 \ln \left (\frac {x^{2}-2 x +1}{x^{2}}\right ) x -13 x^{2}+25 \ln \left (\frac {x^{2}-2 x +1}{x^{2}}\right )+60 x -75}\)	\(80\)

input

int((-52*x^4+390*x^3-1250*x^2+2000*x-1250)*exp(1)/((16*x^6-176*x^5+760*x^4 
-1600*x^3+1625*x^2-625*x)*ln((x^2-2*x+1)/x^2)^2+(-104*x^6+1104*x^5-4650*x^ 
4+9650*x^3-9750*x^2+3750*x)*ln((x^2-2*x+1)/x^2)+169*x^6-1729*x^5+7110*x^4- 
14550*x^3+14625*x^2-5625*x),x,method=_RETURNVERBOSE)
 

output

exp(1)*(-5+2*x)^2/(4*ln((x^2-2*x+1)/x^2)*x^2-20*ln((x^2-2*x+1)/x^2)*x-13*x 
^2+25*ln((x^2-2*x+1)/x^2)+60*x-75)
 

3.12.19.5 Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.59 \[ \int \frac {e \left (-1250+2000 x-1250 x^2+390 x^3-52 x^4\right )}{-5625 x+14625 x^2-14550 x^3+7110 x^4-1729 x^5+169 x^6+\left (3750 x-9750 x^2+9650 x^3-4650 x^4+1104 x^5-104 x^6\right ) \log \left (\frac {1-2 x+x^2}{x^2}\right )+\left (-625 x+1625 x^2-1600 x^3+760 x^4-176 x^5+16 x^6\right ) \log ^2\left (\frac {1-2 x+x^2}{x^2}\right )} \, dx=-\frac {{\left (4 \, x^{2} - 20 \, x + 25\right )} e}{13 \, x^{2} - {\left (4 \, x^{2} - 20 \, x + 25\right )} \log \left (\frac {x^{2} - 2 \, x + 1}{x^{2}}\right ) - 60 \, x + 75} \]

input

integrate((-52*x^4+390*x^3-1250*x^2+2000*x-1250)*exp(1)/((16*x^6-176*x^5+7 
60*x^4-1600*x^3+1625*x^2-625*x)*log((x^2-2*x+1)/x^2)^2+(-104*x^6+1104*x^5- 
4650*x^4+9650*x^3-9750*x^2+3750*x)*log((x^2-2*x+1)/x^2)+169*x^6-1729*x^5+7 
110*x^4-14550*x^3+14625*x^2-5625*x),x, algorithm=\
 

output

-(4*x^2 - 20*x + 25)*e/(13*x^2 - (4*x^2 - 20*x + 25)*log((x^2 - 2*x + 1)/x 
^2) - 60*x + 75)
 

3.12.19.6 Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (26) = 52\).

Time = 0.17 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.66 \[ \int \frac {e \left (-1250+2000 x-1250 x^2+390 x^3-52 x^4\right )}{-5625 x+14625 x^2-14550 x^3+7110 x^4-1729 x^5+169 x^6+\left (3750 x-9750 x^2+9650 x^3-4650 x^4+1104 x^5-104 x^6\right ) \log \left (\frac {1-2 x+x^2}{x^2}\right )+\left (-625 x+1625 x^2-1600 x^3+760 x^4-176 x^5+16 x^6\right ) \log ^2\left (\frac {1-2 x+x^2}{x^2}\right )} \, dx=\frac {4 e x^{2} - 20 e x + 25 e}{- 13 x^{2} + 60 x + \left (4 x^{2} - 20 x + 25\right ) \log {\left (\frac {x^{2} - 2 x + 1}{x^{2}} \right )} - 75} \]

input

integrate((-52*x**4+390*x**3-1250*x**2+2000*x-1250)*exp(1)/((16*x**6-176*x 
**5+760*x**4-1600*x**3+1625*x**2-625*x)*ln((x**2-2*x+1)/x**2)**2+(-104*x** 
6+1104*x**5-4650*x**4+9650*x**3-9750*x**2+3750*x)*ln((x**2-2*x+1)/x**2)+16 
9*x**6-1729*x**5+7110*x**4-14550*x**3+14625*x**2-5625*x),x)
 

output

(4*E*x**2 - 20*E*x + 25*E)/(-13*x**2 + 60*x + (4*x**2 - 20*x + 25)*log((x* 
*2 - 2*x + 1)/x**2) - 75)
 

3.12.19.7 Maxima [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.75 \[ \int \frac {e \left (-1250+2000 x-1250 x^2+390 x^3-52 x^4\right )}{-5625 x+14625 x^2-14550 x^3+7110 x^4-1729 x^5+169 x^6+\left (3750 x-9750 x^2+9650 x^3-4650 x^4+1104 x^5-104 x^6\right ) \log \left (\frac {1-2 x+x^2}{x^2}\right )+\left (-625 x+1625 x^2-1600 x^3+760 x^4-176 x^5+16 x^6\right ) \log ^2\left (\frac {1-2 x+x^2}{x^2}\right )} \, dx=-\frac {{\left (4 \, x^{2} - 20 \, x + 25\right )} e}{13 \, x^{2} - 2 \, {\left (4 \, x^{2} - 20 \, x + 25\right )} \log \left (x - 1\right ) + 2 \, {\left (4 \, x^{2} - 20 \, x + 25\right )} \log \left (x\right ) - 60 \, x + 75} \]

input

integrate((-52*x^4+390*x^3-1250*x^2+2000*x-1250)*exp(1)/((16*x^6-176*x^5+7 
60*x^4-1600*x^3+1625*x^2-625*x)*log((x^2-2*x+1)/x^2)^2+(-104*x^6+1104*x^5- 
4650*x^4+9650*x^3-9750*x^2+3750*x)*log((x^2-2*x+1)/x^2)+169*x^6-1729*x^5+7 
110*x^4-14550*x^3+14625*x^2-5625*x),x, algorithm=\
 

output

-(4*x^2 - 20*x + 25)*e/(13*x^2 - 2*(4*x^2 - 20*x + 25)*log(x - 1) + 2*(4*x 
^2 - 20*x + 25)*log(x) - 60*x + 75)
 

3.12.19.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (33) = 66\).

Time = 0.57 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.31 \[ \int \frac {e \left (-1250+2000 x-1250 x^2+390 x^3-52 x^4\right )}{-5625 x+14625 x^2-14550 x^3+7110 x^4-1729 x^5+169 x^6+\left (3750 x-9750 x^2+9650 x^3-4650 x^4+1104 x^5-104 x^6\right ) \log \left (\frac {1-2 x+x^2}{x^2}\right )+\left (-625 x+1625 x^2-1600 x^3+760 x^4-176 x^5+16 x^6\right ) \log ^2\left (\frac {1-2 x+x^2}{x^2}\right )} \, dx=\frac {{\left (4 \, x^{2} - 20 \, x + 25\right )} e}{4 \, x^{2} \log \left (\frac {x^{2} - 2 \, x + 1}{x^{2}}\right ) - 13 \, x^{2} - 20 \, x \log \left (\frac {x^{2} - 2 \, x + 1}{x^{2}}\right ) + 60 \, x + 25 \, \log \left (\frac {x^{2} - 2 \, x + 1}{x^{2}}\right ) - 75} \]

input

integrate((-52*x^4+390*x^3-1250*x^2+2000*x-1250)*exp(1)/((16*x^6-176*x^5+7 
60*x^4-1600*x^3+1625*x^2-625*x)*log((x^2-2*x+1)/x^2)^2+(-104*x^6+1104*x^5- 
4650*x^4+9650*x^3-9750*x^2+3750*x)*log((x^2-2*x+1)/x^2)+169*x^6-1729*x^5+7 
110*x^4-14550*x^3+14625*x^2-5625*x),x, algorithm=\
 

output

(4*x^2 - 20*x + 25)*e/(4*x^2*log((x^2 - 2*x + 1)/x^2) - 13*x^2 - 20*x*log( 
(x^2 - 2*x + 1)/x^2) + 60*x + 25*log((x^2 - 2*x + 1)/x^2) - 75)
 

3.12.19.9 Mupad [F(-1)]

Timed out. \[ \int \frac {e \left (-1250+2000 x-1250 x^2+390 x^3-52 x^4\right )}{-5625 x+14625 x^2-14550 x^3+7110 x^4-1729 x^5+169 x^6+\left (3750 x-9750 x^2+9650 x^3-4650 x^4+1104 x^5-104 x^6\right ) \log \left (\frac {1-2 x+x^2}{x^2}\right )+\left (-625 x+1625 x^2-1600 x^3+760 x^4-176 x^5+16 x^6\right ) \log ^2\left (\frac {1-2 x+x^2}{x^2}\right )} \, dx=\int \frac {\mathrm {e}\,\left (52\,x^4-390\,x^3+1250\,x^2-2000\,x+1250\right )}{5625\,x-\ln \left (\frac {x^2-2\,x+1}{x^2}\right )\,\left (-104\,x^6+1104\,x^5-4650\,x^4+9650\,x^3-9750\,x^2+3750\,x\right )+{\ln \left (\frac {x^2-2\,x+1}{x^2}\right )}^2\,\left (-16\,x^6+176\,x^5-760\,x^4+1600\,x^3-1625\,x^2+625\,x\right )-14625\,x^2+14550\,x^3-7110\,x^4+1729\,x^5-169\,x^6} \,d x \]

input

int((exp(1)*(1250*x^2 - 2000*x - 390*x^3 + 52*x^4 + 1250))/(5625*x - log(( 
x^2 - 2*x + 1)/x^2)*(3750*x - 9750*x^2 + 9650*x^3 - 4650*x^4 + 1104*x^5 - 
104*x^6) + log((x^2 - 2*x + 1)/x^2)^2*(625*x - 1625*x^2 + 1600*x^3 - 760*x 
^4 + 176*x^5 - 16*x^6) - 14625*x^2 + 14550*x^3 - 7110*x^4 + 1729*x^5 - 169 
*x^6),x)
 

output

int((exp(1)*(1250*x^2 - 2000*x - 390*x^3 + 52*x^4 + 1250))/(5625*x - log(( 
x^2 - 2*x + 1)/x^2)*(3750*x - 9750*x^2 + 9650*x^3 - 4650*x^4 + 1104*x^5 - 
104*x^6) + log((x^2 - 2*x + 1)/x^2)^2*(625*x - 1625*x^2 + 1600*x^3 - 760*x 
^4 + 176*x^5 - 16*x^6) - 14625*x^2 + 14550*x^3 - 7110*x^4 + 1729*x^5 - 169 
*x^6), x)