\(\int \frac {-90 x^2+36 x^3-50 x^5+20 x^6-2 x^7+e^{-5+x} (90 x+72 x^2-36 x^3+50 x^5-20 x^6+2 x^7)+(90-100 x^3+40 x^4-4 x^5+e^{-5+x} (-90+100 x^3-40 x^4+4 x^5)) \log (-1+e^{-5+x})+(-50 x+20 x^2-2 x^3+e^{-5+x} (50 x-20 x^2+2 x^3)) \log ^2(-1+e^{-5+x})}{-25 x^4+10 x^5-x^6+e^{-5+x} (25 x^4-10 x^5+x^6)+(-50 x^2+20 x^3-2 x^4+e^{-5+x} (50 x^2-20 x^3+2 x^4)) \log (-1+e^{-5+x})+(-25+10 x-x^2+e^{-5+x} (25-10 x+x^2)) \log ^2(-1+e^{-5+x})} \, dx\) [745]

Optimal result

Integrand size = 276, antiderivative size = 25 \[ \int \frac {-90 x^2+36 x^3-50 x^5+20 x^6-2 x^7+e^{-5+x} \left (90 x+72 x^2-36 x^3+50 x^5-20 x^6+2 x^7\right )+\left (90-100 x^3+40 x^4-4 x^5+e^{-5+x} \left (-90+100 x^3-40 x^4+4 x^5\right )\right ) \log \left (-1+e^{-5+x}\right )+\left (-50 x+20 x^2-2 x^3+e^{-5+x} \left (50 x-20 x^2+2 x^3\right )\right ) \log ^2\left (-1+e^{-5+x}\right )}{-25 x^4+10 x^5-x^6+e^{-5+x} \left (25 x^4-10 x^5+x^6\right )+\left (-50 x^2+20 x^3-2 x^4+e^{-5+x} \left (50 x^2-20 x^3+2 x^4\right )\right ) \log \left (-1+e^{-5+x}\right )+\left (-25+10 x-x^2+e^{-5+x} \left (25-10 x+x^2\right )\right ) \log ^2\left (-1+e^{-5+x}\right )} \, dx=x \left (x+\frac {18}{(-5+x) \left (x^2+\log \left (-1+e^{-5+x}\right )\right )}\right ) \] Output:

(18/(-5+x)/(ln(exp(-5+x)-1)+x^2)+x)*x
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {-90 x^2+36 x^3-50 x^5+20 x^6-2 x^7+e^{-5+x} \left (90 x+72 x^2-36 x^3+50 x^5-20 x^6+2 x^7\right )+\left (90-100 x^3+40 x^4-4 x^5+e^{-5+x} \left (-90+100 x^3-40 x^4+4 x^5\right )\right ) \log \left (-1+e^{-5+x}\right )+\left (-50 x+20 x^2-2 x^3+e^{-5+x} \left (50 x-20 x^2+2 x^3\right )\right ) \log ^2\left (-1+e^{-5+x}\right )}{-25 x^4+10 x^5-x^6+e^{-5+x} \left (25 x^4-10 x^5+x^6\right )+\left (-50 x^2+20 x^3-2 x^4+e^{-5+x} \left (50 x^2-20 x^3+2 x^4\right )\right ) \log \left (-1+e^{-5+x}\right )+\left (-25+10 x-x^2+e^{-5+x} \left (25-10 x+x^2\right )\right ) \log ^2\left (-1+e^{-5+x}\right )} \, dx=2 \left (\frac {x^2}{2}+\frac {9 x}{(-5+x) \left (x^2+\log \left (-1+e^{-5+x}\right )\right )}\right ) \] Input:

Integrate[(-90*x^2 + 36*x^3 - 50*x^5 + 20*x^6 - 2*x^7 + E^(-5 + x)*(90*x + 
 72*x^2 - 36*x^3 + 50*x^5 - 20*x^6 + 2*x^7) + (90 - 100*x^3 + 40*x^4 - 4*x 
^5 + E^(-5 + x)*(-90 + 100*x^3 - 40*x^4 + 4*x^5))*Log[-1 + E^(-5 + x)] + ( 
-50*x + 20*x^2 - 2*x^3 + E^(-5 + x)*(50*x - 20*x^2 + 2*x^3))*Log[-1 + E^(- 
5 + x)]^2)/(-25*x^4 + 10*x^5 - x^6 + E^(-5 + x)*(25*x^4 - 10*x^5 + x^6) + 
(-50*x^2 + 20*x^3 - 2*x^4 + E^(-5 + x)*(50*x^2 - 20*x^3 + 2*x^4))*Log[-1 + 
 E^(-5 + x)] + (-25 + 10*x - x^2 + E^(-5 + x)*(25 - 10*x + x^2))*Log[-1 + 
E^(-5 + x)]^2),x]
 

Output:

2*(x^2/2 + (9*x)/((-5 + x)*(x^2 + Log[-1 + E^(-5 + x)])))
 

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.

\(\displaystyle \int \frac {-2 x^7+20 x^6-50 x^5+36 x^3-90 x^2+\left (-2 x^3+20 x^2+e^{x-5} \left (2 x^3-20 x^2+50 x\right )-50 x\right ) \log ^2\left (e^{x-5}-1\right )+\left (-4 x^5+40 x^4-100 x^3+e^{x-5} \left (4 x^5-40 x^4+100 x^3-90\right )+90\right ) \log \left (e^{x-5}-1\right )+e^{x-5} \left (2 x^7-20 x^6+50 x^5-36 x^3+72 x^2+90 x\right )}{-x^6+10 x^5-25 x^4+\left (-x^2+e^{x-5} \left (x^2-10 x+25\right )+10 x-25\right ) \log ^2\left (e^{x-5}-1\right )+e^{x-5} \left (x^6-10 x^5+25 x^4\right )+\left (-2 x^4+20 x^3-50 x^2+e^{x-5} \left (2 x^4-20 x^3+50 x^2\right )\right ) \log \left (e^{x-5}-1\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 \left (-\left (e^x-e^5\right ) \left (2 x^5-20 x^4+50 x^3-45\right ) \log \left (e^{x-5}-1\right )-x \left (e^x \left (x^6-10 x^5+25 x^4-18 x^2+36 x+45\right )-e^5 x \left (x^5-10 x^4+25 x^3-18 x+45\right )\right )-\left (\left (e^x-e^5\right ) (x-5)^2 x \log ^2\left (e^{x-5}-1\right )\right )\right )}{\left (e^5-e^x\right ) (5-x)^2 \left (x^2+\log \left (e^{x-5}-1\right )\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \int \frac {\left (e^5-e^x\right ) (5-x)^2 x \log ^2\left (-1+e^{x-5}\right )-\left (e^5-e^x\right ) \left (-2 x^5+20 x^4-50 x^3+45\right ) \log \left (-1+e^{x-5}\right )+x \left (e^5 x \left (x^5-10 x^4+25 x^3-18 x+45\right )-e^x \left (x^6-10 x^5+25 x^4-18 x^2+36 x+45\right )\right )}{\left (e^5-e^x\right ) (5-x)^2 \left (x^2+\log \left (-1+e^{x-5}\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (\frac {9 e^5 x}{\left (e^5-e^x\right ) (x-5) \left (x^2+\log \left (-1+e^{x-5}\right )\right )^2}+\frac {x^7-10 x^6+2 \log \left (-1+e^{x-5}\right ) x^5+25 x^5-20 \log \left (-1+e^{x-5}\right ) x^4+\log ^2\left (-1+e^{x-5}\right ) x^3+50 \log \left (-1+e^{x-5}\right ) x^3-18 x^3-10 \log ^2\left (-1+e^{x-5}\right ) x^2+36 x^2+25 \log ^2\left (-1+e^{x-5}\right ) x+45 x-45 \log \left (-1+e^{x-5}\right )}{(x-5)^2 \left (x^2+\log \left (-1+e^{x-5}\right )\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (-99 \int \frac {1}{\left (x^2+\log \left (-1+e^{x-5}\right )\right )^2}dx-9 e^5 \int \frac {1}{\left (-e^5+e^x\right ) \left (x^2+\log \left (-1+e^{x-5}\right )\right )^2}dx-495 \int \frac {1}{(x-5) \left (x^2+\log \left (-1+e^{x-5}\right )\right )^2}dx-45 e^5 \int \frac {1}{\left (-e^5+e^x\right ) (x-5) \left (x^2+\log \left (-1+e^{x-5}\right )\right )^2}dx-18 \int \frac {x}{\left (x^2+\log \left (-1+e^{x-5}\right )\right )^2}dx-45 \int \frac {1}{(x-5)^2 \left (x^2+\log \left (-1+e^{x-5}\right )\right )}dx+\frac {x^2}{2}\right )\)

Input:

Int[(-90*x^2 + 36*x^3 - 50*x^5 + 20*x^6 - 2*x^7 + E^(-5 + x)*(90*x + 72*x^ 
2 - 36*x^3 + 50*x^5 - 20*x^6 + 2*x^7) + (90 - 100*x^3 + 40*x^4 - 4*x^5 + E 
^(-5 + x)*(-90 + 100*x^3 - 40*x^4 + 4*x^5))*Log[-1 + E^(-5 + x)] + (-50*x 
+ 20*x^2 - 2*x^3 + E^(-5 + x)*(50*x - 20*x^2 + 2*x^3))*Log[-1 + E^(-5 + x) 
]^2)/(-25*x^4 + 10*x^5 - x^6 + E^(-5 + x)*(25*x^4 - 10*x^5 + x^6) + (-50*x 
^2 + 20*x^3 - 2*x^4 + E^(-5 + x)*(50*x^2 - 20*x^3 + 2*x^4))*Log[-1 + E^(-5 
 + x)] + (-25 + 10*x - x^2 + E^(-5 + x)*(25 - 10*x + x^2))*Log[-1 + E^(-5 
+ x)]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 2.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04

method	result	size
risch	\(x^{2}+\frac {18 x}{\left (-5+x \right ) \left (\ln \left ({\mathrm e}^{-5+x}-1\right )+x^{2}\right )}\)	\(26\)
parallelrisch	\(\frac {2 x^{5}-10 x^{4}+2 \ln \left ({\mathrm e}^{-5+x}-1\right ) x^{3}-10 \ln \left ({\mathrm e}^{-5+x}-1\right ) x^{2}+36 x}{2 x^{3}+2 x \ln \left ({\mathrm e}^{-5+x}-1\right )-10 x^{2}-10 \ln \left ({\mathrm e}^{-5+x}-1\right )}\)	\(70\)

Input:

int((((2*x^3-20*x^2+50*x)*exp(-5+x)-2*x^3+20*x^2-50*x)*ln(exp(-5+x)-1)^2+( 
(4*x^5-40*x^4+100*x^3-90)*exp(-5+x)-4*x^5+40*x^4-100*x^3+90)*ln(exp(-5+x)- 
1)+(2*x^7-20*x^6+50*x^5-36*x^3+72*x^2+90*x)*exp(-5+x)-2*x^7+20*x^6-50*x^5+ 
36*x^3-90*x^2)/(((x^2-10*x+25)*exp(-5+x)-x^2+10*x-25)*ln(exp(-5+x)-1)^2+(( 
2*x^4-20*x^3+50*x^2)*exp(-5+x)-2*x^4+20*x^3-50*x^2)*ln(exp(-5+x)-1)+(x^6-1 
0*x^5+25*x^4)*exp(-5+x)-x^6+10*x^5-25*x^4),x,method=_RETURNVERBOSE)
 

Output:

x^2+18*x/(-5+x)/(ln(exp(-5+x)-1)+x^2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (24) = 48\).

Time = 0.10 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.08 \[ \int \frac {-90 x^2+36 x^3-50 x^5+20 x^6-2 x^7+e^{-5+x} \left (90 x+72 x^2-36 x^3+50 x^5-20 x^6+2 x^7\right )+\left (90-100 x^3+40 x^4-4 x^5+e^{-5+x} \left (-90+100 x^3-40 x^4+4 x^5\right )\right ) \log \left (-1+e^{-5+x}\right )+\left (-50 x+20 x^2-2 x^3+e^{-5+x} \left (50 x-20 x^2+2 x^3\right )\right ) \log ^2\left (-1+e^{-5+x}\right )}{-25 x^4+10 x^5-x^6+e^{-5+x} \left (25 x^4-10 x^5+x^6\right )+\left (-50 x^2+20 x^3-2 x^4+e^{-5+x} \left (50 x^2-20 x^3+2 x^4\right )\right ) \log \left (-1+e^{-5+x}\right )+\left (-25+10 x-x^2+e^{-5+x} \left (25-10 x+x^2\right )\right ) \log ^2\left (-1+e^{-5+x}\right )} \, dx=\frac {x^{5} - 5 \, x^{4} + {\left (x^{3} - 5 \, x^{2}\right )} \log \left (e^{\left (x - 5\right )} - 1\right ) + 18 \, x}{x^{3} - 5 \, x^{2} + {\left (x - 5\right )} \log \left (e^{\left (x - 5\right )} - 1\right )} \] Input:

integrate((((2*x^3-20*x^2+50*x)*exp(-5+x)-2*x^3+20*x^2-50*x)*log(exp(-5+x) 
-1)^2+((4*x^5-40*x^4+100*x^3-90)*exp(-5+x)-4*x^5+40*x^4-100*x^3+90)*log(ex 
p(-5+x)-1)+(2*x^7-20*x^6+50*x^5-36*x^3+72*x^2+90*x)*exp(-5+x)-2*x^7+20*x^6 
-50*x^5+36*x^3-90*x^2)/(((x^2-10*x+25)*exp(-5+x)-x^2+10*x-25)*log(exp(-5+x 
)-1)^2+((2*x^4-20*x^3+50*x^2)*exp(-5+x)-2*x^4+20*x^3-50*x^2)*log(exp(-5+x) 
-1)+(x^6-10*x^5+25*x^4)*exp(-5+x)-x^6+10*x^5-25*x^4),x, algorithm="fricas" 
)
 

Output:

(x^5 - 5*x^4 + (x^3 - 5*x^2)*log(e^(x - 5) - 1) + 18*x)/(x^3 - 5*x^2 + (x 
- 5)*log(e^(x - 5) - 1))
 

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {-90 x^2+36 x^3-50 x^5+20 x^6-2 x^7+e^{-5+x} \left (90 x+72 x^2-36 x^3+50 x^5-20 x^6+2 x^7\right )+\left (90-100 x^3+40 x^4-4 x^5+e^{-5+x} \left (-90+100 x^3-40 x^4+4 x^5\right )\right ) \log \left (-1+e^{-5+x}\right )+\left (-50 x+20 x^2-2 x^3+e^{-5+x} \left (50 x-20 x^2+2 x^3\right )\right ) \log ^2\left (-1+e^{-5+x}\right )}{-25 x^4+10 x^5-x^6+e^{-5+x} \left (25 x^4-10 x^5+x^6\right )+\left (-50 x^2+20 x^3-2 x^4+e^{-5+x} \left (50 x^2-20 x^3+2 x^4\right )\right ) \log \left (-1+e^{-5+x}\right )+\left (-25+10 x-x^2+e^{-5+x} \left (25-10 x+x^2\right )\right ) \log ^2\left (-1+e^{-5+x}\right )} \, dx=x^{2} + \frac {18 x}{x^{3} - 5 x^{2} + \left (x - 5\right ) \log {\left (e^{x - 5} - 1 \right )}} \] Input:

integrate((((2*x**3-20*x**2+50*x)*exp(-5+x)-2*x**3+20*x**2-50*x)*ln(exp(-5 
+x)-1)**2+((4*x**5-40*x**4+100*x**3-90)*exp(-5+x)-4*x**5+40*x**4-100*x**3+ 
90)*ln(exp(-5+x)-1)+(2*x**7-20*x**6+50*x**5-36*x**3+72*x**2+90*x)*exp(-5+x 
)-2*x**7+20*x**6-50*x**5+36*x**3-90*x**2)/(((x**2-10*x+25)*exp(-5+x)-x**2+ 
10*x-25)*ln(exp(-5+x)-1)**2+((2*x**4-20*x**3+50*x**2)*exp(-5+x)-2*x**4+20* 
x**3-50*x**2)*ln(exp(-5+x)-1)+(x**6-10*x**5+25*x**4)*exp(-5+x)-x**6+10*x** 
5-25*x**4),x)
 

Output:

x**2 + 18*x/(x**3 - 5*x**2 + (x - 5)*log(exp(x - 5) - 1))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (24) = 48\).

Time = 0.32 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.72 \[ \int \frac {-90 x^2+36 x^3-50 x^5+20 x^6-2 x^7+e^{-5+x} \left (90 x+72 x^2-36 x^3+50 x^5-20 x^6+2 x^7\right )+\left (90-100 x^3+40 x^4-4 x^5+e^{-5+x} \left (-90+100 x^3-40 x^4+4 x^5\right )\right ) \log \left (-1+e^{-5+x}\right )+\left (-50 x+20 x^2-2 x^3+e^{-5+x} \left (50 x-20 x^2+2 x^3\right )\right ) \log ^2\left (-1+e^{-5+x}\right )}{-25 x^4+10 x^5-x^6+e^{-5+x} \left (25 x^4-10 x^5+x^6\right )+\left (-50 x^2+20 x^3-2 x^4+e^{-5+x} \left (50 x^2-20 x^3+2 x^4\right )\right ) \log \left (-1+e^{-5+x}\right )+\left (-25+10 x-x^2+e^{-5+x} \left (25-10 x+x^2\right )\right ) \log ^2\left (-1+e^{-5+x}\right )} \, dx=\frac {x^{5} - 5 \, x^{4} - 5 \, x^{3} + 25 \, x^{2} + {\left (x^{3} - 5 \, x^{2}\right )} \log \left (-e^{5} + e^{x}\right ) + 18 \, x}{x^{3} - 5 \, x^{2} + {\left (x - 5\right )} \log \left (-e^{5} + e^{x}\right ) - 5 \, x + 25} \] Input:

integrate((((2*x^3-20*x^2+50*x)*exp(-5+x)-2*x^3+20*x^2-50*x)*log(exp(-5+x) 
-1)^2+((4*x^5-40*x^4+100*x^3-90)*exp(-5+x)-4*x^5+40*x^4-100*x^3+90)*log(ex 
p(-5+x)-1)+(2*x^7-20*x^6+50*x^5-36*x^3+72*x^2+90*x)*exp(-5+x)-2*x^7+20*x^6 
-50*x^5+36*x^3-90*x^2)/(((x^2-10*x+25)*exp(-5+x)-x^2+10*x-25)*log(exp(-5+x 
)-1)^2+((2*x^4-20*x^3+50*x^2)*exp(-5+x)-2*x^4+20*x^3-50*x^2)*log(exp(-5+x) 
-1)+(x^6-10*x^5+25*x^4)*exp(-5+x)-x^6+10*x^5-25*x^4),x, algorithm="maxima" 
)
 

Output:

(x^5 - 5*x^4 - 5*x^3 + 25*x^2 + (x^3 - 5*x^2)*log(-e^5 + e^x) + 18*x)/(x^3 
 - 5*x^2 + (x - 5)*log(-e^5 + e^x) - 5*x + 25)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (24) = 48\).

Time = 0.35 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.32 \[ \int \frac {-90 x^2+36 x^3-50 x^5+20 x^6-2 x^7+e^{-5+x} \left (90 x+72 x^2-36 x^3+50 x^5-20 x^6+2 x^7\right )+\left (90-100 x^3+40 x^4-4 x^5+e^{-5+x} \left (-90+100 x^3-40 x^4+4 x^5\right )\right ) \log \left (-1+e^{-5+x}\right )+\left (-50 x+20 x^2-2 x^3+e^{-5+x} \left (50 x-20 x^2+2 x^3\right )\right ) \log ^2\left (-1+e^{-5+x}\right )}{-25 x^4+10 x^5-x^6+e^{-5+x} \left (25 x^4-10 x^5+x^6\right )+\left (-50 x^2+20 x^3-2 x^4+e^{-5+x} \left (50 x^2-20 x^3+2 x^4\right )\right ) \log \left (-1+e^{-5+x}\right )+\left (-25+10 x-x^2+e^{-5+x} \left (25-10 x+x^2\right )\right ) \log ^2\left (-1+e^{-5+x}\right )} \, dx=\frac {x^{5} - 5 \, x^{4} + x^{3} \log \left (-e^{5} + e^{x}\right ) - 5 \, x^{3} - 5 \, x^{2} \log \left (-e^{5} + e^{x}\right ) + 25 \, x^{2} + 18 \, x}{x^{3} - 5 \, x^{2} + x \log \left (-e^{5} + e^{x}\right ) - 5 \, x - 5 \, \log \left (-e^{5} + e^{x}\right ) + 25} \] Input:

integrate((((2*x^3-20*x^2+50*x)*exp(-5+x)-2*x^3+20*x^2-50*x)*log(exp(-5+x) 
-1)^2+((4*x^5-40*x^4+100*x^3-90)*exp(-5+x)-4*x^5+40*x^4-100*x^3+90)*log(ex 
p(-5+x)-1)+(2*x^7-20*x^6+50*x^5-36*x^3+72*x^2+90*x)*exp(-5+x)-2*x^7+20*x^6 
-50*x^5+36*x^3-90*x^2)/(((x^2-10*x+25)*exp(-5+x)-x^2+10*x-25)*log(exp(-5+x 
)-1)^2+((2*x^4-20*x^3+50*x^2)*exp(-5+x)-2*x^4+20*x^3-50*x^2)*log(exp(-5+x) 
-1)+(x^6-10*x^5+25*x^4)*exp(-5+x)-x^6+10*x^5-25*x^4),x, algorithm="giac")
 

Output:

(x^5 - 5*x^4 + x^3*log(-e^5 + e^x) - 5*x^3 - 5*x^2*log(-e^5 + e^x) + 25*x^ 
2 + 18*x)/(x^3 - 5*x^2 + x*log(-e^5 + e^x) - 5*x - 5*log(-e^5 + e^x) + 25)
 

Mupad [B] (verification not implemented)

Time = 4.37 (sec) , antiderivative size = 192, normalized size of antiderivative = 7.68 \[ \int \frac {-90 x^2+36 x^3-50 x^5+20 x^6-2 x^7+e^{-5+x} \left (90 x+72 x^2-36 x^3+50 x^5-20 x^6+2 x^7\right )+\left (90-100 x^3+40 x^4-4 x^5+e^{-5+x} \left (-90+100 x^3-40 x^4+4 x^5\right )\right ) \log \left (-1+e^{-5+x}\right )+\left (-50 x+20 x^2-2 x^3+e^{-5+x} \left (50 x-20 x^2+2 x^3\right )\right ) \log ^2\left (-1+e^{-5+x}\right )}{-25 x^4+10 x^5-x^6+e^{-5+x} \left (25 x^4-10 x^5+x^6\right )+\left (-50 x^2+20 x^3-2 x^4+e^{-5+x} \left (50 x^2-20 x^3+2 x^4\right )\right ) \log \left (-1+e^{-5+x}\right )+\left (-25+10 x-x^2+e^{-5+x} \left (25-10 x+x^2\right )\right ) \log ^2\left (-1+e^{-5+x}\right )} \, dx=x^2-\frac {45}{x^3-\frac {19\,x^2}{2}+20\,x+\frac {25}{2}}-\frac {\frac {18\,\left (5\,x\,{\mathrm {e}}^{x-5}+4\,x^2\,{\mathrm {e}}^{x-5}-2\,x^3\,{\mathrm {e}}^{x-5}-5\,x^2+2\,x^3\right )}{{\left (x-5\right )}^2\,\left ({\mathrm {e}}^{x-5}-2\,x+2\,x\,{\mathrm {e}}^{x-5}\right )}-\frac {90\,\ln \left ({\mathrm {e}}^{-5}\,{\mathrm {e}}^x-1\right )\,\left ({\mathrm {e}}^{x-5}-1\right )}{{\left (x-5\right )}^2\,\left ({\mathrm {e}}^{x-5}-2\,x+2\,x\,{\mathrm {e}}^{x-5}\right )}}{\ln \left ({\mathrm {e}}^{-5}\,{\mathrm {e}}^x-1\right )+x^2}+\frac {90\,\left (-2\,x^3+9\,x^2+6\,x-5\right )}{\left (2\,x-{\mathrm {e}}^{x-5}\,\left (2\,x+1\right )\right )\,\left (2\,x+1\right )\,{\left (x-5\right )}^3\,\left (2\,x^2+x-1\right )} \] Input:

int((log(exp(x - 5) - 1)*(exp(x - 5)*(100*x^3 - 40*x^4 + 4*x^5 - 90) - 100 
*x^3 + 40*x^4 - 4*x^5 + 90) - log(exp(x - 5) - 1)^2*(50*x - exp(x - 5)*(50 
*x - 20*x^2 + 2*x^3) - 20*x^2 + 2*x^3) - 90*x^2 + 36*x^3 - 50*x^5 + 20*x^6 
 - 2*x^7 + exp(x - 5)*(90*x + 72*x^2 - 36*x^3 + 50*x^5 - 20*x^6 + 2*x^7))/ 
(log(exp(x - 5) - 1)*(exp(x - 5)*(50*x^2 - 20*x^3 + 2*x^4) - 50*x^2 + 20*x 
^3 - 2*x^4) + exp(x - 5)*(25*x^4 - 10*x^5 + x^6) - 25*x^4 + 10*x^5 - x^6 + 
 log(exp(x - 5) - 1)^2*(10*x + exp(x - 5)*(x^2 - 10*x + 25) - x^2 - 25)),x 
)
 

Output:

x^2 - 45/(20*x - (19*x^2)/2 + x^3 + 25/2) - ((18*(5*x*exp(x - 5) + 4*x^2*e 
xp(x - 5) - 2*x^3*exp(x - 5) - 5*x^2 + 2*x^3))/((x - 5)^2*(exp(x - 5) - 2* 
x + 2*x*exp(x - 5))) - (90*log(exp(-5)*exp(x) - 1)*(exp(x - 5) - 1))/((x - 
 5)^2*(exp(x - 5) - 2*x + 2*x*exp(x - 5))))/(log(exp(-5)*exp(x) - 1) + x^2 
) + (90*(6*x + 9*x^2 - 2*x^3 - 5))/((2*x - exp(x - 5)*(2*x + 1))*(2*x + 1) 
*(x - 5)^3*(x + 2*x^2 - 1))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 174, normalized size of antiderivative = 6.96 \[ \int \frac {-90 x^2+36 x^3-50 x^5+20 x^6-2 x^7+e^{-5+x} \left (90 x+72 x^2-36 x^3+50 x^5-20 x^6+2 x^7\right )+\left (90-100 x^3+40 x^4-4 x^5+e^{-5+x} \left (-90+100 x^3-40 x^4+4 x^5\right )\right ) \log \left (-1+e^{-5+x}\right )+\left (-50 x+20 x^2-2 x^3+e^{-5+x} \left (50 x-20 x^2+2 x^3\right )\right ) \log ^2\left (-1+e^{-5+x}\right )}{-25 x^4+10 x^5-x^6+e^{-5+x} \left (25 x^4-10 x^5+x^6\right )+\left (-50 x^2+20 x^3-2 x^4+e^{-5+x} \left (50 x^2-20 x^3+2 x^4\right )\right ) \log \left (-1+e^{-5+x}\right )+\left (-25+10 x-x^2+e^{-5+x} \left (25-10 x+x^2\right )\right ) \log ^2\left (-1+e^{-5+x}\right )} \, dx=\frac {\mathrm {log}\left (e^{x}-e^{5}\right ) \mathrm {log}\left (\frac {e^{x}-e^{5}}{e^{5}}\right ) x -5 \,\mathrm {log}\left (e^{x}-e^{5}\right ) \mathrm {log}\left (\frac {e^{x}-e^{5}}{e^{5}}\right )+\mathrm {log}\left (e^{x}-e^{5}\right ) x^{3}-5 \,\mathrm {log}\left (e^{x}-e^{5}\right ) x^{2}-\mathrm {log}\left (\frac {e^{x}-e^{5}}{e^{5}}\right )^{2} x +5 \mathrm {log}\left (\frac {e^{x}-e^{5}}{e^{5}}\right )^{2}+x^{5}-5 x^{4}+18 x}{\mathrm {log}\left (\frac {e^{x}-e^{5}}{e^{5}}\right ) x -5 \,\mathrm {log}\left (\frac {e^{x}-e^{5}}{e^{5}}\right )+x^{3}-5 x^{2}} \] Input:

int((((2*x^3-20*x^2+50*x)*exp(-5+x)-2*x^3+20*x^2-50*x)*log(exp(-5+x)-1)^2+ 
((4*x^5-40*x^4+100*x^3-90)*exp(-5+x)-4*x^5+40*x^4-100*x^3+90)*log(exp(-5+x 
)-1)+(2*x^7-20*x^6+50*x^5-36*x^3+72*x^2+90*x)*exp(-5+x)-2*x^7+20*x^6-50*x^ 
5+36*x^3-90*x^2)/(((x^2-10*x+25)*exp(-5+x)-x^2+10*x-25)*log(exp(-5+x)-1)^2 
+((2*x^4-20*x^3+50*x^2)*exp(-5+x)-2*x^4+20*x^3-50*x^2)*log(exp(-5+x)-1)+(x 
^6-10*x^5+25*x^4)*exp(-5+x)-x^6+10*x^5-25*x^4),x)
 

Output:

(log(e**x - e**5)*log((e**x - e**5)/e**5)*x - 5*log(e**x - e**5)*log((e**x 
 - e**5)/e**5) + log(e**x - e**5)*x**3 - 5*log(e**x - e**5)*x**2 - log((e* 
*x - e**5)/e**5)**2*x + 5*log((e**x - e**5)/e**5)**2 + x**5 - 5*x**4 + 18* 
x)/(log((e**x - e**5)/e**5)*x - 5*log((e**x - e**5)/e**5) + x**3 - 5*x**2)