\(\int \frac {1875 x+e^x (750 x-750 x^2)+e^{2 x} (75 x-150 x^2)+(-375+e^{2 x} (-15+30 x)+e^x (-150+150 x)) \log (x^2)+\log (\frac {25+10 e^x+e^{2 x}-2 x}{x}) (750+300 e^x+30 e^{2 x}-60 x+(-375-150 e^x-15 e^{2 x}+30 x) \log (x^2))}{\log ^2(\frac {25+10 e^x+e^{2 x}-2 x}{x}) (625 x^2+250 e^x x^2+25 e^{2 x} x^2-50 x^3+(-250 x-100 e^x x-10 e^{2 x} x+20 x^2) \log (x^2)+(25+10 e^x+e^{2 x}-2 x) \log ^2(x^2))} \, dx\) [1455]

Optimal result

Integrand size = 223, antiderivative size = 35 \[ \int \frac {1875 x+e^x \left (750 x-750 x^2\right )+e^{2 x} \left (75 x-150 x^2\right )+\left (-375+e^{2 x} (-15+30 x)+e^x (-150+150 x)\right ) \log \left (x^2\right )+\log \left (\frac {25+10 e^x+e^{2 x}-2 x}{x}\right ) \left (750+300 e^x+30 e^{2 x}-60 x+\left (-375-150 e^x-15 e^{2 x}+30 x\right ) \log \left (x^2\right )\right )}{\log ^2\left (\frac {25+10 e^x+e^{2 x}-2 x}{x}\right ) \left (625 x^2+250 e^x x^2+25 e^{2 x} x^2-50 x^3+\left (-250 x-100 e^x x-10 e^{2 x} x+20 x^2\right ) \log \left (x^2\right )+\left (25+10 e^x+e^{2 x}-2 x\right ) \log ^2\left (x^2\right )\right )} \, dx=1-\frac {3 x}{\log \left (-2+\frac {\left (5+e^x\right )^2}{x}\right ) \left (-x+\frac {\log \left (x^2\right )}{5}\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \frac {1875 x+e^x \left (750 x-750 x^2\right )+e^{2 x} \left (75 x-150 x^2\right )+\left (-375+e^{2 x} (-15+30 x)+e^x (-150+150 x)\right ) \log \left (x^2\right )+\log \left (\frac {25+10 e^x+e^{2 x}-2 x}{x}\right ) \left (750+300 e^x+30 e^{2 x}-60 x+\left (-375-150 e^x-15 e^{2 x}+30 x\right ) \log \left (x^2\right )\right )}{\log ^2\left (\frac {25+10 e^x+e^{2 x}-2 x}{x}\right ) \left (625 x^2+250 e^x x^2+25 e^{2 x} x^2-50 x^3+\left (-250 x-100 e^x x-10 e^{2 x} x+20 x^2\right ) \log \left (x^2\right )+\left (25+10 e^x+e^{2 x}-2 x\right ) \log ^2\left (x^2\right )\right )} \, dx=\frac {15 x}{\log \left (\frac {25+10 e^x+e^{2 x}-2 x}{x}\right ) \left (5 x-\log \left (x^2\right )\right )} \] Input:

Integrate[(1875*x + E^x*(750*x - 750*x^2) + E^(2*x)*(75*x - 150*x^2) + (-3 
75 + E^(2*x)*(-15 + 30*x) + E^x*(-150 + 150*x))*Log[x^2] + Log[(25 + 10*E^ 
x + E^(2*x) - 2*x)/x]*(750 + 300*E^x + 30*E^(2*x) - 60*x + (-375 - 150*E^x 
 - 15*E^(2*x) + 30*x)*Log[x^2]))/(Log[(25 + 10*E^x + E^(2*x) - 2*x)/x]^2*( 
625*x^2 + 250*E^x*x^2 + 25*E^(2*x)*x^2 - 50*x^3 + (-250*x - 100*E^x*x - 10 
*E^(2*x)*x + 20*x^2)*Log[x^2] + (25 + 10*E^x + E^(2*x) - 2*x)*Log[x^2]^2)) 
,x]
 

Output:

(15*x)/(Log[(25 + 10*E^x + E^(2*x) - 2*x)/x]*(5*x - Log[x^2]))
 

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.

\(\displaystyle \int \frac {e^x \left (750 x-750 x^2\right )+e^{2 x} \left (75 x-150 x^2\right )+\left (e^{2 x} (30 x-15)+e^x (150 x-150)-375\right ) \log \left (x^2\right )+\log \left (\frac {-2 x+10 e^x+e^{2 x}+25}{x}\right ) \left (\left (30 x-150 e^x-15 e^{2 x}-375\right ) \log \left (x^2\right )-60 x+300 e^x+30 e^{2 x}+750\right )+1875 x}{\log ^2\left (\frac {-2 x+10 e^x+e^{2 x}+25}{x}\right ) \left (-50 x^3+250 e^x x^2+25 e^{2 x} x^2+625 x^2+\left (-2 x+10 e^x+e^{2 x}+25\right ) \log ^2\left (x^2\right )+\left (20 x^2-100 e^x x-10 e^{2 x} x-250 x\right ) \log \left (x^2\right )\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {15 \left (-\frac {\left (e^x+5\right ) \left (e^x (2 x-1)-5\right ) \left (5 x-\log \left (x^2\right )\right )}{-2 x+10 e^x+e^{2 x}+25}-\log \left (\frac {-2 x+10 e^x+e^{2 x}+25}{x}\right ) \left (\log \left (x^2\right )-2\right )\right )}{\log ^2\left (\frac {10 e^x}{x}+\frac {e^{2 x}}{x}+\frac {25}{x}-2\right ) \left (5 x-\log \left (x^2\right )\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 15 \int \frac {\log \left (\frac {-2 x+10 e^x+e^{2 x}+25}{x}\right ) \left (2-\log \left (x^2\right )\right )+\frac {\left (5+e^x\right ) \left (e^x (1-2 x)+5\right ) \left (5 x-\log \left (x^2\right )\right )}{-2 x+10 e^x+e^{2 x}+25}}{\log ^2\left (-2+\frac {10 e^x}{x}+\frac {e^{2 x}}{x}+\frac {25}{x}\right ) \left (5 x-\log \left (x^2\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 15 \int \left (\frac {2 \left (-2 x+5 e^x+26\right ) x}{\left (-2 x+10 e^x+e^{2 x}+25\right ) \log ^2\left (-2+\frac {10 e^x}{x}+\frac {e^{2 x}}{x}+\frac {25}{x}\right ) \left (5 x-\log \left (x^2\right )\right )}+\frac {-10 x^2+2 \log \left (x^2\right ) x+5 x+2 \log \left (\frac {-2 x+10 e^x+e^{2 x}+25}{x}\right )-\log \left (\frac {-2 x+10 e^x+e^{2 x}+25}{x}\right ) \log \left (x^2\right )-\log \left (x^2\right )}{\log ^2\left (-2+\frac {10 e^x}{x}+\frac {e^{2 x}}{x}+\frac {25}{x}\right ) \left (5 x-\log \left (x^2\right )\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 15 \left (\int \frac {1}{\log ^2\left (-2+\frac {10 e^x}{x}+\frac {e^{2 x}}{x}+\frac {25}{x}\right ) \left (5 x-\log \left (x^2\right )\right )}dx+52 \int \frac {x}{\left (-2 x+10 e^x+e^{2 x}+25\right ) \log ^2\left (-2+\frac {10 e^x}{x}+\frac {e^{2 x}}{x}+\frac {25}{x}\right ) \left (5 x-\log \left (x^2\right )\right )}dx+10 \int \frac {e^x x}{\left (-2 x+10 e^x+e^{2 x}+25\right ) \log ^2\left (-2+\frac {10 e^x}{x}+\frac {e^{2 x}}{x}+\frac {25}{x}\right ) \left (5 x-\log \left (x^2\right )\right )}dx+4 \int \frac {x^2}{\left (2 x-10 e^x-e^{2 x}-25\right ) \log ^2\left (-2+\frac {10 e^x}{x}+\frac {e^{2 x}}{x}+\frac {25}{x}\right ) \left (5 x-\log \left (x^2\right )\right )}dx+2 \int \frac {x}{\log ^2\left (-2+\frac {10 e^x}{x}+\frac {e^{2 x}}{x}+\frac {25}{x}\right ) \left (\log \left (x^2\right )-5 x\right )}dx+2 \int \frac {1}{\log \left (-2+\frac {10 e^x}{x}+\frac {e^{2 x}}{x}+\frac {25}{x}\right ) \left (5 x-\log \left (x^2\right )\right )^2}dx-5 \int \frac {x}{\log \left (-2+\frac {10 e^x}{x}+\frac {e^{2 x}}{x}+\frac {25}{x}\right ) \left (5 x-\log \left (x^2\right )\right )^2}dx+\int \frac {1}{\log \left (-2+\frac {10 e^x}{x}+\frac {e^{2 x}}{x}+\frac {25}{x}\right ) \left (5 x-\log \left (x^2\right )\right )}dx\right )\)

Input:

Int[(1875*x + E^x*(750*x - 750*x^2) + E^(2*x)*(75*x - 150*x^2) + (-375 + E 
^(2*x)*(-15 + 30*x) + E^x*(-150 + 150*x))*Log[x^2] + Log[(25 + 10*E^x + E^ 
(2*x) - 2*x)/x]*(750 + 300*E^x + 30*E^(2*x) - 60*x + (-375 - 150*E^x - 15* 
E^(2*x) + 30*x)*Log[x^2]))/(Log[(25 + 10*E^x + E^(2*x) - 2*x)/x]^2*(625*x^ 
2 + 250*E^x*x^2 + 25*E^(2*x)*x^2 - 50*x^3 + (-250*x - 100*E^x*x - 10*E^(2* 
x)*x + 20*x^2)*Log[x^2] + (25 + 10*E^x + E^(2*x) - 2*x)*Log[x^2]^2)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 23.88 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.03

method	result	size
parallelrisch	\(\frac {15 x}{\left (5 x -\ln \left (x^{2}\right )\right ) \ln \left (\frac {{\mathrm e}^{2 x}+10 \,{\mathrm e}^{x}+25-2 x}{x}\right )}\)	\(36\)
risch	\(\frac {60 i x}{\left (i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}-2 i \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )+i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+10 x -4 \ln \left (x \right )\right ) \left (\pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i \left (\frac {{\mathrm e}^{2 x}}{2}-x +5 \,{\mathrm e}^{x}+\frac {25}{2}\right )\right ) \operatorname {csgn}\left (\frac {i \left (\frac {{\mathrm e}^{2 x}}{2}-x +5 \,{\mathrm e}^{x}+\frac {25}{2}\right )}{x}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) {\operatorname {csgn}\left (\frac {i \left (\frac {{\mathrm e}^{2 x}}{2}-x +5 \,{\mathrm e}^{x}+\frac {25}{2}\right )}{x}\right )}^{2}+2 \pi {\operatorname {csgn}\left (\frac {i \left (\frac {{\mathrm e}^{2 x}}{2}-x +5 \,{\mathrm e}^{x}+\frac {25}{2}\right )}{x}\right )}^{2}+\pi \,\operatorname {csgn}\left (i \left (\frac {{\mathrm e}^{2 x}}{2}-x +5 \,{\mathrm e}^{x}+\frac {25}{2}\right )\right ) {\operatorname {csgn}\left (\frac {i \left (\frac {{\mathrm e}^{2 x}}{2}-x +5 \,{\mathrm e}^{x}+\frac {25}{2}\right )}{x}\right )}^{2}+\pi {\operatorname {csgn}\left (\frac {i \left (\frac {{\mathrm e}^{2 x}}{2}-x +5 \,{\mathrm e}^{x}+\frac {25}{2}\right )}{x}\right )}^{3}-2 \pi +2 i \ln \left (2\right )-2 i \ln \left (x \right )+2 i \ln \left (-\frac {{\mathrm e}^{2 x}}{2}+x -5 \,{\mathrm e}^{x}-\frac {25}{2}\right )\right )}\)	\(279\)

Input:

int((((-15*exp(x)^2-150*exp(x)+30*x-375)*ln(x^2)+30*exp(x)^2+300*exp(x)-60 
*x+750)*ln((exp(x)^2+10*exp(x)+25-2*x)/x)+((30*x-15)*exp(x)^2+(150*x-150)* 
exp(x)-375)*ln(x^2)+(-150*x^2+75*x)*exp(x)^2+(-750*x^2+750*x)*exp(x)+1875* 
x)/((exp(x)^2+10*exp(x)+25-2*x)*ln(x^2)^2+(-10*x*exp(x)^2-100*exp(x)*x+20* 
x^2-250*x)*ln(x^2)+25*exp(x)^2*x^2+250*exp(x)*x^2-50*x^3+625*x^2)/ln((exp( 
x)^2+10*exp(x)+25-2*x)/x)^2,x,method=_RETURNVERBOSE)
 

Output:

15*x/(5*x-ln(x^2))/ln((exp(x)^2+10*exp(x)+25-2*x)/x)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.09 \[ \int \frac {1875 x+e^x \left (750 x-750 x^2\right )+e^{2 x} \left (75 x-150 x^2\right )+\left (-375+e^{2 x} (-15+30 x)+e^x (-150+150 x)\right ) \log \left (x^2\right )+\log \left (\frac {25+10 e^x+e^{2 x}-2 x}{x}\right ) \left (750+300 e^x+30 e^{2 x}-60 x+\left (-375-150 e^x-15 e^{2 x}+30 x\right ) \log \left (x^2\right )\right )}{\log ^2\left (\frac {25+10 e^x+e^{2 x}-2 x}{x}\right ) \left (625 x^2+250 e^x x^2+25 e^{2 x} x^2-50 x^3+\left (-250 x-100 e^x x-10 e^{2 x} x+20 x^2\right ) \log \left (x^2\right )+\left (25+10 e^x+e^{2 x}-2 x\right ) \log ^2\left (x^2\right )\right )} \, dx=\frac {15 \, x}{{\left (5 \, x - \log \left (x^{2}\right )\right )} \log \left (-\frac {2 \, x - e^{\left (2 \, x\right )} - 10 \, e^{x} - 25}{x}\right )} \] Input:

integrate((((-15*exp(x)^2-150*exp(x)+30*x-375)*log(x^2)+30*exp(x)^2+300*ex 
p(x)-60*x+750)*log((exp(x)^2+10*exp(x)+25-2*x)/x)+((30*x-15)*exp(x)^2+(150 
*x-150)*exp(x)-375)*log(x^2)+(-150*x^2+75*x)*exp(x)^2+(-750*x^2+750*x)*exp 
(x)+1875*x)/((exp(x)^2+10*exp(x)+25-2*x)*log(x^2)^2+(-10*x*exp(x)^2-100*ex 
p(x)*x+20*x^2-250*x)*log(x^2)+25*exp(x)^2*x^2+250*exp(x)*x^2-50*x^3+625*x^ 
2)/log((exp(x)^2+10*exp(x)+25-2*x)/x)^2,x, algorithm="fricas")
 

Output:

15*x/((5*x - log(x^2))*log(-(2*x - e^(2*x) - 10*e^x - 25)/x))
 

Sympy [A] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \frac {1875 x+e^x \left (750 x-750 x^2\right )+e^{2 x} \left (75 x-150 x^2\right )+\left (-375+e^{2 x} (-15+30 x)+e^x (-150+150 x)\right ) \log \left (x^2\right )+\log \left (\frac {25+10 e^x+e^{2 x}-2 x}{x}\right ) \left (750+300 e^x+30 e^{2 x}-60 x+\left (-375-150 e^x-15 e^{2 x}+30 x\right ) \log \left (x^2\right )\right )}{\log ^2\left (\frac {25+10 e^x+e^{2 x}-2 x}{x}\right ) \left (625 x^2+250 e^x x^2+25 e^{2 x} x^2-50 x^3+\left (-250 x-100 e^x x-10 e^{2 x} x+20 x^2\right ) \log \left (x^2\right )+\left (25+10 e^x+e^{2 x}-2 x\right ) \log ^2\left (x^2\right )\right )} \, dx=\frac {15 x}{\left (5 x - \log {\left (x^{2} \right )}\right ) \log {\left (\frac {- 2 x + e^{2 x} + 10 e^{x} + 25}{x} \right )}} \] Input:

integrate((((-15*exp(x)**2-150*exp(x)+30*x-375)*ln(x**2)+30*exp(x)**2+300* 
exp(x)-60*x+750)*ln((exp(x)**2+10*exp(x)+25-2*x)/x)+((30*x-15)*exp(x)**2+( 
150*x-150)*exp(x)-375)*ln(x**2)+(-150*x**2+75*x)*exp(x)**2+(-750*x**2+750* 
x)*exp(x)+1875*x)/((exp(x)**2+10*exp(x)+25-2*x)*ln(x**2)**2+(-10*x*exp(x)* 
*2-100*exp(x)*x+20*x**2-250*x)*ln(x**2)+25*exp(x)**2*x**2+250*exp(x)*x**2- 
50*x**3+625*x**2)/ln((exp(x)**2+10*exp(x)+25-2*x)/x)**2,x)
 

Output:

15*x/((5*x - log(x**2))*log((-2*x + exp(2*x) + 10*exp(x) + 25)/x))
 

Maxima [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.17 \[ \int \frac {1875 x+e^x \left (750 x-750 x^2\right )+e^{2 x} \left (75 x-150 x^2\right )+\left (-375+e^{2 x} (-15+30 x)+e^x (-150+150 x)\right ) \log \left (x^2\right )+\log \left (\frac {25+10 e^x+e^{2 x}-2 x}{x}\right ) \left (750+300 e^x+30 e^{2 x}-60 x+\left (-375-150 e^x-15 e^{2 x}+30 x\right ) \log \left (x^2\right )\right )}{\log ^2\left (\frac {25+10 e^x+e^{2 x}-2 x}{x}\right ) \left (625 x^2+250 e^x x^2+25 e^{2 x} x^2-50 x^3+\left (-250 x-100 e^x x-10 e^{2 x} x+20 x^2\right ) \log \left (x^2\right )+\left (25+10 e^x+e^{2 x}-2 x\right ) \log ^2\left (x^2\right )\right )} \, dx=-\frac {15 \, x}{5 \, x \log \left (x\right ) - 2 \, \log \left (x\right )^{2} - {\left (5 \, x - 2 \, \log \left (x\right )\right )} \log \left (-2 \, x + e^{\left (2 \, x\right )} + 10 \, e^{x} + 25\right )} \] Input:

integrate((((-15*exp(x)^2-150*exp(x)+30*x-375)*log(x^2)+30*exp(x)^2+300*ex 
p(x)-60*x+750)*log((exp(x)^2+10*exp(x)+25-2*x)/x)+((30*x-15)*exp(x)^2+(150 
*x-150)*exp(x)-375)*log(x^2)+(-150*x^2+75*x)*exp(x)^2+(-750*x^2+750*x)*exp 
(x)+1875*x)/((exp(x)^2+10*exp(x)+25-2*x)*log(x^2)^2+(-10*x*exp(x)^2-100*ex 
p(x)*x+20*x^2-250*x)*log(x^2)+25*exp(x)^2*x^2+250*exp(x)*x^2-50*x^3+625*x^ 
2)/log((exp(x)^2+10*exp(x)+25-2*x)/x)^2,x, algorithm="maxima")
 

Output:

-15*x/(5*x*log(x) - 2*log(x)^2 - (5*x - 2*log(x))*log(-2*x + e^(2*x) + 10* 
e^x + 25))
 

Giac [A] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.49 \[ \int \frac {1875 x+e^x \left (750 x-750 x^2\right )+e^{2 x} \left (75 x-150 x^2\right )+\left (-375+e^{2 x} (-15+30 x)+e^x (-150+150 x)\right ) \log \left (x^2\right )+\log \left (\frac {25+10 e^x+e^{2 x}-2 x}{x}\right ) \left (750+300 e^x+30 e^{2 x}-60 x+\left (-375-150 e^x-15 e^{2 x}+30 x\right ) \log \left (x^2\right )\right )}{\log ^2\left (\frac {25+10 e^x+e^{2 x}-2 x}{x}\right ) \left (625 x^2+250 e^x x^2+25 e^{2 x} x^2-50 x^3+\left (-250 x-100 e^x x-10 e^{2 x} x+20 x^2\right ) \log \left (x^2\right )+\left (25+10 e^x+e^{2 x}-2 x\right ) \log ^2\left (x^2\right )\right )} \, dx=-\frac {15 \, x}{5 \, x \log \left (x\right ) - 2 \, \log \left (x\right )^{2} - 5 \, x \log \left (-2 \, x + e^{\left (2 \, x\right )} + 10 \, e^{x} + 25\right ) + 2 \, \log \left (x\right ) \log \left (-2 \, x + e^{\left (2 \, x\right )} + 10 \, e^{x} + 25\right )} \] Input:

integrate((((-15*exp(x)^2-150*exp(x)+30*x-375)*log(x^2)+30*exp(x)^2+300*ex 
p(x)-60*x+750)*log((exp(x)^2+10*exp(x)+25-2*x)/x)+((30*x-15)*exp(x)^2+(150 
*x-150)*exp(x)-375)*log(x^2)+(-150*x^2+75*x)*exp(x)^2+(-750*x^2+750*x)*exp 
(x)+1875*x)/((exp(x)^2+10*exp(x)+25-2*x)*log(x^2)^2+(-10*x*exp(x)^2-100*ex 
p(x)*x+20*x^2-250*x)*log(x^2)+25*exp(x)^2*x^2+250*exp(x)*x^2-50*x^3+625*x^ 
2)/log((exp(x)^2+10*exp(x)+25-2*x)/x)^2,x, algorithm="giac")
 

Output:

-15*x/(5*x*log(x) - 2*log(x)^2 - 5*x*log(-2*x + e^(2*x) + 10*e^x + 25) + 2 
*log(x)*log(-2*x + e^(2*x) + 10*e^x + 25))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1875 x+e^x \left (750 x-750 x^2\right )+e^{2 x} \left (75 x-150 x^2\right )+\left (-375+e^{2 x} (-15+30 x)+e^x (-150+150 x)\right ) \log \left (x^2\right )+\log \left (\frac {25+10 e^x+e^{2 x}-2 x}{x}\right ) \left (750+300 e^x+30 e^{2 x}-60 x+\left (-375-150 e^x-15 e^{2 x}+30 x\right ) \log \left (x^2\right )\right )}{\log ^2\left (\frac {25+10 e^x+e^{2 x}-2 x}{x}\right ) \left (625 x^2+250 e^x x^2+25 e^{2 x} x^2-50 x^3+\left (-250 x-100 e^x x-10 e^{2 x} x+20 x^2\right ) \log \left (x^2\right )+\left (25+10 e^x+e^{2 x}-2 x\right ) \log ^2\left (x^2\right )\right )} \, dx=\int \frac {1875\,x+{\mathrm {e}}^{2\,x}\,\left (75\,x-150\,x^2\right )+\ln \left (x^2\right )\,\left ({\mathrm {e}}^x\,\left (150\,x-150\right )+{\mathrm {e}}^{2\,x}\,\left (30\,x-15\right )-375\right )+\ln \left (\frac {{\mathrm {e}}^{2\,x}-2\,x+10\,{\mathrm {e}}^x+25}{x}\right )\,\left (30\,{\mathrm {e}}^{2\,x}-60\,x+300\,{\mathrm {e}}^x-\ln \left (x^2\right )\,\left (15\,{\mathrm {e}}^{2\,x}-30\,x+150\,{\mathrm {e}}^x+375\right )+750\right )+{\mathrm {e}}^x\,\left (750\,x-750\,x^2\right )}{{\ln \left (\frac {{\mathrm {e}}^{2\,x}-2\,x+10\,{\mathrm {e}}^x+25}{x}\right )}^2\,\left (250\,x^2\,{\mathrm {e}}^x+25\,x^2\,{\mathrm {e}}^{2\,x}+{\ln \left (x^2\right )}^2\,\left ({\mathrm {e}}^{2\,x}-2\,x+10\,{\mathrm {e}}^x+25\right )+625\,x^2-50\,x^3-\ln \left (x^2\right )\,\left (250\,x+10\,x\,{\mathrm {e}}^{2\,x}+100\,x\,{\mathrm {e}}^x-20\,x^2\right )\right )} \,d x \] Input:

int((1875*x + exp(2*x)*(75*x - 150*x^2) + log(x^2)*(exp(x)*(150*x - 150) + 
 exp(2*x)*(30*x - 15) - 375) + log((exp(2*x) - 2*x + 10*exp(x) + 25)/x)*(3 
0*exp(2*x) - 60*x + 300*exp(x) - log(x^2)*(15*exp(2*x) - 30*x + 150*exp(x) 
 + 375) + 750) + exp(x)*(750*x - 750*x^2))/(log((exp(2*x) - 2*x + 10*exp(x 
) + 25)/x)^2*(250*x^2*exp(x) + 25*x^2*exp(2*x) + log(x^2)^2*(exp(2*x) - 2* 
x + 10*exp(x) + 25) + 625*x^2 - 50*x^3 - log(x^2)*(250*x + 10*x*exp(2*x) + 
 100*x*exp(x) - 20*x^2))),x)
 

Output:

int((1875*x + exp(2*x)*(75*x - 150*x^2) + log(x^2)*(exp(x)*(150*x - 150) + 
 exp(2*x)*(30*x - 15) - 375) + log((exp(2*x) - 2*x + 10*exp(x) + 25)/x)*(3 
0*exp(2*x) - 60*x + 300*exp(x) - log(x^2)*(15*exp(2*x) - 30*x + 150*exp(x) 
 + 375) + 750) + exp(x)*(750*x - 750*x^2))/(log((exp(2*x) - 2*x + 10*exp(x 
) + 25)/x)^2*(250*x^2*exp(x) + 25*x^2*exp(2*x) + log(x^2)^2*(exp(2*x) - 2* 
x + 10*exp(x) + 25) + 625*x^2 - 50*x^3 - log(x^2)*(250*x + 10*x*exp(2*x) + 
 100*x*exp(x) - 20*x^2))), x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {1875 x+e^x \left (750 x-750 x^2\right )+e^{2 x} \left (75 x-150 x^2\right )+\left (-375+e^{2 x} (-15+30 x)+e^x (-150+150 x)\right ) \log \left (x^2\right )+\log \left (\frac {25+10 e^x+e^{2 x}-2 x}{x}\right ) \left (750+300 e^x+30 e^{2 x}-60 x+\left (-375-150 e^x-15 e^{2 x}+30 x\right ) \log \left (x^2\right )\right )}{\log ^2\left (\frac {25+10 e^x+e^{2 x}-2 x}{x}\right ) \left (625 x^2+250 e^x x^2+25 e^{2 x} x^2-50 x^3+\left (-250 x-100 e^x x-10 e^{2 x} x+20 x^2\right ) \log \left (x^2\right )+\left (25+10 e^x+e^{2 x}-2 x\right ) \log ^2\left (x^2\right )\right )} \, dx=-\frac {15 x}{\mathrm {log}\left (\frac {e^{2 x}+10 e^{x}-2 x +25}{x}\right ) \left (\mathrm {log}\left (x^{2}\right )-5 x \right )} \] Input:

int((((-15*exp(x)^2-150*exp(x)+30*x-375)*log(x^2)+30*exp(x)^2+300*exp(x)-6 
0*x+750)*log((exp(x)^2+10*exp(x)+25-2*x)/x)+((30*x-15)*exp(x)^2+(150*x-150 
)*exp(x)-375)*log(x^2)+(-150*x^2+75*x)*exp(x)^2+(-750*x^2+750*x)*exp(x)+18 
75*x)/((exp(x)^2+10*exp(x)+25-2*x)*log(x^2)^2+(-10*x*exp(x)^2-100*exp(x)*x 
+20*x^2-250*x)*log(x^2)+25*exp(x)^2*x^2+250*exp(x)*x^2-50*x^3+625*x^2)/log 
((exp(x)^2+10*exp(x)+25-2*x)/x)^2,x)
 

Output:

( - 15*x)/(log((e**(2*x) + 10*e**x - 2*x + 25)/x)*(log(x**2) - 5*x))