\(\int \frac {4-128 x-8 x^2+386 x^3-60 x^4-258 x^5+190 x^6+64 x^7+e^x (-128+256 x^2+64 x^3-128 x^4-64 x^5+192 x^6)+(4 x^2-128 x^3-4 x^4+254 x^5+64 x^6+e^x (-128 x^2+128 x^4+192 x^5)) \log (\frac {1}{256} (x^2+1024 e^{2 x} x^2-64 x^3+1024 x^4+e^x (-64 x^2+2048 x^3)))}{-x^5+32 e^x x^5+32 x^6} \, dx\) [2971]

Optimal result

Integrand size = 179, antiderivative size = 25 \[ \int \frac {4-128 x-8 x^2+386 x^3-60 x^4-258 x^5+190 x^6+64 x^7+e^x \left (-128+256 x^2+64 x^3-128 x^4-64 x^5+192 x^6\right )+\left (4 x^2-128 x^3-4 x^4+254 x^5+64 x^6+e^x \left (-128 x^2+128 x^4+192 x^5\right )\right ) \log \left (\frac {1}{256} \left (x^2+1024 e^{2 x} x^2-64 x^3+1024 x^4+e^x \left (-64 x^2+2048 x^3\right )\right )\right )}{-x^5+32 e^x x^5+32 x^6} \, dx=\left (-1+\frac {1}{x^2}+x+\log \left (\left (\frac {x}{16}-2 x \left (e^x+x\right )\right )^2\right )\right )^2 \] Output:

(1/x^2+ln((1/16*x-2*(exp(x)+x)*x)^2)+x-1)^2
                                                                                    
                                                                                    
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(89\) vs. \(2(25)=50\).

Time = 0.07 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.56 \[ \int \frac {4-128 x-8 x^2+386 x^3-60 x^4-258 x^5+190 x^6+64 x^7+e^x \left (-128+256 x^2+64 x^3-128 x^4-64 x^5+192 x^6\right )+\left (4 x^2-128 x^3-4 x^4+254 x^5+64 x^6+e^x \left (-128 x^2+128 x^4+192 x^5\right )\right ) \log \left (\frac {1}{256} \left (x^2+1024 e^{2 x} x^2-64 x^3+1024 x^4+e^x \left (-64 x^2+2048 x^3\right )\right )\right )}{-x^5+32 e^x x^5+32 x^6} \, dx=\frac {1}{x^4}-\frac {2}{x^2}+\frac {2}{x}-2 x+x^2-4 \log \left (1-32 e^x-32 x\right )-4 \log (x)+\frac {2 \left (1+x^3\right ) \log \left (\frac {1}{256} x^2 \left (-1+32 e^x+32 x\right )^2\right )}{x^2}+\log ^2\left (\frac {1}{256} x^2 \left (-1+32 e^x+32 x\right )^2\right ) \] Input:

Integrate[(4 - 128*x - 8*x^2 + 386*x^3 - 60*x^4 - 258*x^5 + 190*x^6 + 64*x 
^7 + E^x*(-128 + 256*x^2 + 64*x^3 - 128*x^4 - 64*x^5 + 192*x^6) + (4*x^2 - 
 128*x^3 - 4*x^4 + 254*x^5 + 64*x^6 + E^x*(-128*x^2 + 128*x^4 + 192*x^5))* 
Log[(x^2 + 1024*E^(2*x)*x^2 - 64*x^3 + 1024*x^4 + E^x*(-64*x^2 + 2048*x^3) 
)/256])/(-x^5 + 32*E^x*x^5 + 32*x^6),x]
 

Output:

x^(-4) - 2/x^2 + 2/x - 2*x + x^2 - 4*Log[1 - 32*E^x - 32*x] - 4*Log[x] + ( 
2*(1 + x^3)*Log[(x^2*(-1 + 32*E^x + 32*x)^2)/256])/x^2 + Log[(x^2*(-1 + 32 
*E^x + 32*x)^2)/256]^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.

\(\displaystyle \int \frac {64 x^7+190 x^6-258 x^5-60 x^4+386 x^3-8 x^2+e^x \left (192 x^6-64 x^5-128 x^4+64 x^3+256 x^2-128\right )+\left (64 x^6+254 x^5-4 x^4-128 x^3+4 x^2+e^x \left (192 x^5+128 x^4-128 x^2\right )\right ) \log \left (\frac {1}{256} \left (1024 x^4-64 x^3+1024 e^{2 x} x^2+x^2+e^x \left (2048 x^3-64 x^2\right )\right )\right )-128 x+4}{32 x^6+32 e^x x^5-x^5} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 \left (32 x^4+127 x^3-2 x^2+32 e^x \left (3 x^3+2 x^2-2\right )-64 x+2\right ) \left (-x^3+x^2-x^2 \log \left (\frac {1}{256} x^2 \left (32 x+32 e^x-1\right )^2\right )-1\right )}{\left (-32 x-32 e^x+1\right ) x^5}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \int -\frac {\left (32 x^4+127 x^3-2 x^2-64 x-32 e^x \left (-3 x^3-2 x^2+2\right )+2\right ) \left (x^3+\log \left (\frac {1}{256} \left (-32 x-32 e^x+1\right )^2 x^2\right ) x^2-x^2+1\right )}{\left (-32 x-32 e^x+1\right ) x^5}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -2 \int \frac {\left (32 x^4+127 x^3-2 x^2-64 x-32 e^x \left (-3 x^3-2 x^2+2\right )+2\right ) \left (x^3+\log \left (\frac {1}{256} \left (-32 x-32 e^x+1\right )^2 x^2\right ) x^2-x^2+1\right )}{\left (-32 x-32 e^x+1\right ) x^5}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {2 (32 x-33) \left (x^3+\log \left (\frac {1}{256} x^2 \left (32 x+32 e^x-1\right )^2\right ) x^2-x^2+1\right )}{x^2 \left (32 x+32 e^x-1\right )}-\frac {\left (3 x^3+2 x^2-2\right ) \left (x^3+\log \left (\frac {1}{256} x^2 \left (32 x+32 e^x-1\right )^2\right ) x^2-x^2+1\right )}{x^5}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 \left (-128 \int \frac {x^2}{32 x+32 e^x-1}dx-66 \log \left (\frac {1}{256} \left (-32 x-32 e^x+1\right )^2 x^2\right ) \int \frac {1}{32 x+32 e^x-1}dx+64 \log \left (\frac {1}{256} \left (-32 x-32 e^x+1\right )^2 x^2\right ) \int \frac {x}{32 x+32 e^x-1}dx-2 \int \frac {\log \left (\frac {1}{256} x^2 \left (32 x+32 e^x-1\right )^2\right )}{x}dx+66 \int \frac {1}{32 x+32 e^x-1}dx+68 \int \frac {x}{32 x+32 e^x-1}dx+132 \int \int \frac {1}{32 x+32 e^x-1}dxdx+132 \int \frac {\int \frac {1}{32 x+32 e^x-1}dx}{x}dx+4356 \int \frac {\int \frac {1}{32 x+32 e^x-1}dx}{32 x+32 e^x-1}dx-4224 \int \frac {x \int \frac {1}{32 x+32 e^x-1}dx}{32 x+32 e^x-1}dx-128 \int \int \frac {x}{32 x+32 e^x-1}dxdx-128 \int \frac {\int \frac {x}{32 x+32 e^x-1}dx}{x}dx-4224 \int \frac {\int \frac {x}{32 x+32 e^x-1}dx}{32 x+32 e^x-1}dx+4096 \int \frac {x \int \frac {x}{32 x+32 e^x-1}dx}{32 x+32 e^x-1}dx-\frac {1}{2 x^4}-\frac {3 x^2}{2}-\frac {(x+1)^2}{x^2}+\frac {2}{x^2}-3 x \log \left (\frac {1}{256} \left (-32 x-32 e^x+1\right )^2 x^2\right )-\frac {\log \left (\frac {1}{256} \left (-32 x-32 e^x+1\right )^2 x^2\right )}{x^2}+x+3 (x+1)^2+\frac {1}{x}+2 \log (x)\right )\)

Input:

Int[(4 - 128*x - 8*x^2 + 386*x^3 - 60*x^4 - 258*x^5 + 190*x^6 + 64*x^7 + E 
^x*(-128 + 256*x^2 + 64*x^3 - 128*x^4 - 64*x^5 + 192*x^6) + (4*x^2 - 128*x 
^3 - 4*x^4 + 254*x^5 + 64*x^6 + E^x*(-128*x^2 + 128*x^4 + 192*x^5))*Log[(x 
^2 + 1024*E^(2*x)*x^2 - 64*x^3 + 1024*x^4 + E^x*(-64*x^2 + 2048*x^3))/256] 
)/(-x^5 + 32*E^x*x^5 + 32*x^6),x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(196\) vs. \(2(22)=44\).

Time = 1.15 (sec) , antiderivative size = 197, normalized size of antiderivative = 7.88

method	result	size
parallelrisch	\(-\frac {-2048-2048 x^{6}-4096 \ln \left (\frac {x^{2} \left (2048 \,{\mathrm e}^{x} x +1024 x^{2}+1024 \,{\mathrm e}^{2 x}-64 \,{\mathrm e}^{x}-64 x +1\right )}{256}\right ) x^{5}-2048 \ln \left (\frac {x^{2} \left (2048 \,{\mathrm e}^{x} x +1024 x^{2}+1024 \,{\mathrm e}^{2 x}-64 \,{\mathrm e}^{x}-64 x +1\right )}{256}\right )^{2} x^{4}+8448 x^{4} \ln \left (x \right )+8448 \ln \left (x +{\mathrm e}^{x}-\frac {1}{32}\right ) x^{4}+4096 x^{5}-128 \ln \left (\frac {x^{2} \left (2048 \,{\mathrm e}^{x} x +1024 x^{2}+1024 \,{\mathrm e}^{2 x}-64 \,{\mathrm e}^{x}-64 x +1\right )}{256}\right ) x^{4}+128 x^{4}-4096 x^{3}-4096 \ln \left (\frac {x^{2} \left (2048 \,{\mathrm e}^{x} x +1024 x^{2}+1024 \,{\mathrm e}^{2 x}-64 \,{\mathrm e}^{x}-64 x +1\right )}{256}\right ) x^{2}+4096 x^{2}}{2048 x^{4}}\)	\(197\)
risch	\(\text {Expression too large to display}\)	\(1264\)

Input:

int((((192*x^5+128*x^4-128*x^2)*exp(x)+64*x^6+254*x^5-4*x^4-128*x^3+4*x^2) 
*ln(4*exp(x)^2*x^2+1/256*(2048*x^3-64*x^2)*exp(x)+4*x^4-1/4*x^3+1/256*x^2) 
+(192*x^6-64*x^5-128*x^4+64*x^3+256*x^2-128)*exp(x)+64*x^7+190*x^6-258*x^5 
-60*x^4+386*x^3-8*x^2-128*x+4)/(32*x^5*exp(x)+32*x^6-x^5),x,method=_RETURN 
VERBOSE)
 

Output:

-1/2048*(-2048-2048*x^6-4096*ln(1/256*x^2*(1024*exp(x)^2+2048*exp(x)*x+102 
4*x^2-64*exp(x)-64*x+1))*x^5-2048*ln(1/256*x^2*(1024*exp(x)^2+2048*exp(x)* 
x+1024*x^2-64*exp(x)-64*x+1))^2*x^4+8448*x^4*ln(x)+8448*ln(x+exp(x)-1/32)* 
x^4+4096*x^5-128*ln(1/256*x^2*(1024*exp(x)^2+2048*exp(x)*x+1024*x^2-64*exp 
(x)-64*x+1))*x^4+128*x^4-4096*x^3-4096*ln(1/256*x^2*(1024*exp(x)^2+2048*ex 
p(x)*x+1024*x^2-64*exp(x)-64*x+1))*x^2+4096*x^2)/x^4
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (24) = 48\).

Time = 0.08 (sec) , antiderivative size = 126, normalized size of antiderivative = 5.04 \[ \int \frac {4-128 x-8 x^2+386 x^3-60 x^4-258 x^5+190 x^6+64 x^7+e^x \left (-128+256 x^2+64 x^3-128 x^4-64 x^5+192 x^6\right )+\left (4 x^2-128 x^3-4 x^4+254 x^5+64 x^6+e^x \left (-128 x^2+128 x^4+192 x^5\right )\right ) \log \left (\frac {1}{256} \left (x^2+1024 e^{2 x} x^2-64 x^3+1024 x^4+e^x \left (-64 x^2+2048 x^3\right )\right )\right )}{-x^5+32 e^x x^5+32 x^6} \, dx=\frac {x^{6} + x^{4} \log \left (4 \, x^{4} - \frac {1}{4} \, x^{3} + 4 \, x^{2} e^{\left (2 \, x\right )} + \frac {1}{256} \, x^{2} + \frac {1}{4} \, {\left (32 \, x^{3} - x^{2}\right )} e^{x}\right )^{2} - 2 \, x^{5} + 2 \, x^{3} - 2 \, x^{2} + 2 \, {\left (x^{5} - x^{4} + x^{2}\right )} \log \left (4 \, x^{4} - \frac {1}{4} \, x^{3} + 4 \, x^{2} e^{\left (2 \, x\right )} + \frac {1}{256} \, x^{2} + \frac {1}{4} \, {\left (32 \, x^{3} - x^{2}\right )} e^{x}\right ) + 1}{x^{4}} \] Input:

integrate((((192*x^5+128*x^4-128*x^2)*exp(x)+64*x^6+254*x^5-4*x^4-128*x^3+ 
4*x^2)*log(4*exp(x)^2*x^2+1/256*(2048*x^3-64*x^2)*exp(x)+4*x^4-1/4*x^3+1/2 
56*x^2)+(192*x^6-64*x^5-128*x^4+64*x^3+256*x^2-128)*exp(x)+64*x^7+190*x^6- 
258*x^5-60*x^4+386*x^3-8*x^2-128*x+4)/(32*x^5*exp(x)+32*x^6-x^5),x, algori 
thm="fricas")
 

Output:

(x^6 + x^4*log(4*x^4 - 1/4*x^3 + 4*x^2*e^(2*x) + 1/256*x^2 + 1/4*(32*x^3 - 
 x^2)*e^x)^2 - 2*x^5 + 2*x^3 - 2*x^2 + 2*(x^5 - x^4 + x^2)*log(4*x^4 - 1/4 
*x^3 + 4*x^2*e^(2*x) + 1/256*x^2 + 1/4*(32*x^3 - x^2)*e^x) + 1)/x^4
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (26) = 52\).

Time = 0.33 (sec) , antiderivative size = 131, normalized size of antiderivative = 5.24 \[ \int \frac {4-128 x-8 x^2+386 x^3-60 x^4-258 x^5+190 x^6+64 x^7+e^x \left (-128+256 x^2+64 x^3-128 x^4-64 x^5+192 x^6\right )+\left (4 x^2-128 x^3-4 x^4+254 x^5+64 x^6+e^x \left (-128 x^2+128 x^4+192 x^5\right )\right ) \log \left (\frac {1}{256} \left (x^2+1024 e^{2 x} x^2-64 x^3+1024 x^4+e^x \left (-64 x^2+2048 x^3\right )\right )\right )}{-x^5+32 e^x x^5+32 x^6} \, dx=x^{2} - 2 x - 4 \log {\left (x \right )} - 4 \log {\left (x + e^{x} - \frac {1}{32} \right )} + \log {\left (4 x^{4} - \frac {x^{3}}{4} + 4 x^{2} e^{2 x} + \frac {x^{2}}{256} + \left (8 x^{3} - \frac {x^{2}}{4}\right ) e^{x} \right )}^{2} + \frac {\left (2 x^{3} + 2\right ) \log {\left (4 x^{4} - \frac {x^{3}}{4} + 4 x^{2} e^{2 x} + \frac {x^{2}}{256} + \left (8 x^{3} - \frac {x^{2}}{4}\right ) e^{x} \right )}}{x^{2}} + \frac {2 x^{3} - 2 x^{2} + 1}{x^{4}} \] Input:

integrate((((192*x**5+128*x**4-128*x**2)*exp(x)+64*x**6+254*x**5-4*x**4-12 
8*x**3+4*x**2)*ln(4*exp(x)**2*x**2+1/256*(2048*x**3-64*x**2)*exp(x)+4*x**4 
-1/4*x**3+1/256*x**2)+(192*x**6-64*x**5-128*x**4+64*x**3+256*x**2-128)*exp 
(x)+64*x**7+190*x**6-258*x**5-60*x**4+386*x**3-8*x**2-128*x+4)/(32*x**5*ex 
p(x)+32*x**6-x**5),x)
 

Output:

x**2 - 2*x - 4*log(x) - 4*log(x + exp(x) - 1/32) + log(4*x**4 - x**3/4 + 4 
*x**2*exp(2*x) + x**2/256 + (8*x**3 - x**2/4)*exp(x))**2 + (2*x**3 + 2)*lo 
g(4*x**4 - x**3/4 + 4*x**2*exp(2*x) + x**2/256 + (8*x**3 - x**2/4)*exp(x)) 
/x**2 + (2*x**3 - 2*x**2 + 1)/x**4
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (24) = 48\).

Time = 0.16 (sec) , antiderivative size = 121, normalized size of antiderivative = 4.84 \[ \int \frac {4-128 x-8 x^2+386 x^3-60 x^4-258 x^5+190 x^6+64 x^7+e^x \left (-128+256 x^2+64 x^3-128 x^4-64 x^5+192 x^6\right )+\left (4 x^2-128 x^3-4 x^4+254 x^5+64 x^6+e^x \left (-128 x^2+128 x^4+192 x^5\right )\right ) \log \left (\frac {1}{256} \left (x^2+1024 e^{2 x} x^2-64 x^3+1024 x^4+e^x \left (-64 x^2+2048 x^3\right )\right )\right )}{-x^5+32 e^x x^5+32 x^6} \, dx=\frac {x^{6} - 2 \, x^{5} {\left (8 \, \log \left (2\right ) + 1\right )} + 4 \, x^{4} \log \left (32 \, x + 32 \, e^{x} - 1\right )^{2} + 4 \, x^{4} \log \left (x\right )^{2} + 2 \, x^{3} - 2 \, x^{2} {\left (8 \, \log \left (2\right ) + 1\right )} + 4 \, {\left (x^{5} - x^{4} {\left (8 \, \log \left (2\right ) + 1\right )} + 2 \, x^{4} \log \left (x\right ) + x^{2}\right )} \log \left (32 \, x + 32 \, e^{x} - 1\right ) + 4 \, {\left (x^{5} - x^{4} {\left (8 \, \log \left (2\right ) + 1\right )} + x^{2}\right )} \log \left (x\right ) + 1}{x^{4}} \] Input:

integrate((((192*x^5+128*x^4-128*x^2)*exp(x)+64*x^6+254*x^5-4*x^4-128*x^3+ 
4*x^2)*log(4*exp(x)^2*x^2+1/256*(2048*x^3-64*x^2)*exp(x)+4*x^4-1/4*x^3+1/2 
56*x^2)+(192*x^6-64*x^5-128*x^4+64*x^3+256*x^2-128)*exp(x)+64*x^7+190*x^6- 
258*x^5-60*x^4+386*x^3-8*x^2-128*x+4)/(32*x^5*exp(x)+32*x^6-x^5),x, algori 
thm="maxima")
 

Output:

(x^6 - 2*x^5*(8*log(2) + 1) + 4*x^4*log(32*x + 32*e^x - 1)^2 + 4*x^4*log(x 
)^2 + 2*x^3 - 2*x^2*(8*log(2) + 1) + 4*(x^5 - x^4*(8*log(2) + 1) + 2*x^4*l 
og(x) + x^2)*log(32*x + 32*e^x - 1) + 4*(x^5 - x^4*(8*log(2) + 1) + x^2)*l 
og(x) + 1)/x^4
 

Giac [F]

\[ \int \frac {4-128 x-8 x^2+386 x^3-60 x^4-258 x^5+190 x^6+64 x^7+e^x \left (-128+256 x^2+64 x^3-128 x^4-64 x^5+192 x^6\right )+\left (4 x^2-128 x^3-4 x^4+254 x^5+64 x^6+e^x \left (-128 x^2+128 x^4+192 x^5\right )\right ) \log \left (\frac {1}{256} \left (x^2+1024 e^{2 x} x^2-64 x^3+1024 x^4+e^x \left (-64 x^2+2048 x^3\right )\right )\right )}{-x^5+32 e^x x^5+32 x^6} \, dx=\int { \frac {2 \, {\left (32 \, x^{7} + 95 \, x^{6} - 129 \, x^{5} - 30 \, x^{4} + 193 \, x^{3} - 4 \, x^{2} + 32 \, {\left (3 \, x^{6} - x^{5} - 2 \, x^{4} + x^{3} + 4 \, x^{2} - 2\right )} e^{x} + {\left (32 \, x^{6} + 127 \, x^{5} - 2 \, x^{4} - 64 \, x^{3} + 2 \, x^{2} + 32 \, {\left (3 \, x^{5} + 2 \, x^{4} - 2 \, x^{2}\right )} e^{x}\right )} \log \left (4 \, x^{4} - \frac {1}{4} \, x^{3} + 4 \, x^{2} e^{\left (2 \, x\right )} + \frac {1}{256} \, x^{2} + \frac {1}{4} \, {\left (32 \, x^{3} - x^{2}\right )} e^{x}\right ) - 64 \, x + 2\right )}}{32 \, x^{6} + 32 \, x^{5} e^{x} - x^{5}} \,d x } \] Input:

integrate((((192*x^5+128*x^4-128*x^2)*exp(x)+64*x^6+254*x^5-4*x^4-128*x^3+ 
4*x^2)*log(4*exp(x)^2*x^2+1/256*(2048*x^3-64*x^2)*exp(x)+4*x^4-1/4*x^3+1/2 
56*x^2)+(192*x^6-64*x^5-128*x^4+64*x^3+256*x^2-128)*exp(x)+64*x^7+190*x^6- 
258*x^5-60*x^4+386*x^3-8*x^2-128*x+4)/(32*x^5*exp(x)+32*x^6-x^5),x, algori 
thm="giac")
 

Output:

integrate(2*(32*x^7 + 95*x^6 - 129*x^5 - 30*x^4 + 193*x^3 - 4*x^2 + 32*(3* 
x^6 - x^5 - 2*x^4 + x^3 + 4*x^2 - 2)*e^x + (32*x^6 + 127*x^5 - 2*x^4 - 64* 
x^3 + 2*x^2 + 32*(3*x^5 + 2*x^4 - 2*x^2)*e^x)*log(4*x^4 - 1/4*x^3 + 4*x^2* 
e^(2*x) + 1/256*x^2 + 1/4*(32*x^3 - x^2)*e^x) - 64*x + 2)/(32*x^6 + 32*x^5 
*e^x - x^5), x)
 

Mupad [B] (verification not implemented)

Time = 2.85 (sec) , antiderivative size = 129, normalized size of antiderivative = 5.16 \[ \int \frac {4-128 x-8 x^2+386 x^3-60 x^4-258 x^5+190 x^6+64 x^7+e^x \left (-128+256 x^2+64 x^3-128 x^4-64 x^5+192 x^6\right )+\left (4 x^2-128 x^3-4 x^4+254 x^5+64 x^6+e^x \left (-128 x^2+128 x^4+192 x^5\right )\right ) \log \left (\frac {1}{256} \left (x^2+1024 e^{2 x} x^2-64 x^3+1024 x^4+e^x \left (-64 x^2+2048 x^3\right )\right )\right )}{-x^5+32 e^x x^5+32 x^6} \, dx=\frac {2\,x^3-2\,x^2+1}{x^4}-2\,x+{\ln \left (4\,x^2\,{\mathrm {e}}^{2\,x}-\frac {{\mathrm {e}}^x\,\left (64\,x^2-2048\,x^3\right )}{256}+\frac {x^2}{256}-\frac {x^3}{4}+4\,x^4\right )}^2+\ln \left (4\,x^2\,{\mathrm {e}}^{2\,x}-\frac {{\mathrm {e}}^x\,\left (64\,x^2-2048\,x^3\right )}{256}+\frac {x^2}{256}-\frac {x^3}{4}+4\,x^4\right )\,\left (5\,x-\frac {32\,\left (\frac {3\,x^3}{32}+\frac {x^2}{16}-\frac {1}{16}\right )}{x^2}\right )+x^2 \] Input:

int((log(4*x^2*exp(2*x) - (exp(x)*(64*x^2 - 2048*x^3))/256 + x^2/256 - x^3 
/4 + 4*x^4)*(exp(x)*(128*x^4 - 128*x^2 + 192*x^5) + 4*x^2 - 128*x^3 - 4*x^ 
4 + 254*x^5 + 64*x^6) - 128*x + exp(x)*(256*x^2 + 64*x^3 - 128*x^4 - 64*x^ 
5 + 192*x^6 - 128) - 8*x^2 + 386*x^3 - 60*x^4 - 258*x^5 + 190*x^6 + 64*x^7 
 + 4)/(32*x^5*exp(x) - x^5 + 32*x^6),x)
 

Output:

(2*x^3 - 2*x^2 + 1)/x^4 - 2*x + log(4*x^2*exp(2*x) - (exp(x)*(64*x^2 - 204 
8*x^3))/256 + x^2/256 - x^3/4 + 4*x^4)^2 + log(4*x^2*exp(2*x) - (exp(x)*(6 
4*x^2 - 2048*x^3))/256 + x^2/256 - x^3/4 + 4*x^4)*(5*x - (32*(x^2/16 + (3* 
x^3)/32 - 1/16))/x^2) + x^2
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 217, normalized size of antiderivative = 8.68 \[ \int \frac {4-128 x-8 x^2+386 x^3-60 x^4-258 x^5+190 x^6+64 x^7+e^x \left (-128+256 x^2+64 x^3-128 x^4-64 x^5+192 x^6\right )+\left (4 x^2-128 x^3-4 x^4+254 x^5+64 x^6+e^x \left (-128 x^2+128 x^4+192 x^5\right )\right ) \log \left (\frac {1}{256} \left (x^2+1024 e^{2 x} x^2-64 x^3+1024 x^4+e^x \left (-64 x^2+2048 x^3\right )\right )\right )}{-x^5+32 e^x x^5+32 x^6} \, dx=\frac {\mathrm {log}\left (4 e^{2 x} x^{2}+8 e^{x} x^{3}-\frac {e^{x} x^{2}}{4}+4 x^{4}-\frac {x^{3}}{4}+\frac {x^{2}}{256}\right )^{2} x^{4}+2 \,\mathrm {log}\left (4 e^{2 x} x^{2}+8 e^{x} x^{3}-\frac {e^{x} x^{2}}{4}+4 x^{4}-\frac {x^{3}}{4}+\frac {x^{2}}{256}\right ) x^{5}-2 \,\mathrm {log}\left (4 e^{2 x} x^{2}+8 e^{x} x^{3}-\frac {e^{x} x^{2}}{4}+4 x^{4}-\frac {x^{3}}{4}+\frac {x^{2}}{256}\right ) x^{4}+2 \,\mathrm {log}\left (4 e^{2 x} x^{2}+8 e^{x} x^{3}-\frac {e^{x} x^{2}}{4}+4 x^{4}-\frac {x^{3}}{4}+\frac {x^{2}}{256}\right ) x^{2}+x^{6}-2 x^{5}+2 x^{3}-2 x^{2}+1}{x^{4}} \] Input:

int((((192*x^5+128*x^4-128*x^2)*exp(x)+64*x^6+254*x^5-4*x^4-128*x^3+4*x^2) 
*log(4*exp(x)^2*x^2+1/256*(2048*x^3-64*x^2)*exp(x)+4*x^4-1/4*x^3+1/256*x^2 
)+(192*x^6-64*x^5-128*x^4+64*x^3+256*x^2-128)*exp(x)+64*x^7+190*x^6-258*x^ 
5-60*x^4+386*x^3-8*x^2-128*x+4)/(32*x^5*exp(x)+32*x^6-x^5),x)
 

Output:

(log((1024*e**(2*x)*x**2 + 2048*e**x*x**3 - 64*e**x*x**2 + 1024*x**4 - 64* 
x**3 + x**2)/256)**2*x**4 + 2*log((1024*e**(2*x)*x**2 + 2048*e**x*x**3 - 6 
4*e**x*x**2 + 1024*x**4 - 64*x**3 + x**2)/256)*x**5 - 2*log((1024*e**(2*x) 
*x**2 + 2048*e**x*x**3 - 64*e**x*x**2 + 1024*x**4 - 64*x**3 + x**2)/256)*x 
**4 + 2*log((1024*e**(2*x)*x**2 + 2048*e**x*x**3 - 64*e**x*x**2 + 1024*x** 
4 - 64*x**3 + x**2)/256)*x**2 + x**6 - 2*x**5 + 2*x**3 - 2*x**2 + 1)/x**4