\(\int \frac {e^{-2 x} ((-72+48 x-8 x^2+(18-12 x+2 x^2+18 x^4-12 x^5+2 x^6) \log (\frac {2+2 x^4}{x^4})) \log (x \log (\frac {2+2 x^4}{x^4}))+(-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7) \log (\frac {2+2 x^4}{x^4}) \log ^2(x \log (\frac {2+2 x^4}{x^4})))}{(x+x^5) \log (\frac {2+2 x^4}{x^4})} \, dx\) [724]

Optimal result

Integrand size = 151, antiderivative size = 26 \[ \int \frac {e^{-2 x} \left (\left (-72+48 x-8 x^2+\left (18-12 x+2 x^2+18 x^4-12 x^5+2 x^6\right ) \log \left (\frac {2+2 x^4}{x^4}\right )\right ) \log \left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )+\left (-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7\right ) \log \left (\frac {2+2 x^4}{x^4}\right ) \log ^2\left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )\right )}{\left (x+x^5\right ) \log \left (\frac {2+2 x^4}{x^4}\right )} \, dx=3+e^{-2 x} (-3+x)^2 \log ^2\left (x \log \left (2+\frac {2}{x^4}\right )\right ) \] Output:

3+(-3+x)^2*ln(ln(2/x^4+2)*x)^2/exp(x)^2
 

Mathematica [F]

\[ \int \frac {e^{-2 x} \left (\left (-72+48 x-8 x^2+\left (18-12 x+2 x^2+18 x^4-12 x^5+2 x^6\right ) \log \left (\frac {2+2 x^4}{x^4}\right )\right ) \log \left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )+\left (-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7\right ) \log \left (\frac {2+2 x^4}{x^4}\right ) \log ^2\left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )\right )}{\left (x+x^5\right ) \log \left (\frac {2+2 x^4}{x^4}\right )} \, dx=\int \frac {e^{-2 x} \left (\left (-72+48 x-8 x^2+\left (18-12 x+2 x^2+18 x^4-12 x^5+2 x^6\right ) \log \left (\frac {2+2 x^4}{x^4}\right )\right ) \log \left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )+\left (-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7\right ) \log \left (\frac {2+2 x^4}{x^4}\right ) \log ^2\left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )\right )}{\left (x+x^5\right ) \log \left (\frac {2+2 x^4}{x^4}\right )} \, dx \] Input:

Integrate[((-72 + 48*x - 8*x^2 + (18 - 12*x + 2*x^2 + 18*x^4 - 12*x^5 + 2* 
x^6)*Log[(2 + 2*x^4)/x^4])*Log[x*Log[(2 + 2*x^4)/x^4]] + (-24*x + 14*x^2 - 
 2*x^3 - 24*x^5 + 14*x^6 - 2*x^7)*Log[(2 + 2*x^4)/x^4]*Log[x*Log[(2 + 2*x^ 
4)/x^4]]^2)/(E^(2*x)*(x + x^5)*Log[(2 + 2*x^4)/x^4]),x]
 

Output:

Integrate[((-72 + 48*x - 8*x^2 + (18 - 12*x + 2*x^2 + 18*x^4 - 12*x^5 + 2* 
x^6)*Log[(2 + 2*x^4)/x^4])*Log[x*Log[(2 + 2*x^4)/x^4]] + (-24*x + 14*x^2 - 
 2*x^3 - 24*x^5 + 14*x^6 - 2*x^7)*Log[(2 + 2*x^4)/x^4]*Log[x*Log[(2 + 2*x^ 
4)/x^4]]^2)/(E^(2*x)*(x + x^5)*Log[(2 + 2*x^4)/x^4]), 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.

\(\displaystyle \int \frac {e^{-2 x} \left (\left (-8 x^2+\left (2 x^6-12 x^5+18 x^4+2 x^2-12 x+18\right ) \log \left (\frac {2 x^4+2}{x^4}\right )+48 x-72\right ) \log \left (x \log \left (\frac {2 x^4+2}{x^4}\right )\right )+\left (-2 x^7+14 x^6-24 x^5-2 x^3+14 x^2-24 x\right ) \log \left (\frac {2 x^4+2}{x^4}\right ) \log ^2\left (x \log \left (\frac {2 x^4+2}{x^4}\right )\right )\right )}{\left (x^5+x\right ) \log \left (\frac {2 x^4+2}{x^4}\right )} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {e^{-2 x} \left (\left (-8 x^2+\left (2 x^6-12 x^5+18 x^4+2 x^2-12 x+18\right ) \log \left (\frac {2 x^4+2}{x^4}\right )+48 x-72\right ) \log \left (x \log \left (\frac {2 x^4+2}{x^4}\right )\right )+\left (-2 x^7+14 x^6-24 x^5-2 x^3+14 x^2-24 x\right ) \log \left (\frac {2 x^4+2}{x^4}\right ) \log ^2\left (x \log \left (\frac {2 x^4+2}{x^4}\right )\right )\right )}{x \left (x^4+1\right ) \log \left (\frac {2 x^4+2}{x^4}\right )}dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7299

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

Input:

Int[((-72 + 48*x - 8*x^2 + (18 - 12*x + 2*x^2 + 18*x^4 - 12*x^5 + 2*x^6)*L 
og[(2 + 2*x^4)/x^4])*Log[x*Log[(2 + 2*x^4)/x^4]] + (-24*x + 14*x^2 - 2*x^3 
 - 24*x^5 + 14*x^6 - 2*x^7)*Log[(2 + 2*x^4)/x^4]*Log[x*Log[(2 + 2*x^4)/x^4 
]]^2)/(E^(2*x)*(x + x^5)*Log[(2 + 2*x^4)/x^4]),x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(65\) vs. \(2(25)=50\).

Time = 65.48 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.54

method	result	size
parallelrisch	\(\frac {\left (2 x^{2} {\ln \left (x \ln \left (\frac {2 x^{4}+2}{x^{4}}\right )\right )}^{2}-12 {\ln \left (x \ln \left (\frac {2 x^{4}+2}{x^{4}}\right )\right )}^{2} x +18 {\ln \left (x \ln \left (\frac {2 x^{4}+2}{x^{4}}\right )\right )}^{2}\right ) {\mathrm e}^{-2 x}}{2}\)	\(66\)
risch	\(\text {Expression too large to display}\)	\(110816\)

Input:

int(((-2*x^7+14*x^6-24*x^5-2*x^3+14*x^2-24*x)*ln((2*x^4+2)/x^4)*ln(x*ln((2 
*x^4+2)/x^4))^2+((2*x^6-12*x^5+18*x^4+2*x^2-12*x+18)*ln((2*x^4+2)/x^4)-8*x 
^2+48*x-72)*ln(x*ln((2*x^4+2)/x^4)))/(x^5+x)/exp(x)^2/ln((2*x^4+2)/x^4),x, 
method=_RETURNVERBOSE)
 

Output:

1/2*(2*x^2*ln(x*ln(2*(x^4+1)/x^4))^2-12*ln(x*ln(2*(x^4+1)/x^4))^2*x+18*ln( 
x*ln(2*(x^4+1)/x^4))^2)/exp(x)^2
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {e^{-2 x} \left (\left (-72+48 x-8 x^2+\left (18-12 x+2 x^2+18 x^4-12 x^5+2 x^6\right ) \log \left (\frac {2+2 x^4}{x^4}\right )\right ) \log \left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )+\left (-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7\right ) \log \left (\frac {2+2 x^4}{x^4}\right ) \log ^2\left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )\right )}{\left (x+x^5\right ) \log \left (\frac {2+2 x^4}{x^4}\right )} \, dx={\left (x^{2} - 6 \, x + 9\right )} e^{\left (-2 \, x\right )} \log \left (x \log \left (\frac {2 \, {\left (x^{4} + 1\right )}}{x^{4}}\right )\right )^{2} \] Input:

integrate(((-2*x^7+14*x^6-24*x^5-2*x^3+14*x^2-24*x)*log((2*x^4+2)/x^4)*log 
(x*log((2*x^4+2)/x^4))^2+((2*x^6-12*x^5+18*x^4+2*x^2-12*x+18)*log((2*x^4+2 
)/x^4)-8*x^2+48*x-72)*log(x*log((2*x^4+2)/x^4)))/(x^5+x)/exp(x)^2/log((2*x 
^4+2)/x^4),x, algorithm="fricas")
 

Output:

(x^2 - 6*x + 9)*e^(-2*x)*log(x*log(2*(x^4 + 1)/x^4))^2
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{-2 x} \left (\left (-72+48 x-8 x^2+\left (18-12 x+2 x^2+18 x^4-12 x^5+2 x^6\right ) \log \left (\frac {2+2 x^4}{x^4}\right )\right ) \log \left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )+\left (-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7\right ) \log \left (\frac {2+2 x^4}{x^4}\right ) \log ^2\left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )\right )}{\left (x+x^5\right ) \log \left (\frac {2+2 x^4}{x^4}\right )} \, dx=\text {Timed out} \] Input:

integrate(((-2*x**7+14*x**6-24*x**5-2*x**3+14*x**2-24*x)*ln((2*x**4+2)/x** 
4)*ln(x*ln((2*x**4+2)/x**4))**2+((2*x**6-12*x**5+18*x**4+2*x**2-12*x+18)*l 
n((2*x**4+2)/x**4)-8*x**2+48*x-72)*ln(x*ln((2*x**4+2)/x**4)))/(x**5+x)/exp 
(x)**2/ln((2*x**4+2)/x**4),x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (25) = 50\).

Time = 0.20 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.96 \[ \int \frac {e^{-2 x} \left (\left (-72+48 x-8 x^2+\left (18-12 x+2 x^2+18 x^4-12 x^5+2 x^6\right ) \log \left (\frac {2+2 x^4}{x^4}\right )\right ) \log \left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )+\left (-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7\right ) \log \left (\frac {2+2 x^4}{x^4}\right ) \log ^2\left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )\right )}{\left (x+x^5\right ) \log \left (\frac {2+2 x^4}{x^4}\right )} \, dx={\left (x^{2} - 6 \, x + 9\right )} e^{\left (-2 \, x\right )} \log \left (x\right )^{2} + 2 \, {\left (x^{2} - 6 \, x + 9\right )} e^{\left (-2 \, x\right )} \log \left (x\right ) \log \left (\log \left (2\right ) + \log \left (x^{4} + 1\right ) - 4 \, \log \left (x\right )\right ) + {\left (x^{2} - 6 \, x + 9\right )} e^{\left (-2 \, x\right )} \log \left (\log \left (2\right ) + \log \left (x^{4} + 1\right ) - 4 \, \log \left (x\right )\right )^{2} \] Input:

integrate(((-2*x^7+14*x^6-24*x^5-2*x^3+14*x^2-24*x)*log((2*x^4+2)/x^4)*log 
(x*log((2*x^4+2)/x^4))^2+((2*x^6-12*x^5+18*x^4+2*x^2-12*x+18)*log((2*x^4+2 
)/x^4)-8*x^2+48*x-72)*log(x*log((2*x^4+2)/x^4)))/(x^5+x)/exp(x)^2/log((2*x 
^4+2)/x^4),x, algorithm="maxima")
 

Output:

(x^2 - 6*x + 9)*e^(-2*x)*log(x)^2 + 2*(x^2 - 6*x + 9)*e^(-2*x)*log(x)*log( 
log(2) + log(x^4 + 1) - 4*log(x)) + (x^2 - 6*x + 9)*e^(-2*x)*log(log(2) + 
log(x^4 + 1) - 4*log(x))^2
 

Giac [F(-1)]

Timed out. \[ \int \frac {e^{-2 x} \left (\left (-72+48 x-8 x^2+\left (18-12 x+2 x^2+18 x^4-12 x^5+2 x^6\right ) \log \left (\frac {2+2 x^4}{x^4}\right )\right ) \log \left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )+\left (-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7\right ) \log \left (\frac {2+2 x^4}{x^4}\right ) \log ^2\left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )\right )}{\left (x+x^5\right ) \log \left (\frac {2+2 x^4}{x^4}\right )} \, dx=\text {Timed out} \] Input:

integrate(((-2*x^7+14*x^6-24*x^5-2*x^3+14*x^2-24*x)*log((2*x^4+2)/x^4)*log 
(x*log((2*x^4+2)/x^4))^2+((2*x^6-12*x^5+18*x^4+2*x^2-12*x+18)*log((2*x^4+2 
)/x^4)-8*x^2+48*x-72)*log(x*log((2*x^4+2)/x^4)))/(x^5+x)/exp(x)^2/log((2*x 
^4+2)/x^4),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 3.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-2 x} \left (\left (-72+48 x-8 x^2+\left (18-12 x+2 x^2+18 x^4-12 x^5+2 x^6\right ) \log \left (\frac {2+2 x^4}{x^4}\right )\right ) \log \left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )+\left (-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7\right ) \log \left (\frac {2+2 x^4}{x^4}\right ) \log ^2\left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )\right )}{\left (x+x^5\right ) \log \left (\frac {2+2 x^4}{x^4}\right )} \, dx={\ln \left (x\,\ln \left (\frac {2\,\left (x^4+1\right )}{x^4}\right )\right )}^2\,{\mathrm {e}}^{-2\,x}\,{\left (x-3\right )}^2 \] Input:

int((exp(-2*x)*(log(x*log((2*x^4 + 2)/x^4))*(48*x + log((2*x^4 + 2)/x^4)*( 
2*x^2 - 12*x + 18*x^4 - 12*x^5 + 2*x^6 + 18) - 8*x^2 - 72) - log((2*x^4 + 
2)/x^4)*log(x*log((2*x^4 + 2)/x^4))^2*(24*x - 14*x^2 + 2*x^3 + 24*x^5 - 14 
*x^6 + 2*x^7)))/(log((2*x^4 + 2)/x^4)*(x + x^5)),x)
 

Output:

log(x*log((2*(x^4 + 1))/x^4))^2*exp(-2*x)*(x - 3)^2
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27 \[ \int \frac {e^{-2 x} \left (\left (-72+48 x-8 x^2+\left (18-12 x+2 x^2+18 x^4-12 x^5+2 x^6\right ) \log \left (\frac {2+2 x^4}{x^4}\right )\right ) \log \left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )+\left (-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7\right ) \log \left (\frac {2+2 x^4}{x^4}\right ) \log ^2\left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )\right )}{\left (x+x^5\right ) \log \left (\frac {2+2 x^4}{x^4}\right )} \, dx=\frac {{\mathrm {log}\left (\mathrm {log}\left (\frac {2 x^{4}+2}{x^{4}}\right ) x \right )}^{2} \left (x^{2}-6 x +9\right )}{e^{2 x}} \] Input:

int(((-2*x^7+14*x^6-24*x^5-2*x^3+14*x^2-24*x)*log((2*x^4+2)/x^4)*log(x*log 
((2*x^4+2)/x^4))^2+((2*x^6-12*x^5+18*x^4+2*x^2-12*x+18)*log((2*x^4+2)/x^4) 
-8*x^2+48*x-72)*log(x*log((2*x^4+2)/x^4)))/(x^5+x)/exp(x)^2/log((2*x^4+2)/ 
x^4),x)
 

Output:

(log(log((2*x**4 + 2)/x**4)*x)**2*(x**2 - 6*x + 9))/e**(2*x)