\(\int \frac {e^{\frac {1}{3} (9-x \log (\frac {-5 x^2+\log (5)-x^2 \log (-4+2 x+x^2)}{5 x+x \log (-4+2 x+x^2)}))} (100 x^2-50 x^3-25 x^4+(20-12 x-7 x^2) \log (5)+(40 x^2-20 x^3-10 x^4+(4-2 x-x^2) \log (5)) \log (-4+2 x+x^2)+(4 x^2-2 x^3-x^4) \log ^2(-4+2 x+x^2)+(100 x^2-50 x^3-25 x^4+(-20+10 x+5 x^2) \log (5)+(40 x^2-20 x^3-10 x^4+(-4+2 x+x^2) \log (5)) \log (-4+2 x+x^2)+(4 x^2-2 x^3-x^4) \log ^2(-4+2 x+x^2)) \log (\frac {-5 x^2+\log (5)-x^2 \log (-4+2 x+x^2)}{5 x+x \log (-4+2 x+x^2)}))}{-300 x^2+150 x^3+75 x^4+(60-30 x-15 x^2) \log (5)+(-120 x^2+60 x^3+30 x^4+(12-6 x-3 x^2) \log (5)) \log (-4+2 x+x^2)+(-12 x^2+6 x^3+3 x^4) \log ^2(-4+2 x+x^2)} \, dx\) [2263]

Optimal result

Integrand size = 383, antiderivative size = 33 \[ \int \frac {e^{\frac {1}{3} \left (9-x \log \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{5 x+x \log \left (-4+2 x+x^2\right )}\right )\right )} \left (100 x^2-50 x^3-25 x^4+\left (20-12 x-7 x^2\right ) \log (5)+\left (40 x^2-20 x^3-10 x^4+\left (4-2 x-x^2\right ) \log (5)\right ) \log \left (-4+2 x+x^2\right )+\left (4 x^2-2 x^3-x^4\right ) \log ^2\left (-4+2 x+x^2\right )+\left (100 x^2-50 x^3-25 x^4+\left (-20+10 x+5 x^2\right ) \log (5)+\left (40 x^2-20 x^3-10 x^4+\left (-4+2 x+x^2\right ) \log (5)\right ) \log \left (-4+2 x+x^2\right )+\left (4 x^2-2 x^3-x^4\right ) \log ^2\left (-4+2 x+x^2\right )\right ) \log \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{5 x+x \log \left (-4+2 x+x^2\right )}\right )\right )}{-300 x^2+150 x^3+75 x^4+\left (60-30 x-15 x^2\right ) \log (5)+\left (-120 x^2+60 x^3+30 x^4+\left (12-6 x-3 x^2\right ) \log (5)\right ) \log \left (-4+2 x+x^2\right )+\left (-12 x^2+6 x^3+3 x^4\right ) \log ^2\left (-4+2 x+x^2\right )} \, dx=e^{3-\frac {1}{3} x \log \left (-x+\frac {\log (5)}{x \left (5+\log \left (-4+2 x+x^2\right )\right )}\right )} \] Output:

exp(3-1/3*x*ln(ln(5)/x/(ln(x^2+2*x-4)+5)-x))
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48 \[ \int \frac {e^{\frac {1}{3} \left (9-x \log \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{5 x+x \log \left (-4+2 x+x^2\right )}\right )\right )} \left (100 x^2-50 x^3-25 x^4+\left (20-12 x-7 x^2\right ) \log (5)+\left (40 x^2-20 x^3-10 x^4+\left (4-2 x-x^2\right ) \log (5)\right ) \log \left (-4+2 x+x^2\right )+\left (4 x^2-2 x^3-x^4\right ) \log ^2\left (-4+2 x+x^2\right )+\left (100 x^2-50 x^3-25 x^4+\left (-20+10 x+5 x^2\right ) \log (5)+\left (40 x^2-20 x^3-10 x^4+\left (-4+2 x+x^2\right ) \log (5)\right ) \log \left (-4+2 x+x^2\right )+\left (4 x^2-2 x^3-x^4\right ) \log ^2\left (-4+2 x+x^2\right )\right ) \log \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{5 x+x \log \left (-4+2 x+x^2\right )}\right )\right )}{-300 x^2+150 x^3+75 x^4+\left (60-30 x-15 x^2\right ) \log (5)+\left (-120 x^2+60 x^3+30 x^4+\left (12-6 x-3 x^2\right ) \log (5)\right ) \log \left (-4+2 x+x^2\right )+\left (-12 x^2+6 x^3+3 x^4\right ) \log ^2\left (-4+2 x+x^2\right )} \, dx=e^3 \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{x \left (5+\log \left (-4+2 x+x^2\right )\right )}\right )^{-x/3} \] Input:

Integrate[(E^((9 - x*Log[(-5*x^2 + Log[5] - x^2*Log[-4 + 2*x + x^2])/(5*x 
+ x*Log[-4 + 2*x + x^2])])/3)*(100*x^2 - 50*x^3 - 25*x^4 + (20 - 12*x - 7* 
x^2)*Log[5] + (40*x^2 - 20*x^3 - 10*x^4 + (4 - 2*x - x^2)*Log[5])*Log[-4 + 
 2*x + x^2] + (4*x^2 - 2*x^3 - x^4)*Log[-4 + 2*x + x^2]^2 + (100*x^2 - 50* 
x^3 - 25*x^4 + (-20 + 10*x + 5*x^2)*Log[5] + (40*x^2 - 20*x^3 - 10*x^4 + ( 
-4 + 2*x + x^2)*Log[5])*Log[-4 + 2*x + x^2] + (4*x^2 - 2*x^3 - x^4)*Log[-4 
 + 2*x + x^2]^2)*Log[(-5*x^2 + Log[5] - x^2*Log[-4 + 2*x + x^2])/(5*x + x* 
Log[-4 + 2*x + x^2])]))/(-300*x^2 + 150*x^3 + 75*x^4 + (60 - 30*x - 15*x^2 
)*Log[5] + (-120*x^2 + 60*x^3 + 30*x^4 + (12 - 6*x - 3*x^2)*Log[5])*Log[-4 
 + 2*x + x^2] + (-12*x^2 + 6*x^3 + 3*x^4)*Log[-4 + 2*x + x^2]^2),x]
 

Output:

E^3/((-5*x^2 + Log[5] - x^2*Log[-4 + 2*x + x^2])/(x*(5 + Log[-4 + 2*x + x^ 
2])))^(x/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.

\(\displaystyle \int \frac {\left (-25 x^4-50 x^3+100 x^2+\left (-7 x^2-12 x+20\right ) \log (5)+\left (-x^4-2 x^3+4 x^2\right ) \log ^2\left (x^2+2 x-4\right )+\left (-25 x^4-50 x^3+100 x^2+\left (5 x^2+10 x-20\right ) \log (5)+\left (-x^4-2 x^3+4 x^2\right ) \log ^2\left (x^2+2 x-4\right )+\left (-10 x^4-20 x^3+40 x^2+\left (x^2+2 x-4\right ) \log (5)\right ) \log \left (x^2+2 x-4\right )\right ) \log \left (\frac {-5 x^2+x^2 \left (-\log \left (x^2+2 x-4\right )\right )+\log (5)}{x \log \left (x^2+2 x-4\right )+5 x}\right )+\left (-10 x^4-20 x^3+40 x^2+\left (-x^2-2 x+4\right ) \log (5)\right ) \log \left (x^2+2 x-4\right )\right ) \exp \left (\frac {1}{3} \left (9-x \log \left (\frac {-5 x^2+x^2 \left (-\log \left (x^2+2 x-4\right )\right )+\log (5)}{x \log \left (x^2+2 x-4\right )+5 x}\right )\right )\right )}{75 x^4+150 x^3-300 x^2+\left (-15 x^2-30 x+60\right ) \log (5)+\left (3 x^4+6 x^3-12 x^2\right ) \log ^2\left (x^2+2 x-4\right )+\left (30 x^4+60 x^3-120 x^2+\left (-3 x^2-6 x+12\right ) \log (5)\right ) \log \left (x^2+2 x-4\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (25 x^4+50 x^3-100 x^2-\left (-7 x^2-12 x+20\right ) \log (5)-\left (-x^4-2 x^3+4 x^2\right ) \log ^2\left (x^2+2 x-4\right )-\left (-25 x^4-50 x^3+100 x^2+\left (5 x^2+10 x-20\right ) \log (5)+\left (-x^4-2 x^3+4 x^2\right ) \log ^2\left (x^2+2 x-4\right )+\left (-10 x^4-20 x^3+40 x^2+\left (x^2+2 x-4\right ) \log (5)\right ) \log \left (x^2+2 x-4\right )\right ) \log \left (\frac {-5 x^2+x^2 \left (-\log \left (x^2+2 x-4\right )\right )+\log (5)}{x \log \left (x^2+2 x-4\right )+5 x}\right )-\left (-10 x^4-20 x^3+40 x^2+\left (-x^2-2 x+4\right ) \log (5)\right ) \log \left (x^2+2 x-4\right )\right ) \exp \left (\frac {1}{3} \left (9-x \log \left (\frac {-5 x^2+x^2 \left (-\log \left (x^2+2 x-4\right )\right )+\log (5)}{x \log \left (x^2+2 x-4\right )+5 x}\right )\right )\right )}{3 \left (-x^2-2 x+4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (5 x^2+x^2 \log \left (x^2+2 x-4\right )-\log (5)\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \int -\frac {e^3 \left (-\frac {\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)}{\log \left (x^2+2 x-4\right ) x+5 x}\right )^{-x/3} \left (-25 x^4-50 x^3+100 x^2+\left (-x^4-2 x^3+4 x^2\right ) \log ^2\left (x^2+2 x-4\right )+\left (-10 x^4-20 x^3+40 x^2+\left (-x^2-2 x+4\right ) \log (5)\right ) \log \left (x^2+2 x-4\right )+\left (-25 x^4-50 x^3+100 x^2+\left (-x^4-2 x^3+4 x^2\right ) \log ^2\left (x^2+2 x-4\right )+\left (-10 x^4-20 x^3+40 x^2-\left (-x^2-2 x+4\right ) \log (5)\right ) \log \left (x^2+2 x-4\right )-5 \left (-x^2-2 x+4\right ) \log (5)\right ) \log \left (-\frac {\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)}{\log \left (x^2+2 x-4\right ) x+5 x}\right )+\left (-7 x^2-12 x+20\right ) \log (5)\right )}{\left (-x^2-2 x+4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{3} \int \frac {e^3 \left (-\frac {\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)}{\log \left (x^2+2 x-4\right ) x+5 x}\right )^{-x/3} \left (-25 x^4-50 x^3+100 x^2+\left (-x^4-2 x^3+4 x^2\right ) \log ^2\left (x^2+2 x-4\right )+\left (-10 x^4-20 x^3+40 x^2+\left (-x^2-2 x+4\right ) \log (5)\right ) \log \left (x^2+2 x-4\right )+\left (-25 x^4-50 x^3+100 x^2+\left (-x^4-2 x^3+4 x^2\right ) \log ^2\left (x^2+2 x-4\right )+\left (-10 x^4-20 x^3+40 x^2-\left (-x^2-2 x+4\right ) \log (5)\right ) \log \left (x^2+2 x-4\right )-5 \left (-x^2-2 x+4\right ) \log (5)\right ) \log \left (-\frac {\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)}{\log \left (x^2+2 x-4\right ) x+5 x}\right )+\left (-7 x^2-12 x+20\right ) \log (5)\right )}{\left (-x^2-2 x+4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{3} e^3 \int \frac {\left (-\frac {\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)}{\log \left (x^2+2 x-4\right ) x+5 x}\right )^{-x/3} \left (-25 x^4-50 x^3+100 x^2+\left (-x^4-2 x^3+4 x^2\right ) \log ^2\left (x^2+2 x-4\right )+\left (-10 x^4-20 x^3+40 x^2+\left (-x^2-2 x+4\right ) \log (5)\right ) \log \left (x^2+2 x-4\right )+\left (-25 x^4-50 x^3+100 x^2+\left (-x^4-2 x^3+4 x^2\right ) \log ^2\left (x^2+2 x-4\right )+\left (-10 x^4-20 x^3+40 x^2-\left (-x^2-2 x+4\right ) \log (5)\right ) \log \left (x^2+2 x-4\right )-5 \left (-x^2-2 x+4\right ) \log (5)\right ) \log \left (-\frac {\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)}{\log \left (x^2+2 x-4\right ) x+5 x}\right )+\left (-7 x^2-12 x+20\right ) \log (5)\right )}{\left (-x^2-2 x+4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle -\frac {1}{3} e^3 \int \left (\log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right ) \left (-\frac {\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)}{\log \left (x^2+2 x-4\right ) x+5 x}\right )^{-x/3}+\frac {x^2 \log ^2\left (x^2+2 x-4\right ) \left (-\frac {\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)}{\log \left (x^2+2 x-4\right ) x+5 x}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {\left (10 x^2+\log (5)\right ) \log \left (x^2+2 x-4\right ) \left (-\frac {\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)}{\log \left (x^2+2 x-4\right ) x+5 x}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {25 x^4 \left (-\frac {\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)}{\log \left (x^2+2 x-4\right ) x+5 x}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {50 x^3 \left (-\frac {\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)}{\log \left (x^2+2 x-4\right ) x+5 x}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}-\frac {100 x^2 \left (-\frac {\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)}{\log \left (x^2+2 x-4\right ) x+5 x}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {\left (7 x^2+12 x-20\right ) \log (5) \left (-\frac {\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)}{\log \left (x^2+2 x-4\right ) x+5 x}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{3} e^3 \int \frac {\left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3} \left (-25 x^4-50 x^3-\left (x^2+2 x-4\right ) \log ^2\left (x^2+2 x-4\right ) \left (\log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )+1\right ) x^2+100 \left (1-\frac {7 \log (5)}{100}\right ) x^2-12 \log (5) x-5 \left (x^2+2 x-4\right ) \left (5 x^2-\log (5)\right ) \log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )-\left (x^2+2 x-4\right ) \log \left (x^2+2 x-4\right ) \left (10 x^2+\left (10 x^2-\log (5)\right ) \log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )+\log (5)\right )+20 \log (5)\right )}{\left (-x^2-2 x+4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle -\frac {1}{3} e^3 \int \left (\log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}+\frac {x^2 \log ^2\left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {10 x^2 \log \left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {\log (5) \log \left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {25 x^4 \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {50 x^3 \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {x^2 (-100+7 \log (5)) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {12 x \log (5) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}-\frac {20 \log (5) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{3} e^3 \int \frac {\left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3} \left (-25 x^4-50 x^3-\left (x^2+2 x-4\right ) \log ^2\left (x^2+2 x-4\right ) \left (\log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )+1\right ) x^2+100 \left (1-\frac {7 \log (5)}{100}\right ) x^2-12 \log (5) x-5 \left (x^2+2 x-4\right ) \left (5 x^2-\log (5)\right ) \log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )-\left (x^2+2 x-4\right ) \log \left (x^2+2 x-4\right ) \left (10 x^2+\left (10 x^2-\log (5)\right ) \log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )+\log (5)\right )+20 \log (5)\right )}{\left (-x^2-2 x+4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle -\frac {1}{3} e^3 \int \left (\log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}+\frac {x^2 \log ^2\left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {10 x^2 \log \left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {\log (5) \log \left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {25 x^4 \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {50 x^3 \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {x^2 (-100+7 \log (5)) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {12 x \log (5) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}-\frac {20 \log (5) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{3} e^3 \int \frac {\left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3} \left (-25 x^4-50 x^3-\left (x^2+2 x-4\right ) \log ^2\left (x^2+2 x-4\right ) \left (\log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )+1\right ) x^2+100 \left (1-\frac {7 \log (5)}{100}\right ) x^2-12 \log (5) x-5 \left (x^2+2 x-4\right ) \left (5 x^2-\log (5)\right ) \log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )-\left (x^2+2 x-4\right ) \log \left (x^2+2 x-4\right ) \left (10 x^2+\left (10 x^2-\log (5)\right ) \log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )+\log (5)\right )+20 \log (5)\right )}{\left (-x^2-2 x+4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle -\frac {1}{3} e^3 \int \left (\log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}+\frac {x^2 \log ^2\left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {10 x^2 \log \left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {\log (5) \log \left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {25 x^4 \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {50 x^3 \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {x^2 (-100+7 \log (5)) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {12 x \log (5) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}-\frac {20 \log (5) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{3} e^3 \int \frac {\left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3} \left (-25 x^4-50 x^3-\left (x^2+2 x-4\right ) \log ^2\left (x^2+2 x-4\right ) \left (\log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )+1\right ) x^2+100 \left (1-\frac {7 \log (5)}{100}\right ) x^2-12 \log (5) x-5 \left (x^2+2 x-4\right ) \left (5 x^2-\log (5)\right ) \log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )-\left (x^2+2 x-4\right ) \log \left (x^2+2 x-4\right ) \left (10 x^2+\left (10 x^2-\log (5)\right ) \log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )+\log (5)\right )+20 \log (5)\right )}{\left (-x^2-2 x+4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle -\frac {1}{3} e^3 \int \left (\log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}+\frac {x^2 \log ^2\left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {10 x^2 \log \left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {\log (5) \log \left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {25 x^4 \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {50 x^3 \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {x^2 (-100+7 \log (5)) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {12 x \log (5) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}-\frac {20 \log (5) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{3} e^3 \int \frac {\left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3} \left (-25 x^4-50 x^3-\left (x^2+2 x-4\right ) \log ^2\left (x^2+2 x-4\right ) \left (\log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )+1\right ) x^2+100 \left (1-\frac {7 \log (5)}{100}\right ) x^2-12 \log (5) x-5 \left (x^2+2 x-4\right ) \left (5 x^2-\log (5)\right ) \log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )-\left (x^2+2 x-4\right ) \log \left (x^2+2 x-4\right ) \left (10 x^2+\left (10 x^2-\log (5)\right ) \log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )+\log (5)\right )+20 \log (5)\right )}{\left (-x^2-2 x+4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle -\frac {1}{3} e^3 \int \left (\log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}+\frac {x^2 \log ^2\left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {10 x^2 \log \left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {\log (5) \log \left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {25 x^4 \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {50 x^3 \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {x^2 (-100+7 \log (5)) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {12 x \log (5) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}-\frac {20 \log (5) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{3} e^3 \int \frac {\left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3} \left (-25 x^4-50 x^3-\left (x^2+2 x-4\right ) \log ^2\left (x^2+2 x-4\right ) \left (\log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )+1\right ) x^2+100 \left (1-\frac {7 \log (5)}{100}\right ) x^2-12 \log (5) x-5 \left (x^2+2 x-4\right ) \left (5 x^2-\log (5)\right ) \log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )-\left (x^2+2 x-4\right ) \log \left (x^2+2 x-4\right ) \left (10 x^2+\left (10 x^2-\log (5)\right ) \log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )+\log (5)\right )+20 \log (5)\right )}{\left (-x^2-2 x+4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle -\frac {1}{3} e^3 \int \left (\log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}+\frac {x^2 \log ^2\left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {10 x^2 \log \left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {\log (5) \log \left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {25 x^4 \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {50 x^3 \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {x^2 (-100+7 \log (5)) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {12 x \log (5) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}-\frac {20 \log (5) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{3} e^3 \int \frac {\left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3} \left (-25 x^4-50 x^3-\left (x^2+2 x-4\right ) \log ^2\left (x^2+2 x-4\right ) \left (\log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )+1\right ) x^2+100 \left (1-\frac {7 \log (5)}{100}\right ) x^2-12 \log (5) x-5 \left (x^2+2 x-4\right ) \left (5 x^2-\log (5)\right ) \log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )-\left (x^2+2 x-4\right ) \log \left (x^2+2 x-4\right ) \left (10 x^2+\left (10 x^2-\log (5)\right ) \log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )+\log (5)\right )+20 \log (5)\right )}{\left (-x^2-2 x+4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle -\frac {1}{3} e^3 \int \left (\log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}+\frac {x^2 \log ^2\left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {10 x^2 \log \left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {\log (5) \log \left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {25 x^4 \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {50 x^3 \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {x^2 (-100+7 \log (5)) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {12 x \log (5) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}-\frac {20 \log (5) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{3} e^3 \int \frac {\left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3} \left (-25 x^4-50 x^3-\left (x^2+2 x-4\right ) \log ^2\left (x^2+2 x-4\right ) \left (\log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )+1\right ) x^2+100 \left (1-\frac {7 \log (5)}{100}\right ) x^2-12 \log (5) x-5 \left (x^2+2 x-4\right ) \left (5 x^2-\log (5)\right ) \log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )-\left (x^2+2 x-4\right ) \log \left (x^2+2 x-4\right ) \left (10 x^2+\left (10 x^2-\log (5)\right ) \log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )+\log (5)\right )+20 \log (5)\right )}{\left (-x^2-2 x+4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle -\frac {1}{3} e^3 \int \left (\log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}+\frac {x^2 \log ^2\left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {10 x^2 \log \left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {\log (5) \log \left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {25 x^4 \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {50 x^3 \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {x^2 (-100+7 \log (5)) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {12 x \log (5) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}-\frac {20 \log (5) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{3} e^3 \int \frac {\left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3} \left (-25 x^4-50 x^3-\left (x^2+2 x-4\right ) \log ^2\left (x^2+2 x-4\right ) \left (\log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )+1\right ) x^2+100 \left (1-\frac {7 \log (5)}{100}\right ) x^2-12 \log (5) x-5 \left (x^2+2 x-4\right ) \left (5 x^2-\log (5)\right ) \log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )-\left (x^2+2 x-4\right ) \log \left (x^2+2 x-4\right ) \left (10 x^2+\left (10 x^2-\log (5)\right ) \log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )+\log (5)\right )+20 \log (5)\right )}{\left (-x^2-2 x+4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle -\frac {1}{3} e^3 \int \left (\log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}+\frac {x^2 \log ^2\left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {10 x^2 \log \left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {\log (5) \log \left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {25 x^4 \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {50 x^3 \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {x^2 (-100+7 \log (5)) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {12 x \log (5) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}-\frac {20 \log (5) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{3} e^3 \int \frac {\left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3} \left (-25 x^4-50 x^3-\left (x^2+2 x-4\right ) \log ^2\left (x^2+2 x-4\right ) \left (\log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )+1\right ) x^2+100 \left (1-\frac {7 \log (5)}{100}\right ) x^2-12 \log (5) x-5 \left (x^2+2 x-4\right ) \left (5 x^2-\log (5)\right ) \log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )-\left (x^2+2 x-4\right ) \log \left (x^2+2 x-4\right ) \left (10 x^2+\left (10 x^2-\log (5)\right ) \log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )+\log (5)\right )+20 \log (5)\right )}{\left (-x^2-2 x+4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle -\frac {1}{3} e^3 \int \left (\log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}+\frac {x^2 \log ^2\left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {10 x^2 \log \left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {\log (5) \log \left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {25 x^4 \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {50 x^3 \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {x^2 (-100+7 \log (5)) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {12 x \log (5) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}-\frac {20 \log (5) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{3} e^3 \int \frac {\left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3} \left (-25 x^4-50 x^3-\left (x^2+2 x-4\right ) \log ^2\left (x^2+2 x-4\right ) \left (\log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )+1\right ) x^2+100 \left (1-\frac {7 \log (5)}{100}\right ) x^2-12 \log (5) x-5 \left (x^2+2 x-4\right ) \left (5 x^2-\log (5)\right ) \log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )-\left (x^2+2 x-4\right ) \log \left (x^2+2 x-4\right ) \left (10 x^2+\left (10 x^2-\log (5)\right ) \log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )+\log (5)\right )+20 \log (5)\right )}{\left (-x^2-2 x+4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle -\frac {1}{3} e^3 \int \left (\log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}+\frac {x^2 \log ^2\left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {10 x^2 \log \left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {\log (5) \log \left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {25 x^4 \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {50 x^3 \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {x^2 (-100+7 \log (5)) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {12 x \log (5) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}-\frac {20 \log (5) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{3} e^3 \int \frac {\left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3} \left (-25 x^4-50 x^3-\left (x^2+2 x-4\right ) \log ^2\left (x^2+2 x-4\right ) \left (\log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )+1\right ) x^2+100 \left (1-\frac {7 \log (5)}{100}\right ) x^2-12 \log (5) x-5 \left (x^2+2 x-4\right ) \left (5 x^2-\log (5)\right ) \log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )-\left (x^2+2 x-4\right ) \log \left (x^2+2 x-4\right ) \left (10 x^2+\left (10 x^2-\log (5)\right ) \log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )+\log (5)\right )+20 \log (5)\right )}{\left (-x^2-2 x+4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle -\frac {1}{3} e^3 \int \left (\log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}+\frac {x^2 \log ^2\left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {10 x^2 \log \left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {\log (5) \log \left (x^2+2 x-4\right ) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {25 x^4 \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {50 x^3 \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {x^2 (-100+7 \log (5)) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}+\frac {12 x \log (5) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}-\frac {20 \log (5) \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3}}{\left (x^2+2 x-4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {1}{3} e^3 \int \frac {\left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )^{-x/3} \left (-25 x^4-50 x^3-\left (x^2+2 x-4\right ) \log ^2\left (x^2+2 x-4\right ) \left (\log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )+1\right ) x^2+100 \left (1-\frac {7 \log (5)}{100}\right ) x^2-12 \log (5) x-5 \left (x^2+2 x-4\right ) \left (5 x^2-\log (5)\right ) \log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )-\left (x^2+2 x-4\right ) \log \left (x^2+2 x-4\right ) \left (10 x^2+\left (10 x^2-\log (5)\right ) \log \left (\frac {-\log \left (x^2+2 x-4\right ) x^2-5 x^2+\log (5)}{x \left (\log \left (x^2+2 x-4\right )+5\right )}\right )+\log (5)\right )+20 \log (5)\right )}{\left (-x^2-2 x+4\right ) \left (\log \left (x^2+2 x-4\right )+5\right ) \left (\log \left (x^2+2 x-4\right ) x^2+5 x^2-\log (5)\right )}dx\)

Input:

Int[(E^((9 - x*Log[(-5*x^2 + Log[5] - x^2*Log[-4 + 2*x + x^2])/(5*x + x*Lo 
g[-4 + 2*x + x^2])])/3)*(100*x^2 - 50*x^3 - 25*x^4 + (20 - 12*x - 7*x^2)*L 
og[5] + (40*x^2 - 20*x^3 - 10*x^4 + (4 - 2*x - x^2)*Log[5])*Log[-4 + 2*x + 
 x^2] + (4*x^2 - 2*x^3 - x^4)*Log[-4 + 2*x + x^2]^2 + (100*x^2 - 50*x^3 - 
25*x^4 + (-20 + 10*x + 5*x^2)*Log[5] + (40*x^2 - 20*x^3 - 10*x^4 + (-4 + 2 
*x + x^2)*Log[5])*Log[-4 + 2*x + x^2] + (4*x^2 - 2*x^3 - x^4)*Log[-4 + 2*x 
 + x^2]^2)*Log[(-5*x^2 + Log[5] - x^2*Log[-4 + 2*x + x^2])/(5*x + x*Log[-4 
 + 2*x + x^2])]))/(-300*x^2 + 150*x^3 + 75*x^4 + (60 - 30*x - 15*x^2)*Log[ 
5] + (-120*x^2 + 60*x^3 + 30*x^4 + (12 - 6*x - 3*x^2)*Log[5])*Log[-4 + 2*x 
 + x^2] + (-12*x^2 + 6*x^3 + 3*x^4)*Log[-4 + 2*x + x^2]^2),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.20 (sec) , antiderivative size = 566, normalized size of antiderivative = 17.15

\[x^{\frac {x}{3}} {\left (\ln \left (x^{2}+2 x -4\right )+5\right )}^{\frac {x}{3}} {\left (-\left (\ln \left (x^{2}+2 x -4\right )+5\right ) x^{2}+\ln \left (5\right )\right )}^{-\frac {x}{3}} {\mathrm e}^{3+\frac {i x \pi {\operatorname {csgn}\left (\frac {i \left (-\left (\ln \left (x^{2}+2 x -4\right )+5\right ) x^{2}+\ln \left (5\right )\right )}{\ln \left (x^{2}+2 x -4\right )+5}\right )}^{3}}{6}-\frac {i x \pi {\operatorname {csgn}\left (\frac {i \left (-\left (\ln \left (x^{2}+2 x -4\right )+5\right ) x^{2}+\ln \left (5\right )\right )}{\ln \left (x^{2}+2 x -4\right )+5}\right )}^{2} \operatorname {csgn}\left (\frac {i}{\ln \left (x^{2}+2 x -4\right )+5}\right )}{6}-\frac {i x \pi {\operatorname {csgn}\left (\frac {i \left (-\left (\ln \left (x^{2}+2 x -4\right )+5\right ) x^{2}+\ln \left (5\right )\right )}{\ln \left (x^{2}+2 x -4\right )+5}\right )}^{2} \operatorname {csgn}\left (i \left (-\left (\ln \left (x^{2}+2 x -4\right )+5\right ) x^{2}+\ln \left (5\right )\right )\right )}{6}+\frac {i x \pi \,\operatorname {csgn}\left (\frac {i \left (-\left (\ln \left (x^{2}+2 x -4\right )+5\right ) x^{2}+\ln \left (5\right )\right )}{\ln \left (x^{2}+2 x -4\right )+5}\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x^{2}+2 x -4\right )+5}\right ) \operatorname {csgn}\left (i \left (-\left (\ln \left (x^{2}+2 x -4\right )+5\right ) x^{2}+\ln \left (5\right )\right )\right )}{6}-\frac {i x \pi \,\operatorname {csgn}\left (\frac {i \left (-\left (\ln \left (x^{2}+2 x -4\right )+5\right ) x^{2}+\ln \left (5\right )\right )}{\ln \left (x^{2}+2 x -4\right )+5}\right ) {\operatorname {csgn}\left (\frac {i \left (-\left (\ln \left (x^{2}+2 x -4\right )+5\right ) x^{2}+\ln \left (5\right )\right )}{x \left (\ln \left (x^{2}+2 x -4\right )+5\right )}\right )}^{2}}{6}+\frac {i x \pi \,\operatorname {csgn}\left (\frac {i \left (-\left (\ln \left (x^{2}+2 x -4\right )+5\right ) x^{2}+\ln \left (5\right )\right )}{\ln \left (x^{2}+2 x -4\right )+5}\right ) \operatorname {csgn}\left (\frac {i \left (-\left (\ln \left (x^{2}+2 x -4\right )+5\right ) x^{2}+\ln \left (5\right )\right )}{x \left (\ln \left (x^{2}+2 x -4\right )+5\right )}\right ) \operatorname {csgn}\left (\frac {i}{x}\right )}{6}+\frac {i x \pi {\operatorname {csgn}\left (\frac {i \left (-\left (\ln \left (x^{2}+2 x -4\right )+5\right ) x^{2}+\ln \left (5\right )\right )}{x \left (\ln \left (x^{2}+2 x -4\right )+5\right )}\right )}^{3}}{6}-\frac {i x \pi {\operatorname {csgn}\left (\frac {i \left (-\left (\ln \left (x^{2}+2 x -4\right )+5\right ) x^{2}+\ln \left (5\right )\right )}{x \left (\ln \left (x^{2}+2 x -4\right )+5\right )}\right )}^{2} \operatorname {csgn}\left (\frac {i}{x}\right )}{6}}\]

Input:

int((((-x^4-2*x^3+4*x^2)*ln(x^2+2*x-4)^2+((x^2+2*x-4)*ln(5)-10*x^4-20*x^3+ 
40*x^2)*ln(x^2+2*x-4)+(5*x^2+10*x-20)*ln(5)-25*x^4-50*x^3+100*x^2)*ln((-x^ 
2*ln(x^2+2*x-4)+ln(5)-5*x^2)/(x*ln(x^2+2*x-4)+5*x))+(-x^4-2*x^3+4*x^2)*ln( 
x^2+2*x-4)^2+((-x^2-2*x+4)*ln(5)-10*x^4-20*x^3+40*x^2)*ln(x^2+2*x-4)+(-7*x 
^2-12*x+20)*ln(5)-25*x^4-50*x^3+100*x^2)*exp(-1/3*x*ln((-x^2*ln(x^2+2*x-4) 
+ln(5)-5*x^2)/(x*ln(x^2+2*x-4)+5*x))+3)/((3*x^4+6*x^3-12*x^2)*ln(x^2+2*x-4 
)^2+((-3*x^2-6*x+12)*ln(5)+30*x^4+60*x^3-120*x^2)*ln(x^2+2*x-4)+(-15*x^2-3 
0*x+60)*ln(5)+75*x^4+150*x^3-300*x^2),x)
 

Output:

x^(1/3*x)*(ln(x^2+2*x-4)+5)^(1/3*x)*(-(ln(x^2+2*x-4)+5)*x^2+ln(5))^(-1/3*x 
)*exp(3+1/6*I*x*Pi*csgn(I*(-(ln(x^2+2*x-4)+5)*x^2+ln(5))/(ln(x^2+2*x-4)+5) 
)^3-1/6*I*x*Pi*csgn(I*(-(ln(x^2+2*x-4)+5)*x^2+ln(5))/(ln(x^2+2*x-4)+5))^2* 
csgn(I/(ln(x^2+2*x-4)+5))-1/6*I*x*Pi*csgn(I*(-(ln(x^2+2*x-4)+5)*x^2+ln(5)) 
/(ln(x^2+2*x-4)+5))^2*csgn(I*(-(ln(x^2+2*x-4)+5)*x^2+ln(5)))+1/6*I*x*Pi*cs 
gn(I*(-(ln(x^2+2*x-4)+5)*x^2+ln(5))/(ln(x^2+2*x-4)+5))*csgn(I/(ln(x^2+2*x- 
4)+5))*csgn(I*(-(ln(x^2+2*x-4)+5)*x^2+ln(5)))-1/6*I*x*Pi*csgn(I*(-(ln(x^2+ 
2*x-4)+5)*x^2+ln(5))/(ln(x^2+2*x-4)+5))*csgn(I/x*(-(ln(x^2+2*x-4)+5)*x^2+l 
n(5))/(ln(x^2+2*x-4)+5))^2+1/6*I*x*Pi*csgn(I*(-(ln(x^2+2*x-4)+5)*x^2+ln(5) 
)/(ln(x^2+2*x-4)+5))*csgn(I/x*(-(ln(x^2+2*x-4)+5)*x^2+ln(5))/(ln(x^2+2*x-4 
)+5))*csgn(I/x)+1/6*I*x*Pi*csgn(I/x*(-(ln(x^2+2*x-4)+5)*x^2+ln(5))/(ln(x^2 
+2*x-4)+5))^3-1/6*I*x*Pi*csgn(I/x*(-(ln(x^2+2*x-4)+5)*x^2+ln(5))/(ln(x^2+2 
*x-4)+5))^2*csgn(I/x))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48 \[ \int \frac {e^{\frac {1}{3} \left (9-x \log \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{5 x+x \log \left (-4+2 x+x^2\right )}\right )\right )} \left (100 x^2-50 x^3-25 x^4+\left (20-12 x-7 x^2\right ) \log (5)+\left (40 x^2-20 x^3-10 x^4+\left (4-2 x-x^2\right ) \log (5)\right ) \log \left (-4+2 x+x^2\right )+\left (4 x^2-2 x^3-x^4\right ) \log ^2\left (-4+2 x+x^2\right )+\left (100 x^2-50 x^3-25 x^4+\left (-20+10 x+5 x^2\right ) \log (5)+\left (40 x^2-20 x^3-10 x^4+\left (-4+2 x+x^2\right ) \log (5)\right ) \log \left (-4+2 x+x^2\right )+\left (4 x^2-2 x^3-x^4\right ) \log ^2\left (-4+2 x+x^2\right )\right ) \log \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{5 x+x \log \left (-4+2 x+x^2\right )}\right )\right )}{-300 x^2+150 x^3+75 x^4+\left (60-30 x-15 x^2\right ) \log (5)+\left (-120 x^2+60 x^3+30 x^4+\left (12-6 x-3 x^2\right ) \log (5)\right ) \log \left (-4+2 x+x^2\right )+\left (-12 x^2+6 x^3+3 x^4\right ) \log ^2\left (-4+2 x+x^2\right )} \, dx=e^{\left (-\frac {1}{3} \, x \log \left (-\frac {x^{2} \log \left (x^{2} + 2 \, x - 4\right ) + 5 \, x^{2} - \log \left (5\right )}{x \log \left (x^{2} + 2 \, x - 4\right ) + 5 \, x}\right ) + 3\right )} \] Input:

integrate((((-x^4-2*x^3+4*x^2)*log(x^2+2*x-4)^2+((x^2+2*x-4)*log(5)-10*x^4 
-20*x^3+40*x^2)*log(x^2+2*x-4)+(5*x^2+10*x-20)*log(5)-25*x^4-50*x^3+100*x^ 
2)*log((-x^2*log(x^2+2*x-4)+log(5)-5*x^2)/(x*log(x^2+2*x-4)+5*x))+(-x^4-2* 
x^3+4*x^2)*log(x^2+2*x-4)^2+((-x^2-2*x+4)*log(5)-10*x^4-20*x^3+40*x^2)*log 
(x^2+2*x-4)+(-7*x^2-12*x+20)*log(5)-25*x^4-50*x^3+100*x^2)*exp(-1/3*x*log( 
(-x^2*log(x^2+2*x-4)+log(5)-5*x^2)/(x*log(x^2+2*x-4)+5*x))+3)/((3*x^4+6*x^ 
3-12*x^2)*log(x^2+2*x-4)^2+((-3*x^2-6*x+12)*log(5)+30*x^4+60*x^3-120*x^2)* 
log(x^2+2*x-4)+(-15*x^2-30*x+60)*log(5)+75*x^4+150*x^3-300*x^2),x, algorit 
hm="fricas")
 

Output:

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

Sympy [F(-2)]

Exception generated. \[ \int \frac {e^{\frac {1}{3} \left (9-x \log \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{5 x+x \log \left (-4+2 x+x^2\right )}\right )\right )} \left (100 x^2-50 x^3-25 x^4+\left (20-12 x-7 x^2\right ) \log (5)+\left (40 x^2-20 x^3-10 x^4+\left (4-2 x-x^2\right ) \log (5)\right ) \log \left (-4+2 x+x^2\right )+\left (4 x^2-2 x^3-x^4\right ) \log ^2\left (-4+2 x+x^2\right )+\left (100 x^2-50 x^3-25 x^4+\left (-20+10 x+5 x^2\right ) \log (5)+\left (40 x^2-20 x^3-10 x^4+\left (-4+2 x+x^2\right ) \log (5)\right ) \log \left (-4+2 x+x^2\right )+\left (4 x^2-2 x^3-x^4\right ) \log ^2\left (-4+2 x+x^2\right )\right ) \log \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{5 x+x \log \left (-4+2 x+x^2\right )}\right )\right )}{-300 x^2+150 x^3+75 x^4+\left (60-30 x-15 x^2\right ) \log (5)+\left (-120 x^2+60 x^3+30 x^4+\left (12-6 x-3 x^2\right ) \log (5)\right ) \log \left (-4+2 x+x^2\right )+\left (-12 x^2+6 x^3+3 x^4\right ) \log ^2\left (-4+2 x+x^2\right )} \, dx=\text {Exception raised: PolynomialError} \] Input:

integrate((((-x**4-2*x**3+4*x**2)*ln(x**2+2*x-4)**2+((x**2+2*x-4)*ln(5)-10 
*x**4-20*x**3+40*x**2)*ln(x**2+2*x-4)+(5*x**2+10*x-20)*ln(5)-25*x**4-50*x* 
*3+100*x**2)*ln((-x**2*ln(x**2+2*x-4)+ln(5)-5*x**2)/(x*ln(x**2+2*x-4)+5*x) 
)+(-x**4-2*x**3+4*x**2)*ln(x**2+2*x-4)**2+((-x**2-2*x+4)*ln(5)-10*x**4-20* 
x**3+40*x**2)*ln(x**2+2*x-4)+(-7*x**2-12*x+20)*ln(5)-25*x**4-50*x**3+100*x 
**2)*exp(-1/3*x*ln((-x**2*ln(x**2+2*x-4)+ln(5)-5*x**2)/(x*ln(x**2+2*x-4)+5 
*x))+3)/((3*x**4+6*x**3-12*x**2)*ln(x**2+2*x-4)**2+((-3*x**2-6*x+12)*ln(5) 
+30*x**4+60*x**3-120*x**2)*ln(x**2+2*x-4)+(-15*x**2-30*x+60)*ln(5)+75*x**4 
+150*x**3-300*x**2),x)
 

Output:

Exception raised: PolynomialError >> 1/(9*x**5 + 18*x**4 - 36*x**3) contai 
ns an element of the set of generators.
 

Maxima [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.39 \[ \int \frac {e^{\frac {1}{3} \left (9-x \log \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{5 x+x \log \left (-4+2 x+x^2\right )}\right )\right )} \left (100 x^2-50 x^3-25 x^4+\left (20-12 x-7 x^2\right ) \log (5)+\left (40 x^2-20 x^3-10 x^4+\left (4-2 x-x^2\right ) \log (5)\right ) \log \left (-4+2 x+x^2\right )+\left (4 x^2-2 x^3-x^4\right ) \log ^2\left (-4+2 x+x^2\right )+\left (100 x^2-50 x^3-25 x^4+\left (-20+10 x+5 x^2\right ) \log (5)+\left (40 x^2-20 x^3-10 x^4+\left (-4+2 x+x^2\right ) \log (5)\right ) \log \left (-4+2 x+x^2\right )+\left (4 x^2-2 x^3-x^4\right ) \log ^2\left (-4+2 x+x^2\right )\right ) \log \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{5 x+x \log \left (-4+2 x+x^2\right )}\right )\right )}{-300 x^2+150 x^3+75 x^4+\left (60-30 x-15 x^2\right ) \log (5)+\left (-120 x^2+60 x^3+30 x^4+\left (12-6 x-3 x^2\right ) \log (5)\right ) \log \left (-4+2 x+x^2\right )+\left (-12 x^2+6 x^3+3 x^4\right ) \log ^2\left (-4+2 x+x^2\right )} \, dx=e^{\left (-\frac {1}{3} \, x \log \left (-x^{2} {\left (\log \left (x^{2} + 2 \, x - 4\right ) + 5\right )} + \log \left (5\right )\right ) + \frac {1}{3} \, x \log \left (x\right ) + \frac {1}{3} \, x \log \left (\log \left (x^{2} + 2 \, x - 4\right ) + 5\right ) + 3\right )} \] Input:

integrate((((-x^4-2*x^3+4*x^2)*log(x^2+2*x-4)^2+((x^2+2*x-4)*log(5)-10*x^4 
-20*x^3+40*x^2)*log(x^2+2*x-4)+(5*x^2+10*x-20)*log(5)-25*x^4-50*x^3+100*x^ 
2)*log((-x^2*log(x^2+2*x-4)+log(5)-5*x^2)/(x*log(x^2+2*x-4)+5*x))+(-x^4-2* 
x^3+4*x^2)*log(x^2+2*x-4)^2+((-x^2-2*x+4)*log(5)-10*x^4-20*x^3+40*x^2)*log 
(x^2+2*x-4)+(-7*x^2-12*x+20)*log(5)-25*x^4-50*x^3+100*x^2)*exp(-1/3*x*log( 
(-x^2*log(x^2+2*x-4)+log(5)-5*x^2)/(x*log(x^2+2*x-4)+5*x))+3)/((3*x^4+6*x^ 
3-12*x^2)*log(x^2+2*x-4)^2+((-3*x^2-6*x+12)*log(5)+30*x^4+60*x^3-120*x^2)* 
log(x^2+2*x-4)+(-15*x^2-30*x+60)*log(5)+75*x^4+150*x^3-300*x^2),x, algorit 
hm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (30) = 60\).

Time = 1.73 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.45 \[ \int \frac {e^{\frac {1}{3} \left (9-x \log \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{5 x+x \log \left (-4+2 x+x^2\right )}\right )\right )} \left (100 x^2-50 x^3-25 x^4+\left (20-12 x-7 x^2\right ) \log (5)+\left (40 x^2-20 x^3-10 x^4+\left (4-2 x-x^2\right ) \log (5)\right ) \log \left (-4+2 x+x^2\right )+\left (4 x^2-2 x^3-x^4\right ) \log ^2\left (-4+2 x+x^2\right )+\left (100 x^2-50 x^3-25 x^4+\left (-20+10 x+5 x^2\right ) \log (5)+\left (40 x^2-20 x^3-10 x^4+\left (-4+2 x+x^2\right ) \log (5)\right ) \log \left (-4+2 x+x^2\right )+\left (4 x^2-2 x^3-x^4\right ) \log ^2\left (-4+2 x+x^2\right )\right ) \log \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{5 x+x \log \left (-4+2 x+x^2\right )}\right )\right )}{-300 x^2+150 x^3+75 x^4+\left (60-30 x-15 x^2\right ) \log (5)+\left (-120 x^2+60 x^3+30 x^4+\left (12-6 x-3 x^2\right ) \log (5)\right ) \log \left (-4+2 x+x^2\right )+\left (-12 x^2+6 x^3+3 x^4\right ) \log ^2\left (-4+2 x+x^2\right )} \, dx=e^{\left (-\frac {1}{3} \, x \log \left (-\frac {x^{2} \log \left (x^{2} + 2 \, x - 4\right )}{x \log \left (x^{2} + 2 \, x - 4\right ) + 5 \, x} - \frac {5 \, x^{2}}{x \log \left (x^{2} + 2 \, x - 4\right ) + 5 \, x} + \frac {\log \left (5\right )}{x \log \left (x^{2} + 2 \, x - 4\right ) + 5 \, x}\right ) + 3\right )} \] Input:

integrate((((-x^4-2*x^3+4*x^2)*log(x^2+2*x-4)^2+((x^2+2*x-4)*log(5)-10*x^4 
-20*x^3+40*x^2)*log(x^2+2*x-4)+(5*x^2+10*x-20)*log(5)-25*x^4-50*x^3+100*x^ 
2)*log((-x^2*log(x^2+2*x-4)+log(5)-5*x^2)/(x*log(x^2+2*x-4)+5*x))+(-x^4-2* 
x^3+4*x^2)*log(x^2+2*x-4)^2+((-x^2-2*x+4)*log(5)-10*x^4-20*x^3+40*x^2)*log 
(x^2+2*x-4)+(-7*x^2-12*x+20)*log(5)-25*x^4-50*x^3+100*x^2)*exp(-1/3*x*log( 
(-x^2*log(x^2+2*x-4)+log(5)-5*x^2)/(x*log(x^2+2*x-4)+5*x))+3)/((3*x^4+6*x^ 
3-12*x^2)*log(x^2+2*x-4)^2+((-3*x^2-6*x+12)*log(5)+30*x^4+60*x^3-120*x^2)* 
log(x^2+2*x-4)+(-15*x^2-30*x+60)*log(5)+75*x^4+150*x^3-300*x^2),x, algorit 
hm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 4.44 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.55 \[ \int \frac {e^{\frac {1}{3} \left (9-x \log \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{5 x+x \log \left (-4+2 x+x^2\right )}\right )\right )} \left (100 x^2-50 x^3-25 x^4+\left (20-12 x-7 x^2\right ) \log (5)+\left (40 x^2-20 x^3-10 x^4+\left (4-2 x-x^2\right ) \log (5)\right ) \log \left (-4+2 x+x^2\right )+\left (4 x^2-2 x^3-x^4\right ) \log ^2\left (-4+2 x+x^2\right )+\left (100 x^2-50 x^3-25 x^4+\left (-20+10 x+5 x^2\right ) \log (5)+\left (40 x^2-20 x^3-10 x^4+\left (-4+2 x+x^2\right ) \log (5)\right ) \log \left (-4+2 x+x^2\right )+\left (4 x^2-2 x^3-x^4\right ) \log ^2\left (-4+2 x+x^2\right )\right ) \log \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{5 x+x \log \left (-4+2 x+x^2\right )}\right )\right )}{-300 x^2+150 x^3+75 x^4+\left (60-30 x-15 x^2\right ) \log (5)+\left (-120 x^2+60 x^3+30 x^4+\left (12-6 x-3 x^2\right ) \log (5)\right ) \log \left (-4+2 x+x^2\right )+\left (-12 x^2+6 x^3+3 x^4\right ) \log ^2\left (-4+2 x+x^2\right )} \, dx=\frac {{\mathrm {e}}^3}{{\left (-\frac {x^2\,\ln \left (x^2+2\,x-4\right )-\ln \left (5\right )+5\,x^2}{5\,x+x\,\ln \left (x^2+2\,x-4\right )}\right )}^{x/3}} \] Input:

int((exp(3 - (x*log(-(x^2*log(2*x + x^2 - 4) - log(5) + 5*x^2)/(5*x + x*lo 
g(2*x + x^2 - 4))))/3)*(log(5)*(12*x + 7*x^2 - 20) + log(2*x + x^2 - 4)*(2 
0*x^3 - 40*x^2 + 10*x^4 + log(5)*(2*x + x^2 - 4)) - 100*x^2 + 50*x^3 + 25* 
x^4 + log(2*x + x^2 - 4)^2*(2*x^3 - 4*x^2 + x^4) - log(-(x^2*log(2*x + x^2 
 - 4) - log(5) + 5*x^2)/(5*x + x*log(2*x + x^2 - 4)))*(log(5)*(10*x + 5*x^ 
2 - 20) + log(2*x + x^2 - 4)*(40*x^2 - 20*x^3 - 10*x^4 + log(5)*(2*x + x^2 
 - 4)) + 100*x^2 - 50*x^3 - 25*x^4 - log(2*x + x^2 - 4)^2*(2*x^3 - 4*x^2 + 
 x^4))))/(log(2*x + x^2 - 4)*(log(5)*(6*x + 3*x^2 - 12) + 120*x^2 - 60*x^3 
 - 30*x^4) + log(5)*(30*x + 15*x^2 - 60) - log(2*x + x^2 - 4)^2*(6*x^3 - 1 
2*x^2 + 3*x^4) + 300*x^2 - 150*x^3 - 75*x^4),x)
 

Output:

exp(3)/(-(x^2*log(2*x + x^2 - 4) - log(5) + 5*x^2)/(5*x + x*log(2*x + x^2 
- 4)))^(x/3)
 

Reduce [F]

\[ \int \frac {e^{\frac {1}{3} \left (9-x \log \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{5 x+x \log \left (-4+2 x+x^2\right )}\right )\right )} \left (100 x^2-50 x^3-25 x^4+\left (20-12 x-7 x^2\right ) \log (5)+\left (40 x^2-20 x^3-10 x^4+\left (4-2 x-x^2\right ) \log (5)\right ) \log \left (-4+2 x+x^2\right )+\left (4 x^2-2 x^3-x^4\right ) \log ^2\left (-4+2 x+x^2\right )+\left (100 x^2-50 x^3-25 x^4+\left (-20+10 x+5 x^2\right ) \log (5)+\left (40 x^2-20 x^3-10 x^4+\left (-4+2 x+x^2\right ) \log (5)\right ) \log \left (-4+2 x+x^2\right )+\left (4 x^2-2 x^3-x^4\right ) \log ^2\left (-4+2 x+x^2\right )\right ) \log \left (\frac {-5 x^2+\log (5)-x^2 \log \left (-4+2 x+x^2\right )}{5 x+x \log \left (-4+2 x+x^2\right )}\right )\right )}{-300 x^2+150 x^3+75 x^4+\left (60-30 x-15 x^2\right ) \log (5)+\left (-120 x^2+60 x^3+30 x^4+\left (12-6 x-3 x^2\right ) \log (5)\right ) \log \left (-4+2 x+x^2\right )+\left (-12 x^2+6 x^3+3 x^4\right ) \log ^2\left (-4+2 x+x^2\right )} \, dx=\text {too large to display} \] Input:

int((((-x^4-2*x^3+4*x^2)*log(x^2+2*x-4)^2+((x^2+2*x-4)*log(5)-10*x^4-20*x^ 
3+40*x^2)*log(x^2+2*x-4)+(5*x^2+10*x-20)*log(5)-25*x^4-50*x^3+100*x^2)*log 
((-x^2*log(x^2+2*x-4)+log(5)-5*x^2)/(x*log(x^2+2*x-4)+5*x))+(-x^4-2*x^3+4* 
x^2)*log(x^2+2*x-4)^2+((-x^2-2*x+4)*log(5)-10*x^4-20*x^3+40*x^2)*log(x^2+2 
*x-4)+(-7*x^2-12*x+20)*log(5)-25*x^4-50*x^3+100*x^2)*exp(-1/3*x*log((-x^2* 
log(x^2+2*x-4)+log(5)-5*x^2)/(x*log(x^2+2*x-4)+5*x))+3)/((3*x^4+6*x^3-12*x 
^2)*log(x^2+2*x-4)^2+((-3*x^2-6*x+12)*log(5)+30*x^4+60*x^3-120*x^2)*log(x^ 
2+2*x-4)+(-15*x^2-30*x+60)*log(5)+75*x^4+150*x^3-300*x^2),x)
 

Output:

(e**3*(20*int((log(x**2 + 2*x - 4)*x + 5*x)**(x/3)/(( - log(x**2 + 2*x - 4 
)*x**2 + log(5) - 5*x**2)**(x/3)*log(x**2 + 2*x - 4)**2*x**4 + 2*( - log(x 
**2 + 2*x - 4)*x**2 + log(5) - 5*x**2)**(x/3)*log(x**2 + 2*x - 4)**2*x**3 
- 4*( - log(x**2 + 2*x - 4)*x**2 + log(5) - 5*x**2)**(x/3)*log(x**2 + 2*x 
- 4)**2*x**2 - ( - log(x**2 + 2*x - 4)*x**2 + log(5) - 5*x**2)**(x/3)*log( 
x**2 + 2*x - 4)*log(5)*x**2 - 2*( - log(x**2 + 2*x - 4)*x**2 + log(5) - 5* 
x**2)**(x/3)*log(x**2 + 2*x - 4)*log(5)*x + 4*( - log(x**2 + 2*x - 4)*x**2 
 + log(5) - 5*x**2)**(x/3)*log(x**2 + 2*x - 4)*log(5) + 10*( - log(x**2 + 
2*x - 4)*x**2 + log(5) - 5*x**2)**(x/3)*log(x**2 + 2*x - 4)*x**4 + 20*( - 
log(x**2 + 2*x - 4)*x**2 + log(5) - 5*x**2)**(x/3)*log(x**2 + 2*x - 4)*x** 
3 - 40*( - log(x**2 + 2*x - 4)*x**2 + log(5) - 5*x**2)**(x/3)*log(x**2 + 2 
*x - 4)*x**2 - 5*( - log(x**2 + 2*x - 4)*x**2 + log(5) - 5*x**2)**(x/3)*lo 
g(5)*x**2 - 10*( - log(x**2 + 2*x - 4)*x**2 + log(5) - 5*x**2)**(x/3)*log( 
5)*x + 20*( - log(x**2 + 2*x - 4)*x**2 + log(5) - 5*x**2)**(x/3)*log(5) + 
25*( - log(x**2 + 2*x - 4)*x**2 + log(5) - 5*x**2)**(x/3)*x**4 + 50*( - lo 
g(x**2 + 2*x - 4)*x**2 + log(5) - 5*x**2)**(x/3)*x**3 - 100*( - log(x**2 + 
 2*x - 4)*x**2 + log(5) - 5*x**2)**(x/3)*x**2),x)*log(5) - int(((log(x**2 
+ 2*x - 4)*x + 5*x)**(x/3)*log(x**2 + 2*x - 4)**2*x**4)/(( - log(x**2 + 2* 
x - 4)*x**2 + log(5) - 5*x**2)**(x/3)*log(x**2 + 2*x - 4)**2*x**4 + 2*( - 
log(x**2 + 2*x - 4)*x**2 + log(5) - 5*x**2)**(x/3)*log(x**2 + 2*x - 4)*...