3.1.71 \(\int \frac {e^5 (-12+3 x)+(12-3 x) \log (4)+(4 e^{10}-163 e^5 x+1660 x^2+(-8 e^5+163 x) \log (4)+4 \log ^2(4)) \log (\frac {4 e^5-83 x-4 \log (4)}{-5 e^5+100 x+5 \log (4)})}{26560 x^2-13280 x^3+1660 x^4+e^{10} (64-32 x+4 x^2)+e^5 (-2608 x+1304 x^2-163 x^3)+(2608 x-1304 x^2+163 x^3+e^5 (-128+64 x-8 x^2)) \log (4)+(64-32 x+4 x^2) \log ^2(4)} \, dx\) [71]

3.1.71.1 Optimal result

Integrand size = 181, antiderivative size = 31 \[ \int \frac {e^5 (-12+3 x)+(12-3 x) \log (4)+\left (4 e^{10}-163 e^5 x+1660 x^2+\left (-8 e^5+163 x\right ) \log (4)+4 \log ^2(4)\right ) \log \left (\frac {4 e^5-83 x-4 \log (4)}{-5 e^5+100 x+5 \log (4)}\right )}{26560 x^2-13280 x^3+1660 x^4+e^{10} \left (64-32 x+4 x^2\right )+e^5 \left (-2608 x+1304 x^2-163 x^3\right )+\left (2608 x-1304 x^2+163 x^3+e^5 \left (-128+64 x-8 x^2\right )\right ) \log (4)+\left (64-32 x+4 x^2\right ) \log ^2(4)} \, dx=\frac {\log \left (\frac {1}{5} \left (-4+\frac {3 x}{e^5-20 x-\log (4)}\right )\right )}{4-x} \]

output

ln(3/5*x/(exp(5)-2*ln(2)-20*x)-4/5)/(-x+4)
 

3.1.71.2 Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {e^5 (-12+3 x)+(12-3 x) \log (4)+\left (4 e^{10}-163 e^5 x+1660 x^2+\left (-8 e^5+163 x\right ) \log (4)+4 \log ^2(4)\right ) \log \left (\frac {4 e^5-83 x-4 \log (4)}{-5 e^5+100 x+5 \log (4)}\right )}{26560 x^2-13280 x^3+1660 x^4+e^{10} \left (64-32 x+4 x^2\right )+e^5 \left (-2608 x+1304 x^2-163 x^3\right )+\left (2608 x-1304 x^2+163 x^3+e^5 \left (-128+64 x-8 x^2\right )\right ) \log (4)+\left (64-32 x+4 x^2\right ) \log ^2(4)} \, dx=-\frac {\log \left (\frac {-4 e^5+83 x+\log (256)}{5 e^5-100 x-\log (1024)}\right )}{-4+x} \]

input

Integrate[(E^5*(-12 + 3*x) + (12 - 3*x)*Log[4] + (4*E^10 - 163*E^5*x + 166 
0*x^2 + (-8*E^5 + 163*x)*Log[4] + 4*Log[4]^2)*Log[(4*E^5 - 83*x - 4*Log[4] 
)/(-5*E^5 + 100*x + 5*Log[4])])/(26560*x^2 - 13280*x^3 + 1660*x^4 + E^10*( 
64 - 32*x + 4*x^2) + E^5*(-2608*x + 1304*x^2 - 163*x^3) + (2608*x - 1304*x 
^2 + 163*x^3 + E^5*(-128 + 64*x - 8*x^2))*Log[4] + (64 - 32*x + 4*x^2)*Log 
[4]^2),x]
 

output

-(Log[(-4*E^5 + 83*x + Log[256])/(5*E^5 - 100*x - Log[1024])]/(-4 + x))
 

3.1.71.3 Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 3.55 (sec) , antiderivative size = 2367, normalized size of antiderivative = 76.35, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {6, 2463, 2009}

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.

\(\displaystyle \int \frac {\left (1660 x^2-163 e^5 x+\left (163 x-8 e^5\right ) \log (4)+4 e^{10}+4 \log ^2(4)\right ) \log \left (\frac {-83 x+4 e^5-4 \log (4)}{100 x-5 e^5+5 \log (4)}\right )+e^5 (3 x-12)+(12-3 x) \log (4)}{1660 x^4-13280 x^3+26560 x^2+e^{10} \left (4 x^2-32 x+64\right )+\left (4 x^2-32 x+64\right ) \log ^2(4)+e^5 \left (-163 x^3+1304 x^2-2608 x\right )+\left (163 x^3-1304 x^2+e^5 \left (-8 x^2+64 x-128\right )+2608 x\right ) \log (4)} \, dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {\left (1660 x^2-163 e^5 x+\left (163 x-8 e^5\right ) \log (4)+4 e^{10}+4 \log ^2(4)\right ) \log \left (\frac {-83 x+4 e^5-4 \log (4)}{100 x-5 e^5+5 \log (4)}\right )+e^5 (3 x-12)+(12-3 x) \log (4)}{1660 x^4-13280 x^3+26560 x^2+\left (4 x^2-32 x+64\right ) \left (e^{10}+\log ^2(4)\right )+e^5 \left (-163 x^3+1304 x^2-2608 x\right )+\left (163 x^3-1304 x^2+e^5 \left (-8 x^2+64 x-128\right )+2608 x\right ) \log (4)}dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (-\frac {8000 \left (\left (1660 x^2-163 e^5 x+\left (163 x-8 e^5\right ) \log (4)+4 e^{10}+4 \log ^2(4)\right ) \log \left (\frac {-83 x+4 e^5-4 \log (4)}{100 x-5 e^5+5 \log (4)}\right )+e^5 (3 x-12)+(12-3 x) \log (4)\right )}{3 \left (-80+e^5-\log (4)\right )^2 \left (e^5-\log (4)\right ) \left (-20 x+e^5-\log (4)\right )}+\frac {571787 \left (\left (1660 x^2-163 e^5 x+\left (163 x-8 e^5\right ) \log (4)+4 e^{10}+4 \log ^2(4)\right ) \log \left (\frac {-83 x+4 e^5-4 \log (4)}{100 x-5 e^5+5 \log (4)}\right )+e^5 (3 x-12)+(12-3 x) \log (4)\right )}{48 \left (-83+e^5-\log (4)\right )^2 \left (e^5-\log (4)\right ) \left (-83 x+4 e^5-4 \log (4)\right )}+\frac {\left (1660 x^2-163 e^5 x+\left (163 x-8 e^5\right ) \log (4)+4 e^{10}+4 \log ^2(4)\right ) \log \left (\frac {-83 x+4 e^5-4 \log (4)}{100 x-5 e^5+5 \log (4)}\right )+e^5 (3 x-12)+(12-3 x) \log (4)}{4 (x-4)^2 \left (-83+e^5-\log (4)\right ) \left (-80+e^5-\log (4)\right )}+\frac {\left (-13280+163 e^5-163 \log (4)\right ) \left (\left (1660 x^2-163 e^5 x+\left (163 x-8 e^5\right ) \log (4)+4 e^{10}+4 \log ^2(4)\right ) \log \left (\frac {-83 x+4 e^5-4 \log (4)}{100 x-5 e^5+5 \log (4)}\right )+e^5 (3 x-12)+(12-3 x) \log (4)\right )}{16 (x-4) \left (-83+e^5-\log (4)\right )^2 \left (-80+e^5-\log (4)\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {83 \left (13280-163 e^5+163 \log (4)\right ) \log \left (-20 x-\log (4)+e^5\right ) \left (-20 x-\log (4)+e^5\right )^2}{640 \left (80-e^5+\log (4)\right )^2 \left (83-e^5+\log (4)\right )^2}-\frac {571787 \log \left (-\frac {-83 x-\log (256)+4 e^5}{5 \left (-20 x-\log (4)+e^5\right )}\right ) \left (-20 x-\log (4)+e^5\right )^2}{1920 \left (e^5-\log (4)\right ) \left (83-e^5+\log (4)\right )^2}-\frac {83 \left (13280-163 e^5+163 \log (4)\right ) \left (\log \left (-20 x-\log (4)+e^5\right )+\log \left (-\frac {-83 x-\log (256)+4 e^5}{5 \left (-20 x-\log (4)+e^5\right )}\right )-\log \left (83 x+\log (256)-4 e^5\right )\right ) \left (-20 x-\log (4)+e^5\right )^2}{640 \left (80-e^5+\log (4)\right )^2 \left (83-e^5+\log (4)\right )^2}-\frac {83 \left (13280-163 e^5+163 \log (4)\right ) \log \left (83 x+\log (256)-4 e^5\right ) \left (-20 x-\log (4)+e^5\right )^2}{640 \left (80-e^5+\log (4)\right )^2 \left (83-e^5+\log (4)\right )^2}-\frac {\left (13280-163 e^5+163 \log (4)\right ) \left (332-4 e^5+\log (256)\right ) \log \left (-20 x-\log (4)+e^5\right ) \left (-20 x-\log (4)+e^5\right )}{16 \left (80-e^5+\log (4)\right )^2 \left (83-e^5+\log (4)\right )^2}+\frac {83 \log \left (-20 x-\log (4)+e^5\right ) \left (-20 x-\log (4)+e^5\right )}{4 \left (80-e^5+\log (4)\right ) \left (83-e^5+\log (4)\right )}-\frac {5 \left (332-4 e^5+\log (256)\right ) \log (4-x)}{\left (80-e^5+\log (4)\right ) \left (83-e^5+\log (4)\right )}+\frac {3 \left (e^5-\log (4)\right ) \log (4-x)}{4 \left (80-e^5+\log (4)\right ) \left (83-e^5+\log (4)\right )}+\frac {83 \log (4-x)}{4 \left (83-e^5+\log (4)\right )}-\frac {\left (332-4 e^5+\log (256)\right ) \log \left (-20 x-\log (4)+e^5\right )}{4 (4-x) \left (83-e^5+\log (4)\right )}+\frac {5 \left (332-4 e^5+\log (256)\right ) \log \left (-20 x-\log (4)+e^5\right )}{\left (80-e^5+\log (4)\right ) \left (83-e^5+\log (4)\right )}+\frac {10 \left (3 e^5-\log (64)\right )^2 \log \left (-20 x-\log (4)+e^5\right )}{249 \left (e^5-\log (4)\right ) \left (80-e^5+\log (4)\right )^2}-\frac {20 \log \left (-20 x-\log (4)+e^5\right )}{80-e^5+\log (4)}+\frac {\left (163 e^5-83 \log (4)-20 (664+\log (256))\right ) \log \left (-20 x-\log (4)+e^5\right ) \log \left (\frac {20 (4-x)}{80-e^5+\log (4)}\right )}{4 \left (80-e^5+\log (4)\right ) \left (83-e^5+\log (4)\right )}+\frac {\left (13280-163 e^5+163 \log (4)\right ) \left (332-4 e^5+\log (256)\right ) \log \left (-20 x-\log (4)+e^5\right ) \log \left (\frac {20 (4-x)}{80-e^5+\log (4)}\right )}{16 \left (80-e^5+\log (4)\right ) \left (83-e^5+\log (4)\right )^2}+\frac {83 \left (332-4 e^5+\log (256)\right ) \log \left (-83 x-\log (256)+4 e^5\right )}{16 \left (83-e^5+\log (4)\right )^2}+\frac {\left (13280-163 e^5+163 \log (4)\right ) \left (3 e^5-\log (64)\right )^2 \log \left (-83 x-\log (256)+4 e^5\right )}{53120 \left (80-e^5+\log (4)\right )^2 \left (83-e^5+\log (4)\right )^2}+\frac {83 \left (3 e^5-\log (64)\right )^2 \log \left (-83 x-\log (256)+4 e^5\right )}{1920 \left (e^5-\log (4)\right ) \left (83-e^5+\log (4)\right )^2}-\frac {83 \log \left (-83 x-\log (256)+4 e^5\right )}{4 \left (83-e^5+\log (4)\right )}+\frac {4000 \left (-83 x-\log (256)+4 e^5\right )^2 \log \left (-\frac {-83 x-\log (256)+4 e^5}{5 \left (-20 x-\log (4)+e^5\right )}\right )}{249 \left (e^5-\log (4)\right ) \left (80-e^5+\log (4)\right )^2}+\frac {\left (13280-163 e^5+83 \log (4)+20 \log (256)\right ) \log (4-x) \left (\log \left (-20 x-\log (4)+e^5\right )+\log \left (-\frac {-83 x-\log (256)+4 e^5}{5 \left (-20 x-\log (4)+e^5\right )}\right )-\log \left (83 x+\log (256)-4 e^5\right )\right )}{4 \left (80-e^5+\log (4)\right ) \left (83-e^5+\log (4)\right )}-\frac {\left (13280-163 e^5+163 \log (4)\right ) \left (332-4 e^5+\log (256)\right ) \log (4-x) \left (\log \left (-20 x-\log (4)+e^5\right )+\log \left (-\frac {-83 x-\log (256)+4 e^5}{5 \left (-20 x-\log (4)+e^5\right )}\right )-\log \left (83 x+\log (256)-4 e^5\right )\right )}{16 \left (80-e^5+\log (4)\right ) \left (83-e^5+\log (4)\right )^2}-\frac {5 x \left (13280-163 e^5+163 \log (4)\right ) \left (332-4 e^5+\log (256)\right ) \left (\log \left (-20 x-\log (4)+e^5\right )+\log \left (-\frac {-83 x-\log (256)+4 e^5}{5 \left (-20 x-\log (4)+e^5\right )}\right )-\log \left (83 x+\log (256)-4 e^5\right )\right )}{4 \left (80-e^5+\log (4)\right )^2 \left (83-e^5+\log (4)\right )^2}+\frac {\left (332-4 e^5+\log (256)\right ) \left (\log \left (-20 x-\log (4)+e^5\right )+\log \left (-\frac {-83 x-\log (256)+4 e^5}{5 \left (-20 x-\log (4)+e^5\right )}\right )-\log \left (83 x+\log (256)-4 e^5\right )\right )}{4 (4-x) \left (83-e^5+\log (4)\right )}+\frac {415 x \left (\log \left (-20 x-\log (4)+e^5\right )+\log \left (-\frac {-83 x-\log (256)+4 e^5}{5 \left (-20 x-\log (4)+e^5\right )}\right )-\log \left (83 x+\log (256)-4 e^5\right )\right )}{\left (80-e^5+\log (4)\right ) \left (83-e^5+\log (4)\right )}-\frac {5 \left (-83 x-\log (256)+4 e^5\right ) \log \left (83 x+\log (256)-4 e^5\right )}{\left (80-e^5+\log (4)\right ) \left (83-e^5+\log (4)\right )}+\frac {\left (13280-163 e^5+83 \log (4)+20 \log (256)\right ) \log \left (\frac {83 (4-x)}{332-4 e^5+\log (256)}\right ) \log \left (83 x+\log (256)-4 e^5\right )}{4 \left (80-e^5+\log (4)\right ) \left (83-e^5+\log (4)\right )}-\frac {\left (13280-163 e^5+163 \log (4)\right ) \left (332-4 e^5+\log (256)\right ) \log \left (\frac {83 (4-x)}{332-4 e^5+\log (256)}\right ) \log \left (83 x+\log (256)-4 e^5\right )}{16 \left (80-e^5+\log (4)\right ) \left (83-e^5+\log (4)\right )^2}+\frac {5 \left (13280-163 e^5+163 \log (4)\right ) \left (-83 x-\log (256)+4 e^5\right ) \left (332-4 e^5+\log (256)\right ) \log \left (83 x+\log (256)-4 e^5\right )}{332 \left (80-e^5+\log (4)\right )^2 \left (83-e^5+\log (4)\right )^2}+\frac {\left (332-4 e^5+\log (256)\right ) \log \left (83 x+\log (256)-4 e^5\right )}{4 (4-x) \left (83-e^5+\log (4)\right )}+\frac {\left (163 e^5-83 \log (4)-20 (664+\log (256))\right ) \operatorname {PolyLog}\left (2,-\frac {-20 x-\log (4)+e^5}{80-e^5+\log (4)}\right )}{4 \left (80-e^5+\log (4)\right ) \left (83-e^5+\log (4)\right )}+\frac {\left (13280-163 e^5+163 \log (4)\right ) \left (332-4 e^5+\log (256)\right ) \operatorname {PolyLog}\left (2,-\frac {-20 x-\log (4)+e^5}{80-e^5+\log (4)}\right )}{16 \left (80-e^5+\log (4)\right ) \left (83-e^5+\log (4)\right )^2}+\frac {\left (13280-163 e^5+83 \log (4)+20 \log (256)\right ) \operatorname {PolyLog}\left (2,-\frac {-83 x-\log (256)+4 e^5}{332-4 e^5+\log (256)}\right )}{4 \left (80-e^5+\log (4)\right ) \left (83-e^5+\log (4)\right )}-\frac {\left (13280-163 e^5+163 \log (4)\right ) \left (332-4 e^5+\log (256)\right ) \operatorname {PolyLog}\left (2,-\frac {-83 x-\log (256)+4 e^5}{332-4 e^5+\log (256)}\right )}{16 \left (80-e^5+\log (4)\right ) \left (83-e^5+\log (4)\right )^2}-\frac {x \left (13280-163 e^5+163 \log (4)\right ) \left (3 e^5-\log (64)\right )}{32 \left (80-e^5+\log (4)\right )^2 \left (83-e^5+\log (4)\right )^2}-\frac {6889 x \left (3 e^5-\log (64)\right )}{96 \left (e^5-\log (4)\right ) \left (83-e^5+\log (4)\right )^2}+\frac {200 x \left (3 e^5-\log (64)\right )}{3 \left (e^5-\log (4)\right ) \left (80-e^5+\log (4)\right )^2}-\frac {3 x \left (e^5-\log (4)\right ) \left (13280-163 e^5+163 \log (4)\right )}{16 \left (80-e^5+\log (4)\right )^2 \left (83-e^5+\log (4)\right )^2}-\frac {6889 x}{16 \left (83-e^5+\log (4)\right )^2}+\frac {400 x}{\left (80-e^5+\log (4)\right )^2}\)

input

Int[(E^5*(-12 + 3*x) + (12 - 3*x)*Log[4] + (4*E^10 - 163*E^5*x + 1660*x^2 
+ (-8*E^5 + 163*x)*Log[4] + 4*Log[4]^2)*Log[(4*E^5 - 83*x - 4*Log[4])/(-5* 
E^5 + 100*x + 5*Log[4])])/(26560*x^2 - 13280*x^3 + 1660*x^4 + E^10*(64 - 3 
2*x + 4*x^2) + E^5*(-2608*x + 1304*x^2 - 163*x^3) + (2608*x - 1304*x^2 + 1 
63*x^3 + E^5*(-128 + 64*x - 8*x^2))*Log[4] + (64 - 32*x + 4*x^2)*Log[4]^2) 
,x]
 

output

(400*x)/(80 - E^5 + Log[4])^2 - (6889*x)/(16*(83 - E^5 + Log[4])^2) - (3*x 
*(E^5 - Log[4])*(13280 - 163*E^5 + 163*Log[4]))/(16*(80 - E^5 + Log[4])^2* 
(83 - E^5 + Log[4])^2) + (200*x*(3*E^5 - Log[64]))/(3*(E^5 - Log[4])*(80 - 
 E^5 + Log[4])^2) - (6889*x*(3*E^5 - Log[64]))/(96*(E^5 - Log[4])*(83 - E^ 
5 + Log[4])^2) - (x*(13280 - 163*E^5 + 163*Log[4])*(3*E^5 - Log[64]))/(32* 
(80 - E^5 + Log[4])^2*(83 - E^5 + Log[4])^2) + (83*Log[4 - x])/(4*(83 - E^ 
5 + Log[4])) + (3*(E^5 - Log[4])*Log[4 - x])/(4*(80 - E^5 + Log[4])*(83 - 
E^5 + Log[4])) - (5*(332 - 4*E^5 + Log[256])*Log[4 - x])/((80 - E^5 + Log[ 
4])*(83 - E^5 + Log[4])) - (20*Log[E^5 - 20*x - Log[4]])/(80 - E^5 + Log[4 
]) + (83*(E^5 - 20*x - Log[4])*Log[E^5 - 20*x - Log[4]])/(4*(80 - E^5 + Lo 
g[4])*(83 - E^5 + Log[4])) + (83*(E^5 - 20*x - Log[4])^2*(13280 - 163*E^5 
+ 163*Log[4])*Log[E^5 - 20*x - Log[4]])/(640*(80 - E^5 + Log[4])^2*(83 - E 
^5 + Log[4])^2) + (10*(3*E^5 - Log[64])^2*Log[E^5 - 20*x - Log[4]])/(249*( 
E^5 - Log[4])*(80 - E^5 + Log[4])^2) - ((332 - 4*E^5 + Log[256])*Log[E^5 - 
 20*x - Log[4]])/(4*(4 - x)*(83 - E^5 + Log[4])) + (5*(332 - 4*E^5 + Log[2 
56])*Log[E^5 - 20*x - Log[4]])/((80 - E^5 + Log[4])*(83 - E^5 + Log[4])) - 
 ((E^5 - 20*x - Log[4])*(13280 - 163*E^5 + 163*Log[4])*(332 - 4*E^5 + Log[ 
256])*Log[E^5 - 20*x - Log[4]])/(16*(80 - E^5 + Log[4])^2*(83 - E^5 + Log[ 
4])^2) + ((13280 - 163*E^5 + 163*Log[4])*(332 - 4*E^5 + Log[256])*Log[E^5 
- 20*x - Log[4]]*Log[(20*(4 - x))/(80 - E^5 + Log[4])])/(16*(80 - E^5 +...
 

3.1.71.3.1 Defintions of rubi rules used

rule 6

Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v 
+ (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] &&  !FreeQ[Fx, x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2463

Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr 
and[u, Qx^p, x], x] /;  !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt 
Q[Expon[Px, x], 2] &&  !BinomialQ[Px, x] &&  !TrinomialQ[Px, x] && ILtQ[p, 
0]
 

3.1.71.4 Maple [A] (verified)

Time = 1.32 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13

method	result	size
parallelrisch	\(-\frac {\ln \left (-\frac {-8 \ln \left (2\right )+4 \,{\mathrm e}^{5}-83 x}{5 \left ({\mathrm e}^{5}-2 \ln \left (2\right )-20 x \right )}\right )}{x -4}\)	\(35\)
norman	\(-\frac {\ln \left (\frac {-8 \ln \left (2\right )+4 \,{\mathrm e}^{5}-83 x}{10 \ln \left (2\right )-5 \,{\mathrm e}^{5}+100 x}\right )}{x -4}\)	\(36\)
risch	\(-\frac {\ln \left (\frac {-8 \ln \left (2\right )+4 \,{\mathrm e}^{5}-83 x}{10 \ln \left (2\right )-5 \,{\mathrm e}^{5}+100 x}\right )}{x -4}\)	\(36\)
derivativedivides	\(\text {Expression too large to display}\)	\(1298\)
default	\(\text {Expression too large to display}\)	\(1298\)
parts	\(\text {Expression too large to display}\)	\(1304\)

input

int(((16*ln(2)^2+2*(-8*exp(5)+163*x)*ln(2)+4*exp(5)^2-163*x*exp(5)+1660*x^ 
2)*ln((-8*ln(2)+4*exp(5)-83*x)/(10*ln(2)-5*exp(5)+100*x))+2*(-3*x+12)*ln(2 
)+(3*x-12)*exp(5))/(4*(4*x^2-32*x+64)*ln(2)^2+2*((-8*x^2+64*x-128)*exp(5)+ 
163*x^3-1304*x^2+2608*x)*ln(2)+(4*x^2-32*x+64)*exp(5)^2+(-163*x^3+1304*x^2 
-2608*x)*exp(5)+1660*x^4-13280*x^3+26560*x^2),x,method=_RETURNVERBOSE)
 

output

-ln(-1/5*(-8*ln(2)+4*exp(5)-83*x)/(exp(5)-2*ln(2)-20*x))/(x-4)
 

3.1.71.5 Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {e^5 (-12+3 x)+(12-3 x) \log (4)+\left (4 e^{10}-163 e^5 x+1660 x^2+\left (-8 e^5+163 x\right ) \log (4)+4 \log ^2(4)\right ) \log \left (\frac {4 e^5-83 x-4 \log (4)}{-5 e^5+100 x+5 \log (4)}\right )}{26560 x^2-13280 x^3+1660 x^4+e^{10} \left (64-32 x+4 x^2\right )+e^5 \left (-2608 x+1304 x^2-163 x^3\right )+\left (2608 x-1304 x^2+163 x^3+e^5 \left (-128+64 x-8 x^2\right )\right ) \log (4)+\left (64-32 x+4 x^2\right ) \log ^2(4)} \, dx=-\frac {\log \left (-\frac {83 \, x - 4 \, e^{5} + 8 \, \log \left (2\right )}{5 \, {\left (20 \, x - e^{5} + 2 \, \log \left (2\right )\right )}}\right )}{x - 4} \]

input

integrate(((16*log(2)^2+2*(-8*exp(5)+163*x)*log(2)+4*exp(5)^2-163*x*exp(5) 
+1660*x^2)*log((-8*log(2)+4*exp(5)-83*x)/(10*log(2)-5*exp(5)+100*x))+2*(-3 
*x+12)*log(2)+(3*x-12)*exp(5))/(4*(4*x^2-32*x+64)*log(2)^2+2*((-8*x^2+64*x 
-128)*exp(5)+163*x^3-1304*x^2+2608*x)*log(2)+(4*x^2-32*x+64)*exp(5)^2+(-16 
3*x^3+1304*x^2-2608*x)*exp(5)+1660*x^4-13280*x^3+26560*x^2),x, algorithm=\
 

output

-log(-1/5*(83*x - 4*e^5 + 8*log(2))/(20*x - e^5 + 2*log(2)))/(x - 4)
 

3.1.71.6 Sympy [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {e^5 (-12+3 x)+(12-3 x) \log (4)+\left (4 e^{10}-163 e^5 x+1660 x^2+\left (-8 e^5+163 x\right ) \log (4)+4 \log ^2(4)\right ) \log \left (\frac {4 e^5-83 x-4 \log (4)}{-5 e^5+100 x+5 \log (4)}\right )}{26560 x^2-13280 x^3+1660 x^4+e^{10} \left (64-32 x+4 x^2\right )+e^5 \left (-2608 x+1304 x^2-163 x^3\right )+\left (2608 x-1304 x^2+163 x^3+e^5 \left (-128+64 x-8 x^2\right )\right ) \log (4)+\left (64-32 x+4 x^2\right ) \log ^2(4)} \, dx=- \frac {\log {\left (\frac {- 83 x - 8 \log {\left (2 \right )} + 4 e^{5}}{100 x - 5 e^{5} + 10 \log {\left (2 \right )}} \right )}}{x - 4} \]

input

integrate(((16*ln(2)**2+2*(-8*exp(5)+163*x)*ln(2)+4*exp(5)**2-163*x*exp(5) 
+1660*x**2)*ln((-8*ln(2)+4*exp(5)-83*x)/(10*ln(2)-5*exp(5)+100*x))+2*(-3*x 
+12)*ln(2)+(3*x-12)*exp(5))/(4*(4*x**2-32*x+64)*ln(2)**2+2*((-8*x**2+64*x- 
128)*exp(5)+163*x**3-1304*x**2+2608*x)*ln(2)+(4*x**2-32*x+64)*exp(5)**2+(- 
163*x**3+1304*x**2-2608*x)*exp(5)+1660*x**4-13280*x**3+26560*x**2),x)
 

output

-log((-83*x - 8*log(2) + 4*exp(5))/(100*x - 5*exp(5) + 10*log(2)))/(x - 4)
 

3.1.71.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1260 vs. \(2 (27) = 54\).

Time = 0.43 (sec) , antiderivative size = 1260, normalized size of antiderivative = 40.65 \[ \int \frac {e^5 (-12+3 x)+(12-3 x) \log (4)+\left (4 e^{10}-163 e^5 x+1660 x^2+\left (-8 e^5+163 x\right ) \log (4)+4 \log ^2(4)\right ) \log \left (\frac {4 e^5-83 x-4 \log (4)}{-5 e^5+100 x+5 \log (4)}\right )}{26560 x^2-13280 x^3+1660 x^4+e^{10} \left (64-32 x+4 x^2\right )+e^5 \left (-2608 x+1304 x^2-163 x^3\right )+\left (2608 x-1304 x^2+163 x^3+e^5 \left (-128+64 x-8 x^2\right )\right ) \log (4)+\left (64-32 x+4 x^2\right ) \log ^2(4)} \, dx=\text {Too large to display} \]

input

integrate(((16*log(2)^2+2*(-8*exp(5)+163*x)*log(2)+4*exp(5)^2-163*x*exp(5) 
+1660*x^2)*log((-8*log(2)+4*exp(5)-83*x)/(10*log(2)-5*exp(5)+100*x))+2*(-3 
*x+12)*log(2)+(3*x-12)*exp(5))/(4*(4*x^2-32*x+64)*log(2)^2+2*((-8*x^2+64*x 
-128)*exp(5)+163*x^3-1304*x^2+2608*x)*log(2)+(4*x^2-32*x+64)*exp(5)^2+(-16 
3*x^3+1304*x^2-2608*x)*exp(5)+1660*x^4-13280*x^3+26560*x^2),x, algorithm=\
 

output

1/4*(3*(4*e^5*log(2) - 4*log(2)^2 - e^10 + 6640)*log(x - 4)/(16*(2*e^5 - 1 
63)*log(2)^3 - 16*log(2)^4 - 12*(2*e^10 - 326*e^5 + 13283)*log(2)^2 + 4*(2 
*e^15 - 489*e^10 + 39849*e^5 - 1082320)*log(2) - e^20 + 326*e^15 - 39849*e 
^10 + 2164640*e^5 - 44089600) + 83*log(83*x - 4*e^5 + 8*log(2))/(4*(e^5 - 
83)*log(2) - 4*log(2)^2 - e^10 + 166*e^5 - 6889) - 80*log(20*x - e^5 + 2*l 
og(2))/(4*(e^5 - 80)*log(2) - 4*log(2)^2 - e^10 + 160*e^5 - 6400) + 12/((2 
*(2*e^5 - 163)*log(2) - 4*log(2)^2 - e^10 + 163*e^5 - 6640)*x - 8*(2*e^5 - 
 163)*log(2) + 16*log(2)^2 + 4*e^10 - 652*e^5 + 26560))*e^5 + 1/4*(3*(163* 
e^5 - 326*log(2) - 13280)*log(x - 4)/(16*(2*e^5 - 163)*log(2)^3 - 16*log(2 
)^4 - 12*(2*e^10 - 326*e^5 + 13283)*log(2)^2 + 4*(2*e^15 - 489*e^10 + 3984 
9*e^5 - 1082320)*log(2) - e^20 + 326*e^15 - 39849*e^10 + 2164640*e^5 - 440 
89600) + 6889*log(83*x - 4*e^5 + 8*log(2))/(4*(3*e^5 - 166)*log(2)^2 - 8*l 
og(2)^3 - 2*(3*e^10 - 332*e^5 + 6889)*log(2) + e^15 - 166*e^10 + 6889*e^5) 
 - 6400*log(20*x - e^5 + 2*log(2))/(4*(3*e^5 - 160)*log(2)^2 - 8*log(2)^3 
- 2*(3*e^10 - 320*e^5 + 6400)*log(2) + e^15 - 160*e^10 + 6400*e^5) - 12/(( 
2*(2*e^5 - 163)*log(2) - 4*log(2)^2 - e^10 + 163*e^5 - 6640)*x - 8*(2*e^5 
- 163)*log(2) + 16*log(2)^2 + 4*e^10 - 652*e^5 + 26560))*e^5 - 1/2*(3*(4*e 
^5*log(2) - 4*log(2)^2 - e^10 + 6640)*log(x - 4)/(16*(2*e^5 - 163)*log(2)^ 
3 - 16*log(2)^4 - 12*(2*e^10 - 326*e^5 + 13283)*log(2)^2 + 4*(2*e^15 - 489 
*e^10 + 39849*e^5 - 1082320)*log(2) - e^20 + 326*e^15 - 39849*e^10 + 21...
 

3.1.71.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 295 vs. \(2 (27) = 54\).

Time = 0.71 (sec) , antiderivative size = 295, normalized size of antiderivative = 9.52 \[ \int \frac {e^5 (-12+3 x)+(12-3 x) \log (4)+\left (4 e^{10}-163 e^5 x+1660 x^2+\left (-8 e^5+163 x\right ) \log (4)+4 \log ^2(4)\right ) \log \left (\frac {4 e^5-83 x-4 \log (4)}{-5 e^5+100 x+5 \log (4)}\right )}{26560 x^2-13280 x^3+1660 x^4+e^{10} \left (64-32 x+4 x^2\right )+e^5 \left (-2608 x+1304 x^2-163 x^3\right )+\left (2608 x-1304 x^2+163 x^3+e^5 \left (-128+64 x-8 x^2\right )\right ) \log (4)+\left (64-32 x+4 x^2\right ) \log ^2(4)} \, dx=-\frac {\frac {20 \, {\left (83 \, x - 4 \, e^{5} + 8 \, \log \left (2\right )\right )} e^{5} \log \left (-\frac {83 \, x - 4 \, e^{5} + 8 \, \log \left (2\right )}{5 \, {\left (20 \, x - e^{5} + 2 \, \log \left (2\right )\right )}}\right )}{20 \, x - e^{5} + 2 \, \log \left (2\right )} - 83 \, e^{5} \log \left (-\frac {83 \, x - 4 \, e^{5} + 8 \, \log \left (2\right )}{5 \, {\left (20 \, x - e^{5} + 2 \, \log \left (2\right )\right )}}\right ) - \frac {40 \, {\left (83 \, x - 4 \, e^{5} + 8 \, \log \left (2\right )\right )} \log \left (2\right ) \log \left (-\frac {83 \, x - 4 \, e^{5} + 8 \, \log \left (2\right )}{5 \, {\left (20 \, x - e^{5} + 2 \, \log \left (2\right )\right )}}\right )}{20 \, x - e^{5} + 2 \, \log \left (2\right )} + 166 \, \log \left (2\right ) \log \left (-\frac {83 \, x - 4 \, e^{5} + 8 \, \log \left (2\right )}{5 \, {\left (20 \, x - e^{5} + 2 \, \log \left (2\right )\right )}}\right )}{{\left (\frac {{\left (83 \, x - 4 \, e^{5} + 8 \, \log \left (2\right )\right )} e^{5}}{20 \, x - e^{5} + 2 \, \log \left (2\right )} - \frac {2 \, {\left (83 \, x - 4 \, e^{5} + 8 \, \log \left (2\right )\right )} \log \left (2\right )}{20 \, x - e^{5} + 2 \, \log \left (2\right )} - \frac {80 \, {\left (83 \, x - 4 \, e^{5} + 8 \, \log \left (2\right )\right )}}{20 \, x - e^{5} + 2 \, \log \left (2\right )} - 4 \, e^{5} + 8 \, \log \left (2\right ) + 332\right )} {\left (e^{5} - 2 \, \log \left (2\right )\right )}} \]

input

integrate(((16*log(2)^2+2*(-8*exp(5)+163*x)*log(2)+4*exp(5)^2-163*x*exp(5) 
+1660*x^2)*log((-8*log(2)+4*exp(5)-83*x)/(10*log(2)-5*exp(5)+100*x))+2*(-3 
*x+12)*log(2)+(3*x-12)*exp(5))/(4*(4*x^2-32*x+64)*log(2)^2+2*((-8*x^2+64*x 
-128)*exp(5)+163*x^3-1304*x^2+2608*x)*log(2)+(4*x^2-32*x+64)*exp(5)^2+(-16 
3*x^3+1304*x^2-2608*x)*exp(5)+1660*x^4-13280*x^3+26560*x^2),x, algorithm=\
 

output

-(20*(83*x - 4*e^5 + 8*log(2))*e^5*log(-1/5*(83*x - 4*e^5 + 8*log(2))/(20* 
x - e^5 + 2*log(2)))/(20*x - e^5 + 2*log(2)) - 83*e^5*log(-1/5*(83*x - 4*e 
^5 + 8*log(2))/(20*x - e^5 + 2*log(2))) - 40*(83*x - 4*e^5 + 8*log(2))*log 
(2)*log(-1/5*(83*x - 4*e^5 + 8*log(2))/(20*x - e^5 + 2*log(2)))/(20*x - e^ 
5 + 2*log(2)) + 166*log(2)*log(-1/5*(83*x - 4*e^5 + 8*log(2))/(20*x - e^5 
+ 2*log(2))))/(((83*x - 4*e^5 + 8*log(2))*e^5/(20*x - e^5 + 2*log(2)) - 2* 
(83*x - 4*e^5 + 8*log(2))*log(2)/(20*x - e^5 + 2*log(2)) - 80*(83*x - 4*e^ 
5 + 8*log(2))/(20*x - e^5 + 2*log(2)) - 4*e^5 + 8*log(2) + 332)*(e^5 - 2*l 
og(2)))
 

3.1.71.9 Mupad [B] (verification not implemented)

Time = 10.76 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {e^5 (-12+3 x)+(12-3 x) \log (4)+\left (4 e^{10}-163 e^5 x+1660 x^2+\left (-8 e^5+163 x\right ) \log (4)+4 \log ^2(4)\right ) \log \left (\frac {4 e^5-83 x-4 \log (4)}{-5 e^5+100 x+5 \log (4)}\right )}{26560 x^2-13280 x^3+1660 x^4+e^{10} \left (64-32 x+4 x^2\right )+e^5 \left (-2608 x+1304 x^2-163 x^3\right )+\left (2608 x-1304 x^2+163 x^3+e^5 \left (-128+64 x-8 x^2\right )\right ) \log (4)+\left (64-32 x+4 x^2\right ) \log ^2(4)} \, dx=-\frac {\ln \left (-\frac {83\,x-4\,{\mathrm {e}}^5+8\,\ln \left (2\right )}{5\,\left (20\,x-{\mathrm {e}}^5+2\,\ln \left (2\right )\right )}\right )}{x-4} \]

input

int((log(-(83*x - 4*exp(5) + 8*log(2))/(100*x - 5*exp(5) + 10*log(2)))*(4* 
exp(10) - 163*x*exp(5) + 2*log(2)*(163*x - 8*exp(5)) + 16*log(2)^2 + 1660* 
x^2) - 2*log(2)*(3*x - 12) + exp(5)*(3*x - 12))/(exp(10)*(4*x^2 - 32*x + 6 
4) - exp(5)*(2608*x - 1304*x^2 + 163*x^3) + 4*log(2)^2*(4*x^2 - 32*x + 64) 
 + 2*log(2)*(2608*x - exp(5)*(8*x^2 - 64*x + 128) - 1304*x^2 + 163*x^3) + 
26560*x^2 - 13280*x^3 + 1660*x^4),x)
 

output

-log(-(83*x - 4*exp(5) + 8*log(2))/(5*(20*x - exp(5) + 2*log(2))))/(x - 4)