\(\int \frac {4800-4800 x+3600 x^2-4800 x^3+2100 x^4-900 x^5+75 x^6-75 x^7+e^8 (-600 x+600 x^2+150 x^3-150 x^4)+e^4 (1800 x^2-1800 x^3-150 x^4+150 x^5)+(4800-9600 x+3600 x^2-7500 x^3+1500 x^4-1875 x^5+225 x^6-150 x^7+e^8 (-300 x+600 x^2-75 x^3+150 x^4)+e^4 (600 x^2-1200 x^3+150 x^4-300 x^5)) \log (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4})+(-8000+16000 x-6000 x^2+12500 x^3-2500 x^4+3125 x^5-375 x^6+250 x^7+e^8 (500 x-1000 x^2+125 x^3-250 x^4)+e^4 (-1000 x^2+2000 x^3-250 x^4+500 x^5)) \log ^2(\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4})}{-576-432 x^2+36 x^3-108 x^4+9 x^5-9 x^6+e^8 (36 x+9 x^3)+e^4 (-72 x^2-18 x^4)+(1920+1440 x^2-120 x^3+360 x^4-30 x^5+30 x^6+e^8 (-120 x-30 x^3)+e^4 (240 x^2+60 x^4)) \log (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4})+(-1600-1200 x^2+100 x^3-300 x^4+25 x^5-25 x^6+e^8 (100 x+25 x^3)+e^4 (-200 x^2-50 x^4)) \log ^2(\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4})} \, dx\) [6647]

Optimal result

Integrand size = 613, antiderivative size = 40 \[ \int \frac {4800-4800 x+3600 x^2-4800 x^3+2100 x^4-900 x^5+75 x^6-75 x^7+e^8 \left (-600 x+600 x^2+150 x^3-150 x^4\right )+e^4 \left (1800 x^2-1800 x^3-150 x^4+150 x^5\right )+\left (4800-9600 x+3600 x^2-7500 x^3+1500 x^4-1875 x^5+225 x^6-150 x^7+e^8 \left (-300 x+600 x^2-75 x^3+150 x^4\right )+e^4 \left (600 x^2-1200 x^3+150 x^4-300 x^5\right )\right ) \log \left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )+\left (-8000+16000 x-6000 x^2+12500 x^3-2500 x^4+3125 x^5-375 x^6+250 x^7+e^8 \left (500 x-1000 x^2+125 x^3-250 x^4\right )+e^4 \left (-1000 x^2+2000 x^3-250 x^4+500 x^5\right )\right ) \log ^2\left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )}{-576-432 x^2+36 x^3-108 x^4+9 x^5-9 x^6+e^8 \left (36 x+9 x^3\right )+e^4 \left (-72 x^2-18 x^4\right )+\left (1920+1440 x^2-120 x^3+360 x^4-30 x^5+30 x^6+e^8 \left (-120 x-30 x^3\right )+e^4 \left (240 x^2+60 x^4\right )\right ) \log \left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )+\left (-1600-1200 x^2+100 x^3-300 x^4+25 x^5-25 x^6+e^8 \left (100 x+25 x^3\right )+e^4 \left (-200 x^2-50 x^4\right )\right ) \log ^2\left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )} \, dx=\frac {25 \left (-x+x^2\right )}{-5+\frac {3}{\log \left (x-\frac {\left (e^4-x\right )^2}{\left (\frac {4}{x}+x\right )^2}\right )}} \]

[Out]

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

Rubi [F]

\[ \int \frac {4800-4800 x+3600 x^2-4800 x^3+2100 x^4-900 x^5+75 x^6-75 x^7+e^8 \left (-600 x+600 x^2+150 x^3-150 x^4\right )+e^4 \left (1800 x^2-1800 x^3-150 x^4+150 x^5\right )+\left (4800-9600 x+3600 x^2-7500 x^3+1500 x^4-1875 x^5+225 x^6-150 x^7+e^8 \left (-300 x+600 x^2-75 x^3+150 x^4\right )+e^4 \left (600 x^2-1200 x^3+150 x^4-300 x^5\right )\right ) \log \left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )+\left (-8000+16000 x-6000 x^2+12500 x^3-2500 x^4+3125 x^5-375 x^6+250 x^7+e^8 \left (500 x-1000 x^2+125 x^3-250 x^4\right )+e^4 \left (-1000 x^2+2000 x^3-250 x^4+500 x^5\right )\right ) \log ^2\left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )}{-576-432 x^2+36 x^3-108 x^4+9 x^5-9 x^6+e^8 \left (36 x+9 x^3\right )+e^4 \left (-72 x^2-18 x^4\right )+\left (1920+1440 x^2-120 x^3+360 x^4-30 x^5+30 x^6+e^8 \left (-120 x-30 x^3\right )+e^4 \left (240 x^2+60 x^4\right )\right ) \log \left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )+\left (-1600-1200 x^2+100 x^3-300 x^4+25 x^5-25 x^6+e^8 \left (100 x+25 x^3\right )+e^4 \left (-200 x^2-50 x^4\right )\right ) \log ^2\left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )} \, dx=\int \frac {4800-4800 x+3600 x^2-4800 x^3+2100 x^4-900 x^5+75 x^6-75 x^7+e^8 \left (-600 x+600 x^2+150 x^3-150 x^4\right )+e^4 \left (1800 x^2-1800 x^3-150 x^4+150 x^5\right )+\left (4800-9600 x+3600 x^2-7500 x^3+1500 x^4-1875 x^5+225 x^6-150 x^7+e^8 \left (-300 x+600 x^2-75 x^3+150 x^4\right )+e^4 \left (600 x^2-1200 x^3+150 x^4-300 x^5\right )\right ) \log \left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )+\left (-8000+16000 x-6000 x^2+12500 x^3-2500 x^4+3125 x^5-375 x^6+250 x^7+e^8 \left (500 x-1000 x^2+125 x^3-250 x^4\right )+e^4 \left (-1000 x^2+2000 x^3-250 x^4+500 x^5\right )\right ) \log ^2\left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )}{-576-432 x^2+36 x^3-108 x^4+9 x^5-9 x^6+e^8 \left (36 x+9 x^3\right )+e^4 \left (-72 x^2-18 x^4\right )+\left (1920+1440 x^2-120 x^3+360 x^4-30 x^5+30 x^6+e^8 \left (-120 x-30 x^3\right )+e^4 \left (240 x^2+60 x^4\right )\right ) \log \left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )+\left (-1600-1200 x^2+100 x^3-300 x^4+25 x^5-25 x^6+e^8 \left (100 x+25 x^3\right )+e^4 \left (-200 x^2-50 x^4\right )\right ) \log ^2\left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )} \, dx \]

[In]

Int[(4800 - 4800*x + 3600*x^2 - 4800*x^3 + 2100*x^4 - 900*x^5 + 75*x^6 - 75*x^7 + E^8*(-600*x + 600*x^2 + 150*
x^3 - 150*x^4) + E^4*(1800*x^2 - 1800*x^3 - 150*x^4 + 150*x^5) + (4800 - 9600*x + 3600*x^2 - 7500*x^3 + 1500*x
^4 - 1875*x^5 + 225*x^6 - 150*x^7 + E^8*(-300*x + 600*x^2 - 75*x^3 + 150*x^4) + E^4*(600*x^2 - 1200*x^3 + 150*
x^4 - 300*x^5))*Log[(16*x - E^8*x^2 + 8*x^3 + 2*E^4*x^3 - x^4 + x^5)/(16 + 8*x^2 + x^4)] + (-8000 + 16000*x -
6000*x^2 + 12500*x^3 - 2500*x^4 + 3125*x^5 - 375*x^6 + 250*x^7 + E^8*(500*x - 1000*x^2 + 125*x^3 - 250*x^4) +
E^4*(-1000*x^2 + 2000*x^3 - 250*x^4 + 500*x^5))*Log[(16*x - E^8*x^2 + 8*x^3 + 2*E^4*x^3 - x^4 + x^5)/(16 + 8*x
^2 + x^4)]^2)/(-576 - 432*x^2 + 36*x^3 - 108*x^4 + 9*x^5 - 9*x^6 + E^8*(36*x + 9*x^3) + E^4*(-72*x^2 - 18*x^4)
+ (1920 + 1440*x^2 - 120*x^3 + 360*x^4 - 30*x^5 + 30*x^6 + E^8*(-120*x - 30*x^3) + E^4*(240*x^2 + 60*x^4))*Lo
g[(16*x - E^8*x^2 + 8*x^3 + 2*E^4*x^3 - x^4 + x^5)/(16 + 8*x^2 + x^4)] + (-1600 - 1200*x^2 + 100*x^3 - 300*x^4
+ 25*x^5 - 25*x^6 + E^8*(100*x + 25*x^3) + E^4*(-200*x^2 - 50*x^4))*Log[(16*x - E^8*x^2 + 8*x^3 + 2*E^4*x^3 -
x^4 + x^5)/(16 + 8*x^2 + x^4)]^2),x]

[Out]

(-5*(1 - 2*x)^2)/4 + 3600*Defer[Int][1/((16 - E^8*x + 2*(4 + E^4)*x^2 - x^3 + x^4)*(3 - 5*Log[(x*(16 - E^8*x +
8*x^2 + 2*E^4*x^2 - x^3 + x^4))/(4 + x^2)^2])^2), x] - 150*(32 + E^8)*Defer[Int][x/((16 - E^8*x + 2*(4 + E^4)
*x^2 - x^3 + x^4)*(3 - 5*Log[(x*(16 - E^8*x + 8*x^2 + 2*E^4*x^2 - x^3 + x^4))/(4 + x^2)^2])^2), x] + 75*(8 + 2
*E^4 + 3*E^8)*Defer[Int][x^2/((16 - E^8*x + 2*(4 + E^4)*x^2 - x^3 + x^4)*(3 - 5*Log[(x*(16 - E^8*x + 8*x^2 + 2
*E^4*x^2 - x^3 + x^4))/(4 + x^2)^2])^2), x] - 300*(4 + E^4)*Defer[Int][x^3/((16 - E^8*x + 2*(4 + E^4)*x^2 - x^
3 + x^4)*(3 - 5*Log[(x*(16 - E^8*x + 8*x^2 + 2*E^4*x^2 - x^3 + x^4))/(4 + x^2)^2])^2), x] - 15*Defer[Int][(3 -
5*Log[(x*(16 - E^8*x + 8*x^2 + 2*E^4*x^2 - x^3 + x^4))/(4 + x^2)^2])^(-1), x] - (600 + 300*I)*Defer[Int][1/((
2*I - x)*(-3 + 5*Log[(x*(16 - E^8*x + 8*x^2 + 2*E^4*x^2 - x^3 + x^4))/(4 + x^2)^2])^2), x] + 75*Defer[Int][x/(
-3 + 5*Log[(x*(16 - E^8*x + 8*x^2 + 2*E^4*x^2 - x^3 + x^4))/(4 + x^2)^2])^2, x] + (600 - 300*I)*Defer[Int][1/(
(2*I + x)*(-3 + 5*Log[(x*(16 - E^8*x + 8*x^2 + 2*E^4*x^2 - x^3 + x^4))/(4 + x^2)^2])^2), x] - 30*Defer[Int][x/
(-3 + 5*Log[(x*(16 - E^8*x + 8*x^2 + 2*E^4*x^2 - x^3 + x^4))/(4 + x^2)^2]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {25 \left (-192+192 x-144 x^2+192 x^3-84 x^4+36 x^5-3 x^6+3 x^7-6 e^4 x^2 \left (12-12 x-x^2+x^3\right )+6 e^8 x \left (4-4 x-x^2+x^3\right )+3 \left (-4+8 x-x^2+2 x^3\right ) \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right ) \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )-5 \left (-4+8 x-x^2+2 x^3\right ) \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right ) \log ^2\left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )}{\left (4+x^2\right ) \left (16-e^8 x+2 \left (4+e^4\right ) x^2-x^3+x^4\right ) \left (3-5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2} \, dx \\ & = 25 \int \frac {-192+192 x-144 x^2+192 x^3-84 x^4+36 x^5-3 x^6+3 x^7-6 e^4 x^2 \left (12-12 x-x^2+x^3\right )+6 e^8 x \left (4-4 x-x^2+x^3\right )+3 \left (-4+8 x-x^2+2 x^3\right ) \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right ) \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )-5 \left (-4+8 x-x^2+2 x^3\right ) \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right ) \log ^2\left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )}{\left (4+x^2\right ) \left (16-e^8 x+2 \left (4+e^4\right ) x^2-x^3+x^4\right ) \left (3-5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2} \, dx \\ & = 25 \int \left (\frac {1}{5} (1-2 x)+\frac {3 (1-x) \left (-64+8 e^8 x-48 \left (1+\frac {e^4}{2}\right ) x^2+16 \left (1-\frac {e^8}{8}\right ) x^3-12 \left (1-\frac {e^4}{6}\right ) x^4-x^6\right )}{\left (4+x^2\right ) \left (16-e^8 x+2 \left (4+e^4\right ) x^2-x^3+x^4\right ) \left (3-5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2}-\frac {3 (-1+2 x)}{5 \left (-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )}\right ) \, dx \\ & = -\frac {5}{4} (1-2 x)^2-15 \int \frac {-1+2 x}{-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )} \, dx+75 \int \frac {(1-x) \left (-64+8 e^8 x-48 \left (1+\frac {e^4}{2}\right ) x^2+16 \left (1-\frac {e^8}{8}\right ) x^3-12 \left (1-\frac {e^4}{6}\right ) x^4-x^6\right )}{\left (4+x^2\right ) \left (16-e^8 x+2 \left (4+e^4\right ) x^2-x^3+x^4\right ) \left (3-5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2} \, dx \\ & = -\frac {5}{4} (1-2 x)^2-15 \int \left (\frac {1}{3-5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )}+\frac {2 x}{-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )}\right ) \, dx+75 \int \left (\frac {48-2 \left (32+e^8\right ) x+\left (8+2 e^4+3 e^8\right ) x^2-4 \left (4+e^4\right ) x^3}{\left (16-e^8 x+2 \left (4+e^4\right ) x^2-x^3+x^4\right ) \left (3-5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2}+\frac {x}{\left (-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2}+\frac {16 (-1+x)}{\left (4+x^2\right ) \left (-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2}\right ) \, dx \\ & = -\frac {5}{4} (1-2 x)^2-15 \int \frac {1}{3-5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )} \, dx-30 \int \frac {x}{-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )} \, dx+75 \int \frac {48-2 \left (32+e^8\right ) x+\left (8+2 e^4+3 e^8\right ) x^2-4 \left (4+e^4\right ) x^3}{\left (16-e^8 x+2 \left (4+e^4\right ) x^2-x^3+x^4\right ) \left (3-5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2} \, dx+75 \int \frac {x}{\left (-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2} \, dx+1200 \int \frac {-1+x}{\left (4+x^2\right ) \left (-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2} \, dx \\ & = -\frac {5}{4} (1-2 x)^2-15 \int \frac {1}{3-5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )} \, dx-30 \int \frac {x}{-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )} \, dx+75 \int \left (\frac {48}{\left (16-e^8 x+2 \left (4+e^4\right ) x^2-x^3+x^4\right ) \left (3-5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2}+\frac {2 \left (-32-e^8\right ) x}{\left (16-e^8 x+2 \left (4+e^4\right ) x^2-x^3+x^4\right ) \left (3-5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2}+\frac {\left (8+2 e^4+3 e^8\right ) x^2}{\left (16-e^8 x+2 \left (4+e^4\right ) x^2-x^3+x^4\right ) \left (3-5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2}+\frac {4 \left (-4-e^4\right ) x^3}{\left (16-e^8 x+2 \left (4+e^4\right ) x^2-x^3+x^4\right ) \left (3-5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2}\right ) \, dx+75 \int \frac {x}{\left (-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2} \, dx+1200 \int \left (-\frac {1}{\left (4+x^2\right ) \left (-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2}+\frac {x}{\left (4+x^2\right ) \left (-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2}\right ) \, dx \\ & = -\frac {5}{4} (1-2 x)^2-15 \int \frac {1}{3-5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )} \, dx-30 \int \frac {x}{-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )} \, dx+75 \int \frac {x}{\left (-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2} \, dx-1200 \int \frac {1}{\left (4+x^2\right ) \left (-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2} \, dx+1200 \int \frac {x}{\left (4+x^2\right ) \left (-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2} \, dx+3600 \int \frac {1}{\left (16-e^8 x+2 \left (4+e^4\right ) x^2-x^3+x^4\right ) \left (3-5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2} \, dx-\left (300 \left (4+e^4\right )\right ) \int \frac {x^3}{\left (16-e^8 x+2 \left (4+e^4\right ) x^2-x^3+x^4\right ) \left (3-5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2} \, dx-\left (150 \left (32+e^8\right )\right ) \int \frac {x}{\left (16-e^8 x+2 \left (4+e^4\right ) x^2-x^3+x^4\right ) \left (3-5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2} \, dx+\left (75 \left (8+2 e^4+3 e^8\right )\right ) \int \frac {x^2}{\left (16-e^8 x+2 \left (4+e^4\right ) x^2-x^3+x^4\right ) \left (3-5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2} \, dx \\ & = -\frac {5}{4} (1-2 x)^2-15 \int \frac {1}{3-5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )} \, dx-30 \int \frac {x}{-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )} \, dx+75 \int \frac {x}{\left (-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2} \, dx-1200 \int \left (\frac {i}{4 (2 i-x) \left (-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2}+\frac {i}{4 (2 i+x) \left (-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2}\right ) \, dx+1200 \int \left (-\frac {1}{2 (2 i-x) \left (-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2}+\frac {1}{2 (2 i+x) \left (-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2}\right ) \, dx+3600 \int \frac {1}{\left (16-e^8 x+2 \left (4+e^4\right ) x^2-x^3+x^4\right ) \left (3-5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2} \, dx-\left (300 \left (4+e^4\right )\right ) \int \frac {x^3}{\left (16-e^8 x+2 \left (4+e^4\right ) x^2-x^3+x^4\right ) \left (3-5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2} \, dx-\left (150 \left (32+e^8\right )\right ) \int \frac {x}{\left (16-e^8 x+2 \left (4+e^4\right ) x^2-x^3+x^4\right ) \left (3-5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2} \, dx+\left (75 \left (8+2 e^4+3 e^8\right )\right ) \int \frac {x^2}{\left (16-e^8 x+2 \left (4+e^4\right ) x^2-x^3+x^4\right ) \left (3-5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2} \, dx \\ & = -\frac {5}{4} (1-2 x)^2-300 i \int \frac {1}{(2 i-x) \left (-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2} \, dx-300 i \int \frac {1}{(2 i+x) \left (-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2} \, dx-15 \int \frac {1}{3-5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )} \, dx-30 \int \frac {x}{-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )} \, dx+75 \int \frac {x}{\left (-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2} \, dx-600 \int \frac {1}{(2 i-x) \left (-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2} \, dx+600 \int \frac {1}{(2 i+x) \left (-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2} \, dx+3600 \int \frac {1}{\left (16-e^8 x+2 \left (4+e^4\right ) x^2-x^3+x^4\right ) \left (3-5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2} \, dx-\left (300 \left (4+e^4\right )\right ) \int \frac {x^3}{\left (16-e^8 x+2 \left (4+e^4\right ) x^2-x^3+x^4\right ) \left (3-5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2} \, dx-\left (150 \left (32+e^8\right )\right ) \int \frac {x}{\left (16-e^8 x+2 \left (4+e^4\right ) x^2-x^3+x^4\right ) \left (3-5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2} \, dx+\left (75 \left (8+2 e^4+3 e^8\right )\right ) \int \frac {x^2}{\left (16-e^8 x+2 \left (4+e^4\right ) x^2-x^3+x^4\right ) \left (3-5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.40 \[ \int \frac {4800-4800 x+3600 x^2-4800 x^3+2100 x^4-900 x^5+75 x^6-75 x^7+e^8 \left (-600 x+600 x^2+150 x^3-150 x^4\right )+e^4 \left (1800 x^2-1800 x^3-150 x^4+150 x^5\right )+\left (4800-9600 x+3600 x^2-7500 x^3+1500 x^4-1875 x^5+225 x^6-150 x^7+e^8 \left (-300 x+600 x^2-75 x^3+150 x^4\right )+e^4 \left (600 x^2-1200 x^3+150 x^4-300 x^5\right )\right ) \log \left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )+\left (-8000+16000 x-6000 x^2+12500 x^3-2500 x^4+3125 x^5-375 x^6+250 x^7+e^8 \left (500 x-1000 x^2+125 x^3-250 x^4\right )+e^4 \left (-1000 x^2+2000 x^3-250 x^4+500 x^5\right )\right ) \log ^2\left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )}{-576-432 x^2+36 x^3-108 x^4+9 x^5-9 x^6+e^8 \left (36 x+9 x^3\right )+e^4 \left (-72 x^2-18 x^4\right )+\left (1920+1440 x^2-120 x^3+360 x^4-30 x^5+30 x^6+e^8 \left (-120 x-30 x^3\right )+e^4 \left (240 x^2+60 x^4\right )\right ) \log \left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )+\left (-1600-1200 x^2+100 x^3-300 x^4+25 x^5-25 x^6+e^8 \left (100 x+25 x^3\right )+e^4 \left (-200 x^2-50 x^4\right )\right ) \log ^2\left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )} \, dx=-5 x \left (-1+x+\frac {3 (-1+x)}{-3+5 \log \left (\frac {x \left (16-e^8 x+8 x^2+2 e^4 x^2-x^3+x^4\right )}{\left (4+x^2\right )^2}\right )}\right ) \]

[In]

Integrate[(4800 - 4800*x + 3600*x^2 - 4800*x^3 + 2100*x^4 - 900*x^5 + 75*x^6 - 75*x^7 + E^8*(-600*x + 600*x^2
+ 150*x^3 - 150*x^4) + E^4*(1800*x^2 - 1800*x^3 - 150*x^4 + 150*x^5) + (4800 - 9600*x + 3600*x^2 - 7500*x^3 +
1500*x^4 - 1875*x^5 + 225*x^6 - 150*x^7 + E^8*(-300*x + 600*x^2 - 75*x^3 + 150*x^4) + E^4*(600*x^2 - 1200*x^3
+ 150*x^4 - 300*x^5))*Log[(16*x - E^8*x^2 + 8*x^3 + 2*E^4*x^3 - x^4 + x^5)/(16 + 8*x^2 + x^4)] + (-8000 + 1600
0*x - 6000*x^2 + 12500*x^3 - 2500*x^4 + 3125*x^5 - 375*x^6 + 250*x^7 + E^8*(500*x - 1000*x^2 + 125*x^3 - 250*x
^4) + E^4*(-1000*x^2 + 2000*x^3 - 250*x^4 + 500*x^5))*Log[(16*x - E^8*x^2 + 8*x^3 + 2*E^4*x^3 - x^4 + x^5)/(16
+ 8*x^2 + x^4)]^2)/(-576 - 432*x^2 + 36*x^3 - 108*x^4 + 9*x^5 - 9*x^6 + E^8*(36*x + 9*x^3) + E^4*(-72*x^2 - 1
8*x^4) + (1920 + 1440*x^2 - 120*x^3 + 360*x^4 - 30*x^5 + 30*x^6 + E^8*(-120*x - 30*x^3) + E^4*(240*x^2 + 60*x^
4))*Log[(16*x - E^8*x^2 + 8*x^3 + 2*E^4*x^3 - x^4 + x^5)/(16 + 8*x^2 + x^4)] + (-1600 - 1200*x^2 + 100*x^3 - 3
00*x^4 + 25*x^5 - 25*x^6 + E^8*(100*x + 25*x^3) + E^4*(-200*x^2 - 50*x^4))*Log[(16*x - E^8*x^2 + 8*x^3 + 2*E^4
*x^3 - x^4 + x^5)/(16 + 8*x^2 + x^4)]^2),x]

[Out]

-5*x*(-1 + x + (3*(-1 + x))/(-3 + 5*Log[(x*(16 - E^8*x + 8*x^2 + 2*E^4*x^2 - x^3 + x^4))/(4 + x^2)^2]))

Maple [A] (verified)

Time = 41.47 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.68

method	result	size
risch	\(-5 x^{2}+5 x -\frac {15 x \left (-1+x \right )}{5 \ln \left (\frac {-x^{2} {\mathrm e}^{8}+2 x^{3} {\mathrm e}^{4}+x^{5}-x^{4}+8 x^{3}+16 x}{x^{4}+8 x^{2}+16}\right )-3}\)	\(67\)
norman	\(\frac {25 x \ln \left (\frac {-x^{2} {\mathrm e}^{8}+2 x^{3} {\mathrm e}^{4}+x^{5}-x^{4}+8 x^{3}+16 x}{x^{4}+8 x^{2}+16}\right )-25 \ln \left (\frac {-x^{2} {\mathrm e}^{8}+2 x^{3} {\mathrm e}^{4}+x^{5}-x^{4}+8 x^{3}+16 x}{x^{4}+8 x^{2}+16}\right ) x^{2}}{5 \ln \left (\frac {-x^{2} {\mathrm e}^{8}+2 x^{3} {\mathrm e}^{4}+x^{5}-x^{4}+8 x^{3}+16 x}{x^{4}+8 x^{2}+16}\right )-3}\)	\(158\)
parallelrisch	\(\frac {-2025-125 \ln \left (\frac {-x^{2} {\mathrm e}^{8}+2 x^{3} {\mathrm e}^{4}+x^{5}-x^{4}+8 x^{3}+16 x}{x^{4}+8 x^{2}+16}\right ) x^{2}+500 \,{\mathrm e}^{4} \ln \left (\frac {-x^{2} {\mathrm e}^{8}+2 x^{3} {\mathrm e}^{4}+x^{5}-x^{4}+8 x^{3}+16 x}{x^{4}+8 x^{2}+16}\right )+125 x \ln \left (\frac {-x^{2} {\mathrm e}^{8}+2 x^{3} {\mathrm e}^{4}+x^{5}-x^{4}+8 x^{3}+16 x}{x^{4}+8 x^{2}+16}\right )-300 \,{\mathrm e}^{4}+3375 \ln \left (\frac {-x^{2} {\mathrm e}^{8}+2 x^{3} {\mathrm e}^{4}+x^{5}-x^{4}+8 x^{3}+16 x}{x^{4}+8 x^{2}+16}\right )}{25 \ln \left (\frac {-x^{2} {\mathrm e}^{8}+2 x^{3} {\mathrm e}^{4}+x^{5}-x^{4}+8 x^{3}+16 x}{x^{4}+8 x^{2}+16}\right )-15}\)	\(264\)

[In]

int((((-250*x^4+125*x^3-1000*x^2+500*x)*exp(4)^2+(500*x^5-250*x^4+2000*x^3-1000*x^2)*exp(4)+250*x^7-375*x^6+31
25*x^5-2500*x^4+12500*x^3-6000*x^2+16000*x-8000)*ln((-x^2*exp(4)^2+2*x^3*exp(4)+x^5-x^4+8*x^3+16*x)/(x^4+8*x^2
+16))^2+((150*x^4-75*x^3+600*x^2-300*x)*exp(4)^2+(-300*x^5+150*x^4-1200*x^3+600*x^2)*exp(4)-150*x^7+225*x^6-18
75*x^5+1500*x^4-7500*x^3+3600*x^2-9600*x+4800)*ln((-x^2*exp(4)^2+2*x^3*exp(4)+x^5-x^4+8*x^3+16*x)/(x^4+8*x^2+1
6))+(-150*x^4+150*x^3+600*x^2-600*x)*exp(4)^2+(150*x^5-150*x^4-1800*x^3+1800*x^2)*exp(4)-75*x^7+75*x^6-900*x^5
+2100*x^4-4800*x^3+3600*x^2-4800*x+4800)/(((25*x^3+100*x)*exp(4)^2+(-50*x^4-200*x^2)*exp(4)-25*x^6+25*x^5-300*
x^4+100*x^3-1200*x^2-1600)*ln((-x^2*exp(4)^2+2*x^3*exp(4)+x^5-x^4+8*x^3+16*x)/(x^4+8*x^2+16))^2+((-30*x^3-120*
x)*exp(4)^2+(60*x^4+240*x^2)*exp(4)+30*x^6-30*x^5+360*x^4-120*x^3+1440*x^2+1920)*ln((-x^2*exp(4)^2+2*x^3*exp(4
)+x^5-x^4+8*x^3+16*x)/(x^4+8*x^2+16))+(9*x^3+36*x)*exp(4)^2+(-18*x^4-72*x^2)*exp(4)-9*x^6+9*x^5-108*x^4+36*x^3
-432*x^2-576),x,method=_RETURNVERBOSE)

[Out]

-5*x^2+5*x-15*x*(-1+x)/(5*ln((-x^2*exp(8)+2*x^3*exp(4)+x^5-x^4+8*x^3+16*x)/(x^4+8*x^2+16))-3)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (39) = 78\).

Time = 0.39 (sec) , antiderivative size = 105, normalized size of antiderivative = 2.62 \[ \int \frac {4800-4800 x+3600 x^2-4800 x^3+2100 x^4-900 x^5+75 x^6-75 x^7+e^8 \left (-600 x+600 x^2+150 x^3-150 x^4\right )+e^4 \left (1800 x^2-1800 x^3-150 x^4+150 x^5\right )+\left (4800-9600 x+3600 x^2-7500 x^3+1500 x^4-1875 x^5+225 x^6-150 x^7+e^8 \left (-300 x+600 x^2-75 x^3+150 x^4\right )+e^4 \left (600 x^2-1200 x^3+150 x^4-300 x^5\right )\right ) \log \left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )+\left (-8000+16000 x-6000 x^2+12500 x^3-2500 x^4+3125 x^5-375 x^6+250 x^7+e^8 \left (500 x-1000 x^2+125 x^3-250 x^4\right )+e^4 \left (-1000 x^2+2000 x^3-250 x^4+500 x^5\right )\right ) \log ^2\left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )}{-576-432 x^2+36 x^3-108 x^4+9 x^5-9 x^6+e^8 \left (36 x+9 x^3\right )+e^4 \left (-72 x^2-18 x^4\right )+\left (1920+1440 x^2-120 x^3+360 x^4-30 x^5+30 x^6+e^8 \left (-120 x-30 x^3\right )+e^4 \left (240 x^2+60 x^4\right )\right ) \log \left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )+\left (-1600-1200 x^2+100 x^3-300 x^4+25 x^5-25 x^6+e^8 \left (100 x+25 x^3\right )+e^4 \left (-200 x^2-50 x^4\right )\right ) \log ^2\left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )} \, dx=-\frac {25 \, {\left (x^{2} - x\right )} \log \left (\frac {x^{5} - x^{4} + 2 \, x^{3} e^{4} + 8 \, x^{3} - x^{2} e^{8} + 16 \, x}{x^{4} + 8 \, x^{2} + 16}\right )}{5 \, \log \left (\frac {x^{5} - x^{4} + 2 \, x^{3} e^{4} + 8 \, x^{3} - x^{2} e^{8} + 16 \, x}{x^{4} + 8 \, x^{2} + 16}\right ) - 3} \]

[In]

integrate((((-250*x^4+125*x^3-1000*x^2+500*x)*exp(4)^2+(500*x^5-250*x^4+2000*x^3-1000*x^2)*exp(4)+250*x^7-375*
x^6+3125*x^5-2500*x^4+12500*x^3-6000*x^2+16000*x-8000)*log((-x^2*exp(4)^2+2*x^3*exp(4)+x^5-x^4+8*x^3+16*x)/(x^
4+8*x^2+16))^2+((150*x^4-75*x^3+600*x^2-300*x)*exp(4)^2+(-300*x^5+150*x^4-1200*x^3+600*x^2)*exp(4)-150*x^7+225
*x^6-1875*x^5+1500*x^4-7500*x^3+3600*x^2-9600*x+4800)*log((-x^2*exp(4)^2+2*x^3*exp(4)+x^5-x^4+8*x^3+16*x)/(x^4
+8*x^2+16))+(-150*x^4+150*x^3+600*x^2-600*x)*exp(4)^2+(150*x^5-150*x^4-1800*x^3+1800*x^2)*exp(4)-75*x^7+75*x^6
-900*x^5+2100*x^4-4800*x^3+3600*x^2-4800*x+4800)/(((25*x^3+100*x)*exp(4)^2+(-50*x^4-200*x^2)*exp(4)-25*x^6+25*
x^5-300*x^4+100*x^3-1200*x^2-1600)*log((-x^2*exp(4)^2+2*x^3*exp(4)+x^5-x^4+8*x^3+16*x)/(x^4+8*x^2+16))^2+((-30
*x^3-120*x)*exp(4)^2+(60*x^4+240*x^2)*exp(4)+30*x^6-30*x^5+360*x^4-120*x^3+1440*x^2+1920)*log((-x^2*exp(4)^2+2
*x^3*exp(4)+x^5-x^4+8*x^3+16*x)/(x^4+8*x^2+16))+(9*x^3+36*x)*exp(4)^2+(-18*x^4-72*x^2)*exp(4)-9*x^6+9*x^5-108*
x^4+36*x^3-432*x^2-576),x, algorithm="fricas")

[Out]

-25*(x^2 - x)*log((x^5 - x^4 + 2*x^3*e^4 + 8*x^3 - x^2*e^8 + 16*x)/(x^4 + 8*x^2 + 16))/(5*log((x^5 - x^4 + 2*x
^3*e^4 + 8*x^3 - x^2*e^8 + 16*x)/(x^4 + 8*x^2 + 16)) - 3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (27) = 54\).

Time = 0.77 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.52 \[ \int \frac {4800-4800 x+3600 x^2-4800 x^3+2100 x^4-900 x^5+75 x^6-75 x^7+e^8 \left (-600 x+600 x^2+150 x^3-150 x^4\right )+e^4 \left (1800 x^2-1800 x^3-150 x^4+150 x^5\right )+\left (4800-9600 x+3600 x^2-7500 x^3+1500 x^4-1875 x^5+225 x^6-150 x^7+e^8 \left (-300 x+600 x^2-75 x^3+150 x^4\right )+e^4 \left (600 x^2-1200 x^3+150 x^4-300 x^5\right )\right ) \log \left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )+\left (-8000+16000 x-6000 x^2+12500 x^3-2500 x^4+3125 x^5-375 x^6+250 x^7+e^8 \left (500 x-1000 x^2+125 x^3-250 x^4\right )+e^4 \left (-1000 x^2+2000 x^3-250 x^4+500 x^5\right )\right ) \log ^2\left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )}{-576-432 x^2+36 x^3-108 x^4+9 x^5-9 x^6+e^8 \left (36 x+9 x^3\right )+e^4 \left (-72 x^2-18 x^4\right )+\left (1920+1440 x^2-120 x^3+360 x^4-30 x^5+30 x^6+e^8 \left (-120 x-30 x^3\right )+e^4 \left (240 x^2+60 x^4\right )\right ) \log \left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )+\left (-1600-1200 x^2+100 x^3-300 x^4+25 x^5-25 x^6+e^8 \left (100 x+25 x^3\right )+e^4 \left (-200 x^2-50 x^4\right )\right ) \log ^2\left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )} \, dx=- 5 x^{2} + 5 x + \frac {- 15 x^{2} + 15 x}{5 \log {\left (\frac {x^{5} - x^{4} + 8 x^{3} + 2 x^{3} e^{4} - x^{2} e^{8} + 16 x}{x^{4} + 8 x^{2} + 16} \right )} - 3} \]

[In]

integrate((((-250*x**4+125*x**3-1000*x**2+500*x)*exp(4)**2+(500*x**5-250*x**4+2000*x**3-1000*x**2)*exp(4)+250*
x**7-375*x**6+3125*x**5-2500*x**4+12500*x**3-6000*x**2+16000*x-8000)*ln((-x**2*exp(4)**2+2*x**3*exp(4)+x**5-x*
*4+8*x**3+16*x)/(x**4+8*x**2+16))**2+((150*x**4-75*x**3+600*x**2-300*x)*exp(4)**2+(-300*x**5+150*x**4-1200*x**
3+600*x**2)*exp(4)-150*x**7+225*x**6-1875*x**5+1500*x**4-7500*x**3+3600*x**2-9600*x+4800)*ln((-x**2*exp(4)**2+
2*x**3*exp(4)+x**5-x**4+8*x**3+16*x)/(x**4+8*x**2+16))+(-150*x**4+150*x**3+600*x**2-600*x)*exp(4)**2+(150*x**5
-150*x**4-1800*x**3+1800*x**2)*exp(4)-75*x**7+75*x**6-900*x**5+2100*x**4-4800*x**3+3600*x**2-4800*x+4800)/(((2
5*x**3+100*x)*exp(4)**2+(-50*x**4-200*x**2)*exp(4)-25*x**6+25*x**5-300*x**4+100*x**3-1200*x**2-1600)*ln((-x**2
*exp(4)**2+2*x**3*exp(4)+x**5-x**4+8*x**3+16*x)/(x**4+8*x**2+16))**2+((-30*x**3-120*x)*exp(4)**2+(60*x**4+240*
x**2)*exp(4)+30*x**6-30*x**5+360*x**4-120*x**3+1440*x**2+1920)*ln((-x**2*exp(4)**2+2*x**3*exp(4)+x**5-x**4+8*x
**3+16*x)/(x**4+8*x**2+16))+(9*x**3+36*x)*exp(4)**2+(-18*x**4-72*x**2)*exp(4)-9*x**6+9*x**5-108*x**4+36*x**3-4
32*x**2-576),x)

[Out]

-5*x**2 + 5*x + (-15*x**2 + 15*x)/(5*log((x**5 - x**4 + 8*x**3 + 2*x**3*exp(4) - x**2*exp(8) + 16*x)/(x**4 + 8
*x**2 + 16)) - 3)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (39) = 78\).

Time = 0.43 (sec) , antiderivative size = 104, normalized size of antiderivative = 2.60 \[ \int \frac {4800-4800 x+3600 x^2-4800 x^3+2100 x^4-900 x^5+75 x^6-75 x^7+e^8 \left (-600 x+600 x^2+150 x^3-150 x^4\right )+e^4 \left (1800 x^2-1800 x^3-150 x^4+150 x^5\right )+\left (4800-9600 x+3600 x^2-7500 x^3+1500 x^4-1875 x^5+225 x^6-150 x^7+e^8 \left (-300 x+600 x^2-75 x^3+150 x^4\right )+e^4 \left (600 x^2-1200 x^3+150 x^4-300 x^5\right )\right ) \log \left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )+\left (-8000+16000 x-6000 x^2+12500 x^3-2500 x^4+3125 x^5-375 x^6+250 x^7+e^8 \left (500 x-1000 x^2+125 x^3-250 x^4\right )+e^4 \left (-1000 x^2+2000 x^3-250 x^4+500 x^5\right )\right ) \log ^2\left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )}{-576-432 x^2+36 x^3-108 x^4+9 x^5-9 x^6+e^8 \left (36 x+9 x^3\right )+e^4 \left (-72 x^2-18 x^4\right )+\left (1920+1440 x^2-120 x^3+360 x^4-30 x^5+30 x^6+e^8 \left (-120 x-30 x^3\right )+e^4 \left (240 x^2+60 x^4\right )\right ) \log \left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )+\left (-1600-1200 x^2+100 x^3-300 x^4+25 x^5-25 x^6+e^8 \left (100 x+25 x^3\right )+e^4 \left (-200 x^2-50 x^4\right )\right ) \log ^2\left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )} \, dx=-\frac {25 \, {\left ({\left (x^{2} - x\right )} \log \left (x^{4} - x^{3} + 2 \, x^{2} {\left (e^{4} + 4\right )} - x e^{8} + 16\right ) - 2 \, {\left (x^{2} - x\right )} \log \left (x^{2} + 4\right ) + {\left (x^{2} - x\right )} \log \left (x\right )\right )}}{5 \, \log \left (x^{4} - x^{3} + 2 \, x^{2} {\left (e^{4} + 4\right )} - x e^{8} + 16\right ) - 10 \, \log \left (x^{2} + 4\right ) + 5 \, \log \left (x\right ) - 3} \]

[In]

integrate((((-250*x^4+125*x^3-1000*x^2+500*x)*exp(4)^2+(500*x^5-250*x^4+2000*x^3-1000*x^2)*exp(4)+250*x^7-375*
x^6+3125*x^5-2500*x^4+12500*x^3-6000*x^2+16000*x-8000)*log((-x^2*exp(4)^2+2*x^3*exp(4)+x^5-x^4+8*x^3+16*x)/(x^
4+8*x^2+16))^2+((150*x^4-75*x^3+600*x^2-300*x)*exp(4)^2+(-300*x^5+150*x^4-1200*x^3+600*x^2)*exp(4)-150*x^7+225
*x^6-1875*x^5+1500*x^4-7500*x^3+3600*x^2-9600*x+4800)*log((-x^2*exp(4)^2+2*x^3*exp(4)+x^5-x^4+8*x^3+16*x)/(x^4
+8*x^2+16))+(-150*x^4+150*x^3+600*x^2-600*x)*exp(4)^2+(150*x^5-150*x^4-1800*x^3+1800*x^2)*exp(4)-75*x^7+75*x^6
-900*x^5+2100*x^4-4800*x^3+3600*x^2-4800*x+4800)/(((25*x^3+100*x)*exp(4)^2+(-50*x^4-200*x^2)*exp(4)-25*x^6+25*
x^5-300*x^4+100*x^3-1200*x^2-1600)*log((-x^2*exp(4)^2+2*x^3*exp(4)+x^5-x^4+8*x^3+16*x)/(x^4+8*x^2+16))^2+((-30
*x^3-120*x)*exp(4)^2+(60*x^4+240*x^2)*exp(4)+30*x^6-30*x^5+360*x^4-120*x^3+1440*x^2+1920)*log((-x^2*exp(4)^2+2
*x^3*exp(4)+x^5-x^4+8*x^3+16*x)/(x^4+8*x^2+16))+(9*x^3+36*x)*exp(4)^2+(-18*x^4-72*x^2)*exp(4)-9*x^6+9*x^5-108*
x^4+36*x^3-432*x^2-576),x, algorithm="maxima")

[Out]

-25*((x^2 - x)*log(x^4 - x^3 + 2*x^2*(e^4 + 4) - x*e^8 + 16) - 2*(x^2 - x)*log(x^2 + 4) + (x^2 - x)*log(x))/(5
*log(x^4 - x^3 + 2*x^2*(e^4 + 4) - x*e^8 + 16) - 10*log(x^2 + 4) + 5*log(x) - 3)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (39) = 78\).

Time = 9.99 (sec) , antiderivative size = 151, normalized size of antiderivative = 3.78 \[ \int \frac {4800-4800 x+3600 x^2-4800 x^3+2100 x^4-900 x^5+75 x^6-75 x^7+e^8 \left (-600 x+600 x^2+150 x^3-150 x^4\right )+e^4 \left (1800 x^2-1800 x^3-150 x^4+150 x^5\right )+\left (4800-9600 x+3600 x^2-7500 x^3+1500 x^4-1875 x^5+225 x^6-150 x^7+e^8 \left (-300 x+600 x^2-75 x^3+150 x^4\right )+e^4 \left (600 x^2-1200 x^3+150 x^4-300 x^5\right )\right ) \log \left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )+\left (-8000+16000 x-6000 x^2+12500 x^3-2500 x^4+3125 x^5-375 x^6+250 x^7+e^8 \left (500 x-1000 x^2+125 x^3-250 x^4\right )+e^4 \left (-1000 x^2+2000 x^3-250 x^4+500 x^5\right )\right ) \log ^2\left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )}{-576-432 x^2+36 x^3-108 x^4+9 x^5-9 x^6+e^8 \left (36 x+9 x^3\right )+e^4 \left (-72 x^2-18 x^4\right )+\left (1920+1440 x^2-120 x^3+360 x^4-30 x^5+30 x^6+e^8 \left (-120 x-30 x^3\right )+e^4 \left (240 x^2+60 x^4\right )\right ) \log \left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )+\left (-1600-1200 x^2+100 x^3-300 x^4+25 x^5-25 x^6+e^8 \left (100 x+25 x^3\right )+e^4 \left (-200 x^2-50 x^4\right )\right ) \log ^2\left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )} \, dx=-\frac {25 \, {\left (x^{2} \log \left (\frac {x^{5} - x^{4} + 2 \, x^{3} e^{4} + 8 \, x^{3} - x^{2} e^{8} + 16 \, x}{x^{4} + 8 \, x^{2} + 16}\right ) - x \log \left (\frac {x^{5} - x^{4} + 2 \, x^{3} e^{4} + 8 \, x^{3} - x^{2} e^{8} + 16 \, x}{x^{4} + 8 \, x^{2} + 16}\right )\right )}}{5 \, \log \left (\frac {x^{5} - x^{4} + 2 \, x^{3} e^{4} + 8 \, x^{3} - x^{2} e^{8} + 16 \, x}{x^{4} + 8 \, x^{2} + 16}\right ) - 3} \]

[In]

integrate((((-250*x^4+125*x^3-1000*x^2+500*x)*exp(4)^2+(500*x^5-250*x^4+2000*x^3-1000*x^2)*exp(4)+250*x^7-375*
x^6+3125*x^5-2500*x^4+12500*x^3-6000*x^2+16000*x-8000)*log((-x^2*exp(4)^2+2*x^3*exp(4)+x^5-x^4+8*x^3+16*x)/(x^
4+8*x^2+16))^2+((150*x^4-75*x^3+600*x^2-300*x)*exp(4)^2+(-300*x^5+150*x^4-1200*x^3+600*x^2)*exp(4)-150*x^7+225
*x^6-1875*x^5+1500*x^4-7500*x^3+3600*x^2-9600*x+4800)*log((-x^2*exp(4)^2+2*x^3*exp(4)+x^5-x^4+8*x^3+16*x)/(x^4
+8*x^2+16))+(-150*x^4+150*x^3+600*x^2-600*x)*exp(4)^2+(150*x^5-150*x^4-1800*x^3+1800*x^2)*exp(4)-75*x^7+75*x^6
-900*x^5+2100*x^4-4800*x^3+3600*x^2-4800*x+4800)/(((25*x^3+100*x)*exp(4)^2+(-50*x^4-200*x^2)*exp(4)-25*x^6+25*
x^5-300*x^4+100*x^3-1200*x^2-1600)*log((-x^2*exp(4)^2+2*x^3*exp(4)+x^5-x^4+8*x^3+16*x)/(x^4+8*x^2+16))^2+((-30
*x^3-120*x)*exp(4)^2+(60*x^4+240*x^2)*exp(4)+30*x^6-30*x^5+360*x^4-120*x^3+1440*x^2+1920)*log((-x^2*exp(4)^2+2
*x^3*exp(4)+x^5-x^4+8*x^3+16*x)/(x^4+8*x^2+16))+(9*x^3+36*x)*exp(4)^2+(-18*x^4-72*x^2)*exp(4)-9*x^6+9*x^5-108*
x^4+36*x^3-432*x^2-576),x, algorithm="giac")

[Out]

-25*(x^2*log((x^5 - x^4 + 2*x^3*e^4 + 8*x^3 - x^2*e^8 + 16*x)/(x^4 + 8*x^2 + 16)) - x*log((x^5 - x^4 + 2*x^3*e
^4 + 8*x^3 - x^2*e^8 + 16*x)/(x^4 + 8*x^2 + 16)))/(5*log((x^5 - x^4 + 2*x^3*e^4 + 8*x^3 - x^2*e^8 + 16*x)/(x^4
+ 8*x^2 + 16)) - 3)

Mupad [B] (verification not implemented)

Time = 13.33 (sec) , antiderivative size = 460, normalized size of antiderivative = 11.50 \[ \int \frac {4800-4800 x+3600 x^2-4800 x^3+2100 x^4-900 x^5+75 x^6-75 x^7+e^8 \left (-600 x+600 x^2+150 x^3-150 x^4\right )+e^4 \left (1800 x^2-1800 x^3-150 x^4+150 x^5\right )+\left (4800-9600 x+3600 x^2-7500 x^3+1500 x^4-1875 x^5+225 x^6-150 x^7+e^8 \left (-300 x+600 x^2-75 x^3+150 x^4\right )+e^4 \left (600 x^2-1200 x^3+150 x^4-300 x^5\right )\right ) \log \left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )+\left (-8000+16000 x-6000 x^2+12500 x^3-2500 x^4+3125 x^5-375 x^6+250 x^7+e^8 \left (500 x-1000 x^2+125 x^3-250 x^4\right )+e^4 \left (-1000 x^2+2000 x^3-250 x^4+500 x^5\right )\right ) \log ^2\left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )}{-576-432 x^2+36 x^3-108 x^4+9 x^5-9 x^6+e^8 \left (36 x+9 x^3\right )+e^4 \left (-72 x^2-18 x^4\right )+\left (1920+1440 x^2-120 x^3+360 x^4-30 x^5+30 x^6+e^8 \left (-120 x-30 x^3\right )+e^4 \left (240 x^2+60 x^4\right )\right ) \log \left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )+\left (-1600-1200 x^2+100 x^3-300 x^4+25 x^5-25 x^6+e^8 \left (100 x+25 x^3\right )+e^4 \left (-200 x^2-50 x^4\right )\right ) \log ^2\left (\frac {16 x-e^8 x^2+8 x^3+2 e^4 x^3-x^4+x^5}{16+8 x^2+x^4}\right )} \, dx=14\,x+\frac {\left (24\,{\mathrm {e}}^4+18\,{\mathrm {e}}^8-144\right )\,x^5+\left (378\,{\mathrm {e}}^4-69\,{\mathrm {e}}^8+168\right )\,x^4+\left (48\,{\mathrm {e}}^{12}-18\,{\mathrm {e}}^8-576\,{\mathrm {e}}^4-336\right )\,x^3+\left (1224\,{\mathrm {e}}^4+636\,{\mathrm {e}}^8+144\right )\,x^2+\left (-24\,{\mathrm {e}}^8-192\,{\mathrm {e}}^{12}-384\right )\,x+1536\,{\mathrm {e}}^4+192}{x^6+\left (12-2\,{\mathrm {e}}^4\right )\,x^4+\left (2\,{\mathrm {e}}^8-16\right )\,x^3+\left (24\,{\mathrm {e}}^4+48\right )\,x^2-8\,{\mathrm {e}}^8\,x+64}-\frac {\frac {3\,x\,\left (704\,x+52\,x\,{\mathrm {e}}^8-144\,x^2\,{\mathrm {e}}^4+168\,x^3\,{\mathrm {e}}^4+4\,x^4\,{\mathrm {e}}^4+2\,x^5\,{\mathrm {e}}^4-64\,x^2\,{\mathrm {e}}^8-7\,x^3\,{\mathrm {e}}^8+4\,x^4\,{\mathrm {e}}^8-384\,x^2+620\,x^3-200\,x^4+135\,x^5-14\,x^6+11\,x^7-512\right )}{24\,x^2\,{\mathrm {e}}^4-8\,x\,{\mathrm {e}}^8-2\,x^4\,{\mathrm {e}}^4+2\,x^3\,{\mathrm {e}}^8+48\,x^2-16\,x^3+12\,x^4+x^6+64}-\frac {15\,x\,\ln \left (\frac {16\,x+2\,x^3\,{\mathrm {e}}^4-x^2\,{\mathrm {e}}^8+8\,x^3-x^4+x^5}{x^4+8\,x^2+16}\right )\,\left (2\,x-1\right )\,\left (x^2+4\right )\,\left (2\,x^2\,{\mathrm {e}}^4-x\,{\mathrm {e}}^8+8\,x^2-x^3+x^4+16\right )}{24\,x^2\,{\mathrm {e}}^4-8\,x\,{\mathrm {e}}^8-2\,x^4\,{\mathrm {e}}^4+2\,x^3\,{\mathrm {e}}^8+48\,x^2-16\,x^3+12\,x^4+x^6+64}}{5\,\ln \left (\frac {16\,x+2\,x^3\,{\mathrm {e}}^4-x^2\,{\mathrm {e}}^8+8\,x^3-x^4+x^5}{x^4+8\,x^2+16}\right )-3}-11\,x^2 \]

[In]

int((4800*x - log((16*x + 2*x^3*exp(4) - x^2*exp(8) + 8*x^3 - x^4 + x^5)/(8*x^2 + x^4 + 16))^2*(16000*x + exp(
8)*(500*x - 1000*x^2 + 125*x^3 - 250*x^4) - 6000*x^2 + 12500*x^3 - 2500*x^4 + 3125*x^5 - 375*x^6 + 250*x^7 - e
xp(4)*(1000*x^2 - 2000*x^3 + 250*x^4 - 500*x^5) - 8000) + exp(8)*(600*x - 600*x^2 - 150*x^3 + 150*x^4) + log((
16*x + 2*x^3*exp(4) - x^2*exp(8) + 8*x^3 - x^4 + x^5)/(8*x^2 + x^4 + 16))*(9600*x + exp(8)*(300*x - 600*x^2 +
75*x^3 - 150*x^4) - 3600*x^2 + 7500*x^3 - 1500*x^4 + 1875*x^5 - 225*x^6 + 150*x^7 - exp(4)*(600*x^2 - 1200*x^3
+ 150*x^4 - 300*x^5) - 4800) - 3600*x^2 + 4800*x^3 - 2100*x^4 + 900*x^5 - 75*x^6 + 75*x^7 - exp(4)*(1800*x^2
- 1800*x^3 - 150*x^4 + 150*x^5) - 4800)/(log((16*x + 2*x^3*exp(4) - x^2*exp(8) + 8*x^3 - x^4 + x^5)/(8*x^2 + x
^4 + 16))^2*(exp(4)*(200*x^2 + 50*x^4) - exp(8)*(100*x + 25*x^3) + 1200*x^2 - 100*x^3 + 300*x^4 - 25*x^5 + 25*
x^6 + 1600) - exp(8)*(36*x + 9*x^3) + exp(4)*(72*x^2 + 18*x^4) + 432*x^2 - 36*x^3 + 108*x^4 - 9*x^5 + 9*x^6 -
log((16*x + 2*x^3*exp(4) - x^2*exp(8) + 8*x^3 - x^4 + x^5)/(8*x^2 + x^4 + 16))*(exp(4)*(240*x^2 + 60*x^4) - ex
p(8)*(120*x + 30*x^3) + 1440*x^2 - 120*x^3 + 360*x^4 - 30*x^5 + 30*x^6 + 1920) + 576),x)

[Out]

14*x + (1536*exp(4) + x^5*(24*exp(4) + 18*exp(8) - 144) + x^4*(378*exp(4) - 69*exp(8) + 168) + x^2*(1224*exp(4
) + 636*exp(8) + 144) - x*(24*exp(8) + 192*exp(12) + 384) - x^3*(576*exp(4) + 18*exp(8) - 48*exp(12) + 336) +
192)/(x^3*(2*exp(8) - 16) - x^4*(2*exp(4) - 12) - 8*x*exp(8) + x^2*(24*exp(4) + 48) + x^6 + 64) - ((3*x*(704*x
+ 52*x*exp(8) - 144*x^2*exp(4) + 168*x^3*exp(4) + 4*x^4*exp(4) + 2*x^5*exp(4) - 64*x^2*exp(8) - 7*x^3*exp(8)
+ 4*x^4*exp(8) - 384*x^2 + 620*x^3 - 200*x^4 + 135*x^5 - 14*x^6 + 11*x^7 - 512))/(24*x^2*exp(4) - 8*x*exp(8) -
2*x^4*exp(4) + 2*x^3*exp(8) + 48*x^2 - 16*x^3 + 12*x^4 + x^6 + 64) - (15*x*log((16*x + 2*x^3*exp(4) - x^2*exp
(8) + 8*x^3 - x^4 + x^5)/(8*x^2 + x^4 + 16))*(2*x - 1)*(x^2 + 4)*(2*x^2*exp(4) - x*exp(8) + 8*x^2 - x^3 + x^4
+ 16))/(24*x^2*exp(4) - 8*x*exp(8) - 2*x^4*exp(4) + 2*x^3*exp(8) + 48*x^2 - 16*x^3 + 12*x^4 + x^6 + 64))/(5*lo
g((16*x + 2*x^3*exp(4) - x^2*exp(8) + 8*x^3 - x^4 + x^5)/(8*x^2 + x^4 + 16)) - 3) - 11*x^2