Integrand size = 373, antiderivative size = 29 \[ \int \frac {6144-4416 x-462 x^2+681 x^3-24 x^5+e \left (-6144 x+1488 x^2+1167 x^3-96 x^4-48 x^5\right )+e^2 \left (1536 x^2+384 x^3-96 x^4-24 x^5\right )+\left (-3072+3024 x-552 x^2-195 x^3+48 x^4+e^2 \left (-768 x^2+48 x^4\right )+e \left (3072 x-1524 x^2-195 x^3+96 x^4\right )\right ) \log (-4+x)+\left (384-480 x+192 x^2-24 x^3+e \left (-384 x+288 x^2-48 x^3\right )+e^2 \left (96 x^2-24 x^3\right )\right ) \log ^2(-4+x)}{-512+384 x+32 x^2-56 x^3+2 x^5+e^2 \left (-128 x^2-32 x^3+8 x^4+2 x^5\right )+e \left (512 x-128 x^2-96 x^3+8 x^4+4 x^5\right )+\left (256-256 x+48 x^2+16 x^3-4 x^4+e \left (-256 x+128 x^2+16 x^3-8 x^4\right )+e^2 \left (64 x^2-4 x^4\right )\right ) \log (-4+x)+\left (-32+40 x-16 x^2+2 x^3+e^2 \left (-8 x^2+2 x^3\right )+e \left (32 x-24 x^2+4 x^3\right )\right ) \log ^2(-4+x)} \, dx=3 x \left (-4+\frac {x}{(-4+2 (x+e x)) (4+x-\log (-4+x))}\right ) \] Output:
3*(x/(4+x-ln(-4+x))/(2*x*exp(1)+2*x-4)-4)*x
Time = 0.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {6144-4416 x-462 x^2+681 x^3-24 x^5+e \left (-6144 x+1488 x^2+1167 x^3-96 x^4-48 x^5\right )+e^2 \left (1536 x^2+384 x^3-96 x^4-24 x^5\right )+\left (-3072+3024 x-552 x^2-195 x^3+48 x^4+e^2 \left (-768 x^2+48 x^4\right )+e \left (3072 x-1524 x^2-195 x^3+96 x^4\right )\right ) \log (-4+x)+\left (384-480 x+192 x^2-24 x^3+e \left (-384 x+288 x^2-48 x^3\right )+e^2 \left (96 x^2-24 x^3\right )\right ) \log ^2(-4+x)}{-512+384 x+32 x^2-56 x^3+2 x^5+e^2 \left (-128 x^2-32 x^3+8 x^4+2 x^5\right )+e \left (512 x-128 x^2-96 x^3+8 x^4+4 x^5\right )+\left (256-256 x+48 x^2+16 x^3-4 x^4+e \left (-256 x+128 x^2+16 x^3-8 x^4\right )+e^2 \left (64 x^2-4 x^4\right )\right ) \log (-4+x)+\left (-32+40 x-16 x^2+2 x^3+e^2 \left (-8 x^2+2 x^3\right )+e \left (32 x-24 x^2+4 x^3\right )\right ) \log ^2(-4+x)} \, dx=-\frac {3}{2} \left (-32+8 x-\frac {x^2}{(-2+x+e x) (4+x-\log (-4+x))}\right ) \] Input:
Integrate[(6144 - 4416*x - 462*x^2 + 681*x^3 - 24*x^5 + E*(-6144*x + 1488* x^2 + 1167*x^3 - 96*x^4 - 48*x^5) + E^2*(1536*x^2 + 384*x^3 - 96*x^4 - 24* x^5) + (-3072 + 3024*x - 552*x^2 - 195*x^3 + 48*x^4 + E^2*(-768*x^2 + 48*x ^4) + E*(3072*x - 1524*x^2 - 195*x^3 + 96*x^4))*Log[-4 + x] + (384 - 480*x + 192*x^2 - 24*x^3 + E*(-384*x + 288*x^2 - 48*x^3) + E^2*(96*x^2 - 24*x^3 ))*Log[-4 + x]^2)/(-512 + 384*x + 32*x^2 - 56*x^3 + 2*x^5 + E^2*(-128*x^2 - 32*x^3 + 8*x^4 + 2*x^5) + E*(512*x - 128*x^2 - 96*x^3 + 8*x^4 + 4*x^5) + (256 - 256*x + 48*x^2 + 16*x^3 - 4*x^4 + E*(-256*x + 128*x^2 + 16*x^3 - 8 *x^4) + E^2*(64*x^2 - 4*x^4))*Log[-4 + x] + (-32 + 40*x - 16*x^2 + 2*x^3 + E^2*(-8*x^2 + 2*x^3) + E*(32*x - 24*x^2 + 4*x^3))*Log[-4 + x]^2),x]
Output:
(-3*(-32 + 8*x - x^2/((-2 + x + E*x)*(4 + x - Log[-4 + x]))))/2
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.
| \(\displaystyle \int \frac {-24 x^5+681 x^3-462 x^2+\left (-24 x^3+192 x^2+e \left (-48 x^3+288 x^2-384 x\right )+e^2 \left (96 x^2-24 x^3\right )-480 x+384\right ) \log ^2(x-4)+\left (48 x^4-195 x^3-552 x^2+e^2 \left (48 x^4-768 x^2\right )+e \left (96 x^4-195 x^3-1524 x^2+3072 x\right )+3024 x-3072\right ) \log (x-4)+e \left (-48 x^5-96 x^4+1167 x^3+1488 x^2-6144 x\right )+e^2 \left (-24 x^5-96 x^4+384 x^3+1536 x^2\right )-4416 x+6144}{2 x^5-56 x^3+32 x^2+\left (2 x^3-16 x^2+e^2 \left (2 x^3-8 x^2\right )+e \left (4 x^3-24 x^2+32 x\right )+40 x-32\right ) \log ^2(x-4)+\left (-4 x^4+16 x^3+48 x^2+e^2 \left (64 x^2-4 x^4\right )+e \left (-8 x^4+16 x^3+128 x^2-256 x\right )-256 x+256\right ) \log (x-4)+e^2 \left (2 x^5+8 x^4-32 x^3-128 x^2\right )+e \left (4 x^5+8 x^4-96 x^3-128 x^2+512 x\right )+384 x-512} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {3 \left (8 (1+e)^2 x^5+32 e (1+e) x^4-\left (227+389 e+128 e^2\right ) x^3-2 \left (-77+248 e+256 e^2\right ) x^2-(x-4) \left (16 (1+e)^2 x^3+\left (-1+63 e+64 e^2\right ) x^2-4 (47+64 e) x+256\right ) \log (x-4)+64 (23+32 e) x+8 (x-4) (e x+x-2)^2 \log ^2(x-4)-2048\right )}{2 (4-x) (2-(1+e) x)^2 (x-\log (x-4)+4)^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{2} \int -\frac {-8 (1+e)^2 x^5-32 e (1+e) x^4+\left (227+389 e+128 e^2\right ) x^3-2 \left (77-248 e-256 e^2\right ) x^2-64 (23+32 e) x+8 (4-x) (-e x-x+2)^2 \log ^2(x-4)-(4-x) \left (16 (1+e)^2 x^3-(1-64 e) (1+e) x^2-4 (47+64 e) x+256\right ) \log (x-4)+2048}{(4-x) (2-(1+e) x)^2 (x-\log (x-4)+4)^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3}{2} \int \frac {-8 (1+e)^2 x^5-32 e (1+e) x^4+\left (227+389 e+128 e^2\right ) x^3-2 \left (77-248 e-256 e^2\right ) x^2-64 (23+32 e) x+8 (4-x) (-e x-x+2)^2 \log ^2(x-4)-(4-x) \left (16 (1+e)^2 x^3-(1-64 e) (1+e) x^2-4 (47+64 e) x+256\right ) \log (x-4)+2048}{(4-x) (2-(1+e) x)^2 (x-\log (x-4)+4)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3}{2} \int \left (\frac {(x-5) x^2}{(4-x) (2-(1+e) x) (x-\log (x-4)+4)^2}+\frac {(4-(1+e) x) x}{(2-(1+e) x)^2 (x-\log (x-4)+4)}+8\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3}{2} \left (\frac {(1-e) \int \frac {1}{(x-\log (x-4)+4)^2}dx}{(1+e)^2}-\frac {8 \int \frac {1}{(x-4) (x-\log (x-4)+4)^2}dx}{1+2 e}+\frac {\int \frac {x}{(x-\log (x-4)+4)^2}dx}{1+e}-\frac {2 (3+5 e) \int \frac {1}{(2-(1+e) x) (x-\log (x-4)+4)^2}dx}{(1+e)^2 (1+2 e)}-\frac {\int \frac {1}{x-\log (x-4)+4}dx}{1+e}+\frac {4 \int \frac {1}{(2-(1+e) x)^2 (x-\log (x-4)+4)}dx}{1+e}+8 x\right )\) |
Input:
Int[(6144 - 4416*x - 462*x^2 + 681*x^3 - 24*x^5 + E*(-6144*x + 1488*x^2 + 1167*x^3 - 96*x^4 - 48*x^5) + E^2*(1536*x^2 + 384*x^3 - 96*x^4 - 24*x^5) + (-3072 + 3024*x - 552*x^2 - 195*x^3 + 48*x^4 + E^2*(-768*x^2 + 48*x^4) + E*(3072*x - 1524*x^2 - 195*x^3 + 96*x^4))*Log[-4 + x] + (384 - 480*x + 192 *x^2 - 24*x^3 + E*(-384*x + 288*x^2 - 48*x^3) + E^2*(96*x^2 - 24*x^3))*Log [-4 + x]^2)/(-512 + 384*x + 32*x^2 - 56*x^3 + 2*x^5 + E^2*(-128*x^2 - 32*x ^3 + 8*x^4 + 2*x^5) + E*(512*x - 128*x^2 - 96*x^3 + 8*x^4 + 4*x^5) + (256 - 256*x + 48*x^2 + 16*x^3 - 4*x^4 + E*(-256*x + 128*x^2 + 16*x^3 - 8*x^4) + E^2*(64*x^2 - 4*x^4))*Log[-4 + x] + (-32 + 40*x - 16*x^2 + 2*x^3 + E^2*( -8*x^2 + 2*x^3) + E*(32*x - 24*x^2 + 4*x^3))*Log[-4 + x]^2),x]
Output:
$Aborted
Time = 176.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03
| method | result | size |
| risch | \(-12 x +\frac {3 x^{2}}{2 \left (x \,{\mathrm e}+x -2\right ) \left (4+x -\ln \left (x -4\right )\right )}\) | \(30\) |
| norman | \(\frac {\left (-12 \,{\mathrm e}-12\right ) x^{3}+96 x +\left (-\frac {45}{2}-48 \,{\mathrm e}\right ) x^{2}+\left (12 \,{\mathrm e}+12\right ) x^{2} \ln \left (x -4\right )-24 x \ln \left (x -4\right )}{\left (4+x -\ln \left (x -4\right )\right ) \left (x \,{\mathrm e}+x -2\right )}\) | \(67\) |
| parallelrisch | \(-\frac {-24576 x +3072 x^{3} {\mathrm e}+12288 x^{2} {\mathrm e}+5760 x^{2}+3072 x^{3}-3072 x^{2} \ln \left (x -4\right )-3072 \,{\mathrm e} \ln \left (x -4\right ) x^{2}+6144 x \ln \left (x -4\right )}{256 \left (x^{2} {\mathrm e}-{\mathrm e} \ln \left (x -4\right ) x +4 x \,{\mathrm e}+x^{2}-x \ln \left (x -4\right )+2 x +2 \ln \left (x -4\right )-8\right )}\) | \(101\) |
| derivativedivides | \(-\frac {3 \left (-\frac {32 \left (4 \,{\mathrm e}^{2}+4 \,{\mathrm e}+1\right ) \ln \left (x -4\right )}{1+{\mathrm e}}+\frac {8 \left (16 \,{\mathrm e}^{2}+17 \,{\mathrm e}+5\right ) \left (x -4\right )}{1+{\mathrm e}}+\left (-8 \,{\mathrm e}-8\right ) \left (x -4\right )^{3}+\left (-64 \,{\mathrm e}-63\right ) \left (x -4\right )^{2}+\left (8 \,{\mathrm e}+8\right ) \ln \left (x -4\right ) \left (x -4\right )^{2}+\frac {1024 \,{\mathrm e}^{2}+1040 \,{\mathrm e}+272}{1+{\mathrm e}}\right )}{2 \left (\ln \left (x -4\right )-x -4\right ) \left ({\mathrm e} \left (x -4\right )+4 \,{\mathrm e}+x -2\right )}\) | \(137\) |
| default | \(-\frac {3 \left (-\frac {32 \left (4 \,{\mathrm e}^{2}+4 \,{\mathrm e}+1\right ) \ln \left (x -4\right )}{1+{\mathrm e}}+\frac {8 \left (16 \,{\mathrm e}^{2}+17 \,{\mathrm e}+5\right ) \left (x -4\right )}{1+{\mathrm e}}+\left (-8 \,{\mathrm e}-8\right ) \left (x -4\right )^{3}+\left (-64 \,{\mathrm e}-63\right ) \left (x -4\right )^{2}+\left (8 \,{\mathrm e}+8\right ) \ln \left (x -4\right ) \left (x -4\right )^{2}+\frac {1024 \,{\mathrm e}^{2}+1040 \,{\mathrm e}+272}{1+{\mathrm e}}\right )}{2 \left (\ln \left (x -4\right )-x -4\right ) \left ({\mathrm e} \left (x -4\right )+4 \,{\mathrm e}+x -2\right )}\) | \(137\) |
Input:
int((((-24*x^3+96*x^2)*exp(1)^2+(-48*x^3+288*x^2-384*x)*exp(1)-24*x^3+192* x^2-480*x+384)*ln(x-4)^2+((48*x^4-768*x^2)*exp(1)^2+(96*x^4-195*x^3-1524*x ^2+3072*x)*exp(1)+48*x^4-195*x^3-552*x^2+3024*x-3072)*ln(x-4)+(-24*x^5-96* x^4+384*x^3+1536*x^2)*exp(1)^2+(-48*x^5-96*x^4+1167*x^3+1488*x^2-6144*x)*e xp(1)-24*x^5+681*x^3-462*x^2-4416*x+6144)/(((2*x^3-8*x^2)*exp(1)^2+(4*x^3- 24*x^2+32*x)*exp(1)+2*x^3-16*x^2+40*x-32)*ln(x-4)^2+((-4*x^4+64*x^2)*exp(1 )^2+(-8*x^4+16*x^3+128*x^2-256*x)*exp(1)-4*x^4+16*x^3+48*x^2-256*x+256)*ln (x-4)+(2*x^5+8*x^4-32*x^3-128*x^2)*exp(1)^2+(4*x^5+8*x^4-96*x^3-128*x^2+51 2*x)*exp(1)+2*x^5-56*x^3+32*x^2+384*x-512),x,method=_RETURNVERBOSE)
Output:
-12*x+3/2*x^2/(x*exp(1)+x-2)/(4+x-ln(x-4))
Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (27) = 54\).
Time = 0.09 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.79 \[ \int \frac {6144-4416 x-462 x^2+681 x^3-24 x^5+e \left (-6144 x+1488 x^2+1167 x^3-96 x^4-48 x^5\right )+e^2 \left (1536 x^2+384 x^3-96 x^4-24 x^5\right )+\left (-3072+3024 x-552 x^2-195 x^3+48 x^4+e^2 \left (-768 x^2+48 x^4\right )+e \left (3072 x-1524 x^2-195 x^3+96 x^4\right )\right ) \log (-4+x)+\left (384-480 x+192 x^2-24 x^3+e \left (-384 x+288 x^2-48 x^3\right )+e^2 \left (96 x^2-24 x^3\right )\right ) \log ^2(-4+x)}{-512+384 x+32 x^2-56 x^3+2 x^5+e^2 \left (-128 x^2-32 x^3+8 x^4+2 x^5\right )+e \left (512 x-128 x^2-96 x^3+8 x^4+4 x^5\right )+\left (256-256 x+48 x^2+16 x^3-4 x^4+e \left (-256 x+128 x^2+16 x^3-8 x^4\right )+e^2 \left (64 x^2-4 x^4\right )\right ) \log (-4+x)+\left (-32+40 x-16 x^2+2 x^3+e^2 \left (-8 x^2+2 x^3\right )+e \left (32 x-24 x^2+4 x^3\right )\right ) \log ^2(-4+x)} \, dx=-\frac {3 \, {\left (8 \, x^{3} + 15 \, x^{2} + 8 \, {\left (x^{3} + 4 \, x^{2}\right )} e - 8 \, {\left (x^{2} e + x^{2} - 2 \, x\right )} \log \left (x - 4\right ) - 64 \, x\right )}}{2 \, {\left (x^{2} + {\left (x^{2} + 4 \, x\right )} e - {\left (x e + x - 2\right )} \log \left (x - 4\right ) + 2 \, x - 8\right )}} \] Input:
integrate((((-24*x^3+96*x^2)*exp(1)^2+(-48*x^3+288*x^2-384*x)*exp(1)-24*x^ 3+192*x^2-480*x+384)*log(-4+x)^2+((48*x^4-768*x^2)*exp(1)^2+(96*x^4-195*x^ 3-1524*x^2+3072*x)*exp(1)+48*x^4-195*x^3-552*x^2+3024*x-3072)*log(-4+x)+(- 24*x^5-96*x^4+384*x^3+1536*x^2)*exp(1)^2+(-48*x^5-96*x^4+1167*x^3+1488*x^2 -6144*x)*exp(1)-24*x^5+681*x^3-462*x^2-4416*x+6144)/(((2*x^3-8*x^2)*exp(1) ^2+(4*x^3-24*x^2+32*x)*exp(1)+2*x^3-16*x^2+40*x-32)*log(-4+x)^2+((-4*x^4+6 4*x^2)*exp(1)^2+(-8*x^4+16*x^3+128*x^2-256*x)*exp(1)-4*x^4+16*x^3+48*x^2-2 56*x+256)*log(-4+x)+(2*x^5+8*x^4-32*x^3-128*x^2)*exp(1)^2+(4*x^5+8*x^4-96* x^3-128*x^2+512*x)*exp(1)+2*x^5-56*x^3+32*x^2+384*x-512),x, algorithm="fri cas")
Output:
-3/2*(8*x^3 + 15*x^2 + 8*(x^3 + 4*x^2)*e - 8*(x^2*e + x^2 - 2*x)*log(x - 4 ) - 64*x)/(x^2 + (x^2 + 4*x)*e - (x*e + x - 2)*log(x - 4) + 2*x - 8)
Time = 0.18 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.76 \[ \int \frac {6144-4416 x-462 x^2+681 x^3-24 x^5+e \left (-6144 x+1488 x^2+1167 x^3-96 x^4-48 x^5\right )+e^2 \left (1536 x^2+384 x^3-96 x^4-24 x^5\right )+\left (-3072+3024 x-552 x^2-195 x^3+48 x^4+e^2 \left (-768 x^2+48 x^4\right )+e \left (3072 x-1524 x^2-195 x^3+96 x^4\right )\right ) \log (-4+x)+\left (384-480 x+192 x^2-24 x^3+e \left (-384 x+288 x^2-48 x^3\right )+e^2 \left (96 x^2-24 x^3\right )\right ) \log ^2(-4+x)}{-512+384 x+32 x^2-56 x^3+2 x^5+e^2 \left (-128 x^2-32 x^3+8 x^4+2 x^5\right )+e \left (512 x-128 x^2-96 x^3+8 x^4+4 x^5\right )+\left (256-256 x+48 x^2+16 x^3-4 x^4+e \left (-256 x+128 x^2+16 x^3-8 x^4\right )+e^2 \left (64 x^2-4 x^4\right )\right ) \log (-4+x)+\left (-32+40 x-16 x^2+2 x^3+e^2 \left (-8 x^2+2 x^3\right )+e \left (32 x-24 x^2+4 x^3\right )\right ) \log ^2(-4+x)} \, dx=- \frac {3 x^{2}}{- 2 e x^{2} - 2 x^{2} - 8 e x - 4 x + \left (2 x + 2 e x - 4\right ) \log {\left (x - 4 \right )} + 16} - 12 x \] Input:
integrate((((-24*x**3+96*x**2)*exp(1)**2+(-48*x**3+288*x**2-384*x)*exp(1)- 24*x**3+192*x**2-480*x+384)*ln(-4+x)**2+((48*x**4-768*x**2)*exp(1)**2+(96* x**4-195*x**3-1524*x**2+3072*x)*exp(1)+48*x**4-195*x**3-552*x**2+3024*x-30 72)*ln(-4+x)+(-24*x**5-96*x**4+384*x**3+1536*x**2)*exp(1)**2+(-48*x**5-96* x**4+1167*x**3+1488*x**2-6144*x)*exp(1)-24*x**5+681*x**3-462*x**2-4416*x+6 144)/(((2*x**3-8*x**2)*exp(1)**2+(4*x**3-24*x**2+32*x)*exp(1)+2*x**3-16*x* *2+40*x-32)*ln(-4+x)**2+((-4*x**4+64*x**2)*exp(1)**2+(-8*x**4+16*x**3+128* x**2-256*x)*exp(1)-4*x**4+16*x**3+48*x**2-256*x+256)*ln(-4+x)+(2*x**5+8*x* *4-32*x**3-128*x**2)*exp(1)**2+(4*x**5+8*x**4-96*x**3-128*x**2+512*x)*exp( 1)+2*x**5-56*x**3+32*x**2+384*x-512),x)
Output:
-3*x**2/(-2*E*x**2 - 2*x**2 - 8*E*x - 4*x + (2*x + 2*E*x - 4)*log(x - 4) + 16) - 12*x
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (27) = 54\).
Time = 0.15 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.69 \[ \int \frac {6144-4416 x-462 x^2+681 x^3-24 x^5+e \left (-6144 x+1488 x^2+1167 x^3-96 x^4-48 x^5\right )+e^2 \left (1536 x^2+384 x^3-96 x^4-24 x^5\right )+\left (-3072+3024 x-552 x^2-195 x^3+48 x^4+e^2 \left (-768 x^2+48 x^4\right )+e \left (3072 x-1524 x^2-195 x^3+96 x^4\right )\right ) \log (-4+x)+\left (384-480 x+192 x^2-24 x^3+e \left (-384 x+288 x^2-48 x^3\right )+e^2 \left (96 x^2-24 x^3\right )\right ) \log ^2(-4+x)}{-512+384 x+32 x^2-56 x^3+2 x^5+e^2 \left (-128 x^2-32 x^3+8 x^4+2 x^5\right )+e \left (512 x-128 x^2-96 x^3+8 x^4+4 x^5\right )+\left (256-256 x+48 x^2+16 x^3-4 x^4+e \left (-256 x+128 x^2+16 x^3-8 x^4\right )+e^2 \left (64 x^2-4 x^4\right )\right ) \log (-4+x)+\left (-32+40 x-16 x^2+2 x^3+e^2 \left (-8 x^2+2 x^3\right )+e \left (32 x-24 x^2+4 x^3\right )\right ) \log ^2(-4+x)} \, dx=-\frac {3 \, {\left (8 \, x^{3} {\left (e + 1\right )} + x^{2} {\left (32 \, e + 15\right )} - 8 \, {\left (x^{2} {\left (e + 1\right )} - 2 \, x\right )} \log \left (x - 4\right ) - 64 \, x\right )}}{2 \, {\left (x^{2} {\left (e + 1\right )} + 2 \, x {\left (2 \, e + 1\right )} - {\left (x {\left (e + 1\right )} - 2\right )} \log \left (x - 4\right ) - 8\right )}} \] Input:
integrate((((-24*x^3+96*x^2)*exp(1)^2+(-48*x^3+288*x^2-384*x)*exp(1)-24*x^ 3+192*x^2-480*x+384)*log(-4+x)^2+((48*x^4-768*x^2)*exp(1)^2+(96*x^4-195*x^ 3-1524*x^2+3072*x)*exp(1)+48*x^4-195*x^3-552*x^2+3024*x-3072)*log(-4+x)+(- 24*x^5-96*x^4+384*x^3+1536*x^2)*exp(1)^2+(-48*x^5-96*x^4+1167*x^3+1488*x^2 -6144*x)*exp(1)-24*x^5+681*x^3-462*x^2-4416*x+6144)/(((2*x^3-8*x^2)*exp(1) ^2+(4*x^3-24*x^2+32*x)*exp(1)+2*x^3-16*x^2+40*x-32)*log(-4+x)^2+((-4*x^4+6 4*x^2)*exp(1)^2+(-8*x^4+16*x^3+128*x^2-256*x)*exp(1)-4*x^4+16*x^3+48*x^2-2 56*x+256)*log(-4+x)+(2*x^5+8*x^4-32*x^3-128*x^2)*exp(1)^2+(4*x^5+8*x^4-96* x^3-128*x^2+512*x)*exp(1)+2*x^5-56*x^3+32*x^2+384*x-512),x, algorithm="max ima")
Output:
-3/2*(8*x^3*(e + 1) + x^2*(32*e + 15) - 8*(x^2*(e + 1) - 2*x)*log(x - 4) - 64*x)/(x^2*(e + 1) + 2*x*(2*e + 1) - (x*(e + 1) - 2)*log(x - 4) - 8)
Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (27) = 54\).
Time = 0.19 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.45 \[ \int \frac {6144-4416 x-462 x^2+681 x^3-24 x^5+e \left (-6144 x+1488 x^2+1167 x^3-96 x^4-48 x^5\right )+e^2 \left (1536 x^2+384 x^3-96 x^4-24 x^5\right )+\left (-3072+3024 x-552 x^2-195 x^3+48 x^4+e^2 \left (-768 x^2+48 x^4\right )+e \left (3072 x-1524 x^2-195 x^3+96 x^4\right )\right ) \log (-4+x)+\left (384-480 x+192 x^2-24 x^3+e \left (-384 x+288 x^2-48 x^3\right )+e^2 \left (96 x^2-24 x^3\right )\right ) \log ^2(-4+x)}{-512+384 x+32 x^2-56 x^3+2 x^5+e^2 \left (-128 x^2-32 x^3+8 x^4+2 x^5\right )+e \left (512 x-128 x^2-96 x^3+8 x^4+4 x^5\right )+\left (256-256 x+48 x^2+16 x^3-4 x^4+e \left (-256 x+128 x^2+16 x^3-8 x^4\right )+e^2 \left (64 x^2-4 x^4\right )\right ) \log (-4+x)+\left (-32+40 x-16 x^2+2 x^3+e^2 \left (-8 x^2+2 x^3\right )+e \left (32 x-24 x^2+4 x^3\right )\right ) \log ^2(-4+x)} \, dx=-\frac {3 \, {\left (8 \, x^{3} e - 8 \, x^{2} e \log \left (x - 4\right ) + 8 \, x^{3} + 32 \, x^{2} e - 8 \, x^{2} \log \left (x - 4\right ) + 15 \, x^{2} + 16 \, x \log \left (x - 4\right ) - 64 \, x\right )}}{2 \, {\left (x^{2} e - x e \log \left (x - 4\right ) + x^{2} + 4 \, x e - x \log \left (x - 4\right ) + 2 \, x + 2 \, \log \left (x - 4\right ) - 8\right )}} \] Input:
integrate((((-24*x^3+96*x^2)*exp(1)^2+(-48*x^3+288*x^2-384*x)*exp(1)-24*x^ 3+192*x^2-480*x+384)*log(-4+x)^2+((48*x^4-768*x^2)*exp(1)^2+(96*x^4-195*x^ 3-1524*x^2+3072*x)*exp(1)+48*x^4-195*x^3-552*x^2+3024*x-3072)*log(-4+x)+(- 24*x^5-96*x^4+384*x^3+1536*x^2)*exp(1)^2+(-48*x^5-96*x^4+1167*x^3+1488*x^2 -6144*x)*exp(1)-24*x^5+681*x^3-462*x^2-4416*x+6144)/(((2*x^3-8*x^2)*exp(1) ^2+(4*x^3-24*x^2+32*x)*exp(1)+2*x^3-16*x^2+40*x-32)*log(-4+x)^2+((-4*x^4+6 4*x^2)*exp(1)^2+(-8*x^4+16*x^3+128*x^2-256*x)*exp(1)-4*x^4+16*x^3+48*x^2-2 56*x+256)*log(-4+x)+(2*x^5+8*x^4-32*x^3-128*x^2)*exp(1)^2+(4*x^5+8*x^4-96* x^3-128*x^2+512*x)*exp(1)+2*x^5-56*x^3+32*x^2+384*x-512),x, algorithm="gia c")
Output:
-3/2*(8*x^3*e - 8*x^2*e*log(x - 4) + 8*x^3 + 32*x^2*e - 8*x^2*log(x - 4) + 15*x^2 + 16*x*log(x - 4) - 64*x)/(x^2*e - x*e*log(x - 4) + x^2 + 4*x*e - x*log(x - 4) + 2*x + 2*log(x - 4) - 8)
Time = 2.97 (sec) , antiderivative size = 129, normalized size of antiderivative = 4.45 \[ \int \frac {6144-4416 x-462 x^2+681 x^3-24 x^5+e \left (-6144 x+1488 x^2+1167 x^3-96 x^4-48 x^5\right )+e^2 \left (1536 x^2+384 x^3-96 x^4-24 x^5\right )+\left (-3072+3024 x-552 x^2-195 x^3+48 x^4+e^2 \left (-768 x^2+48 x^4\right )+e \left (3072 x-1524 x^2-195 x^3+96 x^4\right )\right ) \log (-4+x)+\left (384-480 x+192 x^2-24 x^3+e \left (-384 x+288 x^2-48 x^3\right )+e^2 \left (96 x^2-24 x^3\right )\right ) \log ^2(-4+x)}{-512+384 x+32 x^2-56 x^3+2 x^5+e^2 \left (-128 x^2-32 x^3+8 x^4+2 x^5\right )+e \left (512 x-128 x^2-96 x^3+8 x^4+4 x^5\right )+\left (256-256 x+48 x^2+16 x^3-4 x^4+e \left (-256 x+128 x^2+16 x^3-8 x^4\right )+e^2 \left (64 x^2-4 x^4\right )\right ) \log (-4+x)+\left (-32+40 x-16 x^2+2 x^3+e^2 \left (-8 x^2+2 x^3\right )+e \left (32 x-24 x^2+4 x^3\right )\right ) \log ^2(-4+x)} \, dx=-\frac {3\,\left (2\,\ln \left (x-4\right )-62\,x+15\,x\,\ln \left (x-4\right )-60\,x\,\mathrm {e}-8\,x^2\,\ln \left (x-4\right )+48\,x^2\,\mathrm {e}+32\,x^2\,{\mathrm {e}}^2+16\,x^3\,\mathrm {e}+8\,x^3\,{\mathrm {e}}^2+16\,x^2+8\,x^3+15\,x\,\ln \left (x-4\right )\,\mathrm {e}-16\,x^2\,\ln \left (x-4\right )\,\mathrm {e}-8\,x^2\,\ln \left (x-4\right )\,{\mathrm {e}}^2-8\right )}{2\,\left (\mathrm {e}+1\right )\,\left (x+x\,\mathrm {e}-2\right )\,\left (x-\ln \left (x-4\right )+4\right )} \] Input:
int(-(4416*x + log(x - 4)^2*(480*x + exp(1)*(384*x - 288*x^2 + 48*x^3) - e xp(2)*(96*x^2 - 24*x^3) - 192*x^2 + 24*x^3 - 384) + log(x - 4)*(exp(2)*(76 8*x^2 - 48*x^4) - 3024*x - exp(1)*(3072*x - 1524*x^2 - 195*x^3 + 96*x^4) + 552*x^2 + 195*x^3 - 48*x^4 + 3072) + exp(1)*(6144*x - 1488*x^2 - 1167*x^3 + 96*x^4 + 48*x^5) + 462*x^2 - 681*x^3 + 24*x^5 - exp(2)*(1536*x^2 + 384* x^3 - 96*x^4 - 24*x^5) - 6144)/(384*x + log(x - 4)^2*(40*x + exp(1)*(32*x - 24*x^2 + 4*x^3) - exp(2)*(8*x^2 - 2*x^3) - 16*x^2 + 2*x^3 - 32) + log(x - 4)*(exp(2)*(64*x^2 - 4*x^4) - 256*x - exp(1)*(256*x - 128*x^2 - 16*x^3 + 8*x^4) + 48*x^2 + 16*x^3 - 4*x^4 + 256) + exp(1)*(512*x - 128*x^2 - 96*x^ 3 + 8*x^4 + 4*x^5) + 32*x^2 - 56*x^3 + 2*x^5 - exp(2)*(128*x^2 + 32*x^3 - 8*x^4 - 2*x^5) - 512),x)
Output:
-(3*(2*log(x - 4) - 62*x + 15*x*log(x - 4) - 60*x*exp(1) - 8*x^2*log(x - 4 ) + 48*x^2*exp(1) + 32*x^2*exp(2) + 16*x^3*exp(1) + 8*x^3*exp(2) + 16*x^2 + 8*x^3 + 15*x*log(x - 4)*exp(1) - 16*x^2*log(x - 4)*exp(1) - 8*x^2*log(x - 4)*exp(2) - 8))/(2*(exp(1) + 1)*(x + x*exp(1) - 2)*(x - log(x - 4) + 4))
Time = 0.18 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.00 \[ \int \frac {6144-4416 x-462 x^2+681 x^3-24 x^5+e \left (-6144 x+1488 x^2+1167 x^3-96 x^4-48 x^5\right )+e^2 \left (1536 x^2+384 x^3-96 x^4-24 x^5\right )+\left (-3072+3024 x-552 x^2-195 x^3+48 x^4+e^2 \left (-768 x^2+48 x^4\right )+e \left (3072 x-1524 x^2-195 x^3+96 x^4\right )\right ) \log (-4+x)+\left (384-480 x+192 x^2-24 x^3+e \left (-384 x+288 x^2-48 x^3\right )+e^2 \left (96 x^2-24 x^3\right )\right ) \log ^2(-4+x)}{-512+384 x+32 x^2-56 x^3+2 x^5+e^2 \left (-128 x^2-32 x^3+8 x^4+2 x^5\right )+e \left (512 x-128 x^2-96 x^3+8 x^4+4 x^5\right )+\left (256-256 x+48 x^2+16 x^3-4 x^4+e \left (-256 x+128 x^2+16 x^3-8 x^4\right )+e^2 \left (64 x^2-4 x^4\right )\right ) \log (-4+x)+\left (-32+40 x-16 x^2+2 x^3+e^2 \left (-8 x^2+2 x^3\right )+e \left (32 x-24 x^2+4 x^3\right )\right ) \log ^2(-4+x)} \, dx=\frac {3 x \left (-8 \,\mathrm {log}\left (x -4\right ) e x -8 \,\mathrm {log}\left (x -4\right ) x +16 \,\mathrm {log}\left (x -4\right )+8 e \,x^{2}+32 e x +8 x^{2}+15 x -64\right )}{2 \,\mathrm {log}\left (x -4\right ) e x +2 \,\mathrm {log}\left (x -4\right ) x -4 \,\mathrm {log}\left (x -4\right )-2 e \,x^{2}-8 e x -2 x^{2}-4 x +16} \] Input:
int((((-24*x^3+96*x^2)*exp(1)^2+(-48*x^3+288*x^2-384*x)*exp(1)-24*x^3+192* x^2-480*x+384)*log(-4+x)^2+((48*x^4-768*x^2)*exp(1)^2+(96*x^4-195*x^3-1524 *x^2+3072*x)*exp(1)+48*x^4-195*x^3-552*x^2+3024*x-3072)*log(-4+x)+(-24*x^5 -96*x^4+384*x^3+1536*x^2)*exp(1)^2+(-48*x^5-96*x^4+1167*x^3+1488*x^2-6144* x)*exp(1)-24*x^5+681*x^3-462*x^2-4416*x+6144)/(((2*x^3-8*x^2)*exp(1)^2+(4* x^3-24*x^2+32*x)*exp(1)+2*x^3-16*x^2+40*x-32)*log(-4+x)^2+((-4*x^4+64*x^2) *exp(1)^2+(-8*x^4+16*x^3+128*x^2-256*x)*exp(1)-4*x^4+16*x^3+48*x^2-256*x+2 56)*log(-4+x)+(2*x^5+8*x^4-32*x^3-128*x^2)*exp(1)^2+(4*x^5+8*x^4-96*x^3-12 8*x^2+512*x)*exp(1)+2*x^5-56*x^3+32*x^2+384*x-512),x)
Output:
(3*x*( - 8*log(x - 4)*e*x - 8*log(x - 4)*x + 16*log(x - 4) + 8*e*x**2 + 32 *e*x + 8*x**2 + 15*x - 64))/(2*(log(x - 4)*e*x + log(x - 4)*x - 2*log(x - 4) - e*x**2 - 4*e*x - x**2 - 2*x + 8))