Integrand size = 153, antiderivative size = 35 \[ \int \frac {3328 x^4+2496 x^5+624 x^6+52 x^7+e^{32-16 x+2 x^2} \left (-66 x^3+4 x^5\right )+\left (-128-96 x-24 x^2-2 x^3\right ) \log ^2(3)+e^{16-8 x+x^2} \left (-32 x^3+248 x^4+64 x^5-16 x^6-4 x^7+\left (-32 x-280 x^2-68 x^3+16 x^4+4 x^5\right ) \log (3)\right )}{1600 x^3+1200 x^4+300 x^5+25 x^6} \, dx=x^2+\frac {1}{25} \left (-x+\frac {e^{(4-x)^2}}{4+x}+\frac {\log (3)}{x}\right )^2 \] Output:
x^2+1/5*(exp((4-x)^2)/(4+x)-x+ln(3)/x)*(1/5*exp((4-x)^2)/(4+x)-1/5*x+1/5*l n(3)/x)
\[ \int \frac {3328 x^4+2496 x^5+624 x^6+52 x^7+e^{32-16 x+2 x^2} \left (-66 x^3+4 x^5\right )+\left (-128-96 x-24 x^2-2 x^3\right ) \log ^2(3)+e^{16-8 x+x^2} \left (-32 x^3+248 x^4+64 x^5-16 x^6-4 x^7+\left (-32 x-280 x^2-68 x^3+16 x^4+4 x^5\right ) \log (3)\right )}{1600 x^3+1200 x^4+300 x^5+25 x^6} \, dx=\int \frac {3328 x^4+2496 x^5+624 x^6+52 x^7+e^{32-16 x+2 x^2} \left (-66 x^3+4 x^5\right )+\left (-128-96 x-24 x^2-2 x^3\right ) \log ^2(3)+e^{16-8 x+x^2} \left (-32 x^3+248 x^4+64 x^5-16 x^6-4 x^7+\left (-32 x-280 x^2-68 x^3+16 x^4+4 x^5\right ) \log (3)\right )}{1600 x^3+1200 x^4+300 x^5+25 x^6} \, dx \] Input:
Integrate[(3328*x^4 + 2496*x^5 + 624*x^6 + 52*x^7 + E^(32 - 16*x + 2*x^2)* (-66*x^3 + 4*x^5) + (-128 - 96*x - 24*x^2 - 2*x^3)*Log[3]^2 + E^(16 - 8*x + x^2)*(-32*x^3 + 248*x^4 + 64*x^5 - 16*x^6 - 4*x^7 + (-32*x - 280*x^2 - 6 8*x^3 + 16*x^4 + 4*x^5)*Log[3]))/(1600*x^3 + 1200*x^4 + 300*x^5 + 25*x^6), x]
Output:
Integrate[(3328*x^4 + 2496*x^5 + 624*x^6 + 52*x^7 + E^(32 - 16*x + 2*x^2)* (-66*x^3 + 4*x^5) + (-128 - 96*x - 24*x^2 - 2*x^3)*Log[3]^2 + E^(16 - 8*x + x^2)*(-32*x^3 + 248*x^4 + 64*x^5 - 16*x^6 - 4*x^7 + (-32*x - 280*x^2 - 6 8*x^3 + 16*x^4 + 4*x^5)*Log[3]))/(1600*x^3 + 1200*x^4 + 300*x^5 + 25*x^6), x]
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.
| \(\displaystyle \int \frac {52 x^7+624 x^6+2496 x^5+3328 x^4+\left (-2 x^3-24 x^2-96 x-128\right ) \log ^2(3)+e^{2 x^2-16 x+32} \left (4 x^5-66 x^3\right )+e^{x^2-8 x+16} \left (-4 x^7-16 x^6+64 x^5+248 x^4-32 x^3+\left (4 x^5+16 x^4-68 x^3-280 x^2-32 x\right ) \log (3)\right )}{25 x^6+300 x^5+1200 x^4+1600 x^3} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {52 x^7+624 x^6+2496 x^5+3328 x^4+\left (-2 x^3-24 x^2-96 x-128\right ) \log ^2(3)+e^{2 x^2-16 x+32} \left (4 x^5-66 x^3\right )+e^{x^2-8 x+16} \left (-4 x^7-16 x^6+64 x^5+248 x^4-32 x^3+\left (4 x^5+16 x^4-68 x^3-280 x^2-32 x\right ) \log (3)\right )}{x^3 \left (25 x^3+300 x^2+1200 x+1600\right )}dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {52 x^7+624 x^6+2496 x^5+3328 x^4+\left (-2 x^3-24 x^2-96 x-128\right ) \log ^2(3)+e^{2 x^2-16 x+32} \left (4 x^5-66 x^3\right )+e^{x^2-8 x+16} \left (-4 x^7-16 x^6+64 x^5+248 x^4-32 x^3+\left (4 x^5+16 x^4-68 x^3-280 x^2-32 x\right ) \log (3)\right )}{x^3 \left (5^{2/3} x+4\ 5^{2/3}\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {52 x^4}{25 (x+4)^3}+\frac {624 x^3}{25 (x+4)^3}-\frac {2 \log ^2(3)}{25 x^3}+\frac {2496 x^2}{25 (x+4)^3}+\frac {2 e^{2 (x-4)^2} \left (2 x^2-33\right )}{25 (x+4)^3}+\frac {4 e^{x^2-8 x+16} \left (-x^5+16 x^3 \left (1+\frac {\log (3)}{16}\right )-2 x^2-17 x \log (3)-\log (9)\right )}{25 (x+4)^2 x^2}+\frac {3328 x}{25 (x+4)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{25} \log (9) \int \frac {e^{x^2-8 x+16}}{x}dx-\frac {4}{25} \log (3) \int \frac {e^{x^2-8 x+16}}{x}dx-\frac {1}{200} (1024-66 \log (3)+\log (9)) \int \frac {e^{x^2-8 x+16}}{x+4}dx+\frac {4}{25} (32-\log (9)) \int \frac {e^{x^2-8 x+16}}{x+4}dx-\frac {8}{25} \sqrt {\pi } \text {erfi}(4-x)+\frac {1}{100} \sqrt {\pi } \log (9) \text {erfi}(4-x)+\frac {1}{100} \sqrt {\pi } (32-\log (9)) \text {erfi}(4-x)+\frac {416 x^2}{25 (x+4)^2}+\frac {26 x^2}{25}-\frac {2}{25} e^{x^2-8 x+16}+\frac {\log ^2(3)}{25 x^2}+\frac {e^{x^2-8 x+16} (32-\log (9))}{100 (x+4)}+\frac {e^{x^2-8 x+16} \log (9)}{100 x}+\frac {3328}{25 (x+4)}+\frac {e^{2 (4-x)^2}}{25 (x+4)^2}-\frac {6656}{25 (x+4)^2}\) |
Input:
Int[(3328*x^4 + 2496*x^5 + 624*x^6 + 52*x^7 + E^(32 - 16*x + 2*x^2)*(-66*x ^3 + 4*x^5) + (-128 - 96*x - 24*x^2 - 2*x^3)*Log[3]^2 + E^(16 - 8*x + x^2) *(-32*x^3 + 248*x^4 + 64*x^5 - 16*x^6 - 4*x^7 + (-32*x - 280*x^2 - 68*x^3 + 16*x^4 + 4*x^5)*Log[3]))/(1600*x^3 + 1200*x^4 + 300*x^5 + 25*x^6),x]
Output:
$Aborted
Time = 4.32 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.57
| method | result | size |
| risch | \(\frac {26 x^{2}}{25}+\frac {\ln \left (3\right )^{2}}{25 x^{2}}+\frac {{\mathrm e}^{2 \left (x -4\right )^{2}}}{25 \left (4+x \right )^{2}}+\frac {2 \left (\ln \left (3\right )-x^{2}\right ) {\mathrm e}^{\left (x -4\right )^{2}}}{25 \left (4+x \right ) x}\) | \(55\) |
| parts | \(\frac {26 x^{2}}{25}+\frac {\ln \left (3\right )^{2}}{25 x^{2}}+\frac {{\mathrm e}^{2 x^{2}-16 x +32}}{25 \left (4+x \right )^{2}}+\frac {-\frac {2 x^{2} {\mathrm e}^{x^{2}-8 x +16}}{25}+\frac {2 \ln \left (3\right ) {\mathrm e}^{x^{2}-8 x +16}}{25}}{x \left (4+x \right )}\) | \(71\) |
| norman | \(\frac {-\frac {3328 x^{3}}{25}+\left (\frac {\ln \left (3\right )^{2}}{25}-\frac {6656}{25}\right ) x^{2}+\frac {208 x^{5}}{25}+\frac {26 x^{6}}{25}+\frac {16 \ln \left (3\right )^{2}}{25}+\frac {8 x \ln \left (3\right )^{2}}{25}+\frac {x^{2} {\mathrm e}^{2 x^{2}-16 x +32}}{25}-\frac {8 \,{\mathrm e}^{x^{2}-8 x +16} x^{3}}{25}-\frac {2 \,{\mathrm e}^{x^{2}-8 x +16} x^{4}}{25}+\frac {8 \,{\mathrm e}^{x^{2}-8 x +16} \ln \left (3\right ) x}{25}+\frac {2 \,{\mathrm e}^{x^{2}-8 x +16} \ln \left (3\right ) x^{2}}{25}}{x^{2} \left (4+x \right )^{2}}\) | \(125\) |
| parallelrisch | \(\frac {26 x^{6}+208 x^{5}-2 \,{\mathrm e}^{x^{2}-8 x +16} x^{4}+x^{2} \ln \left (3\right )^{2}+2 \,{\mathrm e}^{x^{2}-8 x +16} \ln \left (3\right ) x^{2}-8 \,{\mathrm e}^{x^{2}-8 x +16} x^{3}+x^{2} {\mathrm e}^{2 x^{2}-16 x +32}+8 x \ln \left (3\right )^{2}+8 \,{\mathrm e}^{x^{2}-8 x +16} \ln \left (3\right ) x -3328 x^{3}+16 \ln \left (3\right )^{2}-6656 x^{2}}{25 x^{2} \left (x^{2}+8 x +16\right )}\) | \(131\) |
Input:
int(((4*x^5-66*x^3)*exp(x^2-8*x+16)^2+((4*x^5+16*x^4-68*x^3-280*x^2-32*x)* ln(3)-4*x^7-16*x^6+64*x^5+248*x^4-32*x^3)*exp(x^2-8*x+16)+(-2*x^3-24*x^2-9 6*x-128)*ln(3)^2+52*x^7+624*x^6+2496*x^5+3328*x^4)/(25*x^6+300*x^5+1200*x^ 4+1600*x^3),x,method=_RETURNVERBOSE)
Output:
26/25*x^2+1/25/x^2*ln(3)^2+1/25/(4+x)^2*exp(2*(x-4)^2)+2/25*(ln(3)-x^2)/(4 +x)/x*exp((x-4)^2)
Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (30) = 60\).
Time = 0.10 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.66 \[ \int \frac {3328 x^4+2496 x^5+624 x^6+52 x^7+e^{32-16 x+2 x^2} \left (-66 x^3+4 x^5\right )+\left (-128-96 x-24 x^2-2 x^3\right ) \log ^2(3)+e^{16-8 x+x^2} \left (-32 x^3+248 x^4+64 x^5-16 x^6-4 x^7+\left (-32 x-280 x^2-68 x^3+16 x^4+4 x^5\right ) \log (3)\right )}{1600 x^3+1200 x^4+300 x^5+25 x^6} \, dx=\frac {26 \, x^{6} + 208 \, x^{5} + 416 \, x^{4} + x^{2} e^{\left (2 \, x^{2} - 16 \, x + 32\right )} + {\left (x^{2} + 8 \, x + 16\right )} \log \left (3\right )^{2} - 2 \, {\left (x^{4} + 4 \, x^{3} - {\left (x^{2} + 4 \, x\right )} \log \left (3\right )\right )} e^{\left (x^{2} - 8 \, x + 16\right )}}{25 \, {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )}} \] Input:
integrate(((4*x^5-66*x^3)*exp(x^2-8*x+16)^2+((4*x^5+16*x^4-68*x^3-280*x^2- 32*x)*log(3)-4*x^7-16*x^6+64*x^5+248*x^4-32*x^3)*exp(x^2-8*x+16)+(-2*x^3-2 4*x^2-96*x-128)*log(3)^2+52*x^7+624*x^6+2496*x^5+3328*x^4)/(25*x^6+300*x^5 +1200*x^4+1600*x^3),x, algorithm="fricas")
Output:
1/25*(26*x^6 + 208*x^5 + 416*x^4 + x^2*e^(2*x^2 - 16*x + 32) + (x^2 + 8*x + 16)*log(3)^2 - 2*(x^4 + 4*x^3 - (x^2 + 4*x)*log(3))*e^(x^2 - 8*x + 16))/ (x^4 + 8*x^3 + 16*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (26) = 52\).
Time = 0.18 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.86 \[ \int \frac {3328 x^4+2496 x^5+624 x^6+52 x^7+e^{32-16 x+2 x^2} \left (-66 x^3+4 x^5\right )+\left (-128-96 x-24 x^2-2 x^3\right ) \log ^2(3)+e^{16-8 x+x^2} \left (-32 x^3+248 x^4+64 x^5-16 x^6-4 x^7+\left (-32 x-280 x^2-68 x^3+16 x^4+4 x^5\right ) \log (3)\right )}{1600 x^3+1200 x^4+300 x^5+25 x^6} \, dx=\frac {26 x^{2}}{25} + \frac {\left (25 x^{2} + 100 x\right ) e^{2 x^{2} - 16 x + 32} + \left (- 50 x^{4} - 400 x^{3} - 800 x^{2} + 50 x^{2} \log {\left (3 \right )} + 400 x \log {\left (3 \right )} + 800 \log {\left (3 \right )}\right ) e^{x^{2} - 8 x + 16}}{625 x^{4} + 7500 x^{3} + 30000 x^{2} + 40000 x} + \frac {\log {\left (3 \right )}^{2}}{25 x^{2}} \] Input:
integrate(((4*x**5-66*x**3)*exp(x**2-8*x+16)**2+((4*x**5+16*x**4-68*x**3-2 80*x**2-32*x)*ln(3)-4*x**7-16*x**6+64*x**5+248*x**4-32*x**3)*exp(x**2-8*x+ 16)+(-2*x**3-24*x**2-96*x-128)*ln(3)**2+52*x**7+624*x**6+2496*x**5+3328*x* *4)/(25*x**6+300*x**5+1200*x**4+1600*x**3),x)
Output:
26*x**2/25 + ((25*x**2 + 100*x)*exp(2*x**2 - 16*x + 32) + (-50*x**4 - 400* x**3 - 800*x**2 + 50*x**2*log(3) + 400*x*log(3) + 800*log(3))*exp(x**2 - 8 *x + 16))/(625*x**4 + 7500*x**3 + 30000*x**2 + 40000*x) + log(3)**2/(25*x* *2)
Leaf count of result is larger than twice the leaf count of optimal. 277 vs. \(2 (30) = 60\).
Time = 0.18 (sec) , antiderivative size = 277, normalized size of antiderivative = 7.91 \[ \int \frac {3328 x^4+2496 x^5+624 x^6+52 x^7+e^{32-16 x+2 x^2} \left (-66 x^3+4 x^5\right )+\left (-128-96 x-24 x^2-2 x^3\right ) \log ^2(3)+e^{16-8 x+x^2} \left (-32 x^3+248 x^4+64 x^5-16 x^6-4 x^7+\left (-32 x-280 x^2-68 x^3+16 x^4+4 x^5\right ) \log (3)\right )}{1600 x^3+1200 x^4+300 x^5+25 x^6} \, dx=-\frac {1}{100} \, {\left (\frac {4 \, {\left (3 \, x^{3} + 18 \, x^{2} + 16 \, x - 16\right )}}{x^{4} + 8 \, x^{3} + 16 \, x^{2}} - 3 \, \log \left (x + 4\right ) + 3 \, \log \left (x\right )\right )} \log \left (3\right )^{2} + \frac {3}{200} \, {\left (\frac {4 \, {\left (3 \, x^{2} + 18 \, x + 16\right )}}{x^{3} + 8 \, x^{2} + 16 \, x} - 3 \, \log \left (x + 4\right ) + 3 \, \log \left (x\right )\right )} \log \left (3\right )^{2} - \frac {3}{200} \, {\left (\frac {4 \, {\left (x + 6\right )}}{x^{2} + 8 \, x + 16} - \log \left (x + 4\right ) + \log \left (x\right )\right )} \log \left (3\right )^{2} + \frac {26}{25} \, x^{2} + \frac {{\left (x e^{\left (2 \, x^{2} + 32\right )} - 2 \, {\left (x^{3} e^{16} + 4 \, x^{2} e^{16} - x e^{16} \log \left (3\right ) - 4 \, e^{16} \log \left (3\right )\right )} e^{\left (x^{2} + 8 \, x\right )}\right )} e^{\left (-16 \, x\right )}}{25 \, {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )}} + \frac {\log \left (3\right )^{2}}{25 \, {\left (x^{2} + 8 \, x + 16\right )}} - \frac {9984 \, {\left (3 \, x + 10\right )}}{25 \, {\left (x^{2} + 8 \, x + 16\right )}} + \frac {6656 \, {\left (2 \, x + 7\right )}}{25 \, {\left (x^{2} + 8 \, x + 16\right )}} + \frac {19968 \, {\left (x + 3\right )}}{25 \, {\left (x^{2} + 8 \, x + 16\right )}} - \frac {3328 \, {\left (x + 2\right )}}{25 \, {\left (x^{2} + 8 \, x + 16\right )}} \] Input:
integrate(((4*x^5-66*x^3)*exp(x^2-8*x+16)^2+((4*x^5+16*x^4-68*x^3-280*x^2- 32*x)*log(3)-4*x^7-16*x^6+64*x^5+248*x^4-32*x^3)*exp(x^2-8*x+16)+(-2*x^3-2 4*x^2-96*x-128)*log(3)^2+52*x^7+624*x^6+2496*x^5+3328*x^4)/(25*x^6+300*x^5 +1200*x^4+1600*x^3),x, algorithm="maxima")
Output:
-1/100*(4*(3*x^3 + 18*x^2 + 16*x - 16)/(x^4 + 8*x^3 + 16*x^2) - 3*log(x + 4) + 3*log(x))*log(3)^2 + 3/200*(4*(3*x^2 + 18*x + 16)/(x^3 + 8*x^2 + 16*x ) - 3*log(x + 4) + 3*log(x))*log(3)^2 - 3/200*(4*(x + 6)/(x^2 + 8*x + 16) - log(x + 4) + log(x))*log(3)^2 + 26/25*x^2 + 1/25*(x*e^(2*x^2 + 32) - 2*( x^3*e^16 + 4*x^2*e^16 - x*e^16*log(3) - 4*e^16*log(3))*e^(x^2 + 8*x))*e^(- 16*x)/(x^3 + 8*x^2 + 16*x) + 1/25*log(3)^2/(x^2 + 8*x + 16) - 9984/25*(3*x + 10)/(x^2 + 8*x + 16) + 6656/25*(2*x + 7)/(x^2 + 8*x + 16) + 19968/25*(x + 3)/(x^2 + 8*x + 16) - 3328/25*(x + 2)/(x^2 + 8*x + 16)
Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (30) = 60\).
Time = 0.14 (sec) , antiderivative size = 128, normalized size of antiderivative = 3.66 \[ \int \frac {3328 x^4+2496 x^5+624 x^6+52 x^7+e^{32-16 x+2 x^2} \left (-66 x^3+4 x^5\right )+\left (-128-96 x-24 x^2-2 x^3\right ) \log ^2(3)+e^{16-8 x+x^2} \left (-32 x^3+248 x^4+64 x^5-16 x^6-4 x^7+\left (-32 x-280 x^2-68 x^3+16 x^4+4 x^5\right ) \log (3)\right )}{1600 x^3+1200 x^4+300 x^5+25 x^6} \, dx=\frac {26 \, x^{6} + 208 \, x^{5} - 2 \, x^{4} e^{\left (x^{2} - 8 \, x + 16\right )} + 416 \, x^{4} - 8 \, x^{3} e^{\left (x^{2} - 8 \, x + 16\right )} + 2 \, x^{2} e^{\left (x^{2} - 8 \, x + 16\right )} \log \left (3\right ) + x^{2} \log \left (3\right )^{2} + x^{2} e^{\left (2 \, x^{2} - 16 \, x + 32\right )} + 8 \, x e^{\left (x^{2} - 8 \, x + 16\right )} \log \left (3\right ) + 8 \, x \log \left (3\right )^{2} + 16 \, \log \left (3\right )^{2}}{25 \, {\left (x^{4} + 8 \, x^{3} + 16 \, x^{2}\right )}} \] Input:
integrate(((4*x^5-66*x^3)*exp(x^2-8*x+16)^2+((4*x^5+16*x^4-68*x^3-280*x^2- 32*x)*log(3)-4*x^7-16*x^6+64*x^5+248*x^4-32*x^3)*exp(x^2-8*x+16)+(-2*x^3-2 4*x^2-96*x-128)*log(3)^2+52*x^7+624*x^6+2496*x^5+3328*x^4)/(25*x^6+300*x^5 +1200*x^4+1600*x^3),x, algorithm="giac")
Output:
1/25*(26*x^6 + 208*x^5 - 2*x^4*e^(x^2 - 8*x + 16) + 416*x^4 - 8*x^3*e^(x^2 - 8*x + 16) + 2*x^2*e^(x^2 - 8*x + 16)*log(3) + x^2*log(3)^2 + x^2*e^(2*x ^2 - 16*x + 32) + 8*x*e^(x^2 - 8*x + 16)*log(3) + 8*x*log(3)^2 + 16*log(3) ^2)/(x^4 + 8*x^3 + 16*x^2)
Time = 2.58 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.74 \[ \int \frac {3328 x^4+2496 x^5+624 x^6+52 x^7+e^{32-16 x+2 x^2} \left (-66 x^3+4 x^5\right )+\left (-128-96 x-24 x^2-2 x^3\right ) \log ^2(3)+e^{16-8 x+x^2} \left (-32 x^3+248 x^4+64 x^5-16 x^6-4 x^7+\left (-32 x-280 x^2-68 x^3+16 x^4+4 x^5\right ) \log (3)\right )}{1600 x^3+1200 x^4+300 x^5+25 x^6} \, dx=\frac {{\ln \left (3\right )}^2}{25\,x^2}-\frac {2\,{\mathrm {e}}^{x^2-8\,x+16}}{25}+\frac {26\,x^2}{25}+\frac {{\mathrm {e}}^{x^2-8\,x+16}\,\left (\ln \left (9\right )-2\,\ln \left (3\right )+{\mathrm {e}}^{x^2-8\,x+16}\right )}{25\,{\left (x+4\right )}^2}+\frac {{\mathrm {e}}^{x^2-8\,x+16}\,\ln \left (3\right )}{50\,x}-\frac {{\mathrm {e}}^{x^2-8\,x+16}\,\left (\ln \left (3\right )-16\right )}{50\,\left (x+4\right )} \] Input:
int((3328*x^4 - log(3)^2*(96*x + 24*x^2 + 2*x^3 + 128) - exp(x^2 - 8*x + 1 6)*(32*x^3 - 248*x^4 - 64*x^5 + 16*x^6 + 4*x^7 + log(3)*(32*x + 280*x^2 + 68*x^3 - 16*x^4 - 4*x^5)) + 2496*x^5 + 624*x^6 + 52*x^7 - exp(2*x^2 - 16*x + 32)*(66*x^3 - 4*x^5))/(1600*x^3 + 1200*x^4 + 300*x^5 + 25*x^6),x)
Output:
log(3)^2/(25*x^2) - (2*exp(x^2 - 8*x + 16))/25 + (26*x^2)/25 + (exp(x^2 - 8*x + 16)*(log(9) - 2*log(3) + exp(x^2 - 8*x + 16)))/(25*(x + 4)^2) + (exp (x^2 - 8*x + 16)*log(3))/(50*x) - (exp(x^2 - 8*x + 16)*(log(3) - 16))/(50* (x + 4))
Time = 0.20 (sec) , antiderivative size = 173, normalized size of antiderivative = 4.94 \[ \int \frac {3328 x^4+2496 x^5+624 x^6+52 x^7+e^{32-16 x+2 x^2} \left (-66 x^3+4 x^5\right )+\left (-128-96 x-24 x^2-2 x^3\right ) \log ^2(3)+e^{16-8 x+x^2} \left (-32 x^3+248 x^4+64 x^5-16 x^6-4 x^7+\left (-32 x-280 x^2-68 x^3+16 x^4+4 x^5\right ) \log (3)\right )}{1600 x^3+1200 x^4+300 x^5+25 x^6} \, dx=\frac {e^{2 x^{2}} e^{32} x^{2}+2 e^{x^{2}+8 x} \mathrm {log}\left (3\right ) e^{16} x^{2}+8 e^{x^{2}+8 x} \mathrm {log}\left (3\right ) e^{16} x -2 e^{x^{2}+8 x} e^{16} x^{4}-8 e^{x^{2}+8 x} e^{16} x^{3}+e^{16 x} \mathrm {log}\left (3\right )^{2} x^{2}+8 e^{16 x} \mathrm {log}\left (3\right )^{2} x +16 e^{16 x} \mathrm {log}\left (3\right )^{2}+26 e^{16 x} x^{6}+208 e^{16 x} x^{5}+416 e^{16 x} x^{4}}{25 e^{16 x} x^{2} \left (x^{2}+8 x +16\right )} \] Input:
int(((4*x^5-66*x^3)*exp(x^2-8*x+16)^2+((4*x^5+16*x^4-68*x^3-280*x^2-32*x)* log(3)-4*x^7-16*x^6+64*x^5+248*x^4-32*x^3)*exp(x^2-8*x+16)+(-2*x^3-24*x^2- 96*x-128)*log(3)^2+52*x^7+624*x^6+2496*x^5+3328*x^4)/(25*x^6+300*x^5+1200* x^4+1600*x^3),x)
Output:
(e**(2*x**2)*e**32*x**2 + 2*e**(x**2 + 8*x)*log(3)*e**16*x**2 + 8*e**(x**2 + 8*x)*log(3)*e**16*x - 2*e**(x**2 + 8*x)*e**16*x**4 - 8*e**(x**2 + 8*x)* e**16*x**3 + e**(16*x)*log(3)**2*x**2 + 8*e**(16*x)*log(3)**2*x + 16*e**(1 6*x)*log(3)**2 + 26*e**(16*x)*x**6 + 208*e**(16*x)*x**5 + 416*e**(16*x)*x* *4)/(25*e**(16*x)*x**2*(x**2 + 8*x + 16))