Integrand size = 393, antiderivative size = 34 \[ \int \frac {100+560 x+960 x^2+504 x^3-48 x^4-96 x^5-64 x^6+e^{3 x^2} \left (8 x^3-x^6\right )+\left (-130-464 x-384 x^2+12 x^3+48 x^4+48 x^5\right ) \log ^2(5)+\left (56+96 x-6 x^3-12 x^4\right ) \log ^4(5)+\left (-8+x^3\right ) \log ^6(5)+e^{2 x^2} \left (60 x^2+96 x^3+8 x^4-6 x^5-12 x^6+\left (-24 x^2+3 x^5\right ) \log ^2(5)\right )+e^{x^2} \left (140 x+480 x^2+404 x^3+20 x^4-48 x^5-48 x^6+\left (-116 x-192 x^2-8 x^3+12 x^4+24 x^5\right ) \log ^2(5)+\left (24 x-3 x^4\right ) \log ^4(5)\right )}{8 x^3+48 x^4+96 x^5+64 x^6+e^{3 x^2} x^6+\left (-12 x^3-48 x^4-48 x^5\right ) \log ^2(5)+\left (6 x^3+12 x^4\right ) \log ^4(5)-x^3 \log ^6(5)+e^{2 x^2} \left (6 x^5+12 x^6-3 x^5 \log ^2(5)\right )+e^{x^2} \left (12 x^4+48 x^5+48 x^6+\left (-12 x^4-24 x^5\right ) \log ^2(5)+3 x^4 \log ^4(5)\right )} \, dx=5-x-\frac {\left (2+\frac {1}{2+4 x+e^{x^2} x-\log ^2(5)}\right )^2}{x^2} \] Output:
5-x-(2+1/(4*x+2-ln(5)^2+exp(x^2)*x))^2/x^2
Time = 0.16 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.44 \[ \int \frac {100+560 x+960 x^2+504 x^3-48 x^4-96 x^5-64 x^6+e^{3 x^2} \left (8 x^3-x^6\right )+\left (-130-464 x-384 x^2+12 x^3+48 x^4+48 x^5\right ) \log ^2(5)+\left (56+96 x-6 x^3-12 x^4\right ) \log ^4(5)+\left (-8+x^3\right ) \log ^6(5)+e^{2 x^2} \left (60 x^2+96 x^3+8 x^4-6 x^5-12 x^6+\left (-24 x^2+3 x^5\right ) \log ^2(5)\right )+e^{x^2} \left (140 x+480 x^2+404 x^3+20 x^4-48 x^5-48 x^6+\left (-116 x-192 x^2-8 x^3+12 x^4+24 x^5\right ) \log ^2(5)+\left (24 x-3 x^4\right ) \log ^4(5)\right )}{8 x^3+48 x^4+96 x^5+64 x^6+e^{3 x^2} x^6+\left (-12 x^3-48 x^4-48 x^5\right ) \log ^2(5)+\left (6 x^3+12 x^4\right ) \log ^4(5)-x^3 \log ^6(5)+e^{2 x^2} \left (6 x^5+12 x^6-3 x^5 \log ^2(5)\right )+e^{x^2} \left (12 x^4+48 x^5+48 x^6+\left (-12 x^4-24 x^5\right ) \log ^2(5)+3 x^4 \log ^4(5)\right )} \, dx=-\frac {x^3+\frac {\left (5+2 \left (4+e^{x^2}\right ) x-2 \log ^2(5)\right )^2}{\left (2+\left (4+e^{x^2}\right ) x-\log ^2(5)\right )^2}}{x^2} \] Input:
Integrate[(100 + 560*x + 960*x^2 + 504*x^3 - 48*x^4 - 96*x^5 - 64*x^6 + E^ (3*x^2)*(8*x^3 - x^6) + (-130 - 464*x - 384*x^2 + 12*x^3 + 48*x^4 + 48*x^5 )*Log[5]^2 + (56 + 96*x - 6*x^3 - 12*x^4)*Log[5]^4 + (-8 + x^3)*Log[5]^6 + E^(2*x^2)*(60*x^2 + 96*x^3 + 8*x^4 - 6*x^5 - 12*x^6 + (-24*x^2 + 3*x^5)*L og[5]^2) + E^x^2*(140*x + 480*x^2 + 404*x^3 + 20*x^4 - 48*x^5 - 48*x^6 + ( -116*x - 192*x^2 - 8*x^3 + 12*x^4 + 24*x^5)*Log[5]^2 + (24*x - 3*x^4)*Log[ 5]^4))/(8*x^3 + 48*x^4 + 96*x^5 + 64*x^6 + E^(3*x^2)*x^6 + (-12*x^3 - 48*x ^4 - 48*x^5)*Log[5]^2 + (6*x^3 + 12*x^4)*Log[5]^4 - x^3*Log[5]^6 + E^(2*x^ 2)*(6*x^5 + 12*x^6 - 3*x^5*Log[5]^2) + E^x^2*(12*x^4 + 48*x^5 + 48*x^6 + ( -12*x^4 - 24*x^5)*Log[5]^2 + 3*x^4*Log[5]^4)),x]
Output:
-((x^3 + (5 + 2*(4 + E^x^2)*x - 2*Log[5]^2)^2/(2 + (4 + E^x^2)*x - Log[5]^ 2)^2)/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.
| \(\displaystyle \int \frac {-64 x^6-96 x^5-48 x^4+504 x^3+\left (x^3-8\right ) \log ^6(5)+960 x^2+\left (-12 x^4-6 x^3+96 x+56\right ) \log ^4(5)+e^{3 x^2} \left (8 x^3-x^6\right )+\left (48 x^5+48 x^4+12 x^3-384 x^2-464 x-130\right ) \log ^2(5)+e^{2 x^2} \left (-12 x^6-6 x^5+8 x^4+96 x^3+60 x^2+\left (3 x^5-24 x^2\right ) \log ^2(5)\right )+e^{x^2} \left (-48 x^6-48 x^5+20 x^4+\left (24 x-3 x^4\right ) \log ^4(5)+404 x^3+480 x^2+\left (24 x^5+12 x^4-8 x^3-192 x^2-116 x\right ) \log ^2(5)+140 x\right )+560 x+100}{64 x^6+96 x^5+48 x^4+8 x^3-x^3 \log ^6(5)+e^{3 x^2} x^6+\left (12 x^4+6 x^3\right ) \log ^4(5)+e^{2 x^2} \left (12 x^6+6 x^5-3 x^5 \log ^2(5)\right )+\left (-48 x^5-48 x^4-12 x^3\right ) \log ^2(5)+e^{x^2} \left (48 x^6+48 x^5+12 x^4+3 x^4 \log ^4(5)+\left (-24 x^5-12 x^4\right ) \log ^2(5)\right )} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {-64 x^6-96 x^5-48 x^4+504 x^3+\left (x^3-8\right ) \log ^6(5)+960 x^2+\left (-12 x^4-6 x^3+96 x+56\right ) \log ^4(5)+e^{3 x^2} \left (8 x^3-x^6\right )+\left (48 x^5+48 x^4+12 x^3-384 x^2-464 x-130\right ) \log ^2(5)+e^{2 x^2} \left (-12 x^6-6 x^5+8 x^4+96 x^3+60 x^2+\left (3 x^5-24 x^2\right ) \log ^2(5)\right )+e^{x^2} \left (-48 x^6-48 x^5+20 x^4+\left (24 x-3 x^4\right ) \log ^4(5)+404 x^3+480 x^2+\left (24 x^5+12 x^4-8 x^3-192 x^2-116 x\right ) \log ^2(5)+140 x\right )+560 x+100}{64 x^6+96 x^5+48 x^4+x^3 \left (8-\log ^6(5)\right )+e^{3 x^2} x^6+\left (12 x^4+6 x^3\right ) \log ^4(5)+e^{2 x^2} \left (12 x^6+6 x^5-3 x^5 \log ^2(5)\right )+\left (-48 x^5-48 x^4-12 x^3\right ) \log ^2(5)+e^{x^2} \left (48 x^6+48 x^5+12 x^4+3 x^4 \log ^4(5)+\left (-24 x^5-12 x^4\right ) \log ^2(5)\right )}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-12 \left (e^{x^2}+4\right )^2 x^2 \left (2 \log ^2(5)-5\right )+4 \left (e^{x^2}+4\right ) x \left (35+6 \log ^4(5)-29 \log ^2(5)\right )-\left (e^{x^2}+4\right )^3 x^6+3 \left (e^{x^2}+4\right )^2 x^5 \left (\log ^2(5)-2\right )+\left (e^{x^2}+4\right ) x^4 \left (8 e^{x^2}-3 \left (\log ^2(5)-2\right )^2\right )+x^3 \left (96 e^{2 x^2}+8 e^{3 x^2}+e^{x^2} \left (404-8 \log ^2(5)\right )+504+\log ^6(5)-6 \log ^4(5)+12 \log ^2(5)\right )-2 \left (5-2 \log ^2(5)\right )^2 \left (\log ^2(5)-2\right )}{x^3 \left (\left (e^{x^2}+4\right ) x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {8-x^3}{x^3}+\frac {4 \left (2 x^2+3\right )}{x^3 \left (e^{x^2} x+4 x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )}+\frac {4 \left (-8 x^3-x^2 \left (3-2 \log ^2(5)\right )-1+\log ^2(5)\right )}{x^3 \left (e^{x^2} x+4 x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )^2}+\frac {2 \left (-8 x^3-2 x^2 \left (2-\log ^2(5)\right )-2+\log ^2(5)\right )}{x^3 \left (e^{x^2} x+4 x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 16 \int \frac {1}{\left (-e^{x^2} x-4 x-2 \left (1-\frac {\log ^2(5)}{2}\right )\right )^3}dx-4 \left (2-\log ^2(5)\right ) \int \frac {1}{x \left (e^{x^2} x+4 x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )^3}dx-32 \int \frac {1}{\left (e^{x^2} x+4 x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )^2}dx-4 \left (3-2 \log ^2(5)\right ) \int \frac {1}{x \left (e^{x^2} x+4 x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )^2}dx+8 \int \frac {1}{x \left (e^{x^2} x+4 x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )}dx-2 \left (2-\log ^2(5)\right ) \int \frac {1}{x^3 \left (e^{x^2} x+4 x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )^3}dx-4 \left (1-\log ^2(5)\right ) \int \frac {1}{x^3 \left (e^{x^2} x+4 x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )^2}dx+12 \int \frac {1}{x^3 \left (e^{x^2} x+4 x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )}dx-\frac {4}{x^2}-x\) |
Input:
Int[(100 + 560*x + 960*x^2 + 504*x^3 - 48*x^4 - 96*x^5 - 64*x^6 + E^(3*x^2 )*(8*x^3 - x^6) + (-130 - 464*x - 384*x^2 + 12*x^3 + 48*x^4 + 48*x^5)*Log[ 5]^2 + (56 + 96*x - 6*x^3 - 12*x^4)*Log[5]^4 + (-8 + x^3)*Log[5]^6 + E^(2* x^2)*(60*x^2 + 96*x^3 + 8*x^4 - 6*x^5 - 12*x^6 + (-24*x^2 + 3*x^5)*Log[5]^ 2) + E^x^2*(140*x + 480*x^2 + 404*x^3 + 20*x^4 - 48*x^5 - 48*x^6 + (-116*x - 192*x^2 - 8*x^3 + 12*x^4 + 24*x^5)*Log[5]^2 + (24*x - 3*x^4)*Log[5]^4)) /(8*x^3 + 48*x^4 + 96*x^5 + 64*x^6 + E^(3*x^2)*x^6 + (-12*x^3 - 48*x^4 - 4 8*x^5)*Log[5]^2 + (6*x^3 + 12*x^4)*Log[5]^4 - x^3*Log[5]^6 + E^(2*x^2)*(6* x^5 + 12*x^6 - 3*x^5*Log[5]^2) + E^x^2*(12*x^4 + 48*x^5 + 48*x^6 + (-12*x^ 4 - 24*x^5)*Log[5]^2 + 3*x^4*Log[5]^4)),x]
Output:
$Aborted
Time = 19.38 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.47
| method | result | size |
| risch | \(-x -\frac {4}{x^{2}}+\frac {4 \ln \left (5\right )^{2}-4 \,{\mathrm e}^{x^{2}} x -16 x -9}{x^{2} \left (\ln \left (5\right )^{2}-{\mathrm e}^{x^{2}} x -4 x -2\right )^{2}}\) | \(50\) |
| norman | \(\frac {\left (-80+32 \ln \left (5\right )^{2}\right ) x -64 x^{2}+\left (8 \ln \left (5\right )^{2}-16\right ) x^{4}+\left (-\ln \left (5\right )^{4}+4 \ln \left (5\right )^{2}-4\right ) x^{3}+\left (-20+8 \ln \left (5\right )^{2}\right ) x \,{\mathrm e}^{x^{2}}+\left (2 \ln \left (5\right )^{2}-4\right ) x^{4} {\mathrm e}^{x^{2}}-25-16 x^{5}-32 x^{2} {\mathrm e}^{x^{2}}-4 x^{2} {\mathrm e}^{2 x^{2}}-8 \,{\mathrm e}^{x^{2}} x^{5}-{\mathrm e}^{2 x^{2}} x^{5}-4 \ln \left (5\right )^{4}+20 \ln \left (5\right )^{2}}{x^{2} \left (\ln \left (5\right )^{2}-{\mathrm e}^{x^{2}} x -4 x -2\right )^{2}}\) | \(157\) |
| parallelrisch | \(-\frac {x^{3} \ln \left (5\right )^{4}-2 \ln \left (5\right )^{2} {\mathrm e}^{x^{2}} x^{4}+{\mathrm e}^{2 x^{2}} x^{5}-8 x^{4} \ln \left (5\right )^{2}+8 \,{\mathrm e}^{x^{2}} x^{5}-4 x^{3} \ln \left (5\right )^{2}+4 x^{4} {\mathrm e}^{x^{2}}+16 x^{5}+4 \ln \left (5\right )^{4}-8 \ln \left (5\right )^{2} {\mathrm e}^{x^{2}} x +4 x^{2} {\mathrm e}^{2 x^{2}}+16 x^{4}-32 x \ln \left (5\right )^{2}+32 x^{2} {\mathrm e}^{x^{2}}+4 x^{3}-20 \ln \left (5\right )^{2}+20 \,{\mathrm e}^{x^{2}} x +64 x^{2}+80 x +25}{x^{2} \left (\ln \left (5\right )^{4}-2 \ln \left (5\right )^{2} {\mathrm e}^{x^{2}} x +x^{2} {\mathrm e}^{2 x^{2}}-8 x \ln \left (5\right )^{2}+8 x^{2} {\mathrm e}^{x^{2}}-4 \ln \left (5\right )^{2}+16 x^{2}+4 \,{\mathrm e}^{x^{2}} x +16 x +4\right )}\) | \(221\) |
Input:
int(((-x^6+8*x^3)*exp(x^2)^3+((3*x^5-24*x^2)*ln(5)^2-12*x^6-6*x^5+8*x^4+96 *x^3+60*x^2)*exp(x^2)^2+((-3*x^4+24*x)*ln(5)^4+(24*x^5+12*x^4-8*x^3-192*x^ 2-116*x)*ln(5)^2-48*x^6-48*x^5+20*x^4+404*x^3+480*x^2+140*x)*exp(x^2)+(x^3 -8)*ln(5)^6+(-12*x^4-6*x^3+96*x+56)*ln(5)^4+(48*x^5+48*x^4+12*x^3-384*x^2- 464*x-130)*ln(5)^2-64*x^6-96*x^5-48*x^4+504*x^3+960*x^2+560*x+100)/(x^6*ex p(x^2)^3+(-3*x^5*ln(5)^2+12*x^6+6*x^5)*exp(x^2)^2+(3*x^4*ln(5)^4+(-24*x^5- 12*x^4)*ln(5)^2+48*x^6+48*x^5+12*x^4)*exp(x^2)-x^3*ln(5)^6+(12*x^4+6*x^3)* ln(5)^4+(-48*x^5-48*x^4-12*x^3)*ln(5)^2+64*x^6+96*x^5+48*x^4+8*x^3),x,meth od=_RETURNVERBOSE)
Output:
-x-4/x^2+(4*ln(5)^2-4*exp(x^2)*x-16*x-9)/x^2/(ln(5)^2-exp(x^2)*x-4*x-2)^2
Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (33) = 66\).
Time = 0.09 (sec) , antiderivative size = 186, normalized size of antiderivative = 5.47 \[ \int \frac {100+560 x+960 x^2+504 x^3-48 x^4-96 x^5-64 x^6+e^{3 x^2} \left (8 x^3-x^6\right )+\left (-130-464 x-384 x^2+12 x^3+48 x^4+48 x^5\right ) \log ^2(5)+\left (56+96 x-6 x^3-12 x^4\right ) \log ^4(5)+\left (-8+x^3\right ) \log ^6(5)+e^{2 x^2} \left (60 x^2+96 x^3+8 x^4-6 x^5-12 x^6+\left (-24 x^2+3 x^5\right ) \log ^2(5)\right )+e^{x^2} \left (140 x+480 x^2+404 x^3+20 x^4-48 x^5-48 x^6+\left (-116 x-192 x^2-8 x^3+12 x^4+24 x^5\right ) \log ^2(5)+\left (24 x-3 x^4\right ) \log ^4(5)\right )}{8 x^3+48 x^4+96 x^5+64 x^6+e^{3 x^2} x^6+\left (-12 x^3-48 x^4-48 x^5\right ) \log ^2(5)+\left (6 x^3+12 x^4\right ) \log ^4(5)-x^3 \log ^6(5)+e^{2 x^2} \left (6 x^5+12 x^6-3 x^5 \log ^2(5)\right )+e^{x^2} \left (12 x^4+48 x^5+48 x^6+\left (-12 x^4-24 x^5\right ) \log ^2(5)+3 x^4 \log ^4(5)\right )} \, dx=-\frac {16 \, x^{5} + {\left (x^{3} + 4\right )} \log \left (5\right )^{4} + 16 \, x^{4} + 4 \, x^{3} - 4 \, {\left (2 \, x^{4} + x^{3} + 8 \, x + 5\right )} \log \left (5\right )^{2} + 64 \, x^{2} + {\left (x^{5} + 4 \, x^{2}\right )} e^{\left (2 \, x^{2}\right )} + 2 \, {\left (4 \, x^{5} + 2 \, x^{4} - {\left (x^{4} + 4 \, x\right )} \log \left (5\right )^{2} + 16 \, x^{2} + 10 \, x\right )} e^{\left (x^{2}\right )} + 80 \, x + 25}{x^{2} \log \left (5\right )^{4} + x^{4} e^{\left (2 \, x^{2}\right )} + 16 \, x^{4} + 16 \, x^{3} - 4 \, {\left (2 \, x^{3} + x^{2}\right )} \log \left (5\right )^{2} + 4 \, x^{2} - 2 \, {\left (x^{3} \log \left (5\right )^{2} - 4 \, x^{4} - 2 \, x^{3}\right )} e^{\left (x^{2}\right )}} \] Input:
integrate(((-x^6+8*x^3)*exp(x^2)^3+((3*x^5-24*x^2)*log(5)^2-12*x^6-6*x^5+8 *x^4+96*x^3+60*x^2)*exp(x^2)^2+((-3*x^4+24*x)*log(5)^4+(24*x^5+12*x^4-8*x^ 3-192*x^2-116*x)*log(5)^2-48*x^6-48*x^5+20*x^4+404*x^3+480*x^2+140*x)*exp( x^2)+(x^3-8)*log(5)^6+(-12*x^4-6*x^3+96*x+56)*log(5)^4+(48*x^5+48*x^4+12*x ^3-384*x^2-464*x-130)*log(5)^2-64*x^6-96*x^5-48*x^4+504*x^3+960*x^2+560*x+ 100)/(x^6*exp(x^2)^3+(-3*x^5*log(5)^2+12*x^6+6*x^5)*exp(x^2)^2+(3*x^4*log( 5)^4+(-24*x^5-12*x^4)*log(5)^2+48*x^6+48*x^5+12*x^4)*exp(x^2)-x^3*log(5)^6 +(12*x^4+6*x^3)*log(5)^4+(-48*x^5-48*x^4-12*x^3)*log(5)^2+64*x^6+96*x^5+48 *x^4+8*x^3),x, algorithm="fricas")
Output:
-(16*x^5 + (x^3 + 4)*log(5)^4 + 16*x^4 + 4*x^3 - 4*(2*x^4 + x^3 + 8*x + 5) *log(5)^2 + 64*x^2 + (x^5 + 4*x^2)*e^(2*x^2) + 2*(4*x^5 + 2*x^4 - (x^4 + 4 *x)*log(5)^2 + 16*x^2 + 10*x)*e^(x^2) + 80*x + 25)/(x^2*log(5)^4 + x^4*e^( 2*x^2) + 16*x^4 + 16*x^3 - 4*(2*x^3 + x^2)*log(5)^2 + 4*x^2 - 2*(x^3*log(5 )^2 - 4*x^4 - 2*x^3)*e^(x^2))
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (27) = 54\).
Time = 0.23 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.09 \[ \int \frac {100+560 x+960 x^2+504 x^3-48 x^4-96 x^5-64 x^6+e^{3 x^2} \left (8 x^3-x^6\right )+\left (-130-464 x-384 x^2+12 x^3+48 x^4+48 x^5\right ) \log ^2(5)+\left (56+96 x-6 x^3-12 x^4\right ) \log ^4(5)+\left (-8+x^3\right ) \log ^6(5)+e^{2 x^2} \left (60 x^2+96 x^3+8 x^4-6 x^5-12 x^6+\left (-24 x^2+3 x^5\right ) \log ^2(5)\right )+e^{x^2} \left (140 x+480 x^2+404 x^3+20 x^4-48 x^5-48 x^6+\left (-116 x-192 x^2-8 x^3+12 x^4+24 x^5\right ) \log ^2(5)+\left (24 x-3 x^4\right ) \log ^4(5)\right )}{8 x^3+48 x^4+96 x^5+64 x^6+e^{3 x^2} x^6+\left (-12 x^3-48 x^4-48 x^5\right ) \log ^2(5)+\left (6 x^3+12 x^4\right ) \log ^4(5)-x^3 \log ^6(5)+e^{2 x^2} \left (6 x^5+12 x^6-3 x^5 \log ^2(5)\right )+e^{x^2} \left (12 x^4+48 x^5+48 x^6+\left (-12 x^4-24 x^5\right ) \log ^2(5)+3 x^4 \log ^4(5)\right )} \, dx=- x + \frac {- 4 x e^{x^{2}} - 16 x - 9 + 4 \log {\left (5 \right )}^{2}}{x^{4} e^{2 x^{2}} + 16 x^{4} - 8 x^{3} \log {\left (5 \right )}^{2} + 16 x^{3} - 4 x^{2} \log {\left (5 \right )}^{2} + 4 x^{2} + x^{2} \log {\left (5 \right )}^{4} + \left (8 x^{4} - 2 x^{3} \log {\left (5 \right )}^{2} + 4 x^{3}\right ) e^{x^{2}}} - \frac {4}{x^{2}} \] Input:
integrate(((-x**6+8*x**3)*exp(x**2)**3+((3*x**5-24*x**2)*ln(5)**2-12*x**6- 6*x**5+8*x**4+96*x**3+60*x**2)*exp(x**2)**2+((-3*x**4+24*x)*ln(5)**4+(24*x **5+12*x**4-8*x**3-192*x**2-116*x)*ln(5)**2-48*x**6-48*x**5+20*x**4+404*x* *3+480*x**2+140*x)*exp(x**2)+(x**3-8)*ln(5)**6+(-12*x**4-6*x**3+96*x+56)*l n(5)**4+(48*x**5+48*x**4+12*x**3-384*x**2-464*x-130)*ln(5)**2-64*x**6-96*x **5-48*x**4+504*x**3+960*x**2+560*x+100)/(x**6*exp(x**2)**3+(-3*x**5*ln(5) **2+12*x**6+6*x**5)*exp(x**2)**2+(3*x**4*ln(5)**4+(-24*x**5-12*x**4)*ln(5) **2+48*x**6+48*x**5+12*x**4)*exp(x**2)-x**3*ln(5)**6+(12*x**4+6*x**3)*ln(5 )**4+(-48*x**5-48*x**4-12*x**3)*ln(5)**2+64*x**6+96*x**5+48*x**4+8*x**3),x )
Output:
-x + (-4*x*exp(x**2) - 16*x - 9 + 4*log(5)**2)/(x**4*exp(2*x**2) + 16*x**4 - 8*x**3*log(5)**2 + 16*x**3 - 4*x**2*log(5)**2 + 4*x**2 + x**2*log(5)**4 + (8*x**4 - 2*x**3*log(5)**2 + 4*x**3)*exp(x**2)) - 4/x**2
Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (33) = 66\).
Time = 0.23 (sec) , antiderivative size = 186, normalized size of antiderivative = 5.47 \[ \int \frac {100+560 x+960 x^2+504 x^3-48 x^4-96 x^5-64 x^6+e^{3 x^2} \left (8 x^3-x^6\right )+\left (-130-464 x-384 x^2+12 x^3+48 x^4+48 x^5\right ) \log ^2(5)+\left (56+96 x-6 x^3-12 x^4\right ) \log ^4(5)+\left (-8+x^3\right ) \log ^6(5)+e^{2 x^2} \left (60 x^2+96 x^3+8 x^4-6 x^5-12 x^6+\left (-24 x^2+3 x^5\right ) \log ^2(5)\right )+e^{x^2} \left (140 x+480 x^2+404 x^3+20 x^4-48 x^5-48 x^6+\left (-116 x-192 x^2-8 x^3+12 x^4+24 x^5\right ) \log ^2(5)+\left (24 x-3 x^4\right ) \log ^4(5)\right )}{8 x^3+48 x^4+96 x^5+64 x^6+e^{3 x^2} x^6+\left (-12 x^3-48 x^4-48 x^5\right ) \log ^2(5)+\left (6 x^3+12 x^4\right ) \log ^4(5)-x^3 \log ^6(5)+e^{2 x^2} \left (6 x^5+12 x^6-3 x^5 \log ^2(5)\right )+e^{x^2} \left (12 x^4+48 x^5+48 x^6+\left (-12 x^4-24 x^5\right ) \log ^2(5)+3 x^4 \log ^4(5)\right )} \, dx=\frac {8 \, {\left (\log \left (5\right )^{2} - 2\right )} x^{4} - 16 \, x^{5} - {\left (\log \left (5\right )^{4} - 4 \, \log \left (5\right )^{2} + 4\right )} x^{3} - 4 \, \log \left (5\right )^{4} + 16 \, {\left (2 \, \log \left (5\right )^{2} - 5\right )} x - 64 \, x^{2} - {\left (x^{5} + 4 \, x^{2}\right )} e^{\left (2 \, x^{2}\right )} + 2 \, {\left ({\left (\log \left (5\right )^{2} - 2\right )} x^{4} - 4 \, x^{5} + 2 \, {\left (2 \, \log \left (5\right )^{2} - 5\right )} x - 16 \, x^{2}\right )} e^{\left (x^{2}\right )} + 20 \, \log \left (5\right )^{2} - 25}{x^{4} e^{\left (2 \, x^{2}\right )} - 8 \, {\left (\log \left (5\right )^{2} - 2\right )} x^{3} + 16 \, x^{4} + {\left (\log \left (5\right )^{4} - 4 \, \log \left (5\right )^{2} + 4\right )} x^{2} - 2 \, {\left ({\left (\log \left (5\right )^{2} - 2\right )} x^{3} - 4 \, x^{4}\right )} e^{\left (x^{2}\right )}} \] Input:
integrate(((-x^6+8*x^3)*exp(x^2)^3+((3*x^5-24*x^2)*log(5)^2-12*x^6-6*x^5+8 *x^4+96*x^3+60*x^2)*exp(x^2)^2+((-3*x^4+24*x)*log(5)^4+(24*x^5+12*x^4-8*x^ 3-192*x^2-116*x)*log(5)^2-48*x^6-48*x^5+20*x^4+404*x^3+480*x^2+140*x)*exp( x^2)+(x^3-8)*log(5)^6+(-12*x^4-6*x^3+96*x+56)*log(5)^4+(48*x^5+48*x^4+12*x ^3-384*x^2-464*x-130)*log(5)^2-64*x^6-96*x^5-48*x^4+504*x^3+960*x^2+560*x+ 100)/(x^6*exp(x^2)^3+(-3*x^5*log(5)^2+12*x^6+6*x^5)*exp(x^2)^2+(3*x^4*log( 5)^4+(-24*x^5-12*x^4)*log(5)^2+48*x^6+48*x^5+12*x^4)*exp(x^2)-x^3*log(5)^6 +(12*x^4+6*x^3)*log(5)^4+(-48*x^5-48*x^4-12*x^3)*log(5)^2+64*x^6+96*x^5+48 *x^4+8*x^3),x, algorithm="maxima")
Output:
(8*(log(5)^2 - 2)*x^4 - 16*x^5 - (log(5)^4 - 4*log(5)^2 + 4)*x^3 - 4*log(5 )^4 + 16*(2*log(5)^2 - 5)*x - 64*x^2 - (x^5 + 4*x^2)*e^(2*x^2) + 2*((log(5 )^2 - 2)*x^4 - 4*x^5 + 2*(2*log(5)^2 - 5)*x - 16*x^2)*e^(x^2) + 20*log(5)^ 2 - 25)/(x^4*e^(2*x^2) - 8*(log(5)^2 - 2)*x^3 + 16*x^4 + (log(5)^4 - 4*log (5)^2 + 4)*x^2 - 2*((log(5)^2 - 2)*x^3 - 4*x^4)*e^(x^2))
Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (33) = 66\).
Time = 0.45 (sec) , antiderivative size = 240, normalized size of antiderivative = 7.06 \[ \int \frac {100+560 x+960 x^2+504 x^3-48 x^4-96 x^5-64 x^6+e^{3 x^2} \left (8 x^3-x^6\right )+\left (-130-464 x-384 x^2+12 x^3+48 x^4+48 x^5\right ) \log ^2(5)+\left (56+96 x-6 x^3-12 x^4\right ) \log ^4(5)+\left (-8+x^3\right ) \log ^6(5)+e^{2 x^2} \left (60 x^2+96 x^3+8 x^4-6 x^5-12 x^6+\left (-24 x^2+3 x^5\right ) \log ^2(5)\right )+e^{x^2} \left (140 x+480 x^2+404 x^3+20 x^4-48 x^5-48 x^6+\left (-116 x-192 x^2-8 x^3+12 x^4+24 x^5\right ) \log ^2(5)+\left (24 x-3 x^4\right ) \log ^4(5)\right )}{8 x^3+48 x^4+96 x^5+64 x^6+e^{3 x^2} x^6+\left (-12 x^3-48 x^4-48 x^5\right ) \log ^2(5)+\left (6 x^3+12 x^4\right ) \log ^4(5)-x^3 \log ^6(5)+e^{2 x^2} \left (6 x^5+12 x^6-3 x^5 \log ^2(5)\right )+e^{x^2} \left (12 x^4+48 x^5+48 x^6+\left (-12 x^4-24 x^5\right ) \log ^2(5)+3 x^4 \log ^4(5)\right )} \, dx=-\frac {2 \, x^{4} e^{\left (x^{2}\right )} \log \left (5\right )^{2} - x^{3} \log \left (5\right )^{4} - x^{5} e^{\left (2 \, x^{2}\right )} - 8 \, x^{5} e^{\left (x^{2}\right )} + 8 \, x^{4} \log \left (5\right )^{2} - 16 \, x^{5} - 4 \, x^{4} e^{\left (x^{2}\right )} + 4 \, x^{3} \log \left (5\right )^{2} - 16 \, x^{4} + 8 \, x e^{\left (x^{2}\right )} \log \left (5\right )^{2} - 4 \, \log \left (5\right )^{4} - 4 \, x^{3} - 4 \, x^{2} e^{\left (2 \, x^{2}\right )} - 32 \, x^{2} e^{\left (x^{2}\right )} + 32 \, x \log \left (5\right )^{2} - 64 \, x^{2} - 20 \, x e^{\left (x^{2}\right )} + 20 \, \log \left (5\right )^{2} - 80 \, x - 25}{2 \, x^{3} e^{\left (x^{2}\right )} \log \left (5\right )^{2} - x^{2} \log \left (5\right )^{4} - x^{4} e^{\left (2 \, x^{2}\right )} - 8 \, x^{4} e^{\left (x^{2}\right )} + 8 \, x^{3} \log \left (5\right )^{2} - 16 \, x^{4} - 4 \, x^{3} e^{\left (x^{2}\right )} + 4 \, x^{2} \log \left (5\right )^{2} - 16 \, x^{3} - 4 \, x^{2}} \] Input:
integrate(((-x^6+8*x^3)*exp(x^2)^3+((3*x^5-24*x^2)*log(5)^2-12*x^6-6*x^5+8 *x^4+96*x^3+60*x^2)*exp(x^2)^2+((-3*x^4+24*x)*log(5)^4+(24*x^5+12*x^4-8*x^ 3-192*x^2-116*x)*log(5)^2-48*x^6-48*x^5+20*x^4+404*x^3+480*x^2+140*x)*exp( x^2)+(x^3-8)*log(5)^6+(-12*x^4-6*x^3+96*x+56)*log(5)^4+(48*x^5+48*x^4+12*x ^3-384*x^2-464*x-130)*log(5)^2-64*x^6-96*x^5-48*x^4+504*x^3+960*x^2+560*x+ 100)/(x^6*exp(x^2)^3+(-3*x^5*log(5)^2+12*x^6+6*x^5)*exp(x^2)^2+(3*x^4*log( 5)^4+(-24*x^5-12*x^4)*log(5)^2+48*x^6+48*x^5+12*x^4)*exp(x^2)-x^3*log(5)^6 +(12*x^4+6*x^3)*log(5)^4+(-48*x^5-48*x^4-12*x^3)*log(5)^2+64*x^6+96*x^5+48 *x^4+8*x^3),x, algorithm="giac")
Output:
-(2*x^4*e^(x^2)*log(5)^2 - x^3*log(5)^4 - x^5*e^(2*x^2) - 8*x^5*e^(x^2) + 8*x^4*log(5)^2 - 16*x^5 - 4*x^4*e^(x^2) + 4*x^3*log(5)^2 - 16*x^4 + 8*x*e^ (x^2)*log(5)^2 - 4*log(5)^4 - 4*x^3 - 4*x^2*e^(2*x^2) - 32*x^2*e^(x^2) + 3 2*x*log(5)^2 - 64*x^2 - 20*x*e^(x^2) + 20*log(5)^2 - 80*x - 25)/(2*x^3*e^( x^2)*log(5)^2 - x^2*log(5)^4 - x^4*e^(2*x^2) - 8*x^4*e^(x^2) + 8*x^3*log(5 )^2 - 16*x^4 - 4*x^3*e^(x^2) + 4*x^2*log(5)^2 - 16*x^3 - 4*x^2)
Time = 4.34 (sec) , antiderivative size = 213, normalized size of antiderivative = 6.26 \[ \int \frac {100+560 x+960 x^2+504 x^3-48 x^4-96 x^5-64 x^6+e^{3 x^2} \left (8 x^3-x^6\right )+\left (-130-464 x-384 x^2+12 x^3+48 x^4+48 x^5\right ) \log ^2(5)+\left (56+96 x-6 x^3-12 x^4\right ) \log ^4(5)+\left (-8+x^3\right ) \log ^6(5)+e^{2 x^2} \left (60 x^2+96 x^3+8 x^4-6 x^5-12 x^6+\left (-24 x^2+3 x^5\right ) \log ^2(5)\right )+e^{x^2} \left (140 x+480 x^2+404 x^3+20 x^4-48 x^5-48 x^6+\left (-116 x-192 x^2-8 x^3+12 x^4+24 x^5\right ) \log ^2(5)+\left (24 x-3 x^4\right ) \log ^4(5)\right )}{8 x^3+48 x^4+96 x^5+64 x^6+e^{3 x^2} x^6+\left (-12 x^3-48 x^4-48 x^5\right ) \log ^2(5)+\left (6 x^3+12 x^4\right ) \log ^4(5)-x^3 \log ^6(5)+e^{2 x^2} \left (6 x^5+12 x^6-3 x^5 \log ^2(5)\right )+e^{x^2} \left (12 x^4+48 x^5+48 x^6+\left (-12 x^4-24 x^5\right ) \log ^2(5)+3 x^4 \log ^4(5)\right )} \, dx=-x-\frac {4}{x^2}-\frac {4\,\left (4\,x^2-{\ln \left (5\right )}^2-2\,x^2\,{\ln \left (5\right )}^2+8\,x^3+2\right )}{x\,\left ({\mathrm {e}}^{x^2}+\frac {4\,x-{\ln \left (5\right )}^2+2}{x}\right )\,\left (2\,x^2-2\,x^4\,{\ln \left (5\right )}^2-x^2\,{\ln \left (5\right )}^2+4\,x^4+8\,x^5\right )}-\frac {4\,x^2-{\ln \left (5\right )}^2-2\,x^2\,{\ln \left (5\right )}^2+8\,x^3+2}{x^2\,\left ({\mathrm {e}}^{2\,x^2}+\frac {{\left (4\,x-{\ln \left (5\right )}^2+2\right )}^2}{x^2}+\frac {2\,{\mathrm {e}}^{x^2}\,\left (4\,x-{\ln \left (5\right )}^2+2\right )}{x}\right )\,\left (2\,x^2-2\,x^4\,{\ln \left (5\right )}^2-x^2\,{\ln \left (5\right )}^2+4\,x^4+8\,x^5\right )} \] Input:
int((560*x - log(5)^2*(464*x + 384*x^2 - 12*x^3 - 48*x^4 - 48*x^5 + 130) + log(5)^6*(x^3 - 8) + exp(2*x^2)*(60*x^2 + 96*x^3 + 8*x^4 - 6*x^5 - 12*x^6 - log(5)^2*(24*x^2 - 3*x^5)) + exp(x^2)*(140*x - log(5)^2*(116*x + 192*x^ 2 + 8*x^3 - 12*x^4 - 24*x^5) + log(5)^4*(24*x - 3*x^4) + 480*x^2 + 404*x^3 + 20*x^4 - 48*x^5 - 48*x^6) + exp(3*x^2)*(8*x^3 - x^6) + log(5)^4*(96*x - 6*x^3 - 12*x^4 + 56) + 960*x^2 + 504*x^3 - 48*x^4 - 96*x^5 - 64*x^6 + 100 )/(exp(x^2)*(3*x^4*log(5)^4 + 12*x^4 + 48*x^5 + 48*x^6 - log(5)^2*(12*x^4 + 24*x^5)) - log(5)^2*(12*x^3 + 48*x^4 + 48*x^5) - x^3*log(5)^6 + exp(2*x^ 2)*(6*x^5 - 3*x^5*log(5)^2 + 12*x^6) + x^6*exp(3*x^2) + 8*x^3 + 48*x^4 + 9 6*x^5 + 64*x^6 + log(5)^4*(6*x^3 + 12*x^4)),x)
Output:
- x - 4/x^2 - (4*(4*x^2 - log(5)^2 - 2*x^2*log(5)^2 + 8*x^3 + 2))/(x*(exp( x^2) + (4*x - log(5)^2 + 2)/x)*(2*x^2 - 2*x^4*log(5)^2 - x^2*log(5)^2 + 4* x^4 + 8*x^5)) - (4*x^2 - log(5)^2 - 2*x^2*log(5)^2 + 8*x^3 + 2)/(x^2*(exp( 2*x^2) + (4*x - log(5)^2 + 2)^2/x^2 + (2*exp(x^2)*(4*x - log(5)^2 + 2))/x) *(2*x^2 - 2*x^4*log(5)^2 - x^2*log(5)^2 + 4*x^4 + 8*x^5))
\[ \int \frac {100+560 x+960 x^2+504 x^3-48 x^4-96 x^5-64 x^6+e^{3 x^2} \left (8 x^3-x^6\right )+\left (-130-464 x-384 x^2+12 x^3+48 x^4+48 x^5\right ) \log ^2(5)+\left (56+96 x-6 x^3-12 x^4\right ) \log ^4(5)+\left (-8+x^3\right ) \log ^6(5)+e^{2 x^2} \left (60 x^2+96 x^3+8 x^4-6 x^5-12 x^6+\left (-24 x^2+3 x^5\right ) \log ^2(5)\right )+e^{x^2} \left (140 x+480 x^2+404 x^3+20 x^4-48 x^5-48 x^6+\left (-116 x-192 x^2-8 x^3+12 x^4+24 x^5\right ) \log ^2(5)+\left (24 x-3 x^4\right ) \log ^4(5)\right )}{8 x^3+48 x^4+96 x^5+64 x^6+e^{3 x^2} x^6+\left (-12 x^3-48 x^4-48 x^5\right ) \log ^2(5)+\left (6 x^3+12 x^4\right ) \log ^4(5)-x^3 \log ^6(5)+e^{2 x^2} \left (6 x^5+12 x^6-3 x^5 \log ^2(5)\right )+e^{x^2} \left (12 x^4+48 x^5+48 x^6+\left (-12 x^4-24 x^5\right ) \log ^2(5)+3 x^4 \log ^4(5)\right )} \, dx=\int \frac {\left (-x^{6}+8 x^{3}\right ) \left ({\mathrm e}^{x^{2}}\right )^{3}+\left (\left (3 x^{5}-24 x^{2}\right ) \mathrm {log}\left (5\right )^{2}-12 x^{6}-6 x^{5}+8 x^{4}+96 x^{3}+60 x^{2}\right ) \left ({\mathrm e}^{x^{2}}\right )^{2}+\left (\left (-3 x^{4}+24 x \right ) \mathrm {log}\left (5\right )^{4}+\left (24 x^{5}+12 x^{4}-8 x^{3}-192 x^{2}-116 x \right ) \mathrm {log}\left (5\right )^{2}-48 x^{6}-48 x^{5}+20 x^{4}+404 x^{3}+480 x^{2}+140 x \right ) {\mathrm e}^{x^{2}}+\left (x^{3}-8\right ) \mathrm {log}\left (5\right )^{6}+\left (-12 x^{4}-6 x^{3}+96 x +56\right ) \mathrm {log}\left (5\right )^{4}+\left (48 x^{5}+48 x^{4}+12 x^{3}-384 x^{2}-464 x -130\right ) \mathrm {log}\left (5\right )^{2}-64 x^{6}-96 x^{5}-48 x^{4}+504 x^{3}+960 x^{2}+560 x +100}{x^{6} \left ({\mathrm e}^{x^{2}}\right )^{3}+\left (-3 x^{5} \mathrm {log}\left (5\right )^{2}+12 x^{6}+6 x^{5}\right ) \left ({\mathrm e}^{x^{2}}\right )^{2}+\left (3 x^{4} \mathrm {log}\left (5\right )^{4}+\left (-24 x^{5}-12 x^{4}\right ) \mathrm {log}\left (5\right )^{2}+48 x^{6}+48 x^{5}+12 x^{4}\right ) {\mathrm e}^{x^{2}}-x^{3} \mathrm {log}\left (5\right )^{6}+\left (12 x^{4}+6 x^{3}\right ) \mathrm {log}\left (5\right )^{4}+\left (-48 x^{5}-48 x^{4}-12 x^{3}\right ) \mathrm {log}\left (5\right )^{2}+64 x^{6}+96 x^{5}+48 x^{4}+8 x^{3}}d x \] Input:
int(((-x^6+8*x^3)*exp(x^2)^3+((3*x^5-24*x^2)*log(5)^2-12*x^6-6*x^5+8*x^4+9 6*x^3+60*x^2)*exp(x^2)^2+((-3*x^4+24*x)*log(5)^4+(24*x^5+12*x^4-8*x^3-192* x^2-116*x)*log(5)^2-48*x^6-48*x^5+20*x^4+404*x^3+480*x^2+140*x)*exp(x^2)+( x^3-8)*log(5)^6+(-12*x^4-6*x^3+96*x+56)*log(5)^4+(48*x^5+48*x^4+12*x^3-384 *x^2-464*x-130)*log(5)^2-64*x^6-96*x^5-48*x^4+504*x^3+960*x^2+560*x+100)/( x^6*exp(x^2)^3+(-3*x^5*log(5)^2+12*x^6+6*x^5)*exp(x^2)^2+(3*x^4*log(5)^4+( -24*x^5-12*x^4)*log(5)^2+48*x^6+48*x^5+12*x^4)*exp(x^2)-x^3*log(5)^6+(12*x ^4+6*x^3)*log(5)^4+(-48*x^5-48*x^4-12*x^3)*log(5)^2+64*x^6+96*x^5+48*x^4+8 *x^3),x)
Output:
int(((-x^6+8*x^3)*exp(x^2)^3+((3*x^5-24*x^2)*log(5)^2-12*x^6-6*x^5+8*x^4+9 6*x^3+60*x^2)*exp(x^2)^2+((-3*x^4+24*x)*log(5)^4+(24*x^5+12*x^4-8*x^3-192* x^2-116*x)*log(5)^2-48*x^6-48*x^5+20*x^4+404*x^3+480*x^2+140*x)*exp(x^2)+( x^3-8)*log(5)^6+(-12*x^4-6*x^3+96*x+56)*log(5)^4+(48*x^5+48*x^4+12*x^3-384 *x^2-464*x-130)*log(5)^2-64*x^6-96*x^5-48*x^4+504*x^3+960*x^2+560*x+100)/( x^6*exp(x^2)^3+(-3*x^5*log(5)^2+12*x^6+6*x^5)*exp(x^2)^2+(3*x^4*log(5)^4+( -24*x^5-12*x^4)*log(5)^2+48*x^6+48*x^5+12*x^4)*exp(x^2)-x^3*log(5)^6+(12*x ^4+6*x^3)*log(5)^4+(-48*x^5-48*x^4-12*x^3)*log(5)^2+64*x^6+96*x^5+48*x^4+8 *x^3),x)