Integrand size = 98, antiderivative size = 23 \[ \int \frac {1}{625} e^{-x^2} \left (42050-1749712 x-99100 x^2+375150 x^3+10000 x^4-20000 x^5+\left (-336400-20000 x+792800 x^2+20000 x^3-80000 x^4\right ) \log (5)+\left (-5000+456400 x+10000 x^2-120000 x^3\right ) \log ^2(5)+\left (40000-80000 x^2\right ) \log ^3(5)-20000 x \log ^4(5)\right ) \, dx=e^{-x^2} \left (\frac {841}{25}+x-4 (x+\log (5))^2\right )^2 \] Output:
(841/25-4*(ln(5)+x)^2+x)^2/exp(x^2)
Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.52 \[ \int \frac {1}{625} e^{-x^2} \left (42050-1749712 x-99100 x^2+375150 x^3+10000 x^4-20000 x^5+\left (-336400-20000 x+792800 x^2+20000 x^3-80000 x^4\right ) \log (5)+\left (-5000+456400 x+10000 x^2-120000 x^3\right ) \log ^2(5)+\left (40000-80000 x^2\right ) \log ^3(5)-20000 x \log ^4(5)\right ) \, dx=\frac {1}{625} e^{-x^2} \left (-841+100 x^2+100 \log ^2(5)+25 x (-1+8 \log (5))\right )^2 \] Input:
Integrate[(42050 - 1749712*x - 99100*x^2 + 375150*x^3 + 10000*x^4 - 20000* x^5 + (-336400 - 20000*x + 792800*x^2 + 20000*x^3 - 80000*x^4)*Log[5] + (- 5000 + 456400*x + 10000*x^2 - 120000*x^3)*Log[5]^2 + (40000 - 80000*x^2)*L og[5]^3 - 20000*x*Log[5]^4)/(625*E^x^2),x]
Output:
(-841 + 100*x^2 + 100*Log[5]^2 + 25*x*(-1 + 8*Log[5]))^2/(625*E^x^2)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.59 (sec) , antiderivative size = 241, normalized size of antiderivative = 10.48, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {6, 27, 27, 2656, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{625} e^{-x^2} \left (-20000 x^5+10000 x^4+375150 x^3-99100 x^2+\left (40000-80000 x^2\right ) \log ^3(5)+\left (-120000 x^3+10000 x^2+456400 x-5000\right ) \log ^2(5)+\left (-80000 x^4+20000 x^3+792800 x^2-20000 x-336400\right ) \log (5)-1749712 x-20000 x \log ^4(5)+42050\right ) \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {1}{625} e^{-x^2} \left (-20000 x^5+10000 x^4+375150 x^3-99100 x^2+\left (40000-80000 x^2\right ) \log ^3(5)+\left (-120000 x^3+10000 x^2+456400 x-5000\right ) \log ^2(5)+\left (-80000 x^4+20000 x^3+792800 x^2-20000 x-336400\right ) \log (5)+x \left (-1749712-20000 \log ^4(5)\right )+42050\right )dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{625} \int 2 e^{-x^2} \left (-10000 x^5+5000 x^4+187575 x^3-49550 x^2-8 \left (109357+1250 \log ^4(5)\right ) x+20000 \left (1-2 x^2\right ) \log ^3(5)-100 \left (600 x^3-50 x^2-2282 x+25\right ) \log ^2(5)-200 \left (200 x^4-50 x^3-1982 x^2+50 x+841\right ) \log (5)+21025\right )dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{625} \int e^{-x^2} \left (-10000 x^5+5000 x^4+187575 x^3-49550 x^2-8 \left (109357+1250 \log ^4(5)\right ) x+20000 \left (1-2 x^2\right ) \log ^3(5)-100 \left (600 x^3-50 x^2-2282 x+25\right ) \log ^2(5)-200 \left (200 x^4-50 x^3-1982 x^2+50 x+841\right ) \log (5)+21025\right )dx\) |
| \(\Big \downarrow \) 2656 |
| \(\displaystyle \frac {2}{625} \int \left (-10000 e^{-x^2} x^5-5000 e^{-x^2} (-1+8 \log (5)) x^4-25 e^{-x^2} \left (-7503-400 \log (5)+2400 \log ^2(5)\right ) x^3+50 e^{-x^2} (1-8 \log (5)) \left (-991+100 \log ^2(5)\right ) x^2-8 e^{-x^2} \left (109357+1250 \log (5)-28525 \log ^2(5)+1250 \log ^4(5)\right ) x+25 e^{-x^2} \left (841-6728 \log (5)-100 \log ^2(5)+800 \log ^3(5)\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{625} \left (-\frac {25}{2} \sqrt {\pi } (1-8 \log (5)) \left (991-100 \log ^2(5)\right ) \text {erf}(x)+\frac {25}{2} \sqrt {\pi } \left (841+800 \log ^3(5)-100 \log ^2(5)-6728 \log (5)\right ) \text {erf}(x)+1875 \sqrt {\pi } (1-8 \log (5)) \text {erf}(x)+10000 e^{-x^2} x^2+10000 e^{-x^2}-\frac {25}{2} e^{-x^2} x^2 \left (7503-2400 \log ^2(5)+400 \log (5)\right )+25 e^{-x^2} x (1-8 \log (5)) \left (991-100 \log ^2(5)\right )-\frac {25}{2} e^{-x^2} \left (7503-2400 \log ^2(5)+400 \log (5)\right )+4 e^{-x^2} \left (109357+1250 \log ^4(5)-28525 \log ^2(5)+1250 \log (5)\right )-3750 e^{-x^2} x (1-8 \log (5))+5000 e^{-x^2} x^4-2500 e^{-x^2} x^3 (1-8 \log (5))\right )\) |
Input:
Int[(42050 - 1749712*x - 99100*x^2 + 375150*x^3 + 10000*x^4 - 20000*x^5 + (-336400 - 20000*x + 792800*x^2 + 20000*x^3 - 80000*x^4)*Log[5] + (-5000 + 456400*x + 10000*x^2 - 120000*x^3)*Log[5]^2 + (40000 - 80000*x^2)*Log[5]^ 3 - 20000*x*Log[5]^4)/(625*E^x^2),x]
Output:
(2*(10000/E^x^2 + (10000*x^2)/E^x^2 + (5000*x^4)/E^x^2 - (3750*x*(1 - 8*Lo g[5]))/E^x^2 - (2500*x^3*(1 - 8*Log[5]))/E^x^2 + 1875*Sqrt[Pi]*Erf[x]*(1 - 8*Log[5]) - (25*(7503 + 400*Log[5] - 2400*Log[5]^2))/(2*E^x^2) - (25*x^2* (7503 + 400*Log[5] - 2400*Log[5]^2))/(2*E^x^2) + (25*x*(1 - 8*Log[5])*(991 - 100*Log[5]^2))/E^x^2 - (25*Sqrt[Pi]*Erf[x]*(1 - 8*Log[5])*(991 - 100*Lo g[5]^2))/2 + (25*Sqrt[Pi]*Erf[x]*(841 - 6728*Log[5] - 100*Log[5]^2 + 800*L og[5]^3))/2 + (4*(109357 + 1250*Log[5] - 28525*Log[5]^2 + 1250*Log[5]^4))/ E^x^2))/625
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(Px_), x_Symbol] :> Int[ ExpandLinearProduct[F^(a + b*(c + d*x)^n), Px, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[Px, x]
Leaf count of result is larger than twice the leaf count of optimal. \(72\) vs. \(2(20)=40\).
Time = 0.23 (sec) , antiderivative size = 73, normalized size of antiderivative = 3.17
| method | result | size |
| norman | \(\left (\left (64 \ln \left (5\right )-8\right ) x^{3}+\left (-\frac {6703}{25}+96 \ln \left (5\right )^{2}-16 \ln \left (5\right )\right ) x^{2}+\left (64 \ln \left (5\right )^{3}-8 \ln \left (5\right )^{2}-\frac {13456 \ln \left (5\right )}{25}+\frac {1682}{25}\right ) x +16 x^{4}+\frac {707281}{625}-\frac {6728 \ln \left (5\right )^{2}}{25}+16 \ln \left (5\right )^{4}\right ) {\mathrm e}^{-x^{2}}\) | \(73\) |
| gosper | \(\frac {\left (10000 \ln \left (5\right )^{4}+40000 \ln \left (5\right )^{3} x +60000 x^{2} \ln \left (5\right )^{2}+40000 x^{3} \ln \left (5\right )+10000 x^{4}-5000 x \ln \left (5\right )^{2}-10000 x^{2} \ln \left (5\right )-5000 x^{3}-168200 \ln \left (5\right )^{2}-336400 x \ln \left (5\right )-167575 x^{2}+42050 x +707281\right ) {\mathrm e}^{-x^{2}}}{625}\) | \(83\) |
| risch | \(\frac {\left (10000 \ln \left (5\right )^{4}+40000 \ln \left (5\right )^{3} x +60000 x^{2} \ln \left (5\right )^{2}+40000 x^{3} \ln \left (5\right )+10000 x^{4}-5000 x \ln \left (5\right )^{2}-10000 x^{2} \ln \left (5\right )-5000 x^{3}-168200 \ln \left (5\right )^{2}-336400 x \ln \left (5\right )-167575 x^{2}+42050 x +707281\right ) {\mathrm e}^{-x^{2}}}{625}\) | \(83\) |
| parallelrisch | \(\frac {\left (10000 \ln \left (5\right )^{4}+40000 \ln \left (5\right )^{3} x +60000 x^{2} \ln \left (5\right )^{2}+40000 x^{3} \ln \left (5\right )+10000 x^{4}-5000 x \ln \left (5\right )^{2}-10000 x^{2} \ln \left (5\right )-5000 x^{3}-168200 \ln \left (5\right )^{2}-336400 x \ln \left (5\right )-167575 x^{2}+42050 x +707281\right ) {\mathrm e}^{-x^{2}}}{625}\) | \(83\) |
| default | \(\frac {707281 \,{\mathrm e}^{-x^{2}}}{625}+\frac {1682 x \,{\mathrm e}^{-x^{2}}}{25}-\frac {6703 \,{\mathrm e}^{-x^{2}} x^{2}}{25}-8 \,{\mathrm e}^{-x^{2}} x^{3}+16 \,{\mathrm e}^{-x^{2}} x^{4}+48 \ln \left (5\right ) \sqrt {\pi }\, \operatorname {erf}\left (x \right )-\frac {6728 \,{\mathrm e}^{-x^{2}} \ln \left (5\right )^{2}}{25}+16 \,{\mathrm e}^{-x^{2}} \ln \left (5\right )^{4}-\frac {15856 \,{\mathrm e}^{-x^{2}} \ln \left (5\right ) x}{25}-8 \,{\mathrm e}^{-x^{2}} \ln \left (5\right )^{2} x -16 \,{\mathrm e}^{-x^{2}} \ln \left (5\right ) x^{2}-128 \ln \left (5\right ) \left (-\frac {{\mathrm e}^{-x^{2}} x^{3}}{2}-\frac {3 x \,{\mathrm e}^{-x^{2}}}{4}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (x \right )}{8}\right )+96 \,{\mathrm e}^{-x^{2}} \ln \left (5\right )^{2} x^{2}+64 \,{\mathrm e}^{-x^{2}} \ln \left (5\right )^{3} x\) | \(182\) |
| meijerg | \(\frac {841 \sqrt {\pi }\, \operatorname {erf}\left (x \right )}{25}-32+\frac {16 \left (3 x^{4}+6 x^{2}+6\right ) {\mathrm e}^{-x^{2}}}{3}+\frac {\left (-128 \ln \left (5\right )+16\right ) \left (-\frac {x \left (10 x^{2}+15\right ) {\mathrm e}^{-x^{2}}}{10}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (x \right )}{4}\right )}{2}+\frac {\left (-192 \ln \left (5\right )^{2}+32 \ln \left (5\right )+\frac {15006}{25}\right ) \left (1-\frac {\left (2 x^{2}+2\right ) {\mathrm e}^{-x^{2}}}{2}\right )}{2}+\frac {\left (-32 \ln \left (5\right )^{4}+\frac {18256 \ln \left (5\right )^{2}}{25}-32 \ln \left (5\right )-\frac {1749712}{625}\right ) \left (1-{\mathrm e}^{-x^{2}}\right )}{2}+\frac {\left (-128 \ln \left (5\right )^{3}+16 \ln \left (5\right )^{2}+\frac {31712 \ln \left (5\right )}{25}-\frac {3964}{25}\right ) \left (-x \,{\mathrm e}^{-x^{2}}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (x \right )}{2}\right )}{2}+32 \ln \left (5\right )^{3} \sqrt {\pi }\, \operatorname {erf}\left (x \right )-4 \ln \left (5\right )^{2} \sqrt {\pi }\, \operatorname {erf}\left (x \right )-\frac {6728 \ln \left (5\right ) \sqrt {\pi }\, \operatorname {erf}\left (x \right )}{25}\) | \(191\) |
Input:
int(1/625*(-20000*x*ln(5)^4+(-80000*x^2+40000)*ln(5)^3+(-120000*x^3+10000* x^2+456400*x-5000)*ln(5)^2+(-80000*x^4+20000*x^3+792800*x^2-20000*x-336400 )*ln(5)-20000*x^5+10000*x^4+375150*x^3-99100*x^2-1749712*x+42050)/exp(x^2) ,x,method=_RETURNVERBOSE)
Output:
((64*ln(5)-8)*x^3+(-6703/25+96*ln(5)^2-16*ln(5))*x^2+(64*ln(5)^3-8*ln(5)^2 -13456/25*ln(5)+1682/25)*x+16*x^4+707281/625-6728/25*ln(5)^2+16*ln(5)^4)/e xp(x^2)
Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (23) = 46\).
Time = 0.07 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.26 \[ \int \frac {1}{625} e^{-x^2} \left (42050-1749712 x-99100 x^2+375150 x^3+10000 x^4-20000 x^5+\left (-336400-20000 x+792800 x^2+20000 x^3-80000 x^4\right ) \log (5)+\left (-5000+456400 x+10000 x^2-120000 x^3\right ) \log ^2(5)+\left (40000-80000 x^2\right ) \log ^3(5)-20000 x \log ^4(5)\right ) \, dx=\frac {1}{625} \, {\left (10000 \, x^{4} + 40000 \, x \log \left (5\right )^{3} + 10000 \, \log \left (5\right )^{4} - 5000 \, x^{3} + 200 \, {\left (300 \, x^{2} - 25 \, x - 841\right )} \log \left (5\right )^{2} - 167575 \, x^{2} + 400 \, {\left (100 \, x^{3} - 25 \, x^{2} - 841 \, x\right )} \log \left (5\right ) + 42050 \, x + 707281\right )} e^{\left (-x^{2}\right )} \] Input:
integrate(1/625*(-20000*x*log(5)^4+(-80000*x^2+40000)*log(5)^3+(-120000*x^ 3+10000*x^2+456400*x-5000)*log(5)^2+(-80000*x^4+20000*x^3+792800*x^2-20000 *x-336400)*log(5)-20000*x^5+10000*x^4+375150*x^3-99100*x^2-1749712*x+42050 )/exp(x^2),x, algorithm="fricas")
Output:
1/625*(10000*x^4 + 40000*x*log(5)^3 + 10000*log(5)^4 - 5000*x^3 + 200*(300 *x^2 - 25*x - 841)*log(5)^2 - 167575*x^2 + 400*(100*x^3 - 25*x^2 - 841*x)* log(5) + 42050*x + 707281)*e^(-x^2)
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (19) = 38\).
Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.91 \[ \int \frac {1}{625} e^{-x^2} \left (42050-1749712 x-99100 x^2+375150 x^3+10000 x^4-20000 x^5+\left (-336400-20000 x+792800 x^2+20000 x^3-80000 x^4\right ) \log (5)+\left (-5000+456400 x+10000 x^2-120000 x^3\right ) \log ^2(5)+\left (40000-80000 x^2\right ) \log ^3(5)-20000 x \log ^4(5)\right ) \, dx=\frac {\left (10000 x^{4} - 5000 x^{3} + 40000 x^{3} \log {\left (5 \right )} - 167575 x^{2} - 10000 x^{2} \log {\left (5 \right )} + 60000 x^{2} \log {\left (5 \right )}^{2} - 336400 x \log {\left (5 \right )} - 5000 x \log {\left (5 \right )}^{2} + 42050 x + 40000 x \log {\left (5 \right )}^{3} - 168200 \log {\left (5 \right )}^{2} + 10000 \log {\left (5 \right )}^{4} + 707281\right ) e^{- x^{2}}}{625} \] Input:
integrate(1/625*(-20000*x*ln(5)**4+(-80000*x**2+40000)*ln(5)**3+(-120000*x **3+10000*x**2+456400*x-5000)*ln(5)**2+(-80000*x**4+20000*x**3+792800*x**2 -20000*x-336400)*ln(5)-20000*x**5+10000*x**4+375150*x**3-99100*x**2-174971 2*x+42050)/exp(x**2),x)
Output:
(10000*x**4 - 5000*x**3 + 40000*x**3*log(5) - 167575*x**2 - 10000*x**2*log (5) + 60000*x**2*log(5)**2 - 336400*x*log(5) - 5000*x*log(5)**2 + 42050*x + 40000*x*log(5)**3 - 168200*log(5)**2 + 10000*log(5)**4 + 707281)*exp(-x* *2)/625
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.04 (sec) , antiderivative size = 259, normalized size of antiderivative = 11.26 \[ \int \frac {1}{625} e^{-x^2} \left (42050-1749712 x-99100 x^2+375150 x^3+10000 x^4-20000 x^5+\left (-336400-20000 x+792800 x^2+20000 x^3-80000 x^4\right ) \log (5)+\left (-5000+456400 x+10000 x^2-120000 x^3\right ) \log ^2(5)+\left (40000-80000 x^2\right ) \log ^3(5)-20000 x \log ^4(5)\right ) \, dx=16 \, e^{\left (-x^{2}\right )} \log \left (5\right )^{4} + 32 \, \sqrt {\pi } \operatorname {erf}\left (x\right ) \log \left (5\right )^{3} + 96 \, {\left (x^{2} + 1\right )} e^{\left (-x^{2}\right )} \log \left (5\right )^{2} + 32 \, {\left (2 \, x e^{\left (-x^{2}\right )} - \sqrt {\pi } \operatorname {erf}\left (x\right )\right )} \log \left (5\right )^{3} - 4 \, \sqrt {\pi } \operatorname {erf}\left (x\right ) \log \left (5\right )^{2} - 16 \, {\left (x^{2} + 1\right )} e^{\left (-x^{2}\right )} \log \left (5\right ) - 4 \, {\left (2 \, x e^{\left (-x^{2}\right )} - \sqrt {\pi } \operatorname {erf}\left (x\right )\right )} \log \left (5\right )^{2} - \frac {9128}{25} \, e^{\left (-x^{2}\right )} \log \left (5\right )^{2} - \frac {6728}{25} \, \sqrt {\pi } \operatorname {erf}\left (x\right ) \log \left (5\right ) + 16 \, {\left (x^{4} + 2 \, x^{2} + 2\right )} e^{\left (-x^{2}\right )} - 4 \, {\left (2 \, x^{3} + 3 \, x\right )} e^{\left (-x^{2}\right )} - \frac {7503}{25} \, {\left (x^{2} + 1\right )} e^{\left (-x^{2}\right )} + \frac {1982}{25} \, x e^{\left (-x^{2}\right )} + 16 \, {\left (2 \, {\left (2 \, x^{3} + 3 \, x\right )} e^{\left (-x^{2}\right )} - 3 \, \sqrt {\pi } \operatorname {erf}\left (x\right )\right )} \log \left (5\right ) - \frac {7928}{25} \, {\left (2 \, x e^{\left (-x^{2}\right )} - \sqrt {\pi } \operatorname {erf}\left (x\right )\right )} \log \left (5\right ) + 16 \, e^{\left (-x^{2}\right )} \log \left (5\right ) + \frac {874856}{625} \, e^{\left (-x^{2}\right )} \] Input:
integrate(1/625*(-20000*x*log(5)^4+(-80000*x^2+40000)*log(5)^3+(-120000*x^ 3+10000*x^2+456400*x-5000)*log(5)^2+(-80000*x^4+20000*x^3+792800*x^2-20000 *x-336400)*log(5)-20000*x^5+10000*x^4+375150*x^3-99100*x^2-1749712*x+42050 )/exp(x^2),x, algorithm="maxima")
Output:
16*e^(-x^2)*log(5)^4 + 32*sqrt(pi)*erf(x)*log(5)^3 + 96*(x^2 + 1)*e^(-x^2) *log(5)^2 + 32*(2*x*e^(-x^2) - sqrt(pi)*erf(x))*log(5)^3 - 4*sqrt(pi)*erf( x)*log(5)^2 - 16*(x^2 + 1)*e^(-x^2)*log(5) - 4*(2*x*e^(-x^2) - sqrt(pi)*er f(x))*log(5)^2 - 9128/25*e^(-x^2)*log(5)^2 - 6728/25*sqrt(pi)*erf(x)*log(5 ) + 16*(x^4 + 2*x^2 + 2)*e^(-x^2) - 4*(2*x^3 + 3*x)*e^(-x^2) - 7503/25*(x^ 2 + 1)*e^(-x^2) + 1982/25*x*e^(-x^2) + 16*(2*(2*x^3 + 3*x)*e^(-x^2) - 3*sq rt(pi)*erf(x))*log(5) - 7928/25*(2*x*e^(-x^2) - sqrt(pi)*erf(x))*log(5) + 16*e^(-x^2)*log(5) + 874856/625*e^(-x^2)
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (23) = 46\).
Time = 0.12 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.57 \[ \int \frac {1}{625} e^{-x^2} \left (42050-1749712 x-99100 x^2+375150 x^3+10000 x^4-20000 x^5+\left (-336400-20000 x+792800 x^2+20000 x^3-80000 x^4\right ) \log (5)+\left (-5000+456400 x+10000 x^2-120000 x^3\right ) \log ^2(5)+\left (40000-80000 x^2\right ) \log ^3(5)-20000 x \log ^4(5)\right ) \, dx=\frac {1}{625} \, {\left (10000 \, x^{4} + 40000 \, x^{3} \log \left (5\right ) + 60000 \, x^{2} \log \left (5\right )^{2} + 40000 \, x \log \left (5\right )^{3} + 10000 \, \log \left (5\right )^{4} - 5000 \, x^{3} - 10000 \, x^{2} \log \left (5\right ) - 5000 \, x \log \left (5\right )^{2} - 167575 \, x^{2} - 336400 \, x \log \left (5\right ) - 168200 \, \log \left (5\right )^{2} + 42050 \, x + 707281\right )} e^{\left (-x^{2}\right )} \] Input:
integrate(1/625*(-20000*x*log(5)^4+(-80000*x^2+40000)*log(5)^3+(-120000*x^ 3+10000*x^2+456400*x-5000)*log(5)^2+(-80000*x^4+20000*x^3+792800*x^2-20000 *x-336400)*log(5)-20000*x^5+10000*x^4+375150*x^3-99100*x^2-1749712*x+42050 )/exp(x^2),x, algorithm="giac")
Output:
1/625*(10000*x^4 + 40000*x^3*log(5) + 60000*x^2*log(5)^2 + 40000*x*log(5)^ 3 + 10000*log(5)^4 - 5000*x^3 - 10000*x^2*log(5) - 5000*x*log(5)^2 - 16757 5*x^2 - 336400*x*log(5) - 168200*log(5)^2 + 42050*x + 707281)*e^(-x^2)
Time = 0.54 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {1}{625} e^{-x^2} \left (42050-1749712 x-99100 x^2+375150 x^3+10000 x^4-20000 x^5+\left (-336400-20000 x+792800 x^2+20000 x^3-80000 x^4\right ) \log (5)+\left (-5000+456400 x+10000 x^2-120000 x^3\right ) \log ^2(5)+\left (40000-80000 x^2\right ) \log ^3(5)-20000 x \log ^4(5)\right ) \, dx=\frac {{\mathrm {e}}^{-x^2}\,{\left (200\,x\,\ln \left (5\right )-25\,x+100\,{\ln \left (5\right )}^2+100\,x^2-841\right )}^2}{625} \] Input:
int(-exp(-x^2)*((1749712*x)/625 + 32*x*log(5)^4 + (log(5)*(20000*x - 79280 0*x^2 - 20000*x^3 + 80000*x^4 + 336400))/625 + (log(5)^3*(80000*x^2 - 4000 0))/625 - (log(5)^2*(456400*x + 10000*x^2 - 120000*x^3 - 5000))/625 + (396 4*x^2)/25 - (15006*x^3)/25 - 16*x^4 + 32*x^5 - 1682/25),x)
Output:
(exp(-x^2)*(200*x*log(5) - 25*x + 100*log(5)^2 + 100*x^2 - 841)^2)/625
Time = 0.21 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.61 \[ \int \frac {1}{625} e^{-x^2} \left (42050-1749712 x-99100 x^2+375150 x^3+10000 x^4-20000 x^5+\left (-336400-20000 x+792800 x^2+20000 x^3-80000 x^4\right ) \log (5)+\left (-5000+456400 x+10000 x^2-120000 x^3\right ) \log ^2(5)+\left (40000-80000 x^2\right ) \log ^3(5)-20000 x \log ^4(5)\right ) \, dx=\frac {10000 \mathrm {log}\left (5\right )^{4}+40000 \mathrm {log}\left (5\right )^{3} x +60000 \mathrm {log}\left (5\right )^{2} x^{2}-5000 \mathrm {log}\left (5\right )^{2} x -168200 \mathrm {log}\left (5\right )^{2}+40000 \,\mathrm {log}\left (5\right ) x^{3}-10000 \,\mathrm {log}\left (5\right ) x^{2}-336400 \,\mathrm {log}\left (5\right ) x +10000 x^{4}-5000 x^{3}-167575 x^{2}+42050 x +707281}{625 e^{x^{2}}} \] Input:
int(1/625*(-20000*x*log(5)^4+(-80000*x^2+40000)*log(5)^3+(-120000*x^3+1000 0*x^2+456400*x-5000)*log(5)^2+(-80000*x^4+20000*x^3+792800*x^2-20000*x-336 400)*log(5)-20000*x^5+10000*x^4+375150*x^3-99100*x^2-1749712*x+42050)/exp( x^2),x)
Output:
(10000*log(5)**4 + 40000*log(5)**3*x + 60000*log(5)**2*x**2 - 5000*log(5)* *2*x - 168200*log(5)**2 + 40000*log(5)*x**3 - 10000*log(5)*x**2 - 336400*l og(5)*x + 10000*x**4 - 5000*x**3 - 167575*x**2 + 42050*x + 707281)/(625*e* *(x**2))