\(\int \frac {1}{625} e^{-x^2} (42050-1749712 x-99100 x^2+375150 x^3+10000 x^4-20000 x^5+(-336400-20000 x+792800 x^2+20000 x^3-80000 x^4) \log (5)+(-5000+456400 x+10000 x^2-120000 x^3) \log ^2(5)+(40000-80000 x^2) \log ^3(5)-20000 x \log ^4(5)) \, dx\) [449]

Optimal result

Integrand size = 98, antiderivative size = 23 \[ \int \frac {1}{625} e^{-x^2} \left (42050-1749712 x-99100 x^2+375150 x^3+10000 x^4-20000 x^5+\left (-336400-20000 x+792800 x^2+20000 x^3-80000 x^4\right ) \log (5)+\left (-5000+456400 x+10000 x^2-120000 x^3\right ) \log ^2(5)+\left (40000-80000 x^2\right ) \log ^3(5)-20000 x \log ^4(5)\right ) \, dx=e^{-x^2} \left (\frac {841}{25}+x-4 (x+\log (5))^2\right )^2 \]

[Out]

(841/25-4*(ln(5)+x)^2+x)^2/exp(x^2)

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.18 (sec) , antiderivative size = 241, normalized size of antiderivative = 10.48, number of steps used = 16, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {6, 12, 2258, 2243, 2240, 2236} \[ \int \frac {1}{625} e^{-x^2} \left (42050-1749712 x-99100 x^2+375150 x^3+10000 x^4-20000 x^5+\left (-336400-20000 x+792800 x^2+20000 x^3-80000 x^4\right ) \log (5)+\left (-5000+456400 x+10000 x^2-120000 x^3\right ) \log ^2(5)+\left (40000-80000 x^2\right ) \log ^3(5)-20000 x \log ^4(5)\right ) \, dx=-\frac {1}{25} \sqrt {\pi } (1-8 \log (5)) \left (991-100 \log ^2(5)\right ) \text {erf}(x)+\frac {1}{25} \sqrt {\pi } \left (841+800 \log ^3(5)-100 \log ^2(5)-6728 \log (5)\right ) \text {erf}(x)+6 \sqrt {\pi } (1-8 \log (5)) \text {erf}(x)+32 e^{-x^2} x^2+32 e^{-x^2}-\frac {1}{25} e^{-x^2} x^2 \left (7503-2400 \log ^2(5)+400 \log (5)\right )+\frac {2}{25} e^{-x^2} x (1-8 \log (5)) \left (991-100 \log ^2(5)\right )-\frac {1}{25} e^{-x^2} \left (7503-2400 \log ^2(5)+400 \log (5)\right )+\frac {8}{625} e^{-x^2} \left (109357+1250 \log ^4(5)-28525 \log ^2(5)+1250 \log (5)\right )-12 e^{-x^2} x (1-8 \log (5))+16 e^{-x^2} x^4-8 e^{-x^2} x^3 (1-8 \log (5)) \]

[In]

Int[(42050 - 1749712*x - 99100*x^2 + 375150*x^3 + 10000*x^4 - 20000*x^5 + (-336400 - 20000*x + 792800*x^2 + 20
000*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]

[Out]

32/E^x^2 + (32*x^2)/E^x^2 + (16*x^4)/E^x^2 - (12*x*(1 - 8*Log[5]))/E^x^2 - (8*x^3*(1 - 8*Log[5]))/E^x^2 + 6*Sq
rt[Pi]*Erf[x]*(1 - 8*Log[5]) - (7503 + 400*Log[5] - 2400*Log[5]^2)/(25*E^x^2) - (x^2*(7503 + 400*Log[5] - 2400
*Log[5]^2))/(25*E^x^2) + (2*x*(1 - 8*Log[5])*(991 - 100*Log[5]^2))/(25*E^x^2) - (Sqrt[Pi]*Erf[x]*(1 - 8*Log[5]
)*(991 - 100*Log[5]^2))/25 + (Sqrt[Pi]*Erf[x]*(841 - 6728*Log[5] - 100*Log[5]^2 + 800*Log[5]^3))/25 + (8*(1093
57 + 1250*Log[5] - 28525*Log[5]^2 + 1250*Log[5]^4))/(625*E^x^2)

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2236

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F],
2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 2240

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^n*(
F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n*Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2243

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m
- n + 1)*(F^(a + b*(c + d*x)^n)/(b*d*n*Log[F])), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*((m + 1)/n)] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rule 2258

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{625} e^{-x^2} \left (42050-99100 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)+x \left (-1749712-20000 \log ^4(5)\right )\right ) \, dx \\ & = \frac {1}{625} \int e^{-x^2} \left (42050-99100 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)+x \left (-1749712-20000 \log ^4(5)\right )\right ) \, dx \\ & = \frac {1}{625} \int \left (-20000 e^{-x^2} x^5-10000 e^{-x^2} x^4 (-1+8 \log (5))+100 e^{-x^2} x^2 (1-8 \log (5)) \left (-991+100 \log ^2(5)\right )-50 e^{-x^2} x^3 \left (-7503-400 \log (5)+2400 \log ^2(5)\right )+50 e^{-x^2} \left (841-6728 \log (5)-100 \log ^2(5)+800 \log ^3(5)\right )-16 e^{-x^2} x \left (109357+1250 \log (5)-28525 \log ^2(5)+1250 \log ^4(5)\right )\right ) \, dx \\ & = -\left (32 \int e^{-x^2} x^5 \, dx\right )+(16 (1-8 \log (5))) \int e^{-x^2} x^4 \, dx+\frac {1}{25} \left (2 \left (7503+400 \log (5)-2400 \log ^2(5)\right )\right ) \int e^{-x^2} x^3 \, dx-\frac {1}{25} \left (4 (1-8 \log (5)) \left (991-100 \log ^2(5)\right )\right ) \int e^{-x^2} x^2 \, dx+\frac {1}{25} \left (2 \left (841-6728 \log (5)-100 \log ^2(5)+800 \log ^3(5)\right )\right ) \int e^{-x^2} \, dx-\frac {1}{625} \left (16 \left (109357+1250 \log (5)-28525 \log ^2(5)+1250 \log ^4(5)\right )\right ) \int e^{-x^2} x \, dx \\ & = 16 e^{-x^2} x^4-8 e^{-x^2} x^3 (1-8 \log (5))-\frac {1}{25} e^{-x^2} x^2 \left (7503+400 \log (5)-2400 \log ^2(5)\right )+\frac {2}{25} e^{-x^2} x (1-8 \log (5)) \left (991-100 \log ^2(5)\right )+\frac {1}{25} \sqrt {\pi } \text {erf}(x) \left (841-6728 \log (5)-100 \log ^2(5)+800 \log ^3(5)\right )+\frac {8}{625} e^{-x^2} \left (109357+1250 \log (5)-28525 \log ^2(5)+1250 \log ^4(5)\right )-64 \int e^{-x^2} x^3 \, dx+(24 (1-8 \log (5))) \int e^{-x^2} x^2 \, dx+\frac {1}{25} \left (2 \left (7503+400 \log (5)-2400 \log ^2(5)\right )\right ) \int e^{-x^2} x \, dx-\frac {1}{25} \left (2 (1-8 \log (5)) \left (991-100 \log ^2(5)\right )\right ) \int e^{-x^2} \, dx \\ & = 32 e^{-x^2} x^2+16 e^{-x^2} x^4-12 e^{-x^2} x (1-8 \log (5))-8 e^{-x^2} x^3 (1-8 \log (5))-\frac {1}{25} e^{-x^2} \left (7503+400 \log (5)-2400 \log ^2(5)\right )-\frac {1}{25} e^{-x^2} x^2 \left (7503+400 \log (5)-2400 \log ^2(5)\right )+\frac {2}{25} e^{-x^2} x (1-8 \log (5)) \left (991-100 \log ^2(5)\right )-\frac {1}{25} \sqrt {\pi } \text {erf}(x) (1-8 \log (5)) \left (991-100 \log ^2(5)\right )+\frac {1}{25} \sqrt {\pi } \text {erf}(x) \left (841-6728 \log (5)-100 \log ^2(5)+800 \log ^3(5)\right )+\frac {8}{625} e^{-x^2} \left (109357+1250 \log (5)-28525 \log ^2(5)+1250 \log ^4(5)\right )-64 \int e^{-x^2} x \, dx+(12 (1-8 \log (5))) \int e^{-x^2} \, dx \\ & = 32 e^{-x^2}+32 e^{-x^2} x^2+16 e^{-x^2} x^4-12 e^{-x^2} x (1-8 \log (5))-8 e^{-x^2} x^3 (1-8 \log (5))+6 \sqrt {\pi } \text {erf}(x) (1-8 \log (5))-\frac {1}{25} e^{-x^2} \left (7503+400 \log (5)-2400 \log ^2(5)\right )-\frac {1}{25} e^{-x^2} x^2 \left (7503+400 \log (5)-2400 \log ^2(5)\right )+\frac {2}{25} e^{-x^2} x (1-8 \log (5)) \left (991-100 \log ^2(5)\right )-\frac {1}{25} \sqrt {\pi } \text {erf}(x) (1-8 \log (5)) \left (991-100 \log ^2(5)\right )+\frac {1}{25} \sqrt {\pi } \text {erf}(x) \left (841-6728 \log (5)-100 \log ^2(5)+800 \log ^3(5)\right )+\frac {8}{625} e^{-x^2} \left (109357+1250 \log (5)-28525 \log ^2(5)+1250 \log ^4(5)\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.52 \[ \int \frac {1}{625} e^{-x^2} \left (42050-1749712 x-99100 x^2+375150 x^3+10000 x^4-20000 x^5+\left (-336400-20000 x+792800 x^2+20000 x^3-80000 x^4\right ) \log (5)+\left (-5000+456400 x+10000 x^2-120000 x^3\right ) \log ^2(5)+\left (40000-80000 x^2\right ) \log ^3(5)-20000 x \log ^4(5)\right ) \, dx=\frac {1}{625} e^{-x^2} \left (-841+100 x^2+100 \log ^2(5)+25 x (-1+8 \log (5))\right )^2 \]

[In]

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)
*Log[5]^3 - 20000*x*Log[5]^4)/(625*E^x^2),x]

[Out]

(-841 + 100*x^2 + 100*Log[5]^2 + 25*x*(-1 + 8*Log[5]))^2/(625*E^x^2)

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39

method	result	size
gosper	\(\frac {\left (100 \ln \left (5\right )^{2}+200 x \ln \left (5\right )+100 x^{2}-25 x -841\right )^{2} {\mathrm e}^{-x^{2}}}{625}\)	\(32\)
norman	\(\left (\left (64 \ln \left (5\right )-8\right ) x^{3}+\left (-\frac {6703}{25}+96 \ln \left (5\right )^{2}-16 \ln \left (5\right )\right ) x^{2}+\left (64 \ln \left (5\right )^{3}-8 \ln \left (5\right )^{2}-\frac {13456 \ln \left (5\right )}{25}+\frac {1682}{25}\right ) x +16 x^{4}+\frac {707281}{625}-\frac {6728 \ln \left (5\right )^{2}}{25}+16 \ln \left (5\right )^{4}\right ) {\mathrm e}^{-x^{2}}\)	\(73\)
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	\(\frac {707281 \,{\mathrm e}^{-x^{2}}}{625}+\frac {1682 x \,{\mathrm e}^{-x^{2}}}{25}-\frac {6703 \,{\mathrm e}^{-x^{2}} x^{2}}{25}-8 \,{\mathrm e}^{-x^{2}} x^{3}+16 \,{\mathrm e}^{-x^{2}} x^{4}+48 \ln \left (5\right ) \sqrt {\pi }\, \operatorname {erf}\left (x \right )-\frac {6728 \,{\mathrm e}^{-x^{2}} \ln \left (5\right )^{2}}{25}+16 \ln \left (5\right )^{4} {\mathrm e}^{-x^{2}}-\frac {15856 \,{\mathrm e}^{-x^{2}} x \ln \left (5\right )}{25}-8 \,{\mathrm e}^{-x^{2}} x \ln \left (5\right )^{2}+64 \ln \left (5\right )^{3} x \,{\mathrm e}^{-x^{2}}-16 \,{\mathrm e}^{-x^{2}} x^{2} \ln \left (5\right )+96 \,{\mathrm e}^{-x^{2}} x^{2} \ln \left (5\right )^{2}-128 \ln \left (5\right ) \left (-\frac {{\mathrm e}^{-x^{2}} x^{3}}{2}-\frac {3 x \,{\mathrm e}^{-x^{2}}}{4}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (x \right )}{8}\right )\)	\(182\)
meijerg	\(\frac {841 \sqrt {\pi }\, \operatorname {erf}\left (x \right )}{25}+\frac {\left (-128 \ln \left (5\right )+16\right ) \left (-\frac {x \left (10 x^{2}+15\right ) {\mathrm e}^{-x^{2}}}{10}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (x \right )}{4}\right )}{2}+\frac {\left (-128 \ln \left (5\right )^{3}+16 \ln \left (5\right )^{2}+\frac {31712 \ln \left (5\right )}{25}-\frac {3964}{25}\right ) \left (-x \,{\mathrm e}^{-x^{2}}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (x \right )}{2}\right )}{2}+\frac {\left (-192 \ln \left (5\right )^{2}+32 \ln \left (5\right )+\frac {15006}{25}\right ) \left (1-\frac {\left (2 x^{2}+2\right ) {\mathrm e}^{-x^{2}}}{2}\right )}{2}+\frac {\left (-32 \ln \left (5\right )^{4}+\frac {18256 \ln \left (5\right )^{2}}{25}-32 \ln \left (5\right )-\frac {1749712}{625}\right ) \left (1-{\mathrm e}^{-x^{2}}\right )}{2}-32+\frac {16 \left (3 x^{4}+6 x^{2}+6\right ) {\mathrm e}^{-x^{2}}}{3}+32 \ln \left (5\right )^{3} \sqrt {\pi }\, \operatorname {erf}\left (x \right )-4 \ln \left (5\right )^{2} \sqrt {\pi }\, \operatorname {erf}\left (x \right )-\frac {6728 \ln \left (5\right ) \sqrt {\pi }\, \operatorname {erf}\left (x \right )}{25}\)	\(191\)

[In]

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)

[Out]

1/625*(100*ln(5)^2+200*x*ln(5)+100*x^2-25*x-841)^2/exp(x^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (23) = 46\).

Time = 0.24 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.26 \[ \int \frac {1}{625} e^{-x^2} \left (42050-1749712 x-99100 x^2+375150 x^3+10000 x^4-20000 x^5+\left (-336400-20000 x+792800 x^2+20000 x^3-80000 x^4\right ) \log (5)+\left (-5000+456400 x+10000 x^2-120000 x^3\right ) \log ^2(5)+\left (40000-80000 x^2\right ) \log ^3(5)-20000 x \log ^4(5)\right ) \, dx=\frac {1}{625} \, {\left (10000 \, x^{4} + 40000 \, x \log \left (5\right )^{3} + 10000 \, \log \left (5\right )^{4} - 5000 \, x^{3} + 200 \, {\left (300 \, x^{2} - 25 \, x - 841\right )} \log \left (5\right )^{2} - 167575 \, x^{2} + 400 \, {\left (100 \, x^{3} - 25 \, x^{2} - 841 \, x\right )} \log \left (5\right ) + 42050 \, x + 707281\right )} e^{\left (-x^{2}\right )} \]

[In]

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+420
50)/exp(x^2),x, algorithm="fricas")

[Out]

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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (19) = 38\).

Time = 0.11 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.91 \[ \int \frac {1}{625} e^{-x^2} \left (42050-1749712 x-99100 x^2+375150 x^3+10000 x^4-20000 x^5+\left (-336400-20000 x+792800 x^2+20000 x^3-80000 x^4\right ) \log (5)+\left (-5000+456400 x+10000 x^2-120000 x^3\right ) \log ^2(5)+\left (40000-80000 x^2\right ) \log ^3(5)-20000 x \log ^4(5)\right ) \, dx=\frac {\left (10000 x^{4} - 5000 x^{3} + 40000 x^{3} \log {\left (5 \right )} - 167575 x^{2} - 10000 x^{2} \log {\left (5 \right )} + 60000 x^{2} \log {\left (5 \right )}^{2} - 336400 x \log {\left (5 \right )} - 5000 x \log {\left (5 \right )}^{2} + 42050 x + 40000 x \log {\left (5 \right )}^{3} - 168200 \log {\left (5 \right )}^{2} + 10000 \log {\left (5 \right )}^{4} + 707281\right ) e^{- x^{2}}}{625} \]

[In]

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**2-1749
712*x+42050)/exp(x**2),x)

[Out]

(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

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.18 (sec) , antiderivative size = 259, normalized size of antiderivative = 11.26 \[ \int \frac {1}{625} e^{-x^2} \left (42050-1749712 x-99100 x^2+375150 x^3+10000 x^4-20000 x^5+\left (-336400-20000 x+792800 x^2+20000 x^3-80000 x^4\right ) \log (5)+\left (-5000+456400 x+10000 x^2-120000 x^3\right ) \log ^2(5)+\left (40000-80000 x^2\right ) \log ^3(5)-20000 x \log ^4(5)\right ) \, dx=16 \, e^{\left (-x^{2}\right )} \log \left (5\right )^{4} + 32 \, \sqrt {\pi } \operatorname {erf}\left (x\right ) \log \left (5\right )^{3} + 96 \, {\left (x^{2} + 1\right )} e^{\left (-x^{2}\right )} \log \left (5\right )^{2} + 32 \, {\left (2 \, x e^{\left (-x^{2}\right )} - \sqrt {\pi } \operatorname {erf}\left (x\right )\right )} \log \left (5\right )^{3} - 4 \, \sqrt {\pi } \operatorname {erf}\left (x\right ) \log \left (5\right )^{2} - 16 \, {\left (x^{2} + 1\right )} e^{\left (-x^{2}\right )} \log \left (5\right ) - 4 \, {\left (2 \, x e^{\left (-x^{2}\right )} - \sqrt {\pi } \operatorname {erf}\left (x\right )\right )} \log \left (5\right )^{2} - \frac {9128}{25} \, e^{\left (-x^{2}\right )} \log \left (5\right )^{2} - \frac {6728}{25} \, \sqrt {\pi } \operatorname {erf}\left (x\right ) \log \left (5\right ) + 16 \, {\left (x^{4} + 2 \, x^{2} + 2\right )} e^{\left (-x^{2}\right )} - 4 \, {\left (2 \, x^{3} + 3 \, x\right )} e^{\left (-x^{2}\right )} - \frac {7503}{25} \, {\left (x^{2} + 1\right )} e^{\left (-x^{2}\right )} + \frac {1982}{25} \, x e^{\left (-x^{2}\right )} + 16 \, {\left (2 \, {\left (2 \, x^{3} + 3 \, x\right )} e^{\left (-x^{2}\right )} - 3 \, \sqrt {\pi } \operatorname {erf}\left (x\right )\right )} \log \left (5\right ) - \frac {7928}{25} \, {\left (2 \, x e^{\left (-x^{2}\right )} - \sqrt {\pi } \operatorname {erf}\left (x\right )\right )} \log \left (5\right ) + 16 \, e^{\left (-x^{2}\right )} \log \left (5\right ) + \frac {874856}{625} \, e^{\left (-x^{2}\right )} \]

[In]

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+420
50)/exp(x^2),x, algorithm="maxima")

[Out]

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)*
erf(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*sqrt(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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (23) = 46\).

Time = 0.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.57 \[ \int \frac {1}{625} e^{-x^2} \left (42050-1749712 x-99100 x^2+375150 x^3+10000 x^4-20000 x^5+\left (-336400-20000 x+792800 x^2+20000 x^3-80000 x^4\right ) \log (5)+\left (-5000+456400 x+10000 x^2-120000 x^3\right ) \log ^2(5)+\left (40000-80000 x^2\right ) \log ^3(5)-20000 x \log ^4(5)\right ) \, dx=\frac {1}{625} \, {\left (10000 \, x^{4} + 40000 \, x^{3} \log \left (5\right ) + 60000 \, x^{2} \log \left (5\right )^{2} + 40000 \, x \log \left (5\right )^{3} + 10000 \, \log \left (5\right )^{4} - 5000 \, x^{3} - 10000 \, x^{2} \log \left (5\right ) - 5000 \, x \log \left (5\right )^{2} - 167575 \, x^{2} - 336400 \, x \log \left (5\right ) - 168200 \, \log \left (5\right )^{2} + 42050 \, x + 707281\right )} e^{\left (-x^{2}\right )} \]

[In]

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+420
50)/exp(x^2),x, algorithm="giac")

[Out]

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 - 1000
0*x^2*log(5) - 5000*x*log(5)^2 - 167575*x^2 - 336400*x*log(5) - 168200*log(5)^2 + 42050*x + 707281)*e^(-x^2)

Mupad [B] (verification not implemented)

Time = 8.11 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {1}{625} e^{-x^2} \left (42050-1749712 x-99100 x^2+375150 x^3+10000 x^4-20000 x^5+\left (-336400-20000 x+792800 x^2+20000 x^3-80000 x^4\right ) \log (5)+\left (-5000+456400 x+10000 x^2-120000 x^3\right ) \log ^2(5)+\left (40000-80000 x^2\right ) \log ^3(5)-20000 x \log ^4(5)\right ) \, dx=\frac {{\mathrm {e}}^{-x^2}\,{\left (200\,x\,\ln \left (5\right )-25\,x+100\,{\ln \left (5\right )}^2+100\,x^2-841\right )}^2}{625} \]

[In]

int(-exp(-x^2)*((1749712*x)/625 + 32*x*log(5)^4 + (log(5)*(20000*x - 792800*x^2 - 20000*x^3 + 80000*x^4 + 3364
00))/625 + (log(5)^3*(80000*x^2 - 40000))/625 - (log(5)^2*(456400*x + 10000*x^2 - 120000*x^3 - 5000))/625 + (3
964*x^2)/25 - (15006*x^3)/25 - 16*x^4 + 32*x^5 - 1682/25),x)

[Out]

(exp(-x^2)*(200*x*log(5) - 25*x + 100*log(5)^2 + 100*x^2 - 841)^2)/625