Integrand size = 119, antiderivative size = 23 \[ \int e^{-4 x^2-8 x \log ^2(5)-4 \log ^4(5)} \left (-52488 x+e^{4 x^2+8 x \log ^2(5)+4 \log ^4(5)} \left (2 x-6 x^2+4 x^3\right )-52488 \log ^2(5)+e^{2 x^2+4 x \log ^2(5)+2 \log ^4(5)} \left (162-324 x-648 x^2+648 x^3+\left (-648 x+648 x^2\right ) \log ^2(5)\right )\right ) \, dx=\left (81 e^{-2 \left (x+\log ^2(5)\right )^2}+x-x^2\right )^2 \] Output:
(x+81/exp((ln(5)^2+x)^2)^2-x^2)^2
Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48 \[ \int e^{-4 x^2-8 x \log ^2(5)-4 \log ^4(5)} \left (-52488 x+e^{4 x^2+8 x \log ^2(5)+4 \log ^4(5)} \left (2 x-6 x^2+4 x^3\right )-52488 \log ^2(5)+e^{2 x^2+4 x \log ^2(5)+2 \log ^4(5)} \left (162-324 x-648 x^2+648 x^3+\left (-648 x+648 x^2\right ) \log ^2(5)\right )\right ) \, dx=e^{-4 \left (x+\log ^2(5)\right )^2} \left (-81+e^{2 \left (x+\log ^2(5)\right )^2} (-1+x) x\right )^2 \] Input:
Integrate[E^(-4*x^2 - 8*x*Log[5]^2 - 4*Log[5]^4)*(-52488*x + E^(4*x^2 + 8* x*Log[5]^2 + 4*Log[5]^4)*(2*x - 6*x^2 + 4*x^3) - 52488*Log[5]^2 + E^(2*x^2 + 4*x*Log[5]^2 + 2*Log[5]^4)*(162 - 324*x - 648*x^2 + 648*x^3 + (-648*x + 648*x^2)*Log[5]^2)),x]
Output:
(-81 + E^(2*(x + Log[5]^2)^2)*(-1 + x)*x)^2/E^(4*(x + Log[5]^2)^2)
Leaf count is larger than twice the leaf count of optimal. \(129\) vs. \(2(23)=46\).
Time = 1.71 (sec) , antiderivative size = 129, normalized size of antiderivative = 5.61, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {7239, 27, 7293, 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.
\(\displaystyle \int e^{-4 x^2-8 x \log ^2(5)-4 \log ^4(5)} \left (\left (4 x^3-6 x^2+2 x\right ) e^{4 x^2+8 x \log ^2(5)+4 \log ^4(5)}+e^{2 x^2+4 x \log ^2(5)+2 \log ^4(5)} \left (648 x^3-648 x^2+\left (648 x^2-648 x\right ) \log ^2(5)-324 x+162\right )-52488 x-52488 \log ^2(5)\right ) \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int 2 e^{-4 \left (x+\log ^2(5)\right )^2} \left (81-(x-1) x e^{2 \left (x+\log ^2(5)\right )^2}\right ) \left ((2 x-1) \left (-e^{2 \left (x+\log ^2(5)\right )^2}\right )-324 \left (x+\log ^2(5)\right )\right )dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int e^{-4 \left (x+\log ^2(5)\right )^2} \left (e^{2 \left (x+\log ^2(5)\right )^2} (1-x) x+81\right ) \left (e^{2 \left (x+\log ^2(5)\right )^2} (1-2 x)-324 \left (x+\log ^2(5)\right )\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 2 \int \left ((x-1) x (2 x-1)-26244 e^{-4 \left (x+\log ^2(5)\right )^2} \left (x+\log ^2(5)\right )+81 e^{-2 \left (x+\log ^2(5)\right )^2} \left (4 x^3-4 \left (1-\log ^2(5)\right ) x^2-2 \left (1+2 \log ^2(5)\right ) x+1\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {1}{2} (1-x)^2 x^2+\frac {6561}{2} e^{-4 \left (x+\log ^2(5)\right )^2}-\frac {81}{2} e^{-2 \left (x+\log ^2(5)\right )^2}-81 e^{-2 \left (x+\log ^2(5)\right )^2} \left (x+\log ^2(5)\right )^2+81 \left (1+2 \log ^2(5)\right ) e^{-2 \left (x+\log ^2(5)\right )^2} \left (x+\log ^2(5)\right )+\frac {81}{2} \left (1-2 \log ^4(5)-2 \log ^2(5)\right ) e^{-2 \left (x+\log ^2(5)\right )^2}\right )\) |
Input:
Int[E^(-4*x^2 - 8*x*Log[5]^2 - 4*Log[5]^4)*(-52488*x + E^(4*x^2 + 8*x*Log[ 5]^2 + 4*Log[5]^4)*(2*x - 6*x^2 + 4*x^3) - 52488*Log[5]^2 + E^(2*x^2 + 4*x *Log[5]^2 + 2*Log[5]^4)*(162 - 324*x - 648*x^2 + 648*x^3 + (-648*x + 648*x ^2)*Log[5]^2)),x]
Output:
2*(6561/(2*E^(4*(x + Log[5]^2)^2)) - 81/(2*E^(2*(x + Log[5]^2)^2)) + ((1 - x)^2*x^2)/2 - (81*(x + Log[5]^2)^2)/E^(2*(x + Log[5]^2)^2) + (81*(x + Log [5]^2)*(1 + 2*Log[5]^2))/E^(2*(x + Log[5]^2)^2) + (81*(1 - 2*Log[5]^2 - 2* Log[5]^4))/(2*E^(2*(x + Log[5]^2)^2)))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(46\) vs. \(2(22)=44\).
Time = 1.31 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.04
method | result | size |
risch | \(x^{4}-2 x^{3}+x^{2}+\left (-162 x^{2}+162 x \right ) {\mathrm e}^{-2 \left (\ln \left (5\right )^{2}+x \right )^{2}}+6561 \,{\mathrm e}^{-4 \left (\ln \left (5\right )^{2}+x \right )^{2}}\) | \(47\) |
parts | \(6561 \,{\mathrm e}^{-4 \ln \left (5\right )^{4}} {\mathrm e}^{-8 x \ln \left (5\right )^{2}-4 x^{2}}+162 \,{\mathrm e}^{-2 \ln \left (5\right )^{4}} x \,{\mathrm e}^{-4 x \ln \left (5\right )^{2}-2 x^{2}}-162 \,{\mathrm e}^{-2 \ln \left (5\right )^{4}} x^{2} {\mathrm e}^{-4 x \ln \left (5\right )^{2}-2 x^{2}}+x^{2}-2 x^{3}+x^{4}\) | \(86\) |
norman | \(\left (6561+x^{4} {\mathrm e}^{4 \ln \left (5\right )^{4}+8 x \ln \left (5\right )^{2}+4 x^{2}}+{\mathrm e}^{4 \ln \left (5\right )^{4}+8 x \ln \left (5\right )^{2}+4 x^{2}} x^{2}+162 \,{\mathrm e}^{2 \ln \left (5\right )^{4}+4 x \ln \left (5\right )^{2}+2 x^{2}} x -162 \,{\mathrm e}^{2 \ln \left (5\right )^{4}+4 x \ln \left (5\right )^{2}+2 x^{2}} x^{2}-2 \,{\mathrm e}^{4 \ln \left (5\right )^{4}+8 x \ln \left (5\right )^{2}+4 x^{2}} x^{3}\right ) {\mathrm e}^{-4 \ln \left (5\right )^{4}-8 x \ln \left (5\right )^{2}-4 x^{2}}\) | \(133\) |
parallelrisch | \(-\left (-6561-x^{4} {\mathrm e}^{4 \ln \left (5\right )^{4}+8 x \ln \left (5\right )^{2}+4 x^{2}}+2 \,{\mathrm e}^{4 \ln \left (5\right )^{4}+8 x \ln \left (5\right )^{2}+4 x^{2}} x^{3}-{\mathrm e}^{4 \ln \left (5\right )^{4}+8 x \ln \left (5\right )^{2}+4 x^{2}} x^{2}+162 \,{\mathrm e}^{2 \ln \left (5\right )^{4}+4 x \ln \left (5\right )^{2}+2 x^{2}} x^{2}-162 \,{\mathrm e}^{2 \ln \left (5\right )^{4}+4 x \ln \left (5\right )^{2}+2 x^{2}} x \right ) {\mathrm e}^{-4 \ln \left (5\right )^{4}-8 x \ln \left (5\right )^{2}-4 x^{2}}\) | \(136\) |
default | \(\text {Expression too large to display}\) | \(679\) |
Input:
int(((4*x^3-6*x^2+2*x)*exp(ln(5)^4+2*x*ln(5)^2+x^2)^4+((648*x^2-648*x)*ln( 5)^2+648*x^3-648*x^2-324*x+162)*exp(ln(5)^4+2*x*ln(5)^2+x^2)^2-52488*ln(5) ^2-52488*x)/exp(ln(5)^4+2*x*ln(5)^2+x^2)^4,x,method=_RETURNVERBOSE)
Output:
x^4-2*x^3+x^2+(-162*x^2+162*x)*exp(-2*(ln(5)^2+x)^2)+6561*exp(-4*(ln(5)^2+ x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (22) = 44\).
Time = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.70 \[ \int e^{-4 x^2-8 x \log ^2(5)-4 \log ^4(5)} \left (-52488 x+e^{4 x^2+8 x \log ^2(5)+4 \log ^4(5)} \left (2 x-6 x^2+4 x^3\right )-52488 \log ^2(5)+e^{2 x^2+4 x \log ^2(5)+2 \log ^4(5)} \left (162-324 x-648 x^2+648 x^3+\left (-648 x+648 x^2\right ) \log ^2(5)\right )\right ) \, dx={\left ({\left (x^{4} - 2 \, x^{3} + x^{2}\right )} e^{\left (4 \, \log \left (5\right )^{4} + 8 \, x \log \left (5\right )^{2} + 4 \, x^{2}\right )} - 162 \, {\left (x^{2} - x\right )} e^{\left (2 \, \log \left (5\right )^{4} + 4 \, x \log \left (5\right )^{2} + 2 \, x^{2}\right )} + 6561\right )} e^{\left (-4 \, \log \left (5\right )^{4} - 8 \, x \log \left (5\right )^{2} - 4 \, x^{2}\right )} \] Input:
integrate(((4*x^3-6*x^2+2*x)*exp(log(5)^4+2*x*log(5)^2+x^2)^4+((648*x^2-64 8*x)*log(5)^2+648*x^3-648*x^2-324*x+162)*exp(log(5)^4+2*x*log(5)^2+x^2)^2- 52488*log(5)^2-52488*x)/exp(log(5)^4+2*x*log(5)^2+x^2)^4,x, algorithm="fri cas")
Output:
((x^4 - 2*x^3 + x^2)*e^(4*log(5)^4 + 8*x*log(5)^2 + 4*x^2) - 162*(x^2 - x) *e^(2*log(5)^4 + 4*x*log(5)^2 + 2*x^2) + 6561)*e^(-4*log(5)^4 - 8*x*log(5) ^2 - 4*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (19) = 38\).
Time = 0.11 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.96 \[ \int e^{-4 x^2-8 x \log ^2(5)-4 \log ^4(5)} \left (-52488 x+e^{4 x^2+8 x \log ^2(5)+4 \log ^4(5)} \left (2 x-6 x^2+4 x^3\right )-52488 \log ^2(5)+e^{2 x^2+4 x \log ^2(5)+2 \log ^4(5)} \left (162-324 x-648 x^2+648 x^3+\left (-648 x+648 x^2\right ) \log ^2(5)\right )\right ) \, dx=x^{4} - 2 x^{3} + x^{2} + \left (- 162 x^{2} + 162 x\right ) e^{- 2 x^{2} - 4 x \log {\left (5 \right )}^{2} - 2 \log {\left (5 \right )}^{4}} + 6561 e^{- 4 x^{2} - 8 x \log {\left (5 \right )}^{2} - 4 \log {\left (5 \right )}^{4}} \] Input:
integrate(((4*x**3-6*x**2+2*x)*exp(ln(5)**4+2*x*ln(5)**2+x**2)**4+((648*x* *2-648*x)*ln(5)**2+648*x**3-648*x**2-324*x+162)*exp(ln(5)**4+2*x*ln(5)**2+ x**2)**2-52488*ln(5)**2-52488*x)/exp(ln(5)**4+2*x*ln(5)**2+x**2)**4,x)
Output:
x**4 - 2*x**3 + x**2 + (-162*x**2 + 162*x)*exp(-2*x**2 - 4*x*log(5)**2 - 2 *log(5)**4) + 6561*exp(-4*x**2 - 8*x*log(5)**2 - 4*log(5)**4)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.35 (sec) , antiderivative size = 565, normalized size of antiderivative = 24.57 \[ \int e^{-4 x^2-8 x \log ^2(5)-4 \log ^4(5)} \left (-52488 x+e^{4 x^2+8 x \log ^2(5)+4 \log ^4(5)} \left (2 x-6 x^2+4 x^3\right )-52488 \log ^2(5)+e^{2 x^2+4 x \log ^2(5)+2 \log ^4(5)} \left (162-324 x-648 x^2+648 x^3+\left (-648 x+648 x^2\right ) \log ^2(5)\right )\right ) \, dx=\text {Too large to display} \] Input:
integrate(((4*x^3-6*x^2+2*x)*exp(log(5)^4+2*x*log(5)^2+x^2)^4+((648*x^2-64 8*x)*log(5)^2+648*x^3-648*x^2-324*x+162)*exp(log(5)^4+2*x*log(5)^2+x^2)^2- 52488*log(5)^2-52488*x)/exp(log(5)^4+2*x*log(5)^2+x^2)^4,x, algorithm="max ima")
Output:
x^4 + 13122*sqrt(pi)*(log(5)^2 + x)*(erf(2*sqrt((log(5)^2 + x)^2)) - 1)*lo g(5)^2/sqrt((log(5)^2 + x)^2) - 81*I*sqrt(2)*(2*I*sqrt(pi)*(log(5)^2 + x)* (erf(sqrt(2)*sqrt((log(5)^2 + x)^2)) - 1)*log(5)^4/sqrt((log(5)^2 + x)^2) + 2*I*sqrt(2)*e^(-2*(log(5)^2 + x)^2)*log(5)^2 - I*(log(5)^2 + x)^3*gamma( 3/2, 2*(log(5)^2 + x)^2)/((log(5)^2 + x)^2)^(3/2))*log(5)^2 + 81*I*sqrt(2) *(-2*I*sqrt(pi)*(log(5)^2 + x)*(erf(sqrt(2)*sqrt((log(5)^2 + x)^2)) - 1)*l og(5)^2/sqrt((log(5)^2 + x)^2) - I*sqrt(2)*e^(-2*(log(5)^2 + x)^2))*log(5) ^2 - 13122*sqrt(pi)*erf(2*log(5)^2 + 2*x)*log(5)^2 - 2*x^3 + x^2 + 81/2*sq rt(2)*sqrt(pi)*erf(sqrt(2)*log(5)^2 + sqrt(2)*x) - 81/2*I*sqrt(2)*(-4*I*sq rt(pi)*(log(5)^2 + x)*(erf(sqrt(2)*sqrt((log(5)^2 + x)^2)) - 1)*log(5)^6/s qrt((log(5)^2 + x)^2) - 6*I*sqrt(2)*e^(-2*(log(5)^2 + x)^2)*log(5)^4 + 6*I *(log(5)^2 + x)^3*gamma(3/2, 2*(log(5)^2 + x)^2)*log(5)^2/((log(5)^2 + x)^ 2)^(3/2) - I*sqrt(2)*gamma(2, 2*(log(5)^2 + x)^2)) + 81*I*sqrt(2)*(2*I*sqr t(pi)*(log(5)^2 + x)*(erf(sqrt(2)*sqrt((log(5)^2 + x)^2)) - 1)*log(5)^4/sq rt((log(5)^2 + x)^2) + 2*I*sqrt(2)*e^(-2*(log(5)^2 + x)^2)*log(5)^2 - I*(l og(5)^2 + x)^3*gamma(3/2, 2*(log(5)^2 + x)^2)/((log(5)^2 + x)^2)^(3/2)) + 81/2*I*sqrt(2)*(-2*I*sqrt(pi)*(log(5)^2 + x)*(erf(sqrt(2)*sqrt((log(5)^2 + x)^2)) - 1)*log(5)^2/sqrt((log(5)^2 + x)^2) - I*sqrt(2)*e^(-2*(log(5)^2 + x)^2)) + 6561*e^(-4*(log(5)^2 + x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (22) = 44\).
Time = 0.16 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.65 \[ \int e^{-4 x^2-8 x \log ^2(5)-4 \log ^4(5)} \left (-52488 x+e^{4 x^2+8 x \log ^2(5)+4 \log ^4(5)} \left (2 x-6 x^2+4 x^3\right )-52488 \log ^2(5)+e^{2 x^2+4 x \log ^2(5)+2 \log ^4(5)} \left (162-324 x-648 x^2+648 x^3+\left (-648 x+648 x^2\right ) \log ^2(5)\right )\right ) \, dx=x^{4} - 2 \, x^{3} + x^{2} - 162 \, {\left (\log \left (5\right )^{4} - 2 \, {\left (\log \left (5\right )^{2} + x\right )} \log \left (5\right )^{2} + {\left (\log \left (5\right )^{2} + x\right )}^{2} - x\right )} e^{\left (-2 \, \log \left (5\right )^{4} - 4 \, x \log \left (5\right )^{2} - 2 \, x^{2}\right )} + 6561 \, e^{\left (-4 \, \log \left (5\right )^{4} - 8 \, x \log \left (5\right )^{2} - 4 \, x^{2}\right )} \] Input:
integrate(((4*x^3-6*x^2+2*x)*exp(log(5)^4+2*x*log(5)^2+x^2)^4+((648*x^2-64 8*x)*log(5)^2+648*x^3-648*x^2-324*x+162)*exp(log(5)^4+2*x*log(5)^2+x^2)^2- 52488*log(5)^2-52488*x)/exp(log(5)^4+2*x*log(5)^2+x^2)^4,x, algorithm="gia c")
Output:
x^4 - 2*x^3 + x^2 - 162*(log(5)^4 - 2*(log(5)^2 + x)*log(5)^2 + (log(5)^2 + x)^2 - x)*e^(-2*log(5)^4 - 4*x*log(5)^2 - 2*x^2) + 6561*e^(-4*log(5)^4 - 8*x*log(5)^2 - 4*x^2)
Time = 3.31 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.57 \[ \int e^{-4 x^2-8 x \log ^2(5)-4 \log ^4(5)} \left (-52488 x+e^{4 x^2+8 x \log ^2(5)+4 \log ^4(5)} \left (2 x-6 x^2+4 x^3\right )-52488 \log ^2(5)+e^{2 x^2+4 x \log ^2(5)+2 \log ^4(5)} \left (162-324 x-648 x^2+648 x^3+\left (-648 x+648 x^2\right ) \log ^2(5)\right )\right ) \, dx=6561\,{\mathrm {e}}^{-4\,x^2-8\,{\ln \left (5\right )}^2\,x-4\,{\ln \left (5\right )}^4}-162\,x^2\,{\mathrm {e}}^{-2\,x^2-4\,{\ln \left (5\right )}^2\,x-2\,{\ln \left (5\right )}^4}+162\,x\,{\mathrm {e}}^{-2\,x^2-4\,{\ln \left (5\right )}^2\,x-2\,{\ln \left (5\right )}^4}+x^2-2\,x^3+x^4 \] Input:
int(-exp(- 8*x*log(5)^2 - 4*log(5)^4 - 4*x^2)*(52488*x + exp(4*x*log(5)^2 + 2*log(5)^4 + 2*x^2)*(324*x + log(5)^2*(648*x - 648*x^2) + 648*x^2 - 648* x^3 - 162) + 52488*log(5)^2 - exp(8*x*log(5)^2 + 4*log(5)^4 + 4*x^2)*(2*x - 6*x^2 + 4*x^3)),x)
Output:
6561*exp(- 8*x*log(5)^2 - 4*log(5)^4 - 4*x^2) - 162*x^2*exp(- 4*x*log(5)^2 - 2*log(5)^4 - 2*x^2) + 162*x*exp(- 4*x*log(5)^2 - 2*log(5)^4 - 2*x^2) + x^2 - 2*x^3 + x^4
Time = 0.16 (sec) , antiderivative size = 152, normalized size of antiderivative = 6.61 \[ \int e^{-4 x^2-8 x \log ^2(5)-4 \log ^4(5)} \left (-52488 x+e^{4 x^2+8 x \log ^2(5)+4 \log ^4(5)} \left (2 x-6 x^2+4 x^3\right )-52488 \log ^2(5)+e^{2 x^2+4 x \log ^2(5)+2 \log ^4(5)} \left (162-324 x-648 x^2+648 x^3+\left (-648 x+648 x^2\right ) \log ^2(5)\right )\right ) \, dx=\frac {e^{4 \mathrm {log}\left (5\right )^{4}+8 x \mathrm {log}\left (5\right )^{2}+4 x^{2}} x^{4}-2 e^{4 \mathrm {log}\left (5\right )^{4}+8 x \mathrm {log}\left (5\right )^{2}+4 x^{2}} x^{3}+e^{4 \mathrm {log}\left (5\right )^{4}+8 x \mathrm {log}\left (5\right )^{2}+4 x^{2}} x^{2}-162 e^{2 \mathrm {log}\left (5\right )^{4}+4 x \mathrm {log}\left (5\right )^{2}+2 x^{2}} x^{2}+162 e^{2 \mathrm {log}\left (5\right )^{4}+4 x \mathrm {log}\left (5\right )^{2}+2 x^{2}} x +6561}{e^{4 \mathrm {log}\left (5\right )^{4}+8 x \mathrm {log}\left (5\right )^{2}+4 x^{2}}} \] Input:
int(((4*x^3-6*x^2+2*x)*exp(log(5)^4+2*x*log(5)^2+x^2)^4+((648*x^2-648*x)*l og(5)^2+648*x^3-648*x^2-324*x+162)*exp(log(5)^4+2*x*log(5)^2+x^2)^2-52488* log(5)^2-52488*x)/exp(log(5)^4+2*x*log(5)^2+x^2)^4,x)
Output:
(e**(4*log(5)**4 + 8*log(5)**2*x + 4*x**2)*x**4 - 2*e**(4*log(5)**4 + 8*lo g(5)**2*x + 4*x**2)*x**3 + e**(4*log(5)**4 + 8*log(5)**2*x + 4*x**2)*x**2 - 162*e**(2*log(5)**4 + 4*log(5)**2*x + 2*x**2)*x**2 + 162*e**(2*log(5)**4 + 4*log(5)**2*x + 2*x**2)*x + 6561)/e**(4*log(5)**4 + 8*log(5)**2*x + 4*x **2)