Integrand size = 122, antiderivative size = 18 \[ \int \frac {e^{\frac {x^2+2 e x^2+e^2 x^2+\left (-2 e x^2-2 e^2 x^2\right ) \log (2)+e^2 x^2 \log ^2(2)}{1-2 e \log (2)+e^2 \log ^2(2)}} \left (2 x+4 e x+2 e^2 x+\left (-4 e x-4 e^2 x\right ) \log (2)+2 e^2 x \log ^2(2)\right )}{1-2 e \log (2)+e^2 \log ^2(2)} \, dx=e^{\left (x+\frac {x}{\frac {1}{e}-\log (2)}\right )^2} \] Output:
exp((x/(exp(-1)-ln(2))+x)^2)
Leaf count is larger than twice the leaf count of optimal. \(64\) vs. \(2(18)=36\).
Time = 0.89 (sec) , antiderivative size = 64, normalized size of antiderivative = 3.56 \[ \int \frac {e^{\frac {x^2+2 e x^2+e^2 x^2+\left (-2 e x^2-2 e^2 x^2\right ) \log (2)+e^2 x^2 \log ^2(2)}{1-2 e \log (2)+e^2 \log ^2(2)}} \left (2 x+4 e x+2 e^2 x+\left (-4 e x-4 e^2 x\right ) \log (2)+2 e^2 x \log ^2(2)\right )}{1-2 e \log (2)+e^2 \log ^2(2)} \, dx=2^{-1+\frac {1}{(-1+e \log (2))^2}} e^{\frac {e \log ^2(2) (-2+e \log (2))+x^2 \left (1+e^2 \left (1+\log ^2(2)-\log (4)\right )-e (-2+\log (4))\right )}{(-1+e \log (2))^2}} \] Input:
Integrate[(E^((x^2 + 2*E*x^2 + E^2*x^2 + (-2*E*x^2 - 2*E^2*x^2)*Log[2] + E ^2*x^2*Log[2]^2)/(1 - 2*E*Log[2] + E^2*Log[2]^2))*(2*x + 4*E*x + 2*E^2*x + (-4*E*x - 4*E^2*x)*Log[2] + 2*E^2*x*Log[2]^2))/(1 - 2*E*Log[2] + E^2*Log[ 2]^2),x]
Output:
2^(-1 + (-1 + E*Log[2])^(-2))*E^((E*Log[2]^2*(-2 + E*Log[2]) + x^2*(1 + E^ 2*(1 + Log[2]^2 - Log[4]) - E*(-2 + Log[4])))/(-1 + E*Log[2])^2)
Time = 0.43 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.50, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.066, Rules used = {6, 6, 6, 27, 27, 2725, 27, 2638}
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.
| \(\displaystyle \int \frac {\left (2 e^2 x+4 e x+2 x+2 e^2 x \log ^2(2)+\left (-4 e^2 x-4 e x\right ) \log (2)\right ) \exp \left (\frac {e^2 x^2+2 e x^2+x^2+e^2 x^2 \log ^2(2)+\left (-2 e^2 x^2-2 e x^2\right ) \log (2)}{1+e^2 \log ^2(2)-2 e \log (2)}\right )}{1+e^2 \log ^2(2)-2 e \log (2)} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {\left ((2+4 e) x+2 e^2 x+2 e^2 x \log ^2(2)+\left (-4 e^2 x-4 e x\right ) \log (2)\right ) \exp \left (\frac {e^2 x^2+2 e x^2+x^2+e^2 x^2 \log ^2(2)+\left (-2 e^2 x^2-2 e x^2\right ) \log (2)}{1+e^2 \log ^2(2)-2 e \log (2)}\right )}{1+e^2 \log ^2(2)-2 e \log (2)}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {\left (\left (2+4 e+2 e^2\right ) x+2 e^2 x \log ^2(2)+\left (-4 e^2 x-4 e x\right ) \log (2)\right ) \exp \left (\frac {e^2 x^2+2 e x^2+x^2+e^2 x^2 \log ^2(2)+\left (-2 e^2 x^2-2 e x^2\right ) \log (2)}{1+e^2 \log ^2(2)-2 e \log (2)}\right )}{1+e^2 \log ^2(2)-2 e \log (2)}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {\left (x \left (2+4 e+2 e^2+2 e^2 \log ^2(2)\right )+\left (-4 e^2 x-4 e x\right ) \log (2)\right ) \exp \left (\frac {e^2 x^2+2 e x^2+x^2+e^2 x^2 \log ^2(2)+\left (-2 e^2 x^2-2 e x^2\right ) \log (2)}{1+e^2 \log ^2(2)-2 e \log (2)}\right )}{1+e^2 \log ^2(2)-2 e \log (2)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int 2^{1-\frac {2 e^2 x^2+2 e x^2}{1-2 e \log (2)+e^2 \log ^2(2)}} \exp \left (\frac {e^2 \log ^2(2) x^2+e^2 x^2+2 e x^2+x^2}{(1-e \log (2))^2}\right ) x (1+e (1-\log (2)))^2dx}{(1-e \log (2))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(1+e (1-\log (2)))^2 \int 2^{1-\frac {2 \left (e^2 x^2+e x^2\right )}{(1-e \log (2))^2}} \exp \left (\frac {e^2 \log ^2(2) x^2+e^2 x^2+2 e x^2+x^2}{(1-e \log (2))^2}\right ) xdx}{(1-e \log (2))^2}\) |
| \(\Big \downarrow \) 2725 |
| \(\displaystyle \frac {(1+e (1-\log (2)))^2 \int 2 \exp \left (\frac {x^2 (1+e (1-\log (2)))^2}{(1-e \log (2))^2}\right ) xdx}{(1-e \log (2))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (1+e (1-\log (2)))^2 \int \exp \left (\frac {x^2 (1+e (1-\log (2)))^2}{(1-e \log (2))^2}\right ) xdx}{(1-e \log (2))^2}\) |
| \(\Big \downarrow \) 2638 |
| \(\displaystyle \exp \left (\frac {x^2 (1+e (1-\log (2)))^2}{(1-e \log (2))^2}\right )\) |
Input:
Int[(E^((x^2 + 2*E*x^2 + E^2*x^2 + (-2*E*x^2 - 2*E^2*x^2)*Log[2] + E^2*x^2 *Log[2]^2)/(1 - 2*E*Log[2] + E^2*Log[2]^2))*(2*x + 4*E*x + 2*E^2*x + (-4*E *x - 4*E^2*x)*Log[2] + 2*E^2*x*Log[2]^2))/(1 - 2*E*Log[2] + E^2*Log[2]^2), x]
Output:
E^((x^2*(1 + E*(1 - Log[2]))^2)/(1 - E*Log[2])^2)
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_))*((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]
Int[(u_.)*(F_)^(v_)*(G_)^(w_), x_Symbol] :> With[{z = v*Log[F] + w*Log[G]}, Int[u*NormalizeIntegrand[E^z, x], x] /; BinomialQ[z, x] || (PolynomialQ[z, x] && LeQ[Exponent[z, x], 2])] /; FreeQ[{F, G}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(53\) vs. \(2(16)=32\).
Time = 0.66 (sec) , antiderivative size = 54, normalized size of antiderivative = 3.00
| method | result | size |
| risch | \({\mathrm e}^{\frac {x^{2} \left (-1-{\mathrm e}^{2} \ln \left (2\right )^{2}+2 \,{\mathrm e}^{2} \ln \left (2\right )-{\mathrm e}^{2}+2 \,{\mathrm e} \ln \left (2\right )-2 \,{\mathrm e}\right )}{-{\mathrm e}^{2} \ln \left (2\right )^{2}+2 \,{\mathrm e} \ln \left (2\right )-1}}\) | \(54\) |
| gosper | \({\mathrm e}^{\frac {x^{2} \left ({\mathrm e}^{2} \ln \left (2\right )^{2}-2 \,{\mathrm e}^{2} \ln \left (2\right )+{\mathrm e}^{2}-2 \,{\mathrm e} \ln \left (2\right )+2 \,{\mathrm e}+1\right )}{{\mathrm e}^{2} \ln \left (2\right )^{2}-2 \,{\mathrm e} \ln \left (2\right )+1}}\) | \(58\) |
| norman | \({\mathrm e}^{\frac {x^{2} {\mathrm e}^{2} \ln \left (2\right )^{2}+\left (-2 x^{2} {\mathrm e}^{2}-2 x^{2} {\mathrm e}\right ) \ln \left (2\right )+x^{2} {\mathrm e}^{2}+2 x^{2} {\mathrm e}+x^{2}}{{\mathrm e}^{2} \ln \left (2\right )^{2}-2 \,{\mathrm e} \ln \left (2\right )+1}}\) | \(73\) |
| meijerg | \(-\frac {\left (2 \,{\mathrm e}^{2} \ln \left (2\right )^{2}-4 \,{\mathrm e}^{2} \ln \left (2\right )+2 \,{\mathrm e}^{2}-4 \,{\mathrm e} \ln \left (2\right )+4 \,{\mathrm e}+2\right ) \left (1-{\mathrm e}^{\frac {x^{2} \left ({\mathrm e}^{2} \ln \left (2\right )^{2}-2 \,{\mathrm e}^{2} \ln \left (2\right )+{\mathrm e}^{2}-2 \,{\mathrm e} \ln \left (2\right )+2 \,{\mathrm e}+1\right )}{{\mathrm e}^{2} \ln \left (2\right )^{2}-2 \,{\mathrm e} \ln \left (2\right )+1}}\right )}{2 \left ({\mathrm e}^{2} \ln \left (2\right )^{2}-2 \,{\mathrm e}^{2} \ln \left (2\right )+{\mathrm e}^{2}-2 \,{\mathrm e} \ln \left (2\right )+2 \,{\mathrm e}+1\right )}\) | \(115\) |
| default | \(\frac {\left (2 \,{\mathrm e}^{2} \ln \left (2\right )^{2}+\left (-4 \,{\mathrm e}^{2}-4 \,{\mathrm e}\right ) \ln \left (2\right )+2 \,{\mathrm e}^{2}+4 \,{\mathrm e}+2\right ) {\mathrm e}^{\frac {\left ({\mathrm e}^{2} \ln \left (2\right )^{2}+\left (-2 \,{\mathrm e}^{2}-2 \,{\mathrm e}\right ) \ln \left (2\right )+{\mathrm e}^{2}+2 \,{\mathrm e}+1\right ) x^{2}}{{\mathrm e}^{2} \ln \left (2\right )^{2}-2 \,{\mathrm e} \ln \left (2\right )+1}}}{2 \,{\mathrm e}^{2} \ln \left (2\right )^{2}+2 \left (-2 \,{\mathrm e}^{2}-2 \,{\mathrm e}\right ) \ln \left (2\right )+2 \,{\mathrm e}^{2}+4 \,{\mathrm e}+2}\) | \(131\) |
| parallelrisch | \(\frac {{\mathrm e}^{2} \ln \left (2\right )^{2} {\mathrm e}^{\frac {x^{2} \left ({\mathrm e}^{2} \ln \left (2\right )^{2}-2 \,{\mathrm e}^{2} \ln \left (2\right )+{\mathrm e}^{2}-2 \,{\mathrm e} \ln \left (2\right )+2 \,{\mathrm e}+1\right )}{{\mathrm e}^{2} \ln \left (2\right )^{2}-2 \,{\mathrm e} \ln \left (2\right )+1}}-2 \,{\mathrm e} \ln \left (2\right ) {\mathrm e}^{\frac {x^{2} \left ({\mathrm e}^{2} \ln \left (2\right )^{2}-2 \,{\mathrm e}^{2} \ln \left (2\right )+{\mathrm e}^{2}-2 \,{\mathrm e} \ln \left (2\right )+2 \,{\mathrm e}+1\right )}{{\mathrm e}^{2} \ln \left (2\right )^{2}-2 \,{\mathrm e} \ln \left (2\right )+1}}+{\mathrm e}^{\frac {x^{2} \left ({\mathrm e}^{2} \ln \left (2\right )^{2}-2 \,{\mathrm e}^{2} \ln \left (2\right )+{\mathrm e}^{2}-2 \,{\mathrm e} \ln \left (2\right )+2 \,{\mathrm e}+1\right )}{{\mathrm e}^{2} \ln \left (2\right )^{2}-2 \,{\mathrm e} \ln \left (2\right )+1}}}{{\mathrm e}^{2} \ln \left (2\right )^{2}-2 \,{\mathrm e} \ln \left (2\right )+1}\) | \(208\) |
| parts | \(\text {Expression too large to display}\) | \(1125\) |
Input:
int((2*x*exp(1)^2*ln(2)^2+(-4*x*exp(1)^2-4*x*exp(1))*ln(2)+2*x*exp(1)^2+4* x*exp(1)+2*x)*exp((x^2*exp(1)^2*ln(2)^2+(-2*x^2*exp(1)^2-2*x^2*exp(1))*ln( 2)+x^2*exp(1)^2+2*x^2*exp(1)+x^2)/(exp(1)^2*ln(2)^2-2*exp(1)*ln(2)+1))/(ex p(1)^2*ln(2)^2-2*exp(1)*ln(2)+1),x,method=_RETURNVERBOSE)
Output:
exp(x^2*(-1-exp(2)*ln(2)^2+2*exp(2)*ln(2)-exp(2)+2*exp(1)*ln(2)-2*exp(1))/ (-exp(2)*ln(2)^2+2*exp(1)*ln(2)-1))
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (16) = 32\).
Time = 0.10 (sec) , antiderivative size = 63, normalized size of antiderivative = 3.50 \[ \int \frac {e^{\frac {x^2+2 e x^2+e^2 x^2+\left (-2 e x^2-2 e^2 x^2\right ) \log (2)+e^2 x^2 \log ^2(2)}{1-2 e \log (2)+e^2 \log ^2(2)}} \left (2 x+4 e x+2 e^2 x+\left (-4 e x-4 e^2 x\right ) \log (2)+2 e^2 x \log ^2(2)\right )}{1-2 e \log (2)+e^2 \log ^2(2)} \, dx=e^{\left (\frac {x^{2} e^{2} \log \left (2\right )^{2} + x^{2} e^{2} + 2 \, x^{2} e + x^{2} - 2 \, {\left (x^{2} e^{2} + x^{2} e\right )} \log \left (2\right )}{e^{2} \log \left (2\right )^{2} - 2 \, e \log \left (2\right ) + 1}\right )} \] Input:
integrate((2*x*exp(1)^2*log(2)^2+(-4*x*exp(1)^2-4*exp(1)*x)*log(2)+2*x*exp (1)^2+4*exp(1)*x+2*x)*exp((x^2*exp(1)^2*log(2)^2+(-2*x^2*exp(1)^2-2*x^2*ex p(1))*log(2)+x^2*exp(1)^2+2*x^2*exp(1)+x^2)/(exp(1)^2*log(2)^2-2*exp(1)*lo g(2)+1))/(exp(1)^2*log(2)^2-2*exp(1)*log(2)+1),x, algorithm="fricas")
Output:
e^((x^2*e^2*log(2)^2 + x^2*e^2 + 2*x^2*e + x^2 - 2*(x^2*e^2 + x^2*e)*log(2 ))/(e^2*log(2)^2 - 2*e*log(2) + 1))
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (14) = 28\).
Time = 0.11 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.94 \[ \int \frac {e^{\frac {x^2+2 e x^2+e^2 x^2+\left (-2 e x^2-2 e^2 x^2\right ) \log (2)+e^2 x^2 \log ^2(2)}{1-2 e \log (2)+e^2 \log ^2(2)}} \left (2 x+4 e x+2 e^2 x+\left (-4 e x-4 e^2 x\right ) \log (2)+2 e^2 x \log ^2(2)\right )}{1-2 e \log (2)+e^2 \log ^2(2)} \, dx=e^{\frac {x^{2} + x^{2} e^{2} \log {\left (2 \right )}^{2} + 2 e x^{2} + x^{2} e^{2} + \left (- 2 x^{2} e^{2} - 2 e x^{2}\right ) \log {\left (2 \right )}}{- 2 e \log {\left (2 \right )} + 1 + e^{2} \log {\left (2 \right )}^{2}}} \] Input:
integrate((2*x*exp(1)**2*ln(2)**2+(-4*x*exp(1)**2-4*exp(1)*x)*ln(2)+2*x*ex p(1)**2+4*exp(1)*x+2*x)*exp((x**2*exp(1)**2*ln(2)**2+(-2*x**2*exp(1)**2-2* x**2*exp(1))*ln(2)+x**2*exp(1)**2+2*x**2*exp(1)+x**2)/(exp(1)**2*ln(2)**2- 2*exp(1)*ln(2)+1))/(exp(1)**2*ln(2)**2-2*exp(1)*ln(2)+1),x)
Output:
exp((x**2 + x**2*exp(2)*log(2)**2 + 2*E*x**2 + x**2*exp(2) + (-2*x**2*exp( 2) - 2*E*x**2)*log(2))/(-2*E*log(2) + 1 + exp(2)*log(2)**2))
Leaf count of result is larger than twice the leaf count of optimal. 1721 vs. \(2 (16) = 32\).
Time = 0.04 (sec) , antiderivative size = 1721, normalized size of antiderivative = 95.61 \[ \int \frac {e^{\frac {x^2+2 e x^2+e^2 x^2+\left (-2 e x^2-2 e^2 x^2\right ) \log (2)+e^2 x^2 \log ^2(2)}{1-2 e \log (2)+e^2 \log ^2(2)}} \left (2 x+4 e x+2 e^2 x+\left (-4 e x-4 e^2 x\right ) \log (2)+2 e^2 x \log ^2(2)\right )}{1-2 e \log (2)+e^2 \log ^2(2)} \, dx=\text {Too large to display} \] Input:
integrate((2*x*exp(1)^2*log(2)^2+(-4*x*exp(1)^2-4*exp(1)*x)*log(2)+2*x*exp (1)^2+4*exp(1)*x+2*x)*exp((x^2*exp(1)^2*log(2)^2+(-2*x^2*exp(1)^2-2*x^2*ex p(1))*log(2)+x^2*exp(1)^2+2*x^2*exp(1)+x^2)/(exp(1)^2*log(2)^2-2*exp(1)*lo g(2)+1))/(exp(1)^2*log(2)^2-2*exp(1)*log(2)+1),x, algorithm="maxima")
Output:
(e^(x^2*e^2*log(2)^2/(e^2*log(2)^2 - 2*e*log(2) + 1) - 2*x^2*e^2*log(2)/(e ^2*log(2)^2 - 2*e*log(2) + 1) - 2*x^2*e*log(2)/(e^2*log(2)^2 - 2*e*log(2) + 1) + x^2*e^2/(e^2*log(2)^2 - 2*e*log(2) + 1) + 2*x^2*e/(e^2*log(2)^2 - 2 *e*log(2) + 1) + x^2/(e^2*log(2)^2 - 2*e*log(2) + 1) + 2)*log(2)^2/(e^2*lo g(2)^2/(e^2*log(2)^2 - 2*e*log(2) + 1) - 2*e^2*log(2)/(e^2*log(2)^2 - 2*e* log(2) + 1) - 2*e*log(2)/(e^2*log(2)^2 - 2*e*log(2) + 1) + e^2/(e^2*log(2) ^2 - 2*e*log(2) + 1) + 2*e/(e^2*log(2)^2 - 2*e*log(2) + 1) + 1/(e^2*log(2) ^2 - 2*e*log(2) + 1)) - 2*e^(x^2*e^2*log(2)^2/(e^2*log(2)^2 - 2*e*log(2) + 1) - 2*x^2*e^2*log(2)/(e^2*log(2)^2 - 2*e*log(2) + 1) - 2*x^2*e*log(2)/(e ^2*log(2)^2 - 2*e*log(2) + 1) + x^2*e^2/(e^2*log(2)^2 - 2*e*log(2) + 1) + 2*x^2*e/(e^2*log(2)^2 - 2*e*log(2) + 1) + x^2/(e^2*log(2)^2 - 2*e*log(2) + 1) + 2)*log(2)/(e^2*log(2)^2/(e^2*log(2)^2 - 2*e*log(2) + 1) - 2*e^2*log( 2)/(e^2*log(2)^2 - 2*e*log(2) + 1) - 2*e*log(2)/(e^2*log(2)^2 - 2*e*log(2) + 1) + e^2/(e^2*log(2)^2 - 2*e*log(2) + 1) + 2*e/(e^2*log(2)^2 - 2*e*log( 2) + 1) + 1/(e^2*log(2)^2 - 2*e*log(2) + 1)) - 2*e^(x^2*e^2*log(2)^2/(e^2* log(2)^2 - 2*e*log(2) + 1) - 2*x^2*e^2*log(2)/(e^2*log(2)^2 - 2*e*log(2) + 1) - 2*x^2*e*log(2)/(e^2*log(2)^2 - 2*e*log(2) + 1) + x^2*e^2/(e^2*log(2) ^2 - 2*e*log(2) + 1) + 2*x^2*e/(e^2*log(2)^2 - 2*e*log(2) + 1) + x^2/(e^2* log(2)^2 - 2*e*log(2) + 1) + 1)*log(2)/(e^2*log(2)^2/(e^2*log(2)^2 - 2*e*l og(2) + 1) - 2*e^2*log(2)/(e^2*log(2)^2 - 2*e*log(2) + 1) - 2*e*log(2)/...
Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (16) = 32\).
Time = 0.17 (sec) , antiderivative size = 224, normalized size of antiderivative = 12.44 \[ \int \frac {e^{\frac {x^2+2 e x^2+e^2 x^2+\left (-2 e x^2-2 e^2 x^2\right ) \log (2)+e^2 x^2 \log ^2(2)}{1-2 e \log (2)+e^2 \log ^2(2)}} \left (2 x+4 e x+2 e^2 x+\left (-4 e x-4 e^2 x\right ) \log (2)+2 e^2 x \log ^2(2)\right )}{1-2 e \log (2)+e^2 \log ^2(2)} \, dx=\frac {e^{\left (\frac {x^{2} e^{2} \log \left (2\right )^{2} - 2 \, x^{2} e^{2} \log \left (2\right ) - 2 \, x^{2} e \log \left (2\right ) + x^{2} e^{2} + 2 \, x^{2} e + x^{2}}{e^{2} \log \left (2\right )^{2} - 2 \, e \log \left (2\right ) + 1} + 2\right )} \log \left (2\right )^{2} - 2 \, e^{\left (\frac {x^{2} e^{2} \log \left (2\right )^{2} - 2 \, x^{2} e^{2} \log \left (2\right ) - 2 \, x^{2} e \log \left (2\right ) + x^{2} e^{2} + 2 \, x^{2} e + x^{2}}{e^{2} \log \left (2\right )^{2} - 2 \, e \log \left (2\right ) + 1} + 1\right )} \log \left (2\right ) + e^{\left (\frac {x^{2} e^{2} \log \left (2\right )^{2} - 2 \, x^{2} e^{2} \log \left (2\right ) - 2 \, x^{2} e \log \left (2\right ) + x^{2} e^{2} + 2 \, x^{2} e + x^{2}}{e^{2} \log \left (2\right )^{2} - 2 \, e \log \left (2\right ) + 1}\right )}}{e^{2} \log \left (2\right )^{2} - 2 \, e \log \left (2\right ) + 1} \] Input:
integrate((2*x*exp(1)^2*log(2)^2+(-4*x*exp(1)^2-4*exp(1)*x)*log(2)+2*x*exp (1)^2+4*exp(1)*x+2*x)*exp((x^2*exp(1)^2*log(2)^2+(-2*x^2*exp(1)^2-2*x^2*ex p(1))*log(2)+x^2*exp(1)^2+2*x^2*exp(1)+x^2)/(exp(1)^2*log(2)^2-2*exp(1)*lo g(2)+1))/(exp(1)^2*log(2)^2-2*exp(1)*log(2)+1),x, algorithm="giac")
Output:
(e^((x^2*e^2*log(2)^2 - 2*x^2*e^2*log(2) - 2*x^2*e*log(2) + x^2*e^2 + 2*x^ 2*e + x^2)/(e^2*log(2)^2 - 2*e*log(2) + 1) + 2)*log(2)^2 - 2*e^((x^2*e^2*l og(2)^2 - 2*x^2*e^2*log(2) - 2*x^2*e*log(2) + x^2*e^2 + 2*x^2*e + x^2)/(e^ 2*log(2)^2 - 2*e*log(2) + 1) + 1)*log(2) + e^((x^2*e^2*log(2)^2 - 2*x^2*e^ 2*log(2) - 2*x^2*e*log(2) + x^2*e^2 + 2*x^2*e + x^2)/(e^2*log(2)^2 - 2*e*l og(2) + 1)))/(e^2*log(2)^2 - 2*e*log(2) + 1)
Time = 3.97 (sec) , antiderivative size = 133, normalized size of antiderivative = 7.39 \[ \int \frac {e^{\frac {x^2+2 e x^2+e^2 x^2+\left (-2 e x^2-2 e^2 x^2\right ) \log (2)+e^2 x^2 \log ^2(2)}{1-2 e \log (2)+e^2 \log ^2(2)}} \left (2 x+4 e x+2 e^2 x+\left (-4 e x-4 e^2 x\right ) \log (2)+2 e^2 x \log ^2(2)\right )}{1-2 e \log (2)+e^2 \log ^2(2)} \, dx={\left (\frac {1}{4}\right )}^{\frac {x^2\,\mathrm {e}+x^2\,{\mathrm {e}}^2}{{\mathrm {e}}^2\,{\ln \left (2\right )}^2-2\,\mathrm {e}\,\ln \left (2\right )+1}}\,{\mathrm {e}}^{\frac {x^2}{{\mathrm {e}}^2\,{\ln \left (2\right )}^2-2\,\mathrm {e}\,\ln \left (2\right )+1}}\,{\mathrm {e}}^{\frac {x^2\,{\mathrm {e}}^2\,{\ln \left (2\right )}^2}{{\mathrm {e}}^2\,{\ln \left (2\right )}^2-2\,\mathrm {e}\,\ln \left (2\right )+1}}\,{\mathrm {e}}^{\frac {x^2\,{\mathrm {e}}^2}{{\mathrm {e}}^2\,{\ln \left (2\right )}^2-2\,\mathrm {e}\,\ln \left (2\right )+1}}\,{\mathrm {e}}^{\frac {2\,x^2\,\mathrm {e}}{{\mathrm {e}}^2\,{\ln \left (2\right )}^2-2\,\mathrm {e}\,\ln \left (2\right )+1}} \] Input:
int((exp((2*x^2*exp(1) - log(2)*(2*x^2*exp(1) + 2*x^2*exp(2)) + x^2*exp(2) + x^2 + x^2*exp(2)*log(2)^2)/(exp(2)*log(2)^2 - 2*exp(1)*log(2) + 1))*(2* x + 4*x*exp(1) + 2*x*exp(2) - log(2)*(4*x*exp(1) + 4*x*exp(2)) + 2*x*exp(2 )*log(2)^2))/(exp(2)*log(2)^2 - 2*exp(1)*log(2) + 1),x)
Output:
(1/4)^((x^2*exp(1) + x^2*exp(2))/(exp(2)*log(2)^2 - 2*exp(1)*log(2) + 1))* exp(x^2/(exp(2)*log(2)^2 - 2*exp(1)*log(2) + 1))*exp((x^2*exp(2)*log(2)^2) /(exp(2)*log(2)^2 - 2*exp(1)*log(2) + 1))*exp((x^2*exp(2))/(exp(2)*log(2)^ 2 - 2*exp(1)*log(2) + 1))*exp((2*x^2*exp(1))/(exp(2)*log(2)^2 - 2*exp(1)*l og(2) + 1))
Time = 0.16 (sec) , antiderivative size = 88, normalized size of antiderivative = 4.89 \[ \int \frac {e^{\frac {x^2+2 e x^2+e^2 x^2+\left (-2 e x^2-2 e^2 x^2\right ) \log (2)+e^2 x^2 \log ^2(2)}{1-2 e \log (2)+e^2 \log ^2(2)}} \left (2 x+4 e x+2 e^2 x+\left (-4 e x-4 e^2 x\right ) \log (2)+2 e^2 x \log ^2(2)\right )}{1-2 e \log (2)+e^2 \log ^2(2)} \, dx=\frac {e^{\frac {\mathrm {log}\left (2\right )^{2} e^{2} x^{2}-2 \,\mathrm {log}\left (2\right ) e \,x^{2}+e^{2} x^{2}+2 e \,x^{2}+x^{2}}{\mathrm {log}\left (2\right )^{2} e^{2}-2 \,\mathrm {log}\left (2\right ) e +1}}}{e^{\frac {2 \,\mathrm {log}\left (2\right ) e^{2} x^{2}}{\mathrm {log}\left (2\right )^{2} e^{2}-2 \,\mathrm {log}\left (2\right ) e +1}}} \] Input:
int((2*x*exp(1)^2*log(2)^2+(-4*x*exp(1)^2-4*exp(1)*x)*log(2)+2*x*exp(1)^2+ 4*exp(1)*x+2*x)*exp((x^2*exp(1)^2*log(2)^2+(-2*x^2*exp(1)^2-2*x^2*exp(1))* log(2)+x^2*exp(1)^2+2*x^2*exp(1)+x^2)/(exp(1)^2*log(2)^2-2*exp(1)*log(2)+1 ))/(exp(1)^2*log(2)^2-2*exp(1)*log(2)+1),x)
Output:
e**((log(2)**2*e**2*x**2 - 2*log(2)*e*x**2 + e**2*x**2 + 2*e*x**2 + x**2)/ (log(2)**2*e**2 - 2*log(2)*e + 1))/e**((2*log(2)*e**2*x**2)/(log(2)**2*e** 2 - 2*log(2)*e + 1))