Integrand size = 192, antiderivative size = 27 \[ \int \frac {e^4 x^2+e^4 \left (8 e x+5 e^2 x^2\right )+e^6 x^2 \log (4)-e^6 x^2 \log \left (3 e^2\right )}{e^2 x^2+e^2 \left (8 e x+10 e^2 x^2\right )+e^4 \left (16+40 e x+25 e^2 x^2\right )+\left (2 e^4 x^2+e^4 \left (8 e x+10 e^2 x^2\right )\right ) \log (4)+e^6 x^2 \log ^2(4)+\left (-2 e^4 x^2+e^4 \left (-8 e x-10 e^2 x^2\right )-2 e^6 x^2 \log (4)\right ) \log \left (3 e^2\right )+e^6 x^2 \log ^2\left (3 e^2\right )} \, dx=\frac {x}{5+\frac {1}{e^2}+\frac {4}{e x}+\log (4)-\log \left (3 e^2\right )} \]
Leaf count is larger than twice the leaf count of optimal. \(88\) vs. \(2(27)=54\).
Time = 0.05 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.26 \[ \int \frac {e^4 x^2+e^4 \left (8 e x+5 e^2 x^2\right )+e^6 x^2 \log (4)-e^6 x^2 \log \left (3 e^2\right )}{e^2 x^2+e^2 \left (8 e x+10 e^2 x^2\right )+e^4 \left (16+40 e x+25 e^2 x^2\right )+\left (2 e^4 x^2+e^4 \left (8 e x+10 e^2 x^2\right )\right ) \log (4)+e^6 x^2 \log ^2(4)+\left (-2 e^4 x^2+e^4 \left (-8 e x-10 e^2 x^2\right )-2 e^6 x^2 \log (4)\right ) \log \left (3 e^2\right )+e^6 x^2 \log ^2\left (3 e^2\right )} \, dx=\frac {e^2 \left (8 e x+x^2+8 e^3 x \left (3+\log \left (\frac {4}{3}\right )\right )+e^4 x^2 \left (3+\log \left (\frac {4}{3}\right )\right )^2+2 e^2 \left (16+x^2 \left (3+\log \left (\frac {4}{3}\right )\right )\right )\right )}{\left (1+e^2 \left (3+\log \left (\frac {4}{3}\right )\right )\right )^2 \left (4 e+x+e^2 x \left (3+\log \left (\frac {4}{3}\right )\right )\right )} \]
Integrate[(E^4*x^2 + E^4*(8*E*x + 5*E^2*x^2) + E^6*x^2*Log[4] - E^6*x^2*Lo g[3*E^2])/(E^2*x^2 + E^2*(8*E*x + 10*E^2*x^2) + E^4*(16 + 40*E*x + 25*E^2* x^2) + (2*E^4*x^2 + E^4*(8*E*x + 10*E^2*x^2))*Log[4] + E^6*x^2*Log[4]^2 + (-2*E^4*x^2 + E^4*(-8*E*x - 10*E^2*x^2) - 2*E^6*x^2*Log[4])*Log[3*E^2] + E ^6*x^2*Log[3*E^2]^2),x]
(E^2*(8*E*x + x^2 + 8*E^3*x*(3 + Log[4/3]) + E^4*x^2*(3 + Log[4/3])^2 + 2* E^2*(16 + x^2*(3 + Log[4/3]))))/((1 + E^2*(3 + Log[4/3]))^2*(4*E + x + E^2 *x*(3 + Log[4/3])))
Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {6, 6, 6, 6, 2007, 2021}
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.
| \(\displaystyle \int \frac {e^4 x^2+e^4 \left (5 e^2 x^2+8 e x\right )-e^6 x^2 \log \left (3 e^2\right )+e^6 x^2 \log (4)}{e^2 x^2+e^2 \left (10 e^2 x^2+8 e x\right )+e^4 \left (25 e^2 x^2+40 e x+16\right )+e^6 x^2 \log ^2\left (3 e^2\right )+e^6 x^2 \log ^2(4)+\log \left (3 e^2\right ) \left (-2 e^4 x^2+e^4 \left (-10 e^2 x^2-8 e x\right )-2 e^6 x^2 \log (4)\right )+\left (2 e^4 x^2+e^4 \left (10 e^2 x^2+8 e x\right )\right ) \log (4)} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {e^4 \left (5 e^2 x^2+8 e x\right )-e^6 x^2 \log \left (3 e^2\right )+x^2 \left (e^4+e^6 \log (4)\right )}{e^2 x^2+e^2 \left (10 e^2 x^2+8 e x\right )+e^4 \left (25 e^2 x^2+40 e x+16\right )+e^6 x^2 \log ^2\left (3 e^2\right )+e^6 x^2 \log ^2(4)+\log \left (3 e^2\right ) \left (-2 e^4 x^2+e^4 \left (-10 e^2 x^2-8 e x\right )-2 e^6 x^2 \log (4)\right )+\left (2 e^4 x^2+e^4 \left (10 e^2 x^2+8 e x\right )\right ) \log (4)}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {e^4 \left (5 e^2 x^2+8 e x\right )+x^2 \left (e^4+e^6 \log (4)-e^6 \log \left (3 e^2\right )\right )}{e^2 x^2+e^2 \left (10 e^2 x^2+8 e x\right )+e^4 \left (25 e^2 x^2+40 e x+16\right )+e^6 x^2 \log ^2\left (3 e^2\right )+e^6 x^2 \log ^2(4)+\log \left (3 e^2\right ) \left (-2 e^4 x^2+e^4 \left (-10 e^2 x^2-8 e x\right )-2 e^6 x^2 \log (4)\right )+\left (2 e^4 x^2+e^4 \left (10 e^2 x^2+8 e x\right )\right ) \log (4)}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {e^4 \left (5 e^2 x^2+8 e x\right )+x^2 \left (e^4+e^6 \log (4)-e^6 \log \left (3 e^2\right )\right )}{e^2 \left (10 e^2 x^2+8 e x\right )+e^4 \left (25 e^2 x^2+40 e x+16\right )+e^6 x^2 \log ^2\left (3 e^2\right )+x^2 \left (e^2+e^6 \log ^2(4)\right )+\log \left (3 e^2\right ) \left (-2 e^4 x^2+e^4 \left (-10 e^2 x^2-8 e x\right )-2 e^6 x^2 \log (4)\right )+\left (2 e^4 x^2+e^4 \left (10 e^2 x^2+8 e x\right )\right ) \log (4)}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {e^4 \left (5 e^2 x^2+8 e x\right )+x^2 \left (e^4+e^6 \log (4)-e^6 \log \left (3 e^2\right )\right )}{e^2 \left (10 e^2 x^2+8 e x\right )+e^4 \left (25 e^2 x^2+40 e x+16\right )+x^2 \left (e^2+e^6 \log ^2(4)+e^6 \log ^2\left (3 e^2\right )\right )+\log \left (3 e^2\right ) \left (-2 e^4 x^2+e^4 \left (-10 e^2 x^2-8 e x\right )-2 e^6 x^2 \log (4)\right )+\left (2 e^4 x^2+e^4 \left (10 e^2 x^2+8 e x\right )\right ) \log (4)}dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {e^4 \left (5 e^2 x^2+8 e x\right )+x^2 \left (e^4+e^6 \log (4)-e^6 \log \left (3 e^2\right )\right )}{\left (e x \left (1+3 e^2+e^2 \log \left (\frac {4}{3}\right )\right )+4 e^2\right )^2}dx\) |
| \(\Big \downarrow \) 2021 |
| \(\displaystyle \frac {e^2 x^2}{x \left (1+e^2 \left (3+\log \left (\frac {4}{3}\right )\right )\right )+4 e}\) |
Int[(E^4*x^2 + E^4*(8*E*x + 5*E^2*x^2) + E^6*x^2*Log[4] - E^6*x^2*Log[3*E^ 2])/(E^2*x^2 + E^2*(8*E*x + 10*E^2*x^2) + E^4*(16 + 40*E*x + 25*E^2*x^2) + (2*E^4*x^2 + E^4*(8*E*x + 10*E^2*x^2))*Log[4] + E^6*x^2*Log[4]^2 + (-2*E^ 4*x^2 + E^4*(-8*E*x - 10*E^2*x^2) - 2*E^6*x^2*Log[4])*Log[3*E^2] + E^6*x^2 *Log[3*E^2]^2),x]
3.23.11.3.1 Defintions of rubi rules used
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[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x ]}, Simp[Coeff[Pp, x, p]*x^(p - q + 1)*(Qq^(m + 1)/((p + m*q + 1)*Coeff[Qq, x, q])), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x, q]*Pp , Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; Free Q[m, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && NeQ[m, -1]
Time = 0.76 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.67
| method | result | size |
| norman | \(\frac {x^{2} {\mathrm e} \,{\mathrm e}^{2}}{2 \,{\mathrm e}^{2} {\mathrm e} \ln \left (2\right ) x -{\mathrm e}^{2} {\mathrm e} \ln \left (3\right ) x +3 x \,{\mathrm e} \,{\mathrm e}^{2}+x \,{\mathrm e}+4 \,{\mathrm e}^{2}}\) | \(45\) |
| gosper | \(\frac {x^{2} {\mathrm e}^{3}}{2 \,{\mathrm e}^{2} {\mathrm e} \ln \left (2\right ) x -{\mathrm e}^{2} {\mathrm e} \ln \left (3 \,{\mathrm e}^{2}\right ) x +5 x \,{\mathrm e} \,{\mathrm e}^{2}+x \,{\mathrm e}+4 \,{\mathrm e}^{2}}\) | \(46\) |
| parallelrisch | \(\frac {x^{2} {\mathrm e}^{2} {\mathrm e}}{2 \,{\mathrm e}^{2} {\mathrm e} \ln \left (2\right ) x -{\mathrm e}^{2} {\mathrm e} \ln \left (3 \,{\mathrm e}^{2}\right ) x +5 x \,{\mathrm e} \,{\mathrm e}^{2}+x \,{\mathrm e}+4 \,{\mathrm e}^{2}}\) | \(48\) |
| risch | \(\frac {x \,{\mathrm e}^{2}}{2 \,{\mathrm e}^{2} \ln \left (2\right )-{\mathrm e}^{2} \ln \left (3\right )+3 \,{\mathrm e}^{2}+1}+\frac {8 \,{\mathrm e}^{4}}{\left (2 \,{\mathrm e}^{2} \ln \left (2\right )-{\mathrm e}^{2} \ln \left (3\right )+3 \,{\mathrm e}^{2}+1\right )^{2} \left (x \,{\mathrm e}^{2} \ln \left (2\right )-\frac {x \,{\mathrm e}^{2} \ln \left (3\right )}{2}+\frac {3 \,{\mathrm e}^{2} x}{2}+2 \,{\mathrm e}+\frac {x}{2}\right )}\) | \(78\) |
| meijerg | \(\frac {64 \,{\mathrm e} \left (\frac {\ln \left (2\right ) {\mathrm e}^{6}}{8}-\frac {{\mathrm e}^{6} \ln \left (3 \,{\mathrm e}^{2}\right )}{16}+\frac {5 \,{\mathrm e}^{6}}{16}+\frac {{\mathrm e}^{4}}{16}\right ) \left (2 \,{\mathrm e}^{2} \ln \left (2\right )-{\mathrm e}^{2} \ln \left (3 \,{\mathrm e}^{2}\right )+5 \,{\mathrm e}^{2}+1\right )^{2} \left (\frac {x \,{\mathrm e}^{-1} \left (2 \,{\mathrm e}^{2} \ln \left (2\right )-{\mathrm e}^{2} \ln \left (3\right )+3 \,{\mathrm e}^{2}+1\right ) \left (6+\frac {3 x \,{\mathrm e}^{-1} \left (2 \,{\mathrm e}^{2} \ln \left (2\right )-{\mathrm e}^{2} \ln \left (3\right )+3 \,{\mathrm e}^{2}+1\right )}{4}\right )}{12+3 x \,{\mathrm e}^{-1} \left (2 \,{\mathrm e}^{2} \ln \left (2\right )-{\mathrm e}^{2} \ln \left (3\right )+3 \,{\mathrm e}^{2}+1\right )}-2 \ln \left (1+\frac {x \,{\mathrm e}^{-1} \left (2 \,{\mathrm e}^{2} \ln \left (2\right )-{\mathrm e}^{2} \ln \left (3\right )+3 \,{\mathrm e}^{2}+1\right )}{4}\right )\right )}{\left (2 \,{\mathrm e}^{2} \ln \left (2\right )-{\mathrm e}^{2} \ln \left (3\right )+3 \,{\mathrm e}^{2}+1\right )^{3} \left (4 \ln \left (2\right )^{2} {\mathrm e}^{6}-4 \ln \left (2\right ) {\mathrm e}^{6} \ln \left (3 \,{\mathrm e}^{2}\right )+{\mathrm e}^{6} \ln \left (3 \,{\mathrm e}^{2}\right )^{2}+4 \,{\mathrm e}^{4} \ln \left (2\right )-2 \,{\mathrm e}^{4} \ln \left (3 \,{\mathrm e}^{2}\right )+20 \ln \left (2\right ) {\mathrm e}^{6}-10 \,{\mathrm e}^{6} \ln \left (3 \,{\mathrm e}^{2}\right )+{\mathrm e}^{2}+10 \,{\mathrm e}^{4}+25 \,{\mathrm e}^{6}\right )}+\frac {8 \,{\mathrm e}^{5} \left (2 \,{\mathrm e}^{2} \ln \left (2\right )-{\mathrm e}^{2} \ln \left (3 \,{\mathrm e}^{2}\right )+5 \,{\mathrm e}^{2}+1\right )^{2} \left (-\frac {x \,{\mathrm e}^{-1} \left (2 \,{\mathrm e}^{2} \ln \left (2\right )-{\mathrm e}^{2} \ln \left (3\right )+3 \,{\mathrm e}^{2}+1\right )}{4 \left (1+\frac {x \,{\mathrm e}^{-1} \left (2 \,{\mathrm e}^{2} \ln \left (2\right )-{\mathrm e}^{2} \ln \left (3\right )+3 \,{\mathrm e}^{2}+1\right )}{4}\right )}+\ln \left (1+\frac {x \,{\mathrm e}^{-1} \left (2 \,{\mathrm e}^{2} \ln \left (2\right )-{\mathrm e}^{2} \ln \left (3\right )+3 \,{\mathrm e}^{2}+1\right )}{4}\right )\right )}{\left (2 \,{\mathrm e}^{2} \ln \left (2\right )-{\mathrm e}^{2} \ln \left (3\right )+3 \,{\mathrm e}^{2}+1\right )^{2} \left (4 \ln \left (2\right )^{2} {\mathrm e}^{6}-4 \ln \left (2\right ) {\mathrm e}^{6} \ln \left (3 \,{\mathrm e}^{2}\right )+{\mathrm e}^{6} \ln \left (3 \,{\mathrm e}^{2}\right )^{2}+4 \,{\mathrm e}^{4} \ln \left (2\right )-2 \,{\mathrm e}^{4} \ln \left (3 \,{\mathrm e}^{2}\right )+20 \ln \left (2\right ) {\mathrm e}^{6}-10 \,{\mathrm e}^{6} \ln \left (3 \,{\mathrm e}^{2}\right )+{\mathrm e}^{2}+10 \,{\mathrm e}^{4}+25 \,{\mathrm e}^{6}\right )}\) | \(445\) |
int((-x^2*exp(1)^2*exp(2)^2*ln(3*exp(2))+2*x^2*exp(1)^2*exp(2)^2*ln(2)+(5* x^2*exp(1)^2+8*x*exp(1))*exp(2)^2+x^2*exp(1)^2*exp(2))/(x^2*exp(1)^2*exp(2 )^2*ln(3*exp(2))^2+(-4*x^2*exp(1)^2*exp(2)^2*ln(2)+(-10*x^2*exp(1)^2-8*x*e xp(1))*exp(2)^2-2*x^2*exp(1)^2*exp(2))*ln(3*exp(2))+4*x^2*exp(1)^2*exp(2)^ 2*ln(2)^2+2*((10*x^2*exp(1)^2+8*x*exp(1))*exp(2)^2+2*x^2*exp(1)^2*exp(2))* ln(2)+(25*x^2*exp(1)^2+40*x*exp(1)+16)*exp(2)^2+(10*x^2*exp(1)^2+8*x*exp(1 ))*exp(2)+x^2*exp(1)^2),x,method=_RETURNVERBOSE)
x^2*exp(1)*exp(2)/(2*exp(2)*exp(1)*ln(2)*x-exp(2)*exp(1)*ln(3)*x+3*x*exp(1 )*exp(2)+x*exp(1)+4*exp(2))
Leaf count of result is larger than twice the leaf count of optimal. 295 vs. \(2 (26) = 52\).
Time = 0.25 (sec) , antiderivative size = 295, normalized size of antiderivative = 10.93 \[ \int \frac {e^4 x^2+e^4 \left (8 e x+5 e^2 x^2\right )+e^6 x^2 \log (4)-e^6 x^2 \log \left (3 e^2\right )}{e^2 x^2+e^2 \left (8 e x+10 e^2 x^2\right )+e^4 \left (16+40 e x+25 e^2 x^2\right )+\left (2 e^4 x^2+e^4 \left (8 e x+10 e^2 x^2\right )\right ) \log (4)+e^6 x^2 \log ^2(4)+\left (-2 e^4 x^2+e^4 \left (-8 e x-10 e^2 x^2\right )-2 e^6 x^2 \log (4)\right ) \log \left (3 e^2\right )+e^6 x^2 \log ^2\left (3 e^2\right )} \, dx=-\frac {x^{2} e^{6} \log \left (3\right )^{2} + 4 \, x^{2} e^{6} \log \left (2\right )^{2} + 9 \, x^{2} e^{6} + x^{2} e^{2} + 12 \, x e^{5} + 2 \, {\left (3 \, x^{2} + 8\right )} e^{4} + 4 \, x e^{3} - 2 \, {\left (2 \, x^{2} e^{6} \log \left (2\right ) + 3 \, x^{2} e^{6} + x^{2} e^{4} + 2 \, x e^{5}\right )} \log \left (3\right ) + 4 \, {\left (3 \, x^{2} e^{6} + x^{2} e^{4} + 2 \, x e^{5}\right )} \log \left (2\right )}{x e^{6} \log \left (3\right )^{3} - 8 \, x e^{6} \log \left (2\right )^{3} - {\left (6 \, x e^{6} \log \left (2\right ) + 9 \, x e^{6} + 3 \, x e^{4} + 4 \, e^{5}\right )} \log \left (3\right )^{2} - 4 \, {\left (9 \, x e^{6} + 3 \, x e^{4} + 4 \, e^{5}\right )} \log \left (2\right )^{2} - 27 \, x e^{6} - 27 \, x e^{4} - 9 \, x e^{2} + {\left (12 \, x e^{6} \log \left (2\right )^{2} + 27 \, x e^{6} + 18 \, x e^{4} + 3 \, x e^{2} + 4 \, {\left (9 \, x e^{6} + 3 \, x e^{4} + 4 \, e^{5}\right )} \log \left (2\right ) + 24 \, e^{5} + 8 \, e^{3}\right )} \log \left (3\right ) - 2 \, {\left (27 \, x e^{6} + 18 \, x e^{4} + 3 \, x e^{2} + 24 \, e^{5} + 8 \, e^{3}\right )} \log \left (2\right ) - x - 36 \, e^{5} - 24 \, e^{3} - 4 \, e} \]
integrate((-x^2*exp(1)^2*exp(2)^2*log(3*exp(2))+2*x^2*exp(1)^2*exp(2)^2*lo g(2)+(5*x^2*exp(1)^2+8*x*exp(1))*exp(2)^2+x^2*exp(1)^2*exp(2))/(x^2*exp(1) ^2*exp(2)^2*log(3*exp(2))^2+(-4*x^2*exp(1)^2*exp(2)^2*log(2)+(-10*x^2*exp( 1)^2-8*x*exp(1))*exp(2)^2-2*x^2*exp(1)^2*exp(2))*log(3*exp(2))+4*x^2*exp(1 )^2*exp(2)^2*log(2)^2+2*((10*x^2*exp(1)^2+8*x*exp(1))*exp(2)^2+2*x^2*exp(1 )^2*exp(2))*log(2)+(25*x^2*exp(1)^2+40*x*exp(1)+16)*exp(2)^2+(10*x^2*exp(1 )^2+8*x*exp(1))*exp(2)+x^2*exp(1)^2),x, algorithm=\
-(x^2*e^6*log(3)^2 + 4*x^2*e^6*log(2)^2 + 9*x^2*e^6 + x^2*e^2 + 12*x*e^5 + 2*(3*x^2 + 8)*e^4 + 4*x*e^3 - 2*(2*x^2*e^6*log(2) + 3*x^2*e^6 + x^2*e^4 + 2*x*e^5)*log(3) + 4*(3*x^2*e^6 + x^2*e^4 + 2*x*e^5)*log(2))/(x*e^6*log(3) ^3 - 8*x*e^6*log(2)^3 - (6*x*e^6*log(2) + 9*x*e^6 + 3*x*e^4 + 4*e^5)*log(3 )^2 - 4*(9*x*e^6 + 3*x*e^4 + 4*e^5)*log(2)^2 - 27*x*e^6 - 27*x*e^4 - 9*x*e ^2 + (12*x*e^6*log(2)^2 + 27*x*e^6 + 18*x*e^4 + 3*x*e^2 + 4*(9*x*e^6 + 3*x *e^4 + 4*e^5)*log(2) + 24*e^5 + 8*e^3)*log(3) - 2*(27*x*e^6 + 18*x*e^4 + 3 *x*e^2 + 24*e^5 + 8*e^3)*log(2) - x - 36*e^5 - 24*e^3 - 4*e)
Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (26) = 52\).
Time = 0.52 (sec) , antiderivative size = 292, normalized size of antiderivative = 10.81 \[ \int \frac {e^4 x^2+e^4 \left (8 e x+5 e^2 x^2\right )+e^6 x^2 \log (4)-e^6 x^2 \log \left (3 e^2\right )}{e^2 x^2+e^2 \left (8 e x+10 e^2 x^2\right )+e^4 \left (16+40 e x+25 e^2 x^2\right )+\left (2 e^4 x^2+e^4 \left (8 e x+10 e^2 x^2\right )\right ) \log (4)+e^6 x^2 \log ^2(4)+\left (-2 e^4 x^2+e^4 \left (-8 e x-10 e^2 x^2\right )-2 e^6 x^2 \log (4)\right ) \log \left (3 e^2\right )+e^6 x^2 \log ^2\left (3 e^2\right )} \, dx=\frac {x e^{2}}{- e^{2} \log {\left (3 \right )} + 1 + 2 e^{2} \log {\left (2 \right )} + 3 e^{2}} + \frac {16 e^{4}}{x \left (- 27 e^{6} \log {\left (3 \right )} - 36 e^{6} \log {\left (2 \right )} \log {\left (3 \right )} - 12 e^{6} \log {\left (2 \right )}^{2} \log {\left (3 \right )} - 18 e^{4} \log {\left (3 \right )} - e^{6} \log {\left (3 \right )}^{3} - 12 e^{4} \log {\left (2 \right )} \log {\left (3 \right )} - 3 e^{2} \log {\left (3 \right )} + 1 + 6 e^{2} \log {\left (2 \right )} + 9 e^{2} + 3 e^{4} \log {\left (3 \right )}^{2} + 12 e^{4} \log {\left (2 \right )}^{2} + 8 e^{6} \log {\left (2 \right )}^{3} + 36 e^{4} \log {\left (2 \right )} + 27 e^{4} + 6 e^{6} \log {\left (2 \right )} \log {\left (3 \right )}^{2} + 9 e^{6} \log {\left (3 \right )}^{2} + 36 e^{6} \log {\left (2 \right )}^{2} + 27 e^{6} + 54 e^{6} \log {\left (2 \right )}\right ) - 24 e^{5} \log {\left (3 \right )} - 16 e^{5} \log {\left (2 \right )} \log {\left (3 \right )} - 8 e^{3} \log {\left (3 \right )} + 4 e + 16 e^{3} \log {\left (2 \right )} + 24 e^{3} + 4 e^{5} \log {\left (3 \right )}^{2} + 16 e^{5} \log {\left (2 \right )}^{2} + 48 e^{5} \log {\left (2 \right )} + 36 e^{5}} \]
integrate((-x**2*exp(1)**2*exp(2)**2*ln(3*exp(2))+2*x**2*exp(1)**2*exp(2)* *2*ln(2)+(5*x**2*exp(1)**2+8*x*exp(1))*exp(2)**2+x**2*exp(1)**2*exp(2))/(x **2*exp(1)**2*exp(2)**2*ln(3*exp(2))**2+(-4*x**2*exp(1)**2*exp(2)**2*ln(2) +(-10*x**2*exp(1)**2-8*x*exp(1))*exp(2)**2-2*x**2*exp(1)**2*exp(2))*ln(3*e xp(2))+4*x**2*exp(1)**2*exp(2)**2*ln(2)**2+2*((10*x**2*exp(1)**2+8*x*exp(1 ))*exp(2)**2+2*x**2*exp(1)**2*exp(2))*ln(2)+(25*x**2*exp(1)**2+40*x*exp(1) +16)*exp(2)**2+(10*x**2*exp(1)**2+8*x*exp(1))*exp(2)+x**2*exp(1)**2),x)
x*exp(2)/(-exp(2)*log(3) + 1 + 2*exp(2)*log(2) + 3*exp(2)) + 16*exp(4)/(x* (-27*exp(6)*log(3) - 36*exp(6)*log(2)*log(3) - 12*exp(6)*log(2)**2*log(3) - 18*exp(4)*log(3) - exp(6)*log(3)**3 - 12*exp(4)*log(2)*log(3) - 3*exp(2) *log(3) + 1 + 6*exp(2)*log(2) + 9*exp(2) + 3*exp(4)*log(3)**2 + 12*exp(4)* log(2)**2 + 8*exp(6)*log(2)**3 + 36*exp(4)*log(2) + 27*exp(4) + 6*exp(6)*l og(2)*log(3)**2 + 9*exp(6)*log(3)**2 + 36*exp(6)*log(2)**2 + 27*exp(6) + 5 4*exp(6)*log(2)) - 24*exp(5)*log(3) - 16*exp(5)*log(2)*log(3) - 8*exp(3)*l og(3) + 4*E + 16*exp(3)*log(2) + 24*exp(3) + 4*exp(5)*log(3)**2 + 16*exp(5 )*log(2)**2 + 48*exp(5)*log(2) + 36*exp(5))
Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (26) = 52\).
Time = 0.19 (sec) , antiderivative size = 219, normalized size of antiderivative = 8.11 \[ \int \frac {e^4 x^2+e^4 \left (8 e x+5 e^2 x^2\right )+e^6 x^2 \log (4)-e^6 x^2 \log \left (3 e^2\right )}{e^2 x^2+e^2 \left (8 e x+10 e^2 x^2\right )+e^4 \left (16+40 e x+25 e^2 x^2\right )+\left (2 e^4 x^2+e^4 \left (8 e x+10 e^2 x^2\right )\right ) \log (4)+e^6 x^2 \log ^2(4)+\left (-2 e^4 x^2+e^4 \left (-8 e x-10 e^2 x^2\right )-2 e^6 x^2 \log (4)\right ) \log \left (3 e^2\right )+e^6 x^2 \log ^2\left (3 e^2\right )} \, dx=\frac {x e^{2}}{2 \, e^{2} \log \left (2\right ) - e^{2} \log \left (3 \, e^{2}\right ) + 5 \, e^{2} + 1} + \frac {16 \, e^{4}}{16 \, e^{5} \log \left (2\right )^{2} + 4 \, e^{5} \log \left (3 \, e^{2}\right )^{2} + {\left (8 \, e^{6} \log \left (2\right )^{3} - e^{6} \log \left (3 \, e^{2}\right )^{3} + 12 \, {\left (5 \, e^{6} + e^{4}\right )} \log \left (2\right )^{2} + 3 \, {\left (2 \, e^{6} \log \left (2\right ) + 5 \, e^{6} + e^{4}\right )} \log \left (3 \, e^{2}\right )^{2} + 6 \, {\left (25 \, e^{6} + 10 \, e^{4} + e^{2}\right )} \log \left (2\right ) - 3 \, {\left (4 \, e^{6} \log \left (2\right )^{2} + 4 \, {\left (5 \, e^{6} + e^{4}\right )} \log \left (2\right ) + 25 \, e^{6} + 10 \, e^{4} + e^{2}\right )} \log \left (3 \, e^{2}\right ) + 125 \, e^{6} + 75 \, e^{4} + 15 \, e^{2} + 1\right )} x + 16 \, {\left (5 \, e^{5} + e^{3}\right )} \log \left (2\right ) - 8 \, {\left (2 \, e^{5} \log \left (2\right ) + 5 \, e^{5} + e^{3}\right )} \log \left (3 \, e^{2}\right ) + 100 \, e^{5} + 40 \, e^{3} + 4 \, e} \]
integrate((-x^2*exp(1)^2*exp(2)^2*log(3*exp(2))+2*x^2*exp(1)^2*exp(2)^2*lo g(2)+(5*x^2*exp(1)^2+8*x*exp(1))*exp(2)^2+x^2*exp(1)^2*exp(2))/(x^2*exp(1) ^2*exp(2)^2*log(3*exp(2))^2+(-4*x^2*exp(1)^2*exp(2)^2*log(2)+(-10*x^2*exp( 1)^2-8*x*exp(1))*exp(2)^2-2*x^2*exp(1)^2*exp(2))*log(3*exp(2))+4*x^2*exp(1 )^2*exp(2)^2*log(2)^2+2*((10*x^2*exp(1)^2+8*x*exp(1))*exp(2)^2+2*x^2*exp(1 )^2*exp(2))*log(2)+(25*x^2*exp(1)^2+40*x*exp(1)+16)*exp(2)^2+(10*x^2*exp(1 )^2+8*x*exp(1))*exp(2)+x^2*exp(1)^2),x, algorithm=\
x*e^2/(2*e^2*log(2) - e^2*log(3*e^2) + 5*e^2 + 1) + 16*e^4/(16*e^5*log(2)^ 2 + 4*e^5*log(3*e^2)^2 + (8*e^6*log(2)^3 - e^6*log(3*e^2)^3 + 12*(5*e^6 + e^4)*log(2)^2 + 3*(2*e^6*log(2) + 5*e^6 + e^4)*log(3*e^2)^2 + 6*(25*e^6 + 10*e^4 + e^2)*log(2) - 3*(4*e^6*log(2)^2 + 4*(5*e^6 + e^4)*log(2) + 25*e^6 + 10*e^4 + e^2)*log(3*e^2) + 125*e^6 + 75*e^4 + 15*e^2 + 1)*x + 16*(5*e^5 + e^3)*log(2) - 8*(2*e^5*log(2) + 5*e^5 + e^3)*log(3*e^2) + 100*e^5 + 40* e^3 + 4*e)
Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (26) = 52\).
Time = 0.29 (sec) , antiderivative size = 205, normalized size of antiderivative = 7.59 \[ \int \frac {e^4 x^2+e^4 \left (8 e x+5 e^2 x^2\right )+e^6 x^2 \log (4)-e^6 x^2 \log \left (3 e^2\right )}{e^2 x^2+e^2 \left (8 e x+10 e^2 x^2\right )+e^4 \left (16+40 e x+25 e^2 x^2\right )+\left (2 e^4 x^2+e^4 \left (8 e x+10 e^2 x^2\right )\right ) \log (4)+e^6 x^2 \log ^2(4)+\left (-2 e^4 x^2+e^4 \left (-8 e x-10 e^2 x^2\right )-2 e^6 x^2 \log (4)\right ) \log \left (3 e^2\right )+e^6 x^2 \log ^2\left (3 e^2\right )} \, dx=\frac {2 \, x e^{4} \log \left (2\right ) - x e^{4} \log \left (3 \, e^{2}\right ) + 5 \, x e^{4} + x e^{2}}{4 \, e^{4} \log \left (2\right )^{2} - 4 \, e^{4} \log \left (2\right ) \log \left (3 \, e^{2}\right ) + e^{4} \log \left (3 \, e^{2}\right )^{2} + 20 \, e^{4} \log \left (2\right ) + 4 \, e^{2} \log \left (2\right ) - 10 \, e^{4} \log \left (3 \, e^{2}\right ) - 2 \, e^{2} \log \left (3 \, e^{2}\right ) + 25 \, e^{4} + 10 \, e^{2} + 1} + \frac {16 \, e^{4}}{{\left (2 \, x e^{2} \log \left (2\right ) - x e^{2} \log \left (3 \, e^{2}\right ) + 5 \, x e^{2} + x + 4 \, e\right )} {\left (4 \, e^{4} \log \left (2\right )^{2} - 4 \, e^{4} \log \left (2\right ) \log \left (3 \, e^{2}\right ) + e^{4} \log \left (3 \, e^{2}\right )^{2} + 20 \, e^{4} \log \left (2\right ) + 4 \, e^{2} \log \left (2\right ) - 10 \, e^{4} \log \left (3 \, e^{2}\right ) - 2 \, e^{2} \log \left (3 \, e^{2}\right ) + 25 \, e^{4} + 10 \, e^{2} + 1\right )}} \]
integrate((-x^2*exp(1)^2*exp(2)^2*log(3*exp(2))+2*x^2*exp(1)^2*exp(2)^2*lo g(2)+(5*x^2*exp(1)^2+8*x*exp(1))*exp(2)^2+x^2*exp(1)^2*exp(2))/(x^2*exp(1) ^2*exp(2)^2*log(3*exp(2))^2+(-4*x^2*exp(1)^2*exp(2)^2*log(2)+(-10*x^2*exp( 1)^2-8*x*exp(1))*exp(2)^2-2*x^2*exp(1)^2*exp(2))*log(3*exp(2))+4*x^2*exp(1 )^2*exp(2)^2*log(2)^2+2*((10*x^2*exp(1)^2+8*x*exp(1))*exp(2)^2+2*x^2*exp(1 )^2*exp(2))*log(2)+(25*x^2*exp(1)^2+40*x*exp(1)+16)*exp(2)^2+(10*x^2*exp(1 )^2+8*x*exp(1))*exp(2)+x^2*exp(1)^2),x, algorithm=\
(2*x*e^4*log(2) - x*e^4*log(3*e^2) + 5*x*e^4 + x*e^2)/(4*e^4*log(2)^2 - 4* e^4*log(2)*log(3*e^2) + e^4*log(3*e^2)^2 + 20*e^4*log(2) + 4*e^2*log(2) - 10*e^4*log(3*e^2) - 2*e^2*log(3*e^2) + 25*e^4 + 10*e^2 + 1) + 16*e^4/((2*x *e^2*log(2) - x*e^2*log(3*e^2) + 5*x*e^2 + x + 4*e)*(4*e^4*log(2)^2 - 4*e^ 4*log(2)*log(3*e^2) + e^4*log(3*e^2)^2 + 20*e^4*log(2) + 4*e^2*log(2) - 10 *e^4*log(3*e^2) - 2*e^2*log(3*e^2) + 25*e^4 + 10*e^2 + 1))
Timed out. \[ \int \frac {e^4 x^2+e^4 \left (8 e x+5 e^2 x^2\right )+e^6 x^2 \log (4)-e^6 x^2 \log \left (3 e^2\right )}{e^2 x^2+e^2 \left (8 e x+10 e^2 x^2\right )+e^4 \left (16+40 e x+25 e^2 x^2\right )+\left (2 e^4 x^2+e^4 \left (8 e x+10 e^2 x^2\right )\right ) \log (4)+e^6 x^2 \log ^2(4)+\left (-2 e^4 x^2+e^4 \left (-8 e x-10 e^2 x^2\right )-2 e^6 x^2 \log (4)\right ) \log \left (3 e^2\right )+e^6 x^2 \log ^2\left (3 e^2\right )} \, dx=\text {Hanged} \]
int((x^2*exp(4) + exp(4)*(8*x*exp(1) + 5*x^2*exp(2)) + 2*x^2*exp(6)*log(2) - x^2*log(3*exp(2))*exp(6))/(x^2*exp(2) - log(3*exp(2))*(2*x^2*exp(4) + e xp(4)*(8*x*exp(1) + 10*x^2*exp(2)) + 4*x^2*exp(6)*log(2)) + 2*log(2)*(2*x^ 2*exp(4) + exp(4)*(8*x*exp(1) + 10*x^2*exp(2))) + exp(2)*(8*x*exp(1) + 10* x^2*exp(2)) + exp(4)*(40*x*exp(1) + 25*x^2*exp(2) + 16) + x^2*log(3*exp(2) )^2*exp(6) + 4*x^2*exp(6)*log(2)^2),x)