Integrand size = 134, antiderivative size = 26 \[ \int \frac {4+5 e^{12+2 x^2-4 e^3 x^2+2 e^6 x^2}+5 x^2+e^{6+x^2-2 e^3 x^2+e^6 x^2} \left (18 x-16 e^3 x+8 e^6 x\right )}{5 e^{12+2 x^2-4 e^3 x^2+2 e^6 x^2}+10 e^{6+x^2-2 e^3 x^2+e^6 x^2} x+5 x^2} \, dx=8+e+x-\frac {4}{5 \left (e^{6+\left (x-e^3 x\right )^2}+x\right )} \]
Leaf count is larger than twice the leaf count of optimal. \(321\) vs. \(2(26)=52\).
Time = 31.41 (sec) , antiderivative size = 321, normalized size of antiderivative = 12.35 \[ \int \frac {4+5 e^{12+2 x^2-4 e^3 x^2+2 e^6 x^2}+5 x^2+e^{6+x^2-2 e^3 x^2+e^6 x^2} \left (18 x-16 e^3 x+8 e^6 x\right )}{5 e^{12+2 x^2-4 e^3 x^2+2 e^6 x^2}+10 e^{6+x^2-2 e^3 x^2+e^6 x^2} x+5 x^2} \, dx=-\frac {\left (9-8 e^3+4 e^6\right ) x \left (-3+2 x^2-20 e^9 x^2+2 e^{12} x^2+e^3 \left (8-20 x^2\right )+e^6 \left (-3+36 x^2\right )\right )}{10 \left (-1+e^3\right )^2 \left (1-2 \left (-1+e^3\right )^2 x^2\right )^2}+\frac {1}{5} \left (5 x-\frac {\left (9-8 e^3+13 e^6-8 e^9+4 e^{12}\right ) x}{\left (-1+e^3\right )^2 \left (1-2 \left (-1+e^3\right )^2 x^2\right )^2}-\frac {3 \left (9-8 e^3+13 e^6-8 e^9+4 e^{12}\right ) x}{2 \left (-1+e^3\right )^2 \left (-1+2 \left (-1+e^3\right )^2 x^2\right )}+\frac {2 e^{-2 e^3 x^2} \left (9 e^{6+\left (1+e^3\right )^2 x^2} x-8 e^{9+\left (1+e^3\right )^2 x^2} x+4 e^{12+\left (1+e^3\right )^2 x^2} x+e^{4 e^3 x^2} \left (2+5 x^2\right )\right )}{\left (e^{6+\left (1+e^6\right ) x^2}+e^{2 e^3 x^2} x\right ) \left (-1+2 \left (-1+e^3\right )^2 x^2\right )}\right ) \]
Integrate[(4 + 5*E^(12 + 2*x^2 - 4*E^3*x^2 + 2*E^6*x^2) + 5*x^2 + E^(6 + x ^2 - 2*E^3*x^2 + E^6*x^2)*(18*x - 16*E^3*x + 8*E^6*x))/(5*E^(12 + 2*x^2 - 4*E^3*x^2 + 2*E^6*x^2) + 10*E^(6 + x^2 - 2*E^3*x^2 + E^6*x^2)*x + 5*x^2),x ]
-1/10*((9 - 8*E^3 + 4*E^6)*x*(-3 + 2*x^2 - 20*E^9*x^2 + 2*E^12*x^2 + E^3*( 8 - 20*x^2) + E^6*(-3 + 36*x^2)))/((-1 + E^3)^2*(1 - 2*(-1 + E^3)^2*x^2)^2 ) + (5*x - ((9 - 8*E^3 + 13*E^6 - 8*E^9 + 4*E^12)*x)/((-1 + E^3)^2*(1 - 2* (-1 + E^3)^2*x^2)^2) - (3*(9 - 8*E^3 + 13*E^6 - 8*E^9 + 4*E^12)*x)/(2*(-1 + E^3)^2*(-1 + 2*(-1 + E^3)^2*x^2)) + (2*(9*E^(6 + (1 + E^3)^2*x^2)*x - 8* E^(9 + (1 + E^3)^2*x^2)*x + 4*E^(12 + (1 + E^3)^2*x^2)*x + E^(4*E^3*x^2)*( 2 + 5*x^2)))/(E^(2*E^3*x^2)*(E^(6 + (1 + E^6)*x^2) + E^(2*E^3*x^2)*x)*(-1 + 2*(-1 + E^3)^2*x^2)))/5
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.
\(\displaystyle \int \frac {5 x^2+5 e^{2 e^6 x^2-4 e^3 x^2+2 x^2+12}+e^{e^6 x^2-2 e^3 x^2+x^2+6} \left (8 e^6 x-16 e^3 x+18 x\right )+4}{5 x^2+10 e^{e^6 x^2-2 e^3 x^2+x^2+6} x+5 e^{2 e^6 x^2-4 e^3 x^2+2 x^2+12}} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{4 e^3 x^2} \left (5 x^2+5 e^{2 e^6 x^2-4 e^3 x^2+2 x^2+12}+e^{e^6 x^2-2 e^3 x^2+x^2+6} \left (8 e^6 x-16 e^3 x+18 x\right )+4\right )}{5 \left (e^{2 e^3 x^2} x+e^{\left (1+e^6\right ) x^2+6}\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{5} \int \frac {e^{4 e^3 x^2} \left (5 x^2+2 e^{e^6 x^2-2 e^3 x^2+x^2+6} \left (9-8 e^3+4 e^6\right ) x+5 e^{2 e^6 x^2-4 e^3 x^2+2 x^2+12}+4\right )}{\left (e^{2 e^3 x^2} x+e^{\left (1+e^6\right ) x^2+6}\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{5} \int \left (\frac {5 e^{4 e^3 x^2} x^2}{\left (e^{2 e^3 x^2} x+e^{\left (1+e^6\right ) x^2+6}\right )^2}+\frac {2 e^{\left (-1+e^3\right )^2 x^2+4 e^3 x^2+6} \left (9-8 e^3+4 e^6\right ) x}{\left (e^{2 e^3 x^2} x+e^{\left (1+e^6\right ) x^2+6}\right )^2}+\frac {4 e^{4 e^3 x^2}}{\left (e^{2 e^3 x^2} x+e^{\left (1+e^6\right ) x^2+6}\right )^2}+\frac {5 e^{2 \left (-1+e^3\right )^2 x^2+4 e^3 x^2+12}}{\left (e^{2 e^3 x^2} x+e^{\left (1+e^6\right ) x^2+6}\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{5} \left (4 \int \frac {e^{4 e^3 x^2}}{\left (e^{2 e^3 x^2} x+e^{\left (1+e^6\right ) x^2+6}\right )^2}dx+5 \int \frac {e^{2 \left (1+e^6\right ) x^2+12}}{\left (e^{2 e^3 x^2} x+e^{\left (1+e^6\right ) x^2+6}\right )^2}dx+2 \left (9-8 e^3+4 e^6\right ) \int \frac {e^{\left (1+e^3\right )^2 x^2+6} x}{\left (e^{2 e^3 x^2} x+e^{\left (1+e^6\right ) x^2+6}\right )^2}dx+5 \int \frac {e^{4 e^3 x^2} x^2}{\left (e^{2 e^3 x^2} x+e^{\left (1+e^6\right ) x^2+6}\right )^2}dx\right )\) |
Int[(4 + 5*E^(12 + 2*x^2 - 4*E^3*x^2 + 2*E^6*x^2) + 5*x^2 + E^(6 + x^2 - 2 *E^3*x^2 + E^6*x^2)*(18*x - 16*E^3*x + 8*E^6*x))/(5*E^(12 + 2*x^2 - 4*E^3* x^2 + 2*E^6*x^2) + 10*E^(6 + x^2 - 2*E^3*x^2 + E^6*x^2)*x + 5*x^2),x]
3.21.98.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.21 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08
method | result | size |
risch | \(x -\frac {4}{5 \left ({\mathrm e}^{x^{2} {\mathrm e}^{6}-2 x^{2} {\mathrm e}^{3}+x^{2}+6}+x \right )}\) | \(28\) |
norman | \(\frac {-\frac {4}{5}+x^{2}+x \,{\mathrm e}^{x^{2} {\mathrm e}^{6}-2 x^{2} {\mathrm e}^{3}+x^{2}+6}}{{\mathrm e}^{x^{2} {\mathrm e}^{6}-2 x^{2} {\mathrm e}^{3}+x^{2}+6}+x}\) | \(55\) |
parallelrisch | \(\frac {5 x \,{\mathrm e}^{x^{2} {\mathrm e}^{6}-2 x^{2} {\mathrm e}^{3}+x^{2}+6}+5 x^{2}-4}{5 \,{\mathrm e}^{x^{2} {\mathrm e}^{6}-2 x^{2} {\mathrm e}^{3}+x^{2}+6}+5 x}\) | \(59\) |
int((5*exp(x^2*exp(3)^2-2*x^2*exp(3)+x^2+6)^2+(8*x*exp(3)^2-16*x*exp(3)+18 *x)*exp(x^2*exp(3)^2-2*x^2*exp(3)+x^2+6)+5*x^2+4)/(5*exp(x^2*exp(3)^2-2*x^ 2*exp(3)+x^2+6)^2+10*x*exp(x^2*exp(3)^2-2*x^2*exp(3)+x^2+6)+5*x^2),x,metho d=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (24) = 48\).
Time = 0.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.08 \[ \int \frac {4+5 e^{12+2 x^2-4 e^3 x^2+2 e^6 x^2}+5 x^2+e^{6+x^2-2 e^3 x^2+e^6 x^2} \left (18 x-16 e^3 x+8 e^6 x\right )}{5 e^{12+2 x^2-4 e^3 x^2+2 e^6 x^2}+10 e^{6+x^2-2 e^3 x^2+e^6 x^2} x+5 x^2} \, dx=\frac {5 \, x^{2} + 5 \, x e^{\left (x^{2} e^{6} - 2 \, x^{2} e^{3} + x^{2} + 6\right )} - 4}{5 \, {\left (x + e^{\left (x^{2} e^{6} - 2 \, x^{2} e^{3} + x^{2} + 6\right )}\right )}} \]
integrate((5*exp(x^2*exp(3)^2-2*x^2*exp(3)+x^2+6)^2+(8*x*exp(3)^2-16*x*exp (3)+18*x)*exp(x^2*exp(3)^2-2*x^2*exp(3)+x^2+6)+5*x^2+4)/(5*exp(x^2*exp(3)^ 2-2*x^2*exp(3)+x^2+6)^2+10*x*exp(x^2*exp(3)^2-2*x^2*exp(3)+x^2+6)+5*x^2),x , algorithm=\
1/5*(5*x^2 + 5*x*e^(x^2*e^6 - 2*x^2*e^3 + x^2 + 6) - 4)/(x + e^(x^2*e^6 - 2*x^2*e^3 + x^2 + 6))
Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {4+5 e^{12+2 x^2-4 e^3 x^2+2 e^6 x^2}+5 x^2+e^{6+x^2-2 e^3 x^2+e^6 x^2} \left (18 x-16 e^3 x+8 e^6 x\right )}{5 e^{12+2 x^2-4 e^3 x^2+2 e^6 x^2}+10 e^{6+x^2-2 e^3 x^2+e^6 x^2} x+5 x^2} \, dx=x - \frac {4}{5 x + 5 e^{- 2 x^{2} e^{3} + x^{2} + x^{2} e^{6} + 6}} \]
integrate((5*exp(x**2*exp(3)**2-2*x**2*exp(3)+x**2+6)**2+(8*x*exp(3)**2-16 *x*exp(3)+18*x)*exp(x**2*exp(3)**2-2*x**2*exp(3)+x**2+6)+5*x**2+4)/(5*exp( x**2*exp(3)**2-2*x**2*exp(3)+x**2+6)**2+10*x*exp(x**2*exp(3)**2-2*x**2*exp (3)+x**2+6)+5*x**2),x)
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (24) = 48\).
Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.27 \[ \int \frac {4+5 e^{12+2 x^2-4 e^3 x^2+2 e^6 x^2}+5 x^2+e^{6+x^2-2 e^3 x^2+e^6 x^2} \left (18 x-16 e^3 x+8 e^6 x\right )}{5 e^{12+2 x^2-4 e^3 x^2+2 e^6 x^2}+10 e^{6+x^2-2 e^3 x^2+e^6 x^2} x+5 x^2} \, dx=\frac {5 \, x e^{\left (x^{2} e^{6} + x^{2} + 6\right )} + {\left (5 \, x^{2} - 4\right )} e^{\left (2 \, x^{2} e^{3}\right )}}{5 \, {\left (x e^{\left (2 \, x^{2} e^{3}\right )} + e^{\left (x^{2} e^{6} + x^{2} + 6\right )}\right )}} \]
integrate((5*exp(x^2*exp(3)^2-2*x^2*exp(3)+x^2+6)^2+(8*x*exp(3)^2-16*x*exp (3)+18*x)*exp(x^2*exp(3)^2-2*x^2*exp(3)+x^2+6)+5*x^2+4)/(5*exp(x^2*exp(3)^ 2-2*x^2*exp(3)+x^2+6)^2+10*x*exp(x^2*exp(3)^2-2*x^2*exp(3)+x^2+6)+5*x^2),x , algorithm=\
1/5*(5*x*e^(x^2*e^6 + x^2 + 6) + (5*x^2 - 4)*e^(2*x^2*e^3))/(x*e^(2*x^2*e^ 3) + e^(x^2*e^6 + x^2 + 6))
Exception generated. \[ \int \frac {4+5 e^{12+2 x^2-4 e^3 x^2+2 e^6 x^2}+5 x^2+e^{6+x^2-2 e^3 x^2+e^6 x^2} \left (18 x-16 e^3 x+8 e^6 x\right )}{5 e^{12+2 x^2-4 e^3 x^2+2 e^6 x^2}+10 e^{6+x^2-2 e^3 x^2+e^6 x^2} x+5 x^2} \, dx=\text {Exception raised: TypeError} \]
integrate((5*exp(x^2*exp(3)^2-2*x^2*exp(3)+x^2+6)^2+(8*x*exp(3)^2-16*x*exp (3)+18*x)*exp(x^2*exp(3)^2-2*x^2*exp(3)+x^2+6)+5*x^2+4)/(5*exp(x^2*exp(3)^ 2-2*x^2*exp(3)+x^2+6)^2+10*x*exp(x^2*exp(3)^2-2*x^2*exp(3)+x^2+6)+5*x^2),x , algorithm=\
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Time = 13.81 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27 \[ \int \frac {4+5 e^{12+2 x^2-4 e^3 x^2+2 e^6 x^2}+5 x^2+e^{6+x^2-2 e^3 x^2+e^6 x^2} \left (18 x-16 e^3 x+8 e^6 x\right )}{5 e^{12+2 x^2-4 e^3 x^2+2 e^6 x^2}+10 e^{6+x^2-2 e^3 x^2+e^6 x^2} x+5 x^2} \, dx=x-\frac {4}{5\,x+5\,{\mathrm {e}}^{-2\,x^2\,{\mathrm {e}}^3}\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^6}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^6} \]