Integrand size = 59, antiderivative size = 23 \[ \int \frac {2+4 x+e^x \left (-1-x+x^2\right )+e^{2 e^{25} x} \left (1+2 x+e^{25} \left (-2 x-2 x^2\right )\right )}{x^2+2 x^3+x^4} \, dx=\frac {-2+e^x-e^{2 e^{25} x}}{x+x^2} \]
Time = 1.51 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {2+4 x+e^x \left (-1-x+x^2\right )+e^{2 e^{25} x} \left (1+2 x+e^{25} \left (-2 x-2 x^2\right )\right )}{x^2+2 x^3+x^4} \, dx=-\frac {2-e^x+e^{2 e^{25} x}}{x+x^2} \]
Integrate[(2 + 4*x + E^x*(-1 - x + x^2) + E^(2*E^25*x)*(1 + 2*x + E^25*(-2 *x - 2*x^2)))/(x^2 + 2*x^3 + x^4),x]
Leaf count is larger than twice the leaf count of optimal. \(55\) vs. \(2(23)=46\).
Time = 1.16 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.39, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {2026, 2007, 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.
\(\displaystyle \int \frac {e^x \left (x^2-x-1\right )+e^{2 e^{25} x} \left (e^{25} \left (-2 x^2-2 x\right )+2 x+1\right )+4 x+2}{x^4+2 x^3+x^2} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {e^x \left (x^2-x-1\right )+e^{2 e^{25} x} \left (e^{25} \left (-2 x^2-2 x\right )+2 x+1\right )+4 x+2}{x^2 \left (x^2+2 x+1\right )}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {e^x \left (x^2-x-1\right )+e^{2 e^{25} x} \left (e^{25} \left (-2 x^2-2 x\right )+2 x+1\right )+4 x+2}{x^2 (x+1)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e^{2 e^{25} x} \left (-2 e^{25} x^2+2 \left (1-e^{25}\right ) x+1\right )}{x^2 (x+1)^2}+\frac {e^x x^2-e^x x+4 x-e^x+2}{x^2 (x+1)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {e^x}{x+1}+\frac {e^{2 e^{25} x}}{x+1}+\frac {e^x}{x}-\frac {e^{2 e^{25} x}}{x}-\frac {2}{(x+1) x}\) |
Int[(2 + 4*x + E^x*(-1 - x + x^2) + E^(2*E^25*x)*(1 + 2*x + E^25*(-2*x - 2 *x^2)))/(x^2 + 2*x^3 + x^4),x]
3.4.17.3.1 Defintions of rubi rules used
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[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00
method | result | size |
norman | \(\frac {{\mathrm e}^{x}-{\mathrm e}^{2 x \,{\mathrm e}^{25}}-2}{\left (1+x \right ) x}\) | \(23\) |
parallelrisch | \(\frac {{\mathrm e}^{x}-{\mathrm e}^{2 x \,{\mathrm e}^{25}}-2}{\left (1+x \right ) x}\) | \(23\) |
risch | \(-\frac {2}{x \left (1+x \right )}+\frac {{\mathrm e}^{x}}{x \left (1+x \right )}-\frac {{\mathrm e}^{2 x \,{\mathrm e}^{25}}}{x \left (1+x \right )}\) | \(39\) |
parts | \(\frac {{\mathrm e}^{x}}{x}-\frac {{\mathrm e}^{x}}{1+x}+\frac {2}{1+x}-\frac {2}{x}-{\mathrm e}^{25} \left (\frac {{\mathrm e}^{2 x \,{\mathrm e}^{25}} \left (2 x \,{\mathrm e}^{25}+{\mathrm e}^{25}\right ) {\mathrm e}^{-25}}{\left (x \,{\mathrm e}^{25}+{\mathrm e}^{25}\right ) x}+2 \left ({\mathrm e}^{25}-1\right ) {\mathrm e}^{-25} \operatorname {Ei}_{1}\left (-2 x \,{\mathrm e}^{25}\right )+2 \left ({\mathrm e}^{25}+1\right ) {\mathrm e}^{-25} {\mathrm e}^{-2 \,{\mathrm e}^{25}} \operatorname {Ei}_{1}\left (-2 x \,{\mathrm e}^{25}-2 \,{\mathrm e}^{25}\right )+\frac {2 \,{\mathrm e}^{2 x \,{\mathrm e}^{25}} {\mathrm e}^{25}}{x \,{\mathrm e}^{25}+{\mathrm e}^{25}}-2 \,\operatorname {Ei}_{1}\left (-2 x \,{\mathrm e}^{25}\right )-2 \left (-2 \,{\mathrm e}^{25}-1\right ) {\mathrm e}^{-2 \,{\mathrm e}^{25}} \operatorname {Ei}_{1}\left (-2 x \,{\mathrm e}^{25}-2 \,{\mathrm e}^{25}\right )+2 \,{\mathrm e}^{-25} \left (-\frac {{\mathrm e}^{2 x \,{\mathrm e}^{25}} {\mathrm e}^{50}}{x \,{\mathrm e}^{25}+{\mathrm e}^{25}}-2 \,{\mathrm e}^{50} {\mathrm e}^{-2 \,{\mathrm e}^{25}} \operatorname {Ei}_{1}\left (-2 x \,{\mathrm e}^{25}-2 \,{\mathrm e}^{25}\right )\right )-2 \,{\mathrm e}^{-25} \left (\frac {{\mathrm e}^{2 x \,{\mathrm e}^{25}} {\mathrm e}^{25}}{x \,{\mathrm e}^{25}+{\mathrm e}^{25}}-\operatorname {Ei}_{1}\left (-2 x \,{\mathrm e}^{25}\right )-\left (-2 \,{\mathrm e}^{25}-1\right ) {\mathrm e}^{-2 \,{\mathrm e}^{25}} \operatorname {Ei}_{1}\left (-2 x \,{\mathrm e}^{25}-2 \,{\mathrm e}^{25}\right )\right )\right )\) | \(270\) |
default | \({\mathrm e}^{-25} \left (-\frac {{\mathrm e}^{2 x \,{\mathrm e}^{25}} {\mathrm e}^{25} \left (2 x \,{\mathrm e}^{25}+{\mathrm e}^{25}\right )}{\left (x \,{\mathrm e}^{25}+{\mathrm e}^{25}\right ) x}-2 \left (-{\mathrm e}^{50}+{\mathrm e}^{75}\right ) {\mathrm e}^{-25} \operatorname {Ei}_{1}\left (-2 x \,{\mathrm e}^{25}\right )+2 \left (-2 \,{\mathrm e}^{50} {\mathrm e}^{25}+{\mathrm e}^{75}-{\mathrm e}^{50}\right ) {\mathrm e}^{-25} {\mathrm e}^{-2 \,{\mathrm e}^{25}} \operatorname {Ei}_{1}\left (-2 x \,{\mathrm e}^{25}-2 \,{\mathrm e}^{25}\right )\right )+\frac {2}{1+x}-\frac {2}{x}+\frac {{\mathrm e}^{x}}{x}-\frac {{\mathrm e}^{x}}{1+x}+2 \,{\mathrm e}^{-50} \left (\frac {{\mathrm e}^{2 x \,{\mathrm e}^{25}} {\mathrm e}^{75}}{x \,{\mathrm e}^{25}+{\mathrm e}^{25}}-{\mathrm e}^{75} {\mathrm e}^{-25} \operatorname {Ei}_{1}\left (-2 x \,{\mathrm e}^{25}\right )+{\mathrm e}^{75} \left (2 \,{\mathrm e}^{25}+1\right ) {\mathrm e}^{-25} {\mathrm e}^{-2 \,{\mathrm e}^{25}} \operatorname {Ei}_{1}\left (-2 x \,{\mathrm e}^{25}-2 \,{\mathrm e}^{25}\right )\right )-2 \,{\mathrm e}^{-25} \left (\frac {{\mathrm e}^{2 x \,{\mathrm e}^{25}} {\mathrm e}^{75}}{x \,{\mathrm e}^{25}+{\mathrm e}^{25}}-{\mathrm e}^{75} {\mathrm e}^{-25} \operatorname {Ei}_{1}\left (-2 x \,{\mathrm e}^{25}\right )+{\mathrm e}^{75} \left (2 \,{\mathrm e}^{25}+1\right ) {\mathrm e}^{-25} {\mathrm e}^{-2 \,{\mathrm e}^{25}} \operatorname {Ei}_{1}\left (-2 x \,{\mathrm e}^{25}-2 \,{\mathrm e}^{25}\right )\right )-2 \,{\mathrm e}^{-50} \left (-\frac {{\mathrm e}^{2 x \,{\mathrm e}^{25}} {\mathrm e}^{100}}{x \,{\mathrm e}^{25}+{\mathrm e}^{25}}-2 \,{\mathrm e}^{100} {\mathrm e}^{-2 \,{\mathrm e}^{25}} \operatorname {Ei}_{1}\left (-2 x \,{\mathrm e}^{25}-2 \,{\mathrm e}^{25}\right )\right )\) | \(314\) |
int((((-2*x^2-2*x)*exp(25)+2*x+1)*exp(x*exp(25))^2+(x^2-x-1)*exp(x)+4*x+2) /(x^4+2*x^3+x^2),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {2+4 x+e^x \left (-1-x+x^2\right )+e^{2 e^{25} x} \left (1+2 x+e^{25} \left (-2 x-2 x^2\right )\right )}{x^2+2 x^3+x^4} \, dx=-\frac {e^{\left (2 \, x e^{25}\right )} - e^{x} + 2}{x^{2} + x} \]
integrate((((-2*x^2-2*x)*exp(25)+2*x+1)*exp(x*exp(25))^2+(x^2-x-1)*exp(x)+ 4*x+2)/(x^4+2*x^3+x^2),x, algorithm=\
Time = 0.35 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {2+4 x+e^x \left (-1-x+x^2\right )+e^{2 e^{25} x} \left (1+2 x+e^{25} \left (-2 x-2 x^2\right )\right )}{x^2+2 x^3+x^4} \, dx=\frac {e^{x}}{x^{2} + x} - \frac {e^{2 x e^{25}}}{x^{2} + x} - \frac {2}{x^{2} + x} \]
integrate((((-2*x**2-2*x)*exp(25)+2*x+1)*exp(x*exp(25))**2+(x**2-x-1)*exp( x)+4*x+2)/(x**4+2*x**3+x**2),x)
Time = 0.23 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.83 \[ \int \frac {2+4 x+e^x \left (-1-x+x^2\right )+e^{2 e^{25} x} \left (1+2 x+e^{25} \left (-2 x-2 x^2\right )\right )}{x^2+2 x^3+x^4} \, dx=-\frac {2 \, {\left (2 \, x + 1\right )}}{x^{2} + x} - \frac {e^{\left (2 \, x e^{25}\right )} - e^{x}}{x^{2} + x} + \frac {4}{x + 1} \]
integrate((((-2*x^2-2*x)*exp(25)+2*x+1)*exp(x*exp(25))^2+(x^2-x-1)*exp(x)+ 4*x+2)/(x^4+2*x^3+x^2),x, algorithm=\
Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {2+4 x+e^x \left (-1-x+x^2\right )+e^{2 e^{25} x} \left (1+2 x+e^{25} \left (-2 x-2 x^2\right )\right )}{x^2+2 x^3+x^4} \, dx=-\frac {e^{\left (2 \, x e^{25}\right )} - e^{x} + 2}{x^{2} + x} \]
integrate((((-2*x^2-2*x)*exp(25)+2*x+1)*exp(x*exp(25))^2+(x^2-x-1)*exp(x)+ 4*x+2)/(x^4+2*x^3+x^2),x, algorithm=\
Time = 10.99 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {2+4 x+e^x \left (-1-x+x^2\right )+e^{2 e^{25} x} \left (1+2 x+e^{25} \left (-2 x-2 x^2\right )\right )}{x^2+2 x^3+x^4} \, dx=-\frac {{\mathrm {e}}^{2\,x\,{\mathrm {e}}^{25}}-{\mathrm {e}}^x+2}{x\,\left (x+1\right )} \]