Integrand size = 75, antiderivative size = 31 \[ \int e^{25+e^2+e^{2 e^3+2 e^4}-x-4 e^{e^3+2 e^4} x+4 e^{2 e^4} x^2} \left (-1-4 e^{e^3+2 e^4}+8 e^{2 e^4} x\right ) \, dx=e^{25+e^2-x+e^{2 e^4} \left (-e^{e^3}+2 x\right )^2} \]
Time = 1.80 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52 \[ \int e^{25+e^2+e^{2 e^3+2 e^4}-x-4 e^{e^3+2 e^4} x+4 e^{2 e^4} x^2} \left (-1-4 e^{e^3+2 e^4}+8 e^{2 e^4} x\right ) \, dx=e^{25+e^2+e^{2 e^3 (1+e)}-\left (1+4 e^{e^3+2 e^4}\right ) x+4 e^{2 e^4} x^2} \]
Integrate[E^(25 + E^2 + E^(2*E^3 + 2*E^4) - x - 4*E^(E^3 + 2*E^4)*x + 4*E^ (2*E^4)*x^2)*(-1 - 4*E^(E^3 + 2*E^4) + 8*E^(2*E^4)*x),x]
Time = 0.61 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {2674, 2666}
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.
\(\displaystyle \int \left (8 e^{2 e^4} x-4 e^{e^3+2 e^4}-1\right ) \exp \left (4 e^{2 e^4} x^2-4 e^{e^3+2 e^4} x-x+e^{2 e^3+2 e^4}+e^2+25\right ) \, dx\) |
\(\Big \downarrow \) 2674 |
\(\displaystyle \int \left (8 e^{2 e^4} x-4 e^{e^3+2 e^4}-1\right ) \exp \left (4 e^{2 e^4} x^2-\left (1+4 e^{e^3+2 e^4}\right ) x+e^{2 e^3 (1+e)}+e^2+25\right )dx\) |
\(\Big \downarrow \) 2666 |
\(\displaystyle \exp \left (4 e^{2 e^4} x^2-\left (1+4 e^{e^3+2 e^4}\right ) x+e^{2 e^3 (1+e)}+e^2+25\right )\) |
Int[E^(25 + E^2 + E^(2*E^3 + 2*E^4) - x - 4*E^(E^3 + 2*E^4)*x + 4*E^(2*E^4 )*x^2)*(-1 - 4*E^(E^3 + 2*E^4) + 8*E^(2*E^4)*x),x]
3.19.63.3.1 Defintions of rubi rules used
Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol ] :> Simp[e*(F^(a + b*x + c*x^2)/(2*c*Log[F])), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]
Int[(F_)^(v_)*(u_)^(m_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*F^ExpandToSu m[v, x], x] /; FreeQ[{F, m}, x] && LinearQ[u, x] && QuadraticQ[v, x] && !( LinearMatchQ[u, x] && QuadraticMatchQ[v, x])
Time = 0.13 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.29
method | result | size |
risch | \({\mathrm e}^{{\mathrm e}^{2 \,{\mathrm e}^{4}+2 \,{\mathrm e}^{3}}-4 x \,{\mathrm e}^{2 \,{\mathrm e}^{4}+{\mathrm e}^{3}}+4 x^{2} {\mathrm e}^{2 \,{\mathrm e}^{4}}+{\mathrm e}^{2}-x +25}\) | \(40\) |
gosper | \({\mathrm e}^{{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{2 \,{\mathrm e}^{3}}-4 x \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{{\mathrm e}^{3}}+4 x^{2} {\mathrm e}^{2 \,{\mathrm e}^{4}}+{\mathrm e}^{2}-x +25}\) | \(47\) |
derivativedivides | \({\mathrm e}^{{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{2 \,{\mathrm e}^{3}}-4 x \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{{\mathrm e}^{3}}+4 x^{2} {\mathrm e}^{2 \,{\mathrm e}^{4}}+{\mathrm e}^{2}-x +25}\) | \(47\) |
default | \({\mathrm e}^{{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{2 \,{\mathrm e}^{3}}-4 x \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{{\mathrm e}^{3}}+4 x^{2} {\mathrm e}^{2 \,{\mathrm e}^{4}}+{\mathrm e}^{2}-x +25}\) | \(47\) |
norman | \({\mathrm e}^{{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{2 \,{\mathrm e}^{3}}-4 x \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{{\mathrm e}^{3}}+4 x^{2} {\mathrm e}^{2 \,{\mathrm e}^{4}}+{\mathrm e}^{2}-x +25}\) | \(47\) |
parallelrisch | \({\mathrm e}^{{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{2 \,{\mathrm e}^{3}}-4 x \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{{\mathrm e}^{3}}+4 x^{2} {\mathrm e}^{2 \,{\mathrm e}^{4}}+{\mathrm e}^{2}-x +25}\) | \(47\) |
parts | \(i \sqrt {\pi }\, {\mathrm e}^{{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{2 \,{\mathrm e}^{3}}+{\mathrm e}^{2}+25-\frac {\left (-4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{{\mathrm e}^{3}}-1\right )^{2} {\mathrm e}^{-2 \,{\mathrm e}^{4}}}{16}} {\mathrm e}^{{\mathrm e}^{4}} \operatorname {erf}\left (2 i {\mathrm e}^{{\mathrm e}^{4}} x +\frac {i \left (-4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{{\mathrm e}^{3}}-1\right ) {\mathrm e}^{-{\mathrm e}^{4}}}{4}\right ) {\mathrm e}^{{\mathrm e}^{3}}-2 i \sqrt {\pi }\, {\mathrm e}^{{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{2 \,{\mathrm e}^{3}}+{\mathrm e}^{2}+25-\frac {\left (-4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{{\mathrm e}^{3}}-1\right )^{2} {\mathrm e}^{-2 \,{\mathrm e}^{4}}}{16}} {\mathrm e}^{{\mathrm e}^{4}} \operatorname {erf}\left (2 i {\mathrm e}^{{\mathrm e}^{4}} x +\frac {i \left (-4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{{\mathrm e}^{3}}-1\right ) {\mathrm e}^{-{\mathrm e}^{4}}}{4}\right ) x +\frac {i \sqrt {\pi }\, {\mathrm e}^{{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{2 \,{\mathrm e}^{3}}+{\mathrm e}^{2}+25-\frac {\left (-4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{{\mathrm e}^{3}}-1\right )^{2} {\mathrm e}^{-2 \,{\mathrm e}^{4}}}{16}} {\mathrm e}^{-{\mathrm e}^{4}} \operatorname {erf}\left (2 i {\mathrm e}^{{\mathrm e}^{4}} x +\frac {i \left (-4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{{\mathrm e}^{3}}-1\right ) {\mathrm e}^{-{\mathrm e}^{4}}}{4}\right )}{4}+{\mathrm e}^{{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{2 \,{\mathrm e}^{3}}+{\mathrm e}^{2}+25-\frac {\left (-4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{{\mathrm e}^{3}}-1\right )^{2} {\mathrm e}^{-2 \,{\mathrm e}^{4}}}{16}} \left (\left (2 i {\mathrm e}^{{\mathrm e}^{4}} x +\frac {i \left (-4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{{\mathrm e}^{3}}-1\right ) {\mathrm e}^{-{\mathrm e}^{4}}}{4}\right ) \operatorname {erf}\left (2 i {\mathrm e}^{{\mathrm e}^{4}} x +\frac {i \left (-4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{{\mathrm e}^{3}}-1\right ) {\mathrm e}^{-{\mathrm e}^{4}}}{4}\right ) \sqrt {\pi }+{\mathrm e}^{-\left (2 i {\mathrm e}^{{\mathrm e}^{4}} x +\frac {i \left (-4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{{\mathrm e}^{3}}-1\right ) {\mathrm e}^{-{\mathrm e}^{4}}}{4}\right )^{2}}\right )\) | \(432\) |
int((-4*exp(exp(2)^2)^2*exp(exp(3))+8*x*exp(exp(2)^2)^2-1)*exp(exp(exp(2)^ 2)^2*exp(exp(3))^2-4*x*exp(exp(2)^2)^2*exp(exp(3))+4*x^2*exp(exp(2)^2)^2+e xp(2)-x+25),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (25) = 50\).
Time = 0.24 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.19 \[ \int e^{25+e^2+e^{2 e^3+2 e^4}-x-4 e^{e^3+2 e^4} x+4 e^{2 e^4} x^2} \left (-1-4 e^{e^3+2 e^4}+8 e^{2 e^4} x\right ) \, dx=e^{\left ({\left (4 \, x^{2} e^{\left (4 \, e^{4} + 2 \, e^{3}\right )} - {\left (4 \, x e^{\left (2 \, e^{4} + e^{3}\right )} + x - e^{2} - 25\right )} e^{\left (2 \, e^{4} + 2 \, e^{3}\right )} + e^{\left (4 \, e^{4} + 4 \, e^{3}\right )}\right )} e^{\left (-2 \, e^{4} - 2 \, e^{3}\right )}\right )} \]
integrate((-4*exp(exp(2)^2)^2*exp(exp(3))+8*x*exp(exp(2)^2)^2-1)*exp(exp(e xp(2)^2)^2*exp(exp(3))^2-4*x*exp(exp(2)^2)^2*exp(exp(3))+4*x^2*exp(exp(2)^ 2)^2+exp(2)-x+25),x, algorithm=\
e^((4*x^2*e^(4*e^4 + 2*e^3) - (4*x*e^(2*e^4 + e^3) + x - e^2 - 25)*e^(2*e^ 4 + 2*e^3) + e^(4*e^4 + 4*e^3))*e^(-2*e^4 - 2*e^3))
Time = 0.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.55 \[ \int e^{25+e^2+e^{2 e^3+2 e^4}-x-4 e^{e^3+2 e^4} x+4 e^{2 e^4} x^2} \left (-1-4 e^{e^3+2 e^4}+8 e^{2 e^4} x\right ) \, dx=e^{4 x^{2} e^{2 e^{4}} - 4 x e^{e^{3}} e^{2 e^{4}} - x + e^{2} + 25 + e^{2 e^{3}} e^{2 e^{4}}} \]
integrate((-4*exp(exp(2)**2)**2*exp(exp(3))+8*x*exp(exp(2)**2)**2-1)*exp(e xp(exp(2)**2)**2*exp(exp(3))**2-4*x*exp(exp(2)**2)**2*exp(exp(3))+4*x**2*e xp(exp(2)**2)**2+exp(2)-x+25),x)
exp(4*x**2*exp(2*exp(4)) - 4*x*exp(exp(3))*exp(2*exp(4)) - x + exp(2) + 25 + exp(2*exp(3))*exp(2*exp(4)))
Time = 0.19 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26 \[ \int e^{25+e^2+e^{2 e^3+2 e^4}-x-4 e^{e^3+2 e^4} x+4 e^{2 e^4} x^2} \left (-1-4 e^{e^3+2 e^4}+8 e^{2 e^4} x\right ) \, dx=e^{\left (4 \, x^{2} e^{\left (2 \, e^{4}\right )} - 4 \, x e^{\left (2 \, e^{4} + e^{3}\right )} - x + e^{2} + e^{\left (2 \, e^{4} + 2 \, e^{3}\right )} + 25\right )} \]
integrate((-4*exp(exp(2)^2)^2*exp(exp(3))+8*x*exp(exp(2)^2)^2-1)*exp(exp(e xp(2)^2)^2*exp(exp(3))^2-4*x*exp(exp(2)^2)^2*exp(exp(3))+4*x^2*exp(exp(2)^ 2)^2+exp(2)-x+25),x, algorithm=\
Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26 \[ \int e^{25+e^2+e^{2 e^3+2 e^4}-x-4 e^{e^3+2 e^4} x+4 e^{2 e^4} x^2} \left (-1-4 e^{e^3+2 e^4}+8 e^{2 e^4} x\right ) \, dx=e^{\left (4 \, x^{2} e^{\left (2 \, e^{4}\right )} - 4 \, x e^{\left (2 \, e^{4} + e^{3}\right )} - x + e^{2} + e^{\left (2 \, e^{4} + 2 \, e^{3}\right )} + 25\right )} \]
integrate((-4*exp(exp(2)^2)^2*exp(exp(3))+8*x*exp(exp(2)^2)^2-1)*exp(exp(e xp(2)^2)^2*exp(exp(3))^2-4*x*exp(exp(2)^2)^2*exp(exp(3))+4*x^2*exp(exp(2)^ 2)^2+exp(2)-x+25),x, algorithm=\
Time = 0.42 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.45 \[ \int e^{25+e^2+e^{2 e^3+2 e^4}-x-4 e^{e^3+2 e^4} x+4 e^{2 e^4} x^2} \left (-1-4 e^{e^3+2 e^4}+8 e^{2 e^4} x\right ) \, dx={\mathrm {e}}^{{\mathrm {e}}^{2\,{\mathrm {e}}^3}\,{\mathrm {e}}^{2\,{\mathrm {e}}^4}}\,{\mathrm {e}}^{-4\,x\,{\mathrm {e}}^{2\,{\mathrm {e}}^4}\,{\mathrm {e}}^{{\mathrm {e}}^3}}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{25}\,{\mathrm {e}}^{4\,x^2\,{\mathrm {e}}^{2\,{\mathrm {e}}^4}}\,{\mathrm {e}}^{{\mathrm {e}}^2} \]
int(-exp(exp(2) - x + 4*x^2*exp(2*exp(4)) + exp(2*exp(3))*exp(2*exp(4)) - 4*x*exp(2*exp(4))*exp(exp(3)) + 25)*(4*exp(2*exp(4))*exp(exp(3)) - 8*x*exp (2*exp(4)) + 1),x)