Integrand size = 106, antiderivative size = 22 \[ \int \frac {4 e^4+e^{e^{16+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+x}} \left (1+e^{16+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+x} \left (x+32 e^x x+48 e^{2 x} x+24 e^{3 x} x+4 e^{4 x} x\right )\right )}{e^4} \, dx=13+4 x+e^{-4+e^{\left (2+e^x\right )^4+x}} x \] Output:
exp(exp(x+(2+exp(x))^4))*x/exp(4)+4*x+13
Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.77 \[ \int \frac {4 e^4+e^{e^{16+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+x}} \left (1+e^{16+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+x} \left (x+32 e^x x+48 e^{2 x} x+24 e^{3 x} x+4 e^{4 x} x\right )\right )}{e^4} \, dx=4 x+e^{-4+e^{16+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+x}} x \] Input:
Integrate[(4*E^4 + E^E^(16 + 32*E^x + 24*E^(2*x) + 8*E^(3*x) + E^(4*x) + x )*(1 + E^(16 + 32*E^x + 24*E^(2*x) + 8*E^(3*x) + E^(4*x) + x)*(x + 32*E^x* x + 48*E^(2*x)*x + 24*E^(3*x)*x + 4*E^(4*x)*x)))/E^4,x]
Output:
4*x + E^(-4 + E^(16 + 32*E^x + 24*E^(2*x) + 8*E^(3*x) + E^(4*x) + x))*x
Leaf count is larger than twice the leaf count of optimal. \(105\) vs. \(2(22)=44\).
Time = 0.36 (sec) , antiderivative size = 105, normalized size of antiderivative = 4.77, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {27, 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.
\(\displaystyle \int \frac {e^{e^{x+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+16}} \left (e^{x+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+16} \left (32 e^x x+48 e^{2 x} x+24 e^{3 x} x+4 e^{4 x} x+x\right )+1\right )+4 e^4}{e^4} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \left (e^{e^{x+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+16}} \left (e^{x+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+16} \left (32 e^x x+48 e^{2 x} x+24 e^{3 x} x+4 e^{4 x} x+x\right )+1\right )+4 e^4\right )dx}{e^4}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4 e^4 x+\frac {e^{e^{x+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+16}} \left (32 e^x x+48 e^{2 x} x+24 e^{3 x} x+4 e^{4 x} x+x\right )}{32 e^x+48 e^{2 x}+24 e^{3 x}+4 e^{4 x}+1}}{e^4}\) |
Input:
Int[(4*E^4 + E^E^(16 + 32*E^x + 24*E^(2*x) + 8*E^(3*x) + E^(4*x) + x)*(1 + E^(16 + 32*E^x + 24*E^(2*x) + 8*E^(3*x) + E^(4*x) + x)*(x + 32*E^x*x + 48 *E^(2*x)*x + 24*E^(3*x)*x + 4*E^(4*x)*x)))/E^4,x]
Output:
(4*E^4*x + (E^E^(16 + 32*E^x + 24*E^(2*x) + 8*E^(3*x) + E^(4*x) + x)*(x + 32*E^x*x + 48*E^(2*x)*x + 24*E^(3*x)*x + 4*E^(4*x)*x))/(1 + 32*E^x + 48*E^ (2*x) + 24*E^(3*x) + 4*E^(4*x)))/E^4
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.49 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55
method | result | size |
risch | \(4 x +x \,{\mathrm e}^{-4+{\mathrm e}^{{\mathrm e}^{4 x}+8 \,{\mathrm e}^{3 x}+24 \,{\mathrm e}^{2 x}+32 \,{\mathrm e}^{x}+x +16}}\) | \(34\) |
parallelrisch | \({\mathrm e}^{-4} \left (4 x \,{\mathrm e}^{4}+x \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{4 x}+8 \,{\mathrm e}^{3 x}+24 \,{\mathrm e}^{2 x}+32 \,{\mathrm e}^{x}+x +16}}\right )\) | \(39\) |
Input:
int((((4*x*exp(x)^4+24*x*exp(x)^3+48*x*exp(x)^2+32*exp(x)*x+x)*exp(exp(x)^ 4+8*exp(x)^3+24*exp(x)^2+32*exp(x)+x+16)+1)*exp(exp(exp(x)^4+8*exp(x)^3+24 *exp(x)^2+32*exp(x)+x+16))+4*exp(4))/exp(4),x,method=_RETURNVERBOSE)
Output:
4*x+x*exp(-4+exp(exp(4*x)+8*exp(3*x)+24*exp(2*x)+32*exp(x)+x+16))
Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.64 \[ \int \frac {4 e^4+e^{e^{16+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+x}} \left (1+e^{16+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+x} \left (x+32 e^x x+48 e^{2 x} x+24 e^{3 x} x+4 e^{4 x} x\right )\right )}{e^4} \, dx={\left (4 \, x e^{4} + x e^{\left (e^{\left (x + e^{\left (4 \, x\right )} + 8 \, e^{\left (3 \, x\right )} + 24 \, e^{\left (2 \, x\right )} + 32 \, e^{x} + 16\right )}\right )}\right )} e^{\left (-4\right )} \] Input:
integrate((((4*x*exp(x)^4+24*x*exp(x)^3+48*x*exp(x)^2+32*exp(x)*x+x)*exp(e xp(x)^4+8*exp(x)^3+24*exp(x)^2+32*exp(x)+x+16)+1)*exp(exp(exp(x)^4+8*exp(x )^3+24*exp(x)^2+32*exp(x)+x+16))+4*exp(4))/exp(4),x, algorithm="fricas")
Output:
(4*x*e^4 + x*e^(e^(x + e^(4*x) + 8*e^(3*x) + 24*e^(2*x) + 32*e^x + 16)))*e ^(-4)
Timed out. \[ \int \frac {4 e^4+e^{e^{16+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+x}} \left (1+e^{16+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+x} \left (x+32 e^x x+48 e^{2 x} x+24 e^{3 x} x+4 e^{4 x} x\right )\right )}{e^4} \, dx=\text {Timed out} \] Input:
integrate((((4*x*exp(x)**4+24*x*exp(x)**3+48*x*exp(x)**2+32*exp(x)*x+x)*ex p(exp(x)**4+8*exp(x)**3+24*exp(x)**2+32*exp(x)+x+16)+1)*exp(exp(exp(x)**4+ 8*exp(x)**3+24*exp(x)**2+32*exp(x)+x+16))+4*exp(4))/exp(4),x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.64 \[ \int \frac {4 e^4+e^{e^{16+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+x}} \left (1+e^{16+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+x} \left (x+32 e^x x+48 e^{2 x} x+24 e^{3 x} x+4 e^{4 x} x\right )\right )}{e^4} \, dx={\left (4 \, x e^{4} + x e^{\left (e^{\left (x + e^{\left (4 \, x\right )} + 8 \, e^{\left (3 \, x\right )} + 24 \, e^{\left (2 \, x\right )} + 32 \, e^{x} + 16\right )}\right )}\right )} e^{\left (-4\right )} \] Input:
integrate((((4*x*exp(x)^4+24*x*exp(x)^3+48*x*exp(x)^2+32*exp(x)*x+x)*exp(e xp(x)^4+8*exp(x)^3+24*exp(x)^2+32*exp(x)+x+16)+1)*exp(exp(exp(x)^4+8*exp(x )^3+24*exp(x)^2+32*exp(x)+x+16))+4*exp(4))/exp(4),x, algorithm="maxima")
Output:
(4*x*e^4 + x*e^(e^(x + e^(4*x) + 8*e^(3*x) + 24*e^(2*x) + 32*e^x + 16)))*e ^(-4)
\[ \int \frac {4 e^4+e^{e^{16+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+x}} \left (1+e^{16+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+x} \left (x+32 e^x x+48 e^{2 x} x+24 e^{3 x} x+4 e^{4 x} x\right )\right )}{e^4} \, dx=\int { {\left ({\left ({\left (4 \, x e^{\left (4 \, x\right )} + 24 \, x e^{\left (3 \, x\right )} + 48 \, x e^{\left (2 \, x\right )} + 32 \, x e^{x} + x\right )} e^{\left (x + e^{\left (4 \, x\right )} + 8 \, e^{\left (3 \, x\right )} + 24 \, e^{\left (2 \, x\right )} + 32 \, e^{x} + 16\right )} + 1\right )} e^{\left (e^{\left (x + e^{\left (4 \, x\right )} + 8 \, e^{\left (3 \, x\right )} + 24 \, e^{\left (2 \, x\right )} + 32 \, e^{x} + 16\right )}\right )} + 4 \, e^{4}\right )} e^{\left (-4\right )} \,d x } \] Input:
integrate((((4*x*exp(x)^4+24*x*exp(x)^3+48*x*exp(x)^2+32*exp(x)*x+x)*exp(e xp(x)^4+8*exp(x)^3+24*exp(x)^2+32*exp(x)+x+16)+1)*exp(exp(exp(x)^4+8*exp(x )^3+24*exp(x)^2+32*exp(x)+x+16))+4*exp(4))/exp(4),x, algorithm="giac")
Output:
integrate((((4*x*e^(4*x) + 24*x*e^(3*x) + 48*x*e^(2*x) + 32*x*e^x + x)*e^( x + e^(4*x) + 8*e^(3*x) + 24*e^(2*x) + 32*e^x + 16) + 1)*e^(e^(x + e^(4*x) + 8*e^(3*x) + 24*e^(2*x) + 32*e^x + 16)) + 4*e^4)*e^(-4), x)
Time = 0.17 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.77 \[ \int \frac {4 e^4+e^{e^{16+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+x}} \left (1+e^{16+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+x} \left (x+32 e^x x+48 e^{2 x} x+24 e^{3 x} x+4 e^{4 x} x\right )\right )}{e^4} \, dx=x\,{\mathrm {e}}^{-4}\,\left ({\mathrm {e}}^{{\mathrm {e}}^{8\,{\mathrm {e}}^{3\,x}}\,{\mathrm {e}}^{24\,{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^{16}\,{\mathrm {e}}^{{\mathrm {e}}^{4\,x}}\,{\mathrm {e}}^{32\,{\mathrm {e}}^x}\,{\mathrm {e}}^x}+4\,{\mathrm {e}}^4\right ) \] Input:
int(exp(-4)*(4*exp(4) + exp(exp(x + 24*exp(2*x) + 8*exp(3*x) + exp(4*x) + 32*exp(x) + 16))*(exp(x + 24*exp(2*x) + 8*exp(3*x) + exp(4*x) + 32*exp(x) + 16)*(x + 48*x*exp(2*x) + 24*x*exp(3*x) + 4*x*exp(4*x) + 32*x*exp(x)) + 1 )),x)
Output:
x*exp(-4)*(exp(exp(8*exp(3*x))*exp(24*exp(2*x))*exp(16)*exp(exp(4*x))*exp( 32*exp(x))*exp(x)) + 4*exp(4))
Time = 0.18 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.05 \[ \int \frac {4 e^4+e^{e^{16+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+x}} \left (1+e^{16+32 e^x+24 e^{2 x}+8 e^{3 x}+e^{4 x}+x} \left (x+32 e^x x+48 e^{2 x} x+24 e^{3 x} x+4 e^{4 x} x\right )\right )}{e^4} \, dx=\frac {x \left (e^{e^{e^{4 x}+8 e^{3 x}+24 e^{2 x}+32 e^{x}+x} e^{16}}+4 e^{4}\right )}{e^{4}} \] Input:
int((((4*x*exp(x)^4+24*x*exp(x)^3+48*x*exp(x)^2+32*exp(x)*x+x)*exp(exp(x)^ 4+8*exp(x)^3+24*exp(x)^2+32*exp(x)+x+16)+1)*exp(exp(exp(x)^4+8*exp(x)^3+24 *exp(x)^2+32*exp(x)+x+16))+4*exp(4))/exp(4),x)
Output:
(x*(e**(e**(e**(4*x) + 8*e**(3*x) + 24*e**(2*x) + 32*e**x + x)*e**16) + 4* e**4))/e**4