Integrand size = 150, antiderivative size = 26 \[ \int e^{-e^{4 x}} \left (e^{2 e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (2-8 e^{4 x} x\right )+e^{e^{4 x}} \left (6 e^{6 x}+8 x+e^{3 x} (4+12 x)\right )+e^{e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (-2 e^{3 x}+e^{e^{4 x}} \left (-4-6 e^{3 x}\right )-4 x+e^{4 x} \left (8 e^{3 x} x+16 x^2\right )\right )\right ) \, dx=\left (e^{3 x}-2 e^{e^{-e^{4 x}} x}+2 x\right )^2 \]
Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int e^{-e^{4 x}} \left (e^{2 e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (2-8 e^{4 x} x\right )+e^{e^{4 x}} \left (6 e^{6 x}+8 x+e^{3 x} (4+12 x)\right )+e^{e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (-2 e^{3 x}+e^{e^{4 x}} \left (-4-6 e^{3 x}\right )-4 x+e^{4 x} \left (8 e^{3 x} x+16 x^2\right )\right )\right ) \, dx=\left (e^{3 x}-2 e^{e^{-e^{4 x}} x}+2 x\right )^2 \]
Integrate[(E^((2*(x + E^E^(4*x)*Log[2]))/E^E^(4*x))*(2 - 8*E^(4*x)*x) + E^ E^(4*x)*(6*E^(6*x) + 8*x + E^(3*x)*(4 + 12*x)) + E^((x + E^E^(4*x)*Log[2]) /E^E^(4*x))*(-2*E^(3*x) + E^E^(4*x)*(-4 - 6*E^(3*x)) - 4*x + E^(4*x)*(8*E^ (3*x)*x + 16*x^2)))/E^E^(4*x),x]
Time = 1.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {7239, 27, 7237}
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.
\(\displaystyle \int e^{-e^{4 x}} \left (\left (e^{4 x} \left (16 x^2+8 e^{3 x} x\right )+e^{e^{4 x}} \left (-6 e^{3 x}-4\right )-2 e^{3 x}-4 x\right ) e^{e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )}+e^{e^{4 x}} \left (8 x+6 e^{6 x}+e^{3 x} (12 x+4)\right )+\left (2-8 e^{4 x} x\right ) e^{2 e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )}\right ) \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int 2 e^{-e^{4 x}} \left (2 x+e^{3 x}-2 e^{e^{-e^{4 x}} x}\right ) \left (8 e^{\left (e^{-e^{4 x}}+4\right ) x} x+2 e^{e^{4 x}}-2 e^{e^{-e^{4 x}} x}+3 e^{3 x+e^{4 x}}\right )dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int e^{-e^{4 x}} \left (2 x+e^{3 x}-2 e^{e^{-e^{4 x}} x}\right ) \left (8 e^{\left (4+e^{-e^{4 x}}\right ) x} x+2 e^{e^{4 x}}-2 e^{e^{-e^{4 x}} x}+3 e^{3 x+e^{4 x}}\right )dx\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle \left (2 x+e^{3 x}-2 e^{e^{-e^{4 x}} x}\right )^2\) |
Int[(E^((2*(x + E^E^(4*x)*Log[2]))/E^E^(4*x))*(2 - 8*E^(4*x)*x) + E^E^(4*x )*(6*E^(6*x) + 8*x + E^(3*x)*(4 + 12*x)) + E^((x + E^E^(4*x)*Log[2])/E^E^( 4*x))*(-2*E^(3*x) + E^E^(4*x)*(-4 - 6*E^(3*x)) - 4*x + E^(4*x)*(8*E^(3*x)* x + 16*x^2)))/E^E^(4*x),x]
3.26.22.3.1 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]]
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(52\) vs. \(2(25)=50\).
Time = 0.37 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.04
method | result | size |
risch | \({\mathrm e}^{6 x}+4 x \,{\mathrm e}^{3 x}+4 x^{2}+4 \,{\mathrm e}^{2 \,{\mathrm e}^{-{\mathrm e}^{4 x}} x}+2 \left (-2 \,{\mathrm e}^{3 x}-4 x \right ) {\mathrm e}^{{\mathrm e}^{-{\mathrm e}^{4 x}} x}\) | \(53\) |
parallelrisch | \(4 x^{2}+4 x \,{\mathrm e}^{3 x}-4 \,{\mathrm e}^{\left (\ln \left (2\right ) {\mathrm e}^{{\mathrm e}^{4 x}}+x \right ) {\mathrm e}^{-{\mathrm e}^{4 x}}} x +{\mathrm e}^{6 x}-2 \,{\mathrm e}^{3 x} {\mathrm e}^{\left (\ln \left (2\right ) {\mathrm e}^{{\mathrm e}^{4 x}}+x \right ) {\mathrm e}^{-{\mathrm e}^{4 x}}}+{\mathrm e}^{2 \left (\ln \left (2\right ) {\mathrm e}^{{\mathrm e}^{4 x}}+x \right ) {\mathrm e}^{-{\mathrm e}^{4 x}}}\) | \(88\) |
int(((-8*x*exp(4*x)+2)*exp((ln(2)*exp(exp(4*x))+x)/exp(exp(4*x)))^2+((-6*e xp(3*x)-4)*exp(exp(4*x))+(8*x*exp(3*x)+16*x^2)*exp(4*x)-2*exp(3*x)-4*x)*ex p((ln(2)*exp(exp(4*x))+x)/exp(exp(4*x)))+(6*exp(3*x)^2+(12*x+4)*exp(3*x)+8 *x)*exp(exp(4*x)))/exp(exp(4*x)),x,method=_RETURNVERBOSE)
exp(6*x)+4*x*exp(3*x)+4*x^2+4*exp(2*exp(-exp(4*x))*x)+2*(-2*exp(3*x)-4*x)* exp(exp(-exp(4*x))*x)
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (25) = 50\).
Time = 0.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.54 \[ \int e^{-e^{4 x}} \left (e^{2 e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (2-8 e^{4 x} x\right )+e^{e^{4 x}} \left (6 e^{6 x}+8 x+e^{3 x} (4+12 x)\right )+e^{e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (-2 e^{3 x}+e^{e^{4 x}} \left (-4-6 e^{3 x}\right )-4 x+e^{4 x} \left (8 e^{3 x} x+16 x^2\right )\right )\right ) \, dx=4 \, x^{2} - 2 \, {\left (2 \, x + e^{\left (3 \, x\right )}\right )} e^{\left ({\left (e^{\left (e^{\left (4 \, x\right )}\right )} \log \left (2\right ) + x\right )} e^{\left (-e^{\left (4 \, x\right )}\right )}\right )} + 4 \, x e^{\left (3 \, x\right )} + e^{\left (2 \, {\left (e^{\left (e^{\left (4 \, x\right )}\right )} \log \left (2\right ) + x\right )} e^{\left (-e^{\left (4 \, x\right )}\right )}\right )} + e^{\left (6 \, x\right )} \]
integrate(((-8*x*exp(4*x)+2)*exp((log(2)*exp(exp(4*x))+x)/exp(exp(4*x)))^2 +((-6*exp(3*x)-4)*exp(exp(4*x))+(8*x*exp(3*x)+16*x^2)*exp(4*x)-2*exp(3*x)- 4*x)*exp((log(2)*exp(exp(4*x))+x)/exp(exp(4*x)))+(6*exp(3*x)^2+(12*x+4)*ex p(3*x)+8*x)*exp(exp(4*x)))/exp(exp(4*x)),x, algorithm=\
4*x^2 - 2*(2*x + e^(3*x))*e^((e^(e^(4*x))*log(2) + x)*e^(-e^(4*x))) + 4*x* e^(3*x) + e^(2*(e^(e^(4*x))*log(2) + x)*e^(-e^(4*x))) + e^(6*x)
Timed out. \[ \int e^{-e^{4 x}} \left (e^{2 e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (2-8 e^{4 x} x\right )+e^{e^{4 x}} \left (6 e^{6 x}+8 x+e^{3 x} (4+12 x)\right )+e^{e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (-2 e^{3 x}+e^{e^{4 x}} \left (-4-6 e^{3 x}\right )-4 x+e^{4 x} \left (8 e^{3 x} x+16 x^2\right )\right )\right ) \, dx=\text {Timed out} \]
integrate(((-8*x*exp(4*x)+2)*exp((ln(2)*exp(exp(4*x))+x)/exp(exp(4*x)))**2 +((-6*exp(3*x)-4)*exp(exp(4*x))+(8*x*exp(3*x)+16*x**2)*exp(4*x)-2*exp(3*x) -4*x)*exp((ln(2)*exp(exp(4*x))+x)/exp(exp(4*x)))+(6*exp(3*x)**2+(12*x+4)*e xp(3*x)+8*x)*exp(exp(4*x)))/exp(exp(4*x)),x)
\[ \int e^{-e^{4 x}} \left (e^{2 e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (2-8 e^{4 x} x\right )+e^{e^{4 x}} \left (6 e^{6 x}+8 x+e^{3 x} (4+12 x)\right )+e^{e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (-2 e^{3 x}+e^{e^{4 x}} \left (-4-6 e^{3 x}\right )-4 x+e^{4 x} \left (8 e^{3 x} x+16 x^2\right )\right )\right ) \, dx=\int { -2 \, {\left ({\left (4 \, x e^{\left (4 \, x\right )} - 1\right )} e^{\left (2 \, {\left (e^{\left (e^{\left (4 \, x\right )}\right )} \log \left (2\right ) + x\right )} e^{\left (-e^{\left (4 \, x\right )}\right )}\right )} - {\left (4 \, {\left (2 \, x^{2} + x e^{\left (3 \, x\right )}\right )} e^{\left (4 \, x\right )} - {\left (3 \, e^{\left (3 \, x\right )} + 2\right )} e^{\left (e^{\left (4 \, x\right )}\right )} - 2 \, x - e^{\left (3 \, x\right )}\right )} e^{\left ({\left (e^{\left (e^{\left (4 \, x\right )}\right )} \log \left (2\right ) + x\right )} e^{\left (-e^{\left (4 \, x\right )}\right )}\right )} - {\left (2 \, {\left (3 \, x + 1\right )} e^{\left (3 \, x\right )} + 4 \, x + 3 \, e^{\left (6 \, x\right )}\right )} e^{\left (e^{\left (4 \, x\right )}\right )}\right )} e^{\left (-e^{\left (4 \, x\right )}\right )} \,d x } \]
integrate(((-8*x*exp(4*x)+2)*exp((log(2)*exp(exp(4*x))+x)/exp(exp(4*x)))^2 +((-6*exp(3*x)-4)*exp(exp(4*x))+(8*x*exp(3*x)+16*x^2)*exp(4*x)-2*exp(3*x)- 4*x)*exp((log(2)*exp(exp(4*x))+x)/exp(exp(4*x)))+(6*exp(3*x)^2+(12*x+4)*ex p(3*x)+8*x)*exp(exp(4*x)))/exp(exp(4*x)),x, algorithm=\
4*x^2 + 4/3*(3*x - 1)*e^(3*x) + 4*e^(2*x*e^(-e^(4*x))) + e^(6*x) + 4/3*e^( 3*x) - 2*integrate(-2*(8*x^2*e^(4*x) + 4*x*e^(7*x) - (3*e^(3*x) + 2)*e^(e^ (4*x)) - 2*x - e^(3*x))*e^(x*e^(-e^(4*x)) - e^(4*x)), x)
\[ \int e^{-e^{4 x}} \left (e^{2 e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (2-8 e^{4 x} x\right )+e^{e^{4 x}} \left (6 e^{6 x}+8 x+e^{3 x} (4+12 x)\right )+e^{e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (-2 e^{3 x}+e^{e^{4 x}} \left (-4-6 e^{3 x}\right )-4 x+e^{4 x} \left (8 e^{3 x} x+16 x^2\right )\right )\right ) \, dx=\int { -2 \, {\left ({\left (4 \, x e^{\left (4 \, x\right )} - 1\right )} e^{\left (2 \, {\left (e^{\left (e^{\left (4 \, x\right )}\right )} \log \left (2\right ) + x\right )} e^{\left (-e^{\left (4 \, x\right )}\right )}\right )} - {\left (4 \, {\left (2 \, x^{2} + x e^{\left (3 \, x\right )}\right )} e^{\left (4 \, x\right )} - {\left (3 \, e^{\left (3 \, x\right )} + 2\right )} e^{\left (e^{\left (4 \, x\right )}\right )} - 2 \, x - e^{\left (3 \, x\right )}\right )} e^{\left ({\left (e^{\left (e^{\left (4 \, x\right )}\right )} \log \left (2\right ) + x\right )} e^{\left (-e^{\left (4 \, x\right )}\right )}\right )} - {\left (2 \, {\left (3 \, x + 1\right )} e^{\left (3 \, x\right )} + 4 \, x + 3 \, e^{\left (6 \, x\right )}\right )} e^{\left (e^{\left (4 \, x\right )}\right )}\right )} e^{\left (-e^{\left (4 \, x\right )}\right )} \,d x } \]
integrate(((-8*x*exp(4*x)+2)*exp((log(2)*exp(exp(4*x))+x)/exp(exp(4*x)))^2 +((-6*exp(3*x)-4)*exp(exp(4*x))+(8*x*exp(3*x)+16*x^2)*exp(4*x)-2*exp(3*x)- 4*x)*exp((log(2)*exp(exp(4*x))+x)/exp(exp(4*x)))+(6*exp(3*x)^2+(12*x+4)*ex p(3*x)+8*x)*exp(exp(4*x)))/exp(exp(4*x)),x, algorithm=\
integrate(-2*((4*x*e^(4*x) - 1)*e^(2*(e^(e^(4*x))*log(2) + x)*e^(-e^(4*x)) ) - (4*(2*x^2 + x*e^(3*x))*e^(4*x) - (3*e^(3*x) + 2)*e^(e^(4*x)) - 2*x - e ^(3*x))*e^((e^(e^(4*x))*log(2) + x)*e^(-e^(4*x))) - (2*(3*x + 1)*e^(3*x) + 4*x + 3*e^(6*x))*e^(e^(4*x)))*e^(-e^(4*x)), x)
Timed out. \[ \int e^{-e^{4 x}} \left (e^{2 e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (2-8 e^{4 x} x\right )+e^{e^{4 x}} \left (6 e^{6 x}+8 x+e^{3 x} (4+12 x)\right )+e^{e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (-2 e^{3 x}+e^{e^{4 x}} \left (-4-6 e^{3 x}\right )-4 x+e^{4 x} \left (8 e^{3 x} x+16 x^2\right )\right )\right ) \, dx=-\int {\mathrm {e}}^{-{\mathrm {e}}^{4\,x}}\,\left ({\mathrm {e}}^{{\mathrm {e}}^{-{\mathrm {e}}^{4\,x}}\,\left (x+{\mathrm {e}}^{{\mathrm {e}}^{4\,x}}\,\ln \left (2\right )\right )}\,\left (4\,x+2\,{\mathrm {e}}^{3\,x}-{\mathrm {e}}^{4\,x}\,\left (8\,x\,{\mathrm {e}}^{3\,x}+16\,x^2\right )+{\mathrm {e}}^{{\mathrm {e}}^{4\,x}}\,\left (6\,{\mathrm {e}}^{3\,x}+4\right )\right )-{\mathrm {e}}^{{\mathrm {e}}^{4\,x}}\,\left (8\,x+6\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{3\,x}\,\left (12\,x+4\right )\right )+{\mathrm {e}}^{2\,{\mathrm {e}}^{-{\mathrm {e}}^{4\,x}}\,\left (x+{\mathrm {e}}^{{\mathrm {e}}^{4\,x}}\,\ln \left (2\right )\right )}\,\left (8\,x\,{\mathrm {e}}^{4\,x}-2\right )\right ) \,d x \]
int(-exp(-exp(4*x))*(exp(exp(-exp(4*x))*(x + exp(exp(4*x))*log(2)))*(4*x + 2*exp(3*x) - exp(4*x)*(8*x*exp(3*x) + 16*x^2) + exp(exp(4*x))*(6*exp(3*x) + 4)) - exp(exp(4*x))*(8*x + 6*exp(6*x) + exp(3*x)*(12*x + 4)) + exp(2*ex p(-exp(4*x))*(x + exp(exp(4*x))*log(2)))*(8*x*exp(4*x) - 2)),x)