Integrand size = 98, antiderivative size = 30 \[ \int \frac {1}{7} e^{\frac {1}{7} \left (e^{e^x-x} \left (-28+7 e^{4 x}-x\right )+35 x-7 e^{4 x} x+x^2\right )} \left (35+e^{4 x} (-7-28 x)+2 x+e^{e^x-x} \left (27+e^{4 x} \left (21+7 e^x\right )+e^x (-28-x)+x\right )\right ) \, dx=e^{\left (e^{e^x-x}-x\right ) \left (-4+e^{4 x}-\frac {x}{7}\right )+x} \]
Time = 0.18 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43 \[ \int \frac {1}{7} e^{\frac {1}{7} \left (e^{e^x-x} \left (-28+7 e^{4 x}-x\right )+35 x-7 e^{4 x} x+x^2\right )} \left (35+e^{4 x} (-7-28 x)+2 x+e^{e^x-x} \left (27+e^{4 x} \left (21+7 e^x\right )+e^x (-28-x)+x\right )\right ) \, dx=e^{\frac {1}{7} \left (e^{e^x-x} \left (-28+7 e^{4 x}-x\right )+35 x-7 e^{4 x} x+x^2\right )} \]
Integrate[(E^((E^(E^x - x)*(-28 + 7*E^(4*x) - x) + 35*x - 7*E^(4*x)*x + x^ 2)/7)*(35 + E^(4*x)*(-7 - 28*x) + 2*x + E^(E^x - x)*(27 + E^(4*x)*(21 + 7* E^x) + E^x*(-28 - x) + x)))/7,x]
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.
\(\displaystyle \int \frac {1}{7} \left (e^{4 x} (-28 x-7)+2 x+e^{e^x-x} \left (e^{4 x} \left (7 e^x+21\right )+e^x (-x-28)+x+27\right )+35\right ) \exp \left (\frac {1}{7} \left (x^2-7 e^{4 x} x+35 x+e^{e^x-x} \left (-x+7 e^{4 x}-28\right )\right )\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{7} \int \exp \left (\frac {1}{7} \left (x^2-7 e^{4 x} x+35 x-e^{e^x-x} \left (x-7 e^{4 x}+28\right )\right )\right ) \left (2 x-7 e^{4 x} (4 x+1)+e^{e^x-x} \left (7 e^{4 x} \left (3+e^x\right )+x-e^x (x+28)+27\right )+35\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{7} \int \left (2 \exp \left (\frac {1}{7} \left (x^2-7 e^{4 x} x+35 x-e^{e^x-x} \left (x-7 e^{4 x}+28\right )\right )\right ) x+35 \exp \left (\frac {1}{7} \left (x^2-7 e^{4 x} x+35 x-e^{e^x-x} \left (x-7 e^{4 x}+28\right )\right )\right )-7 \exp \left (4 x+\frac {1}{7} \left (x^2-7 e^{4 x} x+35 x-e^{e^x-x} \left (x-7 e^{4 x}+28\right )\right )\right ) (4 x+1)+\exp \left (-x+e^x+\frac {1}{7} \left (x^2-7 e^{4 x} x+35 x-e^{e^x-x} \left (x-7 e^{4 x}+28\right )\right )\right ) \left (-e^x x+x-28 e^x+21 e^{4 x}+7 e^{5 x}+27\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{7} \left (35 \int \exp \left (\frac {1}{7} \left (x^2-7 e^{4 x} x+35 x-e^{e^x-x} \left (x-7 e^{4 x}+28\right )\right )\right )dx-28 \int \exp \left (\frac {1}{7} \left (x^2-7 e^{4 x} x+35 x-e^{e^x-x} \left (x-7 e^{4 x}+28\right )\right )+e^x\right )dx+27 \int \exp \left (-x+e^x+\frac {1}{7} \left (x^2-7 e^{4 x} x+35 x-e^{e^x-x} \left (x-7 e^{4 x}+28\right )\right )\right )dx+21 \int \exp \left (3 x+e^x+\frac {1}{7} \left (x^2-7 e^{4 x} x+35 x-e^{e^x-x} \left (x-7 e^{4 x}+28\right )\right )\right )dx-7 \int \exp \left (4 x+\frac {1}{7} \left (x^2-7 e^{4 x} x+35 x-e^{e^x-x} \left (x-7 e^{4 x}+28\right )\right )\right )dx+7 \int \exp \left (4 x+e^x+\frac {1}{7} \left (x^2-7 e^{4 x} x+35 x-e^{e^x-x} \left (x-7 e^{4 x}+28\right )\right )\right )dx+2 \int \exp \left (\frac {1}{7} \left (x^2-7 e^{4 x} x+35 x-e^{e^x-x} \left (x-7 e^{4 x}+28\right )\right )\right ) xdx-\int \exp \left (\frac {1}{7} \left (x^2-7 e^{4 x} x+35 x-e^{e^x-x} \left (x-7 e^{4 x}+28\right )\right )+e^x\right ) xdx+\int \exp \left (-x+e^x+\frac {1}{7} \left (x^2-7 e^{4 x} x+35 x-e^{e^x-x} \left (x-7 e^{4 x}+28\right )\right )\right ) xdx-28 \int \exp \left (4 x+\frac {1}{7} \left (x^2-7 e^{4 x} x+35 x-e^{e^x-x} \left (x-7 e^{4 x}+28\right )\right )\right ) xdx\right )\) |
Int[(E^((E^(E^x - x)*(-28 + 7*E^(4*x) - x) + 35*x - 7*E^(4*x)*x + x^2)/7)* (35 + E^(4*x)*(-7 - 28*x) + 2*x + E^(E^x - x)*(27 + E^(4*x)*(21 + 7*E^x) + E^x*(-28 - x) + x)))/7,x]
3.27.55.3.1 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 = 0.28 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27
method | result | size |
parallelrisch | \({\mathrm e}^{\frac {\left (7 \,{\mathrm e}^{4 x}-x -28\right ) {\mathrm e}^{{\mathrm e}^{x}-x}}{7}-x \,{\mathrm e}^{4 x}+\frac {x^{2}}{7}+5 x}\) | \(38\) |
risch | \({\mathrm e}^{{\mathrm e}^{3 x +{\mathrm e}^{x}}-\frac {{\mathrm e}^{{\mathrm e}^{x}-x} x}{7}-4 \,{\mathrm e}^{{\mathrm e}^{x}-x}-x \,{\mathrm e}^{4 x}+\frac {x^{2}}{7}+5 x}\) | \(44\) |
int(1/7*(((7*exp(x)+21)*exp(4*x)+(-x-28)*exp(x)+x+27)*exp(exp(x)-x)+(-28*x -7)*exp(4*x)+2*x+35)*exp(1/7*(7*exp(4*x)-x-28)*exp(exp(x)-x)-x*exp(4*x)+1/ 7*x^2+5*x),x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {1}{7} e^{\frac {1}{7} \left (e^{e^x-x} \left (-28+7 e^{4 x}-x\right )+35 x-7 e^{4 x} x+x^2\right )} \left (35+e^{4 x} (-7-28 x)+2 x+e^{e^x-x} \left (27+e^{4 x} \left (21+7 e^x\right )+e^x (-28-x)+x\right )\right ) \, dx=e^{\left (\frac {1}{7} \, x^{2} - x e^{\left (4 \, x\right )} - \frac {1}{7} \, {\left (x - 7 \, e^{\left (4 \, x\right )} + 28\right )} e^{\left (-x + e^{x}\right )} + 5 \, x\right )} \]
integrate(1/7*(((7*exp(x)+21)*exp(4*x)+(-x-28)*exp(x)+x+27)*exp(exp(x)-x)+ (-28*x-7)*exp(4*x)+2*x+35)*exp(1/7*(7*exp(4*x)-x-28)*exp(exp(x)-x)-x*exp(4 *x)+1/7*x^2+5*x),x, algorithm=\
Time = 0.44 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {1}{7} e^{\frac {1}{7} \left (e^{e^x-x} \left (-28+7 e^{4 x}-x\right )+35 x-7 e^{4 x} x+x^2\right )} \left (35+e^{4 x} (-7-28 x)+2 x+e^{e^x-x} \left (27+e^{4 x} \left (21+7 e^x\right )+e^x (-28-x)+x\right )\right ) \, dx=e^{\frac {x^{2}}{7} - x e^{4 x} + 5 x + \left (- \frac {x}{7} + e^{4 x} - 4\right ) e^{- x + e^{x}}} \]
integrate(1/7*(((7*exp(x)+21)*exp(4*x)+(-x-28)*exp(x)+x+27)*exp(exp(x)-x)+ (-28*x-7)*exp(4*x)+2*x+35)*exp(1/7*(7*exp(4*x)-x-28)*exp(exp(x)-x)-x*exp(4 *x)+1/7*x**2+5*x),x)
Time = 0.52 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43 \[ \int \frac {1}{7} e^{\frac {1}{7} \left (e^{e^x-x} \left (-28+7 e^{4 x}-x\right )+35 x-7 e^{4 x} x+x^2\right )} \left (35+e^{4 x} (-7-28 x)+2 x+e^{e^x-x} \left (27+e^{4 x} \left (21+7 e^x\right )+e^x (-28-x)+x\right )\right ) \, dx=e^{\left (\frac {1}{7} \, x^{2} - x e^{\left (4 \, x\right )} - \frac {1}{7} \, x e^{\left (-x + e^{x}\right )} + 5 \, x + e^{\left (3 \, x + e^{x}\right )} - 4 \, e^{\left (-x + e^{x}\right )}\right )} \]
integrate(1/7*(((7*exp(x)+21)*exp(4*x)+(-x-28)*exp(x)+x+27)*exp(exp(x)-x)+ (-28*x-7)*exp(4*x)+2*x+35)*exp(1/7*(7*exp(4*x)-x-28)*exp(exp(x)-x)-x*exp(4 *x)+1/7*x^2+5*x),x, algorithm=\
Time = 0.43 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43 \[ \int \frac {1}{7} e^{\frac {1}{7} \left (e^{e^x-x} \left (-28+7 e^{4 x}-x\right )+35 x-7 e^{4 x} x+x^2\right )} \left (35+e^{4 x} (-7-28 x)+2 x+e^{e^x-x} \left (27+e^{4 x} \left (21+7 e^x\right )+e^x (-28-x)+x\right )\right ) \, dx=e^{\left (\frac {1}{7} \, x^{2} - x e^{\left (4 \, x\right )} - \frac {1}{7} \, x e^{\left (-x + e^{x}\right )} + 5 \, x + e^{\left (3 \, x + e^{x}\right )} - 4 \, e^{\left (-x + e^{x}\right )}\right )} \]
integrate(1/7*(((7*exp(x)+21)*exp(4*x)+(-x-28)*exp(x)+x+27)*exp(exp(x)-x)+ (-28*x-7)*exp(4*x)+2*x+35)*exp(1/7*(7*exp(4*x)-x-28)*exp(exp(x)-x)-x*exp(4 *x)+1/7*x^2+5*x),x, algorithm=\
Time = 14.78 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.63 \[ \int \frac {1}{7} e^{\frac {1}{7} \left (e^{e^x-x} \left (-28+7 e^{4 x}-x\right )+35 x-7 e^{4 x} x+x^2\right )} \left (35+e^{4 x} (-7-28 x)+2 x+e^{e^x-x} \left (27+e^{4 x} \left (21+7 e^x\right )+e^x (-28-x)+x\right )\right ) \, dx={\mathrm {e}}^{5\,x}\,{\mathrm {e}}^{{\mathrm {e}}^{3\,x}\,{\mathrm {e}}^{{\mathrm {e}}^x}}\,{\mathrm {e}}^{-4\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{{\mathrm {e}}^x}}\,{\mathrm {e}}^{-x\,{\mathrm {e}}^{4\,x}}\,{\mathrm {e}}^{\frac {x^2}{7}}\,{\mathrm {e}}^{-\frac {x\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{{\mathrm {e}}^x}}{7}} \]
int((exp(5*x - x*exp(4*x) - (exp(exp(x) - x)*(x - 7*exp(4*x) + 28))/7 + x^ 2/7)*(2*x - exp(4*x)*(28*x + 7) + exp(exp(x) - x)*(x - exp(x)*(x + 28) + e xp(4*x)*(7*exp(x) + 21) + 27) + 35))/7,x)