\(\int e^{1-e^x+e^{1-e^x+x+x \log (2 e^{2 x} x^2)}+x+x \log (2 e^{2 x} x^2)} (3-e^x+2 x+\log (2 e^{2 x} x^2)) \, dx\) [7744]

Optimal result

Integrand size = 68, antiderivative size = 25 \[ \int e^{1-e^x+e^{1-e^x+x+x \log \left (2 e^{2 x} x^2\right )}+x+x \log \left (2 e^{2 x} x^2\right )} \left (3-e^x+2 x+\log \left (2 e^{2 x} x^2\right )\right ) \, dx=e^{e^{1-e^x+x+x \log \left (2 e^{2 x} x^2\right )}} \]

[Out]

exp(exp(x*ln(2*exp(2*x)*x^2)+1-exp(x)+x))

Rubi [F]

\[ \int e^{1-e^x+e^{1-e^x+x+x \log \left (2 e^{2 x} x^2\right )}+x+x \log \left (2 e^{2 x} x^2\right )} \left (3-e^x+2 x+\log \left (2 e^{2 x} x^2\right )\right ) \, dx=\int \exp \left (1-e^x+e^{1-e^x+x+x \log \left (2 e^{2 x} x^2\right )}+x+x \log \left (2 e^{2 x} x^2\right )\right ) \left (3-e^x+2 x+\log \left (2 e^{2 x} x^2\right )\right ) \, dx \]

[In]

Int[E^(1 - E^x + E^(1 - E^x + x + x*Log[2*E^(2*x)*x^2]) + x + x*Log[2*E^(2*x)*x^2])*(3 - E^x + 2*x + Log[2*E^(
2*x)*x^2]),x]

[Out]

3*Defer[Int][E^(1 - E^x + E^(1 - E^x + x + x*Log[2*E^(2*x)*x^2]) + x + x*Log[2*E^(2*x)*x^2]), x] - Defer[Int][
E^(1 - E^x + E^(1 - E^x + x + x*Log[2*E^(2*x)*x^2]) + 2*x + x*Log[2*E^(2*x)*x^2]), x] + 2*Defer[Int][E^(1 - E^
x + E^(1 - E^x + x + x*Log[2*E^(2*x)*x^2]) + x + x*Log[2*E^(2*x)*x^2])*x, x] + Defer[Int][E^(1 - E^x + E^(1 -
E^x + x + x*Log[2*E^(2*x)*x^2]) + x + x*Log[2*E^(2*x)*x^2])*Log[2*E^(2*x)*x^2], x]

Rubi steps \begin{align*} \text {integral}& = \int \left (3 \exp \left (1-e^x+e^{1-e^x+x+x \log \left (2 e^{2 x} x^2\right )}+x+x \log \left (2 e^{2 x} x^2\right )\right )-\exp \left (1-e^x+e^{1-e^x+x+x \log \left (2 e^{2 x} x^2\right )}+2 x+x \log \left (2 e^{2 x} x^2\right )\right )+2 \exp \left (1-e^x+e^{1-e^x+x+x \log \left (2 e^{2 x} x^2\right )}+x+x \log \left (2 e^{2 x} x^2\right )\right ) x+\exp \left (1-e^x+e^{1-e^x+x+x \log \left (2 e^{2 x} x^2\right )}+x+x \log \left (2 e^{2 x} x^2\right )\right ) \log \left (2 e^{2 x} x^2\right )\right ) \, dx \\ & = 2 \int \exp \left (1-e^x+e^{1-e^x+x+x \log \left (2 e^{2 x} x^2\right )}+x+x \log \left (2 e^{2 x} x^2\right )\right ) x \, dx+3 \int \exp \left (1-e^x+e^{1-e^x+x+x \log \left (2 e^{2 x} x^2\right )}+x+x \log \left (2 e^{2 x} x^2\right )\right ) \, dx-\int \exp \left (1-e^x+e^{1-e^x+x+x \log \left (2 e^{2 x} x^2\right )}+2 x+x \log \left (2 e^{2 x} x^2\right )\right ) \, dx+\int \exp \left (1-e^x+e^{1-e^x+x+x \log \left (2 e^{2 x} x^2\right )}+x+x \log \left (2 e^{2 x} x^2\right )\right ) \log \left (2 e^{2 x} x^2\right ) \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 5.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int e^{1-e^x+e^{1-e^x+x+x \log \left (2 e^{2 x} x^2\right )}+x+x \log \left (2 e^{2 x} x^2\right )} \left (3-e^x+2 x+\log \left (2 e^{2 x} x^2\right )\right ) \, dx=e^{2^x e^{1-e^x+x} \left (e^{2 x} x^2\right )^x} \]

[In]

Integrate[E^(1 - E^x + E^(1 - E^x + x + x*Log[2*E^(2*x)*x^2]) + x + x*Log[2*E^(2*x)*x^2])*(3 - E^x + 2*x + Log
[2*E^(2*x)*x^2]),x]

[Out]

E^(2^x*E^(1 - E^x + x)*(E^(2*x)*x^2)^x)

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\({\mathrm e}^{{\mathrm e}^{x \ln \left (2 \,{\mathrm e}^{2 x} x^{2}\right )+1-{\mathrm e}^{x}+x}}\)	\(22\)
default	\({\mathrm e}^{{\mathrm e}^{x \ln \left (2 \,{\mathrm e}^{2 x} x^{2}\right )+1-{\mathrm e}^{x}+x}}\)	\(22\)
parallelrisch	\({\mathrm e}^{{\mathrm e}^{x \ln \left (2 \,{\mathrm e}^{2 x} x^{2}\right )+1-{\mathrm e}^{x}+x}}\)	\(22\)
risch	\({\mathrm e}^{2^{x} x^{2 x} \left ({\mathrm e}^{x}\right )^{2 x} {\mathrm e}^{1-\frac {i x \pi \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )^{3}}{2}+\frac {i x \operatorname {csgn}\left (i x^{2} {\mathrm e}^{2 x}\right )^{2} \operatorname {csgn}\left (i x^{2}\right ) \pi }{2}-\frac {i x \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{x}\right )^{2}}{2}-\frac {i \pi x \operatorname {csgn}\left (i x^{2}\right )^{3}}{2}-\frac {i x \,\operatorname {csgn}\left (i x^{2} {\mathrm e}^{2 x}\right ) \operatorname {csgn}\left (i x^{2}\right ) \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )}{2}-\frac {i \pi x \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )}{2}+i x \pi \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{x}\right )-\frac {i x \operatorname {csgn}\left (i x^{2} {\mathrm e}^{2 x}\right )^{3} \pi }{2}+i x \operatorname {csgn}\left (i x^{2}\right )^{2} \pi \,\operatorname {csgn}\left (i x \right )+\frac {i x \operatorname {csgn}\left (i x^{2} {\mathrm e}^{2 x}\right )^{2} \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )}{2}-{\mathrm e}^{x}+x}}\)	\(234\)

[In]

int((ln(2*exp(2*x)*x^2)-exp(x)+2*x+3)*exp(x*ln(2*exp(2*x)*x^2)+1-exp(x)+x)*exp(exp(x*ln(2*exp(2*x)*x^2)+1-exp(
x)+x)),x,method=_RETURNVERBOSE)

[Out]

exp(exp(x*ln(2*exp(2*x)*x^2)+1-exp(x)+x))

Fricas [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int e^{1-e^x+e^{1-e^x+x+x \log \left (2 e^{2 x} x^2\right )}+x+x \log \left (2 e^{2 x} x^2\right )} \left (3-e^x+2 x+\log \left (2 e^{2 x} x^2\right )\right ) \, dx=e^{\left (e^{\left (x \log \left (2 \, x^{2} e^{\left (2 \, x\right )}\right ) + x - e^{x} + 1\right )}\right )} \]

[In]

integrate((log(2*exp(2*x)*x^2)-exp(x)+2*x+3)*exp(x*log(2*exp(2*x)*x^2)+1-exp(x)+x)*exp(exp(x*log(2*exp(2*x)*x^
2)+1-exp(x)+x)),x, algorithm="fricas")

[Out]

e^(e^(x*log(2*x^2*e^(2*x)) + x - e^x + 1))

Sympy [F(-1)]

Timed out. \[ \int e^{1-e^x+e^{1-e^x+x+x \log \left (2 e^{2 x} x^2\right )}+x+x \log \left (2 e^{2 x} x^2\right )} \left (3-e^x+2 x+\log \left (2 e^{2 x} x^2\right )\right ) \, dx=\text {Timed out} \]

[In]

integrate((ln(2*exp(2*x)*x**2)-exp(x)+2*x+3)*exp(x*ln(2*exp(2*x)*x**2)+1-exp(x)+x)*exp(exp(x*ln(2*exp(2*x)*x**
2)+1-exp(x)+x)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.48 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int e^{1-e^x+e^{1-e^x+x+x \log \left (2 e^{2 x} x^2\right )}+x+x \log \left (2 e^{2 x} x^2\right )} \left (3-e^x+2 x+\log \left (2 e^{2 x} x^2\right )\right ) \, dx=e^{\left (e^{\left (2 \, x^{2} + x \log \left (2\right ) + 2 \, x \log \left (x\right ) + x - e^{x} + 1\right )}\right )} \]

[In]

integrate((log(2*exp(2*x)*x^2)-exp(x)+2*x+3)*exp(x*log(2*exp(2*x)*x^2)+1-exp(x)+x)*exp(exp(x*log(2*exp(2*x)*x^
2)+1-exp(x)+x)),x, algorithm="maxima")

[Out]

e^(e^(2*x^2 + x*log(2) + 2*x*log(x) + x - e^x + 1))

Giac [F]

\[ \int e^{1-e^x+e^{1-e^x+x+x \log \left (2 e^{2 x} x^2\right )}+x+x \log \left (2 e^{2 x} x^2\right )} \left (3-e^x+2 x+\log \left (2 e^{2 x} x^2\right )\right ) \, dx=\int { {\left (2 \, x - e^{x} + \log \left (2 \, x^{2} e^{\left (2 \, x\right )}\right ) + 3\right )} e^{\left (x \log \left (2 \, x^{2} e^{\left (2 \, x\right )}\right ) + x + e^{\left (x \log \left (2 \, x^{2} e^{\left (2 \, x\right )}\right ) + x - e^{x} + 1\right )} - e^{x} + 1\right )} \,d x } \]

[In]

integrate((log(2*exp(2*x)*x^2)-exp(x)+2*x+3)*exp(x*log(2*exp(2*x)*x^2)+1-exp(x)+x)*exp(exp(x*log(2*exp(2*x)*x^
2)+1-exp(x)+x)),x, algorithm="giac")

[Out]

integrate((2*x - e^x + log(2*x^2*e^(2*x)) + 3)*e^(x*log(2*x^2*e^(2*x)) + x + e^(x*log(2*x^2*e^(2*x)) + x - e^x
+ 1) - e^x + 1), x)

Mupad [B] (verification not implemented)

Time = 12.93 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int e^{1-e^x+e^{1-e^x+x+x \log \left (2 e^{2 x} x^2\right )}+x+x \log \left (2 e^{2 x} x^2\right )} \left (3-e^x+2 x+\log \left (2 e^{2 x} x^2\right )\right ) \, dx={\mathrm {e}}^{\mathrm {e}\,{\mathrm {e}}^{2\,x^2}\,{\mathrm {e}}^{-{\mathrm {e}}^x}\,{\mathrm {e}}^x\,{\left (2\,x^2\right )}^x} \]

[In]

int(exp(x - exp(x) + x*log(2*x^2*exp(2*x)) + 1)*exp(exp(x - exp(x) + x*log(2*x^2*exp(2*x)) + 1))*(2*x - exp(x)
+ log(2*x^2*exp(2*x)) + 3),x)

[Out]

exp(exp(1)*exp(2*x^2)*exp(-exp(x))*exp(x)*(2*x^2)^x)