\(\int e^{-4+e^{-4 x^2+2 x^3}+e^{-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)}+x^2-2 x \log (5)+\log ^2(5)} (2 x+e^{-4 x^2+2 x^3} (-8 x+6 x^2)-2 \log (5)) \, dx\) [7459]

Optimal result

Integrand size = 90, antiderivative size = 24 \[ \int e^{-4+e^{-4 x^2+2 x^3}+e^{-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)}+x^2-2 x \log (5)+\log ^2(5)} \left (2 x+e^{-4 x^2+2 x^3} \left (-8 x+6 x^2\right )-2 \log (5)\right ) \, dx=e^{e^{-4+e^{2 (-2+x) x^2}+(x-\log (5))^2}} \]

[Out]

exp(exp((-ln(5)+x)^2+exp(2*(-2+x)*x^2)-4))

Rubi [F]

\[ \int e^{-4+e^{-4 x^2+2 x^3}+e^{-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)}+x^2-2 x \log (5)+\log ^2(5)} \left (2 x+e^{-4 x^2+2 x^3} \left (-8 x+6 x^2\right )-2 \log (5)\right ) \, dx=\int \exp \left (-4+e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2-2 x \log (5)+\log ^2(5)\right ) \left (2 x+e^{-4 x^2+2 x^3} \left (-8 x+6 x^2\right )-2 \log (5)\right ) \, dx \]

[In]

Int[E^(-4 + E^(-4*x^2 + 2*x^3) + E^(-4 + E^(-4*x^2 + 2*x^3) + x^2 - 2*x*Log[5] + Log[5]^2) + x^2 - 2*x*Log[5]
+ Log[5]^2)*(2*x + E^(-4*x^2 + 2*x^3)*(-8*x + 6*x^2) - 2*Log[5]),x]

[Out]

-2*Log[5]*Defer[Int][E^(E^(-4*x^2 + 2*x^3) + E^(-4 + E^(-4*x^2 + 2*x^3) + x^2 - 2*x*Log[5] + Log[5]^2) + x^2 -
2*x*Log[5] - 4*(1 - Log[5]^2/4)), x] + 2*Defer[Int][E^(E^(-4*x^2 + 2*x^3) + E^(-4 + E^(-4*x^2 + 2*x^3) + x^2
- 2*x*Log[5] + Log[5]^2) + x^2 - 2*x*Log[5] - 4*(1 - Log[5]^2/4))*x, x] - 8*Defer[Int][E^(E^(-4*x^2 + 2*x^3) +
E^(-4 + E^(-4*x^2 + 2*x^3) + x^2 - 2*x*Log[5] + Log[5]^2) + x^2 + 2*(-2 + x)*x^2 - 2*x*Log[5] - 4*(1 - Log[5]
^2/4))*x, x] + 6*Defer[Int][E^(E^(-4*x^2 + 2*x^3) + E^(-4 + E^(-4*x^2 + 2*x^3) + x^2 - 2*x*Log[5] + Log[5]^2)
+ x^2 + 2*(-2 + x)*x^2 - 2*x*Log[5] - 4*(1 - Log[5]^2/4))*x^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) \left (2 x+e^{-4 x^2+2 x^3} \left (-8 x+6 x^2\right )-2 \log (5)\right ) \, dx \\ & = \int \left (2 \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) x+2 \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2+2 (-2+x) x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) x (-4+3 x)-2 \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) \log (5)\right ) \, dx \\ & = 2 \int \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) x \, dx+2 \int \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2+2 (-2+x) x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) x (-4+3 x) \, dx-(2 \log (5)) \int \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) \, dx \\ & = 2 \int \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) x \, dx+2 \int \left (-4 \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2+2 (-2+x) x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) x+3 \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2+2 (-2+x) x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) x^2\right ) \, dx-(2 \log (5)) \int \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) \, dx \\ & = 2 \int \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) x \, dx+6 \int \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2+2 (-2+x) x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) x^2 \, dx-8 \int \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2+2 (-2+x) x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) x \, dx-(2 \log (5)) \int \exp \left (e^{-4 x^2+2 x^3}+\exp \left (-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)\right )+x^2-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) \, dx \\ \end{align*}

Mathematica [F]

\[ \int e^{-4+e^{-4 x^2+2 x^3}+e^{-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)}+x^2-2 x \log (5)+\log ^2(5)} \left (2 x+e^{-4 x^2+2 x^3} \left (-8 x+6 x^2\right )-2 \log (5)\right ) \, dx=\int e^{-4+e^{-4 x^2+2 x^3}+e^{-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)}+x^2-2 x \log (5)+\log ^2(5)} \left (2 x+e^{-4 x^2+2 x^3} \left (-8 x+6 x^2\right )-2 \log (5)\right ) \, dx \]

[In]

Integrate[E^(-4 + E^(-4*x^2 + 2*x^3) + E^(-4 + E^(-4*x^2 + 2*x^3) + x^2 - 2*x*Log[5] + Log[5]^2) + x^2 - 2*x*L
og[5] + Log[5]^2)*(2*x + E^(-4*x^2 + 2*x^3)*(-8*x + 6*x^2) - 2*Log[5]),x]

[Out]

Integrate[E^(-4 + E^(-4*x^2 + 2*x^3) + E^(-4 + E^(-4*x^2 + 2*x^3) + x^2 - 2*x*Log[5] + Log[5]^2) + x^2 - 2*x*L
og[5] + Log[5]^2)*(2*x + E^(-4*x^2 + 2*x^3)*(-8*x + 6*x^2) - 2*Log[5]), x]

Maple [A] (verified)

Time = 0.99 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04

method	result	size
risch	\({\mathrm e}^{\left (\frac {1}{25}\right )^{x} {\mathrm e}^{{\mathrm e}^{2 \left (-2+x \right ) x^{2}}+\ln \left (5\right )^{2}-4+x^{2}}}\)	\(25\)
derivativedivides	\({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2 x^{3}-4 x^{2}}+\ln \left (5\right )^{2}-2 x \ln \left (5\right )+x^{2}-4}}\)	\(29\)
default	\({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2 x^{3}-4 x^{2}}+\ln \left (5\right )^{2}-2 x \ln \left (5\right )+x^{2}-4}}\)	\(29\)
norman	\({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2 x^{3}-4 x^{2}}+\ln \left (5\right )^{2}-2 x \ln \left (5\right )+x^{2}-4}}\)	\(29\)
parallelrisch	\({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2 x^{3}-4 x^{2}}+\ln \left (5\right )^{2}-2 x \ln \left (5\right )+x^{2}-4}}\)	\(29\)

[In]

int(((6*x^2-8*x)*exp(2*x^3-4*x^2)-2*ln(5)+2*x)*exp(exp(2*x^3-4*x^2)+ln(5)^2-2*x*ln(5)+x^2-4)*exp(exp(exp(2*x^3
-4*x^2)+ln(5)^2-2*x*ln(5)+x^2-4)),x,method=_RETURNVERBOSE)

[Out]

exp((1/25)^x*exp(exp(2*(-2+x)*x^2)+ln(5)^2-4+x^2))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int e^{-4+e^{-4 x^2+2 x^3}+e^{-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)}+x^2-2 x \log (5)+\log ^2(5)} \left (2 x+e^{-4 x^2+2 x^3} \left (-8 x+6 x^2\right )-2 \log (5)\right ) \, dx=e^{\left (e^{\left (x^{2} - 2 \, x \log \left (5\right ) + \log \left (5\right )^{2} + e^{\left (2 \, x^{3} - 4 \, x^{2}\right )} - 4\right )}\right )} \]

[In]

integrate(((6*x^2-8*x)*exp(2*x^3-4*x^2)-2*log(5)+2*x)*exp(exp(2*x^3-4*x^2)+log(5)^2-2*x*log(5)+x^2-4)*exp(exp(
exp(2*x^3-4*x^2)+log(5)^2-2*x*log(5)+x^2-4)),x, algorithm="fricas")

[Out]

e^(e^(x^2 - 2*x*log(5) + log(5)^2 + e^(2*x^3 - 4*x^2) - 4))

Sympy [A] (verification not implemented)

Time = 0.61 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int e^{-4+e^{-4 x^2+2 x^3}+e^{-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)}+x^2-2 x \log (5)+\log ^2(5)} \left (2 x+e^{-4 x^2+2 x^3} \left (-8 x+6 x^2\right )-2 \log (5)\right ) \, dx=e^{e^{x^{2} - 2 x \log {\left (5 \right )} + e^{2 x^{3} - 4 x^{2}} - 4 + \log {\left (5 \right )}^{2}}} \]

[In]

integrate(((6*x**2-8*x)*exp(2*x**3-4*x**2)-2*ln(5)+2*x)*exp(exp(2*x**3-4*x**2)+ln(5)**2-2*x*ln(5)+x**2-4)*exp(
exp(exp(2*x**3-4*x**2)+ln(5)**2-2*x*ln(5)+x**2-4)),x)

[Out]

exp(exp(x**2 - 2*x*log(5) + exp(2*x**3 - 4*x**2) - 4 + log(5)**2))

Maxima [A] (verification not implemented)

none

Time = 0.84 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int e^{-4+e^{-4 x^2+2 x^3}+e^{-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)}+x^2-2 x \log (5)+\log ^2(5)} \left (2 x+e^{-4 x^2+2 x^3} \left (-8 x+6 x^2\right )-2 \log (5)\right ) \, dx=e^{\left (e^{\left (x^{2} - 2 \, x \log \left (5\right ) + \log \left (5\right )^{2} + e^{\left (2 \, x^{3} - 4 \, x^{2}\right )} - 4\right )}\right )} \]

[In]

integrate(((6*x^2-8*x)*exp(2*x^3-4*x^2)-2*log(5)+2*x)*exp(exp(2*x^3-4*x^2)+log(5)^2-2*x*log(5)+x^2-4)*exp(exp(
exp(2*x^3-4*x^2)+log(5)^2-2*x*log(5)+x^2-4)),x, algorithm="maxima")

[Out]

e^(e^(x^2 - 2*x*log(5) + log(5)^2 + e^(2*x^3 - 4*x^2) - 4))

Giac [F]

\[ \int e^{-4+e^{-4 x^2+2 x^3}+e^{-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)}+x^2-2 x \log (5)+\log ^2(5)} \left (2 x+e^{-4 x^2+2 x^3} \left (-8 x+6 x^2\right )-2 \log (5)\right ) \, dx=\int { 2 \, {\left ({\left (3 \, x^{2} - 4 \, x\right )} e^{\left (2 \, x^{3} - 4 \, x^{2}\right )} + x - \log \left (5\right )\right )} e^{\left (x^{2} - 2 \, x \log \left (5\right ) + \log \left (5\right )^{2} + e^{\left (2 \, x^{3} - 4 \, x^{2}\right )} + e^{\left (x^{2} - 2 \, x \log \left (5\right ) + \log \left (5\right )^{2} + e^{\left (2 \, x^{3} - 4 \, x^{2}\right )} - 4\right )} - 4\right )} \,d x } \]

[In]

integrate(((6*x^2-8*x)*exp(2*x^3-4*x^2)-2*log(5)+2*x)*exp(exp(2*x^3-4*x^2)+log(5)^2-2*x*log(5)+x^2-4)*exp(exp(
exp(2*x^3-4*x^2)+log(5)^2-2*x*log(5)+x^2-4)),x, algorithm="giac")

[Out]

integrate(2*((3*x^2 - 4*x)*e^(2*x^3 - 4*x^2) + x - log(5))*e^(x^2 - 2*x*log(5) + log(5)^2 + e^(2*x^3 - 4*x^2)
+ e^(x^2 - 2*x*log(5) + log(5)^2 + e^(2*x^3 - 4*x^2) - 4) - 4), x)

Mupad [B] (verification not implemented)

Time = 13.60 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42 \[ \int e^{-4+e^{-4 x^2+2 x^3}+e^{-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)}+x^2-2 x \log (5)+\log ^2(5)} \left (2 x+e^{-4 x^2+2 x^3} \left (-8 x+6 x^2\right )-2 \log (5)\right ) \, dx={\mathrm {e}}^{\frac {{\mathrm {e}}^{{\ln \left (5\right )}^2}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x^3}\,{\mathrm {e}}^{-4\,x^2}}}{5^{2\,x}}} \]

[In]

int(-exp(exp(exp(2*x^3 - 4*x^2) - 2*x*log(5) + log(5)^2 + x^2 - 4))*exp(exp(2*x^3 - 4*x^2) - 2*x*log(5) + log(
5)^2 + x^2 - 4)*(2*log(5) - 2*x + exp(2*x^3 - 4*x^2)*(8*x - 6*x^2)),x)

[Out]

exp((exp(log(5)^2)*exp(x^2)*exp(-4)*exp(exp(2*x^3)*exp(-4*x^2)))/5^(2*x))