Integrand size = 90, antiderivative size = 24 \[ \int e^{-4+e^{-4 x^2+2 x^3}+e^{-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)}+x^2-2 x \log (5)+\log ^2(5)} \left (2 x+e^{-4 x^2+2 x^3} \left (-8 x+6 x^2\right )-2 \log (5)\right ) \, dx=e^{e^{-4+e^{2 (-2+x) x^2}+(x-\log (5))^2}} \] Output:
exp(exp((-ln(5)+x)^2+exp(2*(-2+x)*x^2)-4))
\[ \int e^{-4+e^{-4 x^2+2 x^3}+e^{-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)}+x^2-2 x \log (5)+\log ^2(5)} \left (2 x+e^{-4 x^2+2 x^3} \left (-8 x+6 x^2\right )-2 \log (5)\right ) \, dx=\int e^{-4+e^{-4 x^2+2 x^3}+e^{-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)}+x^2-2 x \log (5)+\log ^2(5)} \left (2 x+e^{-4 x^2+2 x^3} \left (-8 x+6 x^2\right )-2 \log (5)\right ) \, dx \] Input:
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*Log[5] + Log[5]^2)*(2*x + E^(-4*x^2 + 2 *x^3)*(-8*x + 6*x^2) - 2*Log[5]),x]
Output:
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*Log[5] + Log[5]^2)*(2*x + E^(-4*x^2 + 2 *x^3)*(-8*x + 6*x^2) - 2*Log[5]), 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.
| \(\displaystyle \int \left (e^{2 x^3-4 x^2} \left (6 x^2-8 x\right )+2 x-2 \log (5)\right ) \exp \left (x^2+e^{2 x^3-4 x^2}+e^{x^2+e^{2 x^3-4 x^2}-2 x \log (5)-4+\log ^2(5)}-2 x \log (5)-4+\log ^2(5)\right ) \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \left (e^{2 x^3-4 x^2} \left (6 x^2-8 x\right )+2 x-2 \log (5)\right ) \exp \left (x^2+e^{2 x^3-4 x^2}+e^{x^2+e^{2 x^3-4 x^2}-2 x \log (5)-4+\log ^2(5)}-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2 x \exp \left (x^2+e^{2 x^3-4 x^2}+e^{x^2+e^{2 x^3-4 x^2}-2 x \log (5)-4+\log ^2(5)}-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right )+2 (3 x-4) x \exp \left (2 (x-2) x^2+x^2+e^{2 x^3-4 x^2}+e^{x^2+e^{2 x^3-4 x^2}-2 x \log (5)-4+\log ^2(5)}-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right )-2 \log (5) \exp \left (x^2+e^{2 x^3-4 x^2}+e^{x^2+e^{2 x^3-4 x^2}-2 x \log (5)-4+\log ^2(5)}-2 x \log (5)-4 \left (1-\frac {\log ^2(5)}{4}\right )\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \log (5) \int \exp \left (x^2-2 \log (5) x+e^{2 x^3-4 x^2}+e^{x^2-2 \log (5) x+e^{2 x^3-4 x^2}+\log ^2(5)-4}-4 \left (1-\frac {\log ^2(5)}{4}\right )\right )dx+2 \int \exp \left (x^2-2 \log (5) x+e^{2 x^3-4 x^2}+e^{x^2-2 \log (5) x+e^{2 x^3-4 x^2}+\log ^2(5)-4}-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) xdx-8 \int \exp \left (2 (x-2) x^2+x^2-2 \log (5) x+e^{2 x^3-4 x^2}+e^{x^2-2 \log (5) x+e^{2 x^3-4 x^2}+\log ^2(5)-4}-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) xdx+6 \int \exp \left (2 (x-2) x^2+x^2-2 \log (5) x+e^{2 x^3-4 x^2}+e^{x^2-2 \log (5) x+e^{2 x^3-4 x^2}+\log ^2(5)-4}-4 \left (1-\frac {\log ^2(5)}{4}\right )\right ) x^2dx\) |
Input:
Int[E^(-4 + E^(-4*x^2 + 2*x^3) + E^(-4 + E^(-4*x^2 + 2*x^3) + x^2 - 2*x*Lo g[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]
Output:
$Aborted
Time = 1.07 (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 | \({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2 x^{3}-4 x^{2}}+\ln \left (5\right )^{2}-2 x \ln \left (5\right )+x^{2}-4}}\) | \(29\) |
Input:
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,me thod=_RETURNVERBOSE)
Output:
exp((1/25)^x*exp(exp(2*(-2+x)*x^2)+ln(5)^2-4+x^2))
Time = 0.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int e^{-4+e^{-4 x^2+2 x^3}+e^{-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)}+x^2-2 x \log (5)+\log ^2(5)} \left (2 x+e^{-4 x^2+2 x^3} \left (-8 x+6 x^2\right )-2 \log (5)\right ) \, dx=e^{\left (e^{\left (x^{2} - 2 \, x \log \left (5\right ) + \log \left (5\right )^{2} + e^{\left (2 \, x^{3} - 4 \, x^{2}\right )} - 4\right )}\right )} \] Input:
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")
Output:
e^(e^(x^2 - 2*x*log(5) + log(5)^2 + e^(2*x^3 - 4*x^2) - 4))
Time = 0.75 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int e^{-4+e^{-4 x^2+2 x^3}+e^{-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)}+x^2-2 x \log (5)+\log ^2(5)} \left (2 x+e^{-4 x^2+2 x^3} \left (-8 x+6 x^2\right )-2 \log (5)\right ) \, dx=e^{e^{x^{2} - 2 x \log {\left (5 \right )} + e^{2 x^{3} - 4 x^{2}} - 4 + \log {\left (5 \right )}^{2}}} \] Input:
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)
Output:
exp(exp(x**2 - 2*x*log(5) + exp(2*x**3 - 4*x**2) - 4 + log(5)**2))
Time = 0.71 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int e^{-4+e^{-4 x^2+2 x^3}+e^{-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)}+x^2-2 x \log (5)+\log ^2(5)} \left (2 x+e^{-4 x^2+2 x^3} \left (-8 x+6 x^2\right )-2 \log (5)\right ) \, dx=e^{\left (e^{\left (x^{2} - 2 \, x \log \left (5\right ) + \log \left (5\right )^{2} + e^{\left (2 \, x^{3} - 4 \, x^{2}\right )} - 4\right )}\right )} \] Input:
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")
Output:
e^(e^(x^2 - 2*x*log(5) + log(5)^2 + e^(2*x^3 - 4*x^2) - 4))
\[ \int e^{-4+e^{-4 x^2+2 x^3}+e^{-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)}+x^2-2 x \log (5)+\log ^2(5)} \left (2 x+e^{-4 x^2+2 x^3} \left (-8 x+6 x^2\right )-2 \log (5)\right ) \, dx=\int { 2 \, {\left ({\left (3 \, x^{2} - 4 \, x\right )} e^{\left (2 \, x^{3} - 4 \, x^{2}\right )} + x - \log \left (5\right )\right )} e^{\left (x^{2} - 2 \, x \log \left (5\right ) + \log \left (5\right )^{2} + e^{\left (2 \, x^{3} - 4 \, x^{2}\right )} + e^{\left (x^{2} - 2 \, x \log \left (5\right ) + \log \left (5\right )^{2} + e^{\left (2 \, x^{3} - 4 \, x^{2}\right )} - 4\right )} - 4\right )} \,d x } \] Input:
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")
Output:
integrate(2*((3*x^2 - 4*x)*e^(2*x^3 - 4*x^2) + x - log(5))*e^(x^2 - 2*x*lo g(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)
Time = 3.13 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42 \[ \int e^{-4+e^{-4 x^2+2 x^3}+e^{-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)}+x^2-2 x \log (5)+\log ^2(5)} \left (2 x+e^{-4 x^2+2 x^3} \left (-8 x+6 x^2\right )-2 \log (5)\right ) \, dx={\mathrm {e}}^{\frac {{\mathrm {e}}^{{\ln \left (5\right )}^2}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x^3}\,{\mathrm {e}}^{-4\,x^2}}}{5^{2\,x}}} \] Input:
int(-exp(exp(exp(2*x^3 - 4*x^2) - 2*x*log(5) + log(5)^2 + x^2 - 4))*exp(ex p(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)
Output:
exp((exp(log(5)^2)*exp(x^2)*exp(-4)*exp(exp(2*x^3)*exp(-4*x^2)))/5^(2*x))
\[ \int e^{-4+e^{-4 x^2+2 x^3}+e^{-4+e^{-4 x^2+2 x^3}+x^2-2 x \log (5)+\log ^2(5)}+x^2-2 x \log (5)+\log ^2(5)} \left (2 x+e^{-4 x^2+2 x^3} \left (-8 x+6 x^2\right )-2 \log (5)\right ) \, dx=\frac {2 e^{\mathrm {log}\left (5\right )^{2}} \left (-\left (\int \frac {e^{\frac {e^{\frac {e^{2 x^{3}}+e^{4 x^{2}} \mathrm {log}\left (5\right )^{2}+5 e^{4 x^{2}} x^{2}}{e^{4 x^{2}}}}+e^{2 x^{3}} 5^{2 x} e^{4}+e^{4 x^{2}} 5^{2 x} e^{4} x^{2}}{e^{4 x^{2}} 5^{2 x} e^{4}}}}{5^{2 x}}d x \right ) \mathrm {log}\left (5\right )+\int \frac {e^{\frac {e^{\frac {e^{2 x^{3}}+e^{4 x^{2}} \mathrm {log}\left (5\right )^{2}+5 e^{4 x^{2}} x^{2}}{e^{4 x^{2}}}}+e^{2 x^{3}} 5^{2 x} e^{4}+e^{4 x^{2}} 5^{2 x} e^{4} x^{2}}{e^{4 x^{2}} 5^{2 x} e^{4}}} x}{5^{2 x}}d x +3 \left (\int \frac {e^{\frac {e^{\frac {e^{2 x^{3}}+e^{4 x^{2}} \mathrm {log}\left (5\right )^{2}+5 e^{4 x^{2}} x^{2}}{e^{4 x^{2}}}}+e^{2 x^{3}} 5^{2 x} e^{4}+2 e^{4 x^{2}} 5^{2 x} e^{4} x^{3}}{e^{4 x^{2}} 5^{2 x} e^{4}}} x^{2}}{e^{3 x^{2}} 5^{2 x}}d x \right )-4 \left (\int \frac {e^{\frac {e^{\frac {e^{2 x^{3}}+e^{4 x^{2}} \mathrm {log}\left (5\right )^{2}+5 e^{4 x^{2}} x^{2}}{e^{4 x^{2}}}}+e^{2 x^{3}} 5^{2 x} e^{4}+2 e^{4 x^{2}} 5^{2 x} e^{4} x^{3}}{e^{4 x^{2}} 5^{2 x} e^{4}}} x}{e^{3 x^{2}} 5^{2 x}}d x \right )\right )}{e^{4}} \] Input:
int(((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)
Output:
(2*e**(log(5)**2)*( - int(e**((e**((e**(2*x**3) + e**(4*x**2)*log(5)**2 + 5*e**(4*x**2)*x**2)/e**(4*x**2)) + e**(2*x**3)*5**(2*x)*e**4 + e**(4*x**2) *5**(2*x)*e**4*x**2)/(e**(4*x**2)*5**(2*x)*e**4))/5**(2*x),x)*log(5) + int ((e**((e**((e**(2*x**3) + e**(4*x**2)*log(5)**2 + 5*e**(4*x**2)*x**2)/e**( 4*x**2)) + e**(2*x**3)*5**(2*x)*e**4 + e**(4*x**2)*5**(2*x)*e**4*x**2)/(e* *(4*x**2)*5**(2*x)*e**4))*x)/5**(2*x),x) + 3*int((e**((e**((e**(2*x**3) + e**(4*x**2)*log(5)**2 + 5*e**(4*x**2)*x**2)/e**(4*x**2)) + e**(2*x**3)*5** (2*x)*e**4 + 2*e**(4*x**2)*5**(2*x)*e**4*x**3)/(e**(4*x**2)*5**(2*x)*e**4) )*x**2)/(e**(3*x**2)*5**(2*x)),x) - 4*int((e**((e**((e**(2*x**3) + e**(4*x **2)*log(5)**2 + 5*e**(4*x**2)*x**2)/e**(4*x**2)) + e**(2*x**3)*5**(2*x)*e **4 + 2*e**(4*x**2)*5**(2*x)*e**4*x**3)/(e**(4*x**2)*5**(2*x)*e**4))*x)/(e **(3*x**2)*5**(2*x)),x)))/e**4