Integrand size = 129, antiderivative size = 22 \[ \int \frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{9+9 x+(-6-6 x) \log (4+4 x)+(1+x) \log ^2(4+4 x)} \, dx=e^{-x+x^4-\frac {4 x}{-3+\log (4+4 x)}} \] Output:
exp(x^4-4/(ln(4+4*x)-3)*x-x)
Time = 0.16 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{9+9 x+(-6-6 x) \log (4+4 x)+(1+x) \log ^2(4+4 x)} \, dx=e^{\frac {x \left (-1-3 x^3+\left (-1+x^3\right ) \log (4 (1+x))\right )}{-3+\log (4 (1+x))}} \] Input:
Integrate[(E^((-x - 3*x^4 + (-x + x^4)*Log[4 + 4*x])/(-3 + Log[4 + 4*x]))* (3 + 7*x + 36*x^3 + 36*x^4 + (2 + 2*x - 24*x^3 - 24*x^4)*Log[4 + 4*x] + (- 1 - x + 4*x^3 + 4*x^4)*Log[4 + 4*x]^2))/(9 + 9*x + (-6 - 6*x)*Log[4 + 4*x] + (1 + x)*Log[4 + 4*x]^2),x]
Output:
E^((x*(-1 - 3*x^3 + (-1 + x^3)*Log[4*(1 + x)]))/(-3 + Log[4*(1 + 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.
| \(\displaystyle \int \frac {\left (36 x^4+36 x^3+\left (4 x^4+4 x^3-x-1\right ) \log ^2(4 x+4)+\left (-24 x^4-24 x^3+2 x+2\right ) \log (4 x+4)+7 x+3\right ) \exp \left (\frac {-3 x^4+\left (x^4-x\right ) \log (4 x+4)-x}{\log (4 x+4)-3}\right )}{9 x+(x+1) \log ^2(4 x+4)+(-6 x-6) \log (4 x+4)+9} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (36 x^4+36 x^3+\left (4 x^4+4 x^3-x-1\right ) \log ^2(4 x+4)+\left (-24 x^4-24 x^3+2 x+2\right ) \log (4 x+4)+7 x+3\right ) \exp \left (\frac {-3 x^4+\left (x^4-x\right ) \log (4 x+4)-x}{\log (4 x+4)-3}\right )}{(x+1) (3-\log (4 (x+1)))^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {36 x^4 \exp \left (\frac {-3 x^4+\left (x^4-x\right ) \log (4 x+4)-x}{\log (4 x+4)-3}\right )}{(x+1) (\log (4 (x+1))-3)^2}+\frac {7 x \exp \left (\frac {-3 x^4+\left (x^4-x\right ) \log (4 x+4)-x}{\log (4 x+4)-3}\right )}{(x+1) (\log (4 (x+1))-3)^2}+\frac {3 \exp \left (\frac {-3 x^4+\left (x^4-x\right ) \log (4 x+4)-x}{\log (4 x+4)-3}\right )}{(x+1) (\log (4 (x+1))-3)^2}+\frac {\left (4 x^3-1\right ) \log ^2(4 x+4) \exp \left (\frac {-3 x^4+\left (x^4-x\right ) \log (4 x+4)-x}{\log (4 x+4)-3}\right )}{(3-\log (4 (x+1)))^2}+\frac {36 x^3 \exp \left (\frac {-3 x^4+\left (x^4-x\right ) \log (4 x+4)-x}{\log (4 x+4)-3}\right )}{(x+1) (\log (4 (x+1))-3)^2}+\frac {2 \left (1-12 x^3\right ) \log (4 x+4) \exp \left (\frac {-3 x^4+\left (x^4-x\right ) \log (4 x+4)-x}{\log (4 x+4)-3}\right )}{(3-\log (4 (x+1)))^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {4^{\frac {x \left (x^3-1\right )}{\log (4 (x+1))-3}} e^{\frac {x \left (3 x^3+1\right )}{3-\log (4 (x+1))}} (x+1)^{\frac {x \left (x^3-1\right )}{\log (4 (x+1))-3}-1} \left (36 x^4+36 x^3+\left (4 x^4+4 x^3-x-1\right ) \log ^2(4 (x+1))+\left (-24 x^4-24 x^3+2 x+2\right ) \log (4 (x+1))+7 x+3\right )}{(3-\log (4 (x+1)))^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-4^{\frac {x \left (x^3-1\right )}{\log (4 (x+1))-3}} e^{\frac {x \left (3 x^3+1\right )}{3-\log (4 (x+1))}} (x+1)^{\frac {x \left (x^3-1\right )}{\log (4 (x+1))-3}-1}+x^3 4^{\frac {x \left (x^3-1\right )}{\log (4 (x+1))-3}+1} e^{\frac {x \left (3 x^3+1\right )}{3-\log (4 (x+1))}} (x+1)^{\frac {x \left (x^3-1\right )}{\log (4 (x+1))-3}-1}-x 4^{\frac {x \left (x^3-1\right )}{\log (4 (x+1))-3}} e^{\frac {x \left (3 x^3+1\right )}{3-\log (4 (x+1))}} (x+1)^{\frac {x \left (x^3-1\right )}{\log (4 (x+1))-3}-1}+\frac {x 4^{\frac {x \left (x^3-1\right )}{\log (4 (x+1))-3}+1} e^{\frac {x \left (3 x^3+1\right )}{3-\log (4 (x+1))}} (x+1)^{\frac {x \left (x^3-1\right )}{\log (4 (x+1))-3}-1}}{(\log (4 (x+1))-3)^2}-\frac {4^{\frac {x \left (x^3-1\right )}{\log (4 (x+1))-3}+1} e^{\frac {x \left (3 x^3+1\right )}{3-\log (4 (x+1))}} (x+1)^{\frac {x \left (x^3-1\right )}{\log (4 (x+1))-3}}}{\log (4 (x+1))-3}+x^4 4^{\frac {x \left (x^3-1\right )}{\log (4 (x+1))-3}+1} e^{\frac {x \left (3 x^3+1\right )}{3-\log (4 (x+1))}} (x+1)^{\frac {x \left (x^3-1\right )}{\log (4 (x+1))-3}-1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\int 4^{\frac {x \left (x^3-1\right )}{\log (4 (x+1))-3}} e^{\frac {x \left (3 x^3+1\right )}{3-\log (4 (x+1))}} (x+1)^{\frac {x \left (x^3-1\right )}{\log (4 (x+1))-3}-1}dx-\int 4^{\frac {x \left (x^3-1\right )}{\log (4 (x+1))-3}} e^{\frac {x \left (3 x^3+1\right )}{3-\log (4 (x+1))}} x (x+1)^{\frac {x \left (x^3-1\right )}{\log (4 (x+1))-3}-1}dx+\int 4^{\frac {x \left (x^3-1\right )}{\log (4 (x+1))-3}+1} e^{\frac {x \left (3 x^3+1\right )}{3-\log (4 (x+1))}} x^3 (x+1)^{\frac {x \left (x^3-1\right )}{\log (4 (x+1))-3}-1}dx-\int \frac {4^{\frac {x \left (x^3-1\right )}{\log (4 (x+1))-3}+1} e^{\frac {x \left (3 x^3+1\right )}{3-\log (4 (x+1))}} (x+1)^{\frac {x \left (x^3-1\right )}{\log (4 (x+1))-3}-1}}{(\log (4 (x+1))-3)^2}dx+\int \frac {4^{\frac {x \left (x^3-1\right )}{\log (4 (x+1))-3}+1} e^{\frac {x \left (3 x^3+1\right )}{3-\log (4 (x+1))}} (x+1)^{\frac {x \left (x^3-1\right )}{\log (4 (x+1))-3}}}{(\log (4 (x+1))-3)^2}dx-\int \frac {4^{\frac {x \left (x^3-1\right )}{\log (4 (x+1))-3}+1} e^{\frac {x \left (3 x^3+1\right )}{3-\log (4 (x+1))}} (x+1)^{\frac {x \left (x^3-1\right )}{\log (4 (x+1))-3}}}{\log (4 (x+1))-3}dx+\int 4^{\frac {x \left (x^3-1\right )}{\log (4 (x+1))-3}+1} e^{\frac {x \left (3 x^3+1\right )}{3-\log (4 (x+1))}} x^4 (x+1)^{\frac {x \left (x^3-1\right )}{\log (4 (x+1))-3}-1}dx\) |
Input:
Int[(E^((-x - 3*x^4 + (-x + x^4)*Log[4 + 4*x])/(-3 + Log[4 + 4*x]))*(3 + 7 *x + 36*x^3 + 36*x^4 + (2 + 2*x - 24*x^3 - 24*x^4)*Log[4 + 4*x] + (-1 - x + 4*x^3 + 4*x^4)*Log[4 + 4*x]^2))/(9 + 9*x + (-6 - 6*x)*Log[4 + 4*x] + (1 + x)*Log[4 + 4*x]^2),x]
Output:
$Aborted
Time = 1.90 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.64
| method | result | size |
| parallelrisch | \({\mathrm e}^{\frac {\left (x^{4}-x \right ) \ln \left (4+4 x \right )-3 x^{4}-x}{\ln \left (4+4 x \right )-3}}\) | \(36\) |
| risch | \({\mathrm e}^{\frac {x \left (\ln \left (4+4 x \right ) x^{3}-3 x^{3}-\ln \left (4+4 x \right )-1\right )}{\ln \left (4+4 x \right )-3}}\) | \(39\) |
Input:
int(((4*x^4+4*x^3-x-1)*ln(4+4*x)^2+(-24*x^4-24*x^3+2*x+2)*ln(4+4*x)+36*x^4 +36*x^3+7*x+3)*exp(((x^4-x)*ln(4+4*x)-3*x^4-x)/(ln(4+4*x)-3))/((1+x)*ln(4+ 4*x)^2+(-6*x-6)*ln(4+4*x)+9*x+9),x,method=_RETURNVERBOSE)
Output:
exp(((x^4-x)*ln(4+4*x)-3*x^4-x)/(ln(4+4*x)-3))
Time = 0.11 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{9+9 x+(-6-6 x) \log (4+4 x)+(1+x) \log ^2(4+4 x)} \, dx=e^{\left (-\frac {3 \, x^{4} - {\left (x^{4} - x\right )} \log \left (4 \, x + 4\right ) + x}{\log \left (4 \, x + 4\right ) - 3}\right )} \] Input:
integrate(((4*x^4+4*x^3-x-1)*log(4+4*x)^2+(-24*x^4-24*x^3+2*x+2)*log(4+4*x )+36*x^4+36*x^3+7*x+3)*exp(((x^4-x)*log(4+4*x)-3*x^4-x)/(log(4+4*x)-3))/(( 1+x)*log(4+4*x)^2+(-6*x-6)*log(4+4*x)+9*x+9),x, algorithm="fricas")
Output:
e^(-(3*x^4 - (x^4 - x)*log(4*x + 4) + x)/(log(4*x + 4) - 3))
Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int \frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{9+9 x+(-6-6 x) \log (4+4 x)+(1+x) \log ^2(4+4 x)} \, dx=e^{\frac {- 3 x^{4} - x + \left (x^{4} - x\right ) \log {\left (4 x + 4 \right )}}{\log {\left (4 x + 4 \right )} - 3}} \] Input:
integrate(((4*x**4+4*x**3-x-1)*ln(4+4*x)**2+(-24*x**4-24*x**3+2*x+2)*ln(4+ 4*x)+36*x**4+36*x**3+7*x+3)*exp(((x**4-x)*ln(4+4*x)-3*x**4-x)/(ln(4+4*x)-3 ))/((1+x)*ln(4+4*x)**2+(-6*x-6)*ln(4+4*x)+9*x+9),x)
Output:
exp((-3*x**4 - x + (x**4 - x)*log(4*x + 4))/(log(4*x + 4) - 3))
Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (21) = 42\).
Time = 0.29 (sec) , antiderivative size = 109, normalized size of antiderivative = 4.95 \[ \int \frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{9+9 x+(-6-6 x) \log (4+4 x)+(1+x) \log ^2(4+4 x)} \, dx=e^{\left (\frac {2 \, x^{4} \log \left (2\right )}{2 \, \log \left (2\right ) + \log \left (x + 1\right ) - 3} + \frac {x^{4} \log \left (x + 1\right )}{2 \, \log \left (2\right ) + \log \left (x + 1\right ) - 3} - \frac {3 \, x^{4}}{2 \, \log \left (2\right ) + \log \left (x + 1\right ) - 3} - \frac {2 \, x \log \left (2\right )}{2 \, \log \left (2\right ) + \log \left (x + 1\right ) - 3} - \frac {x \log \left (x + 1\right )}{2 \, \log \left (2\right ) + \log \left (x + 1\right ) - 3} - \frac {x}{2 \, \log \left (2\right ) + \log \left (x + 1\right ) - 3}\right )} \] Input:
integrate(((4*x^4+4*x^3-x-1)*log(4+4*x)^2+(-24*x^4-24*x^3+2*x+2)*log(4+4*x )+36*x^4+36*x^3+7*x+3)*exp(((x^4-x)*log(4+4*x)-3*x^4-x)/(log(4+4*x)-3))/(( 1+x)*log(4+4*x)^2+(-6*x-6)*log(4+4*x)+9*x+9),x, algorithm="maxima")
Output:
e^(2*x^4*log(2)/(2*log(2) + log(x + 1) - 3) + x^4*log(x + 1)/(2*log(2) + l og(x + 1) - 3) - 3*x^4/(2*log(2) + log(x + 1) - 3) - 2*x*log(2)/(2*log(2) + log(x + 1) - 3) - x*log(x + 1)/(2*log(2) + log(x + 1) - 3) - x/(2*log(2) + log(x + 1) - 3))
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (21) = 42\).
Time = 0.41 (sec) , antiderivative size = 69, normalized size of antiderivative = 3.14 \[ \int \frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{9+9 x+(-6-6 x) \log (4+4 x)+(1+x) \log ^2(4+4 x)} \, dx=e^{\left (\frac {x^{4} \log \left (4 \, x + 4\right )}{\log \left (4 \, x + 4\right ) - 3} - \frac {3 \, x^{4}}{\log \left (4 \, x + 4\right ) - 3} - \frac {x \log \left (4 \, x + 4\right )}{\log \left (4 \, x + 4\right ) - 3} - \frac {x}{\log \left (4 \, x + 4\right ) - 3}\right )} \] Input:
integrate(((4*x^4+4*x^3-x-1)*log(4+4*x)^2+(-24*x^4-24*x^3+2*x+2)*log(4+4*x )+36*x^4+36*x^3+7*x+3)*exp(((x^4-x)*log(4+4*x)-3*x^4-x)/(log(4+4*x)-3))/(( 1+x)*log(4+4*x)^2+(-6*x-6)*log(4+4*x)+9*x+9),x, algorithm="giac")
Output:
e^(x^4*log(4*x + 4)/(log(4*x + 4) - 3) - 3*x^4/(log(4*x + 4) - 3) - x*log( 4*x + 4)/(log(4*x + 4) - 3) - x/(log(4*x + 4) - 3))
Time = 3.46 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.59 \[ \int \frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{9+9 x+(-6-6 x) \log (4+4 x)+(1+x) \log ^2(4+4 x)} \, dx=\frac {{\mathrm {e}}^{-\frac {x}{\ln \left (4\,x+4\right )-3}}\,{\mathrm {e}}^{-\frac {3\,x^4}{\ln \left (4\,x+4\right )-3}}}{{\left (4\,x+4\right )}^{\frac {x-x^4}{\ln \left (4\,x+4\right )-3}}} \] Input:
int((exp(-(x + log(4*x + 4)*(x - x^4) + 3*x^4)/(log(4*x + 4) - 3))*(7*x + log(4*x + 4)*(2*x - 24*x^3 - 24*x^4 + 2) - log(4*x + 4)^2*(x - 4*x^3 - 4*x ^4 + 1) + 36*x^3 + 36*x^4 + 3))/(9*x - log(4*x + 4)*(6*x + 6) + log(4*x + 4)^2*(x + 1) + 9),x)
Output:
(exp(-x/(log(4*x + 4) - 3))*exp(-(3*x^4)/(log(4*x + 4) - 3)))/(4*x + 4)^(( x - x^4)/(log(4*x + 4) - 3))
Time = 0.17 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.41 \[ \int \frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{9+9 x+(-6-6 x) \log (4+4 x)+(1+x) \log ^2(4+4 x)} \, dx=\frac {e^{x^{4}}}{e^{\frac {\mathrm {log}\left (4 x +4\right ) x +x}{\mathrm {log}\left (4 x +4\right )-3}}} \] Input:
int(((4*x^4+4*x^3-x-1)*log(4+4*x)^2+(-24*x^4-24*x^3+2*x+2)*log(4+4*x)+36*x ^4+36*x^3+7*x+3)*exp(((x^4-x)*log(4+4*x)-3*x^4-x)/(log(4+4*x)-3))/((1+x)*l og(4+4*x)^2+(-6*x-6)*log(4+4*x)+9*x+9),x)
Output:
e**(x**4)/e**((log(4*x + 4)*x + x)/(log(4*x + 4) - 3))