Integrand size = 214, antiderivative size = 29 \[ \int \frac {e^{\frac {12+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} (3-3 x)-11 x}{-4+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} (-1+x)+4 x}} \left (-4+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} \left (-1+e^{2 x} \left (2 x-2 x^2\right )+e^x \left (-2 x+2 x^2\right ) \log \left (\frac {13}{4}\right )\right )\right )}{16-32 x+16 x^2+e^{2 e^{2 x}-4 e^x \log \left (\frac {13}{4}\right )+2 \log ^2\left (\frac {13}{4}\right )} \left (1-2 x+x^2\right )+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} \left (8-16 x+8 x^2\right )} \, dx=e^{-3+\frac {x}{\left (4+e^{\left (e^x-\log \left (\frac {13}{4}\right )\right )^2}\right ) (-1+x)}} \] Output:
exp(x/(4+exp((exp(x)-ln(13/4))^2))/(-1+x)-3)
\[ \int \frac {e^{\frac {12+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} (3-3 x)-11 x}{-4+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} (-1+x)+4 x}} \left (-4+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} \left (-1+e^{2 x} \left (2 x-2 x^2\right )+e^x \left (-2 x+2 x^2\right ) \log \left (\frac {13}{4}\right )\right )\right )}{16-32 x+16 x^2+e^{2 e^{2 x}-4 e^x \log \left (\frac {13}{4}\right )+2 \log ^2\left (\frac {13}{4}\right )} \left (1-2 x+x^2\right )+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} \left (8-16 x+8 x^2\right )} \, dx=\int \frac {e^{\frac {12+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} (3-3 x)-11 x}{-4+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} (-1+x)+4 x}} \left (-4+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} \left (-1+e^{2 x} \left (2 x-2 x^2\right )+e^x \left (-2 x+2 x^2\right ) \log \left (\frac {13}{4}\right )\right )\right )}{16-32 x+16 x^2+e^{2 e^{2 x}-4 e^x \log \left (\frac {13}{4}\right )+2 \log ^2\left (\frac {13}{4}\right )} \left (1-2 x+x^2\right )+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} \left (8-16 x+8 x^2\right )} \, dx \] Input:
Integrate[(E^((12 + E^(E^(2*x) - 2*E^x*Log[13/4] + Log[13/4]^2)*(3 - 3*x) - 11*x)/(-4 + E^(E^(2*x) - 2*E^x*Log[13/4] + Log[13/4]^2)*(-1 + x) + 4*x)) *(-4 + E^(E^(2*x) - 2*E^x*Log[13/4] + Log[13/4]^2)*(-1 + E^(2*x)*(2*x - 2* x^2) + E^x*(-2*x + 2*x^2)*Log[13/4])))/(16 - 32*x + 16*x^2 + E^(2*E^(2*x) - 4*E^x*Log[13/4] + 2*Log[13/4]^2)*(1 - 2*x + x^2) + E^(E^(2*x) - 2*E^x*Lo g[13/4] + Log[13/4]^2)*(8 - 16*x + 8*x^2)),x]
Output:
Integrate[(E^((12 + E^(E^(2*x) - 2*E^x*Log[13/4] + Log[13/4]^2)*(3 - 3*x) - 11*x)/(-4 + E^(E^(2*x) - 2*E^x*Log[13/4] + Log[13/4]^2)*(-1 + x) + 4*x)) *(-4 + E^(E^(2*x) - 2*E^x*Log[13/4] + Log[13/4]^2)*(-1 + E^(2*x)*(2*x - 2* x^2) + E^x*(-2*x + 2*x^2)*Log[13/4])))/(16 - 32*x + 16*x^2 + E^(2*E^(2*x) - 4*E^x*Log[13/4] + 2*Log[13/4]^2)*(1 - 2*x + x^2) + E^(E^(2*x) - 2*E^x*Lo g[13/4] + Log[13/4]^2)*(8 - 16*x + 8*x^2)), 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.
| \(\displaystyle \int \frac {\left (e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} \left (e^{2 x} \left (2 x-2 x^2\right )+e^x \left (2 x^2-2 x\right ) \log \left (\frac {13}{4}\right )-1\right )-4\right ) \exp \left (\frac {-11 x+(3-3 x) e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )}+12}{4 x+(x-1) e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )}-4}\right )}{16 x^2+\left (x^2-2 x+1\right ) e^{2 e^{2 x}-4 e^x \log \left (\frac {13}{4}\right )+2 \log ^2\left (\frac {13}{4}\right )}+\left (8 x^2-16 x+8\right ) e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )}-32 x+16} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {13^{4 e^x} \left (e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} \left (e^{2 x} \left (2 x-2 x^2\right )+e^x \left (2 x^2-2 x\right ) \log \left (\frac {13}{4}\right )-1\right )-4\right ) \exp \left (\frac {-11 x+(3-3 x) e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )}+12}{4 x+(x-1) e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )}-4}\right )}{(1-x)^2 \left (4\ 13^{2 e^x}+16^{e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {8\ 13^{4 e^x} x \left (e^x-\log \left (\frac {13}{4}\right )\right ) \exp \left (x+\frac {-11 x+(3-3 x) e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )}+12}{4 x+(x-1) e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )}-4}\right )}{(x-1) \left (4\ 169^{e^x}+16^{e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}\right )^2}-\frac {13^{2 e^x} \left (2 e^{2 x} x^2-2 e^x x^2 \log \left (\frac {13}{4}\right )-2 e^{2 x} x+2 e^x x \log \left (\frac {13}{4}\right )+1\right ) \exp \left (\frac {-11 x+(3-3 x) e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )}+12}{4 x+(x-1) e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )}-4}\right )}{(x-1)^2 \left (4\ 169^{e^x}+16^{e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}\right )}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {169^{e^x} \left (-2 e^{2 x} x^2+2 e^x x^2 \log \left (\frac {13}{4}\right )+2 e^{2 x} x-2 e^x x \log \left (\frac {13}{4}\right )-1\right ) \exp \left (-\frac {11 x+3 \left (\frac {4}{13}\right )^{2 e^x} x e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}-3 \left (\frac {4}{13}\right )^{2 e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}-12}{(x-1) \left (\left (\frac {4}{13}\right )^{2 e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}+4\right )}\right )}{(1-x)^2 \left (4\ 169^{e^x}+16^{e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}\right )}+\frac {8\ 13^{4 e^x} x \left (e^x-\log \left (\frac {13}{4}\right )\right ) \exp \left (x-\frac {11 x}{(x-1) \left (\left (\frac {4}{13}\right )^{2 e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}+4\right )}-\frac {3 \left (\frac {4}{13}\right )^{2 e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}}{\left (\frac {4}{13}\right )^{2 e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}+4}+\frac {12}{(x-1) \left (\left (\frac {4}{13}\right )^{2 e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}+4\right )}\right )}{(x-1) \left (4\ 13^{2 e^x}+16^{e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {8\ 13^{4 e^x} x \left (\log \left (\frac {13}{4}\right )-e^x\right ) \exp \left (\frac {4\ 169^{e^x} x^2+4^{2 e^x} x^2 e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}-15\ 169^{e^x} x+12\ 169^{e^x}-4^{2 e^x+1} x e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}+3\ 4^{2 e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}}{(x-1) \left (4\ 169^{e^x}+4^{2 e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}\right )}\right )}{(1-x) \left (4\ 13^{2 e^x}+16^{e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}\right )^2}+\frac {169^{e^x} \left (-2 e^{2 x} x^2+2 e^x x^2 \log \left (\frac {13}{4}\right )+2 e^{2 x} x-2 e^x x \log \left (\frac {13}{4}\right )-1\right ) \exp \left (-\frac {11 x+3 \left (\frac {4}{13}\right )^{2 e^x} x e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}-3 \left (\frac {4}{13}\right )^{2 e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}-12}{(x-1) \left (\left (\frac {4}{13}\right )^{2 e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}+4\right )}\right )}{(1-x)^2 \left (4\ 169^{e^x}+16^{e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}\right )}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {8\ 13^{4 e^x} x \left (e^x-\log \left (\frac {13}{4}\right )\right ) \exp \left (\frac {16^{e^x} \left (x^2-4 x+3\right ) e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}}{(x-1) \left (4\ 13^{2 e^x}+16^{e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}\right )}+\frac {13^{2 e^x} \left (4 x^2-15 x+12\right )}{(x-1) \left (4\ 13^{2 e^x}+16^{e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}\right )}\right )}{(x-1) \left (4\ 13^{2 e^x}+16^{e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}\right )^2}+\frac {169^{e^x} \left (-2 e^{2 x} x^2+2 e^x x^2 \log \left (\frac {13}{4}\right )+2 e^{2 x} x-2 e^x x \log \left (\frac {13}{4}\right )-1\right ) \exp \left (-\frac {11 x+3 \left (\frac {4}{13}\right )^{2 e^x} x e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}-3 \left (\frac {4}{13}\right )^{2 e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}-12}{(x-1) \left (\left (\frac {4}{13}\right )^{2 e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}+4\right )}\right )}{(1-x)^2 \left (4\ 169^{e^x}+16^{e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {8\ 13^{4 e^x} x \left (e^x-\log \left (\frac {13}{4}\right )\right ) \exp \left (\frac {16^{e^x} \left (x^2-4 x+3\right ) e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}}{(x-1) \left (4\ 13^{2 e^x}+16^{e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}\right )}+\frac {13^{2 e^x} \left (4 x^2-15 x+12\right )}{(x-1) \left (4\ 13^{2 e^x}+16^{e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}\right )}\right )}{(x-1) \left (4\ 13^{2 e^x}+16^{e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}\right )^2}+\frac {169^{e^x} \left (-2 e^{2 x} x^2+2 e^x x^2 \log \left (\frac {13}{4}\right )+2 e^{2 x} x-2 e^x x \log \left (\frac {13}{4}\right )-1\right ) \exp \left (-\frac {11 x+3 \left (\frac {4}{13}\right )^{2 e^x} x e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}-3 \left (\frac {4}{13}\right )^{2 e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}-12}{(x-1) \left (\left (\frac {4}{13}\right )^{2 e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}+4\right )}\right )}{(1-x)^2 \left (4\ 169^{e^x}+16^{e^x} e^{e^{2 x}+\log ^2\left (\frac {13}{4}\right )}\right )}\right )dx\) |
Input:
Int[(E^((12 + E^(E^(2*x) - 2*E^x*Log[13/4] + Log[13/4]^2)*(3 - 3*x) - 11*x )/(-4 + E^(E^(2*x) - 2*E^x*Log[13/4] + Log[13/4]^2)*(-1 + x) + 4*x))*(-4 + E^(E^(2*x) - 2*E^x*Log[13/4] + Log[13/4]^2)*(-1 + E^(2*x)*(2*x - 2*x^2) + E^x*(-2*x + 2*x^2)*Log[13/4])))/(16 - 32*x + 16*x^2 + E^(2*E^(2*x) - 4*E^ x*Log[13/4] + 2*Log[13/4]^2)*(1 - 2*x + x^2) + E^(E^(2*x) - 2*E^x*Log[13/4 ] + Log[13/4]^2)*(8 - 16*x + 8*x^2)),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(72\) vs. \(2(24)=48\).
Time = 143.93 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.52
| method | result | size |
| parallelrisch | \({\mathrm e}^{\frac {\left (-3 x +3\right ) {\mathrm e}^{{\mathrm e}^{2 x}-2 \ln \left (\frac {13}{4}\right ) {\mathrm e}^{x}+\ln \left (\frac {13}{4}\right )^{2}}-11 x +12}{{\mathrm e}^{{\mathrm e}^{2 x}-2 \ln \left (\frac {13}{4}\right ) {\mathrm e}^{x}+\ln \left (\frac {13}{4}\right )^{2}} x -{\mathrm e}^{{\mathrm e}^{2 x}-2 \ln \left (\frac {13}{4}\right ) {\mathrm e}^{x}+\ln \left (\frac {13}{4}\right )^{2}}+4 x -4}}\) | \(73\) |
| risch | \({\mathrm e}^{-\frac {3 \left (\frac {1}{169}\right )^{{\mathrm e}^{x}} \left (\frac {1}{28561}\right )^{\ln \left (2\right )} 16^{{\mathrm e}^{x}} {\mathrm e}^{\ln \left (13\right )^{2}+4 \ln \left (2\right )^{2}+{\mathrm e}^{2 x}} x -3 \left (\frac {1}{169}\right )^{{\mathrm e}^{x}} \left (\frac {1}{28561}\right )^{\ln \left (2\right )} 16^{{\mathrm e}^{x}} {\mathrm e}^{\ln \left (13\right )^{2}+4 \ln \left (2\right )^{2}+{\mathrm e}^{2 x}}+11 x -12}{\left (-1+x \right ) \left (\left (\frac {1}{169}\right )^{{\mathrm e}^{x}} \left (\frac {1}{28561}\right )^{\ln \left (2\right )} 16^{{\mathrm e}^{x}} {\mathrm e}^{\ln \left (13\right )^{2}+4 \ln \left (2\right )^{2}+{\mathrm e}^{2 x}}+4\right )}}\) | \(108\) |
Input:
int((((-2*x^2+2*x)*exp(x)^2+(2*x^2-2*x)*ln(13/4)*exp(x)-1)*exp(exp(x)^2-2* ln(13/4)*exp(x)+ln(13/4)^2)-4)*exp(((-3*x+3)*exp(exp(x)^2-2*ln(13/4)*exp(x )+ln(13/4)^2)-11*x+12)/((-1+x)*exp(exp(x)^2-2*ln(13/4)*exp(x)+ln(13/4)^2)+ 4*x-4))/((x^2-2*x+1)*exp(exp(x)^2-2*ln(13/4)*exp(x)+ln(13/4)^2)^2+(8*x^2-1 6*x+8)*exp(exp(x)^2-2*ln(13/4)*exp(x)+ln(13/4)^2)+16*x^2-32*x+16),x,method =_RETURNVERBOSE)
Output:
exp(((-3*x+3)*exp(exp(x)^2-2*ln(13/4)*exp(x)+ln(13/4)^2)-11*x+12)/(exp(exp (x)^2-2*ln(13/4)*exp(x)+ln(13/4)^2)*x-exp(exp(x)^2-2*ln(13/4)*exp(x)+ln(13 /4)^2)+4*x-4))
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (24) = 48\).
Time = 0.07 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.93 \[ \int \frac {e^{\frac {12+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} (3-3 x)-11 x}{-4+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} (-1+x)+4 x}} \left (-4+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} \left (-1+e^{2 x} \left (2 x-2 x^2\right )+e^x \left (-2 x+2 x^2\right ) \log \left (\frac {13}{4}\right )\right )\right )}{16-32 x+16 x^2+e^{2 e^{2 x}-4 e^x \log \left (\frac {13}{4}\right )+2 \log ^2\left (\frac {13}{4}\right )} \left (1-2 x+x^2\right )+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} \left (8-16 x+8 x^2\right )} \, dx=e^{\left (-\frac {3 \, {\left (x - 1\right )} e^{\left (-2 \, e^{x} \log \left (\frac {13}{4}\right ) + \log \left (\frac {13}{4}\right )^{2} + e^{\left (2 \, x\right )}\right )} + 11 \, x - 12}{{\left (x - 1\right )} e^{\left (-2 \, e^{x} \log \left (\frac {13}{4}\right ) + \log \left (\frac {13}{4}\right )^{2} + e^{\left (2 \, x\right )}\right )} + 4 \, x - 4}\right )} \] Input:
integrate((((-2*x^2+2*x)*exp(x)^2+(2*x^2-2*x)*log(13/4)*exp(x)-1)*exp(exp( x)^2-2*log(13/4)*exp(x)+log(13/4)^2)-4)*exp(((-3*x+3)*exp(exp(x)^2-2*log(1 3/4)*exp(x)+log(13/4)^2)-11*x+12)/((-1+x)*exp(exp(x)^2-2*log(13/4)*exp(x)+ log(13/4)^2)+4*x-4))/((x^2-2*x+1)*exp(exp(x)^2-2*log(13/4)*exp(x)+log(13/4 )^2)^2+(8*x^2-16*x+8)*exp(exp(x)^2-2*log(13/4)*exp(x)+log(13/4)^2)+16*x^2- 32*x+16),x, algorithm="fricas")
Output:
e^(-(3*(x - 1)*e^(-2*e^x*log(13/4) + log(13/4)^2 + e^(2*x)) + 11*x - 12)/( (x - 1)*e^(-2*e^x*log(13/4) + log(13/4)^2 + e^(2*x)) + 4*x - 4))
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (20) = 40\).
Time = 4.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.28 \[ \int \frac {e^{\frac {12+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} (3-3 x)-11 x}{-4+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} (-1+x)+4 x}} \left (-4+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} \left (-1+e^{2 x} \left (2 x-2 x^2\right )+e^x \left (-2 x+2 x^2\right ) \log \left (\frac {13}{4}\right )\right )\right )}{16-32 x+16 x^2+e^{2 e^{2 x}-4 e^x \log \left (\frac {13}{4}\right )+2 \log ^2\left (\frac {13}{4}\right )} \left (1-2 x+x^2\right )+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} \left (8-16 x+8 x^2\right )} \, dx=e^{\frac {- 11 x + \left (3 - 3 x\right ) e^{e^{2 x} - 2 e^{x} \log {\left (\frac {13}{4} \right )} + \log {\left (\frac {13}{4} \right )}^{2}} + 12}{4 x + \left (x - 1\right ) e^{e^{2 x} - 2 e^{x} \log {\left (\frac {13}{4} \right )} + \log {\left (\frac {13}{4} \right )}^{2}} - 4}} \] Input:
integrate((((-2*x**2+2*x)*exp(x)**2+(2*x**2-2*x)*ln(13/4)*exp(x)-1)*exp(ex p(x)**2-2*ln(13/4)*exp(x)+ln(13/4)**2)-4)*exp(((-3*x+3)*exp(exp(x)**2-2*ln (13/4)*exp(x)+ln(13/4)**2)-11*x+12)/((-1+x)*exp(exp(x)**2-2*ln(13/4)*exp(x )+ln(13/4)**2)+4*x-4))/((x**2-2*x+1)*exp(exp(x)**2-2*ln(13/4)*exp(x)+ln(13 /4)**2)**2+(8*x**2-16*x+8)*exp(exp(x)**2-2*ln(13/4)*exp(x)+ln(13/4)**2)+16 *x**2-32*x+16),x)
Output:
exp((-11*x + (3 - 3*x)*exp(exp(2*x) - 2*exp(x)*log(13/4) + log(13/4)**2) + 12)/(4*x + (x - 1)*exp(exp(2*x) - 2*exp(x)*log(13/4) + log(13/4)**2) - 4) )
Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (24) = 48\).
Time = 1.16 (sec) , antiderivative size = 206, normalized size of antiderivative = 7.10 \[ \int \frac {e^{\frac {12+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} (3-3 x)-11 x}{-4+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} (-1+x)+4 x}} \left (-4+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} \left (-1+e^{2 x} \left (2 x-2 x^2\right )+e^x \left (-2 x+2 x^2\right ) \log \left (\frac {13}{4}\right )\right )\right )}{16-32 x+16 x^2+e^{2 e^{2 x}-4 e^x \log \left (\frac {13}{4}\right )+2 \log ^2\left (\frac {13}{4}\right )} \left (1-2 x+x^2\right )+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} \left (8-16 x+8 x^2\right )} \, dx=e^{\left (\frac {13^{2 \, e^{x}} 2^{4 \, \log \left (13\right )}}{{\left (2^{4 \, \log \left (13\right ) + 2} x - 2^{4 \, \log \left (13\right ) + 2}\right )} 13^{2 \, e^{x}} + {\left (x e^{\left (\log \left (13\right )^{2} + 4 \, \log \left (2\right )^{2}\right )} - e^{\left (\log \left (13\right )^{2} + 4 \, \log \left (2\right )^{2}\right )}\right )} e^{\left (4 \, e^{x} \log \left (2\right ) + e^{\left (2 \, x\right )}\right )}} - \frac {11 \cdot 13^{2 \, e^{x}} 2^{4 \, \log \left (13\right )}}{13^{2 \, e^{x}} 2^{4 \, \log \left (13\right ) + 2} + e^{\left (\log \left (13\right )^{2} + 4 \, e^{x} \log \left (2\right ) + 4 \, \log \left (2\right )^{2} + e^{\left (2 \, x\right )}\right )}} - \frac {3 \, e^{\left (\log \left (13\right )^{2} + 4 \, e^{x} \log \left (2\right ) + 4 \, \log \left (2\right )^{2} + e^{\left (2 \, x\right )}\right )}}{13^{2 \, e^{x}} 2^{4 \, \log \left (13\right ) + 2} + e^{\left (\log \left (13\right )^{2} + 4 \, e^{x} \log \left (2\right ) + 4 \, \log \left (2\right )^{2} + e^{\left (2 \, x\right )}\right )}}\right )} \] Input:
integrate((((-2*x^2+2*x)*exp(x)^2+(2*x^2-2*x)*log(13/4)*exp(x)-1)*exp(exp( x)^2-2*log(13/4)*exp(x)+log(13/4)^2)-4)*exp(((-3*x+3)*exp(exp(x)^2-2*log(1 3/4)*exp(x)+log(13/4)^2)-11*x+12)/((-1+x)*exp(exp(x)^2-2*log(13/4)*exp(x)+ log(13/4)^2)+4*x-4))/((x^2-2*x+1)*exp(exp(x)^2-2*log(13/4)*exp(x)+log(13/4 )^2)^2+(8*x^2-16*x+8)*exp(exp(x)^2-2*log(13/4)*exp(x)+log(13/4)^2)+16*x^2- 32*x+16),x, algorithm="maxima")
Output:
e^(13^(2*e^x)*2^(4*log(13))/((2^(4*log(13) + 2)*x - 2^(4*log(13) + 2))*13^ (2*e^x) + (x*e^(log(13)^2 + 4*log(2)^2) - e^(log(13)^2 + 4*log(2)^2))*e^(4 *e^x*log(2) + e^(2*x))) - 11*13^(2*e^x)*2^(4*log(13))/(13^(2*e^x)*2^(4*log (13) + 2) + e^(log(13)^2 + 4*e^x*log(2) + 4*log(2)^2 + e^(2*x))) - 3*e^(lo g(13)^2 + 4*e^x*log(2) + 4*log(2)^2 + e^(2*x))/(13^(2*e^x)*2^(4*log(13) + 2) + e^(log(13)^2 + 4*e^x*log(2) + 4*log(2)^2 + e^(2*x))))
Exception generated. \[ \int \frac {e^{\frac {12+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} (3-3 x)-11 x}{-4+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} (-1+x)+4 x}} \left (-4+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} \left (-1+e^{2 x} \left (2 x-2 x^2\right )+e^x \left (-2 x+2 x^2\right ) \log \left (\frac {13}{4}\right )\right )\right )}{16-32 x+16 x^2+e^{2 e^{2 x}-4 e^x \log \left (\frac {13}{4}\right )+2 \log ^2\left (\frac {13}{4}\right )} \left (1-2 x+x^2\right )+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} \left (8-16 x+8 x^2\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((((-2*x^2+2*x)*exp(x)^2+(2*x^2-2*x)*log(13/4)*exp(x)-1)*exp(exp( x)^2-2*log(13/4)*exp(x)+log(13/4)^2)-4)*exp(((-3*x+3)*exp(exp(x)^2-2*log(1 3/4)*exp(x)+log(13/4)^2)-11*x+12)/((-1+x)*exp(exp(x)^2-2*log(13/4)*exp(x)+ log(13/4)^2)+4*x-4))/((x^2-2*x+1)*exp(exp(x)^2-2*log(13/4)*exp(x)+log(13/4 )^2)^2+(8*x^2-16*x+8)*exp(exp(x)^2-2*log(13/4)*exp(x)+log(13/4)^2)+16*x^2- 32*x+16),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{524288,[0,8,12,22,1,1,8]%%%}+%%%{-7864320,[0,8,12,21,1,1,8 ]%%%}+%%%
Timed out. \[ \int \frac {e^{\frac {12+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} (3-3 x)-11 x}{-4+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} (-1+x)+4 x}} \left (-4+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} \left (-1+e^{2 x} \left (2 x-2 x^2\right )+e^x \left (-2 x+2 x^2\right ) \log \left (\frac {13}{4}\right )\right )\right )}{16-32 x+16 x^2+e^{2 e^{2 x}-4 e^x \log \left (\frac {13}{4}\right )+2 \log ^2\left (\frac {13}{4}\right )} \left (1-2 x+x^2\right )+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} \left (8-16 x+8 x^2\right )} \, dx=\int -\frac {{\mathrm {e}}^{-\frac {11\,x+{\mathrm {e}}^{{\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x\,\ln \left (\frac {13}{4}\right )+{\ln \left (\frac {13}{4}\right )}^2}\,\left (3\,x-3\right )-12}{4\,x+{\mathrm {e}}^{{\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x\,\ln \left (\frac {13}{4}\right )+{\ln \left (\frac {13}{4}\right )}^2}\,\left (x-1\right )-4}}\,\left ({\mathrm {e}}^{{\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x\,\ln \left (\frac {13}{4}\right )+{\ln \left (\frac {13}{4}\right )}^2}\,\left ({\mathrm {e}}^x\,\ln \left (\frac {13}{4}\right )\,\left (2\,x-2\,x^2\right )-{\mathrm {e}}^{2\,x}\,\left (2\,x-2\,x^2\right )+1\right )+4\right )}{{\mathrm {e}}^{{\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x\,\ln \left (\frac {13}{4}\right )+{\ln \left (\frac {13}{4}\right )}^2}\,\left (8\,x^2-16\,x+8\right )-32\,x+{\mathrm {e}}^{2\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^x\,\ln \left (\frac {13}{4}\right )+2\,{\ln \left (\frac {13}{4}\right )}^2}\,\left (x^2-2\,x+1\right )+16\,x^2+16} \,d x \] Input:
int(-(exp(-(11*x + exp(exp(2*x) - 2*exp(x)*log(13/4) + log(13/4)^2)*(3*x - 3) - 12)/(4*x + exp(exp(2*x) - 2*exp(x)*log(13/4) + log(13/4)^2)*(x - 1) - 4))*(exp(exp(2*x) - 2*exp(x)*log(13/4) + log(13/4)^2)*(exp(x)*log(13/4)* (2*x - 2*x^2) - exp(2*x)*(2*x - 2*x^2) + 1) + 4))/(exp(exp(2*x) - 2*exp(x) *log(13/4) + log(13/4)^2)*(8*x^2 - 16*x + 8) - 32*x + exp(2*exp(2*x) - 4*e xp(x)*log(13/4) + 2*log(13/4)^2)*(x^2 - 2*x + 1) + 16*x^2 + 16),x)
Output:
int(-(exp(-(11*x + exp(exp(2*x) - 2*exp(x)*log(13/4) + log(13/4)^2)*(3*x - 3) - 12)/(4*x + exp(exp(2*x) - 2*exp(x)*log(13/4) + log(13/4)^2)*(x - 1) - 4))*(exp(exp(2*x) - 2*exp(x)*log(13/4) + log(13/4)^2)*(exp(x)*log(13/4)* (2*x - 2*x^2) - exp(2*x)*(2*x - 2*x^2) + 1) + 4))/(exp(exp(2*x) - 2*exp(x) *log(13/4) + log(13/4)^2)*(8*x^2 - 16*x + 8) - 32*x + exp(2*exp(2*x) - 4*e xp(x)*log(13/4) + 2*log(13/4)^2)*(x^2 - 2*x + 1) + 16*x^2 + 16), x)
Time = 0.53 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.45 \[ \int \frac {e^{\frac {12+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} (3-3 x)-11 x}{-4+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} (-1+x)+4 x}} \left (-4+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} \left (-1+e^{2 x} \left (2 x-2 x^2\right )+e^x \left (-2 x+2 x^2\right ) \log \left (\frac {13}{4}\right )\right )\right )}{16-32 x+16 x^2+e^{2 e^{2 x}-4 e^x \log \left (\frac {13}{4}\right )+2 \log ^2\left (\frac {13}{4}\right )} \left (1-2 x+x^2\right )+e^{e^{2 x}-2 e^x \log \left (\frac {13}{4}\right )+\log ^2\left (\frac {13}{4}\right )} \left (8-16 x+8 x^2\right )} \, dx=\frac {e^{\frac {e^{2 e^{x} \mathrm {log}\left (\frac {13}{4}\right )} x}{e^{e^{2 x}+\mathrm {log}\left (\frac {13}{4}\right )^{2}} x -e^{e^{2 x}+\mathrm {log}\left (\frac {13}{4}\right )^{2}}+4 e^{2 e^{x} \mathrm {log}\left (\frac {13}{4}\right )} x -4 e^{2 e^{x} \mathrm {log}\left (\frac {13}{4}\right )}}}}{e^{3}} \] Input:
int((((-2*x^2+2*x)*exp(x)^2+(2*x^2-2*x)*log(13/4)*exp(x)-1)*exp(exp(x)^2-2 *log(13/4)*exp(x)+log(13/4)^2)-4)*exp(((-3*x+3)*exp(exp(x)^2-2*log(13/4)*e xp(x)+log(13/4)^2)-11*x+12)/((-1+x)*exp(exp(x)^2-2*log(13/4)*exp(x)+log(13 /4)^2)+4*x-4))/((x^2-2*x+1)*exp(exp(x)^2-2*log(13/4)*exp(x)+log(13/4)^2)^2 +(8*x^2-16*x+8)*exp(exp(x)^2-2*log(13/4)*exp(x)+log(13/4)^2)+16*x^2-32*x+1 6),x)
Output:
e**((e**(2*e**x*log(13/4))*x)/(e**(e**(2*x) + log(13/4)**2)*x - e**(e**(2* x) + log(13/4)**2) + 4*e**(2*e**x*log(13/4))*x - 4*e**(2*e**x*log(13/4)))) /e**3