Integrand size = 85, antiderivative size = 26 \[ \int \frac {e^{-4+x} (-4+4 x)+\left (-36 e^{-3+x}+4 x\right ) \log ^2\left (e^{3-x} \left (-9 e^{-3+x}+x\right )\right )}{\left (9 e^{-7+2 x}-e^{-4+x} x\right ) \log ^2\left (e^{3-x} \left (-9 e^{-3+x}+x\right )\right )} \, dx=4 e^{4-x}-\frac {4}{\log \left (-9+e^{3-x} x\right )} \]
Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {e^{-4+x} (-4+4 x)+\left (-36 e^{-3+x}+4 x\right ) \log ^2\left (e^{3-x} \left (-9 e^{-3+x}+x\right )\right )}{\left (9 e^{-7+2 x}-e^{-4+x} x\right ) \log ^2\left (e^{3-x} \left (-9 e^{-3+x}+x\right )\right )} \, dx=4 e^3 \left (e^{1-x}-\frac {1}{e^3 \log \left (-9+e^{3-x} x\right )}\right ) \]
Integrate[(E^(-4 + x)*(-4 + 4*x) + (-36*E^(-3 + x) + 4*x)*Log[E^(3 - x)*(- 9*E^(-3 + x) + x)]^2)/((9*E^(-7 + 2*x) - E^(-4 + x)*x)*Log[E^(3 - x)*(-9*E ^(-3 + x) + 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.
\(\displaystyle \int \frac {e^{x-4} (4 x-4)+\left (4 x-36 e^{x-3}\right ) \log ^2\left (e^{3-x} \left (x-9 e^{x-3}\right )\right )}{\left (9 e^{2 x-7}-e^{x-4} x\right ) \log ^2\left (e^{3-x} \left (x-9 e^{x-3}\right )\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{7-x} \left (e^{x-4} (4 x-4)+\left (4 x-36 e^{x-3}\right ) \log ^2\left (e^{3-x} \left (x-9 e^{x-3}\right )\right )\right )}{\left (9 e^x-e^3 x\right ) \log ^2\left (e^{3-x} \left (x-9 e^{x-3}\right )\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {4 e^{6-x} (x-1) x}{9 \left (e^3 x-9 e^x\right ) \log ^2\left (e^{3-x} x-9\right )}-\frac {4 e^{3-x} \left (-x+9 e \log ^2\left (e^{3-x} x-9\right )+1\right )}{9 \log ^2\left (e^{3-x} x-9\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4}{9} \int \frac {e^{6-x} x^2}{\left (e^3 x-9 e^x\right ) \log ^2\left (e^{3-x} x-9\right )}dx-\frac {4}{9} \int \frac {e^{3-x}}{\log ^2\left (e^{3-x} x-9\right )}dx+\frac {4}{9} \int \frac {e^{3-x} x}{\log ^2\left (e^{3-x} x-9\right )}dx+\frac {4}{9} \int \frac {e^{6-x} x}{\left (e^3 x-9 e^x\right ) \log ^2\left (e^{3-x} x-9\right )}dx+4 e^{4-x}\) |
Int[(E^(-4 + x)*(-4 + 4*x) + (-36*E^(-3 + x) + 4*x)*Log[E^(3 - x)*(-9*E^(- 3 + x) + x)]^2)/((9*E^(-7 + 2*x) - E^(-4 + x)*x)*Log[E^(3 - x)*(-9*E^(-3 + x) + x)]^2),x]
3.28.76.3.1 Defintions of rubi rules used
Leaf count of result is larger than twice the leaf count of optimal. \(52\) vs. \(2(24)=48\).
Time = 0.43 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.04
method | result | size |
norman | \(\frac {\left (4 \,{\mathrm e} \ln \left (\left (-9 \,{\mathrm e}^{-3+x}+x \right ) {\mathrm e}^{-x +3}\right )-4 \,{\mathrm e}^{-3+x}\right ) {\mathrm e}^{-x +3}}{\ln \left (\left (-9 \,{\mathrm e}^{-3+x}+x \right ) {\mathrm e}^{-x +3}\right )}\) | \(53\) |
parallelrisch | \(\frac {\left (-72 \,{\mathrm e}^{x -4}+72 \ln \left (-\left (9 \,{\mathrm e}^{-3+x}-x \right ) {\mathrm e}^{-x +3}\right )\right ) {\mathrm e}^{-x +4}}{18 \ln \left (-\left (9 \,{\mathrm e}^{-3+x}-x \right ) {\mathrm e}^{-x +3}\right )}\) | \(58\) |
risch | \(4 \,{\mathrm e}^{-x +4}-\frac {8 i}{\pi \,\operatorname {csgn}\left (i \left (-9 \,{\mathrm e}^{-3+x}+x \right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x +3}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x +3} \left (-9 \,{\mathrm e}^{-3+x}+x \right )\right )-\pi \,\operatorname {csgn}\left (i \left (-9 \,{\mathrm e}^{-3+x}+x \right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x +3} \left (-9 \,{\mathrm e}^{-3+x}+x \right )\right )^{2}-\pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x +3}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x +3} \left (-9 \,{\mathrm e}^{-3+x}+x \right )\right )^{2}+\pi \operatorname {csgn}\left (i {\mathrm e}^{-x +3} \left (-9 \,{\mathrm e}^{-3+x}+x \right )\right )^{3}+2 i \ln \left (-9 \,{\mathrm e}^{-3+x}+x \right )-2 i \ln \left ({\mathrm e}^{-3+x}\right )}\) | \(168\) |
int(((-36*exp(-3+x)+4*x)*ln((-9*exp(-3+x)+x)/exp(-3+x))^2+(-4+4*x)*exp(x-4 ))/(9*exp(x-4)*exp(-3+x)-x*exp(x-4))/ln((-9*exp(-3+x)+x)/exp(-3+x))^2,x,me thod=_RETURNVERBOSE)
(4*exp(1)*ln((-9*exp(-3+x)+x)/exp(-3+x))-4*exp(-3+x))/exp(-3+x)/ln((-9*exp (-3+x)+x)/exp(-3+x))
Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (24) = 48\).
Time = 0.25 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.00 \[ \int \frac {e^{-4+x} (-4+4 x)+\left (-36 e^{-3+x}+4 x\right ) \log ^2\left (e^{3-x} \left (-9 e^{-3+x}+x\right )\right )}{\left (9 e^{-7+2 x}-e^{-4+x} x\right ) \log ^2\left (e^{3-x} \left (-9 e^{-3+x}+x\right )\right )} \, dx=\frac {4 \, {\left (e \log \left ({\left (x - 9 \, e^{\left (x - 3\right )}\right )} e^{\left (-x + 3\right )}\right ) - e^{\left (x - 3\right )}\right )} e^{\left (-x + 3\right )}}{\log \left ({\left (x - 9 \, e^{\left (x - 3\right )}\right )} e^{\left (-x + 3\right )}\right )} \]
integrate(((-36*exp(-3+x)+4*x)*log((-9*exp(-3+x)+x)/exp(-3+x))^2+(-4+4*x)* exp(x-4))/(9*exp(x-4)*exp(-3+x)-x*exp(x-4))/log((-9*exp(-3+x)+x)/exp(-3+x) )^2,x, algorithm=\
4*(e*log((x - 9*e^(x - 3))*e^(-x + 3)) - e^(x - 3))*e^(-x + 3)/log((x - 9* e^(x - 3))*e^(-x + 3))
Time = 0.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-4+x} (-4+4 x)+\left (-36 e^{-3+x}+4 x\right ) \log ^2\left (e^{3-x} \left (-9 e^{-3+x}+x\right )\right )}{\left (9 e^{-7+2 x}-e^{-4+x} x\right ) \log ^2\left (e^{3-x} \left (-9 e^{-3+x}+x\right )\right )} \, dx=4 e e^{3 - x} - \frac {4}{\log {\left (\left (x - 9 e^{x - 3}\right ) e^{3 - x} \right )}} \]
integrate(((-36*exp(-3+x)+4*x)*ln((-9*exp(-3+x)+x)/exp(-3+x))**2+(-4+4*x)* exp(x-4))/(9*exp(x-4)*exp(-3+x)-x*exp(x-4))/ln((-9*exp(-3+x)+x)/exp(-3+x)) **2,x)
Time = 0.22 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69 \[ \int \frac {e^{-4+x} (-4+4 x)+\left (-36 e^{-3+x}+4 x\right ) \log ^2\left (e^{3-x} \left (-9 e^{-3+x}+x\right )\right )}{\left (9 e^{-7+2 x}-e^{-4+x} x\right ) \log ^2\left (e^{3-x} \left (-9 e^{-3+x}+x\right )\right )} \, dx=\frac {4 \, {\left (x e^{4} - e^{4} \log \left (x e^{3} - 9 \, e^{x}\right ) + e^{x}\right )}}{x e^{x} - e^{x} \log \left (x e^{3} - 9 \, e^{x}\right )} \]
integrate(((-36*exp(-3+x)+4*x)*log((-9*exp(-3+x)+x)/exp(-3+x))^2+(-4+4*x)* exp(x-4))/(9*exp(x-4)*exp(-3+x)-x*exp(x-4))/log((-9*exp(-3+x)+x)/exp(-3+x) )^2,x, algorithm=\
Time = 0.29 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69 \[ \int \frac {e^{-4+x} (-4+4 x)+\left (-36 e^{-3+x}+4 x\right ) \log ^2\left (e^{3-x} \left (-9 e^{-3+x}+x\right )\right )}{\left (9 e^{-7+2 x}-e^{-4+x} x\right ) \log ^2\left (e^{3-x} \left (-9 e^{-3+x}+x\right )\right )} \, dx=\frac {4 \, {\left (x e^{4} - e^{4} \log \left (x e^{3} - 9 \, e^{x}\right ) + e^{x}\right )}}{x e^{x} - e^{x} \log \left (x e^{3} - 9 \, e^{x}\right )} \]
integrate(((-36*exp(-3+x)+4*x)*log((-9*exp(-3+x)+x)/exp(-3+x))^2+(-4+4*x)* exp(x-4))/(9*exp(x-4)*exp(-3+x)-x*exp(x-4))/log((-9*exp(-3+x)+x)/exp(-3+x) )^2,x, algorithm=\
Time = 12.66 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {e^{-4+x} (-4+4 x)+\left (-36 e^{-3+x}+4 x\right ) \log ^2\left (e^{3-x} \left (-9 e^{-3+x}+x\right )\right )}{\left (9 e^{-7+2 x}-e^{-4+x} x\right ) \log ^2\left (e^{3-x} \left (-9 e^{-3+x}+x\right )\right )} \, dx=4\,{\mathrm {e}}^{4-x}-\frac {4}{\ln \left (x\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^3-9\right )} \]