Integrand size = 81, antiderivative size = 22 \[ \int \frac {-10 x^2+180 e^{-x^2} x^3+\left (-10 x+45 e^{-x^2} x+5 x^2\right ) \log \left (-2+9 e^{-x^2}+x\right )}{\left (-2+9 e^{-x^2}+x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx=\frac {5 x^2}{2 \log ^4\left (-2+9 e^{-x^2}+x\right )} \]
Time = 0.54 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {-10 x^2+180 e^{-x^2} x^3+\left (-10 x+45 e^{-x^2} x+5 x^2\right ) \log \left (-2+9 e^{-x^2}+x\right )}{\left (-2+9 e^{-x^2}+x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx=\frac {5 x^2}{2 \log ^4\left (-2+9 e^{-x^2}+x\right )} \]
Integrate[(-10*x^2 + (180*x^3)/E^x^2 + (-10*x + (45*x)/E^x^2 + 5*x^2)*Log[ -2 + 9/E^x^2 + x])/((-2 + 9/E^x^2 + x)*Log[-2 + 9/E^x^2 + x]^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.
\(\displaystyle \int \frac {-10 x^2+\left (5 x^2+45 e^{-x^2} x-10 x\right ) \log \left (9 e^{-x^2}+x-2\right )+180 e^{-x^2} x^3}{\left (9 e^{-x^2}+x-2\right ) \log ^5\left (9 e^{-x^2}+x-2\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {90 \left (2 x^2-4 x+1\right ) x^2}{(x-2) \left (e^{x^2} x-2 e^{x^2}+9\right ) \log ^5\left (9 e^{-x^2}+x-2\right )}+\frac {5 x \left (x \log \left (9 e^{-x^2}+x-2\right )-2 \log \left (9 e^{-x^2}+x-2\right )-2 x\right )}{(x-2) \log ^5\left (9 e^{-x^2}+x-2\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -20 \int \frac {1}{\log ^5\left (x+9 e^{-x^2}-2\right )}dx-40 \int \frac {1}{(x-2) \log ^5\left (x+9 e^{-x^2}-2\right )}dx-10 \int \frac {x}{\log ^5\left (x+9 e^{-x^2}-2\right )}dx+180 \int \frac {1}{\left (e^{x^2} x-2 e^{x^2}+9\right ) \log ^5\left (x+9 e^{-x^2}-2\right )}dx+360 \int \frac {1}{(x-2) \left (e^{x^2} x-2 e^{x^2}+9\right ) \log ^5\left (x+9 e^{-x^2}-2\right )}dx+90 \int \frac {x}{\left (e^{x^2} x-2 e^{x^2}+9\right ) \log ^5\left (x+9 e^{-x^2}-2\right )}dx+5 \int \frac {x}{\log ^4\left (x+9 e^{-x^2}-2\right )}dx+180 \int \frac {x^3}{\left (e^{x^2} x-2 e^{x^2}+9\right ) \log ^5\left (x+9 e^{-x^2}-2\right )}dx\) |
Int[(-10*x^2 + (180*x^3)/E^x^2 + (-10*x + (45*x)/E^x^2 + 5*x^2)*Log[-2 + 9 /E^x^2 + x])/((-2 + 9/E^x^2 + x)*Log[-2 + 9/E^x^2 + x]^5),x]
3.12.39.3.1 Defintions of rubi rules used
Time = 0.12 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91
method | result | size |
risch | \(\frac {5 x^{2}}{2 \ln \left (9 \,{\mathrm e}^{-x^{2}}+x -2\right )^{4}}\) | \(20\) |
parallelrisch | \(\frac {5 x^{2}}{2 \ln \left ({\mathrm e}^{2 \ln \left (3\right )-x^{2}}+x -2\right )^{4}}\) | \(23\) |
int(((5*x*exp(2*ln(3)-x^2)+5*x^2-10*x)*ln(exp(2*ln(3)-x^2)+x-2)+20*x^3*exp (2*ln(3)-x^2)-10*x^2)/(exp(2*ln(3)-x^2)+x-2)/ln(exp(2*ln(3)-x^2)+x-2)^5,x, method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {-10 x^2+180 e^{-x^2} x^3+\left (-10 x+45 e^{-x^2} x+5 x^2\right ) \log \left (-2+9 e^{-x^2}+x\right )}{\left (-2+9 e^{-x^2}+x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx=\frac {5 \, x^{2}}{2 \, \log \left (x + e^{\left (-x^{2} + 2 \, \log \left (3\right )\right )} - 2\right )^{4}} \]
integrate(((5*x*exp(2*log(3)-x^2)+5*x^2-10*x)*log(exp(2*log(3)-x^2)+x-2)+2 0*x^3*exp(2*log(3)-x^2)-10*x^2)/(exp(2*log(3)-x^2)+x-2)/log(exp(2*log(3)-x ^2)+x-2)^5,x, algorithm=\
Time = 0.11 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {-10 x^2+180 e^{-x^2} x^3+\left (-10 x+45 e^{-x^2} x+5 x^2\right ) \log \left (-2+9 e^{-x^2}+x\right )}{\left (-2+9 e^{-x^2}+x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx=\frac {5 x^{2}}{2 \log {\left (x - 2 + 9 e^{- x^{2}} \right )}^{4}} \]
integrate(((5*x*exp(2*ln(3)-x**2)+5*x**2-10*x)*ln(exp(2*ln(3)-x**2)+x-2)+2 0*x**3*exp(2*ln(3)-x**2)-10*x**2)/(exp(2*ln(3)-x**2)+x-2)/ln(exp(2*ln(3)-x **2)+x-2)**5,x)
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (22) = 44\).
Time = 0.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.45 \[ \int \frac {-10 x^2+180 e^{-x^2} x^3+\left (-10 x+45 e^{-x^2} x+5 x^2\right ) \log \left (-2+9 e^{-x^2}+x\right )}{\left (-2+9 e^{-x^2}+x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx=\frac {5 \, x^{2}}{2 \, {\left (x^{8} - 4 \, x^{6} \log \left ({\left (x - 2\right )} e^{\left (x^{2}\right )} + 9\right ) + 6 \, x^{4} \log \left ({\left (x - 2\right )} e^{\left (x^{2}\right )} + 9\right )^{2} - 4 \, x^{2} \log \left ({\left (x - 2\right )} e^{\left (x^{2}\right )} + 9\right )^{3} + \log \left ({\left (x - 2\right )} e^{\left (x^{2}\right )} + 9\right )^{4}\right )}} \]
integrate(((5*x*exp(2*log(3)-x^2)+5*x^2-10*x)*log(exp(2*log(3)-x^2)+x-2)+2 0*x^3*exp(2*log(3)-x^2)-10*x^2)/(exp(2*log(3)-x^2)+x-2)/log(exp(2*log(3)-x ^2)+x-2)^5,x, algorithm=\
5/2*x^2/(x^8 - 4*x^6*log((x - 2)*e^(x^2) + 9) + 6*x^4*log((x - 2)*e^(x^2) + 9)^2 - 4*x^2*log((x - 2)*e^(x^2) + 9)^3 + log((x - 2)*e^(x^2) + 9)^4)
Time = 0.47 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {-10 x^2+180 e^{-x^2} x^3+\left (-10 x+45 e^{-x^2} x+5 x^2\right ) \log \left (-2+9 e^{-x^2}+x\right )}{\left (-2+9 e^{-x^2}+x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx=\frac {5 \, x^{2}}{2 \, \log \left (x + 9 \, e^{\left (-x^{2}\right )} - 2\right )^{4}} \]
integrate(((5*x*exp(2*log(3)-x^2)+5*x^2-10*x)*log(exp(2*log(3)-x^2)+x-2)+2 0*x^3*exp(2*log(3)-x^2)-10*x^2)/(exp(2*log(3)-x^2)+x-2)/log(exp(2*log(3)-x ^2)+x-2)^5,x, algorithm=\
Time = 13.61 (sec) , antiderivative size = 2169, normalized size of antiderivative = 98.59 \[ \int \frac {-10 x^2+180 e^{-x^2} x^3+\left (-10 x+45 e^{-x^2} x+5 x^2\right ) \log \left (-2+9 e^{-x^2}+x\right )}{\left (-2+9 e^{-x^2}+x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx=\text {Too large to display} \]
int((log(x + exp(2*log(3) - x^2) - 2)*(5*x*exp(2*log(3) - x^2) - 10*x + 5* x^2) - 10*x^2 + 20*x^3*exp(2*log(3) - x^2))/(log(x + exp(2*log(3) - x^2) - 2)^5*(x + exp(2*log(3) - x^2) - 2)),x)
((5*x^2)/2 + (5*x*log(x + 9*exp(-x^2) - 2)*(x + 9*exp(-x^2) - 2))/(4*(18*x *exp(-x^2) - 1)))/log(x + 9*exp(-x^2) - 2)^4 - ((5*(x + 9*exp(-x^2) - 2)*( 4*x + 243*exp(-x^2) - 1944*exp(-2*x^2) + 4374*exp(-3*x^2) - 990*x^2*exp(-x ^2) + 5832*x^2*exp(-2*x^2) + 1080*x^3*exp(-x^2) - 8748*x^2*exp(-3*x^2) - 1 944*x^3*exp(-2*x^2) - 36*x^4*exp(-x^2) - 10368*x^4*exp(-2*x^2) - 288*x^5*e xp(-x^2) + 17496*x^4*exp(-3*x^2) + 15552*x^5*exp(-2*x^2) + 72*x^6*exp(-x^2 ) - 23328*x^5*exp(-3*x^2) - 10368*x^6*exp(-2*x^2) + 11664*x^6*exp(-3*x^2) + 2592*x^7*exp(-2*x^2) - 6))/(24*(18*x*exp(-x^2) - 1)^5) + (5*log(x + 9*ex p(-x^2) - 2)*(x + 9*exp(-x^2) - 2)*(50544*exp(-2*x^2) - 1359*exp(-x^2) - 8 *x - 551124*exp(-3*x^2) + 2361960*exp(-4*x^2) - 3542940*exp(-5*x^2) - 4032 *x*exp(-x^2) + 45360*x*exp(-2*x^2) - 157464*x*exp(-3*x^2) + 157464*x*exp(- 4*x^2) + 11952*x^2*exp(-x^2) - 250776*x^2*exp(-2*x^2) - 3168*x^3*exp(-x^2) + 1872072*x^2*exp(-3*x^2) - 2592*x^3*exp(-2*x^2) - 8280*x^4*exp(-x^2) - 5 668704*x^2*exp(-4*x^2) + 23328*x^3*exp(-3*x^2) + 562464*x^4*exp(-2*x^2) + 4896*x^5*exp(-x^2) + 5668704*x^2*exp(-5*x^2) - 3732480*x^4*exp(-3*x^2) - 6 01344*x^5*exp(-2*x^2) + 576*x^6*exp(-x^2) + 8398080*x^4*exp(-4*x^2) + 2169 504*x^5*exp(-3*x^2) + 90720*x^6*exp(-2*x^2) - 864*x^7*exp(-x^2) - 5668704* x^4*exp(-5*x^2) - 1259712*x^5*exp(-4*x^2) + 3639168*x^6*exp(-3*x^2) + 2073 60*x^7*exp(-2*x^2) + 144*x^8*exp(-x^2) - 13436928*x^6*exp(-4*x^2) - 615859 2*x^7*exp(-3*x^2) - 124416*x^8*exp(-2*x^2) + 7558272*x^6*exp(-5*x^2) + ...