Integrand size = 83, antiderivative size = 26 \[ \int \frac {(-3-3 x) \log (3)+(-3-3 x) \log (3) \log \left (x^2\right )+\left (\left (9+9 x-9 x^2\right ) \log (3)+\left (3+3 x-3 x^2\right ) \log (3) \log \left (x^2\right )\right ) \log \left (\frac {9+3 \log \left (x^2\right )}{x}\right )}{3 e^x+e^x \log \left (x^2\right )} \, dx=3 e^{-x} \left (x+x^2\right ) \log (3) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right ) \] Output:
3*(x^2+x)/exp(x)*ln(3*(ln(x^2)+3)/x)*ln(3)
Time = 5.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {(-3-3 x) \log (3)+(-3-3 x) \log (3) \log \left (x^2\right )+\left (\left (9+9 x-9 x^2\right ) \log (3)+\left (3+3 x-3 x^2\right ) \log (3) \log \left (x^2\right )\right ) \log \left (\frac {9+3 \log \left (x^2\right )}{x}\right )}{3 e^x+e^x \log \left (x^2\right )} \, dx=3 e^{-x} x (1+x) \log (3) \log \left (\frac {3 \left (3+\log \left (x^2\right )\right )}{x}\right ) \] Input:
Integrate[((-3 - 3*x)*Log[3] + (-3 - 3*x)*Log[3]*Log[x^2] + ((9 + 9*x - 9* x^2)*Log[3] + (3 + 3*x - 3*x^2)*Log[3]*Log[x^2])*Log[(9 + 3*Log[x^2])/x])/ (3*E^x + E^x*Log[x^2]),x]
Output:
(3*x*(1 + x)*Log[3]*Log[(3*(3 + Log[x^2]))/x])/E^x
Time = 1.13 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.81, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {7292, 7293, 2009}
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.
\(\displaystyle \int \frac {(-3 x-3) \log (3) \log \left (x^2\right )+\left (\left (-9 x^2+9 x+9\right ) \log (3)+\left (-3 x^2+3 x+3\right ) \log (3) \log \left (x^2\right )\right ) \log \left (\frac {3 \log \left (x^2\right )+9}{x}\right )+(-3 x-3) \log (3)}{e^x \log \left (x^2\right )+3 e^x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{-x} \left ((-3 x-3) \log (3) \log \left (x^2\right )+\left (\left (-9 x^2+9 x+9\right ) \log (3)+\left (-3 x^2+3 x+3\right ) \log (3) \log \left (x^2\right )\right ) \log \left (\frac {3 \log \left (x^2\right )+9}{x}\right )+(-3 x-3) \log (3)\right )}{\log \left (x^2\right )+3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {3 e^{-x} (x+1) \log (3) \left (\log \left (x^2\right )+1\right )}{\log \left (x^2\right )+3}-3 e^{-x} \left (x^2-x-1\right ) \log (3) \log \left (\frac {3 \left (\log \left (x^2\right )+3\right )}{x}\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 e^{-x} x^2 \log (3) \log \left (\frac {3 \left (\log \left (x^2\right )+3\right )}{x}\right )+3 e^{-x} x \log (3) \log \left (\frac {3 \left (\log \left (x^2\right )+3\right )}{x}\right )\) |
Input:
Int[((-3 - 3*x)*Log[3] + (-3 - 3*x)*Log[3]*Log[x^2] + ((9 + 9*x - 9*x^2)*L og[3] + (3 + 3*x - 3*x^2)*Log[3]*Log[x^2])*Log[(9 + 3*Log[x^2])/x])/(3*E^x + E^x*Log[x^2]),x]
Output:
(3*x*Log[3]*Log[(3*(3 + Log[x^2]))/x])/E^x + (3*x^2*Log[3]*Log[(3*(3 + Log [x^2]))/x])/E^x
Time = 1.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69
method | result | size |
parallelrisch | \(\frac {\left (6 \ln \left (3\right ) x^{2} \ln \left (\frac {3 \ln \left (x^{2}\right )+9}{x}\right )+6 \ln \left (3\right ) x \ln \left (\frac {3 \ln \left (x^{2}\right )+9}{x}\right )\right ) {\mathrm e}^{-x}}{2}\) | \(44\) |
risch | \(\text {Expression too large to display}\) | \(1616\) |
Input:
int((((-3*x^2+3*x+3)*ln(3)*ln(x^2)+(-9*x^2+9*x+9)*ln(3))*ln((3*ln(x^2)+9)/ x)+(-3*x-3)*ln(3)*ln(x^2)+(-3*x-3)*ln(3))/(exp(x)*ln(x^2)+3*exp(x)),x,meth od=_RETURNVERBOSE)
Output:
1/2*(6*ln(3)*x^2*ln(3*(ln(x^2)+3)/x)+6*ln(3)*x*ln(3*(ln(x^2)+3)/x))/exp(x)
Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {(-3-3 x) \log (3)+(-3-3 x) \log (3) \log \left (x^2\right )+\left (\left (9+9 x-9 x^2\right ) \log (3)+\left (3+3 x-3 x^2\right ) \log (3) \log \left (x^2\right )\right ) \log \left (\frac {9+3 \log \left (x^2\right )}{x}\right )}{3 e^x+e^x \log \left (x^2\right )} \, dx=3 \, {\left (x^{2} + x\right )} e^{\left (-x\right )} \log \left (3\right ) \log \left (\frac {3 \, {\left (\log \left (x^{2}\right ) + 3\right )}}{x}\right ) \] Input:
integrate((((-3*x^2+3*x+3)*log(3)*log(x^2)+(-9*x^2+9*x+9)*log(3))*log((3*l og(x^2)+9)/x)+(-3*x-3)*log(3)*log(x^2)+(-3*x-3)*log(3))/(exp(x)*log(x^2)+3 *exp(x)),x, algorithm="fricas")
Output:
3*(x^2 + x)*e^(-x)*log(3)*log(3*(log(x^2) + 3)/x)
Time = 2.37 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int \frac {(-3-3 x) \log (3)+(-3-3 x) \log (3) \log \left (x^2\right )+\left (\left (9+9 x-9 x^2\right ) \log (3)+\left (3+3 x-3 x^2\right ) \log (3) \log \left (x^2\right )\right ) \log \left (\frac {9+3 \log \left (x^2\right )}{x}\right )}{3 e^x+e^x \log \left (x^2\right )} \, dx=\left (3 x^{2} \log {\left (3 \right )} \log {\left (\frac {3 \log {\left (x^{2} \right )} + 9}{x} \right )} + 3 x \log {\left (3 \right )} \log {\left (\frac {3 \log {\left (x^{2} \right )} + 9}{x} \right )}\right ) e^{- x} \] Input:
integrate((((-3*x**2+3*x+3)*ln(3)*ln(x**2)+(-9*x**2+9*x+9)*ln(3))*ln((3*ln (x**2)+9)/x)+(-3*x-3)*ln(3)*ln(x**2)+(-3*x-3)*ln(3))/(exp(x)*ln(x**2)+3*ex p(x)),x)
Output:
(3*x**2*log(3)*log((3*log(x**2) + 9)/x) + 3*x*log(3)*log((3*log(x**2) + 9) /x))*exp(-x)
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (25) = 50\).
Time = 0.15 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.35 \[ \int \frac {(-3-3 x) \log (3)+(-3-3 x) \log (3) \log \left (x^2\right )+\left (\left (9+9 x-9 x^2\right ) \log (3)+\left (3+3 x-3 x^2\right ) \log (3) \log \left (x^2\right )\right ) \log \left (\frac {9+3 \log \left (x^2\right )}{x}\right )}{3 e^x+e^x \log \left (x^2\right )} \, dx=3 \, {\left (x^{2} \log \left (3\right ) + x \log \left (3\right )\right )} e^{\left (-x\right )} \log \left (2 \, \log \left (x\right ) + 3\right ) + 3 \, {\left (x^{2} \log \left (3\right )^{2} + x \log \left (3\right )^{2} - {\left (x^{2} \log \left (3\right ) + x \log \left (3\right )\right )} \log \left (x\right )\right )} e^{\left (-x\right )} \] Input:
integrate((((-3*x^2+3*x+3)*log(3)*log(x^2)+(-9*x^2+9*x+9)*log(3))*log((3*l og(x^2)+9)/x)+(-3*x-3)*log(3)*log(x^2)+(-3*x-3)*log(3))/(exp(x)*log(x^2)+3 *exp(x)),x, algorithm="maxima")
Output:
3*(x^2*log(3) + x*log(3))*e^(-x)*log(2*log(x) + 3) + 3*(x^2*log(3)^2 + x*l og(3)^2 - (x^2*log(3) + x*log(3))*log(x))*e^(-x)
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (25) = 50\).
Time = 0.19 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.19 \[ \int \frac {(-3-3 x) \log (3)+(-3-3 x) \log (3) \log \left (x^2\right )+\left (\left (9+9 x-9 x^2\right ) \log (3)+\left (3+3 x-3 x^2\right ) \log (3) \log \left (x^2\right )\right ) \log \left (\frac {9+3 \log \left (x^2\right )}{x}\right )}{3 e^x+e^x \log \left (x^2\right )} \, dx=3 \, x^{2} e^{\left (-x\right )} \log \left (3\right )^{2} - 3 \, x^{2} e^{\left (-x\right )} \log \left (3\right ) \log \left (x\right ) + 3 \, x^{2} e^{\left (-x\right )} \log \left (3\right ) \log \left (\log \left (x^{2}\right ) + 3\right ) + 3 \, x e^{\left (-x\right )} \log \left (3\right )^{2} - 3 \, x e^{\left (-x\right )} \log \left (3\right ) \log \left (x\right ) + 3 \, x e^{\left (-x\right )} \log \left (3\right ) \log \left (\log \left (x^{2}\right ) + 3\right ) \] Input:
integrate((((-3*x^2+3*x+3)*log(3)*log(x^2)+(-9*x^2+9*x+9)*log(3))*log((3*l og(x^2)+9)/x)+(-3*x-3)*log(3)*log(x^2)+(-3*x-3)*log(3))/(exp(x)*log(x^2)+3 *exp(x)),x, algorithm="giac")
Output:
3*x^2*e^(-x)*log(3)^2 - 3*x^2*e^(-x)*log(3)*log(x) + 3*x^2*e^(-x)*log(3)*l og(log(x^2) + 3) + 3*x*e^(-x)*log(3)^2 - 3*x*e^(-x)*log(3)*log(x) + 3*x*e^ (-x)*log(3)*log(log(x^2) + 3)
Timed out. \[ \int \frac {(-3-3 x) \log (3)+(-3-3 x) \log (3) \log \left (x^2\right )+\left (\left (9+9 x-9 x^2\right ) \log (3)+\left (3+3 x-3 x^2\right ) \log (3) \log \left (x^2\right )\right ) \log \left (\frac {9+3 \log \left (x^2\right )}{x}\right )}{3 e^x+e^x \log \left (x^2\right )} \, dx=\int -\frac {\ln \left (3\right )\,\left (3\,x+3\right )-\ln \left (\frac {3\,\ln \left (x^2\right )+9}{x}\right )\,\left (\ln \left (3\right )\,\left (-9\,x^2+9\,x+9\right )+\ln \left (x^2\right )\,\ln \left (3\right )\,\left (-3\,x^2+3\,x+3\right )\right )+\ln \left (x^2\right )\,\ln \left (3\right )\,\left (3\,x+3\right )}{3\,{\mathrm {e}}^x+\ln \left (x^2\right )\,{\mathrm {e}}^x} \,d x \] Input:
int(-(log(3)*(3*x + 3) - log((3*log(x^2) + 9)/x)*(log(3)*(9*x - 9*x^2 + 9) + log(x^2)*log(3)*(3*x - 3*x^2 + 3)) + log(x^2)*log(3)*(3*x + 3))/(3*exp( x) + log(x^2)*exp(x)),x)
Output:
int(-(log(3)*(3*x + 3) - log((3*log(x^2) + 9)/x)*(log(3)*(9*x - 9*x^2 + 9) + log(x^2)*log(3)*(3*x - 3*x^2 + 3)) + log(x^2)*log(3)*(3*x + 3))/(3*exp( x) + log(x^2)*exp(x)), x)
Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {(-3-3 x) \log (3)+(-3-3 x) \log (3) \log \left (x^2\right )+\left (\left (9+9 x-9 x^2\right ) \log (3)+\left (3+3 x-3 x^2\right ) \log (3) \log \left (x^2\right )\right ) \log \left (\frac {9+3 \log \left (x^2\right )}{x}\right )}{3 e^x+e^x \log \left (x^2\right )} \, dx=\frac {3 \,\mathrm {log}\left (\frac {3 \,\mathrm {log}\left (x^{2}\right )+9}{x}\right ) \mathrm {log}\left (3\right ) x \left (x +1\right )}{e^{x}} \] Input:
int((((-3*x^2+3*x+3)*log(3)*log(x^2)+(-9*x^2+9*x+9)*log(3))*log((3*log(x^2 )+9)/x)+(-3*x-3)*log(3)*log(x^2)+(-3*x-3)*log(3))/(exp(x)*log(x^2)+3*exp(x )),x)
Output:
(3*log((3*log(x**2) + 9)/x)*log(3)*x*(x + 1))/e**x