Integrand size = 199, antiderivative size = 25 \[ \int \frac {x^2+\left (-18 x-2 x^2\right ) \log (9+x)+\left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{2 e^x+2 x} \left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{e^x+x} \left (-x^2+\left (18 x-7 x^2-x^3+e^x \left (-9 x^2-x^3\right )\right ) \log (9+x)+\left (-36 x-4 x^2+(-18-2 x) \log (3)\right ) \log ^2(9+x)\right )}{e^{e^x+x} (-18-2 x) \log ^2(9+x)+(9+x) \log ^2(9+x)+e^{2 e^x+2 x} (9+x) \log ^2(9+x)} \, dx=x \left (x+\log (3)+\frac {x}{\left (-1+e^{e^x+x}\right ) \log (9+x)}\right ) \]
Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {x^2+\left (-18 x-2 x^2\right ) \log (9+x)+\left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{2 e^x+2 x} \left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{e^x+x} \left (-x^2+\left (18 x-7 x^2-x^3+e^x \left (-9 x^2-x^3\right )\right ) \log (9+x)+\left (-36 x-4 x^2+(-18-2 x) \log (3)\right ) \log ^2(9+x)\right )}{e^{e^x+x} (-18-2 x) \log ^2(9+x)+(9+x) \log ^2(9+x)+e^{2 e^x+2 x} (9+x) \log ^2(9+x)} \, dx=x \left (x+\log (3)+\frac {x}{\left (-1+e^{e^x+x}\right ) \log (9+x)}\right ) \]
Integrate[(x^2 + (-18*x - 2*x^2)*Log[9 + x] + (18*x + 2*x^2 + (9 + x)*Log[ 3])*Log[9 + x]^2 + E^(2*E^x + 2*x)*(18*x + 2*x^2 + (9 + x)*Log[3])*Log[9 + x]^2 + E^(E^x + x)*(-x^2 + (18*x - 7*x^2 - x^3 + E^x*(-9*x^2 - x^3))*Log[ 9 + x] + (-36*x - 4*x^2 + (-18 - 2*x)*Log[3])*Log[9 + x]^2))/(E^(E^x + x)* (-18 - 2*x)*Log[9 + x]^2 + (9 + x)*Log[9 + x]^2 + E^(2*E^x + 2*x)*(9 + x)* Log[9 + 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.
| \(\displaystyle \int \frac {x^2+e^{2 x+2 e^x} \left (2 x^2+18 x+(x+9) \log (3)\right ) \log ^2(x+9)+\left (2 x^2+18 x+(x+9) \log (3)\right ) \log ^2(x+9)+\left (-2 x^2-18 x\right ) \log (x+9)+e^{x+e^x} \left (-x^2+\left (-4 x^2-36 x+(-2 x-18) \log (3)\right ) \log ^2(x+9)+\left (-x^3-7 x^2+e^x \left (-x^3-9 x^2\right )+18 x\right ) \log (x+9)\right )}{e^{x+e^x} (-2 x-18) \log ^2(x+9)+e^{2 x+2 e^x} (x+9) \log ^2(x+9)+(x+9) \log ^2(x+9)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {x^2+e^{2 x+2 e^x} \left (2 x^2+18 x+(x+9) \log (3)\right ) \log ^2(x+9)+\left (2 x^2+18 x+(x+9) \log (3)\right ) \log ^2(x+9)+\left (-2 x^2-18 x\right ) \log (x+9)+e^{x+e^x} \left (-x^2+\left (-4 x^2-36 x+(-2 x-18) \log (3)\right ) \log ^2(x+9)+\left (-x^3-7 x^2+e^x \left (-x^3-9 x^2\right )+18 x\right ) \log (x+9)\right )}{\left (1-e^{x+e^x}\right )^2 (x+9) \log ^2(x+9)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {e^{-e^x} x \left (e^{e^x} x^2 \log (x+9)+2 x^2 \log (x+9)+e^{e^x} x+7 e^{e^x} x \log (x+9)+18 x \log (x+9)-18 e^{e^x} \log (x+9)\right )}{\left (e^{x+e^x}-1\right ) (x+9) \log ^2(x+9)}-\frac {e^{-e^x} \left (e^{e^x}+1\right ) x^2}{\left (e^{x+e^x}-1\right )^2 \log (x+9)}+\frac {e^{-e^x} \left (-x^2+2 e^{e^x} x \log (x+9)+e^{e^x} \log (3) \log (x+9)\right )}{\log (x+9)}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (-\frac {e^{-e^x} x \left (e^{e^x} x^2 \log (x+9)+2 x^2 \log (x+9)+e^{e^x} x+7 e^{e^x} x \log (x+9)+18 x \log (x+9)-18 e^{e^x} \log (x+9)\right )}{\left (e^{x+e^x}-1\right ) (x+9) \log ^2(x+9)}-\frac {e^{-e^x} \left (e^{e^x}+1\right ) x^2}{\left (e^{x+e^x}-1\right )^2 \log (x+9)}+\frac {e^{-e^x} \left (-x^2+2 e^{e^x} x \log (x+9)+e^{e^x} \log (3) \log (x+9)\right )}{\log (x+9)}\right )dx\) |
Int[(x^2 + (-18*x - 2*x^2)*Log[9 + x] + (18*x + 2*x^2 + (9 + x)*Log[3])*Lo g[9 + x]^2 + E^(2*E^x + 2*x)*(18*x + 2*x^2 + (9 + x)*Log[3])*Log[9 + x]^2 + E^(E^x + x)*(-x^2 + (18*x - 7*x^2 - x^3 + E^x*(-9*x^2 - x^3))*Log[9 + x] + (-36*x - 4*x^2 + (-18 - 2*x)*Log[3])*Log[9 + x]^2))/(E^(E^x + x)*(-18 - 2*x)*Log[9 + x]^2 + (9 + x)*Log[9 + x]^2 + E^(2*E^x + 2*x)*(9 + x)*Log[9 + x]^2),x]
3.26.81.3.1 Defintions of rubi rules used
Time = 4.72 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12
| method | result | size |
| risch | \(x \ln \left (3\right )+x^{2}+\frac {x^{2}}{\ln \left (x +9\right ) \left ({\mathrm e}^{{\mathrm e}^{x}+x}-1\right )}\) | \(28\) |
| parallelrisch | \(\frac {x \ln \left (x +9\right ) {\mathrm e}^{{\mathrm e}^{x}+x} \ln \left (3\right )+\ln \left (x +9\right ) {\mathrm e}^{{\mathrm e}^{x}+x} x^{2}-\ln \left (x +9\right ) x \ln \left (3\right )-18 \ln \left (3\right ) \ln \left (x +9\right ) {\mathrm e}^{{\mathrm e}^{x}+x}-\ln \left (x +9\right ) x^{2}+18 \ln \left (3\right ) \ln \left (x +9\right )+x^{2}-81 \ln \left (x +9\right ) {\mathrm e}^{{\mathrm e}^{x}+x}+81 \ln \left (x +9\right )}{\left ({\mathrm e}^{{\mathrm e}^{x}+x}-1\right ) \ln \left (x +9\right )}\) | \(103\) |
int((((x+9)*ln(3)+2*x^2+18*x)*ln(x+9)^2*exp(exp(x)+x)^2+(((-2*x-18)*ln(3)- 4*x^2-36*x)*ln(x+9)^2+((-x^3-9*x^2)*exp(x)-x^3-7*x^2+18*x)*ln(x+9)-x^2)*ex p(exp(x)+x)+((x+9)*ln(3)+2*x^2+18*x)*ln(x+9)^2+(-2*x^2-18*x)*ln(x+9)+x^2)/ ((x+9)*ln(x+9)^2*exp(exp(x)+x)^2+(-2*x-18)*ln(x+9)^2*exp(exp(x)+x)+(x+9)*l n(x+9)^2),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (23) = 46\).
Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.24 \[ \int \frac {x^2+\left (-18 x-2 x^2\right ) \log (9+x)+\left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{2 e^x+2 x} \left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{e^x+x} \left (-x^2+\left (18 x-7 x^2-x^3+e^x \left (-9 x^2-x^3\right )\right ) \log (9+x)+\left (-36 x-4 x^2+(-18-2 x) \log (3)\right ) \log ^2(9+x)\right )}{e^{e^x+x} (-18-2 x) \log ^2(9+x)+(9+x) \log ^2(9+x)+e^{2 e^x+2 x} (9+x) \log ^2(9+x)} \, dx=\frac {{\left (x^{2} + x \log \left (3\right )\right )} e^{\left (x + e^{x}\right )} \log \left (x + 9\right ) + x^{2} - {\left (x^{2} + x \log \left (3\right )\right )} \log \left (x + 9\right )}{e^{\left (x + e^{x}\right )} \log \left (x + 9\right ) - \log \left (x + 9\right )} \]
integrate((((x+9)*log(3)+2*x^2+18*x)*log(x+9)^2*exp(exp(x)+x)^2+(((-2*x-18 )*log(3)-4*x^2-36*x)*log(x+9)^2+((-x^3-9*x^2)*exp(x)-x^3-7*x^2+18*x)*log(x +9)-x^2)*exp(exp(x)+x)+((x+9)*log(3)+2*x^2+18*x)*log(x+9)^2+(-2*x^2-18*x)* log(x+9)+x^2)/((x+9)*log(x+9)^2*exp(exp(x)+x)^2+(-2*x-18)*log(x+9)^2*exp(e xp(x)+x)+(x+9)*log(x+9)^2),x, algorithm=\
((x^2 + x*log(3))*e^(x + e^x)*log(x + 9) + x^2 - (x^2 + x*log(3))*log(x + 9))/(e^(x + e^x)*log(x + 9) - log(x + 9))
Time = 0.17 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {x^2+\left (-18 x-2 x^2\right ) \log (9+x)+\left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{2 e^x+2 x} \left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{e^x+x} \left (-x^2+\left (18 x-7 x^2-x^3+e^x \left (-9 x^2-x^3\right )\right ) \log (9+x)+\left (-36 x-4 x^2+(-18-2 x) \log (3)\right ) \log ^2(9+x)\right )}{e^{e^x+x} (-18-2 x) \log ^2(9+x)+(9+x) \log ^2(9+x)+e^{2 e^x+2 x} (9+x) \log ^2(9+x)} \, dx=x^{2} + \frac {x^{2}}{e^{x + e^{x}} \log {\left (x + 9 \right )} - \log {\left (x + 9 \right )}} + x \log {\left (3 \right )} \]
integrate((((x+9)*ln(3)+2*x**2+18*x)*ln(x+9)**2*exp(exp(x)+x)**2+(((-2*x-1 8)*ln(3)-4*x**2-36*x)*ln(x+9)**2+((-x**3-9*x**2)*exp(x)-x**3-7*x**2+18*x)* ln(x+9)-x**2)*exp(exp(x)+x)+((x+9)*ln(3)+2*x**2+18*x)*ln(x+9)**2+(-2*x**2- 18*x)*ln(x+9)+x**2)/((x+9)*ln(x+9)**2*exp(exp(x)+x)**2+(-2*x-18)*ln(x+9)** 2*exp(exp(x)+x)+(x+9)*ln(x+9)**2),x)
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (23) = 46\).
Time = 0.33 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.24 \[ \int \frac {x^2+\left (-18 x-2 x^2\right ) \log (9+x)+\left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{2 e^x+2 x} \left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{e^x+x} \left (-x^2+\left (18 x-7 x^2-x^3+e^x \left (-9 x^2-x^3\right )\right ) \log (9+x)+\left (-36 x-4 x^2+(-18-2 x) \log (3)\right ) \log ^2(9+x)\right )}{e^{e^x+x} (-18-2 x) \log ^2(9+x)+(9+x) \log ^2(9+x)+e^{2 e^x+2 x} (9+x) \log ^2(9+x)} \, dx=\frac {{\left (x^{2} + x \log \left (3\right )\right )} e^{\left (x + e^{x}\right )} \log \left (x + 9\right ) + x^{2} - {\left (x^{2} + x \log \left (3\right )\right )} \log \left (x + 9\right )}{e^{\left (x + e^{x}\right )} \log \left (x + 9\right ) - \log \left (x + 9\right )} \]
integrate((((x+9)*log(3)+2*x^2+18*x)*log(x+9)^2*exp(exp(x)+x)^2+(((-2*x-18 )*log(3)-4*x^2-36*x)*log(x+9)^2+((-x^3-9*x^2)*exp(x)-x^3-7*x^2+18*x)*log(x +9)-x^2)*exp(exp(x)+x)+((x+9)*log(3)+2*x^2+18*x)*log(x+9)^2+(-2*x^2-18*x)* log(x+9)+x^2)/((x+9)*log(x+9)^2*exp(exp(x)+x)^2+(-2*x-18)*log(x+9)^2*exp(e xp(x)+x)+(x+9)*log(x+9)^2),x, algorithm=\
((x^2 + x*log(3))*e^(x + e^x)*log(x + 9) + x^2 - (x^2 + x*log(3))*log(x + 9))/(e^(x + e^x)*log(x + 9) - log(x + 9))
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (23) = 46\).
Time = 0.31 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.72 \[ \int \frac {x^2+\left (-18 x-2 x^2\right ) \log (9+x)+\left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{2 e^x+2 x} \left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{e^x+x} \left (-x^2+\left (18 x-7 x^2-x^3+e^x \left (-9 x^2-x^3\right )\right ) \log (9+x)+\left (-36 x-4 x^2+(-18-2 x) \log (3)\right ) \log ^2(9+x)\right )}{e^{e^x+x} (-18-2 x) \log ^2(9+x)+(9+x) \log ^2(9+x)+e^{2 e^x+2 x} (9+x) \log ^2(9+x)} \, dx=\frac {x^{2} e^{\left (x + e^{x}\right )} \log \left (x + 9\right ) + x e^{\left (x + e^{x}\right )} \log \left (3\right ) \log \left (x + 9\right ) - x^{2} \log \left (x + 9\right ) - x \log \left (3\right ) \log \left (x + 9\right ) + x^{2}}{e^{\left (x + e^{x}\right )} \log \left (x + 9\right ) - \log \left (x + 9\right )} \]
integrate((((x+9)*log(3)+2*x^2+18*x)*log(x+9)^2*exp(exp(x)+x)^2+(((-2*x-18 )*log(3)-4*x^2-36*x)*log(x+9)^2+((-x^3-9*x^2)*exp(x)-x^3-7*x^2+18*x)*log(x +9)-x^2)*exp(exp(x)+x)+((x+9)*log(3)+2*x^2+18*x)*log(x+9)^2+(-2*x^2-18*x)* log(x+9)+x^2)/((x+9)*log(x+9)^2*exp(exp(x)+x)^2+(-2*x-18)*log(x+9)^2*exp(e xp(x)+x)+(x+9)*log(x+9)^2),x, algorithm=\
(x^2*e^(x + e^x)*log(x + 9) + x*e^(x + e^x)*log(3)*log(x + 9) - x^2*log(x + 9) - x*log(3)*log(x + 9) + x^2)/(e^(x + e^x)*log(x + 9) - log(x + 9))
Time = 10.81 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.28 \[ \int \frac {x^2+\left (-18 x-2 x^2\right ) \log (9+x)+\left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{2 e^x+2 x} \left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{e^x+x} \left (-x^2+\left (18 x-7 x^2-x^3+e^x \left (-9 x^2-x^3\right )\right ) \log (9+x)+\left (-36 x-4 x^2+(-18-2 x) \log (3)\right ) \log ^2(9+x)\right )}{e^{e^x+x} (-18-2 x) \log ^2(9+x)+(9+x) \log ^2(9+x)+e^{2 e^x+2 x} (9+x) \log ^2(9+x)} \, dx=\frac {x\,\left (x-\ln \left (x+9\right )\,\ln \left (3\right )-x\,\ln \left (x+9\right )+\ln \left (x+9\right )\,{\mathrm {e}}^{x+{\mathrm {e}}^x}\,\ln \left (3\right )+x\,\ln \left (x+9\right )\,{\mathrm {e}}^{x+{\mathrm {e}}^x}\right )}{\ln \left (x+9\right )\,\left ({\mathrm {e}}^{x+{\mathrm {e}}^x}-1\right )} \]
int((log(x + 9)^2*(18*x + log(3)*(x + 9) + 2*x^2) - log(x + 9)*(18*x + 2*x ^2) - exp(x + exp(x))*(log(x + 9)^2*(36*x + log(3)*(2*x + 18) + 4*x^2) + l og(x + 9)*(exp(x)*(9*x^2 + x^3) - 18*x + 7*x^2 + x^3) + x^2) + x^2 + log(x + 9)^2*exp(2*x + 2*exp(x))*(18*x + log(3)*(x + 9) + 2*x^2))/(log(x + 9)^2 *(x + 9) - log(x + 9)^2*exp(x + exp(x))*(2*x + 18) + log(x + 9)^2*exp(2*x + 2*exp(x))*(x + 9)),x)