\(\int \frac {x^2+(-18 x-2 x^2) \log (9+x)+(18 x+2 x^2+(9+x) \log (3)) \log ^2(9+x)+e^{2 e^x+2 x} (18 x+2 x^2+(9+x) \log (3)) \log ^2(9+x)+e^{e^x+x} (-x^2+(18 x-7 x^2-x^3+e^x (-9 x^2-x^3)) \log (9+x)+(-36 x-4 x^2+(-18-2 x) \log (3)) \log ^2(9+x))}{e^{e^x+x} (-18-2 x) \log ^2(9+x)+(9+x) \log ^2(9+x)+e^{2 e^x+2 x} (9+x) \log ^2(9+x)} \, dx\) [5581]

Optimal result

Integrand size = 199, antiderivative size = 25 \[ \int \frac {x^2+\left (-18 x-2 x^2\right ) \log (9+x)+\left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{2 e^x+2 x} \left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{e^x+x} \left (-x^2+\left (18 x-7 x^2-x^3+e^x \left (-9 x^2-x^3\right )\right ) \log (9+x)+\left (-36 x-4 x^2+(-18-2 x) \log (3)\right ) \log ^2(9+x)\right )}{e^{e^x+x} (-18-2 x) \log ^2(9+x)+(9+x) \log ^2(9+x)+e^{2 e^x+2 x} (9+x) \log ^2(9+x)} \, dx=x \left (x+\log (3)+\frac {x}{\left (-1+e^{e^x+x}\right ) \log (9+x)}\right ) \]

[Out]

x*(x+x/(exp(exp(x)+x)-1)/ln(x+9)+ln(3))

Rubi [F]

\[ \int \frac {x^2+\left (-18 x-2 x^2\right ) \log (9+x)+\left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{2 e^x+2 x} \left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{e^x+x} \left (-x^2+\left (18 x-7 x^2-x^3+e^x \left (-9 x^2-x^3\right )\right ) \log (9+x)+\left (-36 x-4 x^2+(-18-2 x) \log (3)\right ) \log ^2(9+x)\right )}{e^{e^x+x} (-18-2 x) \log ^2(9+x)+(9+x) \log ^2(9+x)+e^{2 e^x+2 x} (9+x) \log ^2(9+x)} \, dx=\int \frac {x^2+\left (-18 x-2 x^2\right ) \log (9+x)+\left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{2 e^x+2 x} \left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{e^x+x} \left (-x^2+\left (18 x-7 x^2-x^3+e^x \left (-9 x^2-x^3\right )\right ) \log (9+x)+\left (-36 x-4 x^2+(-18-2 x) \log (3)\right ) \log ^2(9+x)\right )}{e^{e^x+x} (-18-2 x) \log ^2(9+x)+(9+x) \log ^2(9+x)+e^{2 e^x+2 x} (9+x) \log ^2(9+x)} \, dx \]

[In]

Int[(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]

[Out]

x^2 + x*Log[3] + 9*Defer[Int][1/((-1 + E^(E^x + x))*Log[9 + x]^2), x] - Defer[Int][x/((-1 + E^(E^x + x))*Log[9
+ x]^2), x] - 81*Defer[Int][1/((-1 + E^(E^x + x))*(9 + x)*Log[9 + x]^2), x] - 81*Defer[Int][1/(E^E^x*Log[9 +
x]), x] + 2*Defer[Int][x/((-1 + E^(E^x + x))*Log[9 + x]), x] - Defer[Int][x^2/((-1 + E^(E^x + x))^2*Log[9 + x]
), x] - Defer[Int][x^2/(E^E^x*(-1 + E^(E^x + x))^2*Log[9 + x]), x] - Defer[Int][x^2/((-1 + E^(E^x + x))*Log[9
+ x]), x] - 2*Defer[Int][x^2/(E^E^x*(-1 + E^(E^x + x))*Log[9 + x]), x] + 18*Defer[Int][(9 + x)/(E^E^x*Log[9 +
x]), x] - Defer[Int][(9 + x)^2/(E^E^x*Log[9 + x]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-\left (\left (-1+e^{e^x+x}\right ) x^2\right )-x (9+x) \left (2+e^{e^x+x} (-2+x)+e^{e^x+2 x} x\right ) \log (9+x)+\left (2 \left (-1+e^{e^x+x}\right )^2 x^2+9 \left (-1+e^{e^x+x}\right )^2 \log (3)+x \left (18+\log (3)+e^{2 \left (e^x+x\right )} (18+\log (3))-e^{e^x+x} (36+\log (9))\right )\right ) \log ^2(9+x)}{\left (1-e^{e^x+x}\right )^2 (9+x) \log ^2(9+x)} \, dx \\ & = \int \left (-\frac {e^{-e^x} \left (1+e^{e^x}\right ) x^2}{\left (-1+e^{e^x+x}\right )^2 \log (9+x)}-\frac {e^{-e^x} x \left (e^{e^x} x-18 e^{e^x} \log (9+x)+18 x \log (9+x)+7 e^{e^x} x \log (9+x)+2 x^2 \log (9+x)+e^{e^x} x^2 \log (9+x)\right )}{\left (-1+e^{e^x+x}\right ) (9+x) \log ^2(9+x)}+\frac {e^{-e^x} \left (-x^2+2 e^{e^x} x \log (9+x)+e^{e^x} \log (3) \log (9+x)\right )}{\log (9+x)}\right ) \, dx \\ & = -\int \frac {e^{-e^x} \left (1+e^{e^x}\right ) x^2}{\left (-1+e^{e^x+x}\right )^2 \log (9+x)} \, dx-\int \frac {e^{-e^x} x \left (e^{e^x} x-18 e^{e^x} \log (9+x)+18 x \log (9+x)+7 e^{e^x} x \log (9+x)+2 x^2 \log (9+x)+e^{e^x} x^2 \log (9+x)\right )}{\left (-1+e^{e^x+x}\right ) (9+x) \log ^2(9+x)} \, dx+\int \frac {e^{-e^x} \left (-x^2+2 e^{e^x} x \log (9+x)+e^{e^x} \log (3) \log (9+x)\right )}{\log (9+x)} \, dx \\ & = \int \left (2 x+\log (3)-\frac {e^{-e^x} x^2}{\log (9+x)}\right ) \, dx-\int \left (\frac {x^2}{\left (-1+e^{e^x+x}\right )^2 \log (9+x)}+\frac {e^{-e^x} x^2}{\left (-1+e^{e^x+x}\right )^2 \log (9+x)}\right ) \, dx-\int \frac {e^{-e^x} x \left (-e^{e^x} x-(9+x) \left (e^{e^x} (-2+x)+2 x\right ) \log (9+x)\right )}{\left (1-e^{e^x+x}\right ) (9+x) \log ^2(9+x)} \, dx \\ & = x^2+x \log (3)-\int \frac {e^{-e^x} x^2}{\log (9+x)} \, dx-\int \frac {x^2}{\left (-1+e^{e^x+x}\right )^2 \log (9+x)} \, dx-\int \frac {e^{-e^x} x^2}{\left (-1+e^{e^x+x}\right )^2 \log (9+x)} \, dx-\int \left (\frac {e^{-e^x} \left (e^{e^x} x-18 e^{e^x} \log (9+x)+18 x \log (9+x)+7 e^{e^x} x \log (9+x)+2 x^2 \log (9+x)+e^{e^x} x^2 \log (9+x)\right )}{\left (-1+e^{e^x+x}\right ) \log ^2(9+x)}-\frac {9 e^{-e^x} \left (e^{e^x} x-18 e^{e^x} \log (9+x)+18 x \log (9+x)+7 e^{e^x} x \log (9+x)+2 x^2 \log (9+x)+e^{e^x} x^2 \log (9+x)\right )}{\left (-1+e^{e^x+x}\right ) (9+x) \log ^2(9+x)}\right ) \, dx \\ & = x^2+x \log (3)+9 \int \frac {e^{-e^x} \left (e^{e^x} x-18 e^{e^x} \log (9+x)+18 x \log (9+x)+7 e^{e^x} x \log (9+x)+2 x^2 \log (9+x)+e^{e^x} x^2 \log (9+x)\right )}{\left (-1+e^{e^x+x}\right ) (9+x) \log ^2(9+x)} \, dx-\int \left (\frac {81 e^{-e^x}}{\log (9+x)}-\frac {18 e^{-e^x} (9+x)}{\log (9+x)}+\frac {e^{-e^x} (9+x)^2}{\log (9+x)}\right ) \, dx-\int \frac {x^2}{\left (-1+e^{e^x+x}\right )^2 \log (9+x)} \, dx-\int \frac {e^{-e^x} x^2}{\left (-1+e^{e^x+x}\right )^2 \log (9+x)} \, dx-\int \frac {e^{-e^x} \left (e^{e^x} x-18 e^{e^x} \log (9+x)+18 x \log (9+x)+7 e^{e^x} x \log (9+x)+2 x^2 \log (9+x)+e^{e^x} x^2 \log (9+x)\right )}{\left (-1+e^{e^x+x}\right ) \log ^2(9+x)} \, dx \\ & = x^2+x \log (3)+9 \int \frac {e^{-e^x} \left (-e^{e^x} x-(9+x) \left (e^{e^x} (-2+x)+2 x\right ) \log (9+x)\right )}{\left (1-e^{e^x+x}\right ) (9+x) \log ^2(9+x)} \, dx+18 \int \frac {e^{-e^x} (9+x)}{\log (9+x)} \, dx-81 \int \frac {e^{-e^x}}{\log (9+x)} \, dx-\int \frac {x^2}{\left (-1+e^{e^x+x}\right )^2 \log (9+x)} \, dx-\int \frac {e^{-e^x} x^2}{\left (-1+e^{e^x+x}\right )^2 \log (9+x)} \, dx-\int \frac {e^{-e^x} (9+x)^2}{\log (9+x)} \, dx-\int \frac {e^{-e^x} \left (-e^{e^x} x-(9+x) \left (e^{e^x} (-2+x)+2 x\right ) \log (9+x)\right )}{\left (1-e^{e^x+x}\right ) \log ^2(9+x)} \, dx \\ & = x^2+x \log (3)+9 \int \left (\frac {x}{\left (-1+e^{e^x+x}\right ) (9+x) \log ^2(9+x)}-\frac {18}{\left (-1+e^{e^x+x}\right ) (9+x) \log (9+x)}+\frac {7 x}{\left (-1+e^{e^x+x}\right ) (9+x) \log (9+x)}+\frac {18 e^{-e^x} x}{\left (-1+e^{e^x+x}\right ) (9+x) \log (9+x)}+\frac {x^2}{\left (-1+e^{e^x+x}\right ) (9+x) \log (9+x)}+\frac {2 e^{-e^x} x^2}{\left (-1+e^{e^x+x}\right ) (9+x) \log (9+x)}\right ) \, dx+18 \int \frac {e^{-e^x} (9+x)}{\log (9+x)} \, dx-81 \int \frac {e^{-e^x}}{\log (9+x)} \, dx-\int \left (\frac {x}{\left (-1+e^{e^x+x}\right ) \log ^2(9+x)}-\frac {18}{\left (-1+e^{e^x+x}\right ) \log (9+x)}+\frac {7 x}{\left (-1+e^{e^x+x}\right ) \log (9+x)}+\frac {18 e^{-e^x} x}{\left (-1+e^{e^x+x}\right ) \log (9+x)}+\frac {x^2}{\left (-1+e^{e^x+x}\right ) \log (9+x)}+\frac {2 e^{-e^x} x^2}{\left (-1+e^{e^x+x}\right ) \log (9+x)}\right ) \, dx-\int \frac {x^2}{\left (-1+e^{e^x+x}\right )^2 \log (9+x)} \, dx-\int \frac {e^{-e^x} x^2}{\left (-1+e^{e^x+x}\right )^2 \log (9+x)} \, dx-\int \frac {e^{-e^x} (9+x)^2}{\log (9+x)} \, dx \\ & = x^2+x \log (3)-2 \int \frac {e^{-e^x} x^2}{\left (-1+e^{e^x+x}\right ) \log (9+x)} \, dx-7 \int \frac {x}{\left (-1+e^{e^x+x}\right ) \log (9+x)} \, dx+9 \int \frac {x}{\left (-1+e^{e^x+x}\right ) (9+x) \log ^2(9+x)} \, dx+9 \int \frac {x^2}{\left (-1+e^{e^x+x}\right ) (9+x) \log (9+x)} \, dx+18 \int \frac {1}{\left (-1+e^{e^x+x}\right ) \log (9+x)} \, dx-18 \int \frac {e^{-e^x} x}{\left (-1+e^{e^x+x}\right ) \log (9+x)} \, dx+18 \int \frac {e^{-e^x} x^2}{\left (-1+e^{e^x+x}\right ) (9+x) \log (9+x)} \, dx+18 \int \frac {e^{-e^x} (9+x)}{\log (9+x)} \, dx+63 \int \frac {x}{\left (-1+e^{e^x+x}\right ) (9+x) \log (9+x)} \, dx-81 \int \frac {e^{-e^x}}{\log (9+x)} \, dx-162 \int \frac {1}{\left (-1+e^{e^x+x}\right ) (9+x) \log (9+x)} \, dx+162 \int \frac {e^{-e^x} x}{\left (-1+e^{e^x+x}\right ) (9+x) \log (9+x)} \, dx-\int \frac {x}{\left (-1+e^{e^x+x}\right ) \log ^2(9+x)} \, dx-\int \frac {x^2}{\left (-1+e^{e^x+x}\right )^2 \log (9+x)} \, dx-\int \frac {e^{-e^x} x^2}{\left (-1+e^{e^x+x}\right )^2 \log (9+x)} \, dx-\int \frac {x^2}{\left (-1+e^{e^x+x}\right ) \log (9+x)} \, dx-\int \frac {e^{-e^x} (9+x)^2}{\log (9+x)} \, dx \\ & = x^2+x \log (3)-2 \int \frac {e^{-e^x} x^2}{\left (-1+e^{e^x+x}\right ) \log (9+x)} \, dx-7 \int \frac {x}{\left (-1+e^{e^x+x}\right ) \log (9+x)} \, dx+9 \int \left (\frac {1}{\left (-1+e^{e^x+x}\right ) \log ^2(9+x)}-\frac {9}{\left (-1+e^{e^x+x}\right ) (9+x) \log ^2(9+x)}\right ) \, dx+9 \int \left (-\frac {9}{\left (-1+e^{e^x+x}\right ) \log (9+x)}+\frac {x}{\left (-1+e^{e^x+x}\right ) \log (9+x)}+\frac {81}{\left (-1+e^{e^x+x}\right ) (9+x) \log (9+x)}\right ) \, dx+18 \int \left (-\frac {9 e^{-e^x}}{\left (-1+e^{e^x+x}\right ) \log (9+x)}+\frac {e^{-e^x} x}{\left (-1+e^{e^x+x}\right ) \log (9+x)}+\frac {81 e^{-e^x}}{\left (-1+e^{e^x+x}\right ) (9+x) \log (9+x)}\right ) \, dx+18 \int \frac {1}{\left (-1+e^{e^x+x}\right ) \log (9+x)} \, dx-18 \int \frac {e^{-e^x} x}{\left (-1+e^{e^x+x}\right ) \log (9+x)} \, dx+18 \int \frac {e^{-e^x} (9+x)}{\log (9+x)} \, dx+63 \int \left (\frac {1}{\left (-1+e^{e^x+x}\right ) \log (9+x)}-\frac {9}{\left (-1+e^{e^x+x}\right ) (9+x) \log (9+x)}\right ) \, dx-81 \int \frac {e^{-e^x}}{\log (9+x)} \, dx+162 \int \left (\frac {e^{-e^x}}{\left (-1+e^{e^x+x}\right ) \log (9+x)}-\frac {9 e^{-e^x}}{\left (-1+e^{e^x+x}\right ) (9+x) \log (9+x)}\right ) \, dx-162 \int \frac {1}{\left (-1+e^{e^x+x}\right ) (9+x) \log (9+x)} \, dx-\int \frac {x}{\left (-1+e^{e^x+x}\right ) \log ^2(9+x)} \, dx-\int \frac {x^2}{\left (-1+e^{e^x+x}\right )^2 \log (9+x)} \, dx-\int \frac {e^{-e^x} x^2}{\left (-1+e^{e^x+x}\right )^2 \log (9+x)} \, dx-\int \frac {x^2}{\left (-1+e^{e^x+x}\right ) \log (9+x)} \, dx-\int \frac {e^{-e^x} (9+x)^2}{\log (9+x)} \, dx \\ & = x^2+x \log (3)-2 \int \frac {e^{-e^x} x^2}{\left (-1+e^{e^x+x}\right ) \log (9+x)} \, dx-7 \int \frac {x}{\left (-1+e^{e^x+x}\right ) \log (9+x)} \, dx+9 \int \frac {1}{\left (-1+e^{e^x+x}\right ) \log ^2(9+x)} \, dx+9 \int \frac {x}{\left (-1+e^{e^x+x}\right ) \log (9+x)} \, dx+18 \int \frac {1}{\left (-1+e^{e^x+x}\right ) \log (9+x)} \, dx+18 \int \frac {e^{-e^x} (9+x)}{\log (9+x)} \, dx+63 \int \frac {1}{\left (-1+e^{e^x+x}\right ) \log (9+x)} \, dx-81 \int \frac {1}{\left (-1+e^{e^x+x}\right ) (9+x) \log ^2(9+x)} \, dx-81 \int \frac {e^{-e^x}}{\log (9+x)} \, dx-81 \int \frac {1}{\left (-1+e^{e^x+x}\right ) \log (9+x)} \, dx-162 \int \frac {1}{\left (-1+e^{e^x+x}\right ) (9+x) \log (9+x)} \, dx-567 \int \frac {1}{\left (-1+e^{e^x+x}\right ) (9+x) \log (9+x)} \, dx+729 \int \frac {1}{\left (-1+e^{e^x+x}\right ) (9+x) \log (9+x)} \, dx-\int \frac {x}{\left (-1+e^{e^x+x}\right ) \log ^2(9+x)} \, dx-\int \frac {x^2}{\left (-1+e^{e^x+x}\right )^2 \log (9+x)} \, dx-\int \frac {e^{-e^x} x^2}{\left (-1+e^{e^x+x}\right )^2 \log (9+x)} \, dx-\int \frac {x^2}{\left (-1+e^{e^x+x}\right ) \log (9+x)} \, dx-\int \frac {e^{-e^x} (9+x)^2}{\log (9+x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {x^2+\left (-18 x-2 x^2\right ) \log (9+x)+\left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{2 e^x+2 x} \left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{e^x+x} \left (-x^2+\left (18 x-7 x^2-x^3+e^x \left (-9 x^2-x^3\right )\right ) \log (9+x)+\left (-36 x-4 x^2+(-18-2 x) \log (3)\right ) \log ^2(9+x)\right )}{e^{e^x+x} (-18-2 x) \log ^2(9+x)+(9+x) \log ^2(9+x)+e^{2 e^x+2 x} (9+x) \log ^2(9+x)} \, dx=x \left (x+\log (3)+\frac {x}{\left (-1+e^{e^x+x}\right ) \log (9+x)}\right ) \]

[In]

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))*Lo
g[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]

[Out]

x*(x + Log[3] + x/((-1 + E^(E^x + x))*Log[9 + x]))

Maple [A] (verified)

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	\(\frac {x \ln \left (x +9\right ) {\mathrm e}^{{\mathrm e}^{x}+x} \ln \left (3\right )+\ln \left (x +9\right ) {\mathrm e}^{{\mathrm e}^{x}+x} x^{2}-\ln \left (x +9\right ) x \ln \left (3\right )-18 \ln \left (3\right ) \ln \left (x +9\right ) {\mathrm e}^{{\mathrm e}^{x}+x}-\ln \left (x +9\right ) x^{2}+18 \ln \left (3\right ) \ln \left (x +9\right )+x^{2}-81 \ln \left (x +9\right ) {\mathrm e}^{{\mathrm e}^{x}+x}+81 \ln \left (x +9\right )}{\left ({\mathrm e}^{{\mathrm e}^{x}+x}-1\right ) \ln \left (x +9\right )}\)	\(103\)

[In]

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)*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,method=_RETURNVERBOSE)

[Out]

x*ln(3)+x^2+x^2/ln(x+9)/(exp(exp(x)+x)-1)

Fricas [B] (verification not implemented)

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 \[ \int \frac {x^2+\left (-18 x-2 x^2\right ) \log (9+x)+\left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{2 e^x+2 x} \left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{e^x+x} \left (-x^2+\left (18 x-7 x^2-x^3+e^x \left (-9 x^2-x^3\right )\right ) \log (9+x)+\left (-36 x-4 x^2+(-18-2 x) \log (3)\right ) \log ^2(9+x)\right )}{e^{e^x+x} (-18-2 x) \log ^2(9+x)+(9+x) \log ^2(9+x)+e^{2 e^x+2 x} (9+x) \log ^2(9+x)} \, dx=\frac {{\left (x^{2} + x \log \left (3\right )\right )} e^{\left (x + e^{x}\right )} \log \left (x + 9\right ) + x^{2} - {\left (x^{2} + x \log \left (3\right )\right )} \log \left (x + 9\right )}{e^{\left (x + e^{x}\right )} \log \left (x + 9\right ) - \log \left (x + 9\right )} \]

[In]

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(exp(x)+x)+(x+9)*log(x+9)^2),x, algo
rithm="fricas")

[Out]

((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))

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {x^2+\left (-18 x-2 x^2\right ) \log (9+x)+\left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{2 e^x+2 x} \left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{e^x+x} \left (-x^2+\left (18 x-7 x^2-x^3+e^x \left (-9 x^2-x^3\right )\right ) \log (9+x)+\left (-36 x-4 x^2+(-18-2 x) \log (3)\right ) \log ^2(9+x)\right )}{e^{e^x+x} (-18-2 x) \log ^2(9+x)+(9+x) \log ^2(9+x)+e^{2 e^x+2 x} (9+x) \log ^2(9+x)} \, dx=x^{2} + \frac {x^{2}}{e^{x + e^{x}} \log {\left (x + 9 \right )} - \log {\left (x + 9 \right )}} + x \log {\left (3 \right )} \]

[In]

integrate((((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)*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)

[Out]

x**2 + x**2/(exp(x + exp(x))*log(x + 9) - log(x + 9)) + x*log(3)

Maxima [B] (verification not implemented)

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 \[ \int \frac {x^2+\left (-18 x-2 x^2\right ) \log (9+x)+\left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{2 e^x+2 x} \left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{e^x+x} \left (-x^2+\left (18 x-7 x^2-x^3+e^x \left (-9 x^2-x^3\right )\right ) \log (9+x)+\left (-36 x-4 x^2+(-18-2 x) \log (3)\right ) \log ^2(9+x)\right )}{e^{e^x+x} (-18-2 x) \log ^2(9+x)+(9+x) \log ^2(9+x)+e^{2 e^x+2 x} (9+x) \log ^2(9+x)} \, dx=\frac {{\left (x^{2} + x \log \left (3\right )\right )} e^{\left (x + e^{x}\right )} \log \left (x + 9\right ) + x^{2} - {\left (x^{2} + x \log \left (3\right )\right )} \log \left (x + 9\right )}{e^{\left (x + e^{x}\right )} \log \left (x + 9\right ) - \log \left (x + 9\right )} \]

[In]

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(exp(x)+x)+(x+9)*log(x+9)^2),x, algo
rithm="maxima")

[Out]

((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))

Giac [B] (verification not implemented)

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 \[ \int \frac {x^2+\left (-18 x-2 x^2\right ) \log (9+x)+\left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{2 e^x+2 x} \left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{e^x+x} \left (-x^2+\left (18 x-7 x^2-x^3+e^x \left (-9 x^2-x^3\right )\right ) \log (9+x)+\left (-36 x-4 x^2+(-18-2 x) \log (3)\right ) \log ^2(9+x)\right )}{e^{e^x+x} (-18-2 x) \log ^2(9+x)+(9+x) \log ^2(9+x)+e^{2 e^x+2 x} (9+x) \log ^2(9+x)} \, dx=\frac {x^{2} e^{\left (x + e^{x}\right )} \log \left (x + 9\right ) + x e^{\left (x + e^{x}\right )} \log \left (3\right ) \log \left (x + 9\right ) - x^{2} \log \left (x + 9\right ) - x \log \left (3\right ) \log \left (x + 9\right ) + x^{2}}{e^{\left (x + e^{x}\right )} \log \left (x + 9\right ) - \log \left (x + 9\right )} \]

[In]

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(exp(x)+x)+(x+9)*log(x+9)^2),x, algo
rithm="giac")

[Out]

(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))

Mupad [B] (verification not implemented)

Time = 10.81 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.28 \[ \int \frac {x^2+\left (-18 x-2 x^2\right ) \log (9+x)+\left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{2 e^x+2 x} \left (18 x+2 x^2+(9+x) \log (3)\right ) \log ^2(9+x)+e^{e^x+x} \left (-x^2+\left (18 x-7 x^2-x^3+e^x \left (-9 x^2-x^3\right )\right ) \log (9+x)+\left (-36 x-4 x^2+(-18-2 x) \log (3)\right ) \log ^2(9+x)\right )}{e^{e^x+x} (-18-2 x) \log ^2(9+x)+(9+x) \log ^2(9+x)+e^{2 e^x+2 x} (9+x) \log ^2(9+x)} \, dx=\frac {x\,\left (x-\ln \left (x+9\right )\,\ln \left (3\right )-x\,\ln \left (x+9\right )+\ln \left (x+9\right )\,{\mathrm {e}}^{x+{\mathrm {e}}^x}\,\ln \left (3\right )+x\,\ln \left (x+9\right )\,{\mathrm {e}}^{x+{\mathrm {e}}^x}\right )}{\ln \left (x+9\right )\,\left ({\mathrm {e}}^{x+{\mathrm {e}}^x}-1\right )} \]

[In]

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) + log(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 + ex
p(x))*(2*x + 18) + log(x + 9)^2*exp(2*x + 2*exp(x))*(x + 9)),x)

[Out]

(x*(x - log(x + 9)*log(3) - x*log(x + 9) + log(x + 9)*exp(x + exp(x))*log(3) + x*log(x + 9)*exp(x + exp(x))))/
(log(x + 9)*(exp(x + exp(x)) - 1))