\(\int \frac {e^{2 x} (-3-6 x)+9 x^2+e^{-3+e^x} (-3-3 e^x x-3 \log (x))+(-3 e^{2 x}+3 x^2) \log (x)+(-3 e^{-3+e^x}-3 e^{2 x}+3 x^2) \log (-e^{-3+e^x}-e^{2 x}+x^2)}{(-3 e^{2 x} x+3 x^3) \log (x)+(e^{2 x} x^2-x^4) \log ^2(x)+e^{-3+e^x} (-3 x \log (x)+x^2 \log ^2(x))+(-3 e^{2 x} x+3 x^3+(2 e^{2 x} x^2-2 x^4) \log (x)+e^{-3+e^x} (-3 x+2 x^2 \log (x))) \log (-e^{-3+e^x}-e^{2 x}+x^2)+(e^{-3+e^x} x^2+e^{2 x} x^2-x^4) \log ^2(-e^{-3+e^x}-e^{2 x}+x^2)} \, dx\) [8581]

Optimal result

Integrand size = 284, antiderivative size = 35 \[ \int \frac {e^{2 x} (-3-6 x)+9 x^2+e^{-3+e^x} \left (-3-3 e^x x-3 \log (x)\right )+\left (-3 e^{2 x}+3 x^2\right ) \log (x)+\left (-3 e^{-3+e^x}-3 e^{2 x}+3 x^2\right ) \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )}{\left (-3 e^{2 x} x+3 x^3\right ) \log (x)+\left (e^{2 x} x^2-x^4\right ) \log ^2(x)+e^{-3+e^x} \left (-3 x \log (x)+x^2 \log ^2(x)\right )+\left (-3 e^{2 x} x+3 x^3+\left (2 e^{2 x} x^2-2 x^4\right ) \log (x)+e^{-3+e^x} \left (-3 x+2 x^2 \log (x)\right )\right ) \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )+\left (e^{-3+e^x} x^2+e^{2 x} x^2-x^4\right ) \log ^2\left (-e^{-3+e^x}-e^{2 x}+x^2\right )} \, dx=\log \left (\frac {x}{x-\frac {3}{\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )}}\right ) \]

[Out]

ln(x/(x-3/(ln(-exp(exp(x)-3)-exp(x)^2+x^2)+ln(x))))

Rubi [F]

\[ \int \frac {e^{2 x} (-3-6 x)+9 x^2+e^{-3+e^x} \left (-3-3 e^x x-3 \log (x)\right )+\left (-3 e^{2 x}+3 x^2\right ) \log (x)+\left (-3 e^{-3+e^x}-3 e^{2 x}+3 x^2\right ) \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )}{\left (-3 e^{2 x} x+3 x^3\right ) \log (x)+\left (e^{2 x} x^2-x^4\right ) \log ^2(x)+e^{-3+e^x} \left (-3 x \log (x)+x^2 \log ^2(x)\right )+\left (-3 e^{2 x} x+3 x^3+\left (2 e^{2 x} x^2-2 x^4\right ) \log (x)+e^{-3+e^x} \left (-3 x+2 x^2 \log (x)\right )\right ) \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )+\left (e^{-3+e^x} x^2+e^{2 x} x^2-x^4\right ) \log ^2\left (-e^{-3+e^x}-e^{2 x}+x^2\right )} \, dx=\int \frac {e^{2 x} (-3-6 x)+9 x^2+e^{-3+e^x} \left (-3-3 e^x x-3 \log (x)\right )+\left (-3 e^{2 x}+3 x^2\right ) \log (x)+\left (-3 e^{-3+e^x}-3 e^{2 x}+3 x^2\right ) \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )}{\left (-3 e^{2 x} x+3 x^3\right ) \log (x)+\left (e^{2 x} x^2-x^4\right ) \log ^2(x)+e^{-3+e^x} \left (-3 x \log (x)+x^2 \log ^2(x)\right )+\left (-3 e^{2 x} x+3 x^3+\left (2 e^{2 x} x^2-2 x^4\right ) \log (x)+e^{-3+e^x} \left (-3 x+2 x^2 \log (x)\right )\right ) \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )+\left (e^{-3+e^x} x^2+e^{2 x} x^2-x^4\right ) \log ^2\left (-e^{-3+e^x}-e^{2 x}+x^2\right )} \, dx \]

[In]

Int[(E^(2*x)*(-3 - 6*x) + 9*x^2 + E^(-3 + E^x)*(-3 - 3*E^x*x - 3*Log[x]) + (-3*E^(2*x) + 3*x^2)*Log[x] + (-3*E
^(-3 + E^x) - 3*E^(2*x) + 3*x^2)*Log[-E^(-3 + E^x) - E^(2*x) + x^2])/((-3*E^(2*x)*x + 3*x^3)*Log[x] + (E^(2*x)
*x^2 - x^4)*Log[x]^2 + E^(-3 + E^x)*(-3*x*Log[x] + x^2*Log[x]^2) + (-3*E^(2*x)*x + 3*x^3 + (2*E^(2*x)*x^2 - 2*
x^4)*Log[x] + E^(-3 + E^x)*(-3*x + 2*x^2*Log[x]))*Log[-E^(-3 + E^x) - E^(2*x) + x^2] + (E^(-3 + E^x)*x^2 + E^(
2*x)*x^2 - x^4)*Log[-E^(-3 + E^x) - E^(2*x) + x^2]^2),x]

[Out]

2*Defer[Int][(Log[x] + Log[-E^(-3 + E^x) - E^(2*x) + x^2])^(-1), x] + Defer[Int][1/(x*(Log[x] + Log[-E^(-3 + E
^x) - E^(2*x) + x^2])), x] + Defer[Int][(3 - x*Log[x] - x*Log[-E^(-3 + E^x) - E^(2*x) + x^2])^(-1), x] - 3*Def
er[Int][1/(x*(-3 + x*Log[x] + x*Log[-E^(-3 + E^x) - E^(2*x) + x^2])), x] - 2*Defer[Int][x/(-3 + x*Log[x] + x*L
og[-E^(-3 + E^x) - E^(2*x) + x^2]), x] + 6*Defer[Int][E^E^x/((E^E^x + E^(3 + 2*x) - E^3*x^2)*(Log[x] + Log[-E^
(-3 + E^x) - E^(2*x) + x^2])*(-3 + x*Log[x] + x*Log[-E^(-3 + E^x) - E^(2*x) + x^2])), x] - 3*Defer[Int][E^(E^x
+ x)/((E^E^x + E^(3 + 2*x) - E^3*x^2)*(Log[x] + Log[-E^(-3 + E^x) - E^(2*x) + x^2])*(-3 + x*Log[x] + x*Log[-E
^(-3 + E^x) - E^(2*x) + x^2])), x] - 6*E^3*Defer[Int][x/((-E^E^x - E^(3 + 2*x) + E^3*x^2)*(Log[x] + Log[-E^(-3
+ E^x) - E^(2*x) + x^2])*(-3 + x*Log[x] + x*Log[-E^(-3 + E^x) - E^(2*x) + x^2])), x] + 6*E^3*Defer[Int][x^2/(
(-E^E^x - E^(3 + 2*x) + E^3*x^2)*(Log[x] + Log[-E^(-3 + E^x) - E^(2*x) + x^2])*(-3 + x*Log[x] + x*Log[-E^(-3 +
E^x) - E^(2*x) + x^2])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {3 \left (e^{e^x}+e^{3+2 x}+e^{e^x+x} x+2 e^{3+2 x} x-3 e^3 x^2+\left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \log (x)+\left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )}{x \left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (3-x \log (x)-x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx \\ & = 3 \int \frac {e^{e^x}+e^{3+2 x}+e^{e^x+x} x+2 e^{3+2 x} x-3 e^3 x^2+\left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \log (x)+\left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )}{x \left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (3-x \log (x)-x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx \\ & = 3 \int \left (-\frac {-2 e^{e^x}+e^{e^x+x}-2 e^3 x+2 e^3 x^2}{\left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )}+\frac {-1-2 x-\log (x)-\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )}{x \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )}\right ) \, dx \\ & = -\left (3 \int \frac {-2 e^{e^x}+e^{e^x+x}-2 e^3 x+2 e^3 x^2}{\left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx\right )+3 \int \frac {-1-2 x-\log (x)-\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )}{x \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx \\ & = 3 \int \left (\frac {1+2 x}{3 x \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )}+\frac {-3-x-2 x^2}{3 x \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )}\right ) \, dx-3 \int \left (-\frac {2 e^{e^x}}{\left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )}+\frac {e^{e^x+x}}{\left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )}+\frac {2 e^3 x}{\left (-e^{e^x}-e^{3+2 x}+e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )}-\frac {2 e^3 x^2}{\left (-e^{e^x}-e^{3+2 x}+e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )}\right ) \, dx \\ & = -\left (3 \int \frac {e^{e^x+x}}{\left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx\right )+6 \int \frac {e^{e^x}}{\left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx-\left (6 e^3\right ) \int \frac {x}{\left (-e^{e^x}-e^{3+2 x}+e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx+\left (6 e^3\right ) \int \frac {x^2}{\left (-e^{e^x}-e^{3+2 x}+e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx+\int \frac {1+2 x}{x \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx+\int \frac {-3-x-2 x^2}{x \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx \\ & = -\left (3 \int \frac {e^{e^x+x}}{\left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx\right )+6 \int \frac {e^{e^x}}{\left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx-\left (6 e^3\right ) \int \frac {x}{\left (-e^{e^x}-e^{3+2 x}+e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx+\left (6 e^3\right ) \int \frac {x^2}{\left (-e^{e^x}-e^{3+2 x}+e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx+\int \left (\frac {2}{\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )}+\frac {1}{x \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )}\right ) \, dx+\int \left (\frac {1}{3-x \log (x)-x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )}-\frac {3}{x \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )}-\frac {2 x}{-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )}\right ) \, dx \\ & = 2 \int \frac {1}{\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )} \, dx-2 \int \frac {x}{-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )} \, dx-3 \int \frac {1}{x \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx-3 \int \frac {e^{e^x+x}}{\left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx+6 \int \frac {e^{e^x}}{\left (e^{e^x}+e^{3+2 x}-e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx-\left (6 e^3\right ) \int \frac {x}{\left (-e^{e^x}-e^{3+2 x}+e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx+\left (6 e^3\right ) \int \frac {x^2}{\left (-e^{e^x}-e^{3+2 x}+e^3 x^2\right ) \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right ) \left (-3+x \log (x)+x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx+\int \frac {1}{x \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )} \, dx+\int \frac {1}{3-x \log (x)-x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )} \, dx \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(74\) vs. \(2(35)=70\).

Time = 0.24 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.11 \[ \int \frac {e^{2 x} (-3-6 x)+9 x^2+e^{-3+e^x} \left (-3-3 e^x x-3 \log (x)\right )+\left (-3 e^{2 x}+3 x^2\right ) \log (x)+\left (-3 e^{-3+e^x}-3 e^{2 x}+3 x^2\right ) \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )}{\left (-3 e^{2 x} x+3 x^3\right ) \log (x)+\left (e^{2 x} x^2-x^4\right ) \log ^2(x)+e^{-3+e^x} \left (-3 x \log (x)+x^2 \log ^2(x)\right )+\left (-3 e^{2 x} x+3 x^3+\left (2 e^{2 x} x^2-2 x^4\right ) \log (x)+e^{-3+e^x} \left (-3 x+2 x^2 \log (x)\right )\right ) \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )+\left (e^{-3+e^x} x^2+e^{2 x} x^2-x^4\right ) \log ^2\left (-e^{-3+e^x}-e^{2 x}+x^2\right )} \, dx=-3 \left (-\frac {\log (x)}{3}-\frac {1}{3} \log \left (\log (x)+\log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )+\frac {1}{3} \log \left (3-x \log (x)-x \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )\right )\right ) \]

[In]

Integrate[(E^(2*x)*(-3 - 6*x) + 9*x^2 + E^(-3 + E^x)*(-3 - 3*E^x*x - 3*Log[x]) + (-3*E^(2*x) + 3*x^2)*Log[x] +
(-3*E^(-3 + E^x) - 3*E^(2*x) + 3*x^2)*Log[-E^(-3 + E^x) - E^(2*x) + x^2])/((-3*E^(2*x)*x + 3*x^3)*Log[x] + (E
^(2*x)*x^2 - x^4)*Log[x]^2 + E^(-3 + E^x)*(-3*x*Log[x] + x^2*Log[x]^2) + (-3*E^(2*x)*x + 3*x^3 + (2*E^(2*x)*x^
2 - 2*x^4)*Log[x] + E^(-3 + E^x)*(-3*x + 2*x^2*Log[x]))*Log[-E^(-3 + E^x) - E^(2*x) + x^2] + (E^(-3 + E^x)*x^2
+ E^(2*x)*x^2 - x^4)*Log[-E^(-3 + E^x) - E^(2*x) + x^2]^2),x]

[Out]

-3*(-1/3*Log[x] - Log[Log[x] + Log[-E^(-3 + E^x) - E^(2*x) + x^2]]/3 + Log[3 - x*Log[x] - x*Log[-E^(-3 + E^x)
- E^(2*x) + x^2]]/3)

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.60

\[\ln \left (\ln \left (-{\mathrm e}^{{\mathrm e}^{x}-3}-{\mathrm e}^{2 x}+x^{2}\right )+\ln \left (x \right )\right )-\ln \left (\ln \left (-{\mathrm e}^{{\mathrm e}^{x}-3}-{\mathrm e}^{2 x}+x^{2}\right )+\frac {x \ln \left (x \right )-3}{x}\right )\]

[In]

int(((-3*exp(exp(x)-3)-3*exp(x)^2+3*x^2)*ln(-exp(exp(x)-3)-exp(x)^2+x^2)+(-3*ln(x)-3*exp(x)*x-3)*exp(exp(x)-3)
+(-3*exp(x)^2+3*x^2)*ln(x)+(-6*x-3)*exp(x)^2+9*x^2)/((x^2*exp(exp(x)-3)+exp(x)^2*x^2-x^4)*ln(-exp(exp(x)-3)-ex
p(x)^2+x^2)^2+((2*x^2*ln(x)-3*x)*exp(exp(x)-3)+(2*exp(x)^2*x^2-2*x^4)*ln(x)-3*x*exp(x)^2+3*x^3)*ln(-exp(exp(x)
-3)-exp(x)^2+x^2)+(x^2*ln(x)^2-3*x*ln(x))*exp(exp(x)-3)+(exp(x)^2*x^2-x^4)*ln(x)^2+(-3*x*exp(x)^2+3*x^3)*ln(x)
),x)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.60 \[ \int \frac {e^{2 x} (-3-6 x)+9 x^2+e^{-3+e^x} \left (-3-3 e^x x-3 \log (x)\right )+\left (-3 e^{2 x}+3 x^2\right ) \log (x)+\left (-3 e^{-3+e^x}-3 e^{2 x}+3 x^2\right ) \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )}{\left (-3 e^{2 x} x+3 x^3\right ) \log (x)+\left (e^{2 x} x^2-x^4\right ) \log ^2(x)+e^{-3+e^x} \left (-3 x \log (x)+x^2 \log ^2(x)\right )+\left (-3 e^{2 x} x+3 x^3+\left (2 e^{2 x} x^2-2 x^4\right ) \log (x)+e^{-3+e^x} \left (-3 x+2 x^2 \log (x)\right )\right ) \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )+\left (e^{-3+e^x} x^2+e^{2 x} x^2-x^4\right ) \log ^2\left (-e^{-3+e^x}-e^{2 x}+x^2\right )} \, dx=-\log \left (\frac {x \log \left (x^{2} - e^{\left (2 \, x\right )} - e^{\left (e^{x} - 3\right )}\right ) + x \log \left (x\right ) - 3}{x}\right ) + \log \left (\log \left (x^{2} - e^{\left (2 \, x\right )} - e^{\left (e^{x} - 3\right )}\right ) + \log \left (x\right )\right ) \]

[In]

integrate(((-3*exp(exp(x)-3)-3*exp(x)^2+3*x^2)*log(-exp(exp(x)-3)-exp(x)^2+x^2)+(-3*log(x)-3*exp(x)*x-3)*exp(e
xp(x)-3)+(-3*exp(x)^2+3*x^2)*log(x)+(-6*x-3)*exp(x)^2+9*x^2)/((x^2*exp(exp(x)-3)+exp(x)^2*x^2-x^4)*log(-exp(ex
p(x)-3)-exp(x)^2+x^2)^2+((2*x^2*log(x)-3*x)*exp(exp(x)-3)+(2*exp(x)^2*x^2-2*x^4)*log(x)-3*x*exp(x)^2+3*x^3)*lo
g(-exp(exp(x)-3)-exp(x)^2+x^2)+(x^2*log(x)^2-3*x*log(x))*exp(exp(x)-3)+(exp(x)^2*x^2-x^4)*log(x)^2+(-3*x*exp(x
)^2+3*x^3)*log(x)),x, algorithm="fricas")

[Out]

-log((x*log(x^2 - e^(2*x) - e^(e^x - 3)) + x*log(x) - 3)/x) + log(log(x^2 - e^(2*x) - e^(e^x - 3)) + log(x))

Sympy [F(-2)]

Exception generated. \[ \int \frac {e^{2 x} (-3-6 x)+9 x^2+e^{-3+e^x} \left (-3-3 e^x x-3 \log (x)\right )+\left (-3 e^{2 x}+3 x^2\right ) \log (x)+\left (-3 e^{-3+e^x}-3 e^{2 x}+3 x^2\right ) \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )}{\left (-3 e^{2 x} x+3 x^3\right ) \log (x)+\left (e^{2 x} x^2-x^4\right ) \log ^2(x)+e^{-3+e^x} \left (-3 x \log (x)+x^2 \log ^2(x)\right )+\left (-3 e^{2 x} x+3 x^3+\left (2 e^{2 x} x^2-2 x^4\right ) \log (x)+e^{-3+e^x} \left (-3 x+2 x^2 \log (x)\right )\right ) \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )+\left (e^{-3+e^x} x^2+e^{2 x} x^2-x^4\right ) \log ^2\left (-e^{-3+e^x}-e^{2 x}+x^2\right )} \, dx=\text {Exception raised: PolynomialError} \]

[In]

integrate(((-3*exp(exp(x)-3)-3*exp(x)**2+3*x**2)*ln(-exp(exp(x)-3)-exp(x)**2+x**2)+(-3*ln(x)-3*exp(x)*x-3)*exp
(exp(x)-3)+(-3*exp(x)**2+3*x**2)*ln(x)+(-6*x-3)*exp(x)**2+9*x**2)/((x**2*exp(exp(x)-3)+exp(x)**2*x**2-x**4)*ln
(-exp(exp(x)-3)-exp(x)**2+x**2)**2+((2*x**2*ln(x)-3*x)*exp(exp(x)-3)+(2*exp(x)**2*x**2-2*x**4)*ln(x)-3*x*exp(x
)**2+3*x**3)*ln(-exp(exp(x)-3)-exp(x)**2+x**2)+(x**2*ln(x)**2-3*x*ln(x))*exp(exp(x)-3)+(exp(x)**2*x**2-x**4)*l
n(x)**2+(-3*x*exp(x)**2+3*x**3)*ln(x)),x)

[Out]

Exception raised: PolynomialError >> 1/(-_t1**2*x**2 - _t2*x**2 + x**4) contains an element of the set of gene
rators.

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (32) = 64\).

Time = 0.56 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.89 \[ \int \frac {e^{2 x} (-3-6 x)+9 x^2+e^{-3+e^x} \left (-3-3 e^x x-3 \log (x)\right )+\left (-3 e^{2 x}+3 x^2\right ) \log (x)+\left (-3 e^{-3+e^x}-3 e^{2 x}+3 x^2\right ) \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )}{\left (-3 e^{2 x} x+3 x^3\right ) \log (x)+\left (e^{2 x} x^2-x^4\right ) \log ^2(x)+e^{-3+e^x} \left (-3 x \log (x)+x^2 \log ^2(x)\right )+\left (-3 e^{2 x} x+3 x^3+\left (2 e^{2 x} x^2-2 x^4\right ) \log (x)+e^{-3+e^x} \left (-3 x+2 x^2 \log (x)\right )\right ) \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )+\left (e^{-3+e^x} x^2+e^{2 x} x^2-x^4\right ) \log ^2\left (-e^{-3+e^x}-e^{2 x}+x^2\right )} \, dx=-\log \left (\frac {x \log \left (x^{2} e^{3} - e^{\left (2 \, x + 3\right )} - e^{\left (e^{x}\right )}\right ) + x \log \left (x\right ) - 3 \, x - 3}{x}\right ) + \log \left (\log \left (x^{2} e^{3} - e^{\left (2 \, x + 3\right )} - e^{\left (e^{x}\right )}\right ) + \log \left (x\right ) - 3\right ) \]

[In]

integrate(((-3*exp(exp(x)-3)-3*exp(x)^2+3*x^2)*log(-exp(exp(x)-3)-exp(x)^2+x^2)+(-3*log(x)-3*exp(x)*x-3)*exp(e
xp(x)-3)+(-3*exp(x)^2+3*x^2)*log(x)+(-6*x-3)*exp(x)^2+9*x^2)/((x^2*exp(exp(x)-3)+exp(x)^2*x^2-x^4)*log(-exp(ex
p(x)-3)-exp(x)^2+x^2)^2+((2*x^2*log(x)-3*x)*exp(exp(x)-3)+(2*exp(x)^2*x^2-2*x^4)*log(x)-3*x*exp(x)^2+3*x^3)*lo
g(-exp(exp(x)-3)-exp(x)^2+x^2)+(x^2*log(x)^2-3*x*log(x))*exp(exp(x)-3)+(exp(x)^2*x^2-x^4)*log(x)^2+(-3*x*exp(x
)^2+3*x^3)*log(x)),x, algorithm="maxima")

[Out]

-log((x*log(x^2*e^3 - e^(2*x + 3) - e^(e^x)) + x*log(x) - 3*x - 3)/x) + log(log(x^2*e^3 - e^(2*x + 3) - e^(e^x
)) + log(x) - 3)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (32) = 64\).

Time = 1.05 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.34 \[ \int \frac {e^{2 x} (-3-6 x)+9 x^2+e^{-3+e^x} \left (-3-3 e^x x-3 \log (x)\right )+\left (-3 e^{2 x}+3 x^2\right ) \log (x)+\left (-3 e^{-3+e^x}-3 e^{2 x}+3 x^2\right ) \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )}{\left (-3 e^{2 x} x+3 x^3\right ) \log (x)+\left (e^{2 x} x^2-x^4\right ) \log ^2(x)+e^{-3+e^x} \left (-3 x \log (x)+x^2 \log ^2(x)\right )+\left (-3 e^{2 x} x+3 x^3+\left (2 e^{2 x} x^2-2 x^4\right ) \log (x)+e^{-3+e^x} \left (-3 x+2 x^2 \log (x)\right )\right ) \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )+\left (e^{-3+e^x} x^2+e^{2 x} x^2-x^4\right ) \log ^2\left (-e^{-3+e^x}-e^{2 x}+x^2\right )} \, dx=-\log \left (x \log \left ({\left (x^{2} e^{\left (x + 3\right )} - e^{\left (3 \, x + 3\right )} - e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )}\right ) + x \log \left (x\right ) - 3 \, x - 3\right ) + \log \left (x\right ) + \log \left (\log \left ({\left (x^{2} e^{\left (x + 3\right )} - e^{\left (3 \, x + 3\right )} - e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )}\right ) + \log \left (x\right ) - 3\right ) \]

[In]

integrate(((-3*exp(exp(x)-3)-3*exp(x)^2+3*x^2)*log(-exp(exp(x)-3)-exp(x)^2+x^2)+(-3*log(x)-3*exp(x)*x-3)*exp(e
xp(x)-3)+(-3*exp(x)^2+3*x^2)*log(x)+(-6*x-3)*exp(x)^2+9*x^2)/((x^2*exp(exp(x)-3)+exp(x)^2*x^2-x^4)*log(-exp(ex
p(x)-3)-exp(x)^2+x^2)^2+((2*x^2*log(x)-3*x)*exp(exp(x)-3)+(2*exp(x)^2*x^2-2*x^4)*log(x)-3*x*exp(x)^2+3*x^3)*lo
g(-exp(exp(x)-3)-exp(x)^2+x^2)+(x^2*log(x)^2-3*x*log(x))*exp(exp(x)-3)+(exp(x)^2*x^2-x^4)*log(x)^2+(-3*x*exp(x
)^2+3*x^3)*log(x)),x, algorithm="giac")

[Out]

-log(x*log((x^2*e^(x + 3) - e^(3*x + 3) - e^(x + e^x))*e^(-x)) + x*log(x) - 3*x - 3) + log(x) + log(log((x^2*e
^(x + 3) - e^(3*x + 3) - e^(x + e^x))*e^(-x)) + log(x) - 3)

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{2 x} (-3-6 x)+9 x^2+e^{-3+e^x} \left (-3-3 e^x x-3 \log (x)\right )+\left (-3 e^{2 x}+3 x^2\right ) \log (x)+\left (-3 e^{-3+e^x}-3 e^{2 x}+3 x^2\right ) \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )}{\left (-3 e^{2 x} x+3 x^3\right ) \log (x)+\left (e^{2 x} x^2-x^4\right ) \log ^2(x)+e^{-3+e^x} \left (-3 x \log (x)+x^2 \log ^2(x)\right )+\left (-3 e^{2 x} x+3 x^3+\left (2 e^{2 x} x^2-2 x^4\right ) \log (x)+e^{-3+e^x} \left (-3 x+2 x^2 \log (x)\right )\right ) \log \left (-e^{-3+e^x}-e^{2 x}+x^2\right )+\left (e^{-3+e^x} x^2+e^{2 x} x^2-x^4\right ) \log ^2\left (-e^{-3+e^x}-e^{2 x}+x^2\right )} \, dx=\int -\frac {{\mathrm {e}}^{{\mathrm {e}}^x-3}\,\left (3\,\ln \left (x\right )+3\,x\,{\mathrm {e}}^x+3\right )+\ln \left (x\right )\,\left (3\,{\mathrm {e}}^{2\,x}-3\,x^2\right )+{\mathrm {e}}^{2\,x}\,\left (6\,x+3\right )+\ln \left (x^2-{\mathrm {e}}^{{\mathrm {e}}^x-3}-{\mathrm {e}}^{2\,x}\right )\,\left (3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{{\mathrm {e}}^x-3}-3\,x^2\right )-9\,x^2}{{\mathrm {e}}^{{\mathrm {e}}^x-3}\,\left (x^2\,{\ln \left (x\right )}^2-3\,x\,\ln \left (x\right )\right )+{\ln \left (x\right )}^2\,\left (x^2\,{\mathrm {e}}^{2\,x}-x^4\right )+{\ln \left (x^2-{\mathrm {e}}^{{\mathrm {e}}^x-3}-{\mathrm {e}}^{2\,x}\right )}^2\,\left (x^2\,{\mathrm {e}}^{2\,x}-x^4+x^2\,{\mathrm {e}}^{{\mathrm {e}}^x-3}\right )-\ln \left (x^2-{\mathrm {e}}^{{\mathrm {e}}^x-3}-{\mathrm {e}}^{2\,x}\right )\,\left (3\,x\,{\mathrm {e}}^{2\,x}-\ln \left (x\right )\,\left (2\,x^2\,{\mathrm {e}}^{2\,x}-2\,x^4\right )+{\mathrm {e}}^{{\mathrm {e}}^x-3}\,\left (3\,x-2\,x^2\,\ln \left (x\right )\right )-3\,x^3\right )-\ln \left (x\right )\,\left (3\,x\,{\mathrm {e}}^{2\,x}-3\,x^3\right )} \,d x \]

[In]

int(-(exp(exp(x) - 3)*(3*log(x) + 3*x*exp(x) + 3) + log(x)*(3*exp(2*x) - 3*x^2) + exp(2*x)*(6*x + 3) + log(x^2
- exp(exp(x) - 3) - exp(2*x))*(3*exp(2*x) + 3*exp(exp(x) - 3) - 3*x^2) - 9*x^2)/(exp(exp(x) - 3)*(x^2*log(x)^
2 - 3*x*log(x)) + log(x)^2*(x^2*exp(2*x) - x^4) + log(x^2 - exp(exp(x) - 3) - exp(2*x))^2*(x^2*exp(2*x) - x^4
+ x^2*exp(exp(x) - 3)) - log(x^2 - exp(exp(x) - 3) - exp(2*x))*(3*x*exp(2*x) - log(x)*(2*x^2*exp(2*x) - 2*x^4)
+ exp(exp(x) - 3)*(3*x - 2*x^2*log(x)) - 3*x^3) - log(x)*(3*x*exp(2*x) - 3*x^3)),x)

[Out]

int(-(exp(exp(x) - 3)*(3*log(x) + 3*x*exp(x) + 3) + log(x)*(3*exp(2*x) - 3*x^2) + exp(2*x)*(6*x + 3) + log(x^2
- exp(exp(x) - 3) - exp(2*x))*(3*exp(2*x) + 3*exp(exp(x) - 3) - 3*x^2) - 9*x^2)/(exp(exp(x) - 3)*(x^2*log(x)^
2 - 3*x*log(x)) + log(x)^2*(x^2*exp(2*x) - x^4) + log(x^2 - exp(exp(x) - 3) - exp(2*x))^2*(x^2*exp(2*x) - x^4
+ x^2*exp(exp(x) - 3)) - log(x^2 - exp(exp(x) - 3) - exp(2*x))*(3*x*exp(2*x) - log(x)*(2*x^2*exp(2*x) - 2*x^4)
+ exp(exp(x) - 3)*(3*x - 2*x^2*log(x)) - 3*x^3) - log(x)*(3*x*exp(2*x) - 3*x^3)), x)