\(\int \frac {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} (-4 x^3+e^{-5 x^2+x^2 \log (x)} (-9 x^3+9 x^4+(2 x^3-2 x^4) \log (x)))+(-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} (-x+2 x^2)) \log (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x})}{6 x^2-12 x^3+6 x^4+e^{e^{-5 x^2+x^2 \log (x)}} (x^3-2 x^4+x^5)} \, dx\) [7232]

Optimal result

Integrand size = 178, antiderivative size = 33 \[ \int \frac {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} \left (-4 x^3+e^{-5 x^2+x^2 \log (x)} \left (-9 x^3+9 x^4+\left (2 x^3-2 x^4\right ) \log (x)\right )\right )+\left (-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} \left (-x+2 x^2\right )\right ) \log \left (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x}\right )}{6 x^2-12 x^3+6 x^4+e^{e^{-5 x^2+x^2 \log (x)}} \left (x^3-2 x^4+x^5\right )} \, dx=\frac {-4 x+\log \left (e^{e^{x^2 (-5+\log (x))}}+\frac {6}{x}\right )}{x-x^2} \]

[Out]

(ln(exp(exp(x^2*(ln(x)-5)))+6/x)-4*x)/(-x^2+x)

Rubi [F]

\[ \int \frac {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} \left (-4 x^3+e^{-5 x^2+x^2 \log (x)} \left (-9 x^3+9 x^4+\left (2 x^3-2 x^4\right ) \log (x)\right )\right )+\left (-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} \left (-x+2 x^2\right )\right ) \log \left (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x}\right )}{6 x^2-12 x^3+6 x^4+e^{e^{-5 x^2+x^2 \log (x)}} \left (x^3-2 x^4+x^5\right )} \, dx=\int \frac {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} \left (-4 x^3+e^{-5 x^2+x^2 \log (x)} \left (-9 x^3+9 x^4+\left (2 x^3-2 x^4\right ) \log (x)\right )\right )+\left (-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} \left (-x+2 x^2\right )\right ) \log \left (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x}\right )}{6 x^2-12 x^3+6 x^4+e^{e^{-5 x^2+x^2 \log (x)}} \left (x^3-2 x^4+x^5\right )} \, dx \]

[In]

Int[(-6 + 6*x - 24*x^2 + E^E^(-5*x^2 + x^2*Log[x])*(-4*x^3 + E^(-5*x^2 + x^2*Log[x])*(-9*x^3 + 9*x^4 + (2*x^3
- 2*x^4)*Log[x])) + (-6 + 12*x + E^E^(-5*x^2 + x^2*Log[x])*(-x + 2*x^2))*Log[(6 + E^E^(-5*x^2 + x^2*Log[x])*x)
/x])/(6*x^2 - 12*x^3 + 6*x^4 + E^E^(-5*x^2 + x^2*Log[x])*(x^3 - 2*x^4 + x^5)),x]

[Out]

Log[E^(x^x^2/E^(5*x^2)) + 6/x]/((1 - x)*x) - 24*Defer[Int][1/((-1 + x)^2*(6 + E^(x^x^2/E^(5*x^2))*x)), x] - 4*
Defer[Int][E^(x^x^2/E^(5*x^2))/((-1 + x)^2*(6 + E^(x^x^2/E^(5*x^2))*x)), x] - 4*Defer[Int][E^(x^x^2/E^(5*x^2))
/((-1 + x)*(6 + E^(x^x^2/E^(5*x^2))*x)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} \left (-4 x^3+e^{-5 x^2+x^2 \log (x)} \left (-9 x^3+9 x^4+\left (2 x^3-2 x^4\right ) \log (x)\right )\right )+\left (-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} \left (-x+2 x^2\right )\right ) \log \left (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x}\right )}{(1-x)^2 x^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx \\ & = \int \left (-\frac {24}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}-\frac {6}{(-1+x)^2 x^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}+\frac {6}{(-1+x)^2 x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}-\frac {4 e^{e^{-5 x^2} x^{x^2}} x}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}+\frac {(-1+2 x) \log \left (e^{e^{-5 x^2} x^{x^2}}+\frac {6}{x}\right )}{(-1+x)^2 x^2}-\frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2} (-9+2 \log (x))}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}\right ) \, dx \\ & = -\left (4 \int \frac {e^{e^{-5 x^2} x^{x^2}} x}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx\right )-6 \int \frac {1}{(-1+x)^2 x^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx+6 \int \frac {1}{(-1+x)^2 x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-24 \int \frac {1}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx+\int \frac {(-1+2 x) \log \left (e^{e^{-5 x^2} x^{x^2}}+\frac {6}{x}\right )}{(-1+x)^2 x^2} \, dx-\int \frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2} (-9+2 \log (x))}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx \\ & = \frac {\log \left (e^{e^{-5 x^2} x^{x^2}}+\frac {6}{x}\right )}{(1-x) x}-4 \int \left (\frac {e^{e^{-5 x^2} x^{x^2}}}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}+\frac {e^{e^{-5 x^2} x^{x^2}}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}\right ) \, dx+6 \int \left (\frac {1}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}-\frac {1}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}+\frac {1}{x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}\right ) \, dx-6 \int \left (\frac {1}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}-\frac {2}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}+\frac {1}{x^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}+\frac {2}{x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}\right ) \, dx-24 \int \frac {1}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-\int \frac {e^{-5 x^2} \left (-6 e^{5 x^2}-9 e^{e^{-5 x^2} x^{x^2}} x^{3+x^2}+2 e^{e^{-5 x^2} x^{x^2}} x^{3+x^2} \log (x)\right )}{(1-x) x^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-\int \left (-\frac {9 e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}+\frac {2 e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2} \log (x)}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}\right ) \, dx \\ & = \frac {\log \left (e^{e^{-5 x^2} x^{x^2}}+\frac {6}{x}\right )}{(1-x) x}-2 \int \frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2} \log (x)}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-4 \int \frac {e^{e^{-5 x^2} x^{x^2}}}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-4 \int \frac {e^{e^{-5 x^2} x^{x^2}}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-6 \int \frac {1}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-6 \int \frac {1}{x^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx+6 \int \frac {1}{x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx+9 \int \frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx+12 \int \frac {1}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-12 \int \frac {1}{x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-24 \int \frac {1}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-\int \left (\frac {6}{(-1+x) x^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}-\frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2} (-9+2 \log (x))}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}\right ) \, dx \\ & = \frac {\log \left (e^{e^{-5 x^2} x^{x^2}}+\frac {6}{x}\right )}{(1-x) x}+2 \int \frac {\int \frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx}{x} \, dx-4 \int \frac {e^{e^{-5 x^2} x^{x^2}}}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-4 \int \frac {e^{e^{-5 x^2} x^{x^2}}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-6 \int \frac {1}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-6 \int \frac {1}{x^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-6 \int \frac {1}{(-1+x) x^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx+6 \int \frac {1}{x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx+9 \int \frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx+12 \int \frac {1}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-12 \int \frac {1}{x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-24 \int \frac {1}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-(2 \log (x)) \int \frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx+\int \frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2} (-9+2 \log (x))}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx \\ & = \frac {\log \left (e^{e^{-5 x^2} x^{x^2}}+\frac {6}{x}\right )}{(1-x) x}+2 \int \frac {\int \frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx}{x} \, dx-4 \int \frac {e^{e^{-5 x^2} x^{x^2}}}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-4 \int \frac {e^{e^{-5 x^2} x^{x^2}}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-6 \int \frac {1}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-6 \int \frac {1}{x^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx+6 \int \frac {1}{x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-6 \int \left (\frac {1}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}-\frac {1}{x^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}-\frac {1}{x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}\right ) \, dx+9 \int \frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx+12 \int \frac {1}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-12 \int \frac {1}{x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-24 \int \frac {1}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-(2 \log (x)) \int \frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx+\int \left (-\frac {9 e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}+\frac {2 e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2} \log (x)}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}\right ) \, dx \\ & = \frac {\log \left (e^{e^{-5 x^2} x^{x^2}}+\frac {6}{x}\right )}{(1-x) x}+2 \int \frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2} \log (x)}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx+2 \int \frac {\int \frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx}{x} \, dx-4 \int \frac {e^{e^{-5 x^2} x^{x^2}}}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-4 \int \frac {e^{e^{-5 x^2} x^{x^2}}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-2 \left (6 \int \frac {1}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx\right )+2 \left (6 \int \frac {1}{x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx\right )+12 \int \frac {1}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-12 \int \frac {1}{x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-24 \int \frac {1}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-(2 \log (x)) \int \frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx \\ & = \frac {\log \left (e^{e^{-5 x^2} x^{x^2}}+\frac {6}{x}\right )}{(1-x) x}-4 \int \frac {e^{e^{-5 x^2} x^{x^2}}}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-4 \int \frac {e^{e^{-5 x^2} x^{x^2}}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-2 \left (6 \int \frac {1}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx\right )+2 \left (6 \int \frac {1}{x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx\right )+12 \int \frac {1}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-12 \int \frac {1}{x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-24 \int \frac {1}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} \left (-4 x^3+e^{-5 x^2+x^2 \log (x)} \left (-9 x^3+9 x^4+\left (2 x^3-2 x^4\right ) \log (x)\right )\right )+\left (-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} \left (-x+2 x^2\right )\right ) \log \left (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x}\right )}{6 x^2-12 x^3+6 x^4+e^{e^{-5 x^2+x^2 \log (x)}} \left (x^3-2 x^4+x^5\right )} \, dx=\frac {4 x-\log \left (e^{e^{-5 x^2} x^{x^2}}+\frac {6}{x}\right )}{(-1+x) x} \]

[In]

Integrate[(-6 + 6*x - 24*x^2 + E^E^(-5*x^2 + x^2*Log[x])*(-4*x^3 + E^(-5*x^2 + x^2*Log[x])*(-9*x^3 + 9*x^4 + (
2*x^3 - 2*x^4)*Log[x])) + (-6 + 12*x + E^E^(-5*x^2 + x^2*Log[x])*(-x + 2*x^2))*Log[(6 + E^E^(-5*x^2 + x^2*Log[
x])*x)/x])/(6*x^2 - 12*x^3 + 6*x^4 + E^E^(-5*x^2 + x^2*Log[x])*(x^3 - 2*x^4 + x^5)),x]

[Out]

(4*x - Log[E^(x^x^2/E^(5*x^2)) + 6/x])/((-1 + x)*x)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.07 (sec) , antiderivative size = 222, normalized size of antiderivative = 6.73

\[-\frac {\ln \left ({\mathrm e}^{x^{x^{2}} {\mathrm e}^{-5 x^{2}}} x +6\right )}{x \left (-1+x \right )}+\frac {-i \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x^{x^{2}} {\mathrm e}^{-5 x^{2}}} x +6\right )\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{x^{x^{2}} {\mathrm e}^{-5 x^{2}}} x +6\right )}{x}\right )}^{2}+i \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x^{x^{2}} {\mathrm e}^{-5 x^{2}}} x +6\right )\right ) \operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{x^{x^{2}} {\mathrm e}^{-5 x^{2}}} x +6\right )}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right )+i \pi {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{x^{x^{2}} {\mathrm e}^{-5 x^{2}}} x +6\right )}{x}\right )}^{3}-i \pi {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{x^{x^{2}} {\mathrm e}^{-5 x^{2}}} x +6\right )}{x}\right )}^{2} \operatorname {csgn}\left (\frac {i}{x}\right )+8 x +2 \ln \left (x \right )}{2 x \left (-1+x \right )}\]

[In]

int((((2*x^2-x)*exp(exp(x^2*ln(x)-5*x^2))+12*x-6)*ln((x*exp(exp(x^2*ln(x)-5*x^2))+6)/x)+(((-2*x^4+2*x^3)*ln(x)
+9*x^4-9*x^3)*exp(x^2*ln(x)-5*x^2)-4*x^3)*exp(exp(x^2*ln(x)-5*x^2))-24*x^2+6*x-6)/((x^5-2*x^4+x^3)*exp(exp(x^2
*ln(x)-5*x^2))+6*x^4-12*x^3+6*x^2),x)

[Out]

-1/x/(-1+x)*ln(exp(x^(x^2)*exp(-5*x^2))*x+6)+1/2*(-I*Pi*csgn(I*(exp(x^(x^2)*exp(-5*x^2))*x+6))*csgn(I/x*(exp(x
^(x^2)*exp(-5*x^2))*x+6))^2+I*Pi*csgn(I*(exp(x^(x^2)*exp(-5*x^2))*x+6))*csgn(I/x*(exp(x^(x^2)*exp(-5*x^2))*x+6
))*csgn(I/x)+I*Pi*csgn(I/x*(exp(x^(x^2)*exp(-5*x^2))*x+6))^3-I*Pi*csgn(I/x*(exp(x^(x^2)*exp(-5*x^2))*x+6))^2*c
sgn(I/x)+8*x+2*ln(x))/x/(-1+x)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} \left (-4 x^3+e^{-5 x^2+x^2 \log (x)} \left (-9 x^3+9 x^4+\left (2 x^3-2 x^4\right ) \log (x)\right )\right )+\left (-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} \left (-x+2 x^2\right )\right ) \log \left (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x}\right )}{6 x^2-12 x^3+6 x^4+e^{e^{-5 x^2+x^2 \log (x)}} \left (x^3-2 x^4+x^5\right )} \, dx=\frac {4 \, x - \log \left (\frac {x e^{\left (e^{\left (x^{2} \log \left (x\right ) - 5 \, x^{2}\right )}\right )} + 6}{x}\right )}{x^{2} - x} \]

[In]

integrate((((2*x^2-x)*exp(exp(x^2*log(x)-5*x^2))+12*x-6)*log((x*exp(exp(x^2*log(x)-5*x^2))+6)/x)+(((-2*x^4+2*x
^3)*log(x)+9*x^4-9*x^3)*exp(x^2*log(x)-5*x^2)-4*x^3)*exp(exp(x^2*log(x)-5*x^2))-24*x^2+6*x-6)/((x^5-2*x^4+x^3)
*exp(exp(x^2*log(x)-5*x^2))+6*x^4-12*x^3+6*x^2),x, algorithm="fricas")

[Out]

(4*x - log((x*e^(e^(x^2*log(x) - 5*x^2)) + 6)/x))/(x^2 - x)

Sympy [A] (verification not implemented)

Time = 12.50 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} \left (-4 x^3+e^{-5 x^2+x^2 \log (x)} \left (-9 x^3+9 x^4+\left (2 x^3-2 x^4\right ) \log (x)\right )\right )+\left (-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} \left (-x+2 x^2\right )\right ) \log \left (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x}\right )}{6 x^2-12 x^3+6 x^4+e^{e^{-5 x^2+x^2 \log (x)}} \left (x^3-2 x^4+x^5\right )} \, dx=- \frac {\log {\left (\frac {x e^{e^{x^{2} \log {\left (x \right )} - 5 x^{2}}} + 6}{x} \right )}}{x^{2} - x} + \frac {4}{x - 1} \]

[In]

integrate((((2*x**2-x)*exp(exp(x**2*ln(x)-5*x**2))+12*x-6)*ln((x*exp(exp(x**2*ln(x)-5*x**2))+6)/x)+(((-2*x**4+
2*x**3)*ln(x)+9*x**4-9*x**3)*exp(x**2*ln(x)-5*x**2)-4*x**3)*exp(exp(x**2*ln(x)-5*x**2))-24*x**2+6*x-6)/((x**5-
2*x**4+x**3)*exp(exp(x**2*ln(x)-5*x**2))+6*x**4-12*x**3+6*x**2),x)

[Out]

-log((x*exp(exp(x**2*log(x) - 5*x**2)) + 6)/x)/(x**2 - x) + 4/(x - 1)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} \left (-4 x^3+e^{-5 x^2+x^2 \log (x)} \left (-9 x^3+9 x^4+\left (2 x^3-2 x^4\right ) \log (x)\right )\right )+\left (-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} \left (-x+2 x^2\right )\right ) \log \left (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x}\right )}{6 x^2-12 x^3+6 x^4+e^{e^{-5 x^2+x^2 \log (x)}} \left (x^3-2 x^4+x^5\right )} \, dx=\frac {4 \, x - \log \left (x e^{\left (e^{\left (x^{2} \log \left (x\right ) - 5 \, x^{2}\right )}\right )} + 6\right ) + \log \left (x\right )}{x^{2} - x} \]

[In]

integrate((((2*x^2-x)*exp(exp(x^2*log(x)-5*x^2))+12*x-6)*log((x*exp(exp(x^2*log(x)-5*x^2))+6)/x)+(((-2*x^4+2*x
^3)*log(x)+9*x^4-9*x^3)*exp(x^2*log(x)-5*x^2)-4*x^3)*exp(exp(x^2*log(x)-5*x^2))-24*x^2+6*x-6)/((x^5-2*x^4+x^3)
*exp(exp(x^2*log(x)-5*x^2))+6*x^4-12*x^3+6*x^2),x, algorithm="maxima")

[Out]

(4*x - log(x*e^(e^(x^2*log(x) - 5*x^2)) + 6) + log(x))/(x^2 - x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (33) = 66\).

Time = 1.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.36 \[ \int \frac {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} \left (-4 x^3+e^{-5 x^2+x^2 \log (x)} \left (-9 x^3+9 x^4+\left (2 x^3-2 x^4\right ) \log (x)\right )\right )+\left (-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} \left (-x+2 x^2\right )\right ) \log \left (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x}\right )}{6 x^2-12 x^3+6 x^4+e^{e^{-5 x^2+x^2 \log (x)}} \left (x^3-2 x^4+x^5\right )} \, dx=\frac {4 \, x - \log \left ({\left (x e^{\left (x^{2} \log \left (x\right ) - 5 \, x^{2} + e^{\left (x^{2} \log \left (x\right ) - 5 \, x^{2}\right )}\right )} + 6 \, e^{\left (x^{2} \log \left (x\right ) - 5 \, x^{2}\right )}\right )} e^{\left (-x^{2} \log \left (x\right ) + 5 \, x^{2}\right )}\right ) + \log \left (x\right )}{x^{2} - x} \]

[In]

integrate((((2*x^2-x)*exp(exp(x^2*log(x)-5*x^2))+12*x-6)*log((x*exp(exp(x^2*log(x)-5*x^2))+6)/x)+(((-2*x^4+2*x
^3)*log(x)+9*x^4-9*x^3)*exp(x^2*log(x)-5*x^2)-4*x^3)*exp(exp(x^2*log(x)-5*x^2))-24*x^2+6*x-6)/((x^5-2*x^4+x^3)
*exp(exp(x^2*log(x)-5*x^2))+6*x^4-12*x^3+6*x^2),x, algorithm="giac")

[Out]

(4*x - log((x*e^(x^2*log(x) - 5*x^2 + e^(x^2*log(x) - 5*x^2)) + 6*e^(x^2*log(x) - 5*x^2))*e^(-x^2*log(x) + 5*x
^2)) + log(x))/(x^2 - x)

Mupad [B] (verification not implemented)

Time = 13.81 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} \left (-4 x^3+e^{-5 x^2+x^2 \log (x)} \left (-9 x^3+9 x^4+\left (2 x^3-2 x^4\right ) \log (x)\right )\right )+\left (-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} \left (-x+2 x^2\right )\right ) \log \left (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x}\right )}{6 x^2-12 x^3+6 x^4+e^{e^{-5 x^2+x^2 \log (x)}} \left (x^3-2 x^4+x^5\right )} \, dx=\frac {4\,x-\ln \left (\frac {x\,{\mathrm {e}}^{x^{x^2}\,{\mathrm {e}}^{-5\,x^2}}+6}{x}\right )}{x\,\left (x-1\right )} \]

[In]

int(-(log((x*exp(exp(x^2*log(x) - 5*x^2)) + 6)/x)*(exp(exp(x^2*log(x) - 5*x^2))*(x - 2*x^2) - 12*x + 6) - exp(
exp(x^2*log(x) - 5*x^2))*(exp(x^2*log(x) - 5*x^2)*(log(x)*(2*x^3 - 2*x^4) - 9*x^3 + 9*x^4) - 4*x^3) - 6*x + 24
*x^2 + 6)/(exp(exp(x^2*log(x) - 5*x^2))*(x^3 - 2*x^4 + x^5) + 6*x^2 - 12*x^3 + 6*x^4),x)

[Out]

(4*x - log((x*exp(x^(x^2)*exp(-5*x^2)) + 6)/x))/(x*(x - 1))