\(\int e^{16-4 e^3-4 x} ((2-2 x-2 x \log (x)) \log (x^{1-x})+(1-4 x) \log ^2(x^{1-x})) \, dx\) [1306]

Optimal result

Integrand size = 49, antiderivative size = 24 \[ \int e^{16-4 e^3-4 x} \left ((2-2 x-2 x \log (x)) \log \left (x^{1-x}\right )+(1-4 x) \log ^2\left (x^{1-x}\right )\right ) \, dx=e^{16-4 e^3-4 x} x \log ^2\left (x^{1-x}\right ) \]

[Out]

x*ln(x/exp(x*ln(x)))^2/exp(exp(3)+x-4)^4

Rubi [F]

\[ \int e^{16-4 e^3-4 x} \left ((2-2 x-2 x \log (x)) \log \left (x^{1-x}\right )+(1-4 x) \log ^2\left (x^{1-x}\right )\right ) \, dx=\int e^{16-4 e^3-4 x} \left ((2-2 x-2 x \log (x)) \log \left (x^{1-x}\right )+(1-4 x) \log ^2\left (x^{1-x}\right )\right ) \, dx \]

[In]

Int[E^(16 - 4*E^3 - 4*x)*((2 - 2*x - 2*x*Log[x])*Log[x^(1 - x)] + (1 - 4*x)*Log[x^(1 - x)]^2),x]

[Out]

(3*E^(4*(4 - E^3) - 4*x))/32 - (E^(4*(4 - E^3) - 4*x)*x)/8 + (3*E^(16 - 4*E^3)*ExpIntegralEi[-4*x])/16 - E^(16
- 4*E^3)*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -4*x] + (3*E^(4*(4 - E^3) - 4*x)*Log[x])/32 + (E^(16 - 4*E
^3)*EulerGamma*Log[x])/4 - (E^(4*(4 - E^3) - 4*x)*x*Log[x])/8 - (E^(4*(4 - E^3) - 4*x)*(1 + 4*x)*Log[x])/32 -
(E^(16 - 4*E^3)*ExpIntegralEi[-4*x]*Log[x])/8 - (E^(16 - 4*E^3)*x*ExpIntegralEi[-4*x]*Log[x])/8 + (E^(16 - 4*E
^3)*(ExpIntegralE[1, 4*x] + ExpIntegralEi[-4*x])*Log[x])/4 + (E^(16 - 4*E^3)*Log[4*x]^2)/8 - (E^(4*(4 - E^3) -
4*x)*Log[x^(1 - x)])/4 + (E^(4*(4 - E^3) - 4*x)*x*Log[x^(1 - x)])/2 - (E^(16 - 4*E^3)*ExpIntegralEi[-4*x]*Log
[x^(1 - x)])/8 + (E^(4*(4 - E^3) - 4*x)*Log[x]*Log[x^(1 - x)])/8 + (E^(4*(4 - E^3) - 4*x)*x*Log[x]*Log[x^(1 -
x)])/2 + Defer[Int][E^(4*(4 - E^3) - 4*x)*Log[x]^2, x]/8 + Defer[Int][E^(4*(4 - E^3) - 4*x)*x*Log[x]^2, x]/2 +
Defer[Int][E^(16 - 4*E^3 - 4*x)*Log[x^(1 - x)]^2, x] - 4*Defer[Int][E^(16 - 4*E^3 - 4*x)*x*Log[x^(1 - x)]^2,
x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-2 e^{16-4 e^3-4 x} (-1+x+x \log (x)) \log \left (x^{1-x}\right )-e^{16-4 e^3-4 x} (-1+4 x) \log ^2\left (x^{1-x}\right )\right ) \, dx \\ & = -\left (2 \int e^{16-4 e^3-4 x} (-1+x+x \log (x)) \log \left (x^{1-x}\right ) \, dx\right )-\int e^{16-4 e^3-4 x} (-1+4 x) \log ^2\left (x^{1-x}\right ) \, dx \\ & = -\left (2 \int \left (-e^{16-4 e^3-4 x} \log \left (x^{1-x}\right )+e^{16-4 e^3-4 x} x \log \left (x^{1-x}\right )+e^{16-4 e^3-4 x} x \log (x) \log \left (x^{1-x}\right )\right ) \, dx\right )-\int \left (-e^{16-4 e^3-4 x} \log ^2\left (x^{1-x}\right )+4 e^{16-4 e^3-4 x} x \log ^2\left (x^{1-x}\right )\right ) \, dx \\ & = 2 \int e^{16-4 e^3-4 x} \log \left (x^{1-x}\right ) \, dx-2 \int e^{16-4 e^3-4 x} x \log \left (x^{1-x}\right ) \, dx-2 \int e^{16-4 e^3-4 x} x \log (x) \log \left (x^{1-x}\right ) \, dx-4 \int e^{16-4 e^3-4 x} x \log ^2\left (x^{1-x}\right ) \, dx+\int e^{16-4 e^3-4 x} \log ^2\left (x^{1-x}\right ) \, dx \\ & = -\frac {3}{8} e^{4 \left (4-e^3\right )-4 x} \log \left (x^{1-x}\right )+\frac {1}{2} e^{4 \left (4-e^3\right )-4 x} x \log \left (x^{1-x}\right )+\frac {1}{8} e^{4 \left (4-e^3\right )-4 x} \log (x) \log \left (x^{1-x}\right )+\frac {1}{2} e^{4 \left (4-e^3\right )-4 x} x \log (x) \log \left (x^{1-x}\right )-2 \int \frac {e^{4 \left (4-e^3\right )-4 x} (-1+x+x \log (x))}{4 x} \, dx+2 \int \frac {e^{4 \left (4-e^3\right )-4 x} (1+4 x) (-1+x+x \log (x))}{16 x} \, dx+2 \int \frac {e^{4 \left (4-e^3\right )-4 x} (1+4 x) \log (x) (-1+x+x \log (x))}{16 x} \, dx+2 \int \frac {e^{4 \left (4-e^3\right )-4 x} (-1-4 x) \log \left (x^{1-x}\right )}{16 x} \, dx-4 \int e^{16-4 e^3-4 x} x \log ^2\left (x^{1-x}\right ) \, dx+\int e^{16-4 e^3-4 x} \log ^2\left (x^{1-x}\right ) \, dx \\ & = -\frac {3}{8} e^{4 \left (4-e^3\right )-4 x} \log \left (x^{1-x}\right )+\frac {1}{2} e^{4 \left (4-e^3\right )-4 x} x \log \left (x^{1-x}\right )+\frac {1}{8} e^{4 \left (4-e^3\right )-4 x} \log (x) \log \left (x^{1-x}\right )+\frac {1}{2} e^{4 \left (4-e^3\right )-4 x} x \log (x) \log \left (x^{1-x}\right )+\frac {1}{8} \int \frac {e^{4 \left (4-e^3\right )-4 x} (1+4 x) (-1+x+x \log (x))}{x} \, dx+\frac {1}{8} \int \frac {e^{4 \left (4-e^3\right )-4 x} (1+4 x) \log (x) (-1+x+x \log (x))}{x} \, dx+\frac {1}{8} \int \frac {e^{4 \left (4-e^3\right )-4 x} (-1-4 x) \log \left (x^{1-x}\right )}{x} \, dx-\frac {1}{2} \int \frac {e^{4 \left (4-e^3\right )-4 x} (-1+x+x \log (x))}{x} \, dx-4 \int e^{16-4 e^3-4 x} x \log ^2\left (x^{1-x}\right ) \, dx+\int e^{16-4 e^3-4 x} \log ^2\left (x^{1-x}\right ) \, dx \\ & = -\frac {1}{4} e^{4 \left (4-e^3\right )-4 x} \log \left (x^{1-x}\right )+\frac {1}{2} e^{4 \left (4-e^3\right )-4 x} x \log \left (x^{1-x}\right )-\frac {1}{8} e^{16-4 e^3} \operatorname {ExpIntegralEi}(-4 x) \log \left (x^{1-x}\right )+\frac {1}{8} e^{4 \left (4-e^3\right )-4 x} \log (x) \log \left (x^{1-x}\right )+\frac {1}{2} e^{4 \left (4-e^3\right )-4 x} x \log (x) \log \left (x^{1-x}\right )-\frac {1}{8} \int \frac {e^{4 \left (4-e^3\right )-4 x} \left (1-e^{4 x} \operatorname {ExpIntegralEi}(-4 x)\right ) (1-x-x \log (x))}{x} \, dx+\frac {1}{8} \int \left (\frac {e^{4 \left (4-e^3\right )-4 x} \left (-1-3 x+4 x^2\right )}{x}+e^{4 \left (4-e^3\right )-4 x} (1+4 x) \log (x)\right ) \, dx+\frac {1}{8} \int \left (\frac {e^{4 \left (4-e^3\right )-4 x} (-1+x) (1+4 x) \log (x)}{x}+e^{4 \left (4-e^3\right )-4 x} (1+4 x) \log ^2(x)\right ) \, dx-\frac {1}{2} \int \left (\frac {e^{4 \left (4-e^3\right )-4 x} (-1+x)}{x}+e^{4 \left (4-e^3\right )-4 x} \log (x)\right ) \, dx-4 \int e^{16-4 e^3-4 x} x \log ^2\left (x^{1-x}\right ) \, dx+\int e^{16-4 e^3-4 x} \log ^2\left (x^{1-x}\right ) \, dx \\ & = -\frac {1}{4} e^{4 \left (4-e^3\right )-4 x} \log \left (x^{1-x}\right )+\frac {1}{2} e^{4 \left (4-e^3\right )-4 x} x \log \left (x^{1-x}\right )-\frac {1}{8} e^{16-4 e^3} \operatorname {ExpIntegralEi}(-4 x) \log \left (x^{1-x}\right )+\frac {1}{8} e^{4 \left (4-e^3\right )-4 x} \log (x) \log \left (x^{1-x}\right )+\frac {1}{2} e^{4 \left (4-e^3\right )-4 x} x \log (x) \log \left (x^{1-x}\right )+\frac {1}{8} \int \frac {e^{4 \left (4-e^3\right )-4 x} \left (-1-3 x+4 x^2\right )}{x} \, dx+\frac {1}{8} \int e^{4 \left (4-e^3\right )-4 x} (1+4 x) \log (x) \, dx+\frac {1}{8} \int \frac {e^{4 \left (4-e^3\right )-4 x} (-1+x) (1+4 x) \log (x)}{x} \, dx+\frac {1}{8} \int e^{4 \left (4-e^3\right )-4 x} (1+4 x) \log ^2(x) \, dx-\frac {1}{8} \int \left (\frac {e^{4 \left (4-e^3\right )-4 x} (1-x-x \log (x))}{x}+\frac {e^{4 \left (4-e^3\right )} \operatorname {ExpIntegralEi}(-4 x) (-1+x+x \log (x))}{x}\right ) \, dx-\frac {1}{2} \int \frac {e^{4 \left (4-e^3\right )-4 x} (-1+x)}{x} \, dx-\frac {1}{2} \int e^{4 \left (4-e^3\right )-4 x} \log (x) \, dx-4 \int e^{16-4 e^3-4 x} x \log ^2\left (x^{1-x}\right ) \, dx+\int e^{16-4 e^3-4 x} \log ^2\left (x^{1-x}\right ) \, dx \\ & = \frac {5}{32} e^{4 \left (4-e^3\right )-4 x} \log (x)-\frac {1}{8} e^{4 \left (4-e^3\right )-4 x} x \log (x)-\frac {1}{32} e^{4 \left (4-e^3\right )-4 x} (1+4 x) \log (x)-\frac {1}{8} e^{16-4 e^3} \operatorname {ExpIntegralEi}(-4 x) \log (x)-\frac {1}{4} e^{4 \left (4-e^3\right )-4 x} \log \left (x^{1-x}\right )+\frac {1}{2} e^{4 \left (4-e^3\right )-4 x} x \log \left (x^{1-x}\right )-\frac {1}{8} e^{16-4 e^3} \operatorname {ExpIntegralEi}(-4 x) \log \left (x^{1-x}\right )+\frac {1}{8} e^{4 \left (4-e^3\right )-4 x} \log (x) \log \left (x^{1-x}\right )+\frac {1}{2} e^{4 \left (4-e^3\right )-4 x} x \log (x) \log \left (x^{1-x}\right )-\frac {1}{8} \int \frac {e^{4 \left (4-e^3\right )-4 x} (-1-2 x)}{2 x} \, dx+\frac {1}{8} \int \left (-3 e^{4 \left (4-e^3\right )-4 x}-\frac {e^{4 \left (4-e^3\right )-4 x}}{x}+4 e^{4 \left (4-e^3\right )-4 x} x\right ) \, dx-\frac {1}{8} \int \frac {e^{4 \left (4-e^3\right )-4 x} \left (1-2 x-2 e^{4 x} \operatorname {ExpIntegralEi}(-4 x)\right )}{2 x} \, dx-\frac {1}{8} \int \frac {e^{4 \left (4-e^3\right )-4 x} (1-x-x \log (x))}{x} \, dx+\frac {1}{8} \int \left (e^{4 \left (4-e^3\right )-4 x} \log ^2(x)+4 e^{4 \left (4-e^3\right )-4 x} x \log ^2(x)\right ) \, dx-\frac {1}{2} \int \left (e^{4 \left (4-e^3\right )-4 x}-\frac {e^{4 \left (4-e^3\right )-4 x}}{x}\right ) \, dx+\frac {1}{2} \int -\frac {e^{4 \left (4-e^3\right )-4 x}}{4 x} \, dx-4 \int e^{16-4 e^3-4 x} x \log ^2\left (x^{1-x}\right ) \, dx-\frac {1}{8} e^{16-4 e^3} \int \frac {\operatorname {ExpIntegralEi}(-4 x) (-1+x+x \log (x))}{x} \, dx+\int e^{16-4 e^3-4 x} \log ^2\left (x^{1-x}\right ) \, dx \\ & = \frac {5}{32} e^{4 \left (4-e^3\right )-4 x} \log (x)-\frac {1}{8} e^{4 \left (4-e^3\right )-4 x} x \log (x)-\frac {1}{32} e^{4 \left (4-e^3\right )-4 x} (1+4 x) \log (x)-\frac {1}{8} e^{16-4 e^3} \operatorname {ExpIntegralEi}(-4 x) \log (x)-\frac {1}{4} e^{4 \left (4-e^3\right )-4 x} \log \left (x^{1-x}\right )+\frac {1}{2} e^{4 \left (4-e^3\right )-4 x} x \log \left (x^{1-x}\right )-\frac {1}{8} e^{16-4 e^3} \operatorname {ExpIntegralEi}(-4 x) \log \left (x^{1-x}\right )+\frac {1}{8} e^{4 \left (4-e^3\right )-4 x} \log (x) \log \left (x^{1-x}\right )+\frac {1}{2} e^{4 \left (4-e^3\right )-4 x} x \log (x) \log \left (x^{1-x}\right )-\frac {1}{16} \int \frac {e^{4 \left (4-e^3\right )-4 x} (-1-2 x)}{x} \, dx-\frac {1}{16} \int \frac {e^{4 \left (4-e^3\right )-4 x} \left (1-2 x-2 e^{4 x} \operatorname {ExpIntegralEi}(-4 x)\right )}{x} \, dx-2 \left (\frac {1}{8} \int \frac {e^{4 \left (4-e^3\right )-4 x}}{x} \, dx\right )+\frac {1}{8} \int e^{4 \left (4-e^3\right )-4 x} \log ^2(x) \, dx-\frac {1}{8} \int \left (\frac {e^{4 \left (4-e^3\right )-4 x} (1-x)}{x}-e^{4 \left (4-e^3\right )-4 x} \log (x)\right ) \, dx-\frac {3}{8} \int e^{4 \left (4-e^3\right )-4 x} \, dx-\frac {1}{2} \int e^{4 \left (4-e^3\right )-4 x} \, dx+\frac {1}{2} \int \frac {e^{4 \left (4-e^3\right )-4 x}}{x} \, dx+\frac {1}{2} \int e^{4 \left (4-e^3\right )-4 x} x \, dx+\frac {1}{2} \int e^{4 \left (4-e^3\right )-4 x} x \log ^2(x) \, dx-4 \int e^{16-4 e^3-4 x} x \log ^2\left (x^{1-x}\right ) \, dx-\frac {1}{8} e^{16-4 e^3} \int \left (\frac {(-1+x) \operatorname {ExpIntegralEi}(-4 x)}{x}+\operatorname {ExpIntegralEi}(-4 x) \log (x)\right ) \, dx+\int e^{16-4 e^3-4 x} \log ^2\left (x^{1-x}\right ) \, dx \\ & = \frac {7}{32} e^{4 \left (4-e^3\right )-4 x}-\frac {1}{8} e^{4 \left (4-e^3\right )-4 x} x+\frac {1}{4} e^{16-4 e^3} \operatorname {ExpIntegralEi}(-4 x)+\frac {5}{32} e^{4 \left (4-e^3\right )-4 x} \log (x)-\frac {1}{8} e^{4 \left (4-e^3\right )-4 x} x \log (x)-\frac {1}{32} e^{4 \left (4-e^3\right )-4 x} (1+4 x) \log (x)-\frac {1}{8} e^{16-4 e^3} \operatorname {ExpIntegralEi}(-4 x) \log (x)-\frac {1}{4} e^{4 \left (4-e^3\right )-4 x} \log \left (x^{1-x}\right )+\frac {1}{2} e^{4 \left (4-e^3\right )-4 x} x \log \left (x^{1-x}\right )-\frac {1}{8} e^{16-4 e^3} \operatorname {ExpIntegralEi}(-4 x) \log \left (x^{1-x}\right )+\frac {1}{8} e^{4 \left (4-e^3\right )-4 x} \log (x) \log \left (x^{1-x}\right )+\frac {1}{2} e^{4 \left (4-e^3\right )-4 x} x \log (x) \log \left (x^{1-x}\right )-\frac {1}{16} \int \left (-2 e^{4 \left (4-e^3\right )-4 x}-\frac {e^{4 \left (4-e^3\right )-4 x}}{x}\right ) \, dx-\frac {1}{16} \int \left (\frac {e^{4 \left (4-e^3\right )-4 x} (1-2 x)}{x}-\frac {2 e^{4 \left (4-e^3\right )} \operatorname {ExpIntegralEi}(-4 x)}{x}\right ) \, dx+\frac {1}{8} \int e^{4 \left (4-e^3\right )-4 x} \, dx-\frac {1}{8} \int \frac {e^{4 \left (4-e^3\right )-4 x} (1-x)}{x} \, dx+\frac {1}{8} \int e^{4 \left (4-e^3\right )-4 x} \log (x) \, dx+\frac {1}{8} \int e^{4 \left (4-e^3\right )-4 x} \log ^2(x) \, dx+\frac {1}{2} \int e^{4 \left (4-e^3\right )-4 x} x \log ^2(x) \, dx-4 \int e^{16-4 e^3-4 x} x \log ^2\left (x^{1-x}\right ) \, dx-\frac {1}{8} e^{16-4 e^3} \int \frac {(-1+x) \operatorname {ExpIntegralEi}(-4 x)}{x} \, dx-\frac {1}{8} e^{16-4 e^3} \int \operatorname {ExpIntegralEi}(-4 x) \log (x) \, dx+\int e^{16-4 e^3-4 x} \log ^2\left (x^{1-x}\right ) \, dx \\ & = \frac {3}{16} e^{4 \left (4-e^3\right )-4 x}-\frac {1}{8} e^{4 \left (4-e^3\right )-4 x} x+\frac {1}{4} e^{16-4 e^3} \operatorname {ExpIntegralEi}(-4 x)+\frac {3}{32} e^{4 \left (4-e^3\right )-4 x} \log (x)-\frac {1}{8} e^{4 \left (4-e^3\right )-4 x} x \log (x)-\frac {1}{32} e^{4 \left (4-e^3\right )-4 x} (1+4 x) \log (x)-\frac {1}{8} e^{16-4 e^3} \operatorname {ExpIntegralEi}(-4 x) \log (x)-\frac {1}{8} e^{16-4 e^3} x \operatorname {ExpIntegralEi}(-4 x) \log (x)-\frac {1}{4} e^{4 \left (4-e^3\right )-4 x} \log \left (x^{1-x}\right )+\frac {1}{2} e^{4 \left (4-e^3\right )-4 x} x \log \left (x^{1-x}\right )-\frac {1}{8} e^{16-4 e^3} \operatorname {ExpIntegralEi}(-4 x) \log \left (x^{1-x}\right )+\frac {1}{8} e^{4 \left (4-e^3\right )-4 x} \log (x) \log \left (x^{1-x}\right )+\frac {1}{2} e^{4 \left (4-e^3\right )-4 x} x \log (x) \log \left (x^{1-x}\right )+\frac {1}{16} \int \frac {e^{4 \left (4-e^3\right )-4 x}}{x} \, dx-\frac {1}{16} \int \frac {e^{4 \left (4-e^3\right )-4 x} (1-2 x)}{x} \, dx+\frac {1}{8} \int e^{4 \left (4-e^3\right )-4 x} \, dx-\frac {1}{8} \int \left (-e^{4 \left (4-e^3\right )-4 x}+\frac {e^{4 \left (4-e^3\right )-4 x}}{x}\right ) \, dx-\frac {1}{8} \int -\frac {e^{4 \left (4-e^3\right )-4 x}}{4 x} \, dx+\frac {1}{8} \int e^{4 \left (4-e^3\right )-4 x} \log ^2(x) \, dx+\frac {1}{2} \int e^{4 \left (4-e^3\right )-4 x} x \log ^2(x) \, dx-4 \int e^{16-4 e^3-4 x} x \log ^2\left (x^{1-x}\right ) \, dx+\frac {1}{8} e^{16-4 e^3} \int \frac {\operatorname {ExpIntegralEi}(-4 x)}{x} \, dx+\frac {1}{8} e^{16-4 e^3} \int \left (\frac {e^{-4 x}}{4 x}+\operatorname {ExpIntegralEi}(-4 x)\right ) \, dx-\frac {1}{8} e^{16-4 e^3} \int \left (\operatorname {ExpIntegralEi}(-4 x)-\frac {\operatorname {ExpIntegralEi}(-4 x)}{x}\right ) \, dx+\int e^{16-4 e^3-4 x} \log ^2\left (x^{1-x}\right ) \, dx \\ & = \frac {5}{32} e^{4 \left (4-e^3\right )-4 x}-\frac {1}{8} e^{4 \left (4-e^3\right )-4 x} x+\frac {5}{16} e^{16-4 e^3} \operatorname {ExpIntegralEi}(-4 x)+\frac {3}{32} e^{4 \left (4-e^3\right )-4 x} \log (x)-\frac {1}{8} e^{4 \left (4-e^3\right )-4 x} x \log (x)-\frac {1}{32} e^{4 \left (4-e^3\right )-4 x} (1+4 x) \log (x)-\frac {1}{8} e^{16-4 e^3} \operatorname {ExpIntegralEi}(-4 x) \log (x)-\frac {1}{8} e^{16-4 e^3} x \operatorname {ExpIntegralEi}(-4 x) \log (x)+\frac {1}{8} e^{16-4 e^3} (\operatorname {ExpIntegralE}(1,4 x)+\operatorname {ExpIntegralEi}(-4 x)) \log (x)-\frac {1}{4} e^{4 \left (4-e^3\right )-4 x} \log \left (x^{1-x}\right )+\frac {1}{2} e^{4 \left (4-e^3\right )-4 x} x \log \left (x^{1-x}\right )-\frac {1}{8} e^{16-4 e^3} \operatorname {ExpIntegralEi}(-4 x) \log \left (x^{1-x}\right )+\frac {1}{8} e^{4 \left (4-e^3\right )-4 x} \log (x) \log \left (x^{1-x}\right )+\frac {1}{2} e^{4 \left (4-e^3\right )-4 x} x \log (x) \log \left (x^{1-x}\right )+\frac {1}{32} \int \frac {e^{4 \left (4-e^3\right )-4 x}}{x} \, dx-\frac {1}{16} \int \left (-2 e^{4 \left (4-e^3\right )-4 x}+\frac {e^{4 \left (4-e^3\right )-4 x}}{x}\right ) \, dx+\frac {1}{8} \int e^{4 \left (4-e^3\right )-4 x} \, dx-\frac {1}{8} \int \frac {e^{4 \left (4-e^3\right )-4 x}}{x} \, dx+\frac {1}{8} \int e^{4 \left (4-e^3\right )-4 x} \log ^2(x) \, dx+\frac {1}{2} \int e^{4 \left (4-e^3\right )-4 x} x \log ^2(x) \, dx-4 \int e^{16-4 e^3-4 x} x \log ^2\left (x^{1-x}\right ) \, dx+\frac {1}{32} e^{16-4 e^3} \int \frac {e^{-4 x}}{x} \, dx-\frac {1}{8} e^{16-4 e^3} \int \frac {\operatorname {ExpIntegralE}(1,4 x)}{x} \, dx+\frac {1}{8} e^{16-4 e^3} \int \frac {\operatorname {ExpIntegralEi}(-4 x)}{x} \, dx+\int e^{16-4 e^3-4 x} \log ^2\left (x^{1-x}\right ) \, dx \\ & = \frac {1}{8} e^{4 \left (4-e^3\right )-4 x}-\frac {1}{8} e^{4 \left (4-e^3\right )-4 x} x+\frac {1}{4} e^{16-4 e^3} \operatorname {ExpIntegralEi}(-4 x)-\frac {1}{2} e^{16-4 e^3} x \, _3F_3(1,1,1;2,2,2;-4 x)+\frac {3}{32} e^{4 \left (4-e^3\right )-4 x} \log (x)+\frac {1}{8} e^{16-4 e^3} \gamma \log (x)-\frac {1}{8} e^{4 \left (4-e^3\right )-4 x} x \log (x)-\frac {1}{32} e^{4 \left (4-e^3\right )-4 x} (1+4 x) \log (x)-\frac {1}{8} e^{16-4 e^3} \operatorname {ExpIntegralEi}(-4 x) \log (x)-\frac {1}{8} e^{16-4 e^3} x \operatorname {ExpIntegralEi}(-4 x) \log (x)+\frac {1}{4} e^{16-4 e^3} (\operatorname {ExpIntegralE}(1,4 x)+\operatorname {ExpIntegralEi}(-4 x)) \log (x)+\frac {1}{16} e^{16-4 e^3} \log ^2(4 x)-\frac {1}{4} e^{4 \left (4-e^3\right )-4 x} \log \left (x^{1-x}\right )+\frac {1}{2} e^{4 \left (4-e^3\right )-4 x} x \log \left (x^{1-x}\right )-\frac {1}{8} e^{16-4 e^3} \operatorname {ExpIntegralEi}(-4 x) \log \left (x^{1-x}\right )+\frac {1}{8} e^{4 \left (4-e^3\right )-4 x} \log (x) \log \left (x^{1-x}\right )+\frac {1}{2} e^{4 \left (4-e^3\right )-4 x} x \log (x) \log \left (x^{1-x}\right )-\frac {1}{16} \int \frac {e^{4 \left (4-e^3\right )-4 x}}{x} \, dx+\frac {1}{8} \int e^{4 \left (4-e^3\right )-4 x} \, dx+\frac {1}{8} \int e^{4 \left (4-e^3\right )-4 x} \log ^2(x) \, dx+\frac {1}{2} \int e^{4 \left (4-e^3\right )-4 x} x \log ^2(x) \, dx-4 \int e^{16-4 e^3-4 x} x \log ^2\left (x^{1-x}\right ) \, dx-\frac {1}{8} e^{16-4 e^3} \int \frac {\operatorname {ExpIntegralE}(1,4 x)}{x} \, dx+\int e^{16-4 e^3-4 x} \log ^2\left (x^{1-x}\right ) \, dx \\ & = \frac {3}{32} e^{4 \left (4-e^3\right )-4 x}-\frac {1}{8} e^{4 \left (4-e^3\right )-4 x} x+\frac {3}{16} e^{16-4 e^3} \operatorname {ExpIntegralEi}(-4 x)-e^{16-4 e^3} x \, _3F_3(1,1,1;2,2,2;-4 x)+\frac {3}{32} e^{4 \left (4-e^3\right )-4 x} \log (x)+\frac {1}{4} e^{16-4 e^3} \gamma \log (x)-\frac {1}{8} e^{4 \left (4-e^3\right )-4 x} x \log (x)-\frac {1}{32} e^{4 \left (4-e^3\right )-4 x} (1+4 x) \log (x)-\frac {1}{8} e^{16-4 e^3} \operatorname {ExpIntegralEi}(-4 x) \log (x)-\frac {1}{8} e^{16-4 e^3} x \operatorname {ExpIntegralEi}(-4 x) \log (x)+\frac {1}{4} e^{16-4 e^3} (\operatorname {ExpIntegralE}(1,4 x)+\operatorname {ExpIntegralEi}(-4 x)) \log (x)+\frac {1}{8} e^{16-4 e^3} \log ^2(4 x)-\frac {1}{4} e^{4 \left (4-e^3\right )-4 x} \log \left (x^{1-x}\right )+\frac {1}{2} e^{4 \left (4-e^3\right )-4 x} x \log \left (x^{1-x}\right )-\frac {1}{8} e^{16-4 e^3} \operatorname {ExpIntegralEi}(-4 x) \log \left (x^{1-x}\right )+\frac {1}{8} e^{4 \left (4-e^3\right )-4 x} \log (x) \log \left (x^{1-x}\right )+\frac {1}{2} e^{4 \left (4-e^3\right )-4 x} x \log (x) \log \left (x^{1-x}\right )+\frac {1}{8} \int e^{4 \left (4-e^3\right )-4 x} \log ^2(x) \, dx+\frac {1}{2} \int e^{4 \left (4-e^3\right )-4 x} x \log ^2(x) \, dx-4 \int e^{16-4 e^3-4 x} x \log ^2\left (x^{1-x}\right ) \, dx+\int e^{16-4 e^3-4 x} \log ^2\left (x^{1-x}\right ) \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int e^{16-4 e^3-4 x} \left ((2-2 x-2 x \log (x)) \log \left (x^{1-x}\right )+(1-4 x) \log ^2\left (x^{1-x}\right )\right ) \, dx=e^{16-4 e^3-4 x} x \log ^2\left (x^{1-x}\right ) \]

[In]

Integrate[E^(16 - 4*E^3 - 4*x)*((2 - 2*x - 2*x*Log[x])*Log[x^(1 - x)] + (1 - 4*x)*Log[x^(1 - x)]^2),x]

[Out]

E^(16 - 4*E^3 - 4*x)*x*Log[x^(1 - x)]^2

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96

method	result	size
parallelrisch	\(x \ln \left (x \,{\mathrm e}^{-x \ln \left (x \right )}\right )^{2} {\mathrm e}^{-4 \,{\mathrm e}^{3}-4 x +16}\)	\(23\)
risch	\(x \,{\mathrm e}^{-4 \,{\mathrm e}^{3}-4 x +16} \ln \left (x^{x}\right )^{2}-x \left (-i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{-x}\right ) \operatorname {csgn}\left (i x \,x^{-x}\right )+i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \,x^{-x}\right )^{2}+i \pi \,\operatorname {csgn}\left (i x^{-x}\right ) \operatorname {csgn}\left (i x \,x^{-x}\right )^{2}-i \pi \operatorname {csgn}\left (i x \,x^{-x}\right )^{3}+2 \ln \left (x \right )\right ) {\mathrm e}^{-4 \,{\mathrm e}^{3}-4 x +16} \ln \left (x^{x}\right )+\frac {x \left (-4 \pi ^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{-x}\right ) \operatorname {csgn}\left (i x \,x^{-x}\right )^{4}-4 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x \,x^{-x}\right )^{3}+2 \pi ^{2} \operatorname {csgn}\left (i x^{-x}\right ) \operatorname {csgn}\left (i x \,x^{-x}\right )^{5}-\pi ^{2} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x \,x^{-x}\right )^{4}+2 \pi ^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \,x^{-x}\right )^{5}-\pi ^{2} \operatorname {csgn}\left (i x^{-x}\right )^{2} \operatorname {csgn}\left (i x \,x^{-x}\right )^{4}-4 i \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{-x}\right ) \operatorname {csgn}\left (i x \,x^{-x}\right )+4 i \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i x^{-x}\right ) \operatorname {csgn}\left (i x \,x^{-x}\right )^{2}+4 i \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \,x^{-x}\right )^{2}-\pi ^{2} \operatorname {csgn}\left (i x \,x^{-x}\right )^{6}+4 \ln \left (x \right )^{2}-\pi ^{2} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{-x}\right )^{2} \operatorname {csgn}\left (i x \,x^{-x}\right )^{2}+2 \pi ^{2} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{-x}\right ) \operatorname {csgn}\left (i x \,x^{-x}\right )^{3}+2 \pi ^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{-x}\right )^{2} \operatorname {csgn}\left (i x \,x^{-x}\right )^{3}\right ) {\mathrm e}^{-4 \,{\mathrm e}^{3}-4 x +16}}{4}\)	\(499\)

[In]

int(((-4*x+1)*ln(x/exp(x*ln(x)))^2+(-2*x*ln(x)-2*x+2)*ln(x/exp(x*ln(x))))/exp(exp(3)+x-4)^4,x,method=_RETURNVE
RBOSE)

[Out]

x*ln(x/exp(x*ln(x)))^2/exp(exp(3)+x-4)^4

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int e^{16-4 e^3-4 x} \left ((2-2 x-2 x \log (x)) \log \left (x^{1-x}\right )+(1-4 x) \log ^2\left (x^{1-x}\right )\right ) \, dx={\left (x^{3} - 2 \, x^{2} + x\right )} e^{\left (-4 \, x - 4 \, e^{3} + 16\right )} \log \left (x\right )^{2} \]

[In]

integrate(((-4*x+1)*log(x/exp(x*log(x)))^2+(-2*x*log(x)-2*x+2)*log(x/exp(x*log(x))))/exp(exp(3)+x-4)^4,x, algo
rithm="fricas")

[Out]

(x^3 - 2*x^2 + x)*e^(-4*x - 4*e^3 + 16)*log(x)^2

Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int e^{16-4 e^3-4 x} \left ((2-2 x-2 x \log (x)) \log \left (x^{1-x}\right )+(1-4 x) \log ^2\left (x^{1-x}\right )\right ) \, dx=\left (x^{3} \log {\left (x \right )}^{2} - 2 x^{2} \log {\left (x \right )}^{2} + x \log {\left (x \right )}^{2}\right ) e^{- 4 x - 4 e^{3} + 16} \]

[In]

integrate(((-4*x+1)*ln(x/exp(x*ln(x)))**2+(-2*x*ln(x)-2*x+2)*ln(x/exp(x*ln(x))))/exp(exp(3)+x-4)**4,x)

[Out]

(x**3*log(x)**2 - 2*x**2*log(x)**2 + x*log(x)**2)*exp(-4*x - 4*exp(3) + 16)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (22) = 44\).

Time = 0.38 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.12 \[ \int e^{16-4 e^3-4 x} \left ((2-2 x-2 x \log (x)) \log \left (x^{1-x}\right )+(1-4 x) \log ^2\left (x^{1-x}\right )\right ) \, dx=-{\left (2 \, x e^{\left (-4 \, x + 16\right )} \log \left (x\right ) \log \left (x^{x}\right ) - x e^{\left (-4 \, x + 16\right )} \log \left (x^{x}\right )^{2} - x e^{\left (-4 \, x + 16\right )} \log \left (x\right )^{2}\right )} e^{\left (-4 \, e^{3}\right )} \]

[In]

integrate(((-4*x+1)*log(x/exp(x*log(x)))^2+(-2*x*log(x)-2*x+2)*log(x/exp(x*log(x))))/exp(exp(3)+x-4)^4,x, algo
rithm="maxima")

[Out]

-(2*x*e^(-4*x + 16)*log(x)*log(x^x) - x*e^(-4*x + 16)*log(x^x)^2 - x*e^(-4*x + 16)*log(x)^2)*e^(-4*e^3)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (22) = 44\).

Time = 0.25 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.25 \[ \int e^{16-4 e^3-4 x} \left ((2-2 x-2 x \log (x)) \log \left (x^{1-x}\right )+(1-4 x) \log ^2\left (x^{1-x}\right )\right ) \, dx=x^{3} e^{\left (-4 \, x - 4 \, e^{3} + 16\right )} \log \left (x\right )^{2} - 2 \, x^{2} e^{\left (-4 \, x - 4 \, e^{3} + 16\right )} \log \left (x\right )^{2} + x e^{\left (-4 \, x - 4 \, e^{3} + 16\right )} \log \left (x\right )^{2} \]

[In]

integrate(((-4*x+1)*log(x/exp(x*log(x)))^2+(-2*x*log(x)-2*x+2)*log(x/exp(x*log(x))))/exp(exp(3)+x-4)^4,x, algo
rithm="giac")

[Out]

x^3*e^(-4*x - 4*e^3 + 16)*log(x)^2 - 2*x^2*e^(-4*x - 4*e^3 + 16)*log(x)^2 + x*e^(-4*x - 4*e^3 + 16)*log(x)^2

Mupad [F(-1)]

Timed out. \[ \int e^{16-4 e^3-4 x} \left ((2-2 x-2 x \log (x)) \log \left (x^{1-x}\right )+(1-4 x) \log ^2\left (x^{1-x}\right )\right ) \, dx=\int -{\mathrm {e}}^{16-4\,{\mathrm {e}}^3-4\,x}\,\left (\left (4\,x-1\right )\,{\ln \left (x\,{\mathrm {e}}^{-x\,\ln \left (x\right )}\right )}^2+\left (2\,x+2\,x\,\ln \left (x\right )-2\right )\,\ln \left (x\,{\mathrm {e}}^{-x\,\ln \left (x\right )}\right )\right ) \,d x \]

[In]

int(-exp(16 - 4*exp(3) - 4*x)*(log(x*exp(-x*log(x)))^2*(4*x - 1) + log(x*exp(-x*log(x)))*(2*x + 2*x*log(x) - 2
)),x)

[Out]

int(-exp(16 - 4*exp(3) - 4*x)*(log(x*exp(-x*log(x)))^2*(4*x - 1) + log(x*exp(-x*log(x)))*(2*x + 2*x*log(x) - 2
)), x)