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

Optimal. Leaf size=24 \[ e^{16-4 e^3-4 x} x \log ^2\left (x^{1-x}\right ) \]

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Rubi [F]  time = 4.89, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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} \text {Ei}(-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} \text {Ei}(-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} \text {Ei}(-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 )} \text {Ei}(-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} \text {Ei}(-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} \text {Ei}(-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} \text {Ei}(-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 {\text {Ei}(-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} \text {Ei}(-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} \text {Ei}(-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} \text {Ei}(-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) \text {Ei}(-4 x)}{x}+\text {Ei}(-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} \text {Ei}(-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} \text {Ei}(-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} \text {Ei}(-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 )} \text {Ei}(-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) \text {Ei}(-4 x)}{x} \, dx-\frac {1}{8} e^{16-4 e^3} \int \text {Ei}(-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} \text {Ei}(-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} \text {Ei}(-4 x) \log (x)-\frac {1}{8} e^{16-4 e^3} x \text {Ei}(-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} \text {Ei}(-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 {\text {Ei}(-4 x)}{x} \, dx+\frac {1}{8} e^{16-4 e^3} \int \left (\frac {e^{-4 x}}{4 x}+\text {Ei}(-4 x)\right ) \, dx-\frac {1}{8} e^{16-4 e^3} \int \left (\text {Ei}(-4 x)-\frac {\text {Ei}(-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} \text {Ei}(-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} \text {Ei}(-4 x) \log (x)-\frac {1}{8} e^{16-4 e^3} x \text {Ei}(-4 x) \log (x)+\frac {1}{8} e^{16-4 e^3} (E_1(4 x)+\text {Ei}(-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} \text {Ei}(-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 {E_1(4 x)}{x} \, dx+\frac {1}{8} e^{16-4 e^3} \int \frac {\text {Ei}(-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} \text {Ei}(-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} \text {Ei}(-4 x) \log (x)-\frac {1}{8} e^{16-4 e^3} x \text {Ei}(-4 x) \log (x)+\frac {1}{4} e^{16-4 e^3} (E_1(4 x)+\text {Ei}(-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} \text {Ei}(-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 {E_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} \text {Ei}(-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} \text {Ei}(-4 x) \log (x)-\frac {1}{8} e^{16-4 e^3} x \text {Ei}(-4 x) \log (x)+\frac {1}{4} e^{16-4 e^3} (E_1(4 x)+\text {Ei}(-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} \text {Ei}(-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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.21, size = 22, normalized size = 0.92 \begin {gather*} e^{-4 \left (-4+e^3+x\right )} x \log ^2\left (x^{1-x}\right ) \end {gather*}

Antiderivative was successfully verified.

[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]

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

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fricas [A]  time = 1.21, size = 25, normalized size = 1.04 \begin {gather*} {\left (x^{3} - 2 \, x^{2} + x\right )} e^{\left (-4 \, x - 4 \, e^{3} + 16\right )} \log \relax (x)^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [B]  time = 0.30, size = 54, normalized size = 2.25 \begin {gather*} x^{3} e^{\left (-4 \, x - 4 \, e^{3} + 16\right )} \log \relax (x)^{2} - 2 \, x^{2} e^{\left (-4 \, x - 4 \, e^{3} + 16\right )} \log \relax (x)^{2} + x e^{\left (-4 \, x - 4 \, e^{3} + 16\right )} \log \relax (x)^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [C]  time = 0.14, size = 499, normalized size = 20.79

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

Verification of antiderivative is not currently implemented for this CAS.

[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*exp(-4*exp(3)-4*x+16)*ln(x^x)^2-x*(-I*Pi*csgn(I*x)*csgn(I/(x^x))*csgn(I*x/(x^x))+I*Pi*csgn(I*x)*csgn(I*x/(x^
x))^2+I*Pi*csgn(I/(x^x))*csgn(I*x/(x^x))^2-I*Pi*csgn(I*x/(x^x))^3+2*ln(x))*exp(-4*exp(3)-4*x+16)*ln(x^x)+1/4*x
*(-4*I*ln(x)*Pi*csgn(I*x)*csgn(I/(x^x))*csgn(I*x/(x^x))+4*ln(x)^2+4*I*ln(x)*Pi*csgn(I*x)*csgn(I*x/(x^x))^2+4*I
*ln(x)*Pi*csgn(I/(x^x))*csgn(I*x/(x^x))^2-Pi^2*csgn(I*x)^2*csgn(I*x/(x^x))^4+2*Pi^2*csgn(I*x)*csgn(I*x/(x^x))^
5-Pi^2*csgn(I/(x^x))^2*csgn(I*x/(x^x))^4+2*Pi^2*csgn(I/(x^x))*csgn(I*x/(x^x))^5-Pi^2*csgn(I*x/(x^x))^6-Pi^2*cs
gn(I*x)^2*csgn(I/(x^x))^2*csgn(I*x/(x^x))^2+2*Pi^2*csgn(I*x)^2*csgn(I/(x^x))*csgn(I*x/(x^x))^3+2*Pi^2*csgn(I*x
)*csgn(I/(x^x))^2*csgn(I*x/(x^x))^3-4*Pi^2*csgn(I*x)*csgn(I/(x^x))*csgn(I*x/(x^x))^4-4*I*ln(x)*Pi*csgn(I*x/(x^
x))^3)*exp(-4*exp(3)-4*x+16)

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maxima [B]  time = 0.92, size = 51, normalized size = 2.12 \begin {gather*} -{\left (2 \, x e^{\left (-4 \, x + 16\right )} \log \relax (x) \log \left (x^{x}\right ) - x e^{\left (-4 \, x + 16\right )} \log \relax (x)^{2} - x e^{\left (-4 \, x + 16\right )} \log \left (x^{x}\right )^{2}\right )} e^{\left (-4 \, e^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -{\mathrm {e}}^{16-4\,{\mathrm {e}}^3-4\,x}\,\left (\left (4\,x-1\right )\,{\ln \left (x\,{\mathrm {e}}^{-x\,\ln \relax (x)}\right )}^2+\left (2\,x+2\,x\,\ln \relax (x)-2\right )\,\ln \left (x\,{\mathrm {e}}^{-x\,\ln \relax (x)}\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.57, size = 36, normalized size = 1.50 \begin {gather*} \left (x^{3} \log {\relax (x )}^{2} - 2 x^{2} \log {\relax (x )}^{2} + x \log {\relax (x )}^{2}\right ) e^{- 4 x - 4 e^{3} + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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