\(\int \frac {e^{1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)} (e^{2 x}+(-4 x+8 x^2) \log (x))}{21 x} \, dx\) [4650]

Optimal result

Integrand size = 53, antiderivative size = 23 \[ \int \frac {e^{1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)} \left (e^{2 x}+\left (-4 x+8 x^2\right ) \log (x)\right )}{21 x} \, dx=\frac {1}{21} \left (25+e^{1+e^{-4 e^{-2 x} x} \log (x)}\right ) \]

[Out]

25/21+1/21*exp(ln(x)*exp(-4*x/exp(x)^2)+1)

Rubi [F]

\[ \int \frac {e^{1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)} \left (e^{2 x}+\left (-4 x+8 x^2\right ) \log (x)\right )}{21 x} \, dx=\int \frac {\exp \left (1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)\right ) \left (e^{2 x}+\left (-4 x+8 x^2\right ) \log (x)\right )}{21 x} \, dx \]

[In]

Int[(E^(1 - 2*x - (4*x)/E^(2*x) + Log[x]/E^((4*x)/E^(2*x)))*(E^(2*x) + (-4*x + 8*x^2)*Log[x]))/(21*x),x]

[Out]

Defer[Int][E^(1 - (4*x)/E^(2*x) + Log[x]/E^((4*x)/E^(2*x)))/x, x]/21 - (4*Defer[Int][E^(1 - 2*x - (4*x)/E^(2*x
) + Log[x]/E^((4*x)/E^(2*x)))*Log[x], x])/21 + (8*Defer[Int][E^(1 - 2*x - (4*x)/E^(2*x) + Log[x]/E^((4*x)/E^(2
*x)))*x*Log[x], x])/21

Rubi steps \begin{align*} \text {integral}& = \frac {1}{21} \int \frac {\exp \left (1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)\right ) \left (e^{2 x}+\left (-4 x+8 x^2\right ) \log (x)\right )}{x} \, dx \\ & = \frac {1}{21} \int \left (\frac {\exp \left (1-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)\right )}{x}+4 \exp \left (1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)\right ) (-1+2 x) \log (x)\right ) \, dx \\ & = \frac {1}{21} \int \frac {\exp \left (1-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)\right )}{x} \, dx+\frac {4}{21} \int \exp \left (1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)\right ) (-1+2 x) \log (x) \, dx \\ & = \frac {1}{21} \int \frac {\exp \left (1-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)\right )}{x} \, dx+\frac {4}{21} \int \left (-\exp \left (1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)\right ) \log (x)+2 \exp \left (1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)\right ) x \log (x)\right ) \, dx \\ & = \frac {1}{21} \int \frac {\exp \left (1-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)\right )}{x} \, dx-\frac {4}{21} \int \exp \left (1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)\right ) \log (x) \, dx+\frac {8}{21} \int \exp \left (1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)\right ) x \log (x) \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {e^{1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)} \left (e^{2 x}+\left (-4 x+8 x^2\right ) \log (x)\right )}{21 x} \, dx=\frac {1}{21} e x^{e^{-4 e^{-2 x} x}} \]

[In]

Integrate[(E^(1 - 2*x - (4*x)/E^(2*x) + Log[x]/E^((4*x)/E^(2*x)))*(E^(2*x) + (-4*x + 8*x^2)*Log[x]))/(21*x),x]

[Out]

(E*x^E^((-4*x)/E^(2*x)))/21

Maple [A] (verified)

Time = 1.52 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65

method	result	size
risch	\(\frac {x^{{\mathrm e}^{-4 \,{\mathrm e}^{-2 x} x}} {\mathrm e}}{21}\)	\(15\)
parallelrisch	\(\frac {{\mathrm e}^{\ln \left (x \right ) {\mathrm e}^{-4 \,{\mathrm e}^{-2 x} x}+1}}{21}\)	\(17\)

[In]

int(1/21*((8*x^2-4*x)*ln(x)+exp(x)^2)*exp(-4*x/exp(x)^2)*exp(ln(x)*exp(-4*x/exp(x)^2)+1)/x/exp(x)^2,x,method=_
RETURNVERBOSE)

[Out]

1/21*x^exp(-4*exp(-2*x)*x)*exp(1)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (18) = 36\).

Time = 0.23 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.17 \[ \int \frac {e^{1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)} \left (e^{2 x}+\left (-4 x+8 x^2\right ) \log (x)\right )}{21 x} \, dx=\frac {1}{21} \, e^{\left (-{\left ({\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} - e^{\left (-4 \, x e^{\left (-2 \, x\right )} + 2 \, x\right )} \log \left (x\right ) + 4 \, x\right )} e^{\left (-2 \, x\right )} + 4 \, x e^{\left (-2 \, x\right )} + 2 \, x\right )} \]

[In]

integrate(1/21*((8*x^2-4*x)*log(x)+exp(x)^2)*exp(-4*x/exp(x)^2)*exp(log(x)*exp(-4*x/exp(x)^2)+1)/x/exp(x)^2,x,
algorithm="fricas")

[Out]

1/21*e^(-((2*x - 1)*e^(2*x) - e^(-4*x*e^(-2*x) + 2*x)*log(x) + 4*x)*e^(-2*x) + 4*x*e^(-2*x) + 2*x)

Sympy [A] (verification not implemented)

Time = 0.84 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {e^{1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)} \left (e^{2 x}+\left (-4 x+8 x^2\right ) \log (x)\right )}{21 x} \, dx=\frac {e^{1 + e^{- 4 x e^{- 2 x}} \log {\left (x \right )}}}{21} \]

[In]

integrate(1/21*((8*x**2-4*x)*ln(x)+exp(x)**2)*exp(-4*x/exp(x)**2)*exp(ln(x)*exp(-4*x/exp(x)**2)+1)/x/exp(x)**2
,x)

[Out]

exp(1 + exp(-4*x*exp(-2*x))*log(x))/21

Maxima [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.70 \[ \int \frac {e^{1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)} \left (e^{2 x}+\left (-4 x+8 x^2\right ) \log (x)\right )}{21 x} \, dx=\frac {1}{21} \, e^{\left (e^{\left (-4 \, x e^{\left (-2 \, x\right )}\right )} \log \left (x\right ) + 1\right )} \]

[In]

integrate(1/21*((8*x^2-4*x)*log(x)+exp(x)^2)*exp(-4*x/exp(x)^2)*exp(log(x)*exp(-4*x/exp(x)^2)+1)/x/exp(x)^2,x,
algorithm="maxima")

[Out]

1/21*e^(e^(-4*x*e^(-2*x))*log(x) + 1)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.61 \[ \int \frac {e^{1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)} \left (e^{2 x}+\left (-4 x+8 x^2\right ) \log (x)\right )}{21 x} \, dx=\frac {1}{21} \, x^{e^{\left (-4 \, x e^{\left (-2 \, x\right )}\right )}} e \]

[In]

integrate(1/21*((8*x^2-4*x)*log(x)+exp(x)^2)*exp(-4*x/exp(x)^2)*exp(log(x)*exp(-4*x/exp(x)^2)+1)/x/exp(x)^2,x,
algorithm="giac")

[Out]

1/21*x^e^(-4*x*e^(-2*x))*e

Mupad [B] (verification not implemented)

Time = 11.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.61 \[ \int \frac {e^{1-2 x-4 e^{-2 x} x+e^{-4 e^{-2 x} x} \log (x)} \left (e^{2 x}+\left (-4 x+8 x^2\right ) \log (x)\right )}{21 x} \, dx=\frac {x^{{\mathrm {e}}^{-4\,x\,{\mathrm {e}}^{-2\,x}}}\,\mathrm {e}}{21} \]

[In]

int((exp(-2*x)*exp(-4*x*exp(-2*x))*exp(exp(-4*x*exp(-2*x))*log(x) + 1)*(exp(2*x) - log(x)*(4*x - 8*x^2)))/(21*
x),x)

[Out]

(x^exp(-4*x*exp(-2*x))*exp(1))/21