\(\int \frac {(-2+(2-6 x^2) \log (x)-4 x^2 \log ^2(x)) \log (\frac {4 e^{-x^2-x^2 \log (x)} x}{5 \log (x)})}{x \log (x)} \, dx\) [4448]

Optimal result

Integrand size = 54, antiderivative size = 27 \[ \int \frac {\left (-2+\left (2-6 x^2\right ) \log (x)-4 x^2 \log ^2(x)\right ) \log \left (\frac {4 e^{-x^2-x^2 \log (x)} x}{5 \log (x)}\right )}{x \log (x)} \, dx=\log ^2\left (\frac {4 e^{-x^2-x^2 \log (x)} x}{5 \log (x)}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(3*x^2)/2 - (15*x^4)/8 - 3*ExpIntegralEi[2*Log[x]] - (3*x^4*Log[x])/2 - 3*x^2*Log[(4*x^(1 - x^2))/(5*E^x^2*Log
[x])] + 2*Defer[Int][Log[(4*x^(1 - x^2))/(5*E^x^2*Log[x])]/x, x] - 2*Defer[Int][Log[(4*x^(1 - x^2))/(5*E^x^2*L
og[x])]/(x*Log[x]), x] - 4*Defer[Int][x*Log[x]*Log[(4*x^(1 - x^2))/(5*E^x^2*Log[x])], x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x}-6 x \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )-\frac {2 \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x \log (x)}-4 x \log (x) \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )\right ) \, dx \\ & = 2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x} \, dx-2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x \log (x)} \, dx-4 \int x \log (x) \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right ) \, dx-6 \int x \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right ) \, dx \\ & = -3 x^2 \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )+2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x} \, dx-2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x \log (x)} \, dx-4 \int x \log (x) \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right ) \, dx+6 \int \frac {x \left (-1-\left (-1+3 x^2\right ) \log (x)-2 x^2 \log ^2(x)\right )}{2 \log (x)} \, dx \\ & = -3 x^2 \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )+2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x} \, dx-2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x \log (x)} \, dx+3 \int \frac {x \left (-1-\left (-1+3 x^2\right ) \log (x)-2 x^2 \log ^2(x)\right )}{\log (x)} \, dx-4 \int x \log (x) \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right ) \, dx \\ & = -3 x^2 \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )+2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x} \, dx-2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x \log (x)} \, dx+3 \int \left (x-3 x^3-\frac {x}{\log (x)}-2 x^3 \log (x)\right ) \, dx-4 \int x \log (x) \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right ) \, dx \\ & = \frac {3 x^2}{2}-\frac {9 x^4}{4}-3 x^2 \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )+2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x} \, dx-2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x \log (x)} \, dx-3 \int \frac {x}{\log (x)} \, dx-4 \int x \log (x) \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right ) \, dx-6 \int x^3 \log (x) \, dx \\ & = \frac {3 x^2}{2}-\frac {15 x^4}{8}-\frac {3}{2} x^4 \log (x)-3 x^2 \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )+2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x} \, dx-2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x \log (x)} \, dx-3 \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )-4 \int x \log (x) \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right ) \, dx \\ & = \frac {3 x^2}{2}-\frac {15 x^4}{8}-3 \text {Ei}(2 \log (x))-\frac {3}{2} x^4 \log (x)-3 x^2 \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )+2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x} \, dx-2 \int \frac {\log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right )}{x \log (x)} \, dx-4 \int x \log (x) \log \left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right ) \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2+\left (2-6 x^2\right ) \log (x)-4 x^2 \log ^2(x)\right ) \log \left (\frac {4 e^{-x^2-x^2 \log (x)} x}{5 \log (x)}\right )}{x \log (x)} \, dx=\log ^2\left (\frac {4 e^{-x^2} x^{1-x^2}}{5 \log (x)}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.39 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81

method	result	size
parallelrisch	\(\ln \left (\frac {4 x \,{\mathrm e}^{-\left (\ln \left (x \right )+1\right ) x^{2}}}{5 \ln \left (x \right )}\right )^{2}\)	\(22\)
risch	\(\text {Expression too large to display}\)	\(4094\)

[In]

int((-4*x^2*ln(x)^2+(-6*x^2+2)*ln(x)-2)*ln(4/5*x/ln(x)/exp(x^2*ln(x)+x^2))/x/ln(x),x,method=_RETURNVERBOSE)

[Out]

ln(4/5*x/ln(x)/exp((ln(x)+1)*x^2))^2

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-2+\left (2-6 x^2\right ) \log (x)-4 x^2 \log ^2(x)\right ) \log \left (\frac {4 e^{-x^2-x^2 \log (x)} x}{5 \log (x)}\right )}{x \log (x)} \, dx=\log \left (\frac {4 \, x e^{\left (-x^{2} \log \left (x\right ) - x^{2}\right )}}{5 \, \log \left (x\right )}\right )^{2} \]

[In]

integrate((-4*x^2*log(x)^2+(-6*x^2+2)*log(x)-2)*log(4/5*x/log(x)/exp(x^2*log(x)+x^2))/x/log(x),x, algorithm="f
ricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-2+\left (2-6 x^2\right ) \log (x)-4 x^2 \log ^2(x)\right ) \log \left (\frac {4 e^{-x^2-x^2 \log (x)} x}{5 \log (x)}\right )}{x \log (x)} \, dx=\log {\left (\frac {4 x e^{- x^{2} \log {\left (x \right )} - x^{2}}}{5 \log {\left (x \right )}} \right )}^{2} \]

[In]

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

[Out]

log(4*x*exp(-x**2*log(x) - x**2)/(5*log(x)))**2

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (24) = 48\).

Time = 0.25 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.70 \[ \int \frac {\left (-2+\left (2-6 x^2\right ) \log (x)-4 x^2 \log ^2(x)\right ) \log \left (\frac {4 e^{-x^2-x^2 \log (x)} x}{5 \log (x)}\right )}{x \log (x)} \, dx=-x^{4} - {\left (x^{4} - 2 \, x^{2} + 1\right )} \log \left (x\right )^{2} - 2 \, {\left (x^{4} - x^{2}\right )} \log \left (x\right ) - 2 \, {\left (x^{2} \log \left (x\right ) + x^{2} - \log \left (x\right ) + \log \left (\log \left (x\right )\right )\right )} \log \left (\frac {4 \, x e^{\left (-x^{2} \log \left (x\right ) - x^{2}\right )}}{5 \, \log \left (x\right )}\right ) - 2 \, {\left (x^{2} + {\left (x^{2} - 1\right )} \log \left (x\right )\right )} \log \left (\log \left (x\right )\right ) - \log \left (\log \left (x\right )\right )^{2} \]

[In]

integrate((-4*x^2*log(x)^2+(-6*x^2+2)*log(x)-2)*log(4/5*x/log(x)/exp(x^2*log(x)+x^2))/x/log(x),x, algorithm="m
axima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (24) = 48\).

Time = 0.29 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.19 \[ \int \frac {\left (-2+\left (2-6 x^2\right ) \log (x)-4 x^2 \log ^2(x)\right ) \log \left (\frac {4 e^{-x^2-x^2 \log (x)} x}{5 \log (x)}\right )}{x \log (x)} \, dx=x^{4} - 4 \, x^{2} \log \left (2\right ) + {\left (x^{4} - 2 \, x^{2} + 1\right )} \log \left (x\right )^{2} + 2 \, {\left (x^{4} - x^{2} {\left (2 \, \log \left (2\right ) + 1\right )}\right )} \log \left (x\right ) + 4 \, \log \left (2\right ) \log \left (x\right ) + 2 \, {\left (x^{2} \log \left (x\right ) + x^{2} - \log \left (x\right )\right )} \log \left (5 \, \log \left (x\right )\right ) + \log \left (5 \, \log \left (x\right )\right )^{2} - 4 \, \log \left (2\right ) \log \left (\log \left (x\right )\right ) \]

[In]

integrate((-4*x^2*log(x)^2+(-6*x^2+2)*log(x)-2)*log(4/5*x/log(x)/exp(x^2*log(x)+x^2))/x/log(x),x, algorithm="g
iac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 10.78 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {\left (-2+\left (2-6 x^2\right ) \log (x)-4 x^2 \log ^2(x)\right ) \log \left (\frac {4 e^{-x^2-x^2 \log (x)} x}{5 \log (x)}\right )}{x \log (x)} \, dx={\ln \left (\frac {4\,x\,{\mathrm {e}}^{-x^2}}{5\,x^{x^2}\,\ln \left (x\right )}\right )}^2 \]

[In]

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

[Out]

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