3.33.95 $\int \frac{(e^{2/x}(40-20x)-10x+(-2x-4e^{2/x}x)\log (\frac{x}{(x+2e^{2/x}x{)}^5}))\log (\frac{1+\log (\frac{x}{(x+2e^{2/x}x{)}^5})}{x})}{x^2+2e^{2/x}{x}^2+(x^2+2e^{2/x}{x}^2)\log (\frac{x}{(x+2e^{2/x}x{)}^5})}dx$

Optimal. Leaf size=26 $${\log }^2\left(\frac{1+\log \left(\frac{x}{{(x+2e^{2/x}x)}^5}\right)}{x}\right)$$

________________________________________________________________________________________

Rubi [A]  time = 2.07, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 3, integrand size = 126, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.024, Rules used = {6741, 6742, 6686} $$\begin{array}{c}{\log }^2\left(\frac{\log \left(\frac{x}{{(2e^{2/x}x+x)}^5}\right)+1}{x}\right)\end{array}$$

Antiderivative was successfully verified.

[In]

Int[((E^(2/x)*(40 - 20*x) - 10*x + (-2*x - 4*E^(2/x)*x)*Log[x/(x + 2*E^(2/x)*x)^5])*Log[(1 + Log[x/(x + 2*E^(2
/x)*x)^5])/x])/(x^2 + 2*E^(2/x)*x^2 + (x^2 + 2*E^(2/x)*x^2)*Log[x/(x + 2*E^(2/x)*x)^5]),x]

[Out]

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

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& ={\log }^2\left(\frac{1+\log \left(\frac{x}{{(x+2e^{2/x}x)}^5}\right)}{x}\right)\end{array}\end{array}$$

________________________________________________________________________________________

Mathematica [A]  time = 0.12, size = 26, normalized size = 1.00 $$\begin{array}{c}{\log }^2\left(\frac{1+\log \left(\frac{x}{{(x+2e^{2/x}x)}^5}\right)}{x}\right)\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[((E^(2/x)*(40 - 20*x) - 10*x + (-2*x - 4*E^(2/x)*x)*Log[x/(x + 2*E^(2/x)*x)^5])*Log[(1 + Log[x/(x +
2*E^(2/x)*x)^5])/x])/(x^2 + 2*E^(2/x)*x^2 + (x^2 + 2*E^(2/x)*x^2)*Log[x/(x + 2*E^(2/x)*x)^5]),x]

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.60, size = 71, normalized size = 2.73 $$\begin{array}{c}\log {\left(\frac{\log \left(\frac{1}{32x^4{e}^{\frac{10}{x}}+80x^4{e}^{\frac{8}{x}}+80x^4{e}^{\frac{6}{x}}+40x^4{e}^{\frac{4}{x}}+10x^4{e}^{\frac{2}{x}}+x^4}\right)+1}{x}\right)}^2\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x*exp(1/x)^2-2*x)*log(x/(2*x*exp(1/x)^2+x)^5)+(-20*x+40)*exp(1/x)^2-10*x)*log((log(x/(2*x*exp(1
/x)^2+x)^5)+1)/x)/((2*x^2*exp(1/x)^2+x^2)*log(x/(2*x*exp(1/x)^2+x)^5)+2*x^2*exp(1/x)^2+x^2),x, algorithm="fric
as")

[Out]

log((log(1/(32*x^4*e^(10/x) + 80*x^4*e^(8/x) + 80*x^4*e^(6/x) + 40*x^4*e^(4/x) + 10*x^4*e^(2/x) + x^4)) + 1)/x
)^2

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}\int -\frac{2(10(x-2)e^{\frac{2}{x}}+(2x{e}^{\frac{2}{x}}+x)\log \left(\frac{x}{{(2x{e}^{\frac{2}{x}}+x)}^5}\right)+5x)\log \left(\frac{\log \left(\frac{x}{{(2x{e}^{\frac{2}{x}}+x)}^5}\right)+1}{x}\right)}{2x^2{e}^{\frac{2}{x}}+x^2+(2x^2{e}^{\frac{2}{x}}+x^2)\log \left(\frac{x}{{(2x{e}^{\frac{2}{x}}+x)}^5}\right)}dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x*exp(1/x)^2-2*x)*log(x/(2*x*exp(1/x)^2+x)^5)+(-20*x+40)*exp(1/x)^2-10*x)*log((log(x/(2*x*exp(1
/x)^2+x)^5)+1)/x)/((2*x^2*exp(1/x)^2+x^2)*log(x/(2*x*exp(1/x)^2+x)^5)+2*x^2*exp(1/x)^2+x^2),x, algorithm="giac
")

[Out]

integrate(-2*(10*(x - 2)*e^(2/x) + (2*x*e^(2/x) + x)*log(x/(2*x*e^(2/x) + x)^5) + 5*x)*log((log(x/(2*x*e^(2/x)
+ x)^5) + 1)/x)/(2*x^2*e^(2/x) + x^2 + (2*x^2*e^(2/x) + x^2)*log(x/(2*x*e^(2/x) + x)^5)), x)

________________________________________________________________________________________

maple [C]  time = 3.47, size = 28486, normalized size = 1095.62

method	result	size
risch	$\text{Expression too large to display}$	$28486$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x*exp(1/x)^2-2*x)*ln(x/(2*x*exp(1/x)^2+x)^5)+(-20*x+40)*exp(1/x)^2-10*x)*ln((ln(x/(2*x*exp(1/x)^2+x)^
5)+1)/x)/((2*x^2*exp(1/x)^2+x^2)*ln(x/(2*x*exp(1/x)^2+x)^5)+2*x^2*exp(1/x)^2+x^2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

________________________________________________________________________________________

maxima [B]  time = 0.70, size = 112, normalized size = 4.31 $$\begin{array}{c}-2\log (5)\log (x)-\log (x{)}^2-2(\log (x)-\log (\frac{4}{5}\log (x)+\log (2e^{\frac{2}{x}}+1)-\frac{1}{5}))\log \left(\frac{\log \left(\frac{x}{{(2x{e}^{\frac{2}{x}}+x)}^5}\right)+1}{x}\right)+2(\log (5)+\log (x))\log (4\log (x)+5\log (2e^{\frac{2}{x}}+1)-1)-\log {(4\log (x)+5\log (2e^{\frac{2}{x}}+1)-1)}^2\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x*exp(1/x)^2-2*x)*log(x/(2*x*exp(1/x)^2+x)^5)+(-20*x+40)*exp(1/x)^2-10*x)*log((log(x/(2*x*exp(1
/x)^2+x)^5)+1)/x)/((2*x^2*exp(1/x)^2+x^2)*log(x/(2*x*exp(1/x)^2+x)^5)+2*x^2*exp(1/x)^2+x^2),x, algorithm="maxi
ma")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 3.69, size = 25, normalized size = 0.96 $$\begin{array}{c}{\ln \left(\frac{\ln \left(\frac{x}{{(x+2x{\mathrm{e}}^{2/x})}^5}\right)+1}{x}\right)}^2\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log((log(x/(x + 2*x*exp(2/x))^5) + 1)/x)*(10*x + exp(2/x)*(20*x - 40) + log(x/(x + 2*x*exp(2/x))^5)*(2*x
+ 4*x*exp(2/x))))/(log(x/(x + 2*x*exp(2/x))^5)*(2*x^2*exp(2/x) + x^2) + 2*x^2*exp(2/x) + x^2),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 7.24, size = 20, normalized size = 0.77 $$\begin{array}{c}\log {\left(\frac{\log \left(\frac{x}{{(2x{e}^{\frac{2}{x}}+x)}^5}\right)+1}{x}\right)}^2\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x*exp(1/x)**2-2*x)*ln(x/(2*x*exp(1/x)**2+x)**5)+(-20*x+40)*exp(1/x)**2-10*x)*ln((ln(x/(2*x*exp(
1/x)**2+x)**5)+1)/x)/((2*x**2*exp(1/x)**2+x**2)*ln(x/(2*x*exp(1/x)**2+x)**5)+2*x**2*exp(1/x)**2+x**2),x)

[Out]

log((log(x/(2*x*exp(2/x) + x)**5) + 1)/x)**2

________________________________________________________________________________________