∫ −2−20x−20x log(x)__________________________________________ 6480x+3600x2+500x3+(6480x+3600x2+500x3) log(x)+(360x+100x2+(360x+100x2) log(x)) log(1+log(x))+(5x+5x log(x)) log2(1+log(x)) dx [408]

________________________________________________________________________________________

Rubi [A]
time = 0.13, antiderivative size = 16, normalized size of antiderivative = 0.89, number of steps used = 3, number of rules used = 3, integrand size = 88, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {6820, 12, 6818} \begin {gather*} \frac {2}{5 (10 x+\log (\log (x)+1)+36)} \end {gather*}

Int[(-2 - 20*x - 20*x*Log[x])/(6480*x + 3600*x^2 + 500*x^3 + (6480*x + 3600*x^2 + 500*x^3)*Log[x] + (360*x + 1

00*x^2 + (360*x + 100*x^2)*Log[x])*Log[1 + Log[x]] + (5*x + 5*x*Log[x])*Log[1 + Log[x]]^2),x]

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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

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

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 (-1-10 x-10 x \log (x))}{5 x (1+\log (x)) (36+10 x+\log (1+\log (x)))^2} \, dx\\ &=\frac {2}{5} \int \frac {-1-10 x-10 x \log (x)}{x (1+\log (x)) (36+10 x+\log (1+\log (x)))^2} \, dx\\ &=\frac {2}{5 (36+10 x+\log (1+\log (x)))}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 16, normalized size = 0.89 \begin {gather*} \frac {2}{5 (36+10 x+\log (1+\log (x)))} \end {gather*}

Integrate[(-2 - 20*x - 20*x*Log[x])/(6480*x + 3600*x^2 + 500*x^3 + (6480*x + 3600*x^2 + 500*x^3)*Log[x] + (360

*x + 100*x^2 + (360*x + 100*x^2)*Log[x])*Log[1 + Log[x]] + (5*x + 5*x*Log[x])*Log[1 + Log[x]]^2),x]

________________________________________________________________________________________


method	result	size

default	\(\frac {2}{5 \left (\ln \left (\ln \left (x \right )+1\right )+10 x +36\right )}\)	\(15\)
risch	\(\frac {2}{5 \left (\ln \left (\ln \left (x \right )+1\right )+10 x +36\right )}\)	\(15\)

int((-20*x*ln(x)-20*x-2)/((5*x*ln(x)+5*x)*ln(ln(x)+1)^2+((100*x^2+360*x)*ln(x)+100*x^2+360*x)*ln(ln(x)+1)+(500

*x^3+3600*x^2+6480*x)*ln(x)+500*x^3+3600*x^2+6480*x),x,method=_RETURNVERBOSE)

________________________________________________________________________________________

Maxima [A]
time = 0.50, size = 14, normalized size = 0.78 \begin {gather*} \frac {2}{5 \, {\left (10 \, x + \log \left (\log \left (x\right ) + 1\right ) + 36\right )}} \end {gather*}

integrate((-20*x*log(x)-20*x-2)/((5*x*log(x)+5*x)*log(log(x)+1)^2+((100*x^2+360*x)*log(x)+100*x^2+360*x)*log(l

og(x)+1)+(500*x^3+3600*x^2+6480*x)*log(x)+500*x^3+3600*x^2+6480*x),x, algorithm="maxima")

________________________________________________________________________________________

Fricas [A]
time = 0.30, size = 14, normalized size = 0.78 \begin {gather*} \frac {2}{5 \, {\left (10 \, x + \log \left (\log \left (x\right ) + 1\right ) + 36\right )}} \end {gather*}

integrate((-20*x*log(x)-20*x-2)/((5*x*log(x)+5*x)*log(log(x)+1)^2+((100*x^2+360*x)*log(x)+100*x^2+360*x)*log(l

og(x)+1)+(500*x^3+3600*x^2+6480*x)*log(x)+500*x^3+3600*x^2+6480*x),x, algorithm="fricas")

________________________________________________________________________________________

Sympy [A]
time = 0.07, size = 14, normalized size = 0.78 \begin {gather*} \frac {2}{50 x + 5 \log {\left (\log {\left (x \right )} + 1 \right )} + 180} \end {gather*}

integrate((-20*x*ln(x)-20*x-2)/((5*x*ln(x)+5*x)*ln(ln(x)+1)**2+((100*x**2+360*x)*ln(x)+100*x**2+360*x)*ln(ln(x

)+1)+(500*x**3+3600*x**2+6480*x)*ln(x)+500*x**3+3600*x**2+6480*x),x)

________________________________________________________________________________________

Giac [A]
time = 0.41, size = 14, normalized size = 0.78 \begin {gather*} \frac {2}{5 \, {\left (10 \, x + \log \left (\log \left (x\right ) + 1\right ) + 36\right )}} \end {gather*}

integrate((-20*x*log(x)-20*x-2)/((5*x*log(x)+5*x)*log(log(x)+1)^2+((100*x^2+360*x)*log(x)+100*x^2+360*x)*log(l

og(x)+1)+(500*x^3+3600*x^2+6480*x)*log(x)+500*x^3+3600*x^2+6480*x),x, algorithm="giac")

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int -\frac {20\,x+20\,x\,\ln \left (x\right )+2}{6480\,x+\ln \left (\ln \left (x\right )+1\right )\,\left (360\,x+\ln \left (x\right )\,\left (100\,x^2+360\,x\right )+100\,x^2\right )+{\ln \left (\ln \left (x\right )+1\right )}^2\,\left (5\,x+5\,x\,\ln \left (x\right )\right )+3600\,x^2+500\,x^3+\ln \left (x\right )\,\left (500\,x^3+3600\,x^2+6480\,x\right )} \,d x \end {gather*}

int(-(20*x + 20*x*log(x) + 2)/(6480*x + log(log(x) + 1)*(360*x + log(x)*(360*x + 100*x^2) + 100*x^2) + log(log

(x) + 1)^2*(5*x + 5*x*log(x)) + 3600*x^2 + 500*x^3 + log(x)*(6480*x + 3600*x^2 + 500*x^3)),x)

int(-(20*x + 20*x*log(x) + 2)/(6480*x + log(log(x) + 1)*(360*x + log(x)*(360*x + 100*x^2) + 100*x^2) + log(log

(x) + 1)^2*(5*x + 5*x*log(x)) + 3600*x^2 + 500*x^3 + log(x)*(6480*x + 3600*x^2 + 500*x^3)), x)

________________________________________________________________________________________

3.5.8 \(\int \frac {-2-20 x-20 x \log (x)}{6480 x+3600 x^2+500 x^3+(6480 x+3600 x^2+500 x^3) \log (x)+(360 x+100 x^2+(360 x+100 x^2) \log (x)) \log (1+\log (x))+(5 x+5 x \log (x)) \log ^2(1+\log (x))} \, dx\) [408]