∫ −3−9x−6x2+12x3+(−1−3x−2x2+4x3) log(4)+(6x+12x2+(2x+4x2) log(4)) log(x)+(−6x+6x2+(−2x+2x2) log(4)+(6x+2x log(4)) log(x)) log(−1+x+log(x))_______________________________________________________________________________________________________________ −2x+2x3+(2x+2x2) log(x)+(−x+x2+x log(x)) log(−1+x+log(x)) dx

Optimal. Leaf size=26 \[ (3+\log (4)) \left (2 x-\log \left (x+\frac {1}{2} (2+\log (-1+x+\log (x)))\right )\right ) \]

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Rubi [A] time = 0.78, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 139, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {6688, 12, 6742, 6684} \begin {gather*} 2 x (3+\log (4))-(3+\log (4)) \log (2 x+\log (x+\log (x)-1)+2) \end {gather*}

Int[(-3 - 9*x - 6*x^2 + 12*x^3 + (-1 - 3*x - 2*x^2 + 4*x^3)*Log[4] + (6*x + 12*x^2 + (2*x + 4*x^2)*Log[4])*Log

[x] + (-6*x + 6*x^2 + (-2*x + 2*x^2)*Log[4] + (6*x + 2*x*Log[4])*Log[x])*Log[-1 + x + Log[x]])/(-2*x + 2*x^3 +

(2*x + 2*x^2)*Log[x] + (-x + x^2 + x*Log[x])*Log[-1 + x + Log[x]]),x]

2*x*(3 + Log[4]) - (3 + Log[4])*Log[2 + 2*x + Log[-1 + x + Log[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_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /; !Fa

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

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

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(3+\log (4)) \left (1+3 x+2 x^2-4 x^3-2 (-1+x) x \log (-1+x+\log (x))-2 x \log (x) (1+2 x+\log (-1+x+\log (x)))\right )}{x (1-x-\log (x)) (2+2 x+\log (-1+x+\log (x)))} \, dx\\ &=(3+\log (4)) \int \frac {1+3 x+2 x^2-4 x^3-2 (-1+x) x \log (-1+x+\log (x))-2 x \log (x) (1+2 x+\log (-1+x+\log (x)))}{x (1-x-\log (x)) (2+2 x+\log (-1+x+\log (x)))} \, dx\\ &=(3+\log (4)) \int \left (2+\frac {-1+x-2 x^2-2 x \log (x)}{x (-1+x+\log (x)) (2+2 x+\log (-1+x+\log (x)))}\right ) \, dx\\ &=2 x (3+\log (4))+(3+\log (4)) \int \frac {-1+x-2 x^2-2 x \log (x)}{x (-1+x+\log (x)) (2+2 x+\log (-1+x+\log (x)))} \, dx\\ &=2 x (3+\log (4))-(3+\log (4)) \log (2+2 x+\log (-1+x+\log (x)))\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.51, size = 23, normalized size = 0.88 \begin {gather*} (3+\log (4)) (2 x-\log (2+2 x+\log (-1+x+\log (x)))) \end {gather*}

Integrate[(-3 - 9*x - 6*x^2 + 12*x^3 + (-1 - 3*x - 2*x^2 + 4*x^3)*Log[4] + (6*x + 12*x^2 + (2*x + 4*x^2)*Log[4

])*Log[x] + (-6*x + 6*x^2 + (-2*x + 2*x^2)*Log[4] + (6*x + 2*x*Log[4])*Log[x])*Log[-1 + x + Log[x]])/(-2*x + 2

*x^3 + (2*x + 2*x^2)*Log[x] + (-x + x^2 + x*Log[x])*Log[-1 + x + Log[x]]),x]

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

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fricas [A] time = 0.51, size = 29, normalized size = 1.12 \begin {gather*} 4 \, x \log \relax (2) - {\left (2 \, \log \relax (2) + 3\right )} \log \left (2 \, x + \log \left (x + \log \relax (x) - 1\right ) + 2\right ) + 6 \, x \end {gather*}

integrate((((4*x*log(2)+6*x)*log(x)+2*(2*x^2-2*x)*log(2)+6*x^2-6*x)*log(-1+log(x)+x)+(2*(4*x^2+2*x)*log(2)+12*

x^2+6*x)*log(x)+2*(4*x^3-2*x^2-3*x-1)*log(2)+12*x^3-6*x^2-9*x-3)/((x*log(x)+x^2-x)*log(-1+log(x)+x)+(2*x^2+2*x

)*log(x)+2*x^3-2*x),x, algorithm="fricas")

4*x*log(2) - (2*log(2) + 3)*log(2*x + log(x + log(x) - 1) + 2) + 6*x

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giac [A] time = 0.18, size = 30, normalized size = 1.15 \begin {gather*} 2 \, x {\left (2 \, \log \relax (2) + 3\right )} - {\left (2 \, \log \relax (2) + 3\right )} \log \left (2 \, x + \log \left (x + \log \relax (x) - 1\right ) + 2\right ) \end {gather*}

integrate((((4*x*log(2)+6*x)*log(x)+2*(2*x^2-2*x)*log(2)+6*x^2-6*x)*log(-1+log(x)+x)+(2*(4*x^2+2*x)*log(2)+12*

x^2+6*x)*log(x)+2*(4*x^3-2*x^2-3*x-1)*log(2)+12*x^3-6*x^2-9*x-3)/((x*log(x)+x^2-x)*log(-1+log(x)+x)+(2*x^2+2*x

2*x*(2*log(2) + 3) - (2*log(2) + 3)*log(2*x + log(x + log(x) - 1) + 2)

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method	result	size

risch	\(4 x \ln \relax (2)+6 x -2 \ln \left (\ln \left (-1+\ln \relax (x )+x \right )+2 x +2\right ) \ln \relax (2)-3 \ln \left (\ln \left (-1+\ln \relax (x )+x \right )+2 x +2\right )\)	\(40\)

int((((4*x*ln(2)+6*x)*ln(x)+2*(2*x^2-2*x)*ln(2)+6*x^2-6*x)*ln(-1+ln(x)+x)+(2*(4*x^2+2*x)*ln(2)+12*x^2+6*x)*ln(

x)+2*(4*x^3-2*x^2-3*x-1)*ln(2)+12*x^3-6*x^2-9*x-3)/((x*ln(x)+x^2-x)*ln(-1+ln(x)+x)+(2*x^2+2*x)*ln(x)+2*x^3-2*x

4*x*ln(2)+6*x-2*ln(ln(-1+ln(x)+x)+2*x+2)*ln(2)-3*ln(ln(-1+ln(x)+x)+2*x+2)

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maxima [A] time = 0.51, size = 30, normalized size = 1.15 \begin {gather*} 2 \, x {\left (2 \, \log \relax (2) + 3\right )} - {\left (2 \, \log \relax (2) + 3\right )} \log \left (2 \, x + \log \left (x + \log \relax (x) - 1\right ) + 2\right ) \end {gather*}

integrate((((4*x*log(2)+6*x)*log(x)+2*(2*x^2-2*x)*log(2)+6*x^2-6*x)*log(-1+log(x)+x)+(2*(4*x^2+2*x)*log(2)+12*

x^2+6*x)*log(x)+2*(4*x^3-2*x^2-3*x-1)*log(2)+12*x^3-6*x^2-9*x-3)/((x*log(x)+x^2-x)*log(-1+log(x)+x)+(2*x^2+2*x

)*log(x)+2*x^3-2*x),x, algorithm="maxima")

2*x*(2*log(2) + 3) - (2*log(2) + 3)*log(2*x + log(x + log(x) - 1) + 2)

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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} -\int \frac {9\,x+2\,\ln \relax (2)\,\left (-4\,x^3+2\,x^2+3\,x+1\right )+6\,x^2-12\,x^3-\ln \relax (x)\,\left (6\,x+2\,\ln \relax (2)\,\left (4\,x^2+2\,x\right )+12\,x^2\right )+\ln \left (x+\ln \relax (x)-1\right )\,\left (6\,x+2\,\ln \relax (2)\,\left (2\,x-2\,x^2\right )-\ln \relax (x)\,\left (6\,x+4\,x\,\ln \relax (2)\right )-6\,x^2\right )+3}{\ln \left (x+\ln \relax (x)-1\right )\,\left (x\,\ln \relax (x)-x+x^2\right )-2\,x+\ln \relax (x)\,\left (2\,x^2+2\,x\right )+2\,x^3} \,d x \end {gather*}

int(-(9*x + 2*log(2)*(3*x + 2*x^2 - 4*x^3 + 1) + 6*x^2 - 12*x^3 - log(x)*(6*x + 2*log(2)*(2*x + 4*x^2) + 12*x^

2) + log(x + log(x) - 1)*(6*x + 2*log(2)*(2*x - 2*x^2) - log(x)*(6*x + 4*x*log(2)) - 6*x^2) + 3)/(log(x + log(

x) - 1)*(x*log(x) - x + x^2) - 2*x + log(x)*(2*x + 2*x^2) + 2*x^3),x)

-int((9*x + 2*log(2)*(3*x + 2*x^2 - 4*x^3 + 1) + 6*x^2 - 12*x^3 - log(x)*(6*x + 2*log(2)*(2*x + 4*x^2) + 12*x^

2) + log(x + log(x) - 1)*(6*x + 2*log(2)*(2*x - 2*x^2) - log(x)*(6*x + 4*x*log(2)) - 6*x^2) + 3)/(log(x + log(

x) - 1)*(x*log(x) - x + x^2) - 2*x + log(x)*(2*x + 2*x^2) + 2*x^3), x)

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sympy [A] time = 0.60, size = 31, normalized size = 1.19 \begin {gather*} x \left (4 \log {\relax (2 )} + 6\right ) + \left (-3 - 2 \log {\relax (2 )}\right ) \log {\left (2 x + \log {\left (x + \log {\relax (x )} - 1 \right )} + 2 \right )} \end {gather*}

integrate((((4*x*ln(2)+6*x)*ln(x)+2*(2*x**2-2*x)*ln(2)+6*x**2-6*x)*ln(-1+ln(x)+x)+(2*(4*x**2+2*x)*ln(2)+12*x**

2+6*x)*ln(x)+2*(4*x**3-2*x**2-3*x-1)*ln(2)+12*x**3-6*x**2-9*x-3)/((x*ln(x)+x**2-x)*ln(-1+ln(x)+x)+(2*x**2+2*x)

x*(4*log(2) + 6) + (-3 - 2*log(2))*log(2*x + log(x + log(x) - 1) + 2)

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3.42.31 \(\int \frac {-3-9 x-6 x^2+12 x^3+(-1-3 x-2 x^2+4 x^3) \log (4)+(6 x+12 x^2+(2 x+4 x^2) \log (4)) \log (x)+(-6 x+6 x^2+(-2 x+2 x^2) \log (4)+(6 x+2 x \log (4)) \log (x)) \log (-1+x+\log (x))}{-2 x+2 x^3+(2 x+2 x^2) \log (x)+(-x+x^2+x \log (x)) \log (-1+x+\log (x))} \, dx\)