∫ (1−x) log(5)+(−4x+2x2+(4+2x−2x2) log(5)) log(−2+x)+(2−x) log(5) log(−2+x) log(log(−2+x))____________________________________________________________________

Optimal. Leaf size=31 \[ 2-2 x-x^2+\frac {1+x^2}{\log (5)}+(1-x) \log (\log (-2+x)) \]

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Rubi [A] time = 0.27, antiderivative size = 45, normalized size of antiderivative = 1.45, number of steps used = 13, number of rules used = 9, integrand size = 69, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {12, 6688, 6, 2411, 2353, 2298, 2302, 29, 2520} \begin {gather*} \frac {x^2 (2-\log (25))}{2 \log (5)}-\frac {x \log (25)}{\log (5)}+(2-x) \log (\log (x-2))-\log (\log (x-2)) \end {gather*}

Int[((1 - x)*Log[5] + (-4*x + 2*x^2 + (4 + 2*x - 2*x^2)*Log[5])*Log[-2 + x] + (2 - x)*Log[5]*Log[-2 + x]*Log[L

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]

:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[

{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))

^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,

d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[

Int[Log[Log[(d_.)*(x_)^(n_.)]^(p_.)*(c_.)], x_Symbol] :> Simp[x*Log[c*Log[d*x^n]^p], x] - Dist[n*p, Int[1/Log[

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {(1-x) \log (5)+\left (-4 x+2 x^2+\left (4+2 x-2 x^2\right ) \log (5)\right ) \log (-2+x)+(2-x) \log (5) \log (-2+x) \log (\log (-2+x))}{(-2+x) \log (-2+x)} \, dx}{\log (5)}\\ &=\frac {\int \left (2 x-\log (25)-x \log (25)-\frac {(-1+x) \log (5)}{(-2+x) \log (-2+x)}-\log (5) \log (\log (-2+x))\right ) \, dx}{\log (5)}\\ &=\frac {\int \left (x (2-\log (25))-\log (25)-\frac {(-1+x) \log (5)}{(-2+x) \log (-2+x)}-\log (5) \log (\log (-2+x))\right ) \, dx}{\log (5)}\\ &=\frac {x^2 (2-\log (25))}{2 \log (5)}-\frac {x \log (25)}{\log (5)}-\int \frac {-1+x}{(-2+x) \log (-2+x)} \, dx-\int \log (\log (-2+x)) \, dx\\ &=\frac {x^2 (2-\log (25))}{2 \log (5)}-\frac {x \log (25)}{\log (5)}-\operatorname {Subst}\left (\int \frac {1+x}{x \log (x)} \, dx,x,-2+x\right )-\operatorname {Subst}(\int \log (\log (x)) \, dx,x,-2+x)\\ &=\frac {x^2 (2-\log (25))}{2 \log (5)}-\frac {x \log (25)}{\log (5)}-(-2+x) \log (\log (-2+x))-\operatorname {Subst}\left (\int \left (\frac {1}{\log (x)}+\frac {1}{x \log (x)}\right ) \, dx,x,-2+x\right )+\operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,-2+x\right )\\ &=\frac {x^2 (2-\log (25))}{2 \log (5)}-\frac {x \log (25)}{\log (5)}-(-2+x) \log (\log (-2+x))+\text {li}(-2+x)-\operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,-2+x\right )-\operatorname {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,-2+x\right )\\ &=\frac {x^2 (2-\log (25))}{2 \log (5)}-\frac {x \log (25)}{\log (5)}-(-2+x) \log (\log (-2+x))-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (-2+x)\right )\\ &=\frac {x^2 (2-\log (25))}{2 \log (5)}-\frac {x \log (25)}{\log (5)}-\log (\log (-2+x))-(-2+x) \log (\log (-2+x))\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.07, size = 50, normalized size = 1.61 \begin {gather*} -\frac {(-2+x)^2 (-2+\log (25))}{2 \log (5)}-\frac {(-2+x) (-4+3 \log (25))}{\log (5)}-\log (\log (-2+x))-(-2+x) \log (\log (-2+x)) \end {gather*}

Integrate[((1 - x)*Log[5] + (-4*x + 2*x^2 + (4 + 2*x - 2*x^2)*Log[5])*Log[-2 + x] + (2 - x)*Log[5]*Log[-2 + x]

-1/2*((-2 + x)^2*(-2 + Log[25]))/Log[5] - ((-2 + x)*(-4 + 3*Log[25]))/Log[5] - Log[Log[-2 + x]] - (-2 + x)*Log

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fricas [A] time = 1.00, size = 33, normalized size = 1.06 \begin {gather*} -\frac {{\left (x - 1\right )} \log \relax (5) \log \left (\log \left (x - 2\right )\right ) - x^{2} + {\left (x^{2} + 2 \, x\right )} \log \relax (5)}{\log \relax (5)} \end {gather*}

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

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giac [A] time = 0.17, size = 38, normalized size = 1.23 \begin {gather*} -\frac {x^{2} {\left (\log \relax (5) - 1\right )} + x \log \relax (5) \log \left (\log \left (x - 2\right )\right ) + 2 \, x \log \relax (5) - \log \relax (5) \log \left (\log \left (x - 2\right )\right )}{\log \relax (5)} \end {gather*}

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

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method	result	size

norman	\(\ln \left (\ln \left (x -2\right )\right )-2 x -x \ln \left (\ln \left (x -2\right )\right )-\frac {\left (\ln \relax (5)-1\right ) x^{2}}{\ln \relax (5)}\)	\(31\)
risch	\(-x \ln \left (\ln \left (x -2\right )\right )-x^{2}+\ln \left (\ln \left (x -2\right )\right )-2 x +\frac {x^{2}}{\ln \relax (5)}\)	\(31\)
default	\(\frac {-\ln \relax (5) \ln \left (\ln \left (x -2\right )\right ) x -x^{2} \ln \relax (5)+\ln \relax (5) \ln \left (\ln \left (x -2\right )\right )-2 x \ln \relax (5)+x^{2}}{\ln \relax (5)}\)	\(40\)

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

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maxima [B] time = 0.50, size = 90, normalized size = 2.90 \begin {gather*} -\frac {x^{2} {\left (\log \relax (5) - 1\right )} - 6 \, \log \relax (5) \log \left (x - 2\right ) \log \left (\log \left (x - 2\right )\right ) + 4 \, {\left (\log \left (x - 2\right ) \log \left (\log \left (x - 2\right )\right ) - \log \left (x - 2\right )\right )} \log \relax (5) + 2 \, x \log \relax (5) + 4 \, \log \relax (5) \log \left (x - 2\right ) + {\left (x \log \relax (5) + 2 \, \log \relax (5) \log \left (x - 2\right )\right )} \log \left (\log \left (x - 2\right )\right ) - \log \relax (5) \log \left (\log \left (x - 2\right )\right )}{\log \relax (5)} \end {gather*}

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

-(x^2*(log(5) - 1) - 6*log(5)*log(x - 2)*log(log(x - 2)) + 4*(log(x - 2)*log(log(x - 2)) - log(x - 2))*log(5)

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

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mupad [B] time = 4.69, size = 34, normalized size = 1.10 \begin {gather*} -2\,x-\ln \left (\ln \left (x-2\right )\right )-\ln \left (\ln \left (x-2\right )\right )\,\left (x-2\right )-\frac {x^2\,\left (\ln \left (25\right )-2\right )}{2\,\ln \relax (5)} \end {gather*}

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

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sympy [A] time = 0.46, size = 24, normalized size = 0.77 \begin {gather*} \frac {x^{2} \left (1 - \log {\relax (5 )}\right )}{\log {\relax (5 )}} - 2 x + \left (1 - x\right ) \log {\left (\log {\left (x - 2 \right )} \right )} \end {gather*}

integrate(((2-x)*ln(5)*ln(x-2)*ln(ln(x-2))+((-2*x**2+2*x+4)*ln(5)+2*x**2-4*x)*ln(x-2)+(-x+1)*ln(5))/(x-2)/ln(5

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3.70.79 \(\int \frac {(1-x) \log (5)+(-4 x+2 x^2+(4+2 x-2 x^2) \log (5)) \log (-2+x)+(2-x) \log (5) \log (-2+x) \log (\log (-2+x))}{(-2+x) \log (5) \log (-2+x)} \, dx\)