3.26.76 \(\int \frac {-16 x^2+(-16 x-32 x^2) \log (2)+(16 x+32 x^2) \log (3)+(8 x+(8+16 x) \log (2)+(-8-16 x) \log (3)) \log (5)+(8 x^2+(24 x+16 x^2) \log (2)+(-24 x-16 x^2) \log (3)+(-8 x+(-16-16 x) \log (2)+(16+16 x) \log (3)) \log (5)) \log (-x+\log (5))+(-8 x \log (2)+8 x \log (3)+(8 \log (2)-8 \log (3)) \log (5)) \log ^2(-x+\log (5))}{x^4 \log (2)-x^4 \log (3)+(-x^3 \log (2)+x^3 \log (3)) \log (5)} \, dx\)

Optimal. Leaf size=29 \[ \left (-4+\frac {2 \left (-1+\frac {x}{-\log (2)+\log (3)}+\log (-x+\log (5))\right )}{x}\right )^2 \]

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Rubi [B]  time = 1.09, antiderivative size = 317, normalized size of antiderivative = 10.93, number of steps used = 29, number of rules used = 22, integrand size = 181, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {6, 6688, 12, 6742, 142, 2418, 2395, 44, 2392, 2391, 2390, 2301, 36, 29, 31, 2398, 2411, 2347, 2344, 2316, 2315, 2314} \begin {gather*} \frac {4}{x^2}+\frac {4 \log ^2(\log (5)-x)}{x^2}-\frac {4 \log \left (\frac {9}{4}\right ) \log (\log (5)-x)}{x^2 \log \left (\frac {3}{2}\right )}+\frac {8 (x-\log (5)) \log (\log (5)-x)}{x \log ^2(5)}-\frac {8 \left (\log \left (\frac {9}{4}\right ) \log (5)-\log \left (\frac {15}{2}\right )\right ) \log (x)}{\log \left (\frac {3}{2}\right ) \log ^2(5)}-\frac {8 \left (\log \left (\frac {10}{3}\right )-\log \left (\frac {9}{4}\right ) \log (5)\right ) \log (x)}{\log \left (\frac {3}{2}\right ) \log ^2(5)}-\frac {4 \log \left (\frac {9}{4}\right ) \log (x)}{\log \left (\frac {3}{2}\right ) \log ^2(5)}-\frac {8 \log (x)}{\log ^2(5)}+\frac {8 \left (\log \left (\frac {9}{4}\right ) \log (5)-\log \left (\frac {15}{2}\right )\right ) \log (x-\log (5))}{\log \left (\frac {3}{2}\right ) \log ^2(5)}+\frac {8 \left (\log \left (\frac {10}{3}\right )-\log \left (\frac {9}{4}\right ) \log (5)\right ) \log (x-\log (5))}{\log \left (\frac {3}{2}\right ) \log ^2(5)}+\frac {4 \log \left (\frac {9}{4}\right ) \log (x-\log (5))}{\log \left (\frac {3}{2}\right ) \log ^2(5)}-\frac {8 \left (\log \left (\frac {9}{4}\right ) \log (5)-\log \left (\frac {15}{2}\right )\right ) \log (\log (5)-x)}{x \log \left (\frac {3}{2}\right ) \log (5)}+\frac {8 \left (\log \left (\frac {9}{4}\right ) \log (5)-\log \left (\frac {15}{2}\right )\right )}{x \log \left (\frac {3}{2}\right ) \log (5)}+\frac {4 \log \left (\frac {9}{4}\right )}{x \log \left (\frac {3}{2}\right ) \log (5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-16*x^2 + (-16*x - 32*x^2)*Log[2] + (16*x + 32*x^2)*Log[3] + (8*x + (8 + 16*x)*Log[2] + (-8 - 16*x)*Log[3
])*Log[5] + (8*x^2 + (24*x + 16*x^2)*Log[2] + (-24*x - 16*x^2)*Log[3] + (-8*x + (-16 - 16*x)*Log[2] + (16 + 16
*x)*Log[3])*Log[5])*Log[-x + Log[5]] + (-8*x*Log[2] + 8*x*Log[3] + (8*Log[2] - 8*Log[3])*Log[5])*Log[-x + Log[
5]]^2)/(x^4*Log[2] - x^4*Log[3] + (-(x^3*Log[2]) + x^3*Log[3])*Log[5]),x]

[Out]

4/x^2 + (4*Log[9/4])/(x*Log[3/2]*Log[5]) + (8*(Log[9/4]*Log[5] - Log[15/2]))/(x*Log[3/2]*Log[5]) - (8*Log[x])/
Log[5]^2 - (4*Log[9/4]*Log[x])/(Log[3/2]*Log[5]^2) - (8*(Log[10/3] - Log[9/4]*Log[5])*Log[x])/(Log[3/2]*Log[5]
^2) - (8*(Log[9/4]*Log[5] - Log[15/2])*Log[x])/(Log[3/2]*Log[5]^2) + (4*Log[9/4]*Log[x - Log[5]])/(Log[3/2]*Lo
g[5]^2) + (8*(Log[10/3] - Log[9/4]*Log[5])*Log[x - Log[5]])/(Log[3/2]*Log[5]^2) + (8*(Log[9/4]*Log[5] - Log[15
/2])*Log[x - Log[5]])/(Log[3/2]*Log[5]^2) - (4*Log[9/4]*Log[-x + Log[5]])/(x^2*Log[3/2]) + (8*(x - Log[5])*Log
[-x + Log[5]])/(x*Log[5]^2) - (8*(Log[9/4]*Log[5] - Log[15/2])*Log[-x + Log[5]])/(x*Log[3/2]*Log[5]) + (4*Log[
-x + Log[5]]^2)/x^2

Rule 6

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

Rule 12

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 142

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)*(g + h*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h},
x] && (IGtQ[m, 0] || IntegersQ[m, n])

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2314

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[(x*(d + e*x^r)^(q
+ 1)*(a + b*Log[c*x^n]))/d, x] - Dist[(b*n)/d, Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2316

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[((a + b*Log[-((c*d)/e)])*Log[d + e*
x])/e, x] + Dist[b, Int[Log[-((e*x)/d)]/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[-((c*d)/e), 0]

Rule 2344

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Dist[1/d, Int[(a + b*
Log[c*x^n])^p/x, x], x] - Dist[e/d, Int[(a + b*Log[c*x^n])^p/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, n}, x]
&& IGtQ[p, 0]

Rule 2347

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[((
d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p)/x, x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2392

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*d])*Log[x], x] + Dist[
b, Int[Log[1 + (e*x)/d]/x, x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[c*d, 0]

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 2398

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((
f + g*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^p)/(g*(q + 1)), x] - Dist[(b*e*n*p)/(g*(q + 1)), Int[((f + g*x)^(q
+ 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*
f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2411

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[
r, 0]) && IntegerQ[2*r]

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 6688

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-16 x^2+\left (-16 x-32 x^2\right ) \log (2)+\left (16 x+32 x^2\right ) \log (3)+(8 x+(8+16 x) \log (2)+(-8-16 x) \log (3)) \log (5)+\left (8 x^2+\left (24 x+16 x^2\right ) \log (2)+\left (-24 x-16 x^2\right ) \log (3)+(-8 x+(-16-16 x) \log (2)+(16+16 x) \log (3)) \log (5)\right ) \log (-x+\log (5))+(-8 x \log (2)+8 x \log (3)+(8 \log (2)-8 \log (3)) \log (5)) \log ^2(-x+\log (5))}{x^4 (\log (2)-\log (3))+\left (-x^3 \log (2)+x^3 \log (3)\right ) \log (5)} \, dx\\ &=\int \frac {8 \left (\log \left (\frac {3}{2}\right )+x \left (-1+\log \left (\frac {9}{4}\right )\right )-\log \left (\frac {3}{2}\right ) \log (-x+\log (5))\right ) (-2 x+\log (5)+(x-\log (5)) \log (-x+\log (5)))}{x^3 \log \left (\frac {3}{2}\right ) (x-\log (5))} \, dx\\ &=\frac {8 \int \frac {\left (\log \left (\frac {3}{2}\right )+x \left (-1+\log \left (\frac {9}{4}\right )\right )-\log \left (\frac {3}{2}\right ) \log (-x+\log (5))\right ) (-2 x+\log (5)+(x-\log (5)) \log (-x+\log (5)))}{x^3 (x-\log (5))} \, dx}{\log \left (\frac {3}{2}\right )}\\ &=\frac {8 \int \left (\frac {\left (\log \left (\frac {3}{2}\right )-x \left (1-\log \left (\frac {9}{4}\right )\right )\right ) (-2 x+\log (5))}{x^3 (x-\log (5))}+\frac {\left (-x^2 \left (1-\log \left (\frac {9}{4}\right )\right )-\log \left (\frac {9}{4}\right ) \log (5)-x \left (\log \left (\frac {9}{4}\right ) \log (5)-\log \left (\frac {135}{8}\right )\right )\right ) \log (-x+\log (5))}{x^3 (x-\log (5))}-\frac {\log \left (\frac {3}{2}\right ) \log ^2(-x+\log (5))}{x^3}\right ) \, dx}{\log \left (\frac {3}{2}\right )}\\ &=-\left (8 \int \frac {\log ^2(-x+\log (5))}{x^3} \, dx\right )+\frac {8 \int \frac {\left (\log \left (\frac {3}{2}\right )+x \left (-1+\log \left (\frac {9}{4}\right )\right )\right ) (-2 x+\log (5))}{x^3 (x-\log (5))} \, dx}{\log \left (\frac {3}{2}\right )}+\frac {8 \int \frac {\left (-x^2 \left (1-\log \left (\frac {9}{4}\right )\right )-\log \left (\frac {9}{4}\right ) \log (5)-x \left (\log \left (\frac {9}{4}\right ) \log (5)-\log \left (\frac {135}{8}\right )\right )\right ) \log (-x+\log (5))}{x^3 (x-\log (5))} \, dx}{\log \left (\frac {3}{2}\right )}\\ &=\frac {4 \log ^2(-x+\log (5))}{x^2}+8 \int \frac {\log (-x+\log (5))}{x^2 (-x+\log (5))} \, dx+\frac {8 \int \left (-\frac {\log \left (\frac {3}{2}\right )}{x^3}+\frac {\log \left (\frac {10}{3}\right )-\log \left (\frac {9}{4}\right ) \log (5)}{(x-\log (5)) \log ^2(5)}+\frac {-\log \left (\frac {10}{3}\right )+\log \left (\frac {9}{4}\right ) \log (5)}{x \log ^2(5)}+\frac {-\log \left (\frac {9}{4}\right ) \log (5)+\log \left (\frac {15}{2}\right )}{x^2 \log (5)}\right ) \, dx}{\log \left (\frac {3}{2}\right )}+\frac {8 \int \left (\frac {\log \left (\frac {9}{4}\right ) \log (-x+\log (5))}{x^3}-\frac {\log \left (\frac {3}{2}\right ) \log (-x+\log (5))}{x \log ^2(5)}+\frac {\log \left (\frac {3}{2}\right ) \log (-x+\log (5))}{(x-\log (5)) \log ^2(5)}+\frac {\left (\log \left (\frac {9}{4}\right ) \log (5)-\log \left (\frac {15}{2}\right )\right ) \log (-x+\log (5))}{x^2 \log (5)}\right ) \, dx}{\log \left (\frac {3}{2}\right )}\\ &=\frac {4}{x^2}+\frac {8 \left (\log \left (\frac {9}{4}\right ) \log (5)-\log \left (\frac {15}{2}\right )\right )}{x \log \left (\frac {3}{2}\right ) \log (5)}-\frac {8 \left (\log \left (\frac {10}{3}\right )-\log \left (\frac {9}{4}\right ) \log (5)\right ) \log (x)}{\log \left (\frac {3}{2}\right ) \log ^2(5)}+\frac {8 \left (\log \left (\frac {10}{3}\right )-\log \left (\frac {9}{4}\right ) \log (5)\right ) \log (x-\log (5))}{\log \left (\frac {3}{2}\right ) \log ^2(5)}+\frac {4 \log ^2(-x+\log (5))}{x^2}-8 \operatorname {Subst}\left (\int \frac {\log (x)}{x (-x+\log (5))^2} \, dx,x,-x+\log (5)\right )+\frac {\left (8 \log \left (\frac {9}{4}\right )\right ) \int \frac {\log (-x+\log (5))}{x^3} \, dx}{\log \left (\frac {3}{2}\right )}-\frac {8 \int \frac {\log (-x+\log (5))}{x} \, dx}{\log ^2(5)}+\frac {8 \int \frac {\log (-x+\log (5))}{x-\log (5)} \, dx}{\log ^2(5)}+\frac {\left (8 \left (\log \left (\frac {9}{4}\right ) \log (5)-\log \left (\frac {15}{2}\right )\right )\right ) \int \frac {\log (-x+\log (5))}{x^2} \, dx}{\log \left (\frac {3}{2}\right ) \log (5)}\\ &=\frac {4}{x^2}+\frac {8 \left (\log \left (\frac {9}{4}\right ) \log (5)-\log \left (\frac {15}{2}\right )\right )}{x \log \left (\frac {3}{2}\right ) \log (5)}-\frac {8 \left (\log \left (\frac {10}{3}\right )-\log \left (\frac {9}{4}\right ) \log (5)\right ) \log (x)}{\log \left (\frac {3}{2}\right ) \log ^2(5)}+\frac {8 \left (\log \left (\frac {10}{3}\right )-\log \left (\frac {9}{4}\right ) \log (5)\right ) \log (x-\log (5))}{\log \left (\frac {3}{2}\right ) \log ^2(5)}-\frac {8 \log (x) \log (\log (5))}{\log ^2(5)}-\frac {4 \log \left (\frac {9}{4}\right ) \log (-x+\log (5))}{x^2 \log \left (\frac {3}{2}\right )}-\frac {8 \left (\log \left (\frac {9}{4}\right ) \log (5)-\log \left (\frac {15}{2}\right )\right ) \log (-x+\log (5))}{x \log \left (\frac {3}{2}\right ) \log (5)}+\frac {4 \log ^2(-x+\log (5))}{x^2}-\frac {\left (4 \log \left (\frac {9}{4}\right )\right ) \int \frac {1}{x^2 (-x+\log (5))} \, dx}{\log \left (\frac {3}{2}\right )}-\frac {8 \int \frac {\log \left (1-\frac {x}{\log (5)}\right )}{x} \, dx}{\log ^2(5)}+\frac {8 \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-x+\log (5)\right )}{\log ^2(5)}-\frac {8 \operatorname {Subst}\left (\int \frac {\log (x)}{(-x+\log (5))^2} \, dx,x,-x+\log (5)\right )}{\log (5)}-\frac {8 \operatorname {Subst}\left (\int \frac {\log (x)}{x (-x+\log (5))} \, dx,x,-x+\log (5)\right )}{\log (5)}-\frac {\left (8 \left (\log \left (\frac {9}{4}\right ) \log (5)-\log \left (\frac {15}{2}\right )\right )\right ) \int \frac {1}{x (-x+\log (5))} \, dx}{\log \left (\frac {3}{2}\right ) \log (5)}\\ &=\frac {4}{x^2}+\frac {8 \left (\log \left (\frac {9}{4}\right ) \log (5)-\log \left (\frac {15}{2}\right )\right )}{x \log \left (\frac {3}{2}\right ) \log (5)}-\frac {8 \left (\log \left (\frac {10}{3}\right )-\log \left (\frac {9}{4}\right ) \log (5)\right ) \log (x)}{\log \left (\frac {3}{2}\right ) \log ^2(5)}+\frac {8 \left (\log \left (\frac {10}{3}\right )-\log \left (\frac {9}{4}\right ) \log (5)\right ) \log (x-\log (5))}{\log \left (\frac {3}{2}\right ) \log ^2(5)}-\frac {8 \log (x) \log (\log (5))}{\log ^2(5)}-\frac {4 \log \left (\frac {9}{4}\right ) \log (-x+\log (5))}{x^2 \log \left (\frac {3}{2}\right )}+\frac {8 (x-\log (5)) \log (-x+\log (5))}{x \log ^2(5)}-\frac {8 \left (\log \left (\frac {9}{4}\right ) \log (5)-\log \left (\frac {15}{2}\right )\right ) \log (-x+\log (5))}{x \log \left (\frac {3}{2}\right ) \log (5)}+\frac {4 \log ^2(-x+\log (5))}{x^2}+\frac {4 \log ^2(-x+\log (5))}{\log ^2(5)}+\frac {8 \text {Li}_2\left (\frac {x}{\log (5)}\right )}{\log ^2(5)}-\frac {\left (4 \log \left (\frac {9}{4}\right )\right ) \int \left (\frac {1}{x \log ^2(5)}-\frac {1}{(x-\log (5)) \log ^2(5)}+\frac {1}{x^2 \log (5)}\right ) \, dx}{\log \left (\frac {3}{2}\right )}+\frac {8 \operatorname {Subst}\left (\int \frac {1}{-x+\log (5)} \, dx,x,-x+\log (5)\right )}{\log ^2(5)}-\frac {8 \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-x+\log (5)\right )}{\log ^2(5)}-\frac {8 \operatorname {Subst}\left (\int \frac {\log (x)}{-x+\log (5)} \, dx,x,-x+\log (5)\right )}{\log ^2(5)}-\frac {\left (8 \left (\log \left (\frac {9}{4}\right ) \log (5)-\log \left (\frac {15}{2}\right )\right )\right ) \int \frac {1}{x} \, dx}{\log \left (\frac {3}{2}\right ) \log ^2(5)}-\frac {\left (8 \left (\log \left (\frac {9}{4}\right ) \log (5)-\log \left (\frac {15}{2}\right )\right )\right ) \int \frac {1}{-x+\log (5)} \, dx}{\log \left (\frac {3}{2}\right ) \log ^2(5)}\\ &=\frac {4}{x^2}+\frac {4 \log \left (\frac {9}{4}\right )}{x \log \left (\frac {3}{2}\right ) \log (5)}+\frac {8 \left (\log \left (\frac {9}{4}\right ) \log (5)-\log \left (\frac {15}{2}\right )\right )}{x \log \left (\frac {3}{2}\right ) \log (5)}-\frac {8 \log (x)}{\log ^2(5)}-\frac {4 \log \left (\frac {9}{4}\right ) \log (x)}{\log \left (\frac {3}{2}\right ) \log ^2(5)}-\frac {8 \left (\log \left (\frac {10}{3}\right )-\log \left (\frac {9}{4}\right ) \log (5)\right ) \log (x)}{\log \left (\frac {3}{2}\right ) \log ^2(5)}-\frac {8 \left (\log \left (\frac {9}{4}\right ) \log (5)-\log \left (\frac {15}{2}\right )\right ) \log (x)}{\log \left (\frac {3}{2}\right ) \log ^2(5)}+\frac {4 \log \left (\frac {9}{4}\right ) \log (x-\log (5))}{\log \left (\frac {3}{2}\right ) \log ^2(5)}+\frac {8 \left (\log \left (\frac {10}{3}\right )-\log \left (\frac {9}{4}\right ) \log (5)\right ) \log (x-\log (5))}{\log \left (\frac {3}{2}\right ) \log ^2(5)}+\frac {8 \left (\log \left (\frac {9}{4}\right ) \log (5)-\log \left (\frac {15}{2}\right )\right ) \log (x-\log (5))}{\log \left (\frac {3}{2}\right ) \log ^2(5)}-\frac {4 \log \left (\frac {9}{4}\right ) \log (-x+\log (5))}{x^2 \log \left (\frac {3}{2}\right )}+\frac {8 (x-\log (5)) \log (-x+\log (5))}{x \log ^2(5)}-\frac {8 \left (\log \left (\frac {9}{4}\right ) \log (5)-\log \left (\frac {15}{2}\right )\right ) \log (-x+\log (5))}{x \log \left (\frac {3}{2}\right ) \log (5)}+\frac {4 \log ^2(-x+\log (5))}{x^2}+\frac {8 \text {Li}_2\left (\frac {x}{\log (5)}\right )}{\log ^2(5)}-\frac {8 \operatorname {Subst}\left (\int \frac {\log \left (\frac {x}{\log (5)}\right )}{-x+\log (5)} \, dx,x,-x+\log (5)\right )}{\log ^2(5)}\\ &=\frac {4}{x^2}+\frac {4 \log \left (\frac {9}{4}\right )}{x \log \left (\frac {3}{2}\right ) \log (5)}+\frac {8 \left (\log \left (\frac {9}{4}\right ) \log (5)-\log \left (\frac {15}{2}\right )\right )}{x \log \left (\frac {3}{2}\right ) \log (5)}-\frac {8 \log (x)}{\log ^2(5)}-\frac {4 \log \left (\frac {9}{4}\right ) \log (x)}{\log \left (\frac {3}{2}\right ) \log ^2(5)}-\frac {8 \left (\log \left (\frac {10}{3}\right )-\log \left (\frac {9}{4}\right ) \log (5)\right ) \log (x)}{\log \left (\frac {3}{2}\right ) \log ^2(5)}-\frac {8 \left (\log \left (\frac {9}{4}\right ) \log (5)-\log \left (\frac {15}{2}\right )\right ) \log (x)}{\log \left (\frac {3}{2}\right ) \log ^2(5)}+\frac {4 \log \left (\frac {9}{4}\right ) \log (x-\log (5))}{\log \left (\frac {3}{2}\right ) \log ^2(5)}+\frac {8 \left (\log \left (\frac {10}{3}\right )-\log \left (\frac {9}{4}\right ) \log (5)\right ) \log (x-\log (5))}{\log \left (\frac {3}{2}\right ) \log ^2(5)}+\frac {8 \left (\log \left (\frac {9}{4}\right ) \log (5)-\log \left (\frac {15}{2}\right )\right ) \log (x-\log (5))}{\log \left (\frac {3}{2}\right ) \log ^2(5)}-\frac {4 \log \left (\frac {9}{4}\right ) \log (-x+\log (5))}{x^2 \log \left (\frac {3}{2}\right )}+\frac {8 (x-\log (5)) \log (-x+\log (5))}{x \log ^2(5)}-\frac {8 \left (\log \left (\frac {9}{4}\right ) \log (5)-\log \left (\frac {15}{2}\right )\right ) \log (-x+\log (5))}{x \log \left (\frac {3}{2}\right ) \log (5)}+\frac {4 \log ^2(-x+\log (5))}{x^2}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.55, size = 206, normalized size = 7.10 \begin {gather*} \frac {4 \left (\frac {\log \left (\frac {3}{2}\right )}{x^2}+\frac {2 \log \left (\frac {9}{4}\right )}{x}-\frac {\log \left (\frac {3}{2}\right )}{\log ^2(5)}+\frac {2 \log ^2\left (\frac {10}{3}\right )}{\log \left (\frac {9}{4}\right ) \log ^2(5)}+\frac {\log \left (\frac {9}{4}\right )}{x \log (5)}-\frac {4 \log \left (\frac {10}{3}\right )}{\log (5)}+\log \left (\frac {81}{16}\right )-\frac {2 \log \left (\frac {15}{2}\right )}{x \log (5)}+\frac {\left (-1+\log \left (\frac {9}{4}\right )\right ) \log (25) \log (x-\log (5))}{\log ^2(5)}+\left (-\frac {\log \left (\frac {9}{4}\right )}{x^2}+\frac {\log ^2\left (\frac {9}{4}\right )+4 \log \left (\frac {3}{2}\right ) \log \left (\frac {10}{3}\right )-4 \log \left (\frac {3}{2}\right ) \log \left (\frac {9}{4}\right ) \log (5)}{\log \left (\frac {9}{4}\right ) \log ^2(5)}+\frac {-\log \left (\frac {81}{16}\right )+\frac {\log (25)}{\log (5)}}{x}\right ) \log (-x+\log (5))+\frac {\log \left (\frac {3}{2}\right ) \log ^2(-x+\log (5))}{x^2}\right )}{\log \left (\frac {3}{2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-16*x^2 + (-16*x - 32*x^2)*Log[2] + (16*x + 32*x^2)*Log[3] + (8*x + (8 + 16*x)*Log[2] + (-8 - 16*x)
*Log[3])*Log[5] + (8*x^2 + (24*x + 16*x^2)*Log[2] + (-24*x - 16*x^2)*Log[3] + (-8*x + (-16 - 16*x)*Log[2] + (1
6 + 16*x)*Log[3])*Log[5])*Log[-x + Log[5]] + (-8*x*Log[2] + 8*x*Log[3] + (8*Log[2] - 8*Log[3])*Log[5])*Log[-x
+ Log[5]]^2)/(x^4*Log[2] - x^4*Log[3] + (-(x^3*Log[2]) + x^3*Log[3])*Log[5]),x]

[Out]

(4*(Log[3/2]/x^2 + (2*Log[9/4])/x - Log[3/2]/Log[5]^2 + (2*Log[10/3]^2)/(Log[9/4]*Log[5]^2) + Log[9/4]/(x*Log[
5]) - (4*Log[10/3])/Log[5] + Log[81/16] - (2*Log[15/2])/(x*Log[5]) + ((-1 + Log[9/4])*Log[25]*Log[x - Log[5]])
/Log[5]^2 + (-(Log[9/4]/x^2) + (Log[9/4]^2 + 4*Log[3/2]*Log[10/3] - 4*Log[3/2]*Log[9/4]*Log[5])/(Log[9/4]*Log[
5]^2) + (-Log[81/16] + Log[25]/Log[5])/x)*Log[-x + Log[5]] + (Log[3/2]*Log[-x + Log[5]]^2)/x^2))/Log[3/2]

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fricas [B]  time = 0.53, size = 86, normalized size = 2.97 \begin {gather*} \frac {4 \, {\left ({\left (\log \relax (3) - \log \relax (2)\right )} \log \left (-x + \log \relax (5)\right )^{2} + {\left (4 \, x + 1\right )} \log \relax (3) - {\left (4 \, x + 1\right )} \log \relax (2) - 2 \, {\left ({\left (2 \, x + 1\right )} \log \relax (3) - {\left (2 \, x + 1\right )} \log \relax (2) - x\right )} \log \left (-x + \log \relax (5)\right ) - 2 \, x\right )}}{x^{2} \log \relax (3) - x^{2} \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-8*log(3)+8*log(2))*log(5)+8*x*log(3)-8*x*log(2))*log(log(5)-x)^2+(((16*x+16)*log(3)+(-16*x-16)*l
og(2)-8*x)*log(5)+(-16*x^2-24*x)*log(3)+(16*x^2+24*x)*log(2)+8*x^2)*log(log(5)-x)+((-16*x-8)*log(3)+(16*x+8)*l
og(2)+8*x)*log(5)+(32*x^2+16*x)*log(3)+(-32*x^2-16*x)*log(2)-16*x^2)/((x^3*log(3)-x^3*log(2))*log(5)-x^4*log(3
)+x^4*log(2)),x, algorithm="fricas")

[Out]

4*((log(3) - log(2))*log(-x + log(5))^2 + (4*x + 1)*log(3) - (4*x + 1)*log(2) - 2*((2*x + 1)*log(3) - (2*x + 1
)*log(2) - x)*log(-x + log(5)) - 2*x)/(x^2*log(3) - x^2*log(2))

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giac [B]  time = 0.47, size = 98, normalized size = 3.38 \begin {gather*} -\frac {8 \, {\left (2 \, x \log \relax (3) - 2 \, x \log \relax (2) - x + \log \relax (3) - \log \relax (2)\right )} \log \left (-x + \log \relax (5)\right )}{x^{2} \log \relax (3) - x^{2} \log \relax (2)} + \frac {4 \, {\left (4 \, x \log \relax (3) - 4 \, x \log \relax (2) - 2 \, x + \log \relax (3) - \log \relax (2)\right )}}{x^{2} \log \relax (3) - x^{2} \log \relax (2)} + \frac {4 \, \log \left (-x + \log \relax (5)\right )^{2}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-8*log(3)+8*log(2))*log(5)+8*x*log(3)-8*x*log(2))*log(log(5)-x)^2+(((16*x+16)*log(3)+(-16*x-16)*l
og(2)-8*x)*log(5)+(-16*x^2-24*x)*log(3)+(16*x^2+24*x)*log(2)+8*x^2)*log(log(5)-x)+((-16*x-8)*log(3)+(16*x+8)*l
og(2)+8*x)*log(5)+(32*x^2+16*x)*log(3)+(-32*x^2-16*x)*log(2)-16*x^2)/((x^3*log(3)-x^3*log(2))*log(5)-x^4*log(3
)+x^4*log(2)),x, algorithm="giac")

[Out]

-8*(2*x*log(3) - 2*x*log(2) - x + log(3) - log(2))*log(-x + log(5))/(x^2*log(3) - x^2*log(2)) + 4*(4*x*log(3)
- 4*x*log(2) - 2*x + log(3) - log(2))/(x^2*log(3) - x^2*log(2)) + 4*log(-x + log(5))^2/x^2

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maple [B]  time = 0.48, size = 78, normalized size = 2.69

method	result	size
norman	\(\frac {4+4 \ln \left (\ln \relax (5)-x \right )^{2}+\frac {8 \left (-2 \ln \relax (3)+2 \ln \relax (2)+1\right ) x}{-\ln \relax (3)+\ln \relax (2)}-\frac {8 \left (-2 \ln \relax (3)+2 \ln \relax (2)+1\right ) x \ln \left (\ln \relax (5)-x \right )}{-\ln \relax (3)+\ln \relax (2)}-8 \ln \left (\ln \relax (5)-x \right )}{x^{2}}\)	\(78\)
risch	\(\frac {4 \ln \left (\ln \relax (5)-x \right )^{2}}{x^{2}}-\frac {8 \left (2 x \ln \relax (2)-2 x \ln \relax (3)+\ln \relax (2)-\ln \relax (3)+x \right ) \ln \left (\ln \relax (5)-x \right )}{\left (-\ln \relax (3)+\ln \relax (2)\right ) x^{2}}+\frac {16 x \ln \relax (2)-16 x \ln \relax (3)+4 \ln \relax (2)-4 \ln \relax (3)+8 x}{\left (-\ln \relax (3)+\ln \relax (2)\right ) x^{2}}\)	\(89\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-8*ln(3)+8*ln(2))*ln(5)+8*x*ln(3)-8*x*ln(2))*ln(ln(5)-x)^2+(((16*x+16)*ln(3)+(-16*x-16)*ln(2)-8*x)*ln(5
)+(-16*x^2-24*x)*ln(3)+(16*x^2+24*x)*ln(2)+8*x^2)*ln(ln(5)-x)+((-16*x-8)*ln(3)+(16*x+8)*ln(2)+8*x)*ln(5)+(32*x
^2+16*x)*ln(3)+(-32*x^2-16*x)*ln(2)-16*x^2)/((x^3*ln(3)-x^3*ln(2))*ln(5)-x^4*ln(3)+x^4*ln(2)),x,method=_RETURN
VERBOSE)

[Out]

(4+4*ln(ln(5)-x)^2+8*(-2*ln(3)+2*ln(2)+1)/(-ln(3)+ln(2))*x-8*(-2*ln(3)+2*ln(2)+1)/(-ln(3)+ln(2))*x*ln(ln(5)-x)
-8*ln(ln(5)-x))/x^2

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maxima [B]  time = 0.57, size = 295, normalized size = 10.17 \begin {gather*} 4 \, {\left (\frac {2 \, \log \left (x - \log \relax (5)\right )}{{\left (\log \relax (3) - \log \relax (2)\right )} \log \relax (5)^{3}} - \frac {2 \, \log \relax (x)}{{\left (\log \relax (3) - \log \relax (2)\right )} \log \relax (5)^{3}} + \frac {2 \, x + \log \relax (5)}{x^{2} {\left (\log \relax (3) - \log \relax (2)\right )} \log \relax (5)^{2}}\right )} \log \relax (5) \log \relax (3) - 4 \, {\left (\frac {2 \, \log \left (x - \log \relax (5)\right )}{{\left (\log \relax (3) - \log \relax (2)\right )} \log \relax (5)^{3}} - \frac {2 \, \log \relax (x)}{{\left (\log \relax (3) - \log \relax (2)\right )} \log \relax (5)^{3}} + \frac {2 \, x + \log \relax (5)}{x^{2} {\left (\log \relax (3) - \log \relax (2)\right )} \log \relax (5)^{2}}\right )} \log \relax (5) \log \relax (2) + \frac {8 \, \log \relax (x)}{\log \relax (5)^{2}} + \frac {4 \, {\left ({\left (\log \relax (5)^{2} \log \relax (3) - \log \relax (5)^{2} \log \relax (2)\right )} \log \left (-x + \log \relax (5)\right )^{2} - 2 \, {\left (\log \relax (5)^{2} - {\left (2 \, \log \relax (5)^{2} - \log \relax (5)\right )} \log \relax (3) + {\left (2 \, \log \relax (5)^{2} - \log \relax (5)\right )} \log \relax (2)\right )} x - 2 \, {\left (x^{2} {\left (\log \relax (3) - \log \relax (2)\right )} + \log \relax (5)^{2} \log \relax (3) - \log \relax (5)^{2} \log \relax (2) + {\left (2 \, \log \relax (5)^{2} \log \relax (3) - 2 \, \log \relax (5)^{2} \log \relax (2) - \log \relax (5)^{2}\right )} x\right )} \log \left (-x + \log \relax (5)\right )\right )}}{{\left (\log \relax (5)^{2} \log \relax (3) - \log \relax (5)^{2} \log \relax (2)\right )} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-8*log(3)+8*log(2))*log(5)+8*x*log(3)-8*x*log(2))*log(log(5)-x)^2+(((16*x+16)*log(3)+(-16*x-16)*l
og(2)-8*x)*log(5)+(-16*x^2-24*x)*log(3)+(16*x^2+24*x)*log(2)+8*x^2)*log(log(5)-x)+((-16*x-8)*log(3)+(16*x+8)*l
og(2)+8*x)*log(5)+(32*x^2+16*x)*log(3)+(-32*x^2-16*x)*log(2)-16*x^2)/((x^3*log(3)-x^3*log(2))*log(5)-x^4*log(3
)+x^4*log(2)),x, algorithm="maxima")

[Out]

4*(2*log(x - log(5))/((log(3) - log(2))*log(5)^3) - 2*log(x)/((log(3) - log(2))*log(5)^3) + (2*x + log(5))/(x^
2*(log(3) - log(2))*log(5)^2))*log(5)*log(3) - 4*(2*log(x - log(5))/((log(3) - log(2))*log(5)^3) - 2*log(x)/((
log(3) - log(2))*log(5)^3) + (2*x + log(5))/(x^2*(log(3) - log(2))*log(5)^2))*log(5)*log(2) + 8*log(x)/log(5)^
2 + 4*((log(5)^2*log(3) - log(5)^2*log(2))*log(-x + log(5))^2 - 2*(log(5)^2 - (2*log(5)^2 - log(5))*log(3) + (
2*log(5)^2 - log(5))*log(2))*x - 2*(x^2*(log(3) - log(2)) + log(5)^2*log(3) - log(5)^2*log(2) + (2*log(5)^2*lo
g(3) - 2*log(5)^2*log(2) - log(5)^2)*x)*log(-x + log(5)))/((log(5)^2*log(3) - log(5)^2*log(2))*x^2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {\ln \relax (5)\,\left (8\,x+\ln \relax (2)\,\left (16\,x+8\right )-\ln \relax (3)\,\left (16\,x+8\right )\right )-\ln \relax (2)\,\left (32\,x^2+16\,x\right )+\ln \relax (3)\,\left (32\,x^2+16\,x\right )+{\ln \left (\ln \relax (5)-x\right )}^2\,\left (8\,x\,\ln \relax (3)-8\,x\,\ln \relax (2)+\ln \relax (5)\,\left (8\,\ln \relax (2)-8\,\ln \relax (3)\right )\right )-\ln \left (\ln \relax (5)-x\right )\,\left (\ln \relax (5)\,\left (8\,x+\ln \relax (2)\,\left (16\,x+16\right )-\ln \relax (3)\,\left (16\,x+16\right )\right )-\ln \relax (2)\,\left (16\,x^2+24\,x\right )+\ln \relax (3)\,\left (16\,x^2+24\,x\right )-8\,x^2\right )-16\,x^2}{\ln \relax (5)\,\left (x^3\,\ln \relax (2)-x^3\,\ln \relax (3)\right )-x^4\,\ln \relax (2)+x^4\,\ln \relax (3)} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(5)*(8*x + log(2)*(16*x + 8) - log(3)*(16*x + 8)) - log(2)*(16*x + 32*x^2) + log(3)*(16*x + 32*x^2) +
log(log(5) - x)^2*(8*x*log(3) - 8*x*log(2) + log(5)*(8*log(2) - 8*log(3))) - log(log(5) - x)*(log(5)*(8*x + l
og(2)*(16*x + 16) - log(3)*(16*x + 16)) - log(2)*(24*x + 16*x^2) + log(3)*(24*x + 16*x^2) - 8*x^2) - 16*x^2)/(
log(5)*(x^3*log(2) - x^3*log(3)) - x^4*log(2) + x^4*log(3)),x)

[Out]

int(-(log(5)*(8*x + log(2)*(16*x + 8) - log(3)*(16*x + 8)) - log(2)*(16*x + 32*x^2) + log(3)*(16*x + 32*x^2) +
log(log(5) - x)^2*(8*x*log(3) - 8*x*log(2) + log(5)*(8*log(2) - 8*log(3))) - log(log(5) - x)*(log(5)*(8*x + l
og(2)*(16*x + 16) - log(3)*(16*x + 16)) - log(2)*(24*x + 16*x^2) + log(3)*(24*x + 16*x^2) - 8*x^2) - 16*x^2)/(
log(5)*(x^3*log(2) - x^3*log(3)) - x^4*log(2) + x^4*log(3)), x)

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sympy [B]  time = 0.55, size = 94, normalized size = 3.24 \begin {gather*} \frac {\left (- 16 x \log {\relax (2 )} - 8 x + 16 x \log {\relax (3 )} - 8 \log {\relax (2 )} + 8 \log {\relax (3 )}\right ) \log {\left (- x + \log {\relax (5 )} \right )}}{- x^{2} \log {\relax (3 )} + x^{2} \log {\relax (2 )}} - \frac {x \left (- 16 \log {\relax (2 )} - 8 + 16 \log {\relax (3 )}\right ) - 4 \log {\relax (2 )} + 4 \log {\relax (3 )}}{x^{2} \left (- \log {\relax (3 )} + \log {\relax (2 )}\right )} + \frac {4 \log {\left (- x + \log {\relax (5 )} \right )}^{2}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-8*ln(3)+8*ln(2))*ln(5)+8*x*ln(3)-8*x*ln(2))*ln(ln(5)-x)**2+(((16*x+16)*ln(3)+(-16*x-16)*ln(2)-8*
x)*ln(5)+(-16*x**2-24*x)*ln(3)+(16*x**2+24*x)*ln(2)+8*x**2)*ln(ln(5)-x)+((-16*x-8)*ln(3)+(16*x+8)*ln(2)+8*x)*l
n(5)+(32*x**2+16*x)*ln(3)+(-32*x**2-16*x)*ln(2)-16*x**2)/((x**3*ln(3)-x**3*ln(2))*ln(5)-x**4*ln(3)+x**4*ln(2))
,x)

[Out]

(-16*x*log(2) - 8*x + 16*x*log(3) - 8*log(2) + 8*log(3))*log(-x + log(5))/(-x**2*log(3) + x**2*log(2)) - (x*(-
16*log(2) - 8 + 16*log(3)) - 4*log(2) + 4*log(3))/(x**2*(-log(3) + log(2))) + 4*log(-x + log(5))**2/x**2

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