3.86.37 \(\int \frac {4 x^2+(-6 x+2 x^2) \log (\frac {2}{-3+x})+(6 x-2 x^2) \log ^5(\frac {2}{-3+x})}{(-9+3 x) \log ^5(\frac {2}{-3+x})} \, dx\)

Optimal. Leaf size=21 \[ \frac {1}{3} x \left (-x+\frac {x}{\log ^4\left (\frac {2}{-3+x}\right )}\right ) \]

________________________________________________________________________________________

Rubi [B]  time = 0.56, antiderivative size = 262, normalized size of antiderivative = 12.48, number of steps used = 44, number of rules used = 16, integrand size = 62, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {6688, 12, 14, 2411, 2353, 2297, 2299, 2178, 2302, 30, 2306, 2310, 2400, 2399, 2389, 2390} \begin {gather*} -\frac {x^2}{3}+\frac {(3-x)^2}{3 \log ^4\left (-\frac {2}{3-x}\right )}-\frac {2 (3-x)}{\log ^4\left (-\frac {2}{3-x}\right )}+\frac {3}{\log ^4\left (-\frac {2}{3-x}\right )}-\frac {2 (3-x)^2}{9 \log ^3\left (-\frac {2}{3-x}\right )}-\frac {2 x (3-x)}{9 \log ^3\left (-\frac {2}{3-x}\right )}+\frac {2 (3-x)}{3 \log ^3\left (-\frac {2}{3-x}\right )}+\frac {2 (3-x)^2}{9 \log ^2\left (-\frac {2}{3-x}\right )}+\frac {2 x (3-x)}{9 \log ^2\left (-\frac {2}{3-x}\right )}-\frac {2 (3-x)}{3 \log ^2\left (-\frac {2}{3-x}\right )}-\frac {4 (3-x)^2}{9 \log \left (-\frac {2}{3-x}\right )}-\frac {4 x (3-x)}{9 \log \left (-\frac {2}{3-x}\right )}+\frac {4 (3-x)}{3 \log \left (-\frac {2}{3-x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*x^2 + (-6*x + 2*x^2)*Log[2/(-3 + x)] + (6*x - 2*x^2)*Log[2/(-3 + x)]^5)/((-9 + 3*x)*Log[2/(-3 + x)]^5),
x]

[Out]

-1/3*x^2 + 3/Log[-2/(3 - x)]^4 - (2*(3 - x))/Log[-2/(3 - x)]^4 + (3 - x)^2/(3*Log[-2/(3 - x)]^4) + (2*(3 - x))
/(3*Log[-2/(3 - x)]^3) - (2*(3 - x)^2)/(9*Log[-2/(3 - x)]^3) - (2*(3 - x)*x)/(9*Log[-2/(3 - x)]^3) - (2*(3 - x
))/(3*Log[-2/(3 - x)]^2) + (2*(3 - x)^2)/(9*Log[-2/(3 - x)]^2) + (2*(3 - x)*x)/(9*Log[-2/(3 - x)]^2) + (4*(3 -
x))/(3*Log[-2/(3 - x)]) - (4*(3 - x)^2)/(9*Log[-2/(3 - x)]) - (4*(3 - x)*x)/(9*Log[-2/(3 - 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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2297

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

Rule 2299

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[1/(n*c^(1/n)), Subst[Int[E^(x/n)*(a + b*x)^p
, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[1/n]

Rule 2302

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

Rule 2306

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

Rule 2310

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2353

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
]))

Rule 2389

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

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 2399

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

Rule 2400

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

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 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2}{3} x \left (-1+\frac {2 x}{(-3+x) \log ^5\left (\frac {2}{-3+x}\right )}+\frac {1}{\log ^4\left (\frac {2}{-3+x}\right )}\right ) \, dx\\ &=\frac {2}{3} \int x \left (-1+\frac {2 x}{(-3+x) \log ^5\left (\frac {2}{-3+x}\right )}+\frac {1}{\log ^4\left (\frac {2}{-3+x}\right )}\right ) \, dx\\ &=\frac {2}{3} \int \left (-x+\frac {2 x^2}{(-3+x) \log ^5\left (\frac {2}{-3+x}\right )}+\frac {x}{\log ^4\left (\frac {2}{-3+x}\right )}\right ) \, dx\\ &=-\frac {x^2}{3}+\frac {2}{3} \int \frac {x}{\log ^4\left (\frac {2}{-3+x}\right )} \, dx+\frac {4}{3} \int \frac {x^2}{(-3+x) \log ^5\left (\frac {2}{-3+x}\right )} \, dx\\ &=-\frac {x^2}{3}-\frac {2 (3-x) x}{9 \log ^3\left (-\frac {2}{3-x}\right )}-\frac {4}{9} \int \frac {x}{\log ^3\left (\frac {2}{-3+x}\right )} \, dx+\frac {2}{3} \int \frac {1}{\log ^3\left (\frac {2}{-3+x}\right )} \, dx+\frac {4}{3} \operatorname {Subst}\left (\int \frac {(3+x)^2}{x \log ^5\left (\frac {2}{x}\right )} \, dx,x,-3+x\right )\\ &=-\frac {x^2}{3}-\frac {2 (3-x) x}{9 \log ^3\left (-\frac {2}{3-x}\right )}+\frac {2 (3-x) x}{9 \log ^2\left (-\frac {2}{3-x}\right )}+\frac {4}{9} \int \frac {x}{\log ^2\left (\frac {2}{-3+x}\right )} \, dx-\frac {2}{3} \int \frac {1}{\log ^2\left (\frac {2}{-3+x}\right )} \, dx+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{\log ^3\left (\frac {2}{x}\right )} \, dx,x,-3+x\right )+\frac {4}{3} \operatorname {Subst}\left (\int \left (\frac {6}{\log ^5\left (\frac {2}{x}\right )}+\frac {9}{x \log ^5\left (\frac {2}{x}\right )}+\frac {x}{\log ^5\left (\frac {2}{x}\right )}\right ) \, dx,x,-3+x\right )\\ &=-\frac {x^2}{3}-\frac {2 (3-x) x}{9 \log ^3\left (-\frac {2}{3-x}\right )}-\frac {3-x}{3 \log ^2\left (-\frac {2}{3-x}\right )}+\frac {2 (3-x) x}{9 \log ^2\left (-\frac {2}{3-x}\right )}-\frac {4 (3-x) x}{9 \log \left (-\frac {2}{3-x}\right )}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\log ^2\left (\frac {2}{x}\right )} \, dx,x,-3+x\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{\log ^2\left (\frac {2}{x}\right )} \, dx,x,-3+x\right )-\frac {8}{9} \int \frac {x}{\log \left (\frac {2}{-3+x}\right )} \, dx+\frac {4}{3} \int \frac {1}{\log \left (\frac {2}{-3+x}\right )} \, dx+\frac {4}{3} \operatorname {Subst}\left (\int \frac {x}{\log ^5\left (\frac {2}{x}\right )} \, dx,x,-3+x\right )+8 \operatorname {Subst}\left (\int \frac {1}{\log ^5\left (\frac {2}{x}\right )} \, dx,x,-3+x\right )+12 \operatorname {Subst}\left (\int \frac {1}{x \log ^5\left (\frac {2}{x}\right )} \, dx,x,-3+x\right )\\ &=-\frac {x^2}{3}-\frac {2 (3-x)}{\log ^4\left (-\frac {2}{3-x}\right )}+\frac {(3-x)^2}{3 \log ^4\left (-\frac {2}{3-x}\right )}-\frac {2 (3-x) x}{9 \log ^3\left (-\frac {2}{3-x}\right )}-\frac {3-x}{3 \log ^2\left (-\frac {2}{3-x}\right )}+\frac {2 (3-x) x}{9 \log ^2\left (-\frac {2}{3-x}\right )}+\frac {3-x}{\log \left (-\frac {2}{3-x}\right )}-\frac {4 (3-x) x}{9 \log \left (-\frac {2}{3-x}\right )}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\log \left (\frac {2}{x}\right )} \, dx,x,-3+x\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {x}{\log ^4\left (\frac {2}{x}\right )} \, dx,x,-3+x\right )+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{\log \left (\frac {2}{x}\right )} \, dx,x,-3+x\right )-\frac {8}{9} \int \left (\frac {3}{\log \left (\frac {2}{-3+x}\right )}+\frac {-3+x}{\log \left (\frac {2}{-3+x}\right )}\right ) \, dx+\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{\log \left (\frac {2}{x}\right )} \, dx,x,-3+x\right )-2 \operatorname {Subst}\left (\int \frac {1}{\log ^4\left (\frac {2}{x}\right )} \, dx,x,-3+x\right )-12 \operatorname {Subst}\left (\int \frac {1}{x^5} \, dx,x,\log \left (\frac {2}{-3+x}\right )\right )\\ &=-\frac {x^2}{3}+\frac {3}{\log ^4\left (-\frac {2}{3-x}\right )}-\frac {2 (3-x)}{\log ^4\left (-\frac {2}{3-x}\right )}+\frac {(3-x)^2}{3 \log ^4\left (-\frac {2}{3-x}\right )}+\frac {2 (3-x)}{3 \log ^3\left (-\frac {2}{3-x}\right )}-\frac {2 (3-x)^2}{9 \log ^3\left (-\frac {2}{3-x}\right )}-\frac {2 (3-x) x}{9 \log ^3\left (-\frac {2}{3-x}\right )}-\frac {3-x}{3 \log ^2\left (-\frac {2}{3-x}\right )}+\frac {2 (3-x) x}{9 \log ^2\left (-\frac {2}{3-x}\right )}+\frac {3-x}{\log \left (-\frac {2}{3-x}\right )}-\frac {4 (3-x) x}{9 \log \left (-\frac {2}{3-x}\right )}+\frac {4}{9} \operatorname {Subst}\left (\int \frac {x}{\log ^3\left (\frac {2}{x}\right )} \, dx,x,-3+x\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log \left (\frac {2}{-3+x}\right )\right )+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{\log ^3\left (\frac {2}{x}\right )} \, dx,x,-3+x\right )-\frac {8}{9} \int \frac {-3+x}{\log \left (\frac {2}{-3+x}\right )} \, dx-\frac {4}{3} \operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log \left (\frac {2}{-3+x}\right )\right )-\frac {8}{3} \int \frac {1}{\log \left (\frac {2}{-3+x}\right )} \, dx-\frac {8}{3} \operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log \left (\frac {2}{-3+x}\right )\right )\\ &=-\frac {x^2}{3}-\frac {14}{3} \text {Ei}\left (-\log \left (-\frac {2}{3-x}\right )\right )+\frac {3}{\log ^4\left (-\frac {2}{3-x}\right )}-\frac {2 (3-x)}{\log ^4\left (-\frac {2}{3-x}\right )}+\frac {(3-x)^2}{3 \log ^4\left (-\frac {2}{3-x}\right )}+\frac {2 (3-x)}{3 \log ^3\left (-\frac {2}{3-x}\right )}-\frac {2 (3-x)^2}{9 \log ^3\left (-\frac {2}{3-x}\right )}-\frac {2 (3-x) x}{9 \log ^3\left (-\frac {2}{3-x}\right )}-\frac {2 (3-x)}{3 \log ^2\left (-\frac {2}{3-x}\right )}+\frac {2 (3-x)^2}{9 \log ^2\left (-\frac {2}{3-x}\right )}+\frac {2 (3-x) x}{9 \log ^2\left (-\frac {2}{3-x}\right )}+\frac {3-x}{\log \left (-\frac {2}{3-x}\right )}-\frac {4 (3-x) x}{9 \log \left (-\frac {2}{3-x}\right )}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\log ^2\left (\frac {2}{x}\right )} \, dx,x,-3+x\right )-\frac {4}{9} \operatorname {Subst}\left (\int \frac {x}{\log ^2\left (\frac {2}{x}\right )} \, dx,x,-3+x\right )-\frac {8}{9} \operatorname {Subst}\left (\int \frac {x}{\log \left (\frac {2}{x}\right )} \, dx,x,-3+x\right )-\frac {8}{3} \operatorname {Subst}\left (\int \frac {1}{\log \left (\frac {2}{x}\right )} \, dx,x,-3+x\right )\\ &=-\frac {x^2}{3}-\frac {14}{3} \text {Ei}\left (-\log \left (-\frac {2}{3-x}\right )\right )+\frac {3}{\log ^4\left (-\frac {2}{3-x}\right )}-\frac {2 (3-x)}{\log ^4\left (-\frac {2}{3-x}\right )}+\frac {(3-x)^2}{3 \log ^4\left (-\frac {2}{3-x}\right )}+\frac {2 (3-x)}{3 \log ^3\left (-\frac {2}{3-x}\right )}-\frac {2 (3-x)^2}{9 \log ^3\left (-\frac {2}{3-x}\right )}-\frac {2 (3-x) x}{9 \log ^3\left (-\frac {2}{3-x}\right )}-\frac {2 (3-x)}{3 \log ^2\left (-\frac {2}{3-x}\right )}+\frac {2 (3-x)^2}{9 \log ^2\left (-\frac {2}{3-x}\right )}+\frac {2 (3-x) x}{9 \log ^2\left (-\frac {2}{3-x}\right )}+\frac {4 (3-x)}{3 \log \left (-\frac {2}{3-x}\right )}-\frac {4 (3-x)^2}{9 \log \left (-\frac {2}{3-x}\right )}-\frac {4 (3-x) x}{9 \log \left (-\frac {2}{3-x}\right )}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\log \left (\frac {2}{x}\right )} \, dx,x,-3+x\right )+\frac {8}{9} \operatorname {Subst}\left (\int \frac {x}{\log \left (\frac {2}{x}\right )} \, dx,x,-3+x\right )+\frac {32}{9} \operatorname {Subst}\left (\int \frac {e^{-2 x}}{x} \, dx,x,\log \left (\frac {2}{-3+x}\right )\right )+\frac {16}{3} \operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log \left (\frac {2}{-3+x}\right )\right )\\ &=-\frac {x^2}{3}+\frac {32}{9} \text {Ei}\left (-2 \log \left (-\frac {2}{3-x}\right )\right )+\frac {2}{3} \text {Ei}\left (-\log \left (-\frac {2}{3-x}\right )\right )+\frac {3}{\log ^4\left (-\frac {2}{3-x}\right )}-\frac {2 (3-x)}{\log ^4\left (-\frac {2}{3-x}\right )}+\frac {(3-x)^2}{3 \log ^4\left (-\frac {2}{3-x}\right )}+\frac {2 (3-x)}{3 \log ^3\left (-\frac {2}{3-x}\right )}-\frac {2 (3-x)^2}{9 \log ^3\left (-\frac {2}{3-x}\right )}-\frac {2 (3-x) x}{9 \log ^3\left (-\frac {2}{3-x}\right )}-\frac {2 (3-x)}{3 \log ^2\left (-\frac {2}{3-x}\right )}+\frac {2 (3-x)^2}{9 \log ^2\left (-\frac {2}{3-x}\right )}+\frac {2 (3-x) x}{9 \log ^2\left (-\frac {2}{3-x}\right )}+\frac {4 (3-x)}{3 \log \left (-\frac {2}{3-x}\right )}-\frac {4 (3-x)^2}{9 \log \left (-\frac {2}{3-x}\right )}-\frac {4 (3-x) x}{9 \log \left (-\frac {2}{3-x}\right )}-\frac {2}{3} \operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log \left (\frac {2}{-3+x}\right )\right )-\frac {32}{9} \operatorname {Subst}\left (\int \frac {e^{-2 x}}{x} \, dx,x,\log \left (\frac {2}{-3+x}\right )\right )\\ &=-\frac {x^2}{3}+\frac {3}{\log ^4\left (-\frac {2}{3-x}\right )}-\frac {2 (3-x)}{\log ^4\left (-\frac {2}{3-x}\right )}+\frac {(3-x)^2}{3 \log ^4\left (-\frac {2}{3-x}\right )}+\frac {2 (3-x)}{3 \log ^3\left (-\frac {2}{3-x}\right )}-\frac {2 (3-x)^2}{9 \log ^3\left (-\frac {2}{3-x}\right )}-\frac {2 (3-x) x}{9 \log ^3\left (-\frac {2}{3-x}\right )}-\frac {2 (3-x)}{3 \log ^2\left (-\frac {2}{3-x}\right )}+\frac {2 (3-x)^2}{9 \log ^2\left (-\frac {2}{3-x}\right )}+\frac {2 (3-x) x}{9 \log ^2\left (-\frac {2}{3-x}\right )}+\frac {4 (3-x)}{3 \log \left (-\frac {2}{3-x}\right )}-\frac {4 (3-x)^2}{9 \log \left (-\frac {2}{3-x}\right )}-\frac {4 (3-x) x}{9 \log \left (-\frac {2}{3-x}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 25, normalized size = 1.19 \begin {gather*} \frac {1}{3} \left (9-x^2+\frac {x^2}{\log ^4\left (\frac {2}{-3+x}\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*x^2 + (-6*x + 2*x^2)*Log[2/(-3 + x)] + (6*x - 2*x^2)*Log[2/(-3 + x)]^5)/((-9 + 3*x)*Log[2/(-3 + x
)]^5),x]

[Out]

(9 - x^2 + x^2/Log[2/(-3 + x)]^4)/3

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fricas [A]  time = 0.61, size = 32, normalized size = 1.52 \begin {gather*} -\frac {x^{2} \log \left (\frac {2}{x - 3}\right )^{4} - x^{2}}{3 \, \log \left (\frac {2}{x - 3}\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2+6*x)*log(2/(x-3))^5+(2*x^2-6*x)*log(2/(x-3))+4*x^2)/(3*x-9)/log(2/(x-3))^5,x, algorithm="fr
icas")

[Out]

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

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giac [B]  time = 0.18, size = 50, normalized size = 2.38 \begin {gather*} -\frac {1}{3} \, {\left (x - 3\right )}^{2} {\left (\frac {6}{x - 3} + 1\right )} + \frac {{\left (x - 3\right )}^{2} {\left (\frac {6}{x - 3} + \frac {9}{{\left (x - 3\right )}^{2}} + 1\right )}}{3 \, \log \left (\frac {2}{x - 3}\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2+6*x)*log(2/(x-3))^5+(2*x^2-6*x)*log(2/(x-3))+4*x^2)/(3*x-9)/log(2/(x-3))^5,x, algorithm="gi
ac")

[Out]

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

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maple [A]  time = 0.39, size = 22, normalized size = 1.05

method	result	size
risch	\(-\frac {x^{2}}{3}+\frac {x^{2}}{3 \ln \left (\frac {2}{x -3}\right )^{4}}\)	\(22\)
norman	\(\frac {\frac {x^{2}}{3}-\frac {x^{2} \ln \left (\frac {2}{x -3}\right )^{4}}{3}}{\ln \left (\frac {2}{x -3}\right )^{4}}\)	\(33\)
derivativedivides	\(-\frac {\left (x -3\right )^{2}}{3}+\frac {2 x -6}{\ln \left (\frac {2}{x -3}\right )^{4}}+\frac {\left (x -3\right )^{2}}{3 \ln \left (\frac {2}{x -3}\right )^{4}}-2 x +6+\frac {3}{\ln \left (\frac {2}{x -3}\right )^{4}}\)	\(57\)
default	\(-\frac {\left (x -3\right )^{2}}{3}+\frac {2 x -6}{\ln \left (\frac {2}{x -3}\right )^{4}}+\frac {\left (x -3\right )^{2}}{3 \ln \left (\frac {2}{x -3}\right )^{4}}-2 x +6+\frac {3}{\ln \left (\frac {2}{x -3}\right )^{4}}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^2+6*x)*ln(2/(x-3))^5+(2*x^2-6*x)*ln(2/(x-3))+4*x^2)/(3*x-9)/ln(2/(x-3))^5,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.50, size = 111, normalized size = 5.29 \begin {gather*} \frac {4 \, x^{2} \log \relax (2)^{3} \log \left (x - 3\right ) - 6 \, x^{2} \log \relax (2)^{2} \log \left (x - 3\right )^{2} + 4 \, x^{2} \log \relax (2) \log \left (x - 3\right )^{3} - x^{2} \log \left (x - 3\right )^{4} - {\left (\log \relax (2)^{4} - 1\right )} x^{2}}{3 \, {\left (\log \relax (2)^{4} - 4 \, \log \relax (2)^{3} \log \left (x - 3\right ) + 6 \, \log \relax (2)^{2} \log \left (x - 3\right )^{2} - 4 \, \log \relax (2) \log \left (x - 3\right )^{3} + \log \left (x - 3\right )^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2+6*x)*log(2/(x-3))^5+(2*x^2-6*x)*log(2/(x-3))+4*x^2)/(3*x-9)/log(2/(x-3))^5,x, algorithm="ma
xima")

[Out]

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

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mupad [B]  time = 5.33, size = 21, normalized size = 1.00 \begin {gather*} \frac {x^2}{3\,{\ln \left (\frac {2}{x-3}\right )}^4}-\frac {x^2}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(2/(x - 3))^5*(6*x - 2*x^2) - log(2/(x - 3))*(6*x - 2*x^2) + 4*x^2)/(log(2/(x - 3))^5*(3*x - 9)),x)

[Out]

x^2/(3*log(2/(x - 3))^4) - x^2/3

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sympy [A]  time = 0.18, size = 17, normalized size = 0.81 \begin {gather*} - \frac {x^{2}}{3} + \frac {x^{2}}{3 \log {\left (\frac {2}{x - 3} \right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**2+6*x)*ln(2/(x-3))**5+(2*x**2-6*x)*ln(2/(x-3))+4*x**2)/(3*x-9)/ln(2/(x-3))**5,x)

[Out]

-x**2/3 + x**2/(3*log(2/(x - 3))**4)

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