\(\int \frac {-512+88 x+21 x^2-2 x^3+(256-64 x) \log (8 x-x^2)}{-8 x+x^2} \, dx\) [3885]

Optimal result

Integrand size = 41, antiderivative size = 24 \[ \int \frac {-512+88 x+21 x^2-2 x^3+(256-64 x) \log \left (8 x-x^2\right )}{-8 x+x^2} \, dx=4+5 x-x^2-16 (-2+\log ((8-x) x))^2 \]

[Out]

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(83\) vs. \(2(24)=48\).

Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.46, number of steps used = 16, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.244, Rules used = {1607, 6874, 1634, 2594, 2580, 2437, 2338, 2441, 2352, 2353} \[ \int \frac {-512+88 x+21 x^2-2 x^3+(256-64 x) \log \left (8 x-x^2\right )}{-8 x+x^2} \, dx=-x^2+5 x+16 \log ^2(x-8)+16 \log ^2(x)+32 \log (8) \log (8-x)+64 \log (8-x)+32 \log (x-8) \log \left (\frac {x}{8}\right )+64 \log (x)-32 \log (x-8) \log ((8-x) x)-32 \log (x) \log ((8-x) x) \]

[In]

Int[(-512 + 88*x + 21*x^2 - 2*x^3 + (256 - 64*x)*Log[8*x - x^2])/(-8*x + x^2),x]

[Out]

5*x - x^2 + 64*Log[8 - x] + 32*Log[8]*Log[8 - x] + 16*Log[-8 + x]^2 + 32*Log[-8 + x]*Log[x/8] + 64*Log[x] + 16
*Log[x]^2 - 32*Log[-8 + x]*Log[(8 - x)*x] - 32*Log[x]*Log[(8 - x)*x]

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1634

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rule 2338

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 2352

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

Rule 2353

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 2437

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 2441

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

Rule 2580

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]/((g_.) + (h_.)*(x_)), x_Sym
bol] :> Simp[Log[g + h*x]*(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/h), x] + (-Dist[b*p*(r/h), Int[Log[g + h*x]/(a
+ b*x), x], x] - Dist[d*q*(r/h), Int[Log[g + h*x]/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, p, q,
r}, x] && NeQ[b*c - a*d, 0]

Rule 2594

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(RFx_), x_Symbol] :>
With[{u = ExpandIntegrand[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a,
b, c, d, e, f, p, q, r, s}, x] && RationalFunctionQ[RFx, x] && IGtQ[s, 0]

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {-512+88 x+21 x^2-2 x^3+(256-64 x) \log \left (8 x-x^2\right )}{(-8+x) x} \, dx \\ & = \int \left (\frac {-512+88 x+21 x^2-2 x^3}{(-8+x) x}+\frac {64 (-4+x) \log ((8-x) x)}{(8-x) x}\right ) \, dx \\ & = 64 \int \frac {(-4+x) \log ((8-x) x)}{(8-x) x} \, dx+\int \frac {-512+88 x+21 x^2-2 x^3}{(-8+x) x} \, dx \\ & = 64 \int \left (-\frac {\log ((8-x) x)}{2 (-8+x)}-\frac {\log ((8-x) x)}{2 x}\right ) \, dx+\int \left (5+\frac {64}{-8+x}+\frac {64}{x}-2 x\right ) \, dx \\ & = 5 x-x^2+64 \log (8-x)+64 \log (x)-32 \int \frac {\log ((8-x) x)}{-8+x} \, dx-32 \int \frac {\log ((8-x) x)}{x} \, dx \\ & = 5 x-x^2+64 \log (8-x)+64 \log (x)-32 \log (-8+x) \log ((8-x) x)-32 \log (x) \log ((8-x) x)-32 \int \frac {\log (-8+x)}{8-x} \, dx+32 \int \frac {\log (-8+x)}{x} \, dx-32 \int \frac {\log (x)}{8-x} \, dx+32 \int \frac {\log (x)}{x} \, dx \\ & = 5 x-x^2+64 \log (8-x)+32 \log (8) \log (8-x)+32 \log (-8+x) \log \left (\frac {x}{8}\right )+64 \log (x)+16 \log ^2(x)-32 \log (-8+x) \log ((8-x) x)-32 \log (x) \log ((8-x) x)-32 \int \frac {\log \left (\frac {x}{8}\right )}{8-x} \, dx-32 \int \frac {\log \left (\frac {x}{8}\right )}{-8+x} \, dx+32 \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-8+x\right ) \\ & = 5 x-x^2+64 \log (8-x)+32 \log (8) \log (8-x)+16 \log ^2(-8+x)+32 \log (-8+x) \log \left (\frac {x}{8}\right )+64 \log (x)+16 \log ^2(x)-32 \log (-8+x) \log ((8-x) x)-32 \log (x) \log ((8-x) x) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(71\) vs. \(2(24)=48\).

Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.96 \[ \int \frac {-512+88 x+21 x^2-2 x^3+(256-64 x) \log \left (8 x-x^2\right )}{-8 x+x^2} \, dx=5 x-x^2+32 \log (8) \log (8-x)+16 \log ^2(-8+x)+64 \log (x)+16 \log ^2(x)+32 \log (-8+x) \left (2+\log \left (\frac {x}{8}\right )-\log (-((-8+x) x))\right )-32 \log (x) \log (-((-8+x) x)) \]

[In]

Integrate[(-512 + 88*x + 21*x^2 - 2*x^3 + (256 - 64*x)*Log[8*x - x^2])/(-8*x + x^2),x]

[Out]

5*x - x^2 + 32*Log[8]*Log[8 - x] + 16*Log[-8 + x]^2 + 64*Log[x] + 16*Log[x]^2 + 32*Log[-8 + x]*(2 + Log[x/8] -
Log[-((-8 + x)*x)]) - 32*Log[x]*Log[-((-8 + x)*x)]

Maple [A] (verified)

Time = 1.59 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42

method	result	size
risch	\(-16 \ln \left (-x^{2}+8 x \right )^{2}-x^{2}+5 x +64 \ln \left (x^{2}-8 x \right )\)	\(34\)
norman	\(64 \ln \left (-x^{2}+8 x \right )+5 x -x^{2}-16 \ln \left (-x^{2}+8 x \right )^{2}\)	\(36\)
parallelrisch	\(144-x^{2}-16 \ln \left (-x^{2}+8 x \right )^{2}+5 x +64 \ln \left (-x^{2}+8 x \right )\)	\(37\)
default	\(5 x -x^{2}+64 \ln \left (x \right )+64 \ln \left (-8+x \right )-32 \ln \left (x \right ) \ln \left (-x^{2}+8 x \right )+16 \ln \left (x \right )^{2}+32 \left (\ln \left (x \right )-\ln \left (\frac {x}{8}\right )\right ) \ln \left (1-\frac {x}{8}\right )-32 \ln \left (-8+x \right ) \ln \left (-x^{2}+8 x \right )+16 \ln \left (-8+x \right )^{2}+32 \ln \left (-8+x \right ) \ln \left (\frac {x}{8}\right )\)	\(91\)
parts	\(5 x -x^{2}+64 \ln \left (x \right )+64 \ln \left (-8+x \right )-32 \ln \left (x \right ) \ln \left (-x^{2}+8 x \right )+16 \ln \left (x \right )^{2}+32 \left (\ln \left (x \right )-\ln \left (\frac {x}{8}\right )\right ) \ln \left (1-\frac {x}{8}\right )-32 \ln \left (-8+x \right ) \ln \left (-x^{2}+8 x \right )+16 \ln \left (-8+x \right )^{2}+32 \ln \left (-8+x \right ) \ln \left (\frac {x}{8}\right )\)	\(91\)

[In]

int(((-64*x+256)*ln(-x^2+8*x)-2*x^3+21*x^2+88*x-512)/(x^2-8*x),x,method=_RETURNVERBOSE)

[Out]

-16*ln(-x^2+8*x)^2-x^2+5*x+64*ln(x^2-8*x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int \frac {-512+88 x+21 x^2-2 x^3+(256-64 x) \log \left (8 x-x^2\right )}{-8 x+x^2} \, dx=-x^{2} - 16 \, \log \left (-x^{2} + 8 \, x\right )^{2} + 5 \, x + 64 \, \log \left (-x^{2} + 8 \, x\right ) \]

[In]

integrate(((-64*x+256)*log(-x^2+8*x)-2*x^3+21*x^2+88*x-512)/(x^2-8*x),x, algorithm="fricas")

[Out]

-x^2 - 16*log(-x^2 + 8*x)^2 + 5*x + 64*log(-x^2 + 8*x)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {-512+88 x+21 x^2-2 x^3+(256-64 x) \log \left (8 x-x^2\right )}{-8 x+x^2} \, dx=- x^{2} + 5 x - 16 \log {\left (- x^{2} + 8 x \right )}^{2} + 64 \log {\left (x^{2} - 8 x \right )} \]

[In]

integrate(((-64*x+256)*ln(-x**2+8*x)-2*x**3+21*x**2+88*x-512)/(x**2-8*x),x)

[Out]

-x**2 + 5*x - 16*log(-x**2 + 8*x)**2 + 64*log(x**2 - 8*x)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.88 \[ \int \frac {-512+88 x+21 x^2-2 x^3+(256-64 x) \log \left (8 x-x^2\right )}{-8 x+x^2} \, dx=-x^{2} - 16 \, \log \left (x\right )^{2} - 32 \, \log \left (x\right ) \log \left (-x + 8\right ) - 16 \, \log \left (-x + 8\right )^{2} + 5 \, x + 64 \, \log \left (x - 8\right ) + 64 \, \log \left (x\right ) \]

[In]

integrate(((-64*x+256)*log(-x^2+8*x)-2*x^3+21*x^2+88*x-512)/(x^2-8*x),x, algorithm="maxima")

[Out]

-x^2 - 16*log(x)^2 - 32*log(x)*log(-x + 8) - 16*log(-x + 8)^2 + 5*x + 64*log(x - 8) + 64*log(x)

Giac [F]

\[ \int \frac {-512+88 x+21 x^2-2 x^3+(256-64 x) \log \left (8 x-x^2\right )}{-8 x+x^2} \, dx=\int { -\frac {2 \, x^{3} - 21 \, x^{2} + 64 \, {\left (x - 4\right )} \log \left (-x^{2} + 8 \, x\right ) - 88 \, x + 512}{x^{2} - 8 \, x} \,d x } \]

[In]

integrate(((-64*x+256)*log(-x^2+8*x)-2*x^3+21*x^2+88*x-512)/(x^2-8*x),x, algorithm="giac")

[Out]

integrate(-(2*x^3 - 21*x^2 + 64*(x - 4)*log(-x^2 + 8*x) - 88*x + 512)/(x^2 - 8*x), x)

Mupad [B] (verification not implemented)

Time = 9.37 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {-512+88 x+21 x^2-2 x^3+(256-64 x) \log \left (8 x-x^2\right )}{-8 x+x^2} \, dx=5\,x+64\,\ln \left (x\,\left (x-8\right )\right )-16\,{\ln \left (8\,x-x^2\right )}^2-x^2 \]

[In]

int((log(8*x - x^2)*(64*x - 256) - 88*x - 21*x^2 + 2*x^3 + 512)/(8*x - x^2),x)

[Out]

5*x + 64*log(x*(x - 8)) - 16*log(8*x - x^2)^2 - x^2