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

Optimal. Leaf size=24 \[ 4+5 x-x^2-16 (-2+\log ((8-x) x))^2 \]

________________________________________________________________________________________

Rubi [B]  time = 0.38, antiderivative size = 83, normalized size of antiderivative = 3.46, number of steps used = 16, number of rules used = 10, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.244, Rules used = {1593, 6742, 1620, 2514, 2494, 2390, 2301, 2394, 2315, 2316} \begin {gather*} -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) \end {gather*}

Antiderivative was successfully verified.

[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 1593

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 1620

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 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 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 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 2394

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 2494

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 2514

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 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 {-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 \operatorname {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 {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 0.06, size = 71, normalized size = 2.96 \begin {gather*} 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)) \end {gather*}

Antiderivative was successfully verified.

[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)]

________________________________________________________________________________________

fricas [A]  time = 1.10, size = 35, normalized size = 1.46 \begin {gather*} -x^{2} - 16 \, \log \left (-x^{2} + 8 \, x\right )^{2} + 5 \, x + 64 \, \log \left (-x^{2} + 8 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maple [A]  time = 0.15, size = 34, normalized size = 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	\(64 \ln \left (-x^{2}+8 x \right )+5 x -x^{2}-16 \ln \left (-x^{2}+8 x \right )^{2}\)	\(36\)
default	\(-x^{2}+5 x +64 \ln \left (-8+x \right )+64 \ln \relax (x )-32 \ln \left (-8+x \right ) \ln \left (-x^{2}+8 x \right )+32 \ln \left (-8+x \right ) \ln \left (\frac {x}{8}\right )+16 \ln \left (-8+x \right )^{2}-32 \ln \relax (x ) \ln \left (-x^{2}+8 x \right )+32 \left (\ln \relax (x )-\ln \left (\frac {x}{8}\right )\right ) \ln \left (-\frac {x}{8}+1\right )+16 \ln \relax (x )^{2}\)	\(91\)

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maxima [A]  time = 0.39, size = 45, normalized size = 1.88 \begin {gather*} -x^{2} - 16 \, \log \relax (x)^{2} - 32 \, \log \relax (x) \log \left (-x + 8\right ) - 16 \, \log \left (-x + 8\right )^{2} + 5 \, x + 64 \, \log \left (x - 8\right ) + 64 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [B]  time = 2.31, size = 31, normalized size = 1.29 \begin {gather*} 5\,x+64\,\ln \left (x\,\left (x-8\right )\right )-16\,{\ln \left (8\,x-x^2\right )}^2-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

sympy [A]  time = 0.16, size = 27, normalized size = 1.12 \begin {gather*} - x^{2} + 5 x - 16 \log {\left (- x^{2} + 8 x \right )}^{2} + 64 \log {\left (x^{2} - 8 x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________