\(\int \frac {(256-256 x) \log (8-2 x+x^2)}{8-2 x+x^2} \, dx\) [7863]

Optimal result

Integrand size = 25, antiderivative size = 14 \[ \int \frac {(256-256 x) \log \left (8-2 x+x^2\right )}{8-2 x+x^2} \, dx=-64 \log ^2\left (7+(1-x)^2\right ) \]

[Out]

-64*ln(7+(1-x)^2)^2

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.28 (sec) , antiderivative size = 249, normalized size of antiderivative = 17.79, number of steps used = 18, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2608, 2604, 2465, 2437, 2338, 2441, 2440, 2438} \[ \int \frac {(256-256 x) \log \left (8-2 x+x^2\right )}{8-2 x+x^2} \, dx=128 \operatorname {PolyLog}\left (2,-\frac {-i x-\sqrt {7}+i}{2 \sqrt {7}}\right )+128 \operatorname {PolyLog}\left (2,\frac {-i x+\sqrt {7}+i}{2 \sqrt {7}}\right )-128 \log \left (2 x-2 \left (1-i \sqrt {7}\right )\right ) \log \left (x^2-2 x+8\right )-128 \log \left (2 x-2 \left (1+i \sqrt {7}\right )\right ) \log \left (x^2-2 x+8\right )+64 \log ^2\left (-2 \left (-x-i \sqrt {7}+1\right )\right )+64 \log ^2\left (-2 \left (-x+i \sqrt {7}+1\right )\right )+128 \log \left (-\frac {i \left (-x+i \sqrt {7}+1\right )}{2 \sqrt {7}}\right ) \log \left (2 x-2 \left (1-i \sqrt {7}\right )\right )+128 \log \left (\frac {i \left (-x-i \sqrt {7}+1\right )}{2 \sqrt {7}}\right ) \log \left (2 x-2 \left (1+i \sqrt {7}\right )\right ) \]

[In]

Int[((256 - 256*x)*Log[8 - 2*x + x^2])/(8 - 2*x + x^2),x]

[Out]

64*Log[-2*(1 - I*Sqrt[7] - x)]^2 + 64*Log[-2*(1 + I*Sqrt[7] - x)]^2 + 128*Log[((-1/2*I)*(1 + I*Sqrt[7] - x))/S
qrt[7]]*Log[-2*(1 - I*Sqrt[7]) + 2*x] + 128*Log[((I/2)*(1 - I*Sqrt[7] - x))/Sqrt[7]]*Log[-2*(1 + I*Sqrt[7]) +
2*x] - 128*Log[-2*(1 - I*Sqrt[7]) + 2*x]*Log[8 - 2*x + x^2] - 128*Log[-2*(1 + I*Sqrt[7]) + 2*x]*Log[8 - 2*x +
x^2] + 128*PolyLog[2, -1/2*(I - Sqrt[7] - I*x)/Sqrt[7]] + 128*PolyLog[2, (I + Sqrt[7] - I*x)/(2*Sqrt[7])]

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 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 2438

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 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(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 2465

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 2604

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

Rule 2608

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {256 \log \left (8-2 x+x^2\right )}{-2-2 i \sqrt {7}+2 x}-\frac {256 \log \left (8-2 x+x^2\right )}{-2+2 i \sqrt {7}+2 x}\right ) \, dx \\ & = -\left (256 \int \frac {\log \left (8-2 x+x^2\right )}{-2-2 i \sqrt {7}+2 x} \, dx\right )-256 \int \frac {\log \left (8-2 x+x^2\right )}{-2+2 i \sqrt {7}+2 x} \, dx \\ & = -128 \log \left (-2 \left (1-i \sqrt {7}\right )+2 x\right ) \log \left (8-2 x+x^2\right )-128 \log \left (-2 \left (1+i \sqrt {7}\right )+2 x\right ) \log \left (8-2 x+x^2\right )+128 \int \frac {(-2+2 x) \log \left (-2-2 i \sqrt {7}+2 x\right )}{8-2 x+x^2} \, dx+128 \int \frac {(-2+2 x) \log \left (-2+2 i \sqrt {7}+2 x\right )}{8-2 x+x^2} \, dx \\ & = -128 \log \left (-2 \left (1-i \sqrt {7}\right )+2 x\right ) \log \left (8-2 x+x^2\right )-128 \log \left (-2 \left (1+i \sqrt {7}\right )+2 x\right ) \log \left (8-2 x+x^2\right )+128 \int \left (\frac {2 \log \left (-2-2 i \sqrt {7}+2 x\right )}{-2-2 i \sqrt {7}+2 x}+\frac {2 \log \left (-2-2 i \sqrt {7}+2 x\right )}{-2+2 i \sqrt {7}+2 x}\right ) \, dx+128 \int \left (\frac {2 \log \left (-2+2 i \sqrt {7}+2 x\right )}{-2-2 i \sqrt {7}+2 x}+\frac {2 \log \left (-2+2 i \sqrt {7}+2 x\right )}{-2+2 i \sqrt {7}+2 x}\right ) \, dx \\ & = -128 \log \left (-2 \left (1-i \sqrt {7}\right )+2 x\right ) \log \left (8-2 x+x^2\right )-128 \log \left (-2 \left (1+i \sqrt {7}\right )+2 x\right ) \log \left (8-2 x+x^2\right )+256 \int \frac {\log \left (-2-2 i \sqrt {7}+2 x\right )}{-2-2 i \sqrt {7}+2 x} \, dx+256 \int \frac {\log \left (-2-2 i \sqrt {7}+2 x\right )}{-2+2 i \sqrt {7}+2 x} \, dx+256 \int \frac {\log \left (-2+2 i \sqrt {7}+2 x\right )}{-2-2 i \sqrt {7}+2 x} \, dx+256 \int \frac {\log \left (-2+2 i \sqrt {7}+2 x\right )}{-2+2 i \sqrt {7}+2 x} \, dx \\ & = 128 \log \left (-\frac {i \left (1+i \sqrt {7}-x\right )}{2 \sqrt {7}}\right ) \log \left (-2 \left (1-i \sqrt {7}\right )+2 x\right )+128 \log \left (\frac {i \left (1-i \sqrt {7}-x\right )}{2 \sqrt {7}}\right ) \log \left (-2 \left (1+i \sqrt {7}\right )+2 x\right )-128 \log \left (-2 \left (1-i \sqrt {7}\right )+2 x\right ) \log \left (8-2 x+x^2\right )-128 \log \left (-2 \left (1+i \sqrt {7}\right )+2 x\right ) \log \left (8-2 x+x^2\right )+128 \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-2-2 i \sqrt {7}+2 x\right )+128 \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-2+2 i \sqrt {7}+2 x\right )-256 \int \frac {\log \left (\frac {2 \left (-2-2 i \sqrt {7}+2 x\right )}{2 \left (-2-2 i \sqrt {7}\right )-2 \left (-2+2 i \sqrt {7}\right )}\right )}{-2+2 i \sqrt {7}+2 x} \, dx-256 \int \frac {\log \left (\frac {2 \left (-2+2 i \sqrt {7}+2 x\right )}{-2 \left (-2-2 i \sqrt {7}\right )+2 \left (-2+2 i \sqrt {7}\right )}\right )}{-2-2 i \sqrt {7}+2 x} \, dx \\ & = 64 \log ^2\left (-2 \left (1-i \sqrt {7}-x\right )\right )+64 \log ^2\left (-2 \left (1+i \sqrt {7}-x\right )\right )+128 \log \left (-\frac {i \left (1+i \sqrt {7}-x\right )}{2 \sqrt {7}}\right ) \log \left (-2 \left (1-i \sqrt {7}\right )+2 x\right )+128 \log \left (\frac {i \left (1-i \sqrt {7}-x\right )}{2 \sqrt {7}}\right ) \log \left (-2 \left (1+i \sqrt {7}\right )+2 x\right )-128 \log \left (-2 \left (1-i \sqrt {7}\right )+2 x\right ) \log \left (8-2 x+x^2\right )-128 \log \left (-2 \left (1+i \sqrt {7}\right )+2 x\right ) \log \left (8-2 x+x^2\right )-128 \text {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{2 \left (-2-2 i \sqrt {7}\right )-2 \left (-2+2 i \sqrt {7}\right )}\right )}{x} \, dx,x,-2+2 i \sqrt {7}+2 x\right )-128 \text {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{-2 \left (-2-2 i \sqrt {7}\right )+2 \left (-2+2 i \sqrt {7}\right )}\right )}{x} \, dx,x,-2-2 i \sqrt {7}+2 x\right ) \\ & = 64 \log ^2\left (-2 \left (1-i \sqrt {7}-x\right )\right )+64 \log ^2\left (-2 \left (1+i \sqrt {7}-x\right )\right )+128 \log \left (-\frac {i \left (1+i \sqrt {7}-x\right )}{2 \sqrt {7}}\right ) \log \left (-2 \left (1-i \sqrt {7}\right )+2 x\right )+128 \log \left (\frac {i \left (1-i \sqrt {7}-x\right )}{2 \sqrt {7}}\right ) \log \left (-2 \left (1+i \sqrt {7}\right )+2 x\right )-128 \log \left (-2 \left (1-i \sqrt {7}\right )+2 x\right ) \log \left (8-2 x+x^2\right )-128 \log \left (-2 \left (1+i \sqrt {7}\right )+2 x\right ) \log \left (8-2 x+x^2\right )+128 \operatorname {PolyLog}\left (2,-\frac {i-\sqrt {7}-i x}{2 \sqrt {7}}\right )+128 \operatorname {PolyLog}\left (2,\frac {i+\sqrt {7}-i x}{2 \sqrt {7}}\right ) \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.05 (sec) , antiderivative size = 265, normalized size of antiderivative = 18.93 \[ \int \frac {(256-256 x) \log \left (8-2 x+x^2\right )}{8-2 x+x^2} \, dx=-256 \left (-\frac {1}{2} \log \left (-\frac {i \left (1+i \sqrt {7}-x\right )}{2 \sqrt {7}}\right ) \log \left (-2 \left (1-i \sqrt {7}\right )+2 x\right )-\frac {1}{4} \log ^2\left (-2 \left (1-i \sqrt {7}\right )+2 x\right )-\frac {1}{2} \log \left (\frac {i \left (1-i \sqrt {7}-x\right )}{2 \sqrt {7}}\right ) \log \left (-2 \left (1+i \sqrt {7}\right )+2 x\right )-\frac {1}{4} \log ^2\left (-2 \left (1+i \sqrt {7}\right )+2 x\right )+\frac {1}{2} \log \left (-2-2 i \sqrt {7}+2 x\right ) \log \left (8-2 x+x^2\right )+\frac {1}{2} \log \left (-2+2 i \sqrt {7}+2 x\right ) \log \left (8-2 x+x^2\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {i \left (1-i \sqrt {7}-x\right )}{2 \sqrt {7}}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,-\frac {i \left (1+i \sqrt {7}-x\right )}{2 \sqrt {7}}\right )\right ) \]

[In]

Integrate[((256 - 256*x)*Log[8 - 2*x + x^2])/(8 - 2*x + x^2),x]

[Out]

-256*(-1/2*(Log[((-1/2*I)*(1 + I*Sqrt[7] - x))/Sqrt[7]]*Log[-2*(1 - I*Sqrt[7]) + 2*x]) - Log[-2*(1 - I*Sqrt[7]
) + 2*x]^2/4 - (Log[((I/2)*(1 - I*Sqrt[7] - x))/Sqrt[7]]*Log[-2*(1 + I*Sqrt[7]) + 2*x])/2 - Log[-2*(1 + I*Sqrt
[7]) + 2*x]^2/4 + (Log[-2 - (2*I)*Sqrt[7] + 2*x]*Log[8 - 2*x + x^2])/2 + (Log[-2 + (2*I)*Sqrt[7] + 2*x]*Log[8
- 2*x + x^2])/2 - PolyLog[2, ((I/2)*(1 - I*Sqrt[7] - x))/Sqrt[7]]/2 - PolyLog[2, ((-1/2*I)*(1 + I*Sqrt[7] - x)
)/Sqrt[7]]/2)

Maple [A] (verified)

Time = 1.49 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00

method	result	size
norman	\(-64 \ln \left (x^{2}-2 x +8\right )^{2}\)	\(14\)
risch	\(-64 \ln \left (x^{2}-2 x +8\right )^{2}\)	\(14\)
default	\(-128 \ln \left (x -1-i \sqrt {7}\right ) \ln \left (x^{2}-2 x +8\right )+64 \ln \left (x -1-i \sqrt {7}\right )^{2}+128 \operatorname {dilog}\left (-\frac {i \left (-1+i \sqrt {7}+x \right ) \sqrt {7}}{14}\right )+128 \ln \left (x -1-i \sqrt {7}\right ) \ln \left (-\frac {i \left (-1+i \sqrt {7}+x \right ) \sqrt {7}}{14}\right )-128 \ln \left (-1+i \sqrt {7}+x \right ) \ln \left (x^{2}-2 x +8\right )+64 \ln \left (-1+i \sqrt {7}+x \right )^{2}+128 \operatorname {dilog}\left (\frac {i \left (x -1-i \sqrt {7}\right ) \sqrt {7}}{14}\right )+128 \ln \left (-1+i \sqrt {7}+x \right ) \ln \left (\frac {i \left (x -1-i \sqrt {7}\right ) \sqrt {7}}{14}\right )\)	\(164\)
parts	\(-128 \ln \left (x^{2}-2 x +8\right )^{2}+128 \ln \left (x -1-i \sqrt {7}\right ) \ln \left (x^{2}-2 x +8\right )-64 \ln \left (x -1-i \sqrt {7}\right )^{2}-128 \operatorname {dilog}\left (-\frac {i \left (-1+i \sqrt {7}+x \right ) \sqrt {7}}{14}\right )-128 \ln \left (x -1-i \sqrt {7}\right ) \ln \left (-\frac {i \left (-1+i \sqrt {7}+x \right ) \sqrt {7}}{14}\right )+128 \ln \left (-1+i \sqrt {7}+x \right ) \ln \left (x^{2}-2 x +8\right )-64 \ln \left (-1+i \sqrt {7}+x \right )^{2}-128 \operatorname {dilog}\left (\frac {i \left (x -1-i \sqrt {7}\right ) \sqrt {7}}{14}\right )-128 \ln \left (-1+i \sqrt {7}+x \right ) \ln \left (\frac {i \left (x -1-i \sqrt {7}\right ) \sqrt {7}}{14}\right )\)	\(177\)

[In]

int((-256*x+256)*ln(x^2-2*x+8)/(x^2-2*x+8),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \frac {(256-256 x) \log \left (8-2 x+x^2\right )}{8-2 x+x^2} \, dx=-64 \, \log \left (x^{2} - 2 \, x + 8\right )^{2} \]

[In]

integrate((-256*x+256)*log(x^2-2*x+8)/(x^2-2*x+8),x, algorithm="fricas")

[Out]

-64*log(x^2 - 2*x + 8)^2

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {(256-256 x) \log \left (8-2 x+x^2\right )}{8-2 x+x^2} \, dx=- 64 \log {\left (x^{2} - 2 x + 8 \right )}^{2} \]

[In]

integrate((-256*x+256)*ln(x**2-2*x+8)/(x**2-2*x+8),x)

[Out]

-64*log(x**2 - 2*x + 8)**2

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \frac {(256-256 x) \log \left (8-2 x+x^2\right )}{8-2 x+x^2} \, dx=-64 \, \log \left (x^{2} - 2 \, x + 8\right )^{2} \]

[In]

integrate((-256*x+256)*log(x^2-2*x+8)/(x^2-2*x+8),x, algorithm="maxima")

[Out]

-64*log(x^2 - 2*x + 8)^2

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \frac {(256-256 x) \log \left (8-2 x+x^2\right )}{8-2 x+x^2} \, dx=-64 \, \log \left (x^{2} - 2 \, x + 8\right )^{2} \]

[In]

integrate((-256*x+256)*log(x^2-2*x+8)/(x^2-2*x+8),x, algorithm="giac")

[Out]

-64*log(x^2 - 2*x + 8)^2

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \frac {(256-256 x) \log \left (8-2 x+x^2\right )}{8-2 x+x^2} \, dx=-64\,{\ln \left (x^2-2\,x+8\right )}^2 \]

[In]

int(-(log(x^2 - 2*x + 8)*(256*x - 256))/(x^2 - 2*x + 8),x)

[Out]

-64*log(x^2 - 2*x + 8)^2