3.60.3 \(\int \frac {-8+2 x+100 x^4 \log (16-8 x+x^2)+(-200 x^3+50 x^4) \log ^2(16-8 x+x^2)}{(8 x-2 x^2+(-100 x^4+25 x^5) \log ^2(16-8 x+x^2)) \log (\frac {-2+25 x^3 \log ^2(16-8 x+x^2)}{2 x \log ^2(3)})} \, dx\)

Optimal. Leaf size=31 \[ \log \left (\log \left (\frac {-\frac {2}{x}+25 x^2 \log ^2\left ((4-x)^2\right )}{2 \log ^2(3)}\right )\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.12, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 109, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.009, Rules used = {6684} \begin {gather*} \log \left (\log \left (-\frac {2-25 x^3 \log ^2\left (x^2-8 x+16\right )}{2 x \log ^2(3)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-8 + 2*x + 100*x^4*Log[16 - 8*x + x^2] + (-200*x^3 + 50*x^4)*Log[16 - 8*x + x^2]^2)/((8*x - 2*x^2 + (-100
*x^4 + 25*x^5)*Log[16 - 8*x + x^2]^2)*Log[(-2 + 25*x^3*Log[16 - 8*x + x^2]^2)/(2*x*Log[3]^2)]),x]

[Out]

Log[Log[-1/2*(2 - 25*x^3*Log[16 - 8*x + x^2]^2)/(x*Log[3]^2)]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log \left (\log \left (-\frac {2-25 x^3 \log ^2\left (16-8 x+x^2\right )}{2 x \log ^2(3)}\right )\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 28, normalized size = 0.90 \begin {gather*} \log \left (\log \left (\frac {-2+25 x^3 \log ^2\left ((-4+x)^2\right )}{2 x \log ^2(3)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8 + 2*x + 100*x^4*Log[16 - 8*x + x^2] + (-200*x^3 + 50*x^4)*Log[16 - 8*x + x^2]^2)/((8*x - 2*x^2 +
(-100*x^4 + 25*x^5)*Log[16 - 8*x + x^2]^2)*Log[(-2 + 25*x^3*Log[16 - 8*x + x^2]^2)/(2*x*Log[3]^2)]),x]

[Out]

Log[Log[(-2 + 25*x^3*Log[(-4 + x)^2]^2)/(2*x*Log[3]^2)]]

________________________________________________________________________________________

fricas [A]  time = 0.57, size = 29, normalized size = 0.94 \begin {gather*} \log \left (\log \left (\frac {25 \, x^{3} \log \left (x^{2} - 8 \, x + 16\right )^{2} - 2}{2 \, x \log \relax (3)^{2}}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((50*x^4-200*x^3)*log(x^2-8*x+16)^2+100*x^4*log(x^2-8*x+16)+2*x-8)/((25*x^5-100*x^4)*log(x^2-8*x+16)
^2-2*x^2+8*x)/log(1/2*(25*x^3*log(x^2-8*x+16)^2-2)/x/log(3)^2),x, algorithm="fricas")

[Out]

log(log(1/2*(25*x^3*log(x^2 - 8*x + 16)^2 - 2)/(x*log(3)^2)))

________________________________________________________________________________________

giac [A]  time = 0.69, size = 31, normalized size = 1.00 \begin {gather*} \log \left (-\log \left (25 \, x^{3} \log \left (x^{2} - 8 \, x + 16\right )^{2} - 2\right ) + \log \left (2 \, x \log \relax (3)^{2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((50*x^4-200*x^3)*log(x^2-8*x+16)^2+100*x^4*log(x^2-8*x+16)+2*x-8)/((25*x^5-100*x^4)*log(x^2-8*x+16)
^2-2*x^2+8*x)/log(1/2*(25*x^3*log(x^2-8*x+16)^2-2)/x/log(3)^2),x, algorithm="giac")

[Out]

log(-log(25*x^3*log(x^2 - 8*x + 16)^2 - 2) + log(2*x*log(3)^2))

________________________________________________________________________________________

maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (50 x^{4}-200 x^{3}\right ) \ln \left (x^{2}-8 x +16\right )^{2}+100 x^{4} \ln \left (x^{2}-8 x +16\right )+2 x -8}{\left (\left (25 x^{5}-100 x^{4}\right ) \ln \left (x^{2}-8 x +16\right )^{2}-2 x^{2}+8 x \right ) \ln \left (\frac {25 x^{3} \ln \left (x^{2}-8 x +16\right )^{2}-2}{2 x \ln \relax (3)^{2}}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((50*x^4-200*x^3)*ln(x^2-8*x+16)^2+100*x^4*ln(x^2-8*x+16)+2*x-8)/((25*x^5-100*x^4)*ln(x^2-8*x+16)^2-2*x^2+
8*x)/ln(1/2*(25*x^3*ln(x^2-8*x+16)^2-2)/x/ln(3)^2),x)

[Out]

int(((50*x^4-200*x^3)*ln(x^2-8*x+16)^2+100*x^4*ln(x^2-8*x+16)+2*x-8)/((25*x^5-100*x^4)*ln(x^2-8*x+16)^2-2*x^2+
8*x)/ln(1/2*(25*x^3*ln(x^2-8*x+16)^2-2)/x/ln(3)^2),x)

________________________________________________________________________________________

maxima [A]  time = 0.52, size = 25, normalized size = 0.81 \begin {gather*} \log \left (\log \left (50 \, x^{3} \log \left (x - 4\right )^{2} - 1\right ) - \log \relax (x) - 2 \, \log \left (\log \relax (3)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((50*x^4-200*x^3)*log(x^2-8*x+16)^2+100*x^4*log(x^2-8*x+16)+2*x-8)/((25*x^5-100*x^4)*log(x^2-8*x+16)
^2-2*x^2+8*x)/log(1/2*(25*x^3*log(x^2-8*x+16)^2-2)/x/log(3)^2),x, algorithm="maxima")

[Out]

log(log(50*x^3*log(x - 4)^2 - 1) - log(x) - 2*log(log(3)))

________________________________________________________________________________________

mupad [B]  time = 5.91, size = 29, normalized size = 0.94 \begin {gather*} \ln \left (\ln \left (\frac {25\,x^3\,{\ln \left (x^2-8\,x+16\right )}^2-2}{2\,x\,{\ln \relax (3)}^2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - log(x^2 - 8*x + 16)^2*(200*x^3 - 50*x^4) + 100*x^4*log(x^2 - 8*x + 16) - 8)/(log(((25*x^3*log(x^2
- 8*x + 16)^2)/2 - 1)/(x*log(3)^2))*(log(x^2 - 8*x + 16)^2*(100*x^4 - 25*x^5) - 8*x + 2*x^2)),x)

[Out]

log(log((25*x^3*log(x^2 - 8*x + 16)^2 - 2)/(2*x*log(3)^2)))

________________________________________________________________________________________

sympy [A]  time = 0.79, size = 29, normalized size = 0.94 \begin {gather*} \log {\left (\log {\left (\frac {\frac {25 x^{3} \log {\left (x^{2} - 8 x + 16 \right )}^{2}}{2} - 1}{x \log {\relax (3 )}^{2}} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((50*x**4-200*x**3)*ln(x**2-8*x+16)**2+100*x**4*ln(x**2-8*x+16)+2*x-8)/((25*x**5-100*x**4)*ln(x**2-8
*x+16)**2-2*x**2+8*x)/ln(1/2*(25*x**3*ln(x**2-8*x+16)**2-2)/x/ln(3)**2),x)

[Out]

log(log((25*x**3*log(x**2 - 8*x + 16)**2/2 - 1)/(x*log(3)**2)))

________________________________________________________________________________________