\(\int \frac {(-126-447 x-576 x^2-320 x^3-64 x^4) \log (\frac {2}{2+x})-2 \log (\log (\frac {2}{2+x}))}{(-28-140 x-255 x^2-224 x^3-96 x^4-16 x^5) \log (\frac {2}{2+x})+(2+x) \log (\frac {2}{2+x}) \log ^2(\log (\frac {2}{2+x}))} \, dx\) [8010]

Optimal result

Integrand size = 102, antiderivative size = 24 \[ \int \frac {\left (-126-447 x-576 x^2-320 x^3-64 x^4\right ) \log \left (\frac {2}{2+x}\right )-2 \log \left (\log \left (\frac {2}{2+x}\right )\right )}{\left (-28-140 x-255 x^2-224 x^3-96 x^4-16 x^5\right ) \log \left (\frac {2}{2+x}\right )+(2+x) \log \left (\frac {2}{2+x}\right ) \log ^2\left (\log \left (\frac {2}{2+x}\right )\right )} \, dx=\log \left (2+x-(2+2 x)^4+\log ^2\left (\log \left (\frac {2}{2+x}\right )\right )\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {6873, 6816} \[ \int \frac {\left (-126-447 x-576 x^2-320 x^3-64 x^4\right ) \log \left (\frac {2}{2+x}\right )-2 \log \left (\log \left (\frac {2}{2+x}\right )\right )}{\left (-28-140 x-255 x^2-224 x^3-96 x^4-16 x^5\right ) \log \left (\frac {2}{2+x}\right )+(2+x) \log \left (\frac {2}{2+x}\right ) \log ^2\left (\log \left (\frac {2}{2+x}\right )\right )} \, dx=\log \left (16 x^4+64 x^3+96 x^2+63 x-\log ^2\left (\log \left (\frac {2}{x+2}\right )\right )+14\right ) \]

[In]

Int[((-126 - 447*x - 576*x^2 - 320*x^3 - 64*x^4)*Log[2/(2 + x)] - 2*Log[Log[2/(2 + x)]])/((-28 - 140*x - 255*x
^2 - 224*x^3 - 96*x^4 - 16*x^5)*Log[2/(2 + x)] + (2 + x)*Log[2/(2 + x)]*Log[Log[2/(2 + x)]]^2),x]

[Out]

Log[14 + 63*x + 96*x^2 + 64*x^3 + 16*x^4 - Log[Log[2/(2 + x)]]^2]

Rule 6816

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

Rule 6873

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {-\left (\left (-126-447 x-576 x^2-320 x^3-64 x^4\right ) \log \left (\frac {2}{2+x}\right )\right )+2 \log \left (\log \left (\frac {2}{2+x}\right )\right )}{(2+x) \log \left (\frac {2}{2+x}\right ) \left (14+63 x+96 x^2+64 x^3+16 x^4-\log ^2\left (\log \left (\frac {2}{2+x}\right )\right )\right )} \, dx \\ & = \log \left (14+63 x+96 x^2+64 x^3+16 x^4-\log ^2\left (\log \left (\frac {2}{2+x}\right )\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.75 \[ \int \frac {\left (-126-447 x-576 x^2-320 x^3-64 x^4\right ) \log \left (\frac {2}{2+x}\right )-2 \log \left (\log \left (\frac {2}{2+x}\right )\right )}{\left (-28-140 x-255 x^2-224 x^3-96 x^4-16 x^5\right ) \log \left (\frac {2}{2+x}\right )+(2+x) \log \left (\frac {2}{2+x}\right ) \log ^2\left (\log \left (\frac {2}{2+x}\right )\right )} \, dx=\log \left (16-65 (2+x)+96 (2+x)^2-64 (2+x)^3+16 (2+x)^4-\log ^2\left (\log \left (\frac {2}{2+x}\right )\right )\right ) \]

[In]

Integrate[((-126 - 447*x - 576*x^2 - 320*x^3 - 64*x^4)*Log[2/(2 + x)] - 2*Log[Log[2/(2 + x)]])/((-28 - 140*x -
255*x^2 - 224*x^3 - 96*x^4 - 16*x^5)*Log[2/(2 + x)] + (2 + x)*Log[2/(2 + x)]*Log[Log[2/(2 + x)]]^2),x]

[Out]

Log[16 - 65*(2 + x) + 96*(2 + x)^2 - 64*(2 + x)^3 + 16*(2 + x)^4 - Log[Log[2/(2 + x)]]^2]

Maple [A] (verified)

Time = 1.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38

method	result	size
parallelrisch	\(\ln \left (x^{4}+4 x^{3}+6 x^{2}-\frac {\ln \left (\ln \left (\frac {2}{2+x}\right )\right )^{2}}{16}+\frac {63 x}{16}+\frac {7}{8}\right )\)	\(33\)
default	\(\ln \left (114+65 x -96 \left (2+x \right )^{2}+64 \left (2+x \right )^{3}+\ln \left (\ln \left (2\right )+\ln \left (\frac {1}{2+x}\right )\right )^{2}-16 \left (2+x \right )^{4}\right )\)	\(40\)

[In]

int((-2*ln(ln(2/(2+x)))+(-64*x^4-320*x^3-576*x^2-447*x-126)*ln(2/(2+x)))/((2+x)*ln(2/(2+x))*ln(ln(2/(2+x)))^2+
(-16*x^5-96*x^4-224*x^3-255*x^2-140*x-28)*ln(2/(2+x))),x,method=_RETURNVERBOSE)

[Out]

ln(x^4+4*x^3+6*x^2-1/16*ln(ln(2/(2+x)))^2+63/16*x+7/8)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {\left (-126-447 x-576 x^2-320 x^3-64 x^4\right ) \log \left (\frac {2}{2+x}\right )-2 \log \left (\log \left (\frac {2}{2+x}\right )\right )}{\left (-28-140 x-255 x^2-224 x^3-96 x^4-16 x^5\right ) \log \left (\frac {2}{2+x}\right )+(2+x) \log \left (\frac {2}{2+x}\right ) \log ^2\left (\log \left (\frac {2}{2+x}\right )\right )} \, dx=\log \left (-16 \, x^{4} - 64 \, x^{3} - 96 \, x^{2} + \log \left (\log \left (\frac {2}{x + 2}\right )\right )^{2} - 63 \, x - 14\right ) \]

[In]

integrate((-2*log(log(2/(2+x)))+(-64*x^4-320*x^3-576*x^2-447*x-126)*log(2/(2+x)))/((2+x)*log(2/(2+x))*log(log(
2/(2+x)))^2+(-16*x^5-96*x^4-224*x^3-255*x^2-140*x-28)*log(2/(2+x))),x, algorithm="fricas")

[Out]

log(-16*x^4 - 64*x^3 - 96*x^2 + log(log(2/(x + 2)))^2 - 63*x - 14)

Sympy [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {\left (-126-447 x-576 x^2-320 x^3-64 x^4\right ) \log \left (\frac {2}{2+x}\right )-2 \log \left (\log \left (\frac {2}{2+x}\right )\right )}{\left (-28-140 x-255 x^2-224 x^3-96 x^4-16 x^5\right ) \log \left (\frac {2}{2+x}\right )+(2+x) \log \left (\frac {2}{2+x}\right ) \log ^2\left (\log \left (\frac {2}{2+x}\right )\right )} \, dx=\log {\left (- 16 x^{4} - 64 x^{3} - 96 x^{2} - 63 x + \log {\left (\log {\left (\frac {2}{x + 2} \right )} \right )}^{2} - 14 \right )} \]

[In]

integrate((-2*ln(ln(2/(2+x)))+(-64*x**4-320*x**3-576*x**2-447*x-126)*ln(2/(2+x)))/((2+x)*ln(2/(2+x))*ln(ln(2/(
2+x)))**2+(-16*x**5-96*x**4-224*x**3-255*x**2-140*x-28)*ln(2/(2+x))),x)

[Out]

log(-16*x**4 - 64*x**3 - 96*x**2 - 63*x + log(log(2/(x + 2)))**2 - 14)

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38 \[ \int \frac {\left (-126-447 x-576 x^2-320 x^3-64 x^4\right ) \log \left (\frac {2}{2+x}\right )-2 \log \left (\log \left (\frac {2}{2+x}\right )\right )}{\left (-28-140 x-255 x^2-224 x^3-96 x^4-16 x^5\right ) \log \left (\frac {2}{2+x}\right )+(2+x) \log \left (\frac {2}{2+x}\right ) \log ^2\left (\log \left (\frac {2}{2+x}\right )\right )} \, dx=\log \left (-16 \, x^{4} - 64 \, x^{3} - 96 \, x^{2} + \log \left (\log \left (2\right ) - \log \left (x + 2\right )\right )^{2} - 63 \, x - 14\right ) \]

[In]

integrate((-2*log(log(2/(2+x)))+(-64*x^4-320*x^3-576*x^2-447*x-126)*log(2/(2+x)))/((2+x)*log(2/(2+x))*log(log(
2/(2+x)))^2+(-16*x^5-96*x^4-224*x^3-255*x^2-140*x-28)*log(2/(2+x))),x, algorithm="maxima")

[Out]

log(-16*x^4 - 64*x^3 - 96*x^2 + log(log(2) - log(x + 2))^2 - 63*x - 14)

Giac [F]

\[ \int \frac {\left (-126-447 x-576 x^2-320 x^3-64 x^4\right ) \log \left (\frac {2}{2+x}\right )-2 \log \left (\log \left (\frac {2}{2+x}\right )\right )}{\left (-28-140 x-255 x^2-224 x^3-96 x^4-16 x^5\right ) \log \left (\frac {2}{2+x}\right )+(2+x) \log \left (\frac {2}{2+x}\right ) \log ^2\left (\log \left (\frac {2}{2+x}\right )\right )} \, dx=\int { -\frac {{\left (64 \, x^{4} + 320 \, x^{3} + 576 \, x^{2} + 447 \, x + 126\right )} \log \left (\frac {2}{x + 2}\right ) + 2 \, \log \left (\log \left (\frac {2}{x + 2}\right )\right )}{{\left (x + 2\right )} \log \left (\frac {2}{x + 2}\right ) \log \left (\log \left (\frac {2}{x + 2}\right )\right )^{2} - {\left (16 \, x^{5} + 96 \, x^{4} + 224 \, x^{3} + 255 \, x^{2} + 140 \, x + 28\right )} \log \left (\frac {2}{x + 2}\right )} \,d x } \]

[In]

integrate((-2*log(log(2/(2+x)))+(-64*x^4-320*x^3-576*x^2-447*x-126)*log(2/(2+x)))/((2+x)*log(2/(2+x))*log(log(
2/(2+x)))^2+(-16*x^5-96*x^4-224*x^3-255*x^2-140*x-28)*log(2/(2+x))),x, algorithm="giac")

[Out]

integrate(-((64*x^4 + 320*x^3 + 576*x^2 + 447*x + 126)*log(2/(x + 2)) + 2*log(log(2/(x + 2))))/((x + 2)*log(2/
(x + 2))*log(log(2/(x + 2)))^2 - (16*x^5 + 96*x^4 + 224*x^3 + 255*x^2 + 140*x + 28)*log(2/(x + 2))), x)

Mupad [B] (verification not implemented)

Time = 13.46 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {\left (-126-447 x-576 x^2-320 x^3-64 x^4\right ) \log \left (\frac {2}{2+x}\right )-2 \log \left (\log \left (\frac {2}{2+x}\right )\right )}{\left (-28-140 x-255 x^2-224 x^3-96 x^4-16 x^5\right ) \log \left (\frac {2}{2+x}\right )+(2+x) \log \left (\frac {2}{2+x}\right ) \log ^2\left (\log \left (\frac {2}{2+x}\right )\right )} \, dx=\ln \left (-16\,x^4-64\,x^3-96\,x^2-63\,x+{\ln \left (\ln \left (\frac {2}{x+2}\right )\right )}^2-14\right ) \]

[In]

int((2*log(log(2/(x + 2))) + log(2/(x + 2))*(447*x + 576*x^2 + 320*x^3 + 64*x^4 + 126))/(log(2/(x + 2))*(140*x
+ 255*x^2 + 224*x^3 + 96*x^4 + 16*x^5 + 28) - log(2/(x + 2))*log(log(2/(x + 2)))^2*(x + 2)),x)

[Out]

log(log(log(2/(x + 2)))^2 - 63*x - 96*x^2 - 64*x^3 - 16*x^4 - 14)