Integrand size = 102, antiderivative size = 24 \[ \int \frac {\left (-126-447 x-576 x^2-320 x^3-64 x^4\right ) \log \left (\frac {2}{2+x}\right )-2 \log \left (\log \left (\frac {2}{2+x}\right )\right )}{\left (-28-140 x-255 x^2-224 x^3-96 x^4-16 x^5\right ) \log \left (\frac {2}{2+x}\right )+(2+x) \log \left (\frac {2}{2+x}\right ) \log ^2\left (\log \left (\frac {2}{2+x}\right )\right )} \, dx=\log \left (2+x-(2+2 x)^4+\log ^2\left (\log \left (\frac {2}{2+x}\right )\right )\right ) \] Output:
ln(x+2+ln(ln(2/(2+x)))^2-(2+2*x)^4)
Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.75 \[ \int \frac {\left (-126-447 x-576 x^2-320 x^3-64 x^4\right ) \log \left (\frac {2}{2+x}\right )-2 \log \left (\log \left (\frac {2}{2+x}\right )\right )}{\left (-28-140 x-255 x^2-224 x^3-96 x^4-16 x^5\right ) \log \left (\frac {2}{2+x}\right )+(2+x) \log \left (\frac {2}{2+x}\right ) \log ^2\left (\log \left (\frac {2}{2+x}\right )\right )} \, dx=\log \left (16-65 (2+x)+96 (2+x)^2-64 (2+x)^3+16 (2+x)^4-\log ^2\left (\log \left (\frac {2}{2+x}\right )\right )\right ) \] Input:
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]
Output:
Log[16 - 65*(2 + x) + 96*(2 + x)^2 - 64*(2 + x)^3 + 16*(2 + x)^4 - Log[Log [2/(2 + x)]]^2]
Time = 0.58 (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 = {7292, 7235}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (-64 x^4-320 x^3-576 x^2-447 x-126\right ) \log \left (\frac {2}{x+2}\right )-2 \log \left (\log \left (\frac {2}{x+2}\right )\right )}{\left (-16 x^5-96 x^4-224 x^3-255 x^2-140 x-28\right ) \log \left (\frac {2}{x+2}\right )+(x+2) \log \left (\frac {2}{x+2}\right ) \log ^2\left (\log \left (\frac {2}{x+2}\right )\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {2 \log \left (\log \left (\frac {2}{x+2}\right )\right )-\left (-64 x^4-320 x^3-576 x^2-447 x-126\right ) \log \left (\frac {2}{x+2}\right )}{(x+2) \log \left (\frac {2}{x+2}\right ) \left (16 x^4+64 x^3+96 x^2+63 x-\log ^2\left (\log \left (\frac {2}{x+2}\right )\right )+14\right )}dx\) |
\(\Big \downarrow \) 7235 |
\(\displaystyle \log \left (16 x^4+64 x^3+96 x^2+63 x-\log ^2\left (\log \left (\frac {2}{x+2}\right )\right )+14\right )\) |
Input:
Int[((-126 - 447*x - 576*x^2 - 320*x^3 - 64*x^4)*Log[2/(2 + x)] - 2*Log[Lo g[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]
Output:
Log[14 + 63*x + 96*x^2 + 64*x^3 + 16*x^4 - Log[Log[2/(2 + x)]]^2]
Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*L og[RemoveContent[y, x]], x] /; !FalseQ[q]]
Time = 0.98 (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 | \(\ln \left (\ln \left (\ln \left (2\right )+\ln \left (\frac {1}{2+x}\right )\right )^{2}+64 \left (2+x \right )^{3}+114+65 x -96 \left (2+x \right )^{2}-16 \left (2+x \right )^{4}\right )\) | \(40\) |
Input:
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)
Output:
ln(x^4+4*x^3+6*x^2-1/16*ln(ln(2/(2+x)))^2+63/16*x+7/8)
Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {\left (-126-447 x-576 x^2-320 x^3-64 x^4\right ) \log \left (\frac {2}{2+x}\right )-2 \log \left (\log \left (\frac {2}{2+x}\right )\right )}{\left (-28-140 x-255 x^2-224 x^3-96 x^4-16 x^5\right ) \log \left (\frac {2}{2+x}\right )+(2+x) \log \left (\frac {2}{2+x}\right ) \log ^2\left (\log \left (\frac {2}{2+x}\right )\right )} \, dx=\log \left (-16 \, x^{4} - 64 \, x^{3} - 96 \, x^{2} + \log \left (\log \left (\frac {2}{x + 2}\right )\right )^{2} - 63 \, x - 14\right ) \] Input:
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^3-25 5*x^2-140*x-28)*log(2/(2+x))),x, algorithm="fricas")
Output:
log(-16*x^4 - 64*x^3 - 96*x^2 + log(log(2/(x + 2)))^2 - 63*x - 14)
Time = 0.33 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {\left (-126-447 x-576 x^2-320 x^3-64 x^4\right ) \log \left (\frac {2}{2+x}\right )-2 \log \left (\log \left (\frac {2}{2+x}\right )\right )}{\left (-28-140 x-255 x^2-224 x^3-96 x^4-16 x^5\right ) \log \left (\frac {2}{2+x}\right )+(2+x) \log \left (\frac {2}{2+x}\right ) \log ^2\left (\log \left (\frac {2}{2+x}\right )\right )} \, dx=\log {\left (- 16 x^{4} - 64 x^{3} - 96 x^{2} - 63 x + \log {\left (\log {\left (\frac {2}{x + 2} \right )} \right )}^{2} - 14 \right )} \] Input:
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**3-2 55*x**2-140*x-28)*ln(2/(2+x))),x)
Output:
log(-16*x**4 - 64*x**3 - 96*x**2 - 63*x + log(log(2/(x + 2)))**2 - 14)
Time = 0.17 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38 \[ \int \frac {\left (-126-447 x-576 x^2-320 x^3-64 x^4\right ) \log \left (\frac {2}{2+x}\right )-2 \log \left (\log \left (\frac {2}{2+x}\right )\right )}{\left (-28-140 x-255 x^2-224 x^3-96 x^4-16 x^5\right ) \log \left (\frac {2}{2+x}\right )+(2+x) \log \left (\frac {2}{2+x}\right ) \log ^2\left (\log \left (\frac {2}{2+x}\right )\right )} \, dx=\log \left (-16 \, x^{4} - 64 \, x^{3} - 96 \, x^{2} + \log \left (\log \left (2\right ) - \log \left (x + 2\right )\right )^{2} - 63 \, x - 14\right ) \] Input:
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^3-25 5*x^2-140*x-28)*log(2/(2+x))),x, algorithm="maxima")
Output:
log(-16*x^4 - 64*x^3 - 96*x^2 + log(log(2) - log(x + 2))^2 - 63*x - 14)
\[ \int \frac {\left (-126-447 x-576 x^2-320 x^3-64 x^4\right ) \log \left (\frac {2}{2+x}\right )-2 \log \left (\log \left (\frac {2}{2+x}\right )\right )}{\left (-28-140 x-255 x^2-224 x^3-96 x^4-16 x^5\right ) \log \left (\frac {2}{2+x}\right )+(2+x) \log \left (\frac {2}{2+x}\right ) \log ^2\left (\log \left (\frac {2}{2+x}\right )\right )} \, dx=\int { -\frac {{\left (64 \, x^{4} + 320 \, x^{3} + 576 \, x^{2} + 447 \, x + 126\right )} \log \left (\frac {2}{x + 2}\right ) + 2 \, \log \left (\log \left (\frac {2}{x + 2}\right )\right )}{{\left (x + 2\right )} \log \left (\frac {2}{x + 2}\right ) \log \left (\log \left (\frac {2}{x + 2}\right )\right )^{2} - {\left (16 \, x^{5} + 96 \, x^{4} + 224 \, x^{3} + 255 \, x^{2} + 140 \, x + 28\right )} \log \left (\frac {2}{x + 2}\right )} \,d x } \] Input:
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^3-25 5*x^2-140*x-28)*log(2/(2+x))),x, algorithm="giac")
Output:
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)
Time = 3.21 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {\left (-126-447 x-576 x^2-320 x^3-64 x^4\right ) \log \left (\frac {2}{2+x}\right )-2 \log \left (\log \left (\frac {2}{2+x}\right )\right )}{\left (-28-140 x-255 x^2-224 x^3-96 x^4-16 x^5\right ) \log \left (\frac {2}{2+x}\right )+(2+x) \log \left (\frac {2}{2+x}\right ) \log ^2\left (\log \left (\frac {2}{2+x}\right )\right )} \, dx=\ln \left (-16\,x^4-64\,x^3-96\,x^2-63\,x+{\ln \left (\ln \left (\frac {2}{x+2}\right )\right )}^2-14\right ) \] Input:
int((2*log(log(2/(x + 2))) + log(2/(x + 2))*(447*x + 576*x^2 + 320*x^3 + 6 4*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)
Output:
log(log(log(2/(x + 2)))^2 - 63*x - 96*x^2 - 64*x^3 - 16*x^4 - 14)
\[ \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=-64 \left (\int \frac {x^{4}}{\mathrm {log}\left (\mathrm {log}\left (\frac {2}{x +2}\right )\right )^{2} x +2 \mathrm {log}\left (\mathrm {log}\left (\frac {2}{x +2}\right )\right )^{2}-16 x^{5}-96 x^{4}-224 x^{3}-255 x^{2}-140 x -28}d x \right )-320 \left (\int \frac {x^{3}}{\mathrm {log}\left (\mathrm {log}\left (\frac {2}{x +2}\right )\right )^{2} x +2 \mathrm {log}\left (\mathrm {log}\left (\frac {2}{x +2}\right )\right )^{2}-16 x^{5}-96 x^{4}-224 x^{3}-255 x^{2}-140 x -28}d x \right )-576 \left (\int \frac {x^{2}}{\mathrm {log}\left (\mathrm {log}\left (\frac {2}{x +2}\right )\right )^{2} x +2 \mathrm {log}\left (\mathrm {log}\left (\frac {2}{x +2}\right )\right )^{2}-16 x^{5}-96 x^{4}-224 x^{3}-255 x^{2}-140 x -28}d x \right )-2 \left (\int \frac {\mathrm {log}\left (\mathrm {log}\left (\frac {2}{x +2}\right )\right )}{\mathrm {log}\left (\mathrm {log}\left (\frac {2}{x +2}\right )\right )^{2} \mathrm {log}\left (\frac {2}{x +2}\right ) x +2 \mathrm {log}\left (\mathrm {log}\left (\frac {2}{x +2}\right )\right )^{2} \mathrm {log}\left (\frac {2}{x +2}\right )-16 \,\mathrm {log}\left (\frac {2}{x +2}\right ) x^{5}-96 \,\mathrm {log}\left (\frac {2}{x +2}\right ) x^{4}-224 \,\mathrm {log}\left (\frac {2}{x +2}\right ) x^{3}-255 \,\mathrm {log}\left (\frac {2}{x +2}\right ) x^{2}-140 \,\mathrm {log}\left (\frac {2}{x +2}\right ) x -28 \,\mathrm {log}\left (\frac {2}{x +2}\right )}d x \right )-447 \left (\int \frac {x}{\mathrm {log}\left (\mathrm {log}\left (\frac {2}{x +2}\right )\right )^{2} x +2 \mathrm {log}\left (\mathrm {log}\left (\frac {2}{x +2}\right )\right )^{2}-16 x^{5}-96 x^{4}-224 x^{3}-255 x^{2}-140 x -28}d x \right )-126 \left (\int \frac {1}{\mathrm {log}\left (\mathrm {log}\left (\frac {2}{x +2}\right )\right )^{2} x +2 \mathrm {log}\left (\mathrm {log}\left (\frac {2}{x +2}\right )\right )^{2}-16 x^{5}-96 x^{4}-224 x^{3}-255 x^{2}-140 x -28}d x \right ) \] Input:
int((-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)
Output:
- 64*int(x**4/(log(log(2/(x + 2)))**2*x + 2*log(log(2/(x + 2)))**2 - 16*x **5 - 96*x**4 - 224*x**3 - 255*x**2 - 140*x - 28),x) - 320*int(x**3/(log(l og(2/(x + 2)))**2*x + 2*log(log(2/(x + 2)))**2 - 16*x**5 - 96*x**4 - 224*x **3 - 255*x**2 - 140*x - 28),x) - 576*int(x**2/(log(log(2/(x + 2)))**2*x + 2*log(log(2/(x + 2)))**2 - 16*x**5 - 96*x**4 - 224*x**3 - 255*x**2 - 140* x - 28),x) - 2*int(log(log(2/(x + 2)))/(log(log(2/(x + 2)))**2*log(2/(x + 2))*x + 2*log(log(2/(x + 2)))**2*log(2/(x + 2)) - 16*log(2/(x + 2))*x**5 - 96*log(2/(x + 2))*x**4 - 224*log(2/(x + 2))*x**3 - 255*log(2/(x + 2))*x** 2 - 140*log(2/(x + 2))*x - 28*log(2/(x + 2))),x) - 447*int(x/(log(log(2/(x + 2)))**2*x + 2*log(log(2/(x + 2)))**2 - 16*x**5 - 96*x**4 - 224*x**3 - 2 55*x**2 - 140*x - 28),x) - 126*int(1/(log(log(2/(x + 2)))**2*x + 2*log(log (2/(x + 2)))**2 - 16*x**5 - 96*x**4 - 224*x**3 - 255*x**2 - 140*x - 28),x)