Integrand size = 101, antiderivative size = 29 \[ \int \frac {2496 x \log (1-2 x)+(240-480 x) \log ^2(1-2 x)+\left (-768 x \log (1-2 x)+(-192+384 x) \log ^2(1-2 x)\right ) \log \left (x^2\right )}{-169 x^2+338 x^3+\left (104 x^2-208 x^3\right ) \log \left (x^2\right )+\left (-16 x^2+32 x^3\right ) \log ^2\left (x^2\right )} \, dx=\frac {16 \log ^2(1-2 x)}{-x+\frac {4}{3} x \left (4-\log \left (x^2\right )\right )} \]
Time = 0.34 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {2496 x \log (1-2 x)+(240-480 x) \log ^2(1-2 x)+\left (-768 x \log (1-2 x)+(-192+384 x) \log ^2(1-2 x)\right ) \log \left (x^2\right )}{-169 x^2+338 x^3+\left (104 x^2-208 x^3\right ) \log \left (x^2\right )+\left (-16 x^2+32 x^3\right ) \log ^2\left (x^2\right )} \, dx=-\frac {48 \log ^2(1-2 x)}{x \left (-13+4 \log \left (x^2\right )\right )} \]
Integrate[(2496*x*Log[1 - 2*x] + (240 - 480*x)*Log[1 - 2*x]^2 + (-768*x*Lo g[1 - 2*x] + (-192 + 384*x)*Log[1 - 2*x]^2)*Log[x^2])/(-169*x^2 + 338*x^3 + (104*x^2 - 208*x^3)*Log[x^2] + (-16*x^2 + 32*x^3)*Log[x^2]^2),x]
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.
\(\displaystyle \int \frac {\left ((384 x-192) \log ^2(1-2 x)-768 x \log (1-2 x)\right ) \log \left (x^2\right )+(240-480 x) \log ^2(1-2 x)+2496 x \log (1-2 x)}{338 x^3-169 x^2+\left (32 x^3-16 x^2\right ) \log ^2\left (x^2\right )+\left (104 x^2-208 x^3\right ) \log \left (x^2\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-\left ((384 x-192) \log ^2(1-2 x)-768 x \log (1-2 x)\right ) \log \left (x^2\right )-\left ((240-480 x) \log ^2(1-2 x)\right )-2496 x \log (1-2 x)}{(1-2 x) x^2 \left (13-4 \log \left (x^2\right )\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {384 \log ^2(1-2 x)}{x^2 \left (4 \log \left (x^2\right )-13\right )^2}+\frac {48 (-4 x+2 x \log (1-2 x)-\log (1-2 x)) \log (1-2 x)}{x^2 (2 x-1) \left (4 \log \left (x^2\right )-13\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 384 \int \frac {\log ^2(1-2 x)}{x^2 \left (4 \log \left (x^2\right )-13\right )^2}dx+48 \int \frac {\log ^2(1-2 x)}{x^2 \left (4 \log \left (x^2\right )-13\right )}dx+192 \int \frac {\log (1-2 x)}{x \left (4 \log \left (x^2\right )-13\right )}dx-384 \int \frac {\log (1-2 x)}{(2 x-1) \left (4 \log \left (x^2\right )-13\right )}dx\) |
Int[(2496*x*Log[1 - 2*x] + (240 - 480*x)*Log[1 - 2*x]^2 + (-768*x*Log[1 - 2*x] + (-192 + 384*x)*Log[1 - 2*x]^2)*Log[x^2])/(-169*x^2 + 338*x^3 + (104 *x^2 - 208*x^3)*Log[x^2] + (-16*x^2 + 32*x^3)*Log[x^2]^2),x]
3.7.85.3.1 Defintions of rubi rules used
Time = 1.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83
method | result | size |
parallelrisch | \(-\frac {48 \ln \left (1-2 x \right )^{2}}{x \left (-13+4 \ln \left (x^{2}\right )\right )}\) | \(24\) |
risch | \(-\frac {48 i \ln \left (1-2 x \right )^{2}}{x \left (2 \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-4 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+2 \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+8 i \ln \left (x \right )-13 i\right )}\) | \(71\) |
int((((384*x-192)*ln(1-2*x)^2-768*x*ln(1-2*x))*ln(x^2)+(-480*x+240)*ln(1-2 *x)^2+2496*x*ln(1-2*x))/((32*x^3-16*x^2)*ln(x^2)^2+(-208*x^3+104*x^2)*ln(x ^2)+338*x^3-169*x^2),x,method=_RETURNVERBOSE)
Time = 0.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {2496 x \log (1-2 x)+(240-480 x) \log ^2(1-2 x)+\left (-768 x \log (1-2 x)+(-192+384 x) \log ^2(1-2 x)\right ) \log \left (x^2\right )}{-169 x^2+338 x^3+\left (104 x^2-208 x^3\right ) \log \left (x^2\right )+\left (-16 x^2+32 x^3\right ) \log ^2\left (x^2\right )} \, dx=-\frac {48 \, \log \left (-2 \, x + 1\right )^{2}}{4 \, x \log \left (x^{2}\right ) - 13 \, x} \]
integrate((((384*x-192)*log(1-2*x)^2-768*x*log(1-2*x))*log(x^2)+(-480*x+24 0)*log(1-2*x)^2+2496*x*log(1-2*x))/((32*x^3-16*x^2)*log(x^2)^2+(-208*x^3+1 04*x^2)*log(x^2)+338*x^3-169*x^2),x, algorithm=\
Exception generated. \[ \int \frac {2496 x \log (1-2 x)+(240-480 x) \log ^2(1-2 x)+\left (-768 x \log (1-2 x)+(-192+384 x) \log ^2(1-2 x)\right ) \log \left (x^2\right )}{-169 x^2+338 x^3+\left (104 x^2-208 x^3\right ) \log \left (x^2\right )+\left (-16 x^2+32 x^3\right ) \log ^2\left (x^2\right )} \, dx=\text {Exception raised: TypeError} \]
integrate((((384*x-192)*ln(1-2*x)**2-768*x*ln(1-2*x))*ln(x**2)+(-480*x+240 )*ln(1-2*x)**2+2496*x*ln(1-2*x))/((32*x**3-16*x**2)*ln(x**2)**2+(-208*x**3 +104*x**2)*ln(x**2)+338*x**3-169*x**2),x)
Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {2496 x \log (1-2 x)+(240-480 x) \log ^2(1-2 x)+\left (-768 x \log (1-2 x)+(-192+384 x) \log ^2(1-2 x)\right ) \log \left (x^2\right )}{-169 x^2+338 x^3+\left (104 x^2-208 x^3\right ) \log \left (x^2\right )+\left (-16 x^2+32 x^3\right ) \log ^2\left (x^2\right )} \, dx=-\frac {48 \, \log \left (-2 \, x + 1\right )^{2}}{8 \, x \log \left (x\right ) - 13 \, x} \]
integrate((((384*x-192)*log(1-2*x)^2-768*x*log(1-2*x))*log(x^2)+(-480*x+24 0)*log(1-2*x)^2+2496*x*log(1-2*x))/((32*x^3-16*x^2)*log(x^2)^2+(-208*x^3+1 04*x^2)*log(x^2)+338*x^3-169*x^2),x, algorithm=\
Time = 0.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {2496 x \log (1-2 x)+(240-480 x) \log ^2(1-2 x)+\left (-768 x \log (1-2 x)+(-192+384 x) \log ^2(1-2 x)\right ) \log \left (x^2\right )}{-169 x^2+338 x^3+\left (104 x^2-208 x^3\right ) \log \left (x^2\right )+\left (-16 x^2+32 x^3\right ) \log ^2\left (x^2\right )} \, dx=-\frac {48 \, \log \left (-2 \, x + 1\right )^{2}}{4 \, x \log \left (x^{2}\right ) - 13 \, x} \]
integrate((((384*x-192)*log(1-2*x)^2-768*x*log(1-2*x))*log(x^2)+(-480*x+24 0)*log(1-2*x)^2+2496*x*log(1-2*x))/((32*x^3-16*x^2)*log(x^2)^2+(-208*x^3+1 04*x^2)*log(x^2)+338*x^3-169*x^2),x, algorithm=\
Time = 13.47 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {2496 x \log (1-2 x)+(240-480 x) \log ^2(1-2 x)+\left (-768 x \log (1-2 x)+(-192+384 x) \log ^2(1-2 x)\right ) \log \left (x^2\right )}{-169 x^2+338 x^3+\left (104 x^2-208 x^3\right ) \log \left (x^2\right )+\left (-16 x^2+32 x^3\right ) \log ^2\left (x^2\right )} \, dx=-\frac {12\,{\ln \left (1-2\,x\right )}^2}{x\,\left (\ln \left (x^2\right )-\frac {13}{4}\right )} \]