\(\int \frac {2496 x \log (1-2 x)+(240-480 x) \log ^2(1-2 x)+(-768 x \log (1-2 x)+(-192+384 x) \log ^2(1-2 x)) \log (x^2)}{-169 x^2+338 x^3+(104 x^2-208 x^3) \log (x^2)+(-16 x^2+32 x^3) \log ^2(x^2)} \, dx\) [6685]

Optimal result

Integrand size = 101, antiderivative size = 29 \[ \int \frac {2496 x \log (1-2 x)+(240-480 x) \log ^2(1-2 x)+\left (-768 x \log (1-2 x)+(-192+384 x) \log ^2(1-2 x)\right ) \log \left (x^2\right )}{-169 x^2+338 x^3+\left (104 x^2-208 x^3\right ) \log \left (x^2\right )+\left (-16 x^2+32 x^3\right ) \log ^2\left (x^2\right )} \, dx=\frac {16 \log ^2(1-2 x)}{-x+\frac {4}{3} x \left (4-\log \left (x^2\right )\right )} \]

[Out]

16*ln(1-2*x)^2/(x*(16/3-4/3*ln(x^2))-x)

Rubi [F]

\[ \int \frac {2496 x \log (1-2 x)+(240-480 x) \log ^2(1-2 x)+\left (-768 x \log (1-2 x)+(-192+384 x) \log ^2(1-2 x)\right ) \log \left (x^2\right )}{-169 x^2+338 x^3+\left (104 x^2-208 x^3\right ) \log \left (x^2\right )+\left (-16 x^2+32 x^3\right ) \log ^2\left (x^2\right )} \, dx=\int \frac {2496 x \log (1-2 x)+(240-480 x) \log ^2(1-2 x)+\left (-768 x \log (1-2 x)+(-192+384 x) \log ^2(1-2 x)\right ) \log \left (x^2\right )}{-169 x^2+338 x^3+\left (104 x^2-208 x^3\right ) \log \left (x^2\right )+\left (-16 x^2+32 x^3\right ) \log ^2\left (x^2\right )} \, dx \]

[In]

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]

[Out]

384*Defer[Int][Log[1 - 2*x]^2/(x^2*(-13 + 4*Log[x^2])^2), x] + 192*Defer[Int][Log[1 - 2*x]/(x*(-13 + 4*Log[x^2
])), x] - 384*Defer[Int][Log[1 - 2*x]/((-1 + 2*x)*(-13 + 4*Log[x^2])), x] + 48*Defer[Int][Log[1 - 2*x]^2/(x^2*
(-13 + 4*Log[x^2])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {48 \log (1-2 x) \left (-4 x \left (13-4 \log \left (x^2\right )\right )-(-1+2 x) \log (1-2 x) \left (-5+4 \log \left (x^2\right )\right )\right )}{(1-2 x) x^2 \left (13-4 \log \left (x^2\right )\right )^2} \, dx \\ & = 48 \int \frac {\log (1-2 x) \left (-4 x \left (13-4 \log \left (x^2\right )\right )-(-1+2 x) \log (1-2 x) \left (-5+4 \log \left (x^2\right )\right )\right )}{(1-2 x) x^2 \left (13-4 \log \left (x^2\right )\right )^2} \, dx \\ & = 48 \int \left (\frac {8 \log ^2(1-2 x)}{x^2 \left (-13+4 \log \left (x^2\right )\right )^2}+\frac {\log (1-2 x) (-4 x-\log (1-2 x)+2 x \log (1-2 x))}{x^2 (-1+2 x) \left (-13+4 \log \left (x^2\right )\right )}\right ) \, dx \\ & = 48 \int \frac {\log (1-2 x) (-4 x-\log (1-2 x)+2 x \log (1-2 x))}{x^2 (-1+2 x) \left (-13+4 \log \left (x^2\right )\right )} \, dx+384 \int \frac {\log ^2(1-2 x)}{x^2 \left (-13+4 \log \left (x^2\right )\right )^2} \, dx \\ & = 48 \int \left (-\frac {\log (1-2 x) (-4 x-\log (1-2 x)+2 x \log (1-2 x))}{x^2 \left (-13+4 \log \left (x^2\right )\right )}-\frac {2 \log (1-2 x) (-4 x-\log (1-2 x)+2 x \log (1-2 x))}{x \left (-13+4 \log \left (x^2\right )\right )}+\frac {4 \log (1-2 x) (-4 x-\log (1-2 x)+2 x \log (1-2 x))}{(-1+2 x) \left (-13+4 \log \left (x^2\right )\right )}\right ) \, dx+384 \int \frac {\log ^2(1-2 x)}{x^2 \left (-13+4 \log \left (x^2\right )\right )^2} \, dx \\ & = -\left (48 \int \frac {\log (1-2 x) (-4 x-\log (1-2 x)+2 x \log (1-2 x))}{x^2 \left (-13+4 \log \left (x^2\right )\right )} \, dx\right )-96 \int \frac {\log (1-2 x) (-4 x-\log (1-2 x)+2 x \log (1-2 x))}{x \left (-13+4 \log \left (x^2\right )\right )} \, dx+192 \int \frac {\log (1-2 x) (-4 x-\log (1-2 x)+2 x \log (1-2 x))}{(-1+2 x) \left (-13+4 \log \left (x^2\right )\right )} \, dx+384 \int \frac {\log ^2(1-2 x)}{x^2 \left (-13+4 \log \left (x^2\right )\right )^2} \, dx \\ & = -\left (48 \int \left (-\frac {4 \log (1-2 x)}{x \left (-13+4 \log \left (x^2\right )\right )}-\frac {\log ^2(1-2 x)}{x^2 \left (-13+4 \log \left (x^2\right )\right )}+\frac {2 \log ^2(1-2 x)}{x \left (-13+4 \log \left (x^2\right )\right )}\right ) \, dx\right )-96 \int \left (-\frac {4 \log (1-2 x)}{-13+4 \log \left (x^2\right )}+\frac {2 \log ^2(1-2 x)}{-13+4 \log \left (x^2\right )}-\frac {\log ^2(1-2 x)}{x \left (-13+4 \log \left (x^2\right )\right )}\right ) \, dx+192 \int \left (-\frac {4 x \log (1-2 x)}{(-1+2 x) \left (-13+4 \log \left (x^2\right )\right )}-\frac {\log ^2(1-2 x)}{(-1+2 x) \left (-13+4 \log \left (x^2\right )\right )}+\frac {2 x \log ^2(1-2 x)}{(-1+2 x) \left (-13+4 \log \left (x^2\right )\right )}\right ) \, dx+384 \int \frac {\log ^2(1-2 x)}{x^2 \left (-13+4 \log \left (x^2\right )\right )^2} \, dx \\ & = 48 \int \frac {\log ^2(1-2 x)}{x^2 \left (-13+4 \log \left (x^2\right )\right )} \, dx+192 \int \frac {\log (1-2 x)}{x \left (-13+4 \log \left (x^2\right )\right )} \, dx-192 \int \frac {\log ^2(1-2 x)}{-13+4 \log \left (x^2\right )} \, dx-192 \int \frac {\log ^2(1-2 x)}{(-1+2 x) \left (-13+4 \log \left (x^2\right )\right )} \, dx+384 \int \frac {\log ^2(1-2 x)}{x^2 \left (-13+4 \log \left (x^2\right )\right )^2} \, dx+384 \int \frac {\log (1-2 x)}{-13+4 \log \left (x^2\right )} \, dx+384 \int \frac {x \log ^2(1-2 x)}{(-1+2 x) \left (-13+4 \log \left (x^2\right )\right )} \, dx-768 \int \frac {x \log (1-2 x)}{(-1+2 x) \left (-13+4 \log \left (x^2\right )\right )} \, dx \\ & = 48 \int \frac {\log ^2(1-2 x)}{x^2 \left (-13+4 \log \left (x^2\right )\right )} \, dx+192 \int \frac {\log (1-2 x)}{x \left (-13+4 \log \left (x^2\right )\right )} \, dx-192 \int \frac {\log ^2(1-2 x)}{-13+4 \log \left (x^2\right )} \, dx-192 \int \frac {\log ^2(1-2 x)}{(-1+2 x) \left (-13+4 \log \left (x^2\right )\right )} \, dx+384 \int \frac {\log ^2(1-2 x)}{x^2 \left (-13+4 \log \left (x^2\right )\right )^2} \, dx+384 \int \frac {\log (1-2 x)}{-13+4 \log \left (x^2\right )} \, dx+384 \int \left (\frac {\log ^2(1-2 x)}{2 \left (-13+4 \log \left (x^2\right )\right )}+\frac {\log ^2(1-2 x)}{2 (-1+2 x) \left (-13+4 \log \left (x^2\right )\right )}\right ) \, dx-768 \int \left (\frac {\log (1-2 x)}{2 \left (-13+4 \log \left (x^2\right )\right )}+\frac {\log (1-2 x)}{2 (-1+2 x) \left (-13+4 \log \left (x^2\right )\right )}\right ) \, dx \\ & = 48 \int \frac {\log ^2(1-2 x)}{x^2 \left (-13+4 \log \left (x^2\right )\right )} \, dx+192 \int \frac {\log (1-2 x)}{x \left (-13+4 \log \left (x^2\right )\right )} \, dx+384 \int \frac {\log ^2(1-2 x)}{x^2 \left (-13+4 \log \left (x^2\right )\right )^2} \, dx-384 \int \frac {\log (1-2 x)}{(-1+2 x) \left (-13+4 \log \left (x^2\right )\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {2496 x \log (1-2 x)+(240-480 x) \log ^2(1-2 x)+\left (-768 x \log (1-2 x)+(-192+384 x) \log ^2(1-2 x)\right ) \log \left (x^2\right )}{-169 x^2+338 x^3+\left (104 x^2-208 x^3\right ) \log \left (x^2\right )+\left (-16 x^2+32 x^3\right ) \log ^2\left (x^2\right )} \, dx=-\frac {48 \log ^2(1-2 x)}{x \left (-13+4 \log \left (x^2\right )\right )} \]

[In]

Integrate[(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]

[Out]

(-48*Log[1 - 2*x]^2)/(x*(-13 + 4*Log[x^2]))

Maple [A] (verified)

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	\(-\frac {48 i \ln \left (1-2 x \right )^{2}}{x \left (2 \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-4 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+2 \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+8 i \ln \left (x \right )-13 i\right )}\)	\(71\)

[In]

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)

[Out]

-48*ln(1-2*x)^2/x/(-13+4*ln(x^2))

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {2496 x \log (1-2 x)+(240-480 x) \log ^2(1-2 x)+\left (-768 x \log (1-2 x)+(-192+384 x) \log ^2(1-2 x)\right ) \log \left (x^2\right )}{-169 x^2+338 x^3+\left (104 x^2-208 x^3\right ) \log \left (x^2\right )+\left (-16 x^2+32 x^3\right ) \log ^2\left (x^2\right )} \, dx=-\frac {48 \, \log \left (-2 \, x + 1\right )^{2}}{4 \, x \log \left (x^{2}\right ) - 13 \, x} \]

[In]

integrate((((384*x-192)*log(1-2*x)^2-768*x*log(1-2*x))*log(x^2)+(-480*x+240)*log(1-2*x)^2+2496*x*log(1-2*x))/(
(32*x^3-16*x^2)*log(x^2)^2+(-208*x^3+104*x^2)*log(x^2)+338*x^3-169*x^2),x, algorithm="fricas")

[Out]

-48*log(-2*x + 1)^2/(4*x*log(x^2) - 13*x)

Sympy [F(-2)]

Exception generated. \[ \int \frac {2496 x \log (1-2 x)+(240-480 x) \log ^2(1-2 x)+\left (-768 x \log (1-2 x)+(-192+384 x) \log ^2(1-2 x)\right ) \log \left (x^2\right )}{-169 x^2+338 x^3+\left (104 x^2-208 x^3\right ) \log \left (x^2\right )+\left (-16 x^2+32 x^3\right ) \log ^2\left (x^2\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

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))/((3
2*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)

[Out]

Exception raised: TypeError >> '>' not supported between instances of 'Poly' and 'int'

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {2496 x \log (1-2 x)+(240-480 x) \log ^2(1-2 x)+\left (-768 x \log (1-2 x)+(-192+384 x) \log ^2(1-2 x)\right ) \log \left (x^2\right )}{-169 x^2+338 x^3+\left (104 x^2-208 x^3\right ) \log \left (x^2\right )+\left (-16 x^2+32 x^3\right ) \log ^2\left (x^2\right )} \, dx=-\frac {48 \, \log \left (-2 \, x + 1\right )^{2}}{8 \, x \log \left (x\right ) - 13 \, x} \]

[In]

integrate((((384*x-192)*log(1-2*x)^2-768*x*log(1-2*x))*log(x^2)+(-480*x+240)*log(1-2*x)^2+2496*x*log(1-2*x))/(
(32*x^3-16*x^2)*log(x^2)^2+(-208*x^3+104*x^2)*log(x^2)+338*x^3-169*x^2),x, algorithm="maxima")

[Out]

-48*log(-2*x + 1)^2/(8*x*log(x) - 13*x)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {2496 x \log (1-2 x)+(240-480 x) \log ^2(1-2 x)+\left (-768 x \log (1-2 x)+(-192+384 x) \log ^2(1-2 x)\right ) \log \left (x^2\right )}{-169 x^2+338 x^3+\left (104 x^2-208 x^3\right ) \log \left (x^2\right )+\left (-16 x^2+32 x^3\right ) \log ^2\left (x^2\right )} \, dx=-\frac {48 \, \log \left (-2 \, x + 1\right )^{2}}{4 \, x \log \left (x^{2}\right ) - 13 \, x} \]

[In]

integrate((((384*x-192)*log(1-2*x)^2-768*x*log(1-2*x))*log(x^2)+(-480*x+240)*log(1-2*x)^2+2496*x*log(1-2*x))/(
(32*x^3-16*x^2)*log(x^2)^2+(-208*x^3+104*x^2)*log(x^2)+338*x^3-169*x^2),x, algorithm="giac")

[Out]

-48*log(-2*x + 1)^2/(4*x*log(x^2) - 13*x)

Mupad [B] (verification not implemented)

Time = 13.47 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {2496 x \log (1-2 x)+(240-480 x) \log ^2(1-2 x)+\left (-768 x \log (1-2 x)+(-192+384 x) \log ^2(1-2 x)\right ) \log \left (x^2\right )}{-169 x^2+338 x^3+\left (104 x^2-208 x^3\right ) \log \left (x^2\right )+\left (-16 x^2+32 x^3\right ) \log ^2\left (x^2\right )} \, dx=-\frac {12\,{\ln \left (1-2\,x\right )}^2}{x\,\left (\ln \left (x^2\right )-\frac {13}{4}\right )} \]

[In]

int(-(log(x^2)*(768*x*log(1 - 2*x) - log(1 - 2*x)^2*(384*x - 192)) - 2496*x*log(1 - 2*x) + log(1 - 2*x)^2*(480
*x - 240))/(log(x^2)*(104*x^2 - 208*x^3) - log(x^2)^2*(16*x^2 - 32*x^3) - 169*x^2 + 338*x^3),x)

[Out]

-(12*log(1 - 2*x)^2)/(x*(log(x^2) - 13/4))