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

Optimal. Leaf size=29 \[ \frac {16 \log ^2(1-2 x)}{-x+\frac {4}{3} x \left (4-\log \left (x^2\right )\right )} \]

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Rubi [F]  time = 5.95, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.22, size = 23, normalized size = 0.79 \begin {gather*} -\frac {48 \log ^2(1-2 x)}{x \left (-13+4 \log \left (x^2\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[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]))

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fricas [A]  time = 0.47, size = 23, normalized size = 0.79 \begin {gather*} -\frac {48 \, \log \left (-2 \, x + 1\right )^{2}}{4 \, x \log \left (x^{2}\right ) - 13 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.23, size = 23, normalized size = 0.79 \begin {gather*} -\frac {48 \, \log \left (-2 \, x + 1\right )^{2}}{4 \, x \log \left (x^{2}\right ) - 13 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [C]  time = 0.12, size = 71, normalized size = 2.45

method	result	size
risch	\(-\frac {48 i \ln \left (1-2 x \right )^{2}}{x \left (2 \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-4 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+2 \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+8 i \ln \relax (x )-13 i\right )}\)	\(71\)

Verification of antiderivative is not currently implemented for this CAS.

[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*I/x/(2*Pi*csgn(I*x)^2*csgn(I*x^2)-4*Pi*csgn(I*x)*csgn(I*x^2)^2+2*Pi*csgn(I*x^2)^3+8*I*ln(x)-13*I)*ln(1-2*x
)^2

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maxima [A]  time = 0.45, size = 21, normalized size = 0.72 \begin {gather*} -\frac {48 \, \log \left (-2 \, x + 1\right )^{2}}{8 \, x \log \relax (x) - 13 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 4.49, size = 21, normalized size = 0.72 \begin {gather*} -\frac {12\,{\ln \left (1-2\,x\right )}^2}{x\,\left (\ln \left (x^2\right )-\frac {13}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 0.40, size = 22, normalized size = 0.76 \begin {gather*} - \frac {48 \log {\left (1 - 2 x \right )}^{2}}{4 x \log {\left (x^{2} \right )} - 13 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

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

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