3.39.96 \(\int \frac {-24 x^2+3 x \log (4)+(-12 x^2+3 x \log (4)) \log (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)})}{-36 x+9 \log (4)+(-24 x+6 \log (4)) \log (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)})+(-4 x+\log (4)) \log ^2(\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)})} \, dx\)

Optimal. Leaf size=31 \[ 3 \left (5+\frac {x^2}{2 \left (3+\log \left (\frac {x^2}{\left (4-\frac {\log (4)}{x}\right )^2}\right )\right )}\right ) \]

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Rubi [F]  time = 0.72, 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 {-24 x^2+3 x \log (4)+\left (-12 x^2+3 x \log (4)\right ) \log \left (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)}\right )}{-36 x+9 \log (4)+(-24 x+6 \log (4)) \log \left (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)}\right )+(-4 x+\log (4)) \log ^2\left (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-24*x^2 + 3*x*Log[4] + (-12*x^2 + 3*x*Log[4])*Log[x^4/(16*x^2 - 8*x*Log[4] + Log[4]^2)])/(-36*x + 9*Log[4
] + (-24*x + 6*Log[4])*Log[x^4/(16*x^2 - 8*x*Log[4] + Log[4]^2)] + (-4*x + Log[4])*Log[x^4/(16*x^2 - 8*x*Log[4
] + Log[4]^2)]^2),x]

[Out]

(3*Log[4]*Defer[Int][(3 + Log[x^4/(-4*x + Log[4])^2])^(-2), x])/4 - 3*Defer[Int][x/(3 + Log[x^4/(-4*x + Log[4]
)^2])^2, x] + (3*Log[4]^2*Defer[Int][1/((4*x - Log[4])*(3 + Log[x^4/(-4*x + Log[4])^2])^2), x])/4 + 3*Defer[In
t][x/(3 + Log[x^4/(-4*x + Log[4])^2]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 x \left (8 x-\log (4)+(4 x-\log (4)) \log \left (\frac {x^4}{(-4 x+\log (4))^2}\right )\right )}{(4 x-\log (4)) \left (3+\log \left (\frac {x^4}{(-4 x+\log (4))^2}\right )\right )^2} \, dx\\ &=3 \int \frac {x \left (8 x-\log (4)+(4 x-\log (4)) \log \left (\frac {x^4}{(-4 x+\log (4))^2}\right )\right )}{(4 x-\log (4)) \left (3+\log \left (\frac {x^4}{(-4 x+\log (4))^2}\right )\right )^2} \, dx\\ &=3 \int \left (-\frac {2 x (2 x-\log (4))}{(4 x-\log (4)) \left (3+\log \left (\frac {x^4}{(-4 x+\log (4))^2}\right )\right )^2}+\frac {x}{3+\log \left (\frac {x^4}{(-4 x+\log (4))^2}\right )}\right ) \, dx\\ &=3 \int \frac {x}{3+\log \left (\frac {x^4}{(-4 x+\log (4))^2}\right )} \, dx-6 \int \frac {x (2 x-\log (4))}{(4 x-\log (4)) \left (3+\log \left (\frac {x^4}{(-4 x+\log (4))^2}\right )\right )^2} \, dx\\ &=3 \int \frac {x}{3+\log \left (\frac {x^4}{(-4 x+\log (4))^2}\right )} \, dx-6 \int \left (\frac {x}{2 \left (3+\log \left (\frac {x^4}{(-4 x+\log (4))^2}\right )\right )^2}-\frac {\log (4)}{8 \left (3+\log \left (\frac {x^4}{(-4 x+\log (4))^2}\right )\right )^2}-\frac {\log ^2(4)}{8 (4 x-\log (4)) \left (3+\log \left (\frac {x^4}{(-4 x+\log (4))^2}\right )\right )^2}\right ) \, dx\\ &=-\left (3 \int \frac {x}{\left (3+\log \left (\frac {x^4}{(-4 x+\log (4))^2}\right )\right )^2} \, dx\right )+3 \int \frac {x}{3+\log \left (\frac {x^4}{(-4 x+\log (4))^2}\right )} \, dx+\frac {1}{4} (3 \log (4)) \int \frac {1}{\left (3+\log \left (\frac {x^4}{(-4 x+\log (4))^2}\right )\right )^2} \, dx+\frac {1}{4} \left (3 \log ^2(4)\right ) \int \frac {1}{(4 x-\log (4)) \left (3+\log \left (\frac {x^4}{(-4 x+\log (4))^2}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 24, normalized size = 0.77 \begin {gather*} \frac {3 x^2}{2 \left (3+\log \left (\frac {x^4}{(-4 x+\log (4))^2}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-24*x^2 + 3*x*Log[4] + (-12*x^2 + 3*x*Log[4])*Log[x^4/(16*x^2 - 8*x*Log[4] + Log[4]^2)])/(-36*x + 9
*Log[4] + (-24*x + 6*Log[4])*Log[x^4/(16*x^2 - 8*x*Log[4] + Log[4]^2)] + (-4*x + Log[4])*Log[x^4/(16*x^2 - 8*x
*Log[4] + Log[4]^2)]^2),x]

[Out]

(3*x^2)/(2*(3 + Log[x^4/(-4*x + Log[4])^2]))

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fricas [A]  time = 0.49, size = 32, normalized size = 1.03 \begin {gather*} \frac {3 \, x^{2}}{2 \, {\left (\log \left (\frac {x^{4}}{4 \, {\left (4 \, x^{2} - 4 \, x \log \relax (2) + \log \relax (2)^{2}\right )}}\right ) + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x*log(2)-12*x^2)*log(x^4/(4*log(2)^2-16*x*log(2)+16*x^2))+6*x*log(2)-24*x^2)/((2*log(2)-4*x)*log
(x^4/(4*log(2)^2-16*x*log(2)+16*x^2))^2+(12*log(2)-24*x)*log(x^4/(4*log(2)^2-16*x*log(2)+16*x^2))+18*log(2)-36
*x),x, algorithm="fricas")

[Out]

3/2*x^2/(log(1/4*x^4/(4*x^2 - 4*x*log(2) + log(2)^2)) + 3)

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giac [A]  time = 0.40, size = 35, normalized size = 1.13 \begin {gather*} -\frac {3 \, x^{2}}{2 \, {\left (2 \, \log \relax (2) - \log \left (x^{4}\right ) + \log \left (4 \, x^{2} - 4 \, x \log \relax (2) + \log \relax (2)^{2}\right ) - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x*log(2)-12*x^2)*log(x^4/(4*log(2)^2-16*x*log(2)+16*x^2))+6*x*log(2)-24*x^2)/((2*log(2)-4*x)*log
(x^4/(4*log(2)^2-16*x*log(2)+16*x^2))^2+(12*log(2)-24*x)*log(x^4/(4*log(2)^2-16*x*log(2)+16*x^2))+18*log(2)-36
*x),x, algorithm="giac")

[Out]

-3/2*x^2/(2*log(2) - log(x^4) + log(4*x^2 - 4*x*log(2) + log(2)^2) - 3)

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maple [A]  time = 0.15, size = 34, normalized size = 1.10

method	result	size
norman	\(\frac {3 x^{2}}{2 \left (\ln \left (\frac {x^{4}}{4 \ln \relax (2)^{2}-16 x \ln \relax (2)+16 x^{2}}\right )+3\right )}\)	\(34\)
risch	\(\frac {3 x^{2}}{2 \left (\ln \left (\frac {x^{4}}{4 \ln \relax (2)^{2}-16 x \ln \relax (2)+16 x^{2}}\right )+3\right )}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((6*x*ln(2)-12*x^2)*ln(x^4/(4*ln(2)^2-16*x*ln(2)+16*x^2))+6*x*ln(2)-24*x^2)/((2*ln(2)-4*x)*ln(x^4/(4*ln(2)
^2-16*x*ln(2)+16*x^2))^2+(12*ln(2)-24*x)*ln(x^4/(4*ln(2)^2-16*x*ln(2)+16*x^2))+18*ln(2)-36*x),x,method=_RETURN
VERBOSE)

[Out]

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

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maxima [A]  time = 0.49, size = 28, normalized size = 0.90 \begin {gather*} -\frac {3 \, x^{2}}{2 \, {\left (2 \, \log \relax (2) + 2 \, \log \left (2 \, x - \log \relax (2)\right ) - 4 \, \log \relax (x) - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x*log(2)-12*x^2)*log(x^4/(4*log(2)^2-16*x*log(2)+16*x^2))+6*x*log(2)-24*x^2)/((2*log(2)-4*x)*log
(x^4/(4*log(2)^2-16*x*log(2)+16*x^2))^2+(12*log(2)-24*x)*log(x^4/(4*log(2)^2-16*x*log(2)+16*x^2))+18*log(2)-36
*x),x, algorithm="maxima")

[Out]

-3/2*x^2/(2*log(2) + 2*log(2*x - log(2)) - 4*log(x) - 3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int \frac {\ln \left (\frac {x^4}{16\,x^2-16\,\ln \relax (2)\,x+4\,{\ln \relax (2)}^2}\right )\,\left (6\,x\,\ln \relax (2)-12\,x^2\right )+6\,x\,\ln \relax (2)-24\,x^2}{\left (4\,x-2\,\ln \relax (2)\right )\,{\ln \left (\frac {x^4}{16\,x^2-16\,\ln \relax (2)\,x+4\,{\ln \relax (2)}^2}\right )}^2+\left (24\,x-12\,\ln \relax (2)\right )\,\ln \left (\frac {x^4}{16\,x^2-16\,\ln \relax (2)\,x+4\,{\ln \relax (2)}^2}\right )+36\,x-18\,\ln \relax (2)} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x^4/(4*log(2)^2 - 16*x*log(2) + 16*x^2))*(6*x*log(2) - 12*x^2) + 6*x*log(2) - 24*x^2)/(36*x - 18*log
(2) + log(x^4/(4*log(2)^2 - 16*x*log(2) + 16*x^2))*(24*x - 12*log(2)) + log(x^4/(4*log(2)^2 - 16*x*log(2) + 16
*x^2))^2*(4*x - 2*log(2))),x)

[Out]

-int((log(x^4/(4*log(2)^2 - 16*x*log(2) + 16*x^2))*(6*x*log(2) - 12*x^2) + 6*x*log(2) - 24*x^2)/(36*x - 18*log
(2) + log(x^4/(4*log(2)^2 - 16*x*log(2) + 16*x^2))*(24*x - 12*log(2)) + log(x^4/(4*log(2)^2 - 16*x*log(2) + 16
*x^2))^2*(4*x - 2*log(2))), x)

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sympy [A]  time = 0.20, size = 31, normalized size = 1.00 \begin {gather*} \frac {3 x^{2}}{2 \log {\left (\frac {x^{4}}{16 x^{2} - 16 x \log {\relax (2 )} + 4 \log {\relax (2 )}^{2}} \right )} + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x*ln(2)-12*x**2)*ln(x**4/(4*ln(2)**2-16*x*ln(2)+16*x**2))+6*x*ln(2)-24*x**2)/((2*ln(2)-4*x)*ln(x
**4/(4*ln(2)**2-16*x*ln(2)+16*x**2))**2+(12*ln(2)-24*x)*ln(x**4/(4*ln(2)**2-16*x*ln(2)+16*x**2))+18*ln(2)-36*x
),x)

[Out]

3*x**2/(2*log(x**4/(16*x**2 - 16*x*log(2) + 4*log(2)**2)) + 6)

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