3.87.24 \(\int \frac {24-6 x+(12+24 x-6 x^2) \log (\frac {2}{x^2})+(12-3 x) \log (\frac {2}{x^2}) \log (-\frac {x}{(-4+x) \log (\frac {2}{x^2})})}{(-4 x^4+x^5) \log (\frac {2}{x^2})+(-8 x^3+2 x^4) \log (\frac {2}{x^2}) \log (-\frac {x}{(-4+x) \log (\frac {2}{x^2})})+(-4 x^2+x^3) \log (\frac {2}{x^2}) \log ^2(-\frac {x}{(-4+x) \log (\frac {2}{x^2})})} \, dx\)

Optimal. Leaf size=27 \[ \frac {3}{x \left (x+\log \left (\frac {x}{(4-x) \log \left (\frac {2}{x^2}\right )}\right )\right )} \]

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Rubi [F]  time = 4.02, 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 {24-6 x+\left (12+24 x-6 x^2\right ) \log \left (\frac {2}{x^2}\right )+(12-3 x) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )}{\left (-4 x^4+x^5\right ) \log \left (\frac {2}{x^2}\right )+\left (-8 x^3+2 x^4\right ) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )+\left (-4 x^2+x^3\right ) \log \left (\frac {2}{x^2}\right ) \log ^2\left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(24 - 6*x + (12 + 24*x - 6*x^2)*Log[2/x^2] + (12 - 3*x)*Log[2/x^2]*Log[-(x/((-4 + x)*Log[2/x^2]))])/((-4*x
^4 + x^5)*Log[2/x^2] + (-8*x^3 + 2*x^4)*Log[2/x^2]*Log[-(x/((-4 + x)*Log[2/x^2]))] + (-4*x^2 + x^3)*Log[2/x^2]
*Log[-(x/((-4 + x)*Log[2/x^2]))]^2),x]

[Out]

(3*Defer[Int][1/((-4 + x)*(x + Log[-(x/((-4 + x)*Log[2/x^2]))])^2), x])/4 - 3*Defer[Int][1/(x^2*(x + Log[-(x/(
(-4 + x)*Log[2/x^2]))])^2), x] - (15*Defer[Int][1/(x*(x + Log[-(x/((-4 + x)*Log[2/x^2]))])^2), x])/4 - 6*Defer
[Int][1/(x^2*Log[2/x^2]*(x + Log[-(x/((-4 + x)*Log[2/x^2]))])^2), x] - 3*Defer[Int][1/(x^2*(x + Log[-(x/((-4 +
x)*Log[2/x^2]))])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 \left (2 (-4+x)+\log \left (\frac {2}{x^2}\right ) \left (2 \left (-2-4 x+x^2\right )+(-4+x) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )\right )}{(4-x) x^2 \log \left (\frac {2}{x^2}\right ) \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2} \, dx\\ &=3 \int \frac {2 (-4+x)+\log \left (\frac {2}{x^2}\right ) \left (2 \left (-2-4 x+x^2\right )+(-4+x) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )}{(4-x) x^2 \log \left (\frac {2}{x^2}\right ) \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2} \, dx\\ &=3 \int \left (\frac {8-2 x+4 \log \left (\frac {2}{x^2}\right )+4 x \log \left (\frac {2}{x^2}\right )-x^2 \log \left (\frac {2}{x^2}\right )}{(-4+x) x^2 \log \left (\frac {2}{x^2}\right ) \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2}-\frac {1}{x^2 \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )}\right ) \, dx\\ &=3 \int \frac {8-2 x+4 \log \left (\frac {2}{x^2}\right )+4 x \log \left (\frac {2}{x^2}\right )-x^2 \log \left (\frac {2}{x^2}\right )}{(-4+x) x^2 \log \left (\frac {2}{x^2}\right ) \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2} \, dx-3 \int \frac {1}{x^2 \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )} \, dx\\ &=-\left (3 \int \frac {1}{x^2 \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )} \, dx\right )+3 \int \left (\frac {8-2 x+4 \log \left (\frac {2}{x^2}\right )+4 x \log \left (\frac {2}{x^2}\right )-x^2 \log \left (\frac {2}{x^2}\right )}{16 (-4+x) \log \left (\frac {2}{x^2}\right ) \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2}+\frac {-8+2 x-4 \log \left (\frac {2}{x^2}\right )-4 x \log \left (\frac {2}{x^2}\right )+x^2 \log \left (\frac {2}{x^2}\right )}{4 x^2 \log \left (\frac {2}{x^2}\right ) \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2}+\frac {-8+2 x-4 \log \left (\frac {2}{x^2}\right )-4 x \log \left (\frac {2}{x^2}\right )+x^2 \log \left (\frac {2}{x^2}\right )}{16 x \log \left (\frac {2}{x^2}\right ) \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2}\right ) \, dx\\ &=\frac {3}{16} \int \frac {8-2 x+4 \log \left (\frac {2}{x^2}\right )+4 x \log \left (\frac {2}{x^2}\right )-x^2 \log \left (\frac {2}{x^2}\right )}{(-4+x) \log \left (\frac {2}{x^2}\right ) \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2} \, dx+\frac {3}{16} \int \frac {-8+2 x-4 \log \left (\frac {2}{x^2}\right )-4 x \log \left (\frac {2}{x^2}\right )+x^2 \log \left (\frac {2}{x^2}\right )}{x \log \left (\frac {2}{x^2}\right ) \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2} \, dx+\frac {3}{4} \int \frac {-8+2 x-4 \log \left (\frac {2}{x^2}\right )-4 x \log \left (\frac {2}{x^2}\right )+x^2 \log \left (\frac {2}{x^2}\right )}{x^2 \log \left (\frac {2}{x^2}\right ) \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2} \, dx-3 \int \frac {1}{x^2 \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )} \, dx\\ &=\frac {3}{16} \int \frac {2 (-4+x)+\left (-4-4 x+x^2\right ) \log \left (\frac {2}{x^2}\right )}{x \log \left (\frac {2}{x^2}\right ) \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2} \, dx+\frac {3}{16} \int \left (\frac {4}{(-4+x) \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2}+\frac {4 x}{(-4+x) \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2}-\frac {x^2}{(-4+x) \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2}+\frac {8}{(-4+x) \log \left (\frac {2}{x^2}\right ) \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2}-\frac {2 x}{(-4+x) \log \left (\frac {2}{x^2}\right ) \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2}\right ) \, dx+\frac {3}{4} \int \frac {2 (-4+x)+\left (-4-4 x+x^2\right ) \log \left (\frac {2}{x^2}\right )}{x^2 \log \left (\frac {2}{x^2}\right ) \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2} \, dx-3 \int \frac {1}{x^2 \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )} \, dx\\ &=-\left (\frac {3}{16} \int \frac {x^2}{(-4+x) \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2} \, dx\right )+\frac {3}{16} \int \left (-\frac {4}{\left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2}-\frac {4}{x \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2}+\frac {x}{\left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2}+\frac {2}{\log \left (\frac {2}{x^2}\right ) \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2}-\frac {8}{x \log \left (\frac {2}{x^2}\right ) \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2}\right ) \, dx-\frac {3}{8} \int \frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right ) \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2} \, dx+\frac {3}{4} \int \frac {1}{(-4+x) \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2} \, dx+\frac {3}{4} \int \frac {x}{(-4+x) \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2} \, dx+\frac {3}{4} \int \left (\frac {1}{\left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2}-\frac {4}{x^2 \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2}-\frac {4}{x \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2}-\frac {8}{x^2 \log \left (\frac {2}{x^2}\right ) \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2}+\frac {2}{x \log \left (\frac {2}{x^2}\right ) \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2}\right ) \, dx+\frac {3}{2} \int \frac {1}{(-4+x) \log \left (\frac {2}{x^2}\right ) \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )^2} \, dx-3 \int \frac {1}{x^2 \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 26, normalized size = 0.96 \begin {gather*} \frac {3}{x \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(24 - 6*x + (12 + 24*x - 6*x^2)*Log[2/x^2] + (12 - 3*x)*Log[2/x^2]*Log[-(x/((-4 + x)*Log[2/x^2]))])/
((-4*x^4 + x^5)*Log[2/x^2] + (-8*x^3 + 2*x^4)*Log[2/x^2]*Log[-(x/((-4 + x)*Log[2/x^2]))] + (-4*x^2 + x^3)*Log[
2/x^2]*Log[-(x/((-4 + x)*Log[2/x^2]))]^2),x]

[Out]

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

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fricas [A]  time = 0.54, size = 27, normalized size = 1.00 \begin {gather*} \frac {3}{x^{2} + x \log \left (-\frac {x}{{\left (x - 4\right )} \log \left (\frac {2}{x^{2}}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x+12)*log(2/x^2)*log(-x/(x-4)/log(2/x^2))+(-6*x^2+24*x+12)*log(2/x^2)-6*x+24)/((x^3-4*x^2)*log(
2/x^2)*log(-x/(x-4)/log(2/x^2))^2+(2*x^4-8*x^3)*log(2/x^2)*log(-x/(x-4)/log(2/x^2))+(x^5-4*x^4)*log(2/x^2)),x,
algorithm="fricas")

[Out]

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

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giac [C]  time = 4.99, size = 699, normalized size = 25.89 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x+12)*log(2/x^2)*log(-x/(x-4)/log(2/x^2))+(-6*x^2+24*x+12)*log(2/x^2)-6*x+24)/((x^3-4*x^2)*log(
2/x^2)*log(-x/(x-4)/log(2/x^2))^2+(2*x^4-8*x^3)*log(2/x^2)*log(-x/(x-4)/log(2/x^2))+(x^5-4*x^4)*log(2/x^2)),x,
algorithm="giac")

[Out]

-3*(x^2*log(2)*log(2/x^2) - 2*x^2*log(x)*log(2/x^2) - 4*x*log(2)*log(2/x^2) + 8*x*log(x)*log(2/x^2) + 2*x*log(
2/x^2) - 4*log(2)*log(2/x^2) + 8*log(x)*log(2/x^2) - 8*log(2/x^2))/(-I*pi*x^3*log(2)*log(2/x^2) - x^4*log(2)*l
og(2/x^2) + x^3*log(2)*log(x*log(2/x^2) - 4*log(2/x^2))*log(2/x^2) + 2*I*pi*x^3*log(x)*log(2/x^2) + 2*x^4*log(
x)*log(2/x^2) - x^3*log(2)*log(x)*log(2/x^2) - 2*x^3*log(x*log(2/x^2) - 4*log(2/x^2))*log(x)*log(2/x^2) + 2*x^
3*log(x)^2*log(2/x^2) + 4*I*pi*x^2*log(2)*log(2/x^2) + 4*x^3*log(2)*log(2/x^2) - 4*x^2*log(2)*log(x*log(2/x^2)
- 4*log(2/x^2))*log(2/x^2) - 8*I*pi*x^2*log(x)*log(2/x^2) - 8*x^3*log(x)*log(2/x^2) + 4*x^2*log(2)*log(x)*log
(2/x^2) + 8*x^2*log(x*log(2/x^2) - 4*log(2/x^2))*log(x)*log(2/x^2) - 8*x^2*log(x)^2*log(2/x^2) - 2*I*pi*x^2*lo
g(2) - 2*x^3*log(2) + 2*x^2*log(2)*log(x*log(2/x^2) - 4*log(2/x^2)) + 4*I*pi*x^2*log(x) + 4*x^3*log(x) - 2*x^2
*log(2)*log(x) - 4*x^2*log(x*log(2/x^2) - 4*log(2/x^2))*log(x) + 4*x^2*log(x)^2 + 4*I*pi*x*log(2)*log(2/x^2) +
4*x^2*log(2)*log(2/x^2) - 4*x*log(2)*log(x*log(2/x^2) - 4*log(2/x^2))*log(2/x^2) - 8*I*pi*x*log(x)*log(2/x^2)
- 8*x^2*log(x)*log(2/x^2) + 4*x*log(2)*log(x)*log(2/x^2) + 8*x*log(x*log(2/x^2) - 4*log(2/x^2))*log(x)*log(2/
x^2) - 8*x*log(x)^2*log(2/x^2) + 8*I*pi*x*log(2) + 8*x^2*log(2) - 8*x*log(2)*log(x*log(2/x^2) - 4*log(2/x^2))
- 16*I*pi*x*log(x) - 16*x^2*log(x) + 8*x*log(2)*log(x) + 16*x*log(x*log(2/x^2) - 4*log(2/x^2))*log(x) - 16*x*l
og(x)^2)

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (-3 x +12\right ) \ln \left (\frac {2}{x^{2}}\right ) \ln \left (-\frac {x}{\left (x -4\right ) \ln \left (\frac {2}{x^{2}}\right )}\right )+\left (-6 x^{2}+24 x +12\right ) \ln \left (\frac {2}{x^{2}}\right )-6 x +24}{\left (x^{3}-4 x^{2}\right ) \ln \left (\frac {2}{x^{2}}\right ) \ln \left (-\frac {x}{\left (x -4\right ) \ln \left (\frac {2}{x^{2}}\right )}\right )^{2}+\left (2 x^{4}-8 x^{3}\right ) \ln \left (\frac {2}{x^{2}}\right ) \ln \left (-\frac {x}{\left (x -4\right ) \ln \left (\frac {2}{x^{2}}\right )}\right )+\left (x^{5}-4 x^{4}\right ) \ln \left (\frac {2}{x^{2}}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*x+12)*ln(2/x^2)*ln(-x/(x-4)/ln(2/x^2))+(-6*x^2+24*x+12)*ln(2/x^2)-6*x+24)/((x^3-4*x^2)*ln(2/x^2)*ln(-
x/(x-4)/ln(2/x^2))^2+(2*x^4-8*x^3)*ln(2/x^2)*ln(-x/(x-4)/ln(2/x^2))+(x^5-4*x^4)*ln(2/x^2)),x)

[Out]

int(((-3*x+12)*ln(2/x^2)*ln(-x/(x-4)/ln(2/x^2))+(-6*x^2+24*x+12)*ln(2/x^2)-6*x+24)/((x^3-4*x^2)*ln(2/x^2)*ln(-
x/(x-4)/ln(2/x^2))^2+(2*x^4-8*x^3)*ln(2/x^2)*ln(-x/(x-4)/ln(2/x^2))+(x^5-4*x^4)*ln(2/x^2)),x)

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maxima [A]  time = 0.50, size = 32, normalized size = 1.19 \begin {gather*} \frac {3}{x^{2} - x \log \left (x - 4\right ) + x \log \relax (x) - x \log \left (-\log \relax (2) + 2 \, \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x+12)*log(2/x^2)*log(-x/(x-4)/log(2/x^2))+(-6*x^2+24*x+12)*log(2/x^2)-6*x+24)/((x^3-4*x^2)*log(
2/x^2)*log(-x/(x-4)/log(2/x^2))^2+(2*x^4-8*x^3)*log(2/x^2)*log(-x/(x-4)/log(2/x^2))+(x^5-4*x^4)*log(2/x^2)),x,
algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {6\,x-\ln \left (\frac {2}{x^2}\right )\,\left (-6\,x^2+24\,x+12\right )+\ln \left (-\frac {x}{\ln \left (\frac {2}{x^2}\right )\,\left (x-4\right )}\right )\,\ln \left (\frac {2}{x^2}\right )\,\left (3\,x-12\right )-24}{\ln \left (\frac {2}{x^2}\right )\,\left (4\,x^2-x^3\right )\,{\ln \left (-\frac {x}{\ln \left (\frac {2}{x^2}\right )\,\left (x-4\right )}\right )}^2+\ln \left (\frac {2}{x^2}\right )\,\left (8\,x^3-2\,x^4\right )\,\ln \left (-\frac {x}{\ln \left (\frac {2}{x^2}\right )\,\left (x-4\right )}\right )+\ln \left (\frac {2}{x^2}\right )\,\left (4\,x^4-x^5\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x - log(2/x^2)*(24*x - 6*x^2 + 12) + log(-x/(log(2/x^2)*(x - 4)))*log(2/x^2)*(3*x - 12) - 24)/(log(2/x^
2)*(4*x^4 - x^5) + log(-x/(log(2/x^2)*(x - 4)))*log(2/x^2)*(8*x^3 - 2*x^4) + log(-x/(log(2/x^2)*(x - 4)))^2*lo
g(2/x^2)*(4*x^2 - x^3)),x)

[Out]

int((6*x - log(2/x^2)*(24*x - 6*x^2 + 12) + log(-x/(log(2/x^2)*(x - 4)))*log(2/x^2)*(3*x - 12) - 24)/(log(2/x^
2)*(4*x^4 - x^5) + log(-x/(log(2/x^2)*(x - 4)))*log(2/x^2)*(8*x^3 - 2*x^4) + log(-x/(log(2/x^2)*(x - 4)))^2*lo
g(2/x^2)*(4*x^2 - x^3)), x)

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sympy [A]  time = 0.42, size = 20, normalized size = 0.74 \begin {gather*} \frac {3}{x^{2} + x \log {\left (- \frac {x}{\left (x - 4\right ) \log {\left (\frac {2}{x^{2}} \right )}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x+12)*ln(2/x**2)*ln(-x/(x-4)/ln(2/x**2))+(-6*x**2+24*x+12)*ln(2/x**2)-6*x+24)/((x**3-4*x**2)*ln
(2/x**2)*ln(-x/(x-4)/ln(2/x**2))**2+(2*x**4-8*x**3)*ln(2/x**2)*ln(-x/(x-4)/ln(2/x**2))+(x**5-4*x**4)*ln(2/x**2
)),x)

[Out]

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

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