3.84.21 \(\int \frac {-27 x+123 x^2-30 x^3+(576-168 x+126 x^2-30 x^3+(120-30 x) \log (4-x)) \log (\frac {1}{5} (24-x+5 x^2+5 \log (4-x)))}{(-96 x^3+28 x^4-21 x^5+5 x^6+(-20 x^3+5 x^4) \log (4-x)) \log ^2(\frac {1}{5} (24-x+5 x^2+5 \log (4-x)))} \, dx\)

Optimal. Leaf size=26 \[ \frac {3}{x^2 \log \left (\frac {24}{5}-\frac {x}{5}+x^2+\log (4-x)\right )} \]

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Rubi [F]  time = 2.12, 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 {-27 x+123 x^2-30 x^3+\left (576-168 x+126 x^2-30 x^3+(120-30 x) \log (4-x)\right ) \log \left (\frac {1}{5} \left (24-x+5 x^2+5 \log (4-x)\right )\right )}{\left (-96 x^3+28 x^4-21 x^5+5 x^6+\left (-20 x^3+5 x^4\right ) \log (4-x)\right ) \log ^2\left (\frac {1}{5} \left (24-x+5 x^2+5 \log (4-x)\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-27*x + 123*x^2 - 30*x^3 + (576 - 168*x + 126*x^2 - 30*x^3 + (120 - 30*x)*Log[4 - x])*Log[(24 - x + 5*x^2
+ 5*Log[4 - x])/5])/((-96*x^3 + 28*x^4 - 21*x^5 + 5*x^6 + (-20*x^3 + 5*x^4)*Log[4 - x])*Log[(24 - x + 5*x^2 +
5*Log[4 - x])/5]^2),x]

[Out]

(-15*Defer[Int][1/((-4 + x)*(24 - x + 5*x^2 + 5*Log[4 - x])*Log[24/5 - x/5 + x^2 + Log[4 - x]]^2), x])/16 + (2
7*Defer[Int][1/(x^2*(24 - x + 5*x^2 + 5*Log[4 - x])*Log[24/5 - x/5 + x^2 + Log[4 - x]]^2), x])/4 - (465*Defer[
Int][1/(x*(24 - x + 5*x^2 + 5*Log[4 - x])*Log[24/5 - x/5 + x^2 + Log[4 - x]]^2), x])/16 - 6*Defer[Int][1/(x^3*
Log[24/5 - x/5 + x^2 + Log[4 - x]]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 \left (\frac {x \left (-9+41 x-10 x^2\right )}{(-4+x) \left (24-x+5 x^2+5 \log (4-x)\right )}-2 \log \left (\frac {24}{5}-\frac {x}{5}+x^2+\log (4-x)\right )\right )}{x^3 \log ^2\left (\frac {24}{5}-\frac {x}{5}+x^2+\log (4-x)\right )} \, dx\\ &=3 \int \frac {\frac {x \left (-9+41 x-10 x^2\right )}{(-4+x) \left (24-x+5 x^2+5 \log (4-x)\right )}-2 \log \left (\frac {24}{5}-\frac {x}{5}+x^2+\log (4-x)\right )}{x^3 \log ^2\left (\frac {24}{5}-\frac {x}{5}+x^2+\log (4-x)\right )} \, dx\\ &=3 \int \left (\frac {-9+41 x-10 x^2}{(-4+x) x^2 \left (24-x+5 x^2+5 \log (4-x)\right ) \log ^2\left (\frac {24}{5}-\frac {x}{5}+x^2+\log (4-x)\right )}-\frac {2}{x^3 \log \left (\frac {24}{5}-\frac {x}{5}+x^2+\log (4-x)\right )}\right ) \, dx\\ &=3 \int \frac {-9+41 x-10 x^2}{(-4+x) x^2 \left (24-x+5 x^2+5 \log (4-x)\right ) \log ^2\left (\frac {24}{5}-\frac {x}{5}+x^2+\log (4-x)\right )} \, dx-6 \int \frac {1}{x^3 \log \left (\frac {24}{5}-\frac {x}{5}+x^2+\log (4-x)\right )} \, dx\\ &=3 \int \left (-\frac {5}{16 (-4+x) \left (24-x+5 x^2+5 \log (4-x)\right ) \log ^2\left (\frac {24}{5}-\frac {x}{5}+x^2+\log (4-x)\right )}+\frac {9}{4 x^2 \left (24-x+5 x^2+5 \log (4-x)\right ) \log ^2\left (\frac {24}{5}-\frac {x}{5}+x^2+\log (4-x)\right )}-\frac {155}{16 x \left (24-x+5 x^2+5 \log (4-x)\right ) \log ^2\left (\frac {24}{5}-\frac {x}{5}+x^2+\log (4-x)\right )}\right ) \, dx-6 \int \frac {1}{x^3 \log \left (\frac {24}{5}-\frac {x}{5}+x^2+\log (4-x)\right )} \, dx\\ &=-\left (\frac {15}{16} \int \frac {1}{(-4+x) \left (24-x+5 x^2+5 \log (4-x)\right ) \log ^2\left (\frac {24}{5}-\frac {x}{5}+x^2+\log (4-x)\right )} \, dx\right )-6 \int \frac {1}{x^3 \log \left (\frac {24}{5}-\frac {x}{5}+x^2+\log (4-x)\right )} \, dx+\frac {27}{4} \int \frac {1}{x^2 \left (24-x+5 x^2+5 \log (4-x)\right ) \log ^2\left (\frac {24}{5}-\frac {x}{5}+x^2+\log (4-x)\right )} \, dx-\frac {465}{16} \int \frac {1}{x \left (24-x+5 x^2+5 \log (4-x)\right ) \log ^2\left (\frac {24}{5}-\frac {x}{5}+x^2+\log (4-x)\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 26, normalized size = 1.00 \begin {gather*} \frac {3}{x^2 \log \left (\frac {24}{5}-\frac {x}{5}+x^2+\log (4-x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-27*x + 123*x^2 - 30*x^3 + (576 - 168*x + 126*x^2 - 30*x^3 + (120 - 30*x)*Log[4 - x])*Log[(24 - x +
5*x^2 + 5*Log[4 - x])/5])/((-96*x^3 + 28*x^4 - 21*x^5 + 5*x^6 + (-20*x^3 + 5*x^4)*Log[4 - x])*Log[(24 - x + 5
*x^2 + 5*Log[4 - x])/5]^2),x]

[Out]

3/(x^2*Log[24/5 - x/5 + x^2 + Log[4 - x]])

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fricas [A]  time = 0.59, size = 22, normalized size = 0.85 \begin {gather*} \frac {3}{x^{2} \log \left (x^{2} - \frac {1}{5} \, x + \log \left (-x + 4\right ) + \frac {24}{5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-30*x+120)*log(-x+4)-30*x^3+126*x^2-168*x+576)*log(log(-x+4)+x^2-1/5*x+24/5)-30*x^3+123*x^2-27*x)
/((5*x^4-20*x^3)*log(-x+4)+5*x^6-21*x^5+28*x^4-96*x^3)/log(log(-x+4)+x^2-1/5*x+24/5)^2,x, algorithm="fricas")

[Out]

3/(x^2*log(x^2 - 1/5*x + log(-x + 4) + 24/5))

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giac [A]  time = 0.34, size = 35, normalized size = 1.35 \begin {gather*} -\frac {3}{x^{2} \log \relax (5) - x^{2} \log \left (5 \, x^{2} - x + 5 \, \log \left (-x + 4\right ) + 24\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-30*x+120)*log(-x+4)-30*x^3+126*x^2-168*x+576)*log(log(-x+4)+x^2-1/5*x+24/5)-30*x^3+123*x^2-27*x)
/((5*x^4-20*x^3)*log(-x+4)+5*x^6-21*x^5+28*x^4-96*x^3)/log(log(-x+4)+x^2-1/5*x+24/5)^2,x, algorithm="giac")

[Out]

-3/(x^2*log(5) - x^2*log(5*x^2 - x + 5*log(-x + 4) + 24))

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maple [A]  time = 0.04, size = 23, normalized size = 0.88

method	result	size
risch	\(\frac {3}{\ln \left (\ln \left (-x +4\right )+x^{2}-\frac {x}{5}+\frac {24}{5}\right ) x^{2}}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-30*x+120)*ln(-x+4)-30*x^3+126*x^2-168*x+576)*ln(ln(-x+4)+x^2-1/5*x+24/5)-30*x^3+123*x^2-27*x)/((5*x^4-
20*x^3)*ln(-x+4)+5*x^6-21*x^5+28*x^4-96*x^3)/ln(ln(-x+4)+x^2-1/5*x+24/5)^2,x,method=_RETURNVERBOSE)

[Out]

3/ln(ln(-x+4)+x^2-1/5*x+24/5)/x^2

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maxima [A]  time = 0.51, size = 35, normalized size = 1.35 \begin {gather*} -\frac {3}{x^{2} \log \relax (5) - x^{2} \log \left (5 \, x^{2} - x + 5 \, \log \left (-x + 4\right ) + 24\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-30*x+120)*log(-x+4)-30*x^3+126*x^2-168*x+576)*log(log(-x+4)+x^2-1/5*x+24/5)-30*x^3+123*x^2-27*x)
/((5*x^4-20*x^3)*log(-x+4)+5*x^6-21*x^5+28*x^4-96*x^3)/log(log(-x+4)+x^2-1/5*x+24/5)^2,x, algorithm="maxima")

[Out]

-3/(x^2*log(5) - x^2*log(5*x^2 - x + 5*log(-x + 4) + 24))

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mupad [B]  time = 5.52, size = 22, normalized size = 0.85 \begin {gather*} \frac {3}{x^2\,\ln \left (\ln \left (4-x\right )-\frac {x}{5}+x^2+\frac {24}{5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((27*x + log(log(4 - x) - x/5 + x^2 + 24/5)*(168*x + log(4 - x)*(30*x - 120) - 126*x^2 + 30*x^3 - 576) - 12
3*x^2 + 30*x^3)/(log(log(4 - x) - x/5 + x^2 + 24/5)^2*(log(4 - x)*(20*x^3 - 5*x^4) + 96*x^3 - 28*x^4 + 21*x^5
- 5*x^6)),x)

[Out]

3/(x^2*log(log(4 - x) - x/5 + x^2 + 24/5))

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sympy [A]  time = 0.43, size = 20, normalized size = 0.77 \begin {gather*} \frac {3}{x^{2} \log {\left (x^{2} - \frac {x}{5} + \log {\left (4 - x \right )} + \frac {24}{5} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-30*x+120)*ln(-x+4)-30*x**3+126*x**2-168*x+576)*ln(ln(-x+4)+x**2-1/5*x+24/5)-30*x**3+123*x**2-27*
x)/((5*x**4-20*x**3)*ln(-x+4)+5*x**6-21*x**5+28*x**4-96*x**3)/ln(ln(-x+4)+x**2-1/5*x+24/5)**2,x)

[Out]

3/(x**2*log(x**2 - x/5 + log(4 - x) + 24/5))

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