3.26.26 \(\int \frac {(-4 x^2+x^3) \log ^2(3) \log (4-x)+(-8+2 x+(64-16 x) \log (3)+(-128+32 x) \log ^2(3)) \log ^2(4-x)-x^3 \log ^2(3) \log (x)}{(-4 x^3+x^4) \log ^2(3) \log ^2(4-x)} \, dx\)

Optimal. Leaf size=25 \[ -\frac {\left (-4+\frac {1}{\log (3)}\right )^2}{x^2}+\frac {\log (x)}{\log (4-x)} \]

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Rubi [F]  time = 0.50, 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 {\left (-4 x^2+x^3\right ) \log ^2(3) \log (4-x)+\left (-8+2 x+(64-16 x) \log (3)+(-128+32 x) \log ^2(3)\right ) \log ^2(4-x)-x^3 \log ^2(3) \log (x)}{\left (-4 x^3+x^4\right ) \log ^2(3) \log ^2(4-x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-4*x^2 + x^3)*Log[3]^2*Log[4 - x] + (-8 + 2*x + (64 - 16*x)*Log[3] + (-128 + 32*x)*Log[3]^2)*Log[4 - x]^
2 - x^3*Log[3]^2*Log[x])/((-4*x^3 + x^4)*Log[3]^2*Log[4 - x]^2),x]

[Out]

-((1 - Log[81])^2/(x^2*Log[3]^2)) + Defer[Int][1/(x*Log[4 - x]), x] - Defer[Subst][Defer[Int][Log[4 - x]/(x*Lo
g[x]^2), x], x, 4 - x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {\left (-4 x^2+x^3\right ) \log ^2(3) \log (4-x)+\left (-8+2 x+(64-16 x) \log (3)+(-128+32 x) \log ^2(3)\right ) \log ^2(4-x)-x^3 \log ^2(3) \log (x)}{\left (-4 x^3+x^4\right ) \log ^2(4-x)} \, dx}{\log ^2(3)}\\ &=\frac {\int \frac {\left (-4 x^2+x^3\right ) \log ^2(3) \log (4-x)+\left (-8+2 x+(64-16 x) \log (3)+(-128+32 x) \log ^2(3)\right ) \log ^2(4-x)-x^3 \log ^2(3) \log (x)}{(-4+x) x^3 \log ^2(4-x)} \, dx}{\log ^2(3)}\\ &=\frac {\int \left (\frac {2 (-1+\log (81))^2}{x^3}+\frac {\log ^2(3)}{x \log (4-x)}-\frac {\log ^2(3) \log (x)}{(-4+x) \log ^2(4-x)}\right ) \, dx}{\log ^2(3)}\\ &=-\frac {(1-\log (81))^2}{x^2 \log ^2(3)}+\int \frac {1}{x \log (4-x)} \, dx-\int \frac {\log (x)}{(-4+x) \log ^2(4-x)} \, dx\\ &=-\frac {(1-\log (81))^2}{x^2 \log ^2(3)}+\int \frac {1}{x \log (4-x)} \, dx-\operatorname {Subst}\left (\int \frac {\log (4-x)}{x \log ^2(x)} \, dx,x,4-x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 27, normalized size = 1.08 \begin {gather*} -\frac {(-1+\log (81))^2}{x^2 \log ^2(3)}+\frac {\log (x)}{\log (4-x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-4*x^2 + x^3)*Log[3]^2*Log[4 - x] + (-8 + 2*x + (64 - 16*x)*Log[3] + (-128 + 32*x)*Log[3]^2)*Log[4
- x]^2 - x^3*Log[3]^2*Log[x])/((-4*x^3 + x^4)*Log[3]^2*Log[4 - x]^2),x]

[Out]

-((-1 + Log[81])^2/(x^2*Log[3]^2)) + Log[x]/Log[4 - x]

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fricas [A]  time = 1.02, size = 47, normalized size = 1.88 \begin {gather*} \frac {x^{2} \log \relax (3)^{2} \log \relax (x) - {\left (16 \, \log \relax (3)^{2} - 8 \, \log \relax (3) + 1\right )} \log \left (-x + 4\right )}{x^{2} \log \relax (3)^{2} \log \left (-x + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^3*log(3)^2*log(x)+((32*x-128)*log(3)^2+(-16*x+64)*log(3)+2*x-8)*log(-x+4)^2+(x^3-4*x^2)*log(3)^2
*log(-x+4))/(x^4-4*x^3)/log(3)^2/log(-x+4)^2,x, algorithm="fricas")

[Out]

(x^2*log(3)^2*log(x) - (16*log(3)^2 - 8*log(3) + 1)*log(-x + 4))/(x^2*log(3)^2*log(-x + 4))

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giac [A]  time = 0.43, size = 38, normalized size = 1.52 \begin {gather*} \frac {\frac {\log \relax (3)^{2} \log \relax (x)}{\log \left (-x + 4\right )} - \frac {16 \, \log \relax (3)^{2} - 8 \, \log \relax (3) + 1}{x^{2}}}{\log \relax (3)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^3*log(3)^2*log(x)+((32*x-128)*log(3)^2+(-16*x+64)*log(3)+2*x-8)*log(-x+4)^2+(x^3-4*x^2)*log(3)^2
*log(-x+4))/(x^4-4*x^3)/log(3)^2/log(-x+4)^2,x, algorithm="giac")

[Out]

(log(3)^2*log(x)/log(-x + 4) - (16*log(3)^2 - 8*log(3) + 1)/x^2)/log(3)^2

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maple [A]  time = 0.41, size = 36, normalized size = 1.44

method	result	size
risch	\(-\frac {16}{x^{2}}+\frac {8}{\ln \relax (3) x^{2}}-\frac {1}{\ln \relax (3)^{2} x^{2}}+\frac {\ln \relax (x )}{\ln \left (-x +4\right )}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^3*ln(3)^2*ln(x)+((32*x-128)*ln(3)^2+(-16*x+64)*ln(3)+2*x-8)*ln(-x+4)^2+(x^3-4*x^2)*ln(3)^2*ln(-x+4))/(
x^4-4*x^3)/ln(3)^2/ln(-x+4)^2,x,method=_RETURNVERBOSE)

[Out]

-16/x^2+8/ln(3)/x^2-1/ln(3)^2/x^2+1/ln(-x+4)*ln(x)

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maxima [B]  time = 0.64, size = 117, normalized size = 4.68 \begin {gather*} -\frac {4 \, {\left (\frac {4 \, {\left (x + 2\right )}}{x^{2}} + \log \left (x - 4\right ) - \log \relax (x)\right )} \log \relax (3)^{2} - 4 \, {\left (\frac {4}{x} + \log \left (x - 4\right ) - \log \relax (x)\right )} \log \relax (3)^{2} - 2 \, {\left (\frac {4 \, {\left (x + 2\right )}}{x^{2}} + \log \left (x - 4\right ) - \log \relax (x)\right )} \log \relax (3) + 2 \, {\left (\frac {4}{x} + \log \left (x - 4\right ) - \log \relax (x)\right )} \log \relax (3) - \frac {2 \, \log \relax (3)^{2} \log \relax (x)}{\log \left (-x + 4\right )} + \frac {x + 2}{x^{2}} - \frac {1}{x}}{2 \, \log \relax (3)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^3*log(3)^2*log(x)+((32*x-128)*log(3)^2+(-16*x+64)*log(3)+2*x-8)*log(-x+4)^2+(x^3-4*x^2)*log(3)^2
*log(-x+4))/(x^4-4*x^3)/log(3)^2/log(-x+4)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.74, size = 55, normalized size = 2.20 \begin {gather*} \frac {\ln \relax (x)-\frac {\ln \left (4-x\right )\,\left (x-4\right )}{x}}{\ln \left (4-x\right )}-\frac {4\,x\,{\ln \relax (3)}^2-8\,\ln \relax (3)+16\,{\ln \relax (3)}^2+1}{x^2\,{\ln \relax (3)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(3)^2*log(4 - x)*(4*x^2 - x^3) - log(4 - x)^2*(2*x - log(3)*(16*x - 64) + log(3)^2*(32*x - 128) - 8) +
x^3*log(3)^2*log(x))/(log(3)^2*log(4 - x)^2*(4*x^3 - x^4)),x)

[Out]

(log(x) - (log(4 - x)*(x - 4))/x)/log(4 - x) - (4*x*log(3)^2 - 8*log(3) + 16*log(3)^2 + 1)/(x^2*log(3)^2)

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sympy [A]  time = 0.28, size = 31, normalized size = 1.24 \begin {gather*} \frac {\log {\relax (x )}}{\log {\left (4 - x \right )}} - \frac {- 16 \log {\relax (3 )} + 2 + 32 \log {\relax (3 )}^{2}}{2 x^{2} \log {\relax (3 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**3*ln(3)**2*ln(x)+((32*x-128)*ln(3)**2+(-16*x+64)*ln(3)+2*x-8)*ln(-x+4)**2+(x**3-4*x**2)*ln(3)**
2*ln(-x+4))/(x**4-4*x**3)/ln(3)**2/ln(-x+4)**2,x)

[Out]

log(x)/log(4 - x) - (-16*log(3) + 2 + 32*log(3)**2)/(2*x**2*log(3)**2)

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