3.83.80 \(\int \frac {(32 x^3-32 x^4 \log (x)) \log (-4+x-\log (\log (x)))+((256 x^3-64 x^4) \log (x)+64 x^3 \log (x) \log (\log (x))) \log ^2(-4+x-\log (\log (x)))}{(4-x) \log (x)+\log (x) \log (\log (x))} \, dx\)

Optimal. Leaf size=16 \[ 16 x^4 \log ^2(-4+x-\log (\log (x))) \]

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Rubi [F]  time = 1.12, 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 {\left (32 x^3-32 x^4 \log (x)\right ) \log (-4+x-\log (\log (x)))+\left (\left (256 x^3-64 x^4\right ) \log (x)+64 x^3 \log (x) \log (\log (x))\right ) \log ^2(-4+x-\log (\log (x)))}{(4-x) \log (x)+\log (x) \log (\log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((32*x^3 - 32*x^4*Log[x])*Log[-4 + x - Log[Log[x]]] + ((256*x^3 - 64*x^4)*Log[x] + 64*x^3*Log[x]*Log[Log[x
]])*Log[-4 + x - Log[Log[x]]]^2)/((4 - x)*Log[x] + Log[x]*Log[Log[x]]),x]

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.64, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (32 x^3-32 x^4 \log (x)\right ) \log (-4+x-\log (\log (x)))+\left (\left (256 x^3-64 x^4\right ) \log (x)+64 x^3 \log (x) \log (\log (x))\right ) \log ^2(-4+x-\log (\log (x)))}{(4-x) \log (x)+\log (x) \log (\log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((32*x^3 - 32*x^4*Log[x])*Log[-4 + x - Log[Log[x]]] + ((256*x^3 - 64*x^4)*Log[x] + 64*x^3*Log[x]*Log
[Log[x]])*Log[-4 + x - Log[Log[x]]]^2)/((4 - x)*Log[x] + Log[x]*Log[Log[x]]),x]

[Out]

Integrate[((32*x^3 - 32*x^4*Log[x])*Log[-4 + x - Log[Log[x]]] + ((256*x^3 - 64*x^4)*Log[x] + 64*x^3*Log[x]*Log
[Log[x]])*Log[-4 + x - Log[Log[x]]]^2)/((4 - x)*Log[x] + Log[x]*Log[Log[x]]), x]

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fricas [A]  time = 0.69, size = 16, normalized size = 1.00 \begin {gather*} 16 \, x^{4} \log \left (x - \log \left (\log \relax (x)\right ) - 4\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((64*x^3*log(x)*log(log(x))+(-64*x^4+256*x^3)*log(x))*log(-log(log(x))+x-4)^2+(-32*x^4*log(x)+32*x^3
)*log(-log(log(x))+x-4))/(log(x)*log(log(x))+(-x+4)*log(x)),x, algorithm="fricas")

[Out]

16*x^4*log(x - log(log(x)) - 4)^2

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((64*x^3*log(x)*log(log(x))+(-64*x^4+256*x^3)*log(x))*log(-log(log(x))+x-4)^2+(-32*x^4*log(x)+32*x^3
)*log(-log(log(x))+x-4))/(log(x)*log(log(x))+(-x+4)*log(x)),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Sign error %%%{ln(` w`),0%%%}Sign error %%%{ln(` w`),0%%%}Sign error %%%{ln(` w`),0%%%}Done

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maple [A]  time = 0.03, size = 17, normalized size = 1.06

method	result	size
risch	\(16 x^{4} \ln \left (-\ln \left (\ln \relax (x )\right )+x -4\right )^{2}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((64*x^3*ln(x)*ln(ln(x))+(-64*x^4+256*x^3)*ln(x))*ln(-ln(ln(x))+x-4)^2+(-32*x^4*ln(x)+32*x^3)*ln(-ln(ln(x)
)+x-4))/(ln(x)*ln(ln(x))+(-x+4)*ln(x)),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.41, size = 16, normalized size = 1.00 \begin {gather*} 16 \, x^{4} \log \left (x - \log \left (\log \relax (x)\right ) - 4\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((64*x^3*log(x)*log(log(x))+(-64*x^4+256*x^3)*log(x))*log(-log(log(x))+x-4)^2+(-32*x^4*log(x)+32*x^3
)*log(-log(log(x))+x-4))/(log(x)*log(log(x))+(-x+4)*log(x)),x, algorithm="maxima")

[Out]

16*x^4*log(x - log(log(x)) - 4)^2

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mupad [B]  time = 5.92, size = 16, normalized size = 1.00 \begin {gather*} 16\,x^4\,{\ln \left (x-\ln \left (\ln \relax (x)\right )-4\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x - log(log(x)) - 4)*(32*x^4*log(x) - 32*x^3) - log(x - log(log(x)) - 4)^2*(log(x)*(256*x^3 - 64*x^4)
+ 64*x^3*log(log(x))*log(x)))/(log(x)*(x - 4) - log(log(x))*log(x)),x)

[Out]

16*x^4*log(x - log(log(x)) - 4)^2

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sympy [A]  time = 0.95, size = 15, normalized size = 0.94 \begin {gather*} 16 x^{4} \log {\left (x - \log {\left (\log {\relax (x )} \right )} - 4 \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((64*x**3*ln(x)*ln(ln(x))+(-64*x**4+256*x**3)*ln(x))*ln(-ln(ln(x))+x-4)**2+(-32*x**4*ln(x)+32*x**3)*
ln(-ln(ln(x))+x-4))/(ln(x)*ln(ln(x))+(-x+4)*ln(x)),x)

[Out]

16*x**4*log(x - log(log(x)) - 4)**2

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