3.70.54 \(\int \frac {(2 x^3+4 x^2 \log (4)+2 x \log ^2(4)) \log (\log (x) \log (\log (4)))+(4 x^3+6 x^2 \log (4)+2 x \log ^2(4)) \log (x) \log ^2(\log (x) \log (\log (4)))}{\log (x)} \, dx\) [6954]

Optimal. Leaf size=19 \[ x^2 (x+\log (4))^2 \log ^2(\log (x) \log (\log (4))) \]

[Out]

ln(ln(x)*ln(2*ln(2)))^2*(x+2*ln(2))^2*x^2

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Rubi [F]
time = 0.83, 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 (2 x^3+4 x^2 \log (4)+2 x \log ^2(4)\right ) \log (\log (x) \log (\log (4)))+\left (4 x^3+6 x^2 \log (4)+2 x \log ^2(4)\right ) \log (x) \log ^2(\log (x) \log (\log (4)))}{\log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.06, size = 19, normalized size = 1.00 \begin {gather*} x^2 (x+\log (4))^2 \log ^2(\log (x) \log (\log (4))) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((2*x^3 + 4*x^2*Log[4] + 2*x*Log[4]^2)*Log[Log[x]*Log[Log[4]]] + (4*x^3 + 6*x^2*Log[4] + 2*x*Log[4]^
2)*Log[x]*Log[Log[x]*Log[Log[4]]]^2)/Log[x],x]

[Out]

x^2*(x + Log[4])^2*Log[Log[x]*Log[Log[4]]]^2

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Maple [A]
time = 1.87, size = 34, normalized size = 1.79

method	result	size
risch	\(\left (4 x^{2} \ln \left (2\right )^{2}+4 x^{3} \ln \left (2\right )+x^{4}\right ) \ln \left (\ln \left (x \right ) \left (\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )\right )\right )^{2}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x*ln(2)^2+12*x^2*ln(2)+4*x^3)*ln(x)*ln(ln(x)*ln(2*ln(2)))^2+(8*x*ln(2)^2+8*x^2*ln(2)+2*x^3)*ln(ln(x)*l
n(2*ln(2))))/ln(x),x,method=_RETURNVERBOSE)

[Out]

(4*x^2*ln(2)^2+4*x^3*ln(2)+x^4)*ln(ln(x)*(ln(2)+ln(ln(2))))^2

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (23) = 46\).
time = 0.55, size = 121, normalized size = 6.37 \begin {gather*} x^{4} \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )^{2} + 4 \, x^{3} \log \left (2\right ) \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )^{2} + 4 \, x^{2} \log \left (2\right )^{2} \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )^{2} + {\left (x^{4} + 4 \, x^{3} \log \left (2\right ) + 4 \, x^{2} \log \left (2\right )^{2}\right )} \log \left (\log \left (x\right )\right )^{2} + 2 \, {\left (x^{4} \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right ) + 4 \, x^{3} \log \left (2\right ) \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right ) + 4 \, x^{2} \log \left (2\right )^{2} \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )\right )} \log \left (\log \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^4*log(log(2) + log(log(2)))^2 + 4*x^3*log(2)*log(log(2) + log(log(2)))^2 + 4*x^2*log(2)^2*log(log(2) + log(l
og(2)))^2 + (x^4 + 4*x^3*log(2) + 4*x^2*log(2)^2)*log(log(x))^2 + 2*(x^4*log(log(2) + log(log(2))) + 4*x^3*log
(2)*log(log(2) + log(log(2))) + 4*x^2*log(2)^2*log(log(2) + log(log(2))))*log(log(x))

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Fricas [A]
time = 0.40, size = 32, normalized size = 1.68 \begin {gather*} {\left (x^{4} + 4 \, x^{3} \log \left (2\right ) + 4 \, x^{2} \log \left (2\right )^{2}\right )} \log \left (\log \left (x\right ) \log \left (2 \, \log \left (2\right )\right )\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.16, size = 34, normalized size = 1.79 \begin {gather*} \left (x^{4} + 4 x^{3} \log {\left (2 \right )} + 4 x^{2} \log {\left (2 \right )}^{2}\right ) \log {\left (\log {\left (x \right )} \log {\left (2 \log {\left (2 \right )} \right )} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x*ln(2)**2+12*x**2*ln(2)+4*x**3)*ln(x)*ln(ln(x)*ln(2*ln(2)))**2+(8*x*ln(2)**2+8*x**2*ln(2)+2*x**
3)*ln(ln(x)*ln(2*ln(2))))/ln(x),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (23) = 46\).
time = 0.50, size = 134, normalized size = 7.05 \begin {gather*} x^{4} \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )^{2} + 4 \, x^{3} \log \left (2\right ) \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )^{2} + 4 \, x^{2} \log \left (2\right )^{2} \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )^{2} + 2 \, x^{4} \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right ) \log \left (\log \left (x\right )\right ) + 8 \, x^{3} \log \left (2\right ) \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right ) \log \left (\log \left (x\right )\right ) + 8 \, x^{2} \log \left (2\right )^{2} \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right ) \log \left (\log \left (x\right )\right ) + x^{4} \log \left (\log \left (x\right )\right )^{2} + 4 \, x^{3} \log \left (2\right ) \log \left (\log \left (x\right )\right )^{2} + 4 \, x^{2} \log \left (2\right )^{2} \log \left (\log \left (x\right )\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^4*log(log(2) + log(log(2)))^2 + 4*x^3*log(2)*log(log(2) + log(log(2)))^2 + 4*x^2*log(2)^2*log(log(2) + log(l
og(2)))^2 + 2*x^4*log(log(2) + log(log(2)))*log(log(x)) + 8*x^3*log(2)*log(log(2) + log(log(2)))*log(log(x)) +
8*x^2*log(2)^2*log(log(2) + log(log(2)))*log(log(x)) + x^4*log(log(x))^2 + 4*x^3*log(2)*log(log(x))^2 + 4*x^2
*log(2)^2*log(log(x))^2

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Mupad [B]
time = 4.33, size = 30, normalized size = 1.58 \begin {gather*} {\ln \left (\ln \left (\ln \left (4\right )\right )\,\ln \left (x\right )\right )}^2\,\left (x^4+4\,\ln \left (2\right )\,x^3+4\,{\ln \left (2\right )}^2\,x^2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(log(2*log(2))*log(x))*(8*x*log(2)^2 + 8*x^2*log(2) + 2*x^3) + log(x)*log(log(2*log(2))*log(x))^2*(8*x
*log(2)^2 + 12*x^2*log(2) + 4*x^3))/log(x),x)

[Out]

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

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