3.89.40 \(\int \frac {(-19 x-12 x^2+12 x^3+4 x^4) \log (x)+(-20+8 x+4 x^2) \log ^2(x)+(4 x+4 x^2) \log ^3(x)+4 \log ^4(x)+(-80+32 x+16 x^2+16 \log ^2(x)) \log (\log ^4(x))+((4 x+4 x^2) \log (x)+4 \log ^2(x)) \log ^2(\log ^4(x))+16 \log ^3(\log ^4(x))}{x \log (x)} \, dx\)

Optimal. Leaf size=23 \[ x+\left (-5+2 x+x^2+\log ^2(x)+\log ^2\left (\log ^4(x)\right )\right )^2 \]

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Rubi [F]  time = 0.79, 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 (-19 x-12 x^2+12 x^3+4 x^4\right ) \log (x)+\left (-20+8 x+4 x^2\right ) \log ^2(x)+\left (4 x+4 x^2\right ) \log ^3(x)+4 \log ^4(x)+\left (-80+32 x+16 x^2+16 \log ^2(x)\right ) \log \left (\log ^4(x)\right )+\left (\left (4 x+4 x^2\right ) \log (x)+4 \log ^2(x)\right ) \log ^2\left (\log ^4(x)\right )+16 \log ^3\left (\log ^4(x)\right )}{x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-19*x - 12*x^2 + 12*x^3 + 4*x^4)*Log[x] + (-20 + 8*x + 4*x^2)*Log[x]^2 + (4*x + 4*x^2)*Log[x]^3 + 4*Log[
x]^4 + (-80 + 32*x + 16*x^2 + 16*Log[x]^2)*Log[Log[x]^4] + ((4*x + 4*x^2)*Log[x] + 4*Log[x]^2)*Log[Log[x]^4]^2
+ 16*Log[Log[x]^4]^3)/(x*Log[x]),x]

[Out]

-19*x - 6*x^2 + 4*x^3 + x^4 - 10*Log[x]^2 + 4*x*Log[x]^2 + 2*x^2*Log[x]^2 + Log[x]^4 - 10*Log[Log[x]^4]^2 + 2*
Log[x]^2*Log[Log[x]^4]^2 + Log[Log[x]^4]^4 + 32*Defer[Int][Log[Log[x]^4]/Log[x], x] + 16*Defer[Int][(x*Log[Log
[x]^4])/Log[x], x] + 4*Defer[Int][Log[Log[x]^4]^2, x] + 4*Defer[Int][x*Log[Log[x]^4]^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-19 x-12 x^2+12 x^3+4 x^4-20 \log (x)+8 x \log (x)+4 x^2 \log (x)+4 x \log ^2(x)+4 x^2 \log ^2(x)+4 \log ^3(x)}{x}+\frac {16 \left (-5+2 x+x^2+\log ^2(x)\right ) \log \left (\log ^4(x)\right )}{x \log (x)}+\frac {4 \left (x+x^2+\log (x)\right ) \log ^2\left (\log ^4(x)\right )}{x}+\frac {16 \log ^3\left (\log ^4(x)\right )}{x \log (x)}\right ) \, dx\\ &=4 \int \frac {\left (x+x^2+\log (x)\right ) \log ^2\left (\log ^4(x)\right )}{x} \, dx+16 \int \frac {\left (-5+2 x+x^2+\log ^2(x)\right ) \log \left (\log ^4(x)\right )}{x \log (x)} \, dx+16 \int \frac {\log ^3\left (\log ^4(x)\right )}{x \log (x)} \, dx+\int \frac {-19 x-12 x^2+12 x^3+4 x^4-20 \log (x)+8 x \log (x)+4 x^2 \log (x)+4 x \log ^2(x)+4 x^2 \log ^2(x)+4 \log ^3(x)}{x} \, dx\\ &=4 \int \left (\log ^2\left (\log ^4(x)\right )+x \log ^2\left (\log ^4(x)\right )+\frac {\log (x) \log ^2\left (\log ^4(x)\right )}{x}\right ) \, dx+16 \int \left (\frac {2 \log \left (\log ^4(x)\right )}{\log (x)}-\frac {5 \log \left (\log ^4(x)\right )}{x \log (x)}+\frac {x \log \left (\log ^4(x)\right )}{\log (x)}+\frac {\log (x) \log \left (\log ^4(x)\right )}{x}\right ) \, dx+16 \operatorname {Subst}\left (\int \frac {\log ^3\left (x^4\right )}{x} \, dx,x,\log (x)\right )+\int \left (-19-12 x+12 x^2+4 x^3+\frac {4 \left (-5+2 x+x^2\right ) \log (x)}{x}+4 (1+x) \log ^2(x)+\frac {4 \log ^3(x)}{x}\right ) \, dx\\ &=-19 x-6 x^2+4 x^3+x^4+4 \int \frac {\left (-5+2 x+x^2\right ) \log (x)}{x} \, dx+4 \int (1+x) \log ^2(x) \, dx+4 \int \frac {\log ^3(x)}{x} \, dx+4 \int \log ^2\left (\log ^4(x)\right ) \, dx+4 \int x \log ^2\left (\log ^4(x)\right ) \, dx+4 \int \frac {\log (x) \log ^2\left (\log ^4(x)\right )}{x} \, dx+4 \operatorname {Subst}\left (\int x^3 \, dx,x,\log \left (\log ^4(x)\right )\right )+16 \int \frac {x \log \left (\log ^4(x)\right )}{\log (x)} \, dx+16 \int \frac {\log (x) \log \left (\log ^4(x)\right )}{x} \, dx+32 \int \frac {\log \left (\log ^4(x)\right )}{\log (x)} \, dx-80 \int \frac {\log \left (\log ^4(x)\right )}{x \log (x)} \, dx\\ &=-19 x-6 x^2+4 x^3+x^4+\log ^4\left (\log ^4(x)\right )+4 \int \left (2 \log (x)-\frac {5 \log (x)}{x}+x \log (x)\right ) \, dx+4 \int \left (\log ^2(x)+x \log ^2(x)\right ) \, dx+4 \int \log ^2\left (\log ^4(x)\right ) \, dx+4 \int x \log ^2\left (\log ^4(x)\right ) \, dx+4 \operatorname {Subst}\left (\int x^3 \, dx,x,\log (x)\right )+4 \operatorname {Subst}\left (\int x \log ^2\left (x^4\right ) \, dx,x,\log (x)\right )+16 \int \frac {x \log \left (\log ^4(x)\right )}{\log (x)} \, dx+16 \operatorname {Subst}\left (\int x \log \left (x^4\right ) \, dx,x,\log (x)\right )+32 \int \frac {\log \left (\log ^4(x)\right )}{\log (x)} \, dx-80 \operatorname {Subst}\left (\int \frac {\log \left (x^4\right )}{x} \, dx,x,\log (x)\right )\\ &=-19 x-6 x^2+4 x^3+x^4-16 \log ^2(x)+\log ^4(x)+8 \log ^2(x) \log \left (\log ^4(x)\right )-10 \log ^2\left (\log ^4(x)\right )+2 \log ^2(x) \log ^2\left (\log ^4(x)\right )+\log ^4\left (\log ^4(x)\right )+4 \int x \log (x) \, dx+4 \int \log ^2(x) \, dx+4 \int x \log ^2(x) \, dx+4 \int \log ^2\left (\log ^4(x)\right ) \, dx+4 \int x \log ^2\left (\log ^4(x)\right ) \, dx+8 \int \log (x) \, dx+16 \int \frac {x \log \left (\log ^4(x)\right )}{\log (x)} \, dx-16 \operatorname {Subst}\left (\int x \log \left (x^4\right ) \, dx,x,\log (x)\right )-20 \int \frac {\log (x)}{x} \, dx+32 \int \frac {\log \left (\log ^4(x)\right )}{\log (x)} \, dx\\ &=-27 x-7 x^2+4 x^3+x^4+8 x \log (x)+2 x^2 \log (x)-10 \log ^2(x)+4 x \log ^2(x)+2 x^2 \log ^2(x)+\log ^4(x)-10 \log ^2\left (\log ^4(x)\right )+2 \log ^2(x) \log ^2\left (\log ^4(x)\right )+\log ^4\left (\log ^4(x)\right )-4 \int x \log (x) \, dx+4 \int \log ^2\left (\log ^4(x)\right ) \, dx+4 \int x \log ^2\left (\log ^4(x)\right ) \, dx-8 \int \log (x) \, dx+16 \int \frac {x \log \left (\log ^4(x)\right )}{\log (x)} \, dx+32 \int \frac {\log \left (\log ^4(x)\right )}{\log (x)} \, dx\\ &=-19 x-6 x^2+4 x^3+x^4-10 \log ^2(x)+4 x \log ^2(x)+2 x^2 \log ^2(x)+\log ^4(x)-10 \log ^2\left (\log ^4(x)\right )+2 \log ^2(x) \log ^2\left (\log ^4(x)\right )+\log ^4\left (\log ^4(x)\right )+4 \int \log ^2\left (\log ^4(x)\right ) \, dx+4 \int x \log ^2\left (\log ^4(x)\right ) \, dx+16 \int \frac {x \log \left (\log ^4(x)\right )}{\log (x)} \, dx+32 \int \frac {\log \left (\log ^4(x)\right )}{\log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.07, size = 63, normalized size = 2.74 \begin {gather*} -19 x-6 x^2+4 x^3+x^4+2 \left (-5+2 x+x^2\right ) \log ^2(x)+\log ^4(x)+2 \left (-5+2 x+x^2+\log ^2(x)\right ) \log ^2\left (\log ^4(x)\right )+\log ^4\left (\log ^4(x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-19*x - 12*x^2 + 12*x^3 + 4*x^4)*Log[x] + (-20 + 8*x + 4*x^2)*Log[x]^2 + (4*x + 4*x^2)*Log[x]^3 +
4*Log[x]^4 + (-80 + 32*x + 16*x^2 + 16*Log[x]^2)*Log[Log[x]^4] + ((4*x + 4*x^2)*Log[x] + 4*Log[x]^2)*Log[Log[x
]^4]^2 + 16*Log[Log[x]^4]^3)/(x*Log[x]),x]

[Out]

-19*x - 6*x^2 + 4*x^3 + x^4 + 2*(-5 + 2*x + x^2)*Log[x]^2 + Log[x]^4 + 2*(-5 + 2*x + x^2 + Log[x]^2)*Log[Log[x
]^4]^2 + Log[Log[x]^4]^4

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fricas [B]  time = 0.57, size = 63, normalized size = 2.74 \begin {gather*} x^{4} + \log \left (\log \relax (x)^{4}\right )^{4} + \log \relax (x)^{4} + 4 \, x^{3} + 2 \, {\left (x^{2} + \log \relax (x)^{2} + 2 \, x - 5\right )} \log \left (\log \relax (x)^{4}\right )^{2} + 2 \, {\left (x^{2} + 2 \, x - 5\right )} \log \relax (x)^{2} - 6 \, x^{2} - 19 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*log(log(x)^4)^3+(4*log(x)^2+(4*x^2+4*x)*log(x))*log(log(x)^4)^2+(16*log(x)^2+16*x^2+32*x-80)*log
(log(x)^4)+4*log(x)^4+(4*x^2+4*x)*log(x)^3+(4*x^2+8*x-20)*log(x)^2+(4*x^4+12*x^3-12*x^2-19*x)*log(x))/x/log(x)
,x, algorithm="fricas")

[Out]

x^4 + log(log(x)^4)^4 + log(x)^4 + 4*x^3 + 2*(x^2 + log(x)^2 + 2*x - 5)*log(log(x)^4)^2 + 2*(x^2 + 2*x - 5)*lo
g(x)^2 - 6*x^2 - 19*x

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giac [B]  time = 0.33, size = 63, normalized size = 2.74 \begin {gather*} x^{4} + \log \left (\log \relax (x)^{4}\right )^{4} + \log \relax (x)^{4} + 4 \, x^{3} + 2 \, {\left (x^{2} + \log \relax (x)^{2} + 2 \, x - 5\right )} \log \left (\log \relax (x)^{4}\right )^{2} + 2 \, {\left (x^{2} + 2 \, x - 5\right )} \log \relax (x)^{2} - 6 \, x^{2} - 19 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*log(log(x)^4)^3+(4*log(x)^2+(4*x^2+4*x)*log(x))*log(log(x)^4)^2+(16*log(x)^2+16*x^2+32*x-80)*log
(log(x)^4)+4*log(x)^4+(4*x^2+4*x)*log(x)^3+(4*x^2+8*x-20)*log(x)^2+(4*x^4+12*x^3-12*x^2-19*x)*log(x))/x/log(x)
,x, algorithm="giac")

[Out]

x^4 + log(log(x)^4)^4 + log(x)^4 + 4*x^3 + 2*(x^2 + log(x)^2 + 2*x - 5)*log(log(x)^4)^2 + 2*(x^2 + 2*x - 5)*lo
g(x)^2 - 6*x^2 - 19*x

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maple [C]  time = 0.97, size = 12852, normalized size = 558.78

method	result	size
risch	\(\text {Expression too large to display}\)	\(12852\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*ln(ln(x)^4)^3+(4*ln(x)^2+(4*x^2+4*x)*ln(x))*ln(ln(x)^4)^2+(16*ln(x)^2+16*x^2+32*x-80)*ln(ln(x)^4)+4*ln
(x)^4+(4*x^2+4*x)*ln(x)^3+(4*x^2+8*x-20)*ln(x)^2+(4*x^4+12*x^3-12*x^2-19*x)*ln(x))/x/ln(x),x,method=_RETURNVER
BOSE)

[Out]

result too large to display

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maxima [B]  time = 0.42, size = 94, normalized size = 4.09 \begin {gather*} x^{4} + \log \relax (x)^{4} + 256 \, \log \left (\log \relax (x)\right )^{4} + {\left (2 \, \log \relax (x)^{2} - 2 \, \log \relax (x) + 1\right )} x^{2} + 4 \, x^{3} + 2 \, x^{2} \log \relax (x) + 32 \, {\left (x^{2} + \log \relax (x)^{2} + 2 \, x - 5\right )} \log \left (\log \relax (x)\right )^{2} + 4 \, {\left (\log \relax (x)^{2} - 2 \, \log \relax (x) + 2\right )} x - 7 \, x^{2} + 8 \, x \log \relax (x) - 10 \, \log \relax (x)^{2} - 27 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*log(log(x)^4)^3+(4*log(x)^2+(4*x^2+4*x)*log(x))*log(log(x)^4)^2+(16*log(x)^2+16*x^2+32*x-80)*log
(log(x)^4)+4*log(x)^4+(4*x^2+4*x)*log(x)^3+(4*x^2+8*x-20)*log(x)^2+(4*x^4+12*x^3-12*x^2-19*x)*log(x))/x/log(x)
,x, algorithm="maxima")

[Out]

x^4 + log(x)^4 + 256*log(log(x))^4 + (2*log(x)^2 - 2*log(x) + 1)*x^2 + 4*x^3 + 2*x^2*log(x) + 32*(x^2 + log(x)
^2 + 2*x - 5)*log(log(x))^2 + 4*(log(x)^2 - 2*log(x) + 2)*x - 7*x^2 + 8*x*log(x) - 10*log(x)^2 - 27*x

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mupad [B]  time = 5.28, size = 67, normalized size = 2.91 \begin {gather*} {\ln \left ({\ln \relax (x)}^4\right )}^4-19\,x+{\ln \relax (x)}^2\,\left (2\,x^2+4\,x-10\right )+{\ln \relax (x)}^4+{\ln \left ({\ln \relax (x)}^4\right )}^2\,\left (2\,x^2+4\,x+2\,{\ln \relax (x)}^2-10\right )-6\,x^2+4\,x^3+x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*log(log(x)^4)^3 + log(x)^3*(4*x + 4*x^2) + log(x)^2*(8*x + 4*x^2 - 20) - log(x)*(19*x + 12*x^2 - 12*x^
3 - 4*x^4) + log(log(x)^4)*(32*x + 16*log(x)^2 + 16*x^2 - 80) + log(log(x)^4)^2*(4*log(x)^2 + log(x)*(4*x + 4*
x^2)) + 4*log(x)^4)/(x*log(x)),x)

[Out]

log(log(x)^4)^4 - 19*x + log(x)^2*(4*x + 2*x^2 - 10) + log(x)^4 + log(log(x)^4)^2*(4*x + 2*log(x)^2 + 2*x^2 -
10) - 6*x^2 + 4*x^3 + x^4

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sympy [B]  time = 0.50, size = 70, normalized size = 3.04 \begin {gather*} x^{4} + 4 x^{3} - 6 x^{2} - 19 x + \left (2 x^{2} + 4 x - 10\right ) \log {\relax (x )}^{2} + \left (2 x^{2} + 4 x + 2 \log {\relax (x )}^{2} - 10\right ) \log {\left (\log {\relax (x )}^{4} \right )}^{2} + \log {\relax (x )}^{4} + \log {\left (\log {\relax (x )}^{4} \right )}^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*ln(ln(x)**4)**3+(4*ln(x)**2+(4*x**2+4*x)*ln(x))*ln(ln(x)**4)**2+(16*ln(x)**2+16*x**2+32*x-80)*ln
(ln(x)**4)+4*ln(x)**4+(4*x**2+4*x)*ln(x)**3+(4*x**2+8*x-20)*ln(x)**2+(4*x**4+12*x**3-12*x**2-19*x)*ln(x))/x/ln
(x),x)

[Out]

x**4 + 4*x**3 - 6*x**2 - 19*x + (2*x**2 + 4*x - 10)*log(x)**2 + (2*x**2 + 4*x + 2*log(x)**2 - 10)*log(log(x)**
4)**2 + log(x)**4 + log(log(x)**4)**4

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