3.37.19 \(\int \frac {5-x+(5-51 x+20 x^2-2 x^3) \log (x) \log (\log (x))+5 \log (x) \log (\log (x)) \log (\frac {1}{5} x \log (\log (x)))}{(50 x^4-20 x^5+2 x^6) \log (x) \log (\log (x))+(-20 x^3+4 x^4) \log (x) \log (\log (x)) \log (\frac {1}{5} x \log (\log (x)))+2 x^2 \log (x) \log (\log (x)) \log ^2(\frac {1}{5} x \log (\log (x)))} \, dx\)

Optimal. Leaf size=34 \[ \frac {5-x}{2 x \left (5 x-x^2-\log \left (\frac {1}{5} x \log (\log (x))\right )\right )} \]

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Rubi [F]  time = 4.09, 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 {5-x+\left (5-51 x+20 x^2-2 x^3\right ) \log (x) \log (\log (x))+5 \log (x) \log (\log (x)) \log \left (\frac {1}{5} x \log (\log (x))\right )}{\left (50 x^4-20 x^5+2 x^6\right ) \log (x) \log (\log (x))+\left (-20 x^3+4 x^4\right ) \log (x) \log (\log (x)) \log \left (\frac {1}{5} x \log (\log (x))\right )+2 x^2 \log (x) \log (\log (x)) \log ^2\left (\frac {1}{5} x \log (\log (x))\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(5 - x + (5 - 51*x + 20*x^2 - 2*x^3)*Log[x]*Log[Log[x]] + 5*Log[x]*Log[Log[x]]*Log[(x*Log[Log[x]])/5])/((5
0*x^4 - 20*x^5 + 2*x^6)*Log[x]*Log[Log[x]] + (-20*x^3 + 4*x^4)*Log[x]*Log[Log[x]]*Log[(x*Log[Log[x]])/5] + 2*x
^2*Log[x]*Log[Log[x]]*Log[(x*Log[Log[x]])/5]^2),x]

[Out]

-1/2*1/(5*x - x^2 - Log[(x*Log[Log[x]])/5]) + 5*Defer[Int][(-5*x + x^2 + Log[(x*Log[Log[x]])/5])^(-2), x] + (5
*Defer[Int][1/(x^2*(-5*x + x^2 + Log[(x*Log[Log[x]])/5])^2), x])/2 - (25*Defer[Int][1/(x*(-5*x + x^2 + Log[(x*
Log[Log[x]])/5])^2), x])/2 + (5*Defer[Int][1/(x^2*Log[x]*Log[Log[x]]*(-5*x + x^2 + Log[(x*Log[Log[x]])/5])^2),
x])/2 + (5*Defer[Int][1/(x^2*(-5*x + x^2 + Log[(x*Log[Log[x]])/5])), x])/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5-x+\log (x) \log (\log (x)) \left (5-51 x+20 x^2-2 x^3+5 \log \left (\frac {1}{5} x \log (\log (x))\right )\right )}{2 x^2 \log (x) \log (\log (x)) \left ((-5+x) x+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {5-x+\log (x) \log (\log (x)) \left (5-51 x+20 x^2-2 x^3+5 \log \left (\frac {1}{5} x \log (\log (x))\right )\right )}{x^2 \log (x) \log (\log (x)) \left ((-5+x) x+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2} \, dx\\ &=\frac {1}{2} \int \left (-\frac {(-5+x) \left (1+\log (x) \log (\log (x))-5 x \log (x) \log (\log (x))+2 x^2 \log (x) \log (\log (x))\right )}{x^2 \log (x) \log (\log (x)) \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2}+\frac {5}{x^2 \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {(-5+x) \left (1+\log (x) \log (\log (x))-5 x \log (x) \log (\log (x))+2 x^2 \log (x) \log (\log (x))\right )}{x^2 \log (x) \log (\log (x)) \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2} \, dx\right )+\frac {5}{2} \int \frac {1}{x^2 \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )} \, dx\\ &=-\left (\frac {1}{2} \int \left (-\frac {5 \left (1+\log (x) \log (\log (x))-5 x \log (x) \log (\log (x))+2 x^2 \log (x) \log (\log (x))\right )}{x^2 \log (x) \log (\log (x)) \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2}+\frac {1+\log (x) \log (\log (x))-5 x \log (x) \log (\log (x))+2 x^2 \log (x) \log (\log (x))}{x \log (x) \log (\log (x)) \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2}\right ) \, dx\right )+\frac {5}{2} \int \frac {1}{x^2 \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )} \, dx\\ &=-\left (\frac {1}{2} \int \frac {1+\log (x) \log (\log (x))-5 x \log (x) \log (\log (x))+2 x^2 \log (x) \log (\log (x))}{x \log (x) \log (\log (x)) \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2} \, dx\right )+\frac {5}{2} \int \frac {1+\log (x) \log (\log (x))-5 x \log (x) \log (\log (x))+2 x^2 \log (x) \log (\log (x))}{x^2 \log (x) \log (\log (x)) \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2} \, dx+\frac {5}{2} \int \frac {1}{x^2 \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )} \, dx\\ &=-\frac {1}{2 \left (5 x-x^2-\log \left (\frac {1}{5} x \log (\log (x))\right )\right )}+\frac {5}{2} \int \frac {1}{x^2 \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )} \, dx+\frac {5}{2} \int \left (\frac {2}{\left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2}+\frac {1}{x^2 \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2}-\frac {5}{x \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2}+\frac {1}{x^2 \log (x) \log (\log (x)) \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2}\right ) \, dx\\ &=-\frac {1}{2 \left (5 x-x^2-\log \left (\frac {1}{5} x \log (\log (x))\right )\right )}+\frac {5}{2} \int \frac {1}{x^2 \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2} \, dx+\frac {5}{2} \int \frac {1}{x^2 \log (x) \log (\log (x)) \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2} \, dx+\frac {5}{2} \int \frac {1}{x^2 \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )} \, dx+5 \int \frac {1}{\left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2} \, dx-\frac {25}{2} \int \frac {1}{x \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.20, size = 28, normalized size = 0.82 \begin {gather*} \frac {-5+x}{2 x \left (-5 x+x^2+\log \left (\frac {1}{5} x \log (\log (x))\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 - x + (5 - 51*x + 20*x^2 - 2*x^3)*Log[x]*Log[Log[x]] + 5*Log[x]*Log[Log[x]]*Log[(x*Log[Log[x]])/5
])/((50*x^4 - 20*x^5 + 2*x^6)*Log[x]*Log[Log[x]] + (-20*x^3 + 4*x^4)*Log[x]*Log[Log[x]]*Log[(x*Log[Log[x]])/5]
+ 2*x^2*Log[x]*Log[Log[x]]*Log[(x*Log[Log[x]])/5]^2),x]

[Out]

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

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fricas [A]  time = 0.72, size = 25, normalized size = 0.74 \begin {gather*} \frac {x - 5}{2 \, {\left (x^{3} - 5 \, x^{2} + x \log \left (\frac {1}{5} \, x \log \left (\log \relax (x)\right )\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*log(x)*log(log(x))*log(1/5*x*log(log(x)))+(-2*x^3+20*x^2-51*x+5)*log(x)*log(log(x))+5-x)/(2*x^2*l
og(x)*log(log(x))*log(1/5*x*log(log(x)))^2+(4*x^4-20*x^3)*log(x)*log(log(x))*log(1/5*x*log(log(x)))+(2*x^6-20*
x^5+50*x^4)*log(x)*log(log(x))),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 1.35, size = 31, normalized size = 0.91 \begin {gather*} \frac {x - 5}{2 \, {\left (x^{3} - 5 \, x^{2} - x \log \relax (5) + x \log \relax (x) + x \log \left (\log \left (\log \relax (x)\right )\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*log(x)*log(log(x))*log(1/5*x*log(log(x)))+(-2*x^3+20*x^2-51*x+5)*log(x)*log(log(x))+5-x)/(2*x^2*l
og(x)*log(log(x))*log(1/5*x*log(log(x)))^2+(4*x^4-20*x^3)*log(x)*log(log(x))*log(1/5*x*log(log(x)))+(2*x^6-20*
x^5+50*x^4)*log(x)*log(log(x))),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.14, size = 111, normalized size = 3.26

method	result	size
risch	\(\frac {x -5}{x \left (-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \left (\ln \relax (x )\right )\right ) \mathrm {csgn}\left (i x \ln \left (\ln \relax (x )\right )\right )+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \left (\ln \relax (x )\right )\right )^{2}+i \pi \,\mathrm {csgn}\left (i \ln \left (\ln \relax (x )\right )\right ) \mathrm {csgn}\left (i x \ln \left (\ln \relax (x )\right )\right )^{2}-i \pi \mathrm {csgn}\left (i x \ln \left (\ln \relax (x )\right )\right )^{3}+2 x^{2}-2 \ln \relax (5)-10 x +2 \ln \relax (x )+2 \ln \left (\ln \left (\ln \relax (x )\right )\right )\right )}\)	\(111\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*ln(x)*ln(ln(x))*ln(1/5*x*ln(ln(x)))+(-2*x^3+20*x^2-51*x+5)*ln(x)*ln(ln(x))+5-x)/(2*x^2*ln(x)*ln(ln(x))*
ln(1/5*x*ln(ln(x)))^2+(4*x^4-20*x^3)*ln(x)*ln(ln(x))*ln(1/5*x*ln(ln(x)))+(2*x^6-20*x^5+50*x^4)*ln(x)*ln(ln(x))
),x,method=_RETURNVERBOSE)

[Out]

(x-5)/x/(-I*Pi*csgn(I*x)*csgn(I*ln(ln(x)))*csgn(I*x*ln(ln(x)))+I*Pi*csgn(I*x)*csgn(I*x*ln(ln(x)))^2+I*Pi*csgn(
I*ln(ln(x)))*csgn(I*x*ln(ln(x)))^2-I*Pi*csgn(I*x*ln(ln(x)))^3+2*x^2-2*ln(5)-10*x+2*ln(x)+2*ln(ln(ln(x))))

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maxima [A]  time = 0.92, size = 31, normalized size = 0.91 \begin {gather*} \frac {x - 5}{2 \, {\left (x^{3} - 5 \, x^{2} - x \log \relax (5) + x \log \relax (x) + x \log \left (\log \left (\log \relax (x)\right )\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*log(x)*log(log(x))*log(1/5*x*log(log(x)))+(-2*x^3+20*x^2-51*x+5)*log(x)*log(log(x))+5-x)/(2*x^2*l
og(x)*log(log(x))*log(1/5*x*log(log(x)))^2+(4*x^4-20*x^3)*log(x)*log(log(x))*log(1/5*x*log(log(x)))+(2*x^6-20*
x^5+50*x^4)*log(x)*log(log(x))),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int \frac {x-5\,\ln \left (\ln \relax (x)\right )\,\ln \left (\frac {x\,\ln \left (\ln \relax (x)\right )}{5}\right )\,\ln \relax (x)+\ln \left (\ln \relax (x)\right )\,\ln \relax (x)\,\left (2\,x^3-20\,x^2+51\,x-5\right )-5}{\ln \left (\ln \relax (x)\right )\,\ln \relax (x)\,\left (2\,x^6-20\,x^5+50\,x^4\right )+2\,x^2\,\ln \left (\ln \relax (x)\right )\,{\ln \left (\frac {x\,\ln \left (\ln \relax (x)\right )}{5}\right )}^2\,\ln \relax (x)-\ln \left (\ln \relax (x)\right )\,\ln \left (\frac {x\,\ln \left (\ln \relax (x)\right )}{5}\right )\,\ln \relax (x)\,\left (20\,x^3-4\,x^4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x - 5*log(log(x))*log((x*log(log(x)))/5)*log(x) + log(log(x))*log(x)*(51*x - 20*x^2 + 2*x^3 - 5) - 5)/(l
og(log(x))*log(x)*(50*x^4 - 20*x^5 + 2*x^6) + 2*x^2*log(log(x))*log((x*log(log(x)))/5)^2*log(x) - log(log(x))*
log((x*log(log(x)))/5)*log(x)*(20*x^3 - 4*x^4)),x)

[Out]

-int((x - 5*log(log(x))*log((x*log(log(x)))/5)*log(x) + log(log(x))*log(x)*(51*x - 20*x^2 + 2*x^3 - 5) - 5)/(l
og(log(x))*log(x)*(50*x^4 - 20*x^5 + 2*x^6) + 2*x^2*log(log(x))*log((x*log(log(x)))/5)^2*log(x) - log(log(x))*
log((x*log(log(x)))/5)*log(x)*(20*x^3 - 4*x^4)), x)

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sympy [A]  time = 0.40, size = 26, normalized size = 0.76 \begin {gather*} \frac {x - 5}{2 x^{3} - 10 x^{2} + 2 x \log {\left (\frac {x \log {\left (\log {\relax (x )} \right )}}{5} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*ln(x)*ln(ln(x))*ln(1/5*x*ln(ln(x)))+(-2*x**3+20*x**2-51*x+5)*ln(x)*ln(ln(x))+5-x)/(2*x**2*ln(x)*l
n(ln(x))*ln(1/5*x*ln(ln(x)))**2+(4*x**4-20*x**3)*ln(x)*ln(ln(x))*ln(1/5*x*ln(ln(x)))+(2*x**6-20*x**5+50*x**4)*
ln(x)*ln(ln(x))),x)

[Out]

(x - 5)/(2*x**3 - 10*x**2 + 2*x*log(x*log(log(x))/5))

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