3.51.22 \(\int \frac {500-5 x^2+125 \log (3)+(-625-125 \log (3)) \log (x)+(-125+125 \log (x)) \log (\log (x))}{25 x^2+10 x^3+x^4+(-1250 x-250 x^2+(-250 x-50 x^2) \log (3)) \log (x)+(15625+6250 \log (3)+625 \log ^2(3)) \log ^2(x)+((250 x+50 x^2) \log (x)+(-6250-1250 \log (3)) \log ^2(x)) \log (\log (x))+625 \log ^2(x) \log ^2(\log (x))} \, dx\)

Optimal. Leaf size=26 \[ \frac {x}{\frac {1}{5} x (5+x)+5 \log (x) (-5-\log (3)+\log (\log (x)))} \]

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Rubi [F]  time = 0.69, 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 {500-5 x^2+125 \log (3)+(-625-125 \log (3)) \log (x)+(-125+125 \log (x)) \log (\log (x))}{25 x^2+10 x^3+x^4+\left (-1250 x-250 x^2+\left (-250 x-50 x^2\right ) \log (3)\right ) \log (x)+\left (15625+6250 \log (3)+625 \log ^2(3)\right ) \log ^2(x)+\left (\left (250 x+50 x^2\right ) \log (x)+(-6250-1250 \log (3)) \log ^2(x)\right ) \log (\log (x))+625 \log ^2(x) \log ^2(\log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(500 - 5*x^2 + 125*Log[3] + (-625 - 125*Log[3])*Log[x] + (-125 + 125*Log[x])*Log[Log[x]])/(25*x^2 + 10*x^3
+ x^4 + (-1250*x - 250*x^2 + (-250*x - 50*x^2)*Log[3])*Log[x] + (15625 + 6250*Log[3] + 625*Log[3]^2)*Log[x]^2
+ ((250*x + 50*x^2)*Log[x] + (-6250 - 1250*Log[3])*Log[x]^2)*Log[Log[x]] + 625*Log[x]^2*Log[Log[x]]^2),x]

[Out]

125*(4 + Log[3])*Defer[Int][(5*x + x^2 - 125*Log[x] + 25*Log[x]*Log[Log[x]/3])^(-2), x] - 5*Defer[Int][x^2/(5*
x + x^2 - 125*Log[x] + 25*Log[x]*Log[Log[x]/3])^2, x] - 125*(5 + Log[3])*Defer[Int][Log[x]/(5*x + x^2 - 125*Lo
g[x] + 25*Log[x]*Log[Log[x]/3])^2, x] - 125*Defer[Int][Log[Log[x]]/(5*x + x^2 - 125*Log[x] + 25*Log[x]*Log[Log
[x]/3])^2, x] + 125*Defer[Int][(Log[x]*Log[Log[x]])/(5*x + x^2 - 125*Log[x] + 25*Log[x]*Log[Log[x]/3])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 \left (-x^2+100 \left (1+\frac {\log (3)}{4}\right )-25 \log (x) (5+\log (3)-\log (\log (x)))-25 \log (\log (x))\right )}{\left (x (5+x)+25 \log (x) \left (-5+\log \left (\frac {\log (x)}{3}\right )\right )\right )^2} \, dx\\ &=5 \int \frac {-x^2+100 \left (1+\frac {\log (3)}{4}\right )-25 \log (x) (5+\log (3)-\log (\log (x)))-25 \log (\log (x))}{\left (x (5+x)+25 \log (x) \left (-5+\log \left (\frac {\log (x)}{3}\right )\right )\right )^2} \, dx\\ &=5 \int \left (-\frac {x^2}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2}+\frac {25 (4+\log (3))}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2}-\frac {125 \left (1+\frac {\log (3)}{5}\right ) \log (x)}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2}+\frac {25 (-1+\log (x)) \log (\log (x))}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2}\right ) \, dx\\ &=-\left (5 \int \frac {x^2}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx\right )+125 \int \frac {(-1+\log (x)) \log (\log (x))}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx+(125 (4+\log (3))) \int \frac {1}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx-(125 (5+\log (3))) \int \frac {\log (x)}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx\\ &=-\left (5 \int \frac {x^2}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx\right )+125 \int \left (-\frac {\log (\log (x))}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2}+\frac {\log (x) \log (\log (x))}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2}\right ) \, dx+(125 (4+\log (3))) \int \frac {1}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx-(125 (5+\log (3))) \int \frac {\log (x)}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx\\ &=-\left (5 \int \frac {x^2}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx\right )-125 \int \frac {\log (\log (x))}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx+125 \int \frac {\log (x) \log (\log (x))}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx+(125 (4+\log (3))) \int \frac {1}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx-(125 (5+\log (3))) \int \frac {\log (x)}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.29, size = 24, normalized size = 0.92 \begin {gather*} \frac {5 x}{x (5+x)+25 \log (x) \left (-5+\log \left (\frac {\log (x)}{3}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(500 - 5*x^2 + 125*Log[3] + (-625 - 125*Log[3])*Log[x] + (-125 + 125*Log[x])*Log[Log[x]])/(25*x^2 +
10*x^3 + x^4 + (-1250*x - 250*x^2 + (-250*x - 50*x^2)*Log[3])*Log[x] + (15625 + 6250*Log[3] + 625*Log[3]^2)*Lo
g[x]^2 + ((250*x + 50*x^2)*Log[x] + (-6250 - 1250*Log[3])*Log[x]^2)*Log[Log[x]] + 625*Log[x]^2*Log[Log[x]]^2),
x]

[Out]

(5*x)/(x*(5 + x) + 25*Log[x]*(-5 + Log[Log[x]/3]))

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fricas [A]  time = 0.59, size = 27, normalized size = 1.04 \begin {gather*} \frac {5 \, x}{x^{2} - 25 \, {\left (\log \relax (3) + 5\right )} \log \relax (x) + 25 \, \log \relax (x) \log \left (\log \relax (x)\right ) + 5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((125*log(x)-125)*log(log(x))+(-125*log(3)-625)*log(x)+125*log(3)-5*x^2+500)/(625*log(x)^2*log(log(x
))^2+((-1250*log(3)-6250)*log(x)^2+(50*x^2+250*x)*log(x))*log(log(x))+(625*log(3)^2+6250*log(3)+15625)*log(x)^
2+((-50*x^2-250*x)*log(3)-250*x^2-1250*x)*log(x)+x^4+10*x^3+25*x^2),x, algorithm="fricas")

[Out]

5*x/(x^2 - 25*(log(3) + 5)*log(x) + 25*log(x)*log(log(x)) + 5*x)

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giac [A]  time = 0.24, size = 29, normalized size = 1.12 \begin {gather*} \frac {5 \, x}{x^{2} - 25 \, \log \relax (3) \log \relax (x) + 25 \, \log \relax (x) \log \left (\log \relax (x)\right ) + 5 \, x - 125 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((125*log(x)-125)*log(log(x))+(-125*log(3)-625)*log(x)+125*log(3)-5*x^2+500)/(625*log(x)^2*log(log(x
))^2+((-1250*log(3)-6250)*log(x)^2+(50*x^2+250*x)*log(x))*log(log(x))+(625*log(3)^2+6250*log(3)+15625)*log(x)^
2+((-50*x^2-250*x)*log(3)-250*x^2-1250*x)*log(x)+x^4+10*x^3+25*x^2),x, algorithm="giac")

[Out]

5*x/(x^2 - 25*log(3)*log(x) + 25*log(x)*log(log(x)) + 5*x - 125*log(x))

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maple [A]  time = 0.05, size = 32, normalized size = 1.23

method	result	size
risch	\(-\frac {5 x}{25 \ln \relax (3) \ln \relax (x )-x^{2}-25 \ln \relax (x ) \ln \left (\ln \relax (x )\right )-5 x +125 \ln \relax (x )}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((125*ln(x)-125)*ln(ln(x))+(-125*ln(3)-625)*ln(x)+125*ln(3)-5*x^2+500)/(625*ln(x)^2*ln(ln(x))^2+((-1250*ln
(3)-6250)*ln(x)^2+(50*x^2+250*x)*ln(x))*ln(ln(x))+(625*ln(3)^2+6250*ln(3)+15625)*ln(x)^2+((-50*x^2-250*x)*ln(3
)-250*x^2-1250*x)*ln(x)+x^4+10*x^3+25*x^2),x,method=_RETURNVERBOSE)

[Out]

-5*x/(25*ln(3)*ln(x)-x^2-25*ln(x)*ln(ln(x))-5*x+125*ln(x))

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maxima [A]  time = 0.47, size = 27, normalized size = 1.04 \begin {gather*} \frac {5 \, x}{x^{2} - 25 \, {\left (\log \relax (3) + 5\right )} \log \relax (x) + 25 \, \log \relax (x) \log \left (\log \relax (x)\right ) + 5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((125*log(x)-125)*log(log(x))+(-125*log(3)-625)*log(x)+125*log(3)-5*x^2+500)/(625*log(x)^2*log(log(x
))^2+((-1250*log(3)-6250)*log(x)^2+(50*x^2+250*x)*log(x))*log(log(x))+(625*log(3)^2+6250*log(3)+15625)*log(x)^
2+((-50*x^2-250*x)*log(3)-250*x^2-1250*x)*log(x)+x^4+10*x^3+25*x^2),x, algorithm="maxima")

[Out]

5*x/(x^2 - 25*(log(3) + 5)*log(x) + 25*log(x)*log(log(x)) + 5*x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {125\,\ln \relax (3)+\ln \left (\ln \relax (x)\right )\,\left (125\,\ln \relax (x)-125\right )-\ln \relax (x)\,\left (125\,\ln \relax (3)+625\right )-5\,x^2+500}{625\,{\ln \left (\ln \relax (x)\right )}^2\,{\ln \relax (x)}^2+\ln \left (\ln \relax (x)\right )\,\left (\ln \relax (x)\,\left (50\,x^2+250\,x\right )-{\ln \relax (x)}^2\,\left (1250\,\ln \relax (3)+6250\right )\right )+25\,x^2+10\,x^3+x^4-\ln \relax (x)\,\left (1250\,x+\ln \relax (3)\,\left (50\,x^2+250\,x\right )+250\,x^2\right )+{\ln \relax (x)}^2\,\left (6250\,\ln \relax (3)+625\,{\ln \relax (3)}^2+15625\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((125*log(3) + log(log(x))*(125*log(x) - 125) - log(x)*(125*log(3) + 625) - 5*x^2 + 500)/(625*log(log(x))^2
*log(x)^2 + log(log(x))*(log(x)*(250*x + 50*x^2) - log(x)^2*(1250*log(3) + 6250)) + 25*x^2 + 10*x^3 + x^4 - lo
g(x)*(1250*x + log(3)*(250*x + 50*x^2) + 250*x^2) + log(x)^2*(6250*log(3) + 625*log(3)^2 + 15625)),x)

[Out]

int((125*log(3) + log(log(x))*(125*log(x) - 125) - log(x)*(125*log(3) + 625) - 5*x^2 + 500)/(625*log(log(x))^2
*log(x)^2 + log(log(x))*(log(x)*(250*x + 50*x^2) - log(x)^2*(1250*log(3) + 6250)) + 25*x^2 + 10*x^3 + x^4 - lo
g(x)*(1250*x + log(3)*(250*x + 50*x^2) + 250*x^2) + log(x)^2*(6250*log(3) + 625*log(3)^2 + 15625)), x)

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sympy [A]  time = 0.39, size = 32, normalized size = 1.23 \begin {gather*} \frac {5 x}{x^{2} + 5 x + 25 \log {\relax (x )} \log {\left (\log {\relax (x )} \right )} - 125 \log {\relax (x )} - 25 \log {\relax (3 )} \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((125*ln(x)-125)*ln(ln(x))+(-125*ln(3)-625)*ln(x)+125*ln(3)-5*x**2+500)/(625*ln(x)**2*ln(ln(x))**2+(
(-1250*ln(3)-6250)*ln(x)**2+(50*x**2+250*x)*ln(x))*ln(ln(x))+(625*ln(3)**2+6250*ln(3)+15625)*ln(x)**2+((-50*x*
*2-250*x)*ln(3)-250*x**2-1250*x)*ln(x)+x**4+10*x**3+25*x**2),x)

[Out]

5*x/(x**2 + 5*x + 25*log(x)*log(log(x)) - 125*log(x) - 25*log(3)*log(x))

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