3.69.28 \(\int \frac {x-x \log (x)+(-4 x^6 \log (x)+4 x^5 \log ^2(x)) \log (\frac {-5 x+5 \log (x)}{\log (x)})+(x \log (x)-\log ^2(x)) \log (\frac {-5 x+5 \log (x)}{\log (x)}) \log (\log (\frac {-5 x+5 \log (x)}{\log (x)}))}{(-x^3 \log (x)+x^2 \log ^2(x)) \log (\frac {-5 x+5 \log (x)}{\log (x)})} \, dx\)

Optimal. Leaf size=21 \[ x^4+\frac {\log \left (\log \left (5 \left (1-\frac {x}{\log (x)}\right )\right )\right )}{x} \]

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Rubi [F]  time = 2.49, 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 {x-x \log (x)+\left (-4 x^6 \log (x)+4 x^5 \log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )+\left (x \log (x)-\log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right ) \log \left (\log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )\right )}{\left (-x^3 \log (x)+x^2 \log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x - x*Log[x] + (-4*x^6*Log[x] + 4*x^5*Log[x]^2)*Log[(-5*x + 5*Log[x])/Log[x]] + (x*Log[x] - Log[x]^2)*Log
[(-5*x + 5*Log[x])/Log[x]]*Log[Log[(-5*x + 5*Log[x])/Log[x]]])/((-(x^3*Log[x]) + x^2*Log[x]^2)*Log[(-5*x + 5*L
og[x])/Log[x]]),x]

[Out]

x^4 + Defer[Int][1/(x*(x - Log[x])*Log[5 - (5*x)/Log[x]]), x] - Defer[Int][1/(x*(x - Log[x])*Log[x]*Log[5 - (5
*x)/Log[x]]), x] - Defer[Int][Log[Log[5 - (5*x)/Log[x]]]/x^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x+x \log (x)-\left (-4 x^6 \log (x)+4 x^5 \log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )-\left (x \log (x)-\log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right ) \log \left (\log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )\right )}{x^2 (x-\log (x)) \log (x) \log \left (-\frac {5 (x-\log (x))}{\log (x)}\right )} \, dx\\ &=\int \left (\frac {-1+\log (x)+4 x^5 \log (x) \log \left (5-\frac {5 x}{\log (x)}\right )-4 x^4 \log ^2(x) \log \left (5-\frac {5 x}{\log (x)}\right )}{x (x-\log (x)) \log (x) \log \left (5-\frac {5 x}{\log (x)}\right )}-\frac {\log \left (\log \left (5-\frac {5 x}{\log (x)}\right )\right )}{x^2}\right ) \, dx\\ &=\int \frac {-1+\log (x)+4 x^5 \log (x) \log \left (5-\frac {5 x}{\log (x)}\right )-4 x^4 \log ^2(x) \log \left (5-\frac {5 x}{\log (x)}\right )}{x (x-\log (x)) \log (x) \log \left (5-\frac {5 x}{\log (x)}\right )} \, dx-\int \frac {\log \left (\log \left (5-\frac {5 x}{\log (x)}\right )\right )}{x^2} \, dx\\ &=\int \left (4 x^3+\frac {-1+\log (x)}{x (x-\log (x)) \log (x) \log \left (5-\frac {5 x}{\log (x)}\right )}\right ) \, dx-\int \frac {\log \left (\log \left (5-\frac {5 x}{\log (x)}\right )\right )}{x^2} \, dx\\ &=x^4+\int \frac {-1+\log (x)}{x (x-\log (x)) \log (x) \log \left (5-\frac {5 x}{\log (x)}\right )} \, dx-\int \frac {\log \left (\log \left (5-\frac {5 x}{\log (x)}\right )\right )}{x^2} \, dx\\ &=x^4+\int \left (\frac {1}{x (x-\log (x)) \log \left (5-\frac {5 x}{\log (x)}\right )}-\frac {1}{x (x-\log (x)) \log (x) \log \left (5-\frac {5 x}{\log (x)}\right )}\right ) \, dx-\int \frac {\log \left (\log \left (5-\frac {5 x}{\log (x)}\right )\right )}{x^2} \, dx\\ &=x^4+\int \frac {1}{x (x-\log (x)) \log \left (5-\frac {5 x}{\log (x)}\right )} \, dx-\int \frac {1}{x (x-\log (x)) \log (x) \log \left (5-\frac {5 x}{\log (x)}\right )} \, dx-\int \frac {\log \left (\log \left (5-\frac {5 x}{\log (x)}\right )\right )}{x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 19, normalized size = 0.90 \begin {gather*} x^4+\frac {\log \left (\log \left (5-\frac {5 x}{\log (x)}\right )\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x^4 + Log[Log[5 - (5*x)/Log[x]]]/x

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fricas [A]  time = 0.47, size = 22, normalized size = 1.05 \begin {gather*} \frac {x^{5} + \log \left (\log \left (-\frac {5 \, {\left (x - \log \relax (x)\right )}}{\log \relax (x)}\right )\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-log(x)^2+x*log(x))*log((5*log(x)-5*x)/log(x))*log(log((5*log(x)-5*x)/log(x)))+(4*x^5*log(x)^2-4*x
^6*log(x))*log((5*log(x)-5*x)/log(x))+x-x*log(x))/(x^2*log(x)^2-x^3*log(x))/log((5*log(x)-5*x)/log(x)),x, algo
rithm="fricas")

[Out]

(x^5 + log(log(-5*(x - log(x))/log(x))))/x

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giac [A]  time = 0.30, size = 24, normalized size = 1.14 \begin {gather*} x^{4} + \frac {\log \left (\log \left (-5 \, x + 5 \, \log \relax (x)\right ) - \log \left (\log \relax (x)\right )\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-log(x)^2+x*log(x))*log((5*log(x)-5*x)/log(x))*log(log((5*log(x)-5*x)/log(x)))+(4*x^5*log(x)^2-4*x
^6*log(x))*log((5*log(x)-5*x)/log(x))+x-x*log(x))/(x^2*log(x)^2-x^3*log(x))/log((5*log(x)-5*x)/log(x)),x, algo
rithm="giac")

[Out]

x^4 + log(log(-5*x + 5*log(x)) - log(log(x)))/x

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maple [C]  time = 0.14, size = 135, normalized size = 6.43

method	result	size
risch	\(\frac {\ln \left (\ln \relax (5)+i \pi -\ln \left (\ln \relax (x )\right )+\ln \left (x -\ln \relax (x )\right )+\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left (\ln \relax (x )-x \right )}{\ln \relax (x )}\right ) \left (\mathrm {csgn}\left (\frac {i \left (\ln \relax (x )-x \right )}{\ln \relax (x )}\right )+\mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right )\right ) \left (\mathrm {csgn}\left (\frac {i \left (\ln \relax (x )-x \right )}{\ln \relax (x )}\right )-\mathrm {csgn}\left (i \left (\ln \relax (x )-x \right )\right )\right )}{2}+i \pi \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )-x \right )}{\ln \relax (x )}\right )^{2} \left (-\mathrm {csgn}\left (\frac {i \left (\ln \relax (x )-x \right )}{\ln \relax (x )}\right )-1\right )\right )}{x}+x^{4}\)	\(135\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/x*ln(ln(5)+I*Pi-ln(ln(x))+ln(x-ln(x))+1/2*I*Pi*csgn(I*(ln(x)-x)/ln(x))*(csgn(I*(ln(x)-x)/ln(x))+csgn(I/ln(x)
))*(csgn(I*(ln(x)-x)/ln(x))-csgn(I*(ln(x)-x)))+I*Pi*csgn(I*(ln(x)-x)/ln(x))^2*(-csgn(I*(ln(x)-x)/ln(x))-1))+x^
4

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maxima [A]  time = 0.53, size = 24, normalized size = 1.14 \begin {gather*} \frac {x^{5} + \log \left (\log \relax (5) + \log \left (-x + \log \relax (x)\right ) - \log \left (\log \relax (x)\right )\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-log(x)^2+x*log(x))*log((5*log(x)-5*x)/log(x))*log(log((5*log(x)-5*x)/log(x)))+(4*x^5*log(x)^2-4*x
^6*log(x))*log((5*log(x)-5*x)/log(x))+x-x*log(x))/(x^2*log(x)^2-x^3*log(x))/log((5*log(x)-5*x)/log(x)),x, algo
rithm="maxima")

[Out]

(x^5 + log(log(5) + log(-x + log(x)) - log(log(x))))/x

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mupad [B]  time = 4.61, size = 22, normalized size = 1.05 \begin {gather*} \frac {\ln \left (\ln \left (-\frac {5\,\left (x-\ln \relax (x)\right )}{\ln \relax (x)}\right )\right )}{x}+x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*log(x) - x + log(-(5*x - 5*log(x))/log(x))*(4*x^6*log(x) - 4*x^5*log(x)^2) + log(-(5*x - 5*log(x))/log(
x))*log(log(-(5*x - 5*log(x))/log(x)))*(log(x)^2 - x*log(x)))/(log(-(5*x - 5*log(x))/log(x))*(x^3*log(x) - x^2
*log(x)^2)),x)

[Out]

log(log(-(5*(x - log(x)))/log(x)))/x + x^4

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sympy [A]  time = 1.55, size = 19, normalized size = 0.90 \begin {gather*} x^{4} + \frac {\log {\left (\log {\left (\frac {- 5 x + 5 \log {\relax (x )}}{\log {\relax (x )}} \right )} \right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-ln(x)**2+x*ln(x))*ln((5*ln(x)-5*x)/ln(x))*ln(ln((5*ln(x)-5*x)/ln(x)))+(4*x**5*ln(x)**2-4*x**6*ln(
x))*ln((5*ln(x)-5*x)/ln(x))+x-x*ln(x))/(x**2*ln(x)**2-x**3*ln(x))/ln((5*ln(x)-5*x)/ln(x)),x)

[Out]

x**4 + log(log((-5*x + 5*log(x))/log(x)))/x

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