3.54.3 \(\int \frac {81-\log (x)+e^{-5+x \log (\frac {81-\log (x)}{x})} (82-\log (x)+(-81+\log (x)) \log (\frac {81-\log (x)}{x}))}{\log (4+e^{-5+x \log (\frac {81-\log (x)}{x})}-x) (-1296+324 x+(16-4 x) \log (x)+e^{-5+x \log (\frac {81-\log (x)}{x})} (-324+4 \log (x)))} \, dx\)

Optimal. Leaf size=28 \[ \frac {1}{4} \log \left (\log \left (4+e^{-5+x \log \left (\frac {81-\log (x)}{x}\right )}-x\right )\right ) \]

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Rubi [A]  time = 0.29, antiderivative size = 27, normalized size of antiderivative = 0.96, number of steps used = 1, number of rules used = 1, integrand size = 111, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.009, Rules used = {6684} \begin {gather*} \frac {1}{4} \log \left (\log \left (-x+\frac {\left (\frac {81-\log (x)}{x}\right )^x}{e^5}+4\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(81 - Log[x] + E^(-5 + x*Log[(81 - Log[x])/x])*(82 - Log[x] + (-81 + Log[x])*Log[(81 - Log[x])/x]))/(Log[4
+ E^(-5 + x*Log[(81 - Log[x])/x]) - x]*(-1296 + 324*x + (16 - 4*x)*Log[x] + E^(-5 + x*Log[(81 - Log[x])/x])*(
-324 + 4*Log[x]))),x]

[Out]

Log[Log[4 - x + ((81 - Log[x])/x)^x/E^5]]/4

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \log \left (\log \left (4-x+\frac {\left (\frac {81-\log (x)}{x}\right )^x}{e^5}\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 27, normalized size = 0.96 \begin {gather*} \frac {1}{4} \log \left (\log \left (4-x+\frac {\left (\frac {81-\log (x)}{x}\right )^x}{e^5}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(81 - Log[x] + E^(-5 + x*Log[(81 - Log[x])/x])*(82 - Log[x] + (-81 + Log[x])*Log[(81 - Log[x])/x]))/
(Log[4 + E^(-5 + x*Log[(81 - Log[x])/x]) - x]*(-1296 + 324*x + (16 - 4*x)*Log[x] + E^(-5 + x*Log[(81 - Log[x])
/x])*(-324 + 4*Log[x]))),x]

[Out]

Log[Log[4 - x + ((81 - Log[x])/x)^x/E^5]]/4

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fricas [A]  time = 0.60, size = 24, normalized size = 0.86 \begin {gather*} \frac {1}{4} \, \log \left (\log \left (-x + e^{\left (x \log \left (-\frac {\log \relax (x) - 81}{x}\right ) - 5\right )} + 4\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((log(x)-81)*log((-log(x)+81)/x)-log(x)+82)*exp(x*log((-log(x)+81)/x)-5)-log(x)+81)/((4*log(x)-324)
*exp(x*log((-log(x)+81)/x)-5)+(-4*x+16)*log(x)+324*x-1296)/log(exp(x*log((-log(x)+81)/x)-5)-x+4),x, algorithm=
"fricas")

[Out]

1/4*log(log(-x + e^(x*log(-(log(x) - 81)/x) - 5) + 4))

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giac [A]  time = 3.34, size = 32, normalized size = 1.14 \begin {gather*} \frac {1}{4} \, \log \left (\log \left (-x e^{5} + 4 \, e^{5} + e^{\left (-x \log \relax (x) + x \log \left (-\log \relax (x) + 81\right )\right )}\right ) - 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((log(x)-81)*log((-log(x)+81)/x)-log(x)+82)*exp(x*log((-log(x)+81)/x)-5)-log(x)+81)/((4*log(x)-324)
*exp(x*log((-log(x)+81)/x)-5)+(-4*x+16)*log(x)+324*x-1296)/log(exp(x*log((-log(x)+81)/x)-5)-x+4),x, algorithm=
"giac")

[Out]

1/4*log(log(-x*e^5 + 4*e^5 + e^(-x*log(x) + x*log(-log(x) + 81))) - 5)

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maple [C]  time = 0.24, size = 153, normalized size = 5.46

method	result	size
risch	\(\frac {\ln \left (\ln \left (x^{-x} \left (\ln \relax (x )-81\right )^{x} {\mathrm e}^{-5} {\mathrm e}^{i x \pi } {\mathrm e}^{\frac {i x \pi \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )-81\right )}{x}\right )^{3}}{2}} {\mathrm e}^{\frac {i x \pi \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )-81\right )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )}{2}} {\mathrm e}^{\frac {i x \pi \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )-81\right )}{x}\right )^{2} \mathrm {csgn}\left (i \left (\ln \relax (x )-81\right )\right )}{2}} {\mathrm e}^{-\frac {i x \pi \,\mathrm {csgn}\left (\frac {i \left (\ln \relax (x )-81\right )}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (\ln \relax (x )-81\right )\right )}{2}} {\mathrm e}^{-i x \pi \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )-81\right )}{x}\right )^{2}}-x +4\right )\right )}{4}\)	\(153\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((ln(x)-81)*ln((-ln(x)+81)/x)-ln(x)+82)*exp(x*ln((-ln(x)+81)/x)-5)-ln(x)+81)/((4*ln(x)-324)*exp(x*ln((-ln
(x)+81)/x)-5)+(-4*x+16)*ln(x)+324*x-1296)/ln(exp(x*ln((-ln(x)+81)/x)-5)-x+4),x,method=_RETURNVERBOSE)

[Out]

1/4*ln(ln(x^(-x)*(ln(x)-81)^x*exp(-5)*exp(I*Pi*x)*exp(1/2*I*x*Pi*csgn(I*(ln(x)-81)/x)^3)*exp(1/2*I*x*Pi*csgn(I
*(ln(x)-81)/x)^2*csgn(I/x))*exp(1/2*I*x*Pi*csgn(I*(ln(x)-81)/x)^2*csgn(I*(ln(x)-81)))*exp(-1/2*I*x*Pi*csgn(I*(
ln(x)-81)/x)*csgn(I/x)*csgn(I*(ln(x)-81)))*exp(-I*x*Pi*csgn(I*(ln(x)-81)/x)^2)-x+4))

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maxima [A]  time = 0.47, size = 35, normalized size = 1.25 \begin {gather*} \frac {1}{4} \, \log \left (\log \left (-{\left (x e^{5} - 4 \, e^{5}\right )} x^{x} + {\left (-\log \relax (x) + 81\right )}^{x}\right ) - \log \left (x^{x}\right ) - 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((log(x)-81)*log((-log(x)+81)/x)-log(x)+82)*exp(x*log((-log(x)+81)/x)-5)-log(x)+81)/((4*log(x)-324)
*exp(x*log((-log(x)+81)/x)-5)+(-4*x+16)*log(x)+324*x-1296)/log(exp(x*log((-log(x)+81)/x)-5)-x+4),x, algorithm=
"maxima")

[Out]

1/4*log(log(-(x*e^5 - 4*e^5)*x^x + (-log(x) + 81)^x) - log(x^x) - 5)

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mupad [B]  time = 5.67, size = 23, normalized size = 0.82 \begin {gather*} \frac {\ln \left (\ln \left ({\mathrm {e}}^{-5}\,{\left (-\frac {\ln \relax (x)-81}{x}\right )}^x-x+4\right )\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x*log(-(log(x) - 81)/x) - 5)*(log(-(log(x) - 81)/x)*(log(x) - 81) - log(x) + 82) - log(x) + 81)/(log(
exp(x*log(-(log(x) - 81)/x) - 5) - x + 4)*(324*x - log(x)*(4*x - 16) + exp(x*log(-(log(x) - 81)/x) - 5)*(4*log
(x) - 324) - 1296)),x)

[Out]

log(log(exp(-5)*(-(log(x) - 81)/x)^x - x + 4))/4

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((ln(x)-81)*ln((-ln(x)+81)/x)-ln(x)+82)*exp(x*ln((-ln(x)+81)/x)-5)-ln(x)+81)/((4*ln(x)-324)*exp(x*l
n((-ln(x)+81)/x)-5)+(-4*x+16)*ln(x)+324*x-1296)/ln(exp(x*ln((-ln(x)+81)/x)-5)-x+4),x)

[Out]

Timed out

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