\(\int \frac {-3+\log (\frac {x^3}{3+\log (21)}) \log (\log (\frac {x^3}{3+\log (21)}))}{\log (\frac {x^3}{3+\log (21)}) \log ^2(\log (\frac {x^3}{3+\log (21)}))} \, dx\) [9291]

Optimal result

Integrand size = 54, antiderivative size = 18 \[ \int \frac {-3+\log \left (\frac {x^3}{3+\log (21)}\right ) \log \left (\log \left (\frac {x^3}{3+\log (21)}\right )\right )}{\log \left (\frac {x^3}{3+\log (21)}\right ) \log ^2\left (\log \left (\frac {x^3}{3+\log (21)}\right )\right )} \, dx=25+\frac {x}{\log \left (\log \left (\frac {x^3}{3+\log (21)}\right )\right )} \]

[Out]

25+x/ln(ln(x^3/(ln(21)+3)))

Rubi [F]

\[ \int \frac {-3+\log \left (\frac {x^3}{3+\log (21)}\right ) \log \left (\log \left (\frac {x^3}{3+\log (21)}\right )\right )}{\log \left (\frac {x^3}{3+\log (21)}\right ) \log ^2\left (\log \left (\frac {x^3}{3+\log (21)}\right )\right )} \, dx=\int \frac {-3+\log \left (\frac {x^3}{3+\log (21)}\right ) \log \left (\log \left (\frac {x^3}{3+\log (21)}\right )\right )}{\log \left (\frac {x^3}{3+\log (21)}\right ) \log ^2\left (\log \left (\frac {x^3}{3+\log (21)}\right )\right )} \, dx \]

[In]

Int[(-3 + Log[x^3/(3 + Log[21])]*Log[Log[x^3/(3 + Log[21])]])/(Log[x^3/(3 + Log[21])]*Log[Log[x^3/(3 + Log[21]
)]]^2),x]

[Out]

-3*Defer[Int][1/(Log[x^3/(3 + Log[21])]*Log[Log[x^3/(3 + Log[21])]]^2), x] + Defer[Int][Log[Log[x^3/(3 + Log[2
1])]]^(-1), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-\frac {3}{\log \left (\frac {x^3}{3+\log (21)}\right )}+\log \left (\log \left (\frac {x^3}{3+\log (21)}\right )\right )}{\log ^2\left (\log \left (\frac {x^3}{3+\log (21)}\right )\right )} \, dx \\ & = \int \left (-\frac {3}{\log \left (\frac {x^3}{3+\log (21)}\right ) \log ^2\left (\log \left (\frac {x^3}{3+\log (21)}\right )\right )}+\frac {1}{\log \left (\log \left (\frac {x^3}{3+\log (21)}\right )\right )}\right ) \, dx \\ & = -\left (3 \int \frac {1}{\log \left (\frac {x^3}{3+\log (21)}\right ) \log ^2\left (\log \left (\frac {x^3}{3+\log (21)}\right )\right )} \, dx\right )+\int \frac {1}{\log \left (\log \left (\frac {x^3}{3+\log (21)}\right )\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-3+\log \left (\frac {x^3}{3+\log (21)}\right ) \log \left (\log \left (\frac {x^3}{3+\log (21)}\right )\right )}{\log \left (\frac {x^3}{3+\log (21)}\right ) \log ^2\left (\log \left (\frac {x^3}{3+\log (21)}\right )\right )} \, dx=\frac {x}{\log \left (\log \left (\frac {x^3}{3+\log (21)}\right )\right )} \]

[In]

Integrate[(-3 + Log[x^3/(3 + Log[21])]*Log[Log[x^3/(3 + Log[21])]])/(Log[x^3/(3 + Log[21])]*Log[Log[x^3/(3 + L
og[21])]]^2),x]

[Out]

x/Log[Log[x^3/(3 + Log[21])]]

Maple [A] (verified)

Time = 1.05 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94

method	result	size
norman	\(\frac {x}{\ln \left (\ln \left (\frac {x^{3}}{\ln \left (21\right )+3}\right )\right )}\)	\(17\)
parallelrisch	\(\frac {x}{\ln \left (\ln \left (\frac {x^{3}}{\ln \left (21\right )+3}\right )\right )}\)	\(17\)

[In]

int((ln(x^3/(ln(21)+3))*ln(ln(x^3/(ln(21)+3)))-3)/ln(x^3/(ln(21)+3))/ln(ln(x^3/(ln(21)+3)))^2,x,method=_RETURN
VERBOSE)

[Out]

x/ln(ln(x^3/(ln(21)+3)))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-3+\log \left (\frac {x^3}{3+\log (21)}\right ) \log \left (\log \left (\frac {x^3}{3+\log (21)}\right )\right )}{\log \left (\frac {x^3}{3+\log (21)}\right ) \log ^2\left (\log \left (\frac {x^3}{3+\log (21)}\right )\right )} \, dx=\frac {x}{\log \left (\log \left (\frac {x^{3}}{\log \left (21\right ) + 3}\right )\right )} \]

[In]

integrate((log(x^3/(log(21)+3))*log(log(x^3/(log(21)+3)))-3)/log(x^3/(log(21)+3))/log(log(x^3/(log(21)+3)))^2,
x, algorithm="fricas")

[Out]

x/log(log(x^3/(log(21) + 3)))

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67 \[ \int \frac {-3+\log \left (\frac {x^3}{3+\log (21)}\right ) \log \left (\log \left (\frac {x^3}{3+\log (21)}\right )\right )}{\log \left (\frac {x^3}{3+\log (21)}\right ) \log ^2\left (\log \left (\frac {x^3}{3+\log (21)}\right )\right )} \, dx=\frac {x}{\log {\left (\log {\left (\frac {x^{3}}{3 + \log {\left (21 \right )}} \right )} \right )}} \]

[In]

integrate((ln(x**3/(ln(21)+3))*ln(ln(x**3/(ln(21)+3)))-3)/ln(x**3/(ln(21)+3))/ln(ln(x**3/(ln(21)+3)))**2,x)

[Out]

x/log(log(x**3/(3 + log(21))))

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {-3+\log \left (\frac {x^3}{3+\log (21)}\right ) \log \left (\log \left (\frac {x^3}{3+\log (21)}\right )\right )}{\log \left (\frac {x^3}{3+\log (21)}\right ) \log ^2\left (\log \left (\frac {x^3}{3+\log (21)}\right )\right )} \, dx=\frac {x}{\log \left (3 \, \log \left (x\right ) - \log \left (\log \left (7\right ) + \log \left (3\right ) + 3\right )\right )} \]

[In]

integrate((log(x^3/(log(21)+3))*log(log(x^3/(log(21)+3)))-3)/log(x^3/(log(21)+3))/log(log(x^3/(log(21)+3)))^2,
x, algorithm="maxima")

[Out]

x/log(3*log(x) - log(log(7) + log(3) + 3))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (18) = 36\).

Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.44 \[ \int \frac {-3+\log \left (\frac {x^3}{3+\log (21)}\right ) \log \left (\log \left (\frac {x^3}{3+\log (21)}\right )\right )}{\log \left (\frac {x^3}{3+\log (21)}\right ) \log ^2\left (\log \left (\frac {x^3}{3+\log (21)}\right )\right )} \, dx=\frac {x \log \left (x^{3}\right ) - x \log \left (\log \left (21\right ) + 3\right )}{\log \left (\frac {x^{3}}{\log \left (21\right ) + 3}\right ) \log \left (\log \left (x^{3}\right ) - \log \left (\log \left (21\right ) + 3\right )\right )} \]

[In]

integrate((log(x^3/(log(21)+3))*log(log(x^3/(log(21)+3)))-3)/log(x^3/(log(21)+3))/log(log(x^3/(log(21)+3)))^2,
x, algorithm="giac")

[Out]

(x*log(x^3) - x*log(log(21) + 3))/(log(x^3/(log(21) + 3))*log(log(x^3) - log(log(21) + 3)))

Mupad [B] (verification not implemented)

Time = 13.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {-3+\log \left (\frac {x^3}{3+\log (21)}\right ) \log \left (\log \left (\frac {x^3}{3+\log (21)}\right )\right )}{\log \left (\frac {x^3}{3+\log (21)}\right ) \log ^2\left (\log \left (\frac {x^3}{3+\log (21)}\right )\right )} \, dx=\frac {x}{\ln \left (\ln \left (x^3\right )-\ln \left (\ln \left (21\right )+3\right )\right )} \]

[In]

int((log(log(x^3/(log(21) + 3)))*log(x^3/(log(21) + 3)) - 3)/(log(log(x^3/(log(21) + 3)))^2*log(x^3/(log(21) +
3))),x)

[Out]

x/log(log(x^3) - log(log(21) + 3))