3.103.56 \(\int \frac {-\log (3)+\log (3) \log (x) \log (\log (x))+(-1+4 \log (3)+\log (3) \log (4)) \log (x) \log ^2(\log (x))}{\log (x) \log ^2(\log (x))} \, dx\)

Optimal. Leaf size=18 \[ -5-x+x \log (3) \left (4+\log (4)+\frac {1}{\log (\log (x))}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 0.13, 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 {-\log (3)+\log (3) \log (x) \log (\log (x))+(-1+4 \log (3)+\log (3) \log (4)) \log (x) \log ^2(\log (x))}{\log (x) \log ^2(\log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-Log[3] + Log[3]*Log[x]*Log[Log[x]] + (-1 + 4*Log[3] + Log[3]*Log[4])*Log[x]*Log[Log[x]]^2)/(Log[x]*Log[L
og[x]]^2),x]

[Out]

-(x*(1 - Log[3]*Log[4] - Log[81])) - Log[3]*Defer[Int][1/(Log[x]*Log[Log[x]]^2), x] + Log[3]*Defer[Int][Log[Lo
g[x]]^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1+\log (3) \log (4)+\log (81)-\frac {\log (3)}{\log (x) \log ^2(\log (x))}+\frac {\log (3)}{\log (\log (x))}\right ) \, dx\\ &=-x (1-\log (3) \log (4)-\log (81))-\log (3) \int \frac {1}{\log (x) \log ^2(\log (x))} \, dx+\log (3) \int \frac {1}{\log (\log (x))} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 21, normalized size = 1.17 \begin {gather*} x (-1+\log (3) \log (4)+\log (81))+\frac {x \log (3)}{\log (\log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-Log[3] + Log[3]*Log[x]*Log[Log[x]] + (-1 + 4*Log[3] + Log[3]*Log[4])*Log[x]*Log[Log[x]]^2)/(Log[x]
*Log[Log[x]]^2),x]

[Out]

x*(-1 + Log[3]*Log[4] + Log[81]) + (x*Log[3])/Log[Log[x]]

________________________________________________________________________________________

fricas [A]  time = 1.10, size = 31, normalized size = 1.72 \begin {gather*} \frac {x \log \relax (3) + {\left (2 \, {\left (x \log \relax (2) + 2 \, x\right )} \log \relax (3) - x\right )} \log \left (\log \relax (x)\right )}{\log \left (\log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*log(2)*log(3)+4*log(3)-1)*log(x)*log(log(x))^2+log(3)*log(x)*log(log(x))-log(3))/log(x)/log(log(
x))^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.26, size = 25, normalized size = 1.39 \begin {gather*} 2 \, x \log \relax (3) \log \relax (2) + 4 \, x \log \relax (3) - x + \frac {x \log \relax (3)}{\log \left (\log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*log(2)*log(3)+4*log(3)-1)*log(x)*log(log(x))^2+log(3)*log(x)*log(log(x))-log(3))/log(x)/log(log(
x))^2,x, algorithm="giac")

[Out]

2*x*log(3)*log(2) + 4*x*log(3) - x + x*log(3)/log(log(x))

________________________________________________________________________________________

maple [A]  time = 0.06, size = 26, normalized size = 1.44

method	result	size
risch	\(2 x \ln \relax (2) \ln \relax (3)+4 x \ln \relax (3)-x +\frac {x \ln \relax (3)}{\ln \left (\ln \relax (x )\right )}\)	\(26\)
norman	\(\frac {x \ln \relax (3)+\left (2 \ln \relax (2) \ln \relax (3)+4 \ln \relax (3)-1\right ) x \ln \left (\ln \relax (x )\right )}{\ln \left (\ln \relax (x )\right )}\)	\(29\)
default	\(-x +\frac {x \ln \relax (3)+4 x \ln \relax (3) \ln \left (\ln \relax (x )\right )}{\ln \left (\ln \relax (x )\right )}+2 x \ln \relax (2) \ln \relax (3)\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*ln(2)*ln(3)+4*ln(3)-1)*ln(x)*ln(ln(x))^2+ln(3)*ln(x)*ln(ln(x))-ln(3))/ln(x)/ln(ln(x))^2,x,method=_RETU
RNVERBOSE)

[Out]

2*x*ln(2)*ln(3)+4*x*ln(3)-x+x*ln(3)/ln(ln(x))

________________________________________________________________________________________

maxima [A]  time = 0.49, size = 25, normalized size = 1.39 \begin {gather*} 2 \, x \log \relax (3) \log \relax (2) + 4 \, x \log \relax (3) - x + \frac {x \log \relax (3)}{\log \left (\log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*log(2)*log(3)+4*log(3)-1)*log(x)*log(log(x))^2+log(3)*log(x)*log(log(x))-log(3))/log(x)/log(log(
x))^2,x, algorithm="maxima")

[Out]

2*x*log(3)*log(2) + 4*x*log(3) - x + x*log(3)/log(log(x))

________________________________________________________________________________________

mupad [B]  time = 7.05, size = 24, normalized size = 1.33 \begin {gather*} x\,\left (4\,\ln \relax (3)+2\,\ln \relax (2)\,\ln \relax (3)-1\right )+\frac {x\,\ln \relax (3)}{\ln \left (\ln \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(log(x))*log(3)*log(x) - log(3) + log(log(x))^2*log(x)*(4*log(3) + 2*log(2)*log(3) - 1))/(log(log(x))^
2*log(x)),x)

[Out]

x*(4*log(3) + 2*log(2)*log(3) - 1) + (x*log(3))/log(log(x))

________________________________________________________________________________________

sympy [A]  time = 0.26, size = 26, normalized size = 1.44 \begin {gather*} x \left (-1 + 2 \log {\relax (2 )} \log {\relax (3 )} + 4 \log {\relax (3 )}\right ) + \frac {x \log {\relax (3 )}}{\log {\left (\log {\relax (x )} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*ln(2)*ln(3)+4*ln(3)-1)*ln(x)*ln(ln(x))**2+ln(3)*ln(x)*ln(ln(x))-ln(3))/ln(x)/ln(ln(x))**2,x)

[Out]

x*(-1 + 2*log(2)*log(3) + 4*log(3)) + x*log(3)/log(log(x))

________________________________________________________________________________________