3.29.60 \(\int \frac {-15+5 \log (3)}{48 x-15 x \log (4)+(-15 x+5 x \log (3)) \log (x)} \, dx\)

Optimal. Leaf size=20 \[ \log \left (\frac {3 \left (\frac {16}{5}-\log (4)\right )}{-3+\log (3)}+\log (x)\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6, 12, 31} \begin {gather*} \log (3 (16-5 \log (4))-5 (3-\log (3)) \log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-15 + 5*Log[3])/(48*x - 15*x*Log[4] + (-15*x + 5*x*Log[3])*Log[x]),x]

[Out]

Log[3*(16 - 5*Log[4]) - 5*(3 - Log[3])*Log[x]]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-15+5 \log (3)}{x (48-15 \log (4))+(-15 x+5 x \log (3)) \log (x)} \, dx\\ &=-\left ((5 (3-\log (3))) \int \frac {1}{x (48-15 \log (4))+(-15 x+5 x \log (3)) \log (x)} \, dx\right )\\ &=(5 (3-\log (3))) \operatorname {Subst}\left (\int \frac {1}{-48+15 x-5 x \log (3)+15 \log (4)} \, dx,x,\log (x)\right )\\ &=(5 (3-\log (3))) \operatorname {Subst}\left (\int \frac {1}{-48+x (15-5 \log (3))+15 \log (4)} \, dx,x,\log (x)\right )\\ &=\log (3 (16-5 \log (4))-5 (3-\log (3)) \log (x))\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 31, normalized size = 1.55 \begin {gather*} -\frac {5 (-3+\log (3)) \log (-48+15 \log (4)+15 \log (x)-5 \log (3) \log (x))}{15-5 \log (3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-15 + 5*Log[3])/(48*x - 15*x*Log[4] + (-15*x + 5*x*Log[3])*Log[x]),x]

[Out]

(-5*(-3 + Log[3])*Log[-48 + 15*Log[4] + 15*Log[x] - 5*Log[3]*Log[x]])/(15 - 5*Log[3])

________________________________________________________________________________________

fricas [A]  time = 0.68, size = 15, normalized size = 0.75 \begin {gather*} \log \left (5 \, {\left (\log \relax (3) - 3\right )} \log \relax (x) - 30 \, \log \relax (2) + 48\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*log(3)-15)/((5*x*log(3)-15*x)*log(x)-30*x*log(2)+48*x),x, algorithm="fricas")

[Out]

log(5*(log(3) - 3)*log(x) - 30*log(2) + 48)

________________________________________________________________________________________

giac [B]  time = 0.28, size = 44, normalized size = 2.20 \begin {gather*} \frac {1}{2} \, \log \left (\frac {25}{4} \, {\left (\pi {\left (\mathrm {sgn}\relax (x) - 1\right )} \log \relax (3) - 3 \, \pi {\left (\mathrm {sgn}\relax (x) - 1\right )}\right )}^{2} + {\left (5 \, \log \relax (3) \log \left ({\left | x \right |}\right ) - 30 \, \log \relax (2) - 15 \, \log \left ({\left | x \right |}\right ) + 48\right )}^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*log(3)-15)/((5*x*log(3)-15*x)*log(x)-30*x*log(2)+48*x),x, algorithm="giac")

[Out]

1/2*log(25/4*(pi*(sgn(x) - 1)*log(3) - 3*pi*(sgn(x) - 1))^2 + (5*log(3)*log(abs(x)) - 30*log(2) - 15*log(abs(x
)) + 48)^2)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 17, normalized size = 0.85

method	result	size
default	\(\ln \left (\left (5 \ln \relax (3)-15\right ) \ln \relax (x )-30 \ln \relax (2)+48\right )\)	\(17\)
norman	\(\ln \left (-5 \ln \relax (3) \ln \relax (x )+30 \ln \relax (2)+15 \ln \relax (x )-48\right )\)	\(18\)
risch	\(\frac {\ln \left (\ln \relax (x )-\frac {6 \left (5 \ln \relax (2)-8\right )}{5 \left (\ln \relax (3)-3\right )}\right ) \ln \relax (3)}{\ln \relax (3)-3}-\frac {3 \ln \left (\ln \relax (x )-\frac {6 \left (5 \ln \relax (2)-8\right )}{5 \left (\ln \relax (3)-3\right )}\right )}{\ln \relax (3)-3}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*ln(3)-15)/((5*x*ln(3)-15*x)*ln(x)-30*x*ln(2)+48*x),x,method=_RETURNVERBOSE)

[Out]

ln((5*ln(3)-15)*ln(x)-30*ln(2)+48)

________________________________________________________________________________________

maxima [A]  time = 0.52, size = 23, normalized size = 1.15 \begin {gather*} \log \left (\frac {5 \, {\left (\log \relax (3) - 3\right )} \log \relax (x) - 30 \, \log \relax (2) + 48}{5 \, {\left (\log \relax (3) - 3\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*log(3)-15)/((5*x*log(3)-15*x)*log(x)-30*x*log(2)+48*x),x, algorithm="maxima")

[Out]

log(1/5*(5*(log(3) - 3)*log(x) - 30*log(2) + 48)/(log(3) - 3))

________________________________________________________________________________________

mupad [B]  time = 2.46, size = 16, normalized size = 0.80 \begin {gather*} \ln \left (\ln \relax (x)\,\left (5\,\ln \relax (3)-15\right )-30\,\ln \relax (2)+48\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(5*log(3) - 15)/(30*x*log(2) - 48*x + log(x)*(15*x - 5*x*log(3))),x)

[Out]

log(log(x)*(5*log(3) - 15) - 30*log(2) + 48)

________________________________________________________________________________________

sympy [A]  time = 0.15, size = 17, normalized size = 0.85 \begin {gather*} \log {\left (\log {\relax (x )} + \frac {48 - 30 \log {\relax (2 )}}{-15 + 5 \log {\relax (3 )}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*ln(3)-15)/((5*x*ln(3)-15*x)*ln(x)-30*x*ln(2)+48*x),x)

[Out]

log(log(x) + (48 - 30*log(2))/(-15 + 5*log(3)))

________________________________________________________________________________________