3.79.84 \(\int \frac {-147 x+24 \log ^2(4)+42 x \log (6 x)-3 x \log ^2(6 x)}{49 x \log (4)-14 x \log (4) \log (6 x)+x \log (4) \log ^2(6 x)} \, dx\)

Optimal. Leaf size=24 \[ 3 \left (-\frac {x}{\log (4)}+\frac {8 \log (4)}{7-\log (6 x)}\right ) \]

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Rubi [A]  time = 0.33, antiderivative size = 22, normalized size of antiderivative = 0.92, number of steps used = 6, number of rules used = 5, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {6741, 12, 6742, 2302, 30} \begin {gather*} \frac {24 \log (4)}{7-\log (6 x)}-\frac {3 x}{\log (4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-147*x + 24*Log[4]^2 + 42*x*Log[6*x] - 3*x*Log[6*x]^2)/(49*x*Log[4] - 14*x*Log[4]*Log[6*x] + x*Log[4]*Log
[6*x]^2),x]

[Out]

(-3*x)/Log[4] + (24*Log[4])/(7 - Log[6*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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 \left (-49 x+8 \log ^2(4)+14 x \log (6 x)-x \log ^2(6 x)\right )}{x \log (4) (7-\log (6 x))^2} \, dx\\ &=\frac {3 \int \frac {-49 x+8 \log ^2(4)+14 x \log (6 x)-x \log ^2(6 x)}{x (7-\log (6 x))^2} \, dx}{\log (4)}\\ &=\frac {3 \int \left (-1+\frac {8 \log ^2(4)}{x (-7+\log (6 x))^2}\right ) \, dx}{\log (4)}\\ &=-\frac {3 x}{\log (4)}+(24 \log (4)) \int \frac {1}{x (-7+\log (6 x))^2} \, dx\\ &=-\frac {3 x}{\log (4)}+(24 \log (4)) \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,-7+\log (6 x)\right )\\ &=-\frac {3 x}{\log (4)}+\frac {24 \log (4)}{7-\log (6 x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 24, normalized size = 1.00 \begin {gather*} -\frac {3 \left (x-\frac {8 \log ^2(4)}{7-\log (6 x)}\right )}{\log (4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-147*x + 24*Log[4]^2 + 42*x*Log[6*x] - 3*x*Log[6*x]^2)/(49*x*Log[4] - 14*x*Log[4]*Log[6*x] + x*Log[
4]*Log[6*x]^2),x]

[Out]

(-3*(x - (8*Log[4]^2)/(7 - Log[6*x])))/Log[4]

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fricas [A]  time = 0.60, size = 32, normalized size = 1.33 \begin {gather*} -\frac {3 \, {\left (32 \, \log \relax (2)^{2} + x \log \left (6 \, x\right ) - 7 \, x\right )}}{2 \, {\left (\log \relax (2) \log \left (6 \, x\right ) - 7 \, \log \relax (2)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x*log(6*x)^2+42*x*log(6*x)+96*log(2)^2-147*x)/(2*x*log(2)*log(6*x)^2-28*x*log(2)*log(6*x)+98*x*l
og(2)),x, algorithm="fricas")

[Out]

-3/2*(32*log(2)^2 + x*log(6*x) - 7*x)/(log(2)*log(6*x) - 7*log(2))

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giac [A]  time = 0.23, size = 22, normalized size = 0.92 \begin {gather*} -\frac {3 \, x}{2 \, \log \relax (2)} - \frac {48 \, \log \relax (2)}{\log \relax (2) + \log \left (3 \, x\right ) - 7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x*log(6*x)^2+42*x*log(6*x)+96*log(2)^2-147*x)/(2*x*log(2)*log(6*x)^2-28*x*log(2)*log(6*x)+98*x*l
og(2)),x, algorithm="giac")

[Out]

-3/2*x/log(2) - 48*log(2)/(log(2) + log(3*x) - 7)

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maple [A]  time = 0.06, size = 21, normalized size = 0.88

method	result	size
derivativedivides	\(-\frac {3 x}{2 \ln \relax (2)}-\frac {48 \ln \relax (2)}{\ln \left (6 x \right )-7}\)	\(21\)
default	\(-\frac {3 x}{2 \ln \relax (2)}-\frac {48 \ln \relax (2)}{\ln \left (6 x \right )-7}\)	\(21\)
risch	\(-\frac {3 x}{2 \ln \relax (2)}-\frac {48 \ln \relax (2)}{\ln \left (6 x \right )-7}\)	\(21\)
norman	\(\frac {\frac {21 x}{2 \ln \relax (2)}-\frac {3 x \ln \left (6 x \right )}{2 \ln \relax (2)}-48 \ln \relax (2)}{\ln \left (6 x \right )-7}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3*x*ln(6*x)^2+42*x*ln(6*x)+96*ln(2)^2-147*x)/(2*x*ln(2)*ln(6*x)^2-28*x*ln(2)*ln(6*x)+98*x*ln(2)),x,metho
d=_RETURNVERBOSE)

[Out]

-3/2*x/ln(2)-48*ln(2)/(ln(6*x)-7)

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maxima [A]  time = 0.49, size = 33, normalized size = 1.38 \begin {gather*} -\frac {48 \, \log \relax (2)^{2}}{{\left (\log \relax (3) - 7\right )} \log \relax (2) + \log \relax (2)^{2} + \log \relax (2) \log \relax (x)} - \frac {3 \, x}{2 \, \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x*log(6*x)^2+42*x*log(6*x)+96*log(2)^2-147*x)/(2*x*log(2)*log(6*x)^2-28*x*log(2)*log(6*x)+98*x*l
og(2)),x, algorithm="maxima")

[Out]

-48*log(2)^2/((log(3) - 7)*log(2) + log(2)^2 + log(2)*log(x)) - 3/2*x/log(2)

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mupad [B]  time = 5.41, size = 20, normalized size = 0.83 \begin {gather*} -\frac {3\,x}{2\,\ln \relax (2)}-\frac {48\,\ln \relax (2)}{\ln \left (6\,x\right )-7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(147*x - 42*x*log(6*x) + 3*x*log(6*x)^2 - 96*log(2)^2)/(98*x*log(2) - 28*x*log(6*x)*log(2) + 2*x*log(6*x)
^2*log(2)),x)

[Out]

- (3*x)/(2*log(2)) - (48*log(2))/(log(6*x) - 7)

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sympy [A]  time = 0.13, size = 20, normalized size = 0.83 \begin {gather*} - \frac {3 x}{2 \log {\relax (2 )}} - \frac {48 \log {\relax (2 )}}{\log {\left (6 x \right )} - 7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x*ln(6*x)**2+42*x*ln(6*x)+96*ln(2)**2-147*x)/(2*x*ln(2)*ln(6*x)**2-28*x*ln(2)*ln(6*x)+98*x*ln(2)
),x)

[Out]

-3*x/(2*log(2)) - 48*log(2)/(log(6*x) - 7)

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