3.22.27 \(\int \frac {-3-6 x \log (84)}{x \log ^2(6)+x^2 \log ^2(6) \log (84)+(-2 x \log (6)-2 x^2 \log (6) \log (84)) \log (x+x^2 \log (84))+(x+x^2 \log (84)) \log ^2(x+x^2 \log (84))} \, dx\)

Optimal. Leaf size=18 \[ \frac {3}{-\log (6)+\log \left (x+x^2 \log (84)\right )} \]

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Rubi [A]  time = 0.25, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 72, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {6688, 12, 6686} \begin {gather*} -\frac {3}{\log (6)-\log (x (x \log (84)+1))} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 - 6*x*Log[84])/(x*Log[6]^2 + x^2*Log[6]^2*Log[84] + (-2*x*Log[6] - 2*x^2*Log[6]*Log[84])*Log[x + x^2*L
og[84]] + (x + x^2*Log[84])*Log[x + x^2*Log[84]]^2),x]

[Out]

-3/(Log[6] - Log[x*(1 + x*Log[84])])

Rule 12

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

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 (-1-2 x \log (84))}{x (1+x \log (84)) (\log (6)-\log (x (1+x \log (84))))^2} \, dx\\ &=3 \int \frac {-1-2 x \log (84)}{x (1+x \log (84)) (\log (6)-\log (x (1+x \log (84))))^2} \, dx\\ &=-\frac {3}{\log (6)-\log (x (1+x \log (84)))}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} -\frac {3}{\log (6)-\log (x (1+x \log (84)))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 - 6*x*Log[84])/(x*Log[6]^2 + x^2*Log[6]^2*Log[84] + (-2*x*Log[6] - 2*x^2*Log[6]*Log[84])*Log[x +
x^2*Log[84]] + (x + x^2*Log[84])*Log[x + x^2*Log[84]]^2),x]

[Out]

-3/(Log[6] - Log[x*(1 + x*Log[84])])

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fricas [A]  time = 0.66, size = 18, normalized size = 1.00 \begin {gather*} -\frac {3}{\log \relax (6) - \log \left (x^{2} \log \left (84\right ) + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x*log(84)-3)/((x^2*log(84)+x)*log(x^2*log(84)+x)^2+(-2*x^2*log(6)*log(84)-2*x*log(6))*log(x^2*lo
g(84)+x)+x^2*log(6)^2*log(84)+x*log(6)^2),x, algorithm="fricas")

[Out]

-3/(log(6) - log(x^2*log(84) + x))

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giac [B]  time = 0.57, size = 279, normalized size = 15.50 \begin {gather*} -\frac {3 \, {\left (2 \, x^{2} \log \left (84\right ) \log \relax (7) + 2 \, x^{2} \log \left (84\right ) \log \relax (3) + 4 \, x^{2} \log \left (84\right ) \log \relax (2) + x \log \left (84\right ) + 2 \, x \log \relax (7) + 2 \, x \log \relax (3) + 4 \, x \log \relax (2) + 1\right )}}{2 \, x^{2} \log \left (84\right ) \log \relax (7) \log \relax (3) + 2 \, x^{2} \log \left (84\right ) \log \relax (3)^{2} + 2 \, x^{2} \log \left (84\right ) \log \relax (7) \log \relax (2) + 6 \, x^{2} \log \left (84\right ) \log \relax (3) \log \relax (2) + 4 \, x^{2} \log \left (84\right ) \log \relax (2)^{2} - 2 \, x^{2} \log \left (84\right ) \log \relax (7) \log \left (x^{2} \log \left (84\right ) + x\right ) - 2 \, x^{2} \log \left (84\right ) \log \relax (3) \log \left (x^{2} \log \left (84\right ) + x\right ) - 4 \, x^{2} \log \left (84\right ) \log \relax (2) \log \left (x^{2} \log \left (84\right ) + x\right ) + 2 \, x \log \left (84\right ) \log \relax (3) + x \log \relax (7) \log \relax (3) + x \log \relax (3)^{2} + 2 \, x \log \left (84\right ) \log \relax (2) + x \log \relax (7) \log \relax (2) + 3 \, x \log \relax (3) \log \relax (2) + 2 \, x \log \relax (2)^{2} - 2 \, x \log \left (84\right ) \log \left (x^{2} \log \left (84\right ) + x\right ) - x \log \relax (7) \log \left (x^{2} \log \left (84\right ) + x\right ) - x \log \relax (3) \log \left (x^{2} \log \left (84\right ) + x\right ) - 2 \, x \log \relax (2) \log \left (x^{2} \log \left (84\right ) + x\right ) + \log \relax (3) + \log \relax (2) - \log \left (x^{2} \log \left (84\right ) + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x*log(84)-3)/((x^2*log(84)+x)*log(x^2*log(84)+x)^2+(-2*x^2*log(6)*log(84)-2*x*log(6))*log(x^2*lo
g(84)+x)+x^2*log(6)^2*log(84)+x*log(6)^2),x, algorithm="giac")

[Out]

-3*(2*x^2*log(84)*log(7) + 2*x^2*log(84)*log(3) + 4*x^2*log(84)*log(2) + x*log(84) + 2*x*log(7) + 2*x*log(3) +
4*x*log(2) + 1)/(2*x^2*log(84)*log(7)*log(3) + 2*x^2*log(84)*log(3)^2 + 2*x^2*log(84)*log(7)*log(2) + 6*x^2*l
og(84)*log(3)*log(2) + 4*x^2*log(84)*log(2)^2 - 2*x^2*log(84)*log(7)*log(x^2*log(84) + x) - 2*x^2*log(84)*log(
3)*log(x^2*log(84) + x) - 4*x^2*log(84)*log(2)*log(x^2*log(84) + x) + 2*x*log(84)*log(3) + x*log(7)*log(3) + x
*log(3)^2 + 2*x*log(84)*log(2) + x*log(7)*log(2) + 3*x*log(3)*log(2) + 2*x*log(2)^2 - 2*x*log(84)*log(x^2*log(
84) + x) - x*log(7)*log(x^2*log(84) + x) - x*log(3)*log(x^2*log(84) + x) - 2*x*log(2)*log(x^2*log(84) + x) + l
og(3) + log(2) - log(x^2*log(84) + x))

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maple [A]  time = 0.13, size = 19, normalized size = 1.06

method	result	size
norman	\(-\frac {3}{\ln \relax (6)-\ln \left (x^{2} \ln \left (84\right )+x \right )}\)	\(19\)
risch	\(-\frac {3}{\ln \relax (3)+\ln \relax (2)-\ln \left (x^{2} \left (2 \ln \relax (2)+\ln \relax (3)+\ln \relax (7)\right )+x \right )}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-6*x*ln(84)-3)/((x^2*ln(84)+x)*ln(x^2*ln(84)+x)^2+(-2*x^2*ln(6)*ln(84)-2*x*ln(6))*ln(x^2*ln(84)+x)+x^2*ln
(6)^2*ln(84)+x*ln(6)^2),x,method=_RETURNVERBOSE)

[Out]

-3/(ln(6)-ln(x^2*ln(84)+x))

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maxima [A]  time = 0.84, size = 29, normalized size = 1.61 \begin {gather*} -\frac {3}{\log \relax (3) + \log \relax (2) - \log \left (x {\left (\log \relax (7) + \log \relax (3) + 2 \, \log \relax (2)\right )} + 1\right ) - \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x*log(84)-3)/((x^2*log(84)+x)*log(x^2*log(84)+x)^2+(-2*x^2*log(6)*log(84)-2*x*log(6))*log(x^2*lo
g(84)+x)+x^2*log(6)^2*log(84)+x*log(6)^2),x, algorithm="maxima")

[Out]

-3/(log(3) + log(2) - log(x*(log(7) + log(3) + 2*log(2)) + 1) - log(x))

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mupad [B]  time = 1.61, size = 16, normalized size = 0.89 \begin {gather*} \frac {3}{\ln \left (\frac {\ln \left (84\right )\,x^2}{6}+\frac {x}{6}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(6*x*log(84) + 3)/(log(x + x^2*log(84))^2*(x + x^2*log(84)) + x*log(6)^2 - log(x + x^2*log(84))*(2*x*log(
6) + 2*x^2*log(6)*log(84)) + x^2*log(6)^2*log(84)),x)

[Out]

3/log(x/6 + (x^2*log(84))/6)

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sympy [A]  time = 0.15, size = 14, normalized size = 0.78 \begin {gather*} \frac {3}{\log {\left (x^{2} \log {\left (84 \right )} + x \right )} - \log {\relax (6 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x*ln(84)-3)/((x**2*ln(84)+x)*ln(x**2*ln(84)+x)**2+(-2*x**2*ln(6)*ln(84)-2*x*ln(6))*ln(x**2*ln(84
)+x)+x**2*ln(6)**2*ln(84)+x*ln(6)**2),x)

[Out]

3/(log(x**2*log(84) + x) - log(6))

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