Integrand size = 72, antiderivative size = 18 \[ \int \frac {-3-6 x \log (84)}{x \log ^2(6)+x^2 \log ^2(6) \log (84)+\left (-2 x \log (6)-2 x^2 \log (6) \log (84)\right ) \log \left (x+x^2 \log (84)\right )+\left (x+x^2 \log (84)\right ) \log ^2\left (x+x^2 \log (84)\right )} \, dx=\frac {3}{-\log (6)+\log \left (x+x^2 \log (84)\right )} \]
Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {-3-6 x \log (84)}{x \log ^2(6)+x^2 \log ^2(6) \log (84)+\left (-2 x \log (6)-2 x^2 \log (6) \log (84)\right ) \log \left (x+x^2 \log (84)\right )+\left (x+x^2 \log (84)\right ) \log ^2\left (x+x^2 \log (84)\right )} \, dx=-\frac {3}{\log (6)-\log (x (1+x \log (84)))} \]
Integrate[(-3 - 6*x*Log[84])/(x*Log[6]^2 + x^2*Log[6]^2*Log[84] + (-2*x*Lo g[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]
Time = 0.45 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {7239, 27, 25, 7237}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {-6 x \log (84)-3}{x^2 \log ^2(6) \log (84)+\left (x^2 \log (84)+x\right ) \log ^2\left (x^2 \log (84)+x\right )+\left (-2 x^2 \log (6) \log (84)-2 x \log (6)\right ) \log \left (x^2 \log (84)+x\right )+x \log ^2(6)} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {3 (-2 x \log (84)-1)}{x (x \log (84)+1) (\log (6)-\log (x (x \log (84)+1)))^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 3 \int -\frac {2 \log (84) x+1}{x (\log (84) x+1) (\log (6)-\log (x (\log (84) x+1)))^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -3 \int \frac {2 \log (84) x+1}{x (\log (84) x+1) (\log (6)-\log (x (\log (84) x+1)))^2}dx\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle -\frac {3}{\log (6)-\log (x (x \log (84)+1))}\) |
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*Log[84]] + (x + x^2*Log[84])*Log[x + x^ 2*Log[84]]^2),x]
3.22.27.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06
method | result | size |
default | \(\frac {3}{\ln \left (x^{2} \ln \left (84\right )+x \right )-\ln \left (6\right )}\) | \(19\) |
norman | \(-\frac {3}{\ln \left (6\right )-\ln \left (x^{2} \ln \left (84\right )+x \right )}\) | \(19\) |
parallelrisch | \(-\frac {3}{\ln \left (6\right )-\ln \left (x^{2} \ln \left (84\right )+x \right )}\) | \(19\) |
risch | \(-\frac {3}{\ln \left (3\right )+\ln \left (2\right )-\ln \left (x^{2} \left (2 \ln \left (2\right )+\ln \left (3\right )+\ln \left (7\right )\right )+x \right )}\) | \(28\) |
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=_RETU RNVERBOSE)
Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {-3-6 x \log (84)}{x \log ^2(6)+x^2 \log ^2(6) \log (84)+\left (-2 x \log (6)-2 x^2 \log (6) \log (84)\right ) \log \left (x+x^2 \log (84)\right )+\left (x+x^2 \log (84)\right ) \log ^2\left (x+x^2 \log (84)\right )} \, dx=-\frac {3}{\log \left (6\right ) - \log \left (x^{2} \log \left (84\right ) + x\right )} \]
integrate((-6*x*log(84)-3)/((x^2*log(84)+x)*log(x^2*log(84)+x)^2+(-2*x^2*l og(6)*log(84)-2*x*log(6))*log(x^2*log(84)+x)+x^2*log(6)^2*log(84)+x*log(6) ^2),x, algorithm=\
Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {-3-6 x \log (84)}{x \log ^2(6)+x^2 \log ^2(6) \log (84)+\left (-2 x \log (6)-2 x^2 \log (6) \log (84)\right ) \log \left (x+x^2 \log (84)\right )+\left (x+x^2 \log (84)\right ) \log ^2\left (x+x^2 \log (84)\right )} \, dx=\frac {3}{\log {\left (x^{2} \log {\left (84 \right )} + x \right )} - \log {\left (6 \right )}} \]
integrate((-6*x*ln(84)-3)/((x**2*ln(84)+x)*ln(x**2*ln(84)+x)**2+(-2*x**2*l n(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)
Time = 0.32 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.61 \[ \int \frac {-3-6 x \log (84)}{x \log ^2(6)+x^2 \log ^2(6) \log (84)+\left (-2 x \log (6)-2 x^2 \log (6) \log (84)\right ) \log \left (x+x^2 \log (84)\right )+\left (x+x^2 \log (84)\right ) \log ^2\left (x+x^2 \log (84)\right )} \, dx=-\frac {3}{\log \left (3\right ) + \log \left (2\right ) - \log \left (x {\left (\log \left (7\right ) + \log \left (3\right ) + 2 \, \log \left (2\right )\right )} + 1\right ) - \log \left (x\right )} \]
integrate((-6*x*log(84)-3)/((x^2*log(84)+x)*log(x^2*log(84)+x)^2+(-2*x^2*l og(6)*log(84)-2*x*log(6))*log(x^2*log(84)+x)+x^2*log(6)^2*log(84)+x*log(6) ^2),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (18) = 36\).
Time = 0.30 (sec) , antiderivative size = 279, normalized size of antiderivative = 15.50 \[ \int \frac {-3-6 x \log (84)}{x \log ^2(6)+x^2 \log ^2(6) \log (84)+\left (-2 x \log (6)-2 x^2 \log (6) \log (84)\right ) \log \left (x+x^2 \log (84)\right )+\left (x+x^2 \log (84)\right ) \log ^2\left (x+x^2 \log (84)\right )} \, dx=-\frac {3 \, {\left (2 \, x^{2} \log \left (84\right ) \log \left (7\right ) + 2 \, x^{2} \log \left (84\right ) \log \left (3\right ) + 4 \, x^{2} \log \left (84\right ) \log \left (2\right ) + x \log \left (84\right ) + 2 \, x \log \left (7\right ) + 2 \, x \log \left (3\right ) + 4 \, x \log \left (2\right ) + 1\right )}}{2 \, x^{2} \log \left (84\right ) \log \left (7\right ) \log \left (3\right ) + 2 \, x^{2} \log \left (84\right ) \log \left (3\right )^{2} + 2 \, x^{2} \log \left (84\right ) \log \left (7\right ) \log \left (2\right ) + 6 \, x^{2} \log \left (84\right ) \log \left (3\right ) \log \left (2\right ) + 4 \, x^{2} \log \left (84\right ) \log \left (2\right )^{2} - 2 \, x^{2} \log \left (84\right ) \log \left (7\right ) \log \left (x^{2} \log \left (84\right ) + x\right ) - 2 \, x^{2} \log \left (84\right ) \log \left (3\right ) \log \left (x^{2} \log \left (84\right ) + x\right ) - 4 \, x^{2} \log \left (84\right ) \log \left (2\right ) \log \left (x^{2} \log \left (84\right ) + x\right ) + 2 \, x \log \left (84\right ) \log \left (3\right ) + x \log \left (7\right ) \log \left (3\right ) + x \log \left (3\right )^{2} + 2 \, x \log \left (84\right ) \log \left (2\right ) + x \log \left (7\right ) \log \left (2\right ) + 3 \, x \log \left (3\right ) \log \left (2\right ) + 2 \, x \log \left (2\right )^{2} - 2 \, x \log \left (84\right ) \log \left (x^{2} \log \left (84\right ) + x\right ) - x \log \left (7\right ) \log \left (x^{2} \log \left (84\right ) + x\right ) - x \log \left (3\right ) \log \left (x^{2} \log \left (84\right ) + x\right ) - 2 \, x \log \left (2\right ) \log \left (x^{2} \log \left (84\right ) + x\right ) + \log \left (3\right ) + \log \left (2\right ) - \log \left (x^{2} \log \left (84\right ) + x\right )} \]
integrate((-6*x*log(84)-3)/((x^2*log(84)+x)*log(x^2*log(84)+x)^2+(-2*x^2*l og(6)*log(84)-2*x*log(6))*log(x^2*log(84)+x)+x^2*log(6)^2*log(84)+x*log(6) ^2),x, algorithm=\
-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*log (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) + log(3) + log(2) - log(x^2*log(84) + x))
Time = 10.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-3-6 x \log (84)}{x \log ^2(6)+x^2 \log ^2(6) \log (84)+\left (-2 x \log (6)-2 x^2 \log (6) \log (84)\right ) \log \left (x+x^2 \log (84)\right )+\left (x+x^2 \log (84)\right ) \log ^2\left (x+x^2 \log (84)\right )} \, dx=\frac {3}{\ln \left (\frac {\ln \left (84\right )\,x^2}{6}+\frac {x}{6}\right )} \]