Integrand size = 59, antiderivative size = 26 \[ \int \frac {-60+140 \log \left (\frac {\log (5)}{3}\right )+85 \log ^2\left (\frac {\log (5)}{3}\right )}{36-240 x+400 x^2+(-204+680 x) \log \left (\frac {\log (5)}{3}\right )+289 \log ^2\left (\frac {\log (5)}{3}\right )} \, dx=-1+\frac {x}{\frac {17}{5}+\frac {4 (-2+x)}{2+\log \left (\frac {\log (5)}{3}\right )}} \]
Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62 \[ \int \frac {-60+140 \log \left (\frac {\log (5)}{3}\right )+85 \log ^2\left (\frac {\log (5)}{3}\right )}{36-240 x+400 x^2+(-204+680 x) \log \left (\frac {\log (5)}{3}\right )+289 \log ^2\left (\frac {\log (5)}{3}\right )} \, dx=-\frac {-12+28 \log \left (\frac {\log (5)}{3}\right )+17 \log ^2\left (\frac {\log (5)}{3}\right )}{4 \left (-6+20 x+17 \log \left (\frac {\log (5)}{3}\right )\right )} \]
Integrate[(-60 + 140*Log[Log[5]/3] + 85*Log[Log[5]/3]^2)/(36 - 240*x + 400 *x^2 + (-204 + 680*x)*Log[Log[5]/3] + 289*Log[Log[5]/3]^2),x]
Time = 0.17 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.54, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {27, 2007, 17}
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.
\(\displaystyle \int \frac {-60+85 \log ^2\left (\frac {\log (5)}{3}\right )+140 \log \left (\frac {\log (5)}{3}\right )}{400 x^2-240 x+(680 x-204) \log \left (\frac {\log (5)}{3}\right )+36+289 \log ^2\left (\frac {\log (5)}{3}\right )} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -5 \left (6-17 \log \left (\frac {\log (5)}{3}\right )\right ) \left (2+\log \left (\frac {\log (5)}{3}\right )\right ) \int \frac {1}{400 x^2-240 x+289 \log ^2\left (\frac {\log (5)}{3}\right )-68 (3-10 x) \log \left (\frac {\log (5)}{3}\right )+36}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle -5 \left (6-17 \log \left (\frac {\log (5)}{3}\right )\right ) \left (2+\log \left (\frac {\log (5)}{3}\right )\right ) \int \frac {1}{\left (20 x+17 \log \left (\frac {\log (5)}{3}\right )-6\right )^2}dx\) |
\(\Big \downarrow \) 17 |
\(\displaystyle -\frac {\left (6-17 \log \left (\frac {\log (5)}{3}\right )\right ) \left (2+\log \left (\frac {\log (5)}{3}\right )\right )}{4 \left (-20 x+6-17 \log \left (\frac {\log (5)}{3}\right )\right )}\) |
Int[(-60 + 140*Log[Log[5]/3] + 85*Log[Log[5]/3]^2)/(36 - 240*x + 400*x^2 + (-204 + 680*x)*Log[Log[5]/3] + 289*Log[Log[5]/3]^2),x]
3.17.90.3.1 Defintions of rubi rules used
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Time = 0.31 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.35
method | result | size |
gosper | \(-\frac {17 \ln \left (\frac {\ln \left (5\right )}{3}\right )^{2}+28 \ln \left (\frac {\ln \left (5\right )}{3}\right )-12}{4 \left (17 \ln \left (\frac {\ln \left (5\right )}{3}\right )+20 x -6\right )}\) | \(35\) |
default | \(-\frac {85 \ln \left (\frac {\ln \left (5\right )}{3}\right )^{2}+140 \ln \left (\frac {\ln \left (5\right )}{3}\right )-60}{20 \left (17 \ln \left (\frac {\ln \left (5\right )}{3}\right )+20 x -6\right )}\) | \(35\) |
parallelrisch | \(-\frac {85 \ln \left (\frac {\ln \left (5\right )}{3}\right )^{2}+140 \ln \left (\frac {\ln \left (5\right )}{3}\right )-60}{20 \left (17 \ln \left (\frac {\ln \left (5\right )}{3}\right )+20 x -6\right )}\) | \(35\) |
norman | \(\frac {3-\frac {17 \ln \left (3\right )^{2}}{4}+\frac {17 \ln \left (3\right ) \ln \left (\ln \left (5\right )\right )}{2}-\frac {17 \ln \left (\ln \left (5\right )\right )^{2}}{4}+7 \ln \left (3\right )-7 \ln \left (\ln \left (5\right )\right )}{17 \ln \left (\frac {\ln \left (5\right )}{3}\right )+20 x -6}\) | \(47\) |
risch | \(\frac {\ln \left (3\right )^{2}}{4 \ln \left (3\right )-4 \ln \left (\ln \left (5\right )\right )-\frac {80 x}{17}+\frac {24}{17}}-\frac {\ln \left (3\right ) \ln \left (\ln \left (5\right )\right )}{2 \left (\ln \left (3\right )-\ln \left (\ln \left (5\right )\right )-\frac {20 x}{17}+\frac {6}{17}\right )}+\frac {\ln \left (\ln \left (5\right )\right )^{2}}{4 \ln \left (3\right )-4 \ln \left (\ln \left (5\right )\right )-\frac {80 x}{17}+\frac {24}{17}}-\frac {7 \ln \left (3\right )}{17 \left (\ln \left (3\right )-\ln \left (\ln \left (5\right )\right )-\frac {20 x}{17}+\frac {6}{17}\right )}+\frac {7 \ln \left (\ln \left (5\right )\right )}{17 \left (\ln \left (3\right )-\ln \left (\ln \left (5\right )\right )-\frac {20 x}{17}+\frac {6}{17}\right )}-\frac {3}{17 \left (\ln \left (3\right )-\ln \left (\ln \left (5\right )\right )-\frac {20 x}{17}+\frac {6}{17}\right )}\) | \(117\) |
meijerg | \(\frac {3 x}{\left (\frac {17 \ln \left (\frac {\ln \left (5\right )}{3}\right )}{20}-\frac {3}{10}\right ) \left (-17 \ln \left (\frac {\ln \left (5\right )}{3}\right )+6\right ) \left (1-\frac {20 x}{-17 \ln \left (\frac {\ln \left (5\right )}{3}\right )+6}\right )}-\frac {17 \ln \left (\frac {\ln \left (5\right )}{3}\right )^{2} x}{4 \left (\frac {17 \ln \left (\frac {\ln \left (5\right )}{3}\right )}{20}-\frac {3}{10}\right ) \left (-17 \ln \left (\frac {\ln \left (5\right )}{3}\right )+6\right ) \left (1-\frac {20 x}{-17 \ln \left (\frac {\ln \left (5\right )}{3}\right )+6}\right )}-\frac {7 \ln \left (\frac {\ln \left (5\right )}{3}\right ) x}{\left (\frac {17 \ln \left (\frac {\ln \left (5\right )}{3}\right )}{20}-\frac {3}{10}\right ) \left (-17 \ln \left (\frac {\ln \left (5\right )}{3}\right )+6\right ) \left (1-\frac {20 x}{-17 \ln \left (\frac {\ln \left (5\right )}{3}\right )+6}\right )}\) | \(143\) |
int((85*ln(1/3*ln(5))^2+140*ln(1/3*ln(5))-60)/(289*ln(1/3*ln(5))^2+(680*x- 204)*ln(1/3*ln(5))+400*x^2-240*x+36),x,method=_RETURNVERBOSE)
Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {-60+140 \log \left (\frac {\log (5)}{3}\right )+85 \log ^2\left (\frac {\log (5)}{3}\right )}{36-240 x+400 x^2+(-204+680 x) \log \left (\frac {\log (5)}{3}\right )+289 \log ^2\left (\frac {\log (5)}{3}\right )} \, dx=-\frac {17 \, \log \left (\frac {1}{3} \, \log \left (5\right )\right )^{2} + 28 \, \log \left (\frac {1}{3} \, \log \left (5\right )\right ) - 12}{4 \, {\left (20 \, x + 17 \, \log \left (\frac {1}{3} \, \log \left (5\right )\right ) - 6\right )}} \]
integrate((85*log(1/3*log(5))^2+140*log(1/3*log(5))-60)/(289*log(1/3*log(5 ))^2+(680*x-204)*log(1/3*log(5))+400*x^2-240*x+36),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (20) = 40\).
Time = 0.15 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.15 \[ \int \frac {-60+140 \log \left (\frac {\log (5)}{3}\right )+85 \log ^2\left (\frac {\log (5)}{3}\right )}{36-240 x+400 x^2+(-204+680 x) \log \left (\frac {\log (5)}{3}\right )+289 \log ^2\left (\frac {\log (5)}{3}\right )} \, dx=- \frac {- 140 \log {\left (3 \right )} - 170 \log {\left (3 \right )} \log {\left (\log {\left (5 \right )} \right )} - 60 + 85 \log {\left (\log {\left (5 \right )} \right )}^{2} + 140 \log {\left (\log {\left (5 \right )} \right )} + 85 \log {\left (3 \right )}^{2}}{400 x - 340 \log {\left (3 \right )} - 120 + 340 \log {\left (\log {\left (5 \right )} \right )}} \]
integrate((85*ln(1/3*ln(5))**2+140*ln(1/3*ln(5))-60)/(289*ln(1/3*ln(5))**2 +(680*x-204)*ln(1/3*ln(5))+400*x**2-240*x+36),x)
-(-140*log(3) - 170*log(3)*log(log(5)) - 60 + 85*log(log(5))**2 + 140*log( log(5)) + 85*log(3)**2)/(400*x - 340*log(3) - 120 + 340*log(log(5)))
Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {-60+140 \log \left (\frac {\log (5)}{3}\right )+85 \log ^2\left (\frac {\log (5)}{3}\right )}{36-240 x+400 x^2+(-204+680 x) \log \left (\frac {\log (5)}{3}\right )+289 \log ^2\left (\frac {\log (5)}{3}\right )} \, dx=-\frac {17 \, \log \left (\frac {1}{3} \, \log \left (5\right )\right )^{2} + 28 \, \log \left (\frac {1}{3} \, \log \left (5\right )\right ) - 12}{4 \, {\left (20 \, x + 17 \, \log \left (\frac {1}{3} \, \log \left (5\right )\right ) - 6\right )}} \]
integrate((85*log(1/3*log(5))^2+140*log(1/3*log(5))-60)/(289*log(1/3*log(5 ))^2+(680*x-204)*log(1/3*log(5))+400*x^2-240*x+36),x, algorithm=\
Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {-60+140 \log \left (\frac {\log (5)}{3}\right )+85 \log ^2\left (\frac {\log (5)}{3}\right )}{36-240 x+400 x^2+(-204+680 x) \log \left (\frac {\log (5)}{3}\right )+289 \log ^2\left (\frac {\log (5)}{3}\right )} \, dx=-\frac {17 \, \log \left (\frac {1}{3} \, \log \left (5\right )\right )^{2} + 28 \, \log \left (\frac {1}{3} \, \log \left (5\right )\right ) - 12}{4 \, {\left (20 \, x + 17 \, \log \left (\frac {1}{3} \, \log \left (5\right )\right ) - 6\right )}} \]
integrate((85*log(1/3*log(5))^2+140*log(1/3*log(5))-60)/(289*log(1/3*log(5 ))^2+(680*x-204)*log(1/3*log(5))+400*x^2-240*x+36),x, algorithm=\
Time = 0.24 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.77 \[ \int \frac {-60+140 \log \left (\frac {\log (5)}{3}\right )+85 \log ^2\left (\frac {\log (5)}{3}\right )}{36-240 x+400 x^2+(-204+680 x) \log \left (\frac {\log (5)}{3}\right )+289 \log ^2\left (\frac {\log (5)}{3}\right )} \, dx=\frac {7\,\ln \left (3\right )-7\,\ln \left (\ln \left (5\right )\right )-\frac {17\,{\ln \left (\ln \left (5\right )\right )}^2}{4}+\frac {17\,\ln \left (3\right )\,\ln \left (\ln \left (5\right )\right )}{2}-\frac {17\,{\ln \left (3\right )}^2}{4}+3}{20\,x+\ln \left (\frac {{\ln \left (5\right )}^{17}}{129140163}\right )-6} \]
int((140*log(log(5)/3) + 85*log(log(5)/3)^2 - 60)/(289*log(log(5)/3)^2 - 2 40*x + log(log(5)/3)*(680*x - 204) + 400*x^2 + 36),x)