Integrand size = 71, antiderivative size = 20 \[ \int \frac {-2+x+((-12+7 x) \log (3)+(12-7 x) \log (x)) \log (-\log (3)+\log (x))}{\left (-256 x^7+256 x^8-64 x^9\right ) \log (3)+\left (256 x^7-256 x^8+64 x^9\right ) \log (x)} \, dx=\frac {\log (-\log (3)+\log (x))}{64 (-2+x) x^6} \]
\[ \int \frac {-2+x+((-12+7 x) \log (3)+(12-7 x) \log (x)) \log (-\log (3)+\log (x))}{\left (-256 x^7+256 x^8-64 x^9\right ) \log (3)+\left (256 x^7-256 x^8+64 x^9\right ) \log (x)} \, dx=\int \frac {-2+x+((-12+7 x) \log (3)+(12-7 x) \log (x)) \log (-\log (3)+\log (x))}{\left (-256 x^7+256 x^8-64 x^9\right ) \log (3)+\left (256 x^7-256 x^8+64 x^9\right ) \log (x)} \, dx \]
Integrate[(-2 + x + ((-12 + 7*x)*Log[3] + (12 - 7*x)*Log[x])*Log[-Log[3] + Log[x]])/((-256*x^7 + 256*x^8 - 64*x^9)*Log[3] + (256*x^7 - 256*x^8 + 64* x^9)*Log[x]),x]
Integrate[(-2 + x + ((-12 + 7*x)*Log[3] + (12 - 7*x)*Log[x])*Log[-Log[3] + Log[x]])/((-256*x^7 + 256*x^8 - 64*x^9)*Log[3] + (256*x^7 - 256*x^8 + 64* x^9)*Log[x]), x]
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.
\(\displaystyle \int \frac {x+((7 x-12) \log (3)+(12-7 x) \log (x)) \log (\log (x)-\log (3))-2}{\left (-64 x^9+256 x^8-256 x^7\right ) \log (3)+\left (64 x^9-256 x^8+256 x^7\right ) \log (x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-x-((7 x-12) \log (3)+(12-7 x) \log (x)) \log (\log (x)-\log (3))+2}{64 (2-x)^2 x^7 (\log (3)-\log (x))}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{64} \int \frac {-x+((12-7 x) \log (3)-(12-7 x) \log (x)) \log (\log (x)-\log (3))+2}{(2-x)^2 x^7 (\log (3)-\log (x))}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{64} \int \left (-\frac {(7 x-12) \log \left (\log \left (\frac {x}{3}\right )\right )}{(x-2)^2 x^7}-\frac {1}{(x-2) x^7 (\log (3)-\log (x))}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{64} \left (-\frac {3}{64} \text {Subst}\left (\int \frac {\log (\log (x))}{(3 x-2)^2}dx,x,\frac {x}{3}\right )-\int \frac {1}{(x-2) x^7 (\log (3)-\log (x))}dx+\frac {\operatorname {ExpIntegralEi}\left (-6 \log \left (\frac {x}{3}\right )\right )}{1458}+\frac {1}{972} \operatorname {ExpIntegralEi}\left (-5 \log \left (\frac {x}{3}\right )\right )+\frac {1}{648} \operatorname {ExpIntegralEi}\left (-4 \log \left (\frac {x}{3}\right )\right )+\frac {1}{432} \operatorname {ExpIntegralEi}\left (-3 \log \left (\frac {x}{3}\right )\right )+\frac {1}{288} \operatorname {ExpIntegralEi}\left (-2 \log \left (\frac {x}{3}\right )\right )+\frac {1}{192} \operatorname {ExpIntegralEi}\left (-\log \left (\frac {x}{3}\right )\right )-\frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{2 x^6}-\frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{4 x^5}-\frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{8 x^4}-\frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{16 x^3}-\frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{32 x^2}-\frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{64 x}\right )\) |
Int[(-2 + x + ((-12 + 7*x)*Log[3] + (12 - 7*x)*Log[x])*Log[-Log[3] + Log[x ]])/((-256*x^7 + 256*x^8 - 64*x^9)*Log[3] + (256*x^7 - 256*x^8 + 64*x^9)*L og[x]),x]
3.29.18.3.1 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]]
Time = 7.54 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95
method | result | size |
risch | \(\frac {\ln \left (\ln \left (x \right )-\ln \left (3\right )\right )}{64 x^{6} \left (-2+x \right )}\) | \(19\) |
parallelrisch | \(\frac {\ln \left (\ln \left (x \right )-\ln \left (3\right )\right )}{64 x^{6} \left (-2+x \right )}\) | \(19\) |
int((((-7*x+12)*ln(x)+(7*x-12)*ln(3))*ln(ln(x)-ln(3))+x-2)/((64*x^9-256*x^ 8+256*x^7)*ln(x)+(-64*x^9+256*x^8-256*x^7)*ln(3)),x,method=_RETURNVERBOSE)
Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {-2+x+((-12+7 x) \log (3)+(12-7 x) \log (x)) \log (-\log (3)+\log (x))}{\left (-256 x^7+256 x^8-64 x^9\right ) \log (3)+\left (256 x^7-256 x^8+64 x^9\right ) \log (x)} \, dx=\frac {\log \left (-\log \left (3\right ) + \log \left (x\right )\right )}{64 \, {\left (x^{7} - 2 \, x^{6}\right )}} \]
integrate((((-7*x+12)*log(x)+(7*x-12)*log(3))*log(log(x)-log(3))+x-2)/((64 *x^9-256*x^8+256*x^7)*log(x)+(-64*x^9+256*x^8-256*x^7)*log(3)),x, algorith m=\
Time = 0.16 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {-2+x+((-12+7 x) \log (3)+(12-7 x) \log (x)) \log (-\log (3)+\log (x))}{\left (-256 x^7+256 x^8-64 x^9\right ) \log (3)+\left (256 x^7-256 x^8+64 x^9\right ) \log (x)} \, dx=\frac {\log {\left (\log {\left (x \right )} - \log {\left (3 \right )} \right )}}{64 x^{7} - 128 x^{6}} \]
integrate((((-7*x+12)*ln(x)+(7*x-12)*ln(3))*ln(ln(x)-ln(3))+x-2)/((64*x**9 -256*x**8+256*x**7)*ln(x)+(-64*x**9+256*x**8-256*x**7)*ln(3)),x)
Time = 0.31 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {-2+x+((-12+7 x) \log (3)+(12-7 x) \log (x)) \log (-\log (3)+\log (x))}{\left (-256 x^7+256 x^8-64 x^9\right ) \log (3)+\left (256 x^7-256 x^8+64 x^9\right ) \log (x)} \, dx=\frac {\log \left (-\log \left (3\right ) + \log \left (x\right )\right )}{64 \, {\left (x^{7} - 2 \, x^{6}\right )}} \]
integrate((((-7*x+12)*log(x)+(7*x-12)*log(3))*log(log(x)-log(3))+x-2)/((64 *x^9-256*x^8+256*x^7)*log(x)+(-64*x^9+256*x^8-256*x^7)*log(3)),x, algorith m=\
Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (18) = 36\).
Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.20 \[ \int \frac {-2+x+((-12+7 x) \log (3)+(12-7 x) \log (x)) \log (-\log (3)+\log (x))}{\left (-256 x^7+256 x^8-64 x^9\right ) \log (3)+\left (256 x^7-256 x^8+64 x^9\right ) \log (x)} \, dx=\frac {1}{4096} \, {\left (\frac {1}{x - 2} - \frac {x^{5} + 2 \, x^{4} + 4 \, x^{3} + 8 \, x^{2} + 16 \, x + 32}{x^{6}}\right )} \log \left (-\log \left (3\right ) + \log \left (x\right )\right ) \]
integrate((((-7*x+12)*log(x)+(7*x-12)*log(3))*log(log(x)-log(3))+x-2)/((64 *x^9-256*x^8+256*x^7)*log(x)+(-64*x^9+256*x^8-256*x^7)*log(3)),x, algorith m=\
Time = 15.62 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {-2+x+((-12+7 x) \log (3)+(12-7 x) \log (x)) \log (-\log (3)+\log (x))}{\left (-256 x^7+256 x^8-64 x^9\right ) \log (3)+\left (256 x^7-256 x^8+64 x^9\right ) \log (x)} \, dx=-\frac {\ln \left (\ln \left (\frac {x}{3}\right )\right )}{128\,x^6-64\,x^7} \]