Integrand size = 77, antiderivative size = 23 \[ \int \frac {-21 x^2+9 \log (4)+7 x^2 \log (x)+((27-63 x) \log (4)+(-9+21 x) \log (4) \log (x)) \log (-3+\log (x))}{-9 x^2+3 x^2 \log (x)+(-27 x \log (4)+9 x \log (4) \log (x)) \log (-3+\log (x))} \, dx=-5+\frac {7 x}{3}-\log (x)+\log (x+3 \log (4) \log (-3+\log (x))) \]
Time = 1.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.91 \[ \int \frac {-21 x^2+9 \log (4)+7 x^2 \log (x)+((27-63 x) \log (4)+(-9+21 x) \log (4) \log (x)) \log (-3+\log (x))}{-9 x^2+3 x^2 \log (x)+(-27 x \log (4)+9 x \log (4) \log (x)) \log (-3+\log (x))} \, dx=\frac {7 x \log (4)}{\log (64)}+\frac {1}{3} \left (-\frac {9 \log (4) \log (x)}{\log (64)}+\frac {9 \log (4) \log (x+\log (64) \log (-3+\log (x)))}{\log (64)}\right ) \]
Integrate[(-21*x^2 + 9*Log[4] + 7*x^2*Log[x] + ((27 - 63*x)*Log[4] + (-9 + 21*x)*Log[4]*Log[x])*Log[-3 + Log[x]])/(-9*x^2 + 3*x^2*Log[x] + (-27*x*Lo g[4] + 9*x*Log[4]*Log[x])*Log[-3 + Log[x]]),x]
(7*x*Log[4])/Log[64] + ((-9*Log[4]*Log[x])/Log[64] + (9*Log[4]*Log[x + Log [64]*Log[-3 + Log[x]]])/Log[64])/3
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.
\(\displaystyle \int \frac {-21 x^2+7 x^2 \log (x)+((27-63 x) \log (4)+(21 x-9) \log (4) \log (x)) \log (\log (x)-3)+9 \log (4)}{-9 x^2+3 x^2 \log (x)+(9 x \log (4) \log (x)-27 x \log (4)) \log (\log (x)-3)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {21 x^2-7 x^2 \log (x)-((27-63 x) \log (4)+(21 x-9) \log (4) \log (x)) \log (\log (x)-3)-9 \log (4)}{3 x (3-\log (x)) (x+\log (64) \log (\log (x)-3))}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int \frac {-7 \log (x) x^2+21 x^2-3 (3 (3-7 x) \log (4)-(3-7 x) \log (4) \log (x)) \log (\log (x)-3)-9 \log (4)}{x (3-\log (x)) (x+\log (64) \log (\log (x)-3))}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{3} \int \left (\frac {3 \log (4) (7 x-3)}{x \log (64)}+\frac {\log (262144) \log (x) x-\log (18014398509481984) x+\log (64) \log (262144)}{x \log (64) (\log (x)-3) (x+\log (64) \log (\log (x)-3))}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (-\frac {\log (18014398509481984) \int \frac {1}{(\log (x)-3) (x+\log (64) \log (\log (x)-3))}dx}{\log (64)}+\log (262144) \int \frac {1}{x (\log (x)-3) (x+\log (64) \log (\log (x)-3))}dx+\frac {\log (262144) \int \frac {\log (x)}{(\log (x)-3) (x+\log (64) \log (\log (x)-3))}dx}{\log (64)}+\frac {21 x \log (4)}{\log (64)}-\frac {9 \log (4) \log (x)}{\log (64)}\right )\) |
Int[(-21*x^2 + 9*Log[4] + 7*x^2*Log[x] + ((27 - 63*x)*Log[4] + (-9 + 21*x) *Log[4]*Log[x])*Log[-3 + Log[x]])/(-9*x^2 + 3*x^2*Log[x] + (-27*x*Log[4] + 9*x*Log[4]*Log[x])*Log[-3 + Log[x]]),x]
3.9.23.3.1 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 = 4.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91
method | result | size |
default | \(-\ln \left (x \right )+\frac {7 x}{3}+\ln \left (6 \ln \left (2\right ) \ln \left (\ln \left (x \right )-3\right )+x \right )\) | \(21\) |
norman | \(-\ln \left (x \right )+\frac {7 x}{3}+\ln \left (6 \ln \left (2\right ) \ln \left (\ln \left (x \right )-3\right )+x \right )\) | \(21\) |
parallelrisch | \(\ln \left (6 \ln \left (2\right ) \ln \left (\ln \left (x \right )-3\right )+x \right )+\frac {7 x}{3}-\ln \left (x \right )-6\) | \(22\) |
risch | \(\frac {7 x}{3}-\ln \left (x \right )+\ln \left (\ln \left (\ln \left (x \right )-3\right )+\frac {x}{6 \ln \left (2\right )}\right )\) | \(23\) |
int(((2*(21*x-9)*ln(2)*ln(x)+2*(-63*x+27)*ln(2))*ln(ln(x)-3)+7*x^2*ln(x)+1 8*ln(2)-21*x^2)/((18*x*ln(2)*ln(x)-54*x*ln(2))*ln(ln(x)-3)+3*x^2*ln(x)-9*x ^2),x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {-21 x^2+9 \log (4)+7 x^2 \log (x)+((27-63 x) \log (4)+(-9+21 x) \log (4) \log (x)) \log (-3+\log (x))}{-9 x^2+3 x^2 \log (x)+(-27 x \log (4)+9 x \log (4) \log (x)) \log (-3+\log (x))} \, dx=\frac {7}{3} \, x + \log \left (6 \, \log \left (2\right ) \log \left (\log \left (x\right ) - 3\right ) + x\right ) - \log \left (x\right ) \]
integrate(((2*(21*x-9)*log(2)*log(x)+2*(-63*x+27)*log(2))*log(log(x)-3)+7* x^2*log(x)+18*log(2)-21*x^2)/((18*x*log(2)*log(x)-54*x*log(2))*log(log(x)- 3)+3*x^2*log(x)-9*x^2),x, algorithm=\
Time = 0.20 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {-21 x^2+9 \log (4)+7 x^2 \log (x)+((27-63 x) \log (4)+(-9+21 x) \log (4) \log (x)) \log (-3+\log (x))}{-9 x^2+3 x^2 \log (x)+(-27 x \log (4)+9 x \log (4) \log (x)) \log (-3+\log (x))} \, dx=\frac {7 x}{3} - \log {\left (x \right )} + \log {\left (\frac {x}{6 \log {\left (2 \right )}} + \log {\left (\log {\left (x \right )} - 3 \right )} \right )} \]
integrate(((2*(21*x-9)*ln(2)*ln(x)+2*(-63*x+27)*ln(2))*ln(ln(x)-3)+7*x**2* ln(x)+18*ln(2)-21*x**2)/((18*x*ln(2)*ln(x)-54*x*ln(2))*ln(ln(x)-3)+3*x**2* ln(x)-9*x**2),x)
Time = 0.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {-21 x^2+9 \log (4)+7 x^2 \log (x)+((27-63 x) \log (4)+(-9+21 x) \log (4) \log (x)) \log (-3+\log (x))}{-9 x^2+3 x^2 \log (x)+(-27 x \log (4)+9 x \log (4) \log (x)) \log (-3+\log (x))} \, dx=\frac {7}{3} \, x - \log \left (x\right ) + \log \left (\frac {6 \, \log \left (2\right ) \log \left (\log \left (x\right ) - 3\right ) + x}{6 \, \log \left (2\right )}\right ) \]
integrate(((2*(21*x-9)*log(2)*log(x)+2*(-63*x+27)*log(2))*log(log(x)-3)+7* x^2*log(x)+18*log(2)-21*x^2)/((18*x*log(2)*log(x)-54*x*log(2))*log(log(x)- 3)+3*x^2*log(x)-9*x^2),x, algorithm=\
Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {-21 x^2+9 \log (4)+7 x^2 \log (x)+((27-63 x) \log (4)+(-9+21 x) \log (4) \log (x)) \log (-3+\log (x))}{-9 x^2+3 x^2 \log (x)+(-27 x \log (4)+9 x \log (4) \log (x)) \log (-3+\log (x))} \, dx=\frac {7}{3} \, x + \log \left (6 \, \log \left (2\right ) \log \left (\log \left (x\right ) - 3\right ) + x\right ) - \log \left (x\right ) \]
integrate(((2*(21*x-9)*log(2)*log(x)+2*(-63*x+27)*log(2))*log(log(x)-3)+7* x^2*log(x)+18*log(2)-21*x^2)/((18*x*log(2)*log(x)-54*x*log(2))*log(log(x)- 3)+3*x^2*log(x)-9*x^2),x, algorithm=\
Timed out. \[ \int \frac {-21 x^2+9 \log (4)+7 x^2 \log (x)+((27-63 x) \log (4)+(-9+21 x) \log (4) \log (x)) \log (-3+\log (x))}{-9 x^2+3 x^2 \log (x)+(-27 x \log (4)+9 x \log (4) \log (x)) \log (-3+\log (x))} \, dx=\int -\frac {18\,\ln \left (2\right )+7\,x^2\,\ln \left (x\right )-\ln \left (\ln \left (x\right )-3\right )\,\left (2\,\ln \left (2\right )\,\left (63\,x-27\right )-2\,\ln \left (2\right )\,\ln \left (x\right )\,\left (21\,x-9\right )\right )-21\,x^2}{9\,x^2-3\,x^2\,\ln \left (x\right )+\ln \left (\ln \left (x\right )-3\right )\,\left (54\,x\,\ln \left (2\right )-18\,x\,\ln \left (2\right )\,\ln \left (x\right )\right )} \,d x \]
int(-(18*log(2) + 7*x^2*log(x) - log(log(x) - 3)*(2*log(2)*(63*x - 27) - 2 *log(2)*log(x)*(21*x - 9)) - 21*x^2)/(9*x^2 - 3*x^2*log(x) + log(log(x) - 3)*(54*x*log(2) - 18*x*log(2)*log(x))),x)