Integrand size = 174, antiderivative size = 20 \[ \int \frac {47+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )}{\left (10 x^2+47 x^3\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )+\left (-20 x-94 x^2\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log \left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log ^2\left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )} \, dx=\frac {1}{-x+\log \left (\log \left (\frac {2}{\log \left (5+\frac {47 x}{2}\right )}\right )\right )} \]
Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {47+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )}{\left (10 x^2+47 x^3\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )+\left (-20 x-94 x^2\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log \left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log ^2\left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )} \, dx=-\frac {1}{x-\log \left (\log \left (\frac {2}{\log \left (5+\frac {47 x}{2}\right )}\right )\right )} \]
Integrate[(47 + (10 + 47*x)*Log[(10 + 47*x)/2]*Log[2/Log[(10 + 47*x)/2]])/ ((10*x^2 + 47*x^3)*Log[(10 + 47*x)/2]*Log[2/Log[(10 + 47*x)/2]] + (-20*x - 94*x^2)*Log[(10 + 47*x)/2]*Log[2/Log[(10 + 47*x)/2]]*Log[Log[2/Log[(10 + 47*x)/2]]] + (10 + 47*x)*Log[(10 + 47*x)/2]*Log[2/Log[(10 + 47*x)/2]]*Log[ Log[2/Log[(10 + 47*x)/2]]]^2),x]
Time = 0.42 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {7239, 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.
| \(\displaystyle \int \frac {(47 x+10) \log \left (\frac {1}{2} (47 x+10)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (47 x+10)\right )}\right )+47}{\left (-94 x^2-20 x\right ) \log \left (\frac {1}{2} (47 x+10)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (47 x+10)\right )}\right ) \log \left (\log \left (\frac {2}{\log \left (\frac {1}{2} (47 x+10)\right )}\right )\right )+\left (47 x^3+10 x^2\right ) \log \left (\frac {1}{2} (47 x+10)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (47 x+10)\right )}\right )+(47 x+10) \log \left (\frac {1}{2} (47 x+10)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (47 x+10)\right )}\right ) \log ^2\left (\log \left (\frac {2}{\log \left (\frac {1}{2} (47 x+10)\right )}\right )\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\frac {47}{(47 x+10) \log \left (\frac {47 x}{2}+5\right ) \log \left (\frac {2}{\log \left (\frac {47 x}{2}+5\right )}\right )}+1}{\left (x-\log \left (\log \left (\frac {2}{\log \left (\frac {47 x}{2}+5\right )}\right )\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7237 |
| \(\displaystyle -\frac {1}{x-\log \left (\log \left (\frac {2}{\log \left (\frac {47 x}{2}+5\right )}\right )\right )}\) |
Int[(47 + (10 + 47*x)*Log[(10 + 47*x)/2]*Log[2/Log[(10 + 47*x)/2]])/((10*x ^2 + 47*x^3)*Log[(10 + 47*x)/2]*Log[2/Log[(10 + 47*x)/2]] + (-20*x - 94*x^ 2)*Log[(10 + 47*x)/2]*Log[2/Log[(10 + 47*x)/2]]*Log[Log[2/Log[(10 + 47*x)/ 2]]] + (10 + 47*x)*Log[(10 + 47*x)/2]*Log[2/Log[(10 + 47*x)/2]]*Log[Log[2/ Log[(10 + 47*x)/2]]]^2),x]
3.8.80.3.1 Defintions of rubi rules used
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 = 8.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10
| method | result | size |
| risch | \(-\frac {1}{x -\ln \left (\ln \left (2\right )-\ln \left (\ln \left (\frac {47 x}{2}+5\right )\right )\right )}\) | \(22\) |
| parallelrisch | \(\frac {-940+2209 x -2209 \ln \left (\ln \left (\frac {2}{\ln \left (\frac {47 x}{2}+5\right )}\right )\right )}{940 x -940 \ln \left (\ln \left (\frac {2}{\ln \left (\frac {47 x}{2}+5\right )}\right )\right )}\) | \(40\) |
| default | \(-\frac {47}{47 x -47 \ln \left (\ln \left (2\right )+i \pi -\ln \left (\ln \left (2\right )-\ln \left (47 x +10\right )\right )+i \pi \operatorname {csgn}\left (\frac {i}{\ln \left (2\right )-\ln \left (47 x +10\right )}\right )^{2} \left (\operatorname {csgn}\left (\frac {i}{\ln \left (2\right )-\ln \left (47 x +10\right )}\right )-1\right )\right )}\) | \(75\) |
| parts | \(\text {Expression too large to display}\) | \(695\) |
int(((47*x+10)*ln(47/2*x+5)*ln(2/ln(47/2*x+5))+47)/((47*x+10)*ln(47/2*x+5) *ln(2/ln(47/2*x+5))*ln(ln(2/ln(47/2*x+5)))^2+(-94*x^2-20*x)*ln(47/2*x+5)*l n(2/ln(47/2*x+5))*ln(ln(2/ln(47/2*x+5)))+(47*x^3+10*x^2)*ln(47/2*x+5)*ln(2 /ln(47/2*x+5))),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {47+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )}{\left (10 x^2+47 x^3\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )+\left (-20 x-94 x^2\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log \left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log ^2\left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )} \, dx=-\frac {1}{x - \log \left (\log \left (\frac {2}{\log \left (\frac {47}{2} \, x + 5\right )}\right )\right )} \]
integrate(((47*x+10)*log(47/2*x+5)*log(2/log(47/2*x+5))+47)/((47*x+10)*log (47/2*x+5)*log(2/log(47/2*x+5))*log(log(2/log(47/2*x+5)))^2+(-94*x^2-20*x) *log(47/2*x+5)*log(2/log(47/2*x+5))*log(log(2/log(47/2*x+5)))+(47*x^3+10*x ^2)*log(47/2*x+5)*log(2/log(47/2*x+5))),x, algorithm=\
Time = 0.16 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \frac {47+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )}{\left (10 x^2+47 x^3\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )+\left (-20 x-94 x^2\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log \left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log ^2\left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )} \, dx=\frac {1}{- x + \log {\left (\log {\left (\frac {2}{\log {\left (\frac {47 x}{2} + 5 \right )}} \right )} \right )}} \]
integrate(((47*x+10)*ln(47/2*x+5)*ln(2/ln(47/2*x+5))+47)/((47*x+10)*ln(47/ 2*x+5)*ln(2/ln(47/2*x+5))*ln(ln(2/ln(47/2*x+5)))**2+(-94*x**2-20*x)*ln(47/ 2*x+5)*ln(2/ln(47/2*x+5))*ln(ln(2/ln(47/2*x+5)))+(47*x**3+10*x**2)*ln(47/2 *x+5)*ln(2/ln(47/2*x+5))),x)
Time = 0.35 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int \frac {47+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )}{\left (10 x^2+47 x^3\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )+\left (-20 x-94 x^2\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log \left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log ^2\left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )} \, dx=-\frac {1}{x - \log \left (\log \left (2\right ) - \log \left (-\log \left (2\right ) + \log \left (47 \, x + 10\right )\right )\right )} \]
integrate(((47*x+10)*log(47/2*x+5)*log(2/log(47/2*x+5))+47)/((47*x+10)*log (47/2*x+5)*log(2/log(47/2*x+5))*log(log(2/log(47/2*x+5)))^2+(-94*x^2-20*x) *log(47/2*x+5)*log(2/log(47/2*x+5))*log(log(2/log(47/2*x+5)))+(47*x^3+10*x ^2)*log(47/2*x+5)*log(2/log(47/2*x+5))),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 590 vs. \(2 (20) = 40\).
Time = 2.73 (sec) , antiderivative size = 590, normalized size of antiderivative = 29.50 \[ \int \frac {47+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )}{\left (10 x^2+47 x^3\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )+\left (-20 x-94 x^2\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log \left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log ^2\left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )} \, dx=\text {Too large to display} \]
integrate(((47*x+10)*log(47/2*x+5)*log(2/log(47/2*x+5))+47)/((47*x+10)*log (47/2*x+5)*log(2/log(47/2*x+5))*log(log(2/log(47/2*x+5)))^2+(-94*x^2-20*x) *log(47/2*x+5)*log(2/log(47/2*x+5))*log(log(2/log(47/2*x+5)))+(47*x^3+10*x ^2)*log(47/2*x+5)*log(2/log(47/2*x+5))),x, algorithm=\
-(47*x*log(2)^2*log(47/2*x + 5) - 47*x*log(2)*log(47*x + 10)*log(47/2*x + 5) - 47*x*log(2)*log(47/2*x + 5)*log(log(47/2*x + 5)) + 47*x*log(47*x + 10 )*log(47/2*x + 5)*log(log(47/2*x + 5)) + 10*log(2)^2*log(47/2*x + 5) - 10* log(2)*log(47*x + 10)*log(47/2*x + 5) - 10*log(2)*log(47/2*x + 5)*log(log( 47/2*x + 5)) + 10*log(47*x + 10)*log(47/2*x + 5)*log(log(47/2*x + 5)) - 47 *log(47/2*x + 5))/(47*x^2*log(2)^2*log(47/2*x + 5) - 47*x^2*log(2)*log(47* x + 10)*log(47/2*x + 5) - 47*x*log(2)^2*log(47/2*x + 5)*log(log(2) - log(l og(47/2*x + 5))) + 47*x*log(2)*log(47*x + 10)*log(47/2*x + 5)*log(log(2) - log(log(47/2*x + 5))) - 47*x^2*log(2)*log(47/2*x + 5)*log(log(47/2*x + 5) ) + 47*x^2*log(47*x + 10)*log(47/2*x + 5)*log(log(47/2*x + 5)) + 47*x*log( 2)*log(47/2*x + 5)*log(log(2) - log(log(47/2*x + 5)))*log(log(47/2*x + 5)) - 47*x*log(47*x + 10)*log(47/2*x + 5)*log(log(2) - log(log(47/2*x + 5)))* log(log(47/2*x + 5)) + 10*x*log(2)^2*log(47/2*x + 5) - 10*x*log(2)*log(47* x + 10)*log(47/2*x + 5) - 10*log(2)^2*log(47/2*x + 5)*log(log(2) - log(log (47/2*x + 5))) + 10*log(2)*log(47*x + 10)*log(47/2*x + 5)*log(log(2) - log (log(47/2*x + 5))) - 10*x*log(2)*log(47/2*x + 5)*log(log(47/2*x + 5)) + 10 *x*log(47*x + 10)*log(47/2*x + 5)*log(log(47/2*x + 5)) + 10*log(2)*log(47/ 2*x + 5)*log(log(2) - log(log(47/2*x + 5)))*log(log(47/2*x + 5)) - 10*log( 47*x + 10)*log(47/2*x + 5)*log(log(2) - log(log(47/2*x + 5)))*log(log(47/2 *x + 5)) + 47*x*log(2) - 47*x*log(47*x + 10) - 47*log(2)*log(log(2) - l...
Time = 12.92 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {47+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )}{\left (10 x^2+47 x^3\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )+\left (-20 x-94 x^2\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log \left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log ^2\left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )} \, dx=-\frac {1}{x-\ln \left (\ln \left (\frac {2}{\ln \left (\frac {47\,x}{2}+5\right )}\right )\right )} \]
int((log((47*x)/2 + 5)*log(2/log((47*x)/2 + 5))*(47*x + 10) + 47)/(log((47 *x)/2 + 5)*log(2/log((47*x)/2 + 5))*(10*x^2 + 47*x^3) - log(log(2/log((47* x)/2 + 5)))*log((47*x)/2 + 5)*log(2/log((47*x)/2 + 5))*(20*x + 94*x^2) + l og(log(2/log((47*x)/2 + 5)))^2*log((47*x)/2 + 5)*log(2/log((47*x)/2 + 5))* (47*x + 10)),x)