Integrand size = 106, antiderivative size = 19 \[ \int \frac {-60+169 x^2+\left (-5+26 x^2\right ) \log ^2(4)+x^2 \log ^4(4)+\left (5-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)}{169 x^2+26 x^2 \log ^2(4)+x^2 \log ^4(4)+\left (-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)} \, dx=x+\frac {5}{x \left (13+\log ^2(4)-\log (x)\right )} \]
Time = 0.36 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-60+169 x^2+\left (-5+26 x^2\right ) \log ^2(4)+x^2 \log ^4(4)+\left (5-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)}{169 x^2+26 x^2 \log ^2(4)+x^2 \log ^4(4)+\left (-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)} \, dx=x-\frac {5}{x \left (-13-\log ^2(4)+\log (x)\right )} \]
Integrate[(-60 + 169*x^2 + (-5 + 26*x^2)*Log[4]^2 + x^2*Log[4]^4 + (5 - 26 *x^2 - 2*x^2*Log[4]^2)*Log[x] + x^2*Log[x]^2)/(169*x^2 + 26*x^2*Log[4]^2 + x^2*Log[4]^4 + (-26*x^2 - 2*x^2*Log[4]^2)*Log[x] + x^2*Log[x]^2),x]
Time = 0.74 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {6, 6, 6, 7292, 7293, 2009}
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.
\(\displaystyle \int \frac {169 x^2+x^2 \log ^4(4)+x^2 \log ^2(x)+\left (-26 x^2-2 x^2 \log ^2(4)+5\right ) \log (x)+\left (26 x^2-5\right ) \log ^2(4)-60}{169 x^2+x^2 \log ^4(4)+x^2 \log ^2(x)+26 x^2 \log ^2(4)+\left (-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {169 x^2+x^2 \log ^4(4)+x^2 \log ^2(x)+\left (-26 x^2-2 x^2 \log ^2(4)+5\right ) \log (x)+\left (26 x^2-5\right ) \log ^2(4)-60}{x^2 \log ^4(4)+x^2 \log ^2(x)+x^2 \left (169+26 \log ^2(4)\right )+\left (-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)}dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {169 x^2+x^2 \log ^4(4)+x^2 \log ^2(x)+\left (-26 x^2-2 x^2 \log ^2(4)+5\right ) \log (x)+\left (26 x^2-5\right ) \log ^2(4)-60}{x^2 \log ^2(x)+\left (-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \left (169+\log ^4(4)+26 \log ^2(4)\right )}dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {x^2 \left (169+\log ^4(4)\right )+x^2 \log ^2(x)+\left (-26 x^2-2 x^2 \log ^2(4)+5\right ) \log (x)+\left (26 x^2-5\right ) \log ^2(4)-60}{x^2 \log ^2(x)+\left (-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \left (169+\log ^4(4)+26 \log ^2(4)\right )}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {x^2 \left (169+\log ^4(4)\right )+x^2 \log ^2(x)+\left (-26 x^2-2 x^2 \log ^2(4)+5\right ) \log (x)+\left (26 x^2-5\right ) \log ^2(4)-60}{x^2 \left (13 \left (1+\frac {\log ^2(4)}{13}\right )-\log (x)\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {5}{x^2 \left (13 \left (1+\frac {\log ^2(4)}{13}\right )-\log (x)\right )^2}+\frac {5}{x^2 \left (\log (x)-13 \left (1+\frac {\log ^2(4)}{13}\right )\right )}+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x+\frac {5}{x \left (-\log (x)+13+\log ^2(4)\right )}\) |
Int[(-60 + 169*x^2 + (-5 + 26*x^2)*Log[4]^2 + x^2*Log[4]^4 + (5 - 26*x^2 - 2*x^2*Log[4]^2)*Log[x] + x^2*Log[x]^2)/(169*x^2 + 26*x^2*Log[4]^2 + x^2*L og[4]^4 + (-26*x^2 - 2*x^2*Log[4]^2)*Log[x] + x^2*Log[x]^2),x]
3.20.52.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Time = 0.52 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16
method | result | size |
risch | \(x +\frac {5}{\left (13-\ln \left (x \right )+4 \ln \left (2\right )^{2}\right ) x}\) | \(22\) |
norman | \(\frac {5+\left (4 \ln \left (2\right )^{2}+13\right ) x^{2}-x^{2} \ln \left (x \right )}{x \left (13-\ln \left (x \right )+4 \ln \left (2\right )^{2}\right )}\) | \(40\) |
parallelrisch | \(-\frac {-5-4 x^{2} \ln \left (2\right )^{2}+x^{2} \ln \left (x \right )-13 x^{2}}{x \left (13-\ln \left (x \right )+4 \ln \left (2\right )^{2}\right )}\) | \(42\) |
default | \(-\frac {20 \ln \left (2\right )^{2}}{x \left (13-\ln \left (x \right )+4 \ln \left (2\right )^{2}\right )}+\frac {20 \ln \left (2\right )^{2}+65}{x \left (13-\ln \left (x \right )+4 \ln \left (2\right )^{2}\right )}-\frac {60}{\left (13-\ln \left (x \right )+4 \ln \left (2\right )^{2}\right ) x}+x\) | \(72\) |
int((x^2*ln(x)^2+(-8*x^2*ln(2)^2-26*x^2+5)*ln(x)+16*x^2*ln(2)^4+4*(26*x^2- 5)*ln(2)^2+169*x^2-60)/(x^2*ln(x)^2+(-8*x^2*ln(2)^2-26*x^2)*ln(x)+16*x^2*l n(2)^4+104*x^2*ln(2)^2+169*x^2),x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.21 \[ \int \frac {-60+169 x^2+\left (-5+26 x^2\right ) \log ^2(4)+x^2 \log ^4(4)+\left (5-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)}{169 x^2+26 x^2 \log ^2(4)+x^2 \log ^4(4)+\left (-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)} \, dx=\frac {4 \, x^{2} \log \left (2\right )^{2} - x^{2} \log \left (x\right ) + 13 \, x^{2} + 5}{4 \, x \log \left (2\right )^{2} - x \log \left (x\right ) + 13 \, x} \]
integrate((x^2*log(x)^2+(-8*x^2*log(2)^2-26*x^2+5)*log(x)+16*x^2*log(2)^4+ 4*(26*x^2-5)*log(2)^2+169*x^2-60)/(x^2*log(x)^2+(-8*x^2*log(2)^2-26*x^2)*l og(x)+16*x^2*log(2)^4+104*x^2*log(2)^2+169*x^2),x, algorithm=\
Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-60+169 x^2+\left (-5+26 x^2\right ) \log ^2(4)+x^2 \log ^4(4)+\left (5-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)}{169 x^2+26 x^2 \log ^2(4)+x^2 \log ^4(4)+\left (-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)} \, dx=x - \frac {5}{x \log {\left (x \right )} - 13 x - 4 x \log {\left (2 \right )}^{2}} \]
integrate((x**2*ln(x)**2+(-8*x**2*ln(2)**2-26*x**2+5)*ln(x)+16*x**2*ln(2)* *4+4*(26*x**2-5)*ln(2)**2+169*x**2-60)/(x**2*ln(x)**2+(-8*x**2*ln(2)**2-26 *x**2)*ln(x)+16*x**2*ln(2)**4+104*x**2*ln(2)**2+169*x**2),x)
Time = 0.31 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.11 \[ \int \frac {-60+169 x^2+\left (-5+26 x^2\right ) \log ^2(4)+x^2 \log ^4(4)+\left (5-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)}{169 x^2+26 x^2 \log ^2(4)+x^2 \log ^4(4)+\left (-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)} \, dx=\frac {{\left (4 \, \log \left (2\right )^{2} + 13\right )} x^{2} - x^{2} \log \left (x\right ) + 5}{{\left (4 \, \log \left (2\right )^{2} + 13\right )} x - x \log \left (x\right )} \]
integrate((x^2*log(x)^2+(-8*x^2*log(2)^2-26*x^2+5)*log(x)+16*x^2*log(2)^4+ 4*(26*x^2-5)*log(2)^2+169*x^2-60)/(x^2*log(x)^2+(-8*x^2*log(2)^2-26*x^2)*l og(x)+16*x^2*log(2)^4+104*x^2*log(2)^2+169*x^2),x, algorithm=\
Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {-60+169 x^2+\left (-5+26 x^2\right ) \log ^2(4)+x^2 \log ^4(4)+\left (5-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)}{169 x^2+26 x^2 \log ^2(4)+x^2 \log ^4(4)+\left (-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)} \, dx=x + \frac {5}{4 \, x \log \left (2\right )^{2} - x \log \left (x\right ) + 13 \, x} \]
integrate((x^2*log(x)^2+(-8*x^2*log(2)^2-26*x^2+5)*log(x)+16*x^2*log(2)^4+ 4*(26*x^2-5)*log(2)^2+169*x^2-60)/(x^2*log(x)^2+(-8*x^2*log(2)^2-26*x^2)*l og(x)+16*x^2*log(2)^4+104*x^2*log(2)^2+169*x^2),x, algorithm=\
Time = 11.14 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {-60+169 x^2+\left (-5+26 x^2\right ) \log ^2(4)+x^2 \log ^4(4)+\left (5-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)}{169 x^2+26 x^2 \log ^2(4)+x^2 \log ^4(4)+\left (-26 x^2-2 x^2 \log ^2(4)\right ) \log (x)+x^2 \log ^2(x)} \, dx=x+\frac {5}{x\,\left (4\,{\ln \left (2\right )}^2-\ln \left (x\right )+13\right )} \]