Integrand size = 45, antiderivative size = 29 \[ \int \frac {3-x-x \log (x)+\left (3+x-5 x^2+\left (-2+x-10 x^2\right ) \log (x)-\log \left (5 x^2\right )\right ) \log (\log (4))}{x} \, dx=\log (x) \left (3-x-\left (-3-x+5 x^2+\log \left (5 x^2\right )\right ) \log (\log (4))\right ) \]
Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.00 \[ \int \frac {3-x-x \log (x)+\left (3+x-5 x^2+\left (-2+x-10 x^2\right ) \log (x)-\log \left (5 x^2\right )\right ) \log (\log (4))}{x} \, dx=3 \log (x)-x \log (x)+3 \log (x) \log (\log (4))+x \log (x) \log (\log (4))-5 x^2 \log (x) \log (\log (4))-\log ^2(x) \log (\log (4))-\frac {1}{4} \log ^2\left (5 x^2\right ) \log (\log (4)) \]
Integrate[(3 - x - x*Log[x] + (3 + x - 5*x^2 + (-2 + x - 10*x^2)*Log[x] - Log[5*x^2])*Log[Log[4]])/x,x]
3*Log[x] - x*Log[x] + 3*Log[x]*Log[Log[4]] + x*Log[x]*Log[Log[4]] - 5*x^2* Log[x]*Log[Log[4]] - Log[x]^2*Log[Log[4]] - (Log[5*x^2]^2*Log[Log[4]])/4
Time = 0.31 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.93, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {2010, 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.
\(\displaystyle \int \frac {\log (\log (4)) \left (-5 x^2+\left (-10 x^2+x-2\right ) \log (x)-\log \left (5 x^2\right )+x+3\right )-x+x (-\log (x))+3}{x} \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (\frac {-10 x^2 \log (\log (4)) \log (x)-5 x^2 \log (\log (4))-x (1-\log (\log (4))) \log (x)-x (1-\log (\log (4)))-2 \log (\log (4)) \log (x)+3 (1+\log (\log (4)))}{x}-\frac {\log (\log (4)) \log \left (5 x^2\right )}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{4} \log (\log (4)) \log ^2\left (5 x^2\right )-5 x^2 \log (\log (4)) \log (x)-\log (\log (4)) \log ^2(x)-x (1-\log (\log (4))) \log (x)+3 (1+\log (\log (4))) \log (x)\) |
Int[(3 - x - x*Log[x] + (3 + x - 5*x^2 + (-2 + x - 10*x^2)*Log[x] - Log[5* x^2])*Log[Log[4]])/x,x]
-(x*Log[x]*(1 - Log[Log[4]])) - 5*x^2*Log[x]*Log[Log[4]] - Log[x]^2*Log[Lo g[4]] - (Log[5*x^2]^2*Log[Log[4]])/4 + 3*Log[x]*(1 + Log[Log[4]])
3.9.24.3.1 Defintions of rubi rules used
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Time = 1.67 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.93
method | result | size |
parallelrisch | \(-5 \ln \left (x \right ) \ln \left (2 \ln \left (2\right )\right ) x^{2}+\ln \left (x \right ) \ln \left (2 \ln \left (2\right )\right ) x -\ln \left (x \right ) \ln \left (5 x^{2}\right ) \ln \left (2 \ln \left (2\right )\right )+3 \ln \left (x \right ) \ln \left (2 \ln \left (2\right )\right )-x \ln \left (x \right )+3 \ln \left (x \right )\) | \(56\) |
norman | \(\left (\frac {3}{2}+\frac {3 \ln \left (2\right )}{2}+\frac {3 \ln \left (\ln \left (2\right )\right )}{2}\right ) \ln \left (5 x^{2}\right )+\left (-5 \ln \left (2\right )-5 \ln \left (\ln \left (2\right )\right )\right ) x^{2} \ln \left (x \right )+\left (-\ln \left (2\right )-\ln \left (\ln \left (2\right )\right )\right ) \ln \left (x \right ) \ln \left (5 x^{2}\right )+\left (\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )-1\right ) x \ln \left (x \right )\) | \(66\) |
default | \(-10 \ln \left (\ln \left (2\right )\right ) \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )+\ln \left (\ln \left (2\right )\right ) \left (x \ln \left (x \right )-x \right )-\ln \left (x \right )^{2} \ln \left (\ln \left (2\right )\right )-x \ln \left (x \right )+\ln \left (2\right ) \left (3 \ln \left (x \right )-\ln \left (5\right ) \ln \left (x \right )-\frac {\ln \left (x^{2}\right )^{2}}{4}-5 x^{2} \ln \left (x \right )+x \ln \left (x \right )-\ln \left (x \right )^{2}\right )+3 \ln \left (x \right )-\ln \left (5\right ) \ln \left (\ln \left (2\right )\right ) \ln \left (x \right )-\frac {\ln \left (\ln \left (2\right )\right ) \ln \left (x^{2}\right )^{2}}{4}-\ln \left (\ln \left (2\right )\right ) \left (\frac {5 x^{2}}{2}-x -3 \ln \left (x \right )\right )\) | \(127\) |
parts | \(-\frac {5 x^{2} \ln \left (2 \ln \left (2\right )\right )}{2}+x \ln \left (2 \ln \left (2\right )\right )-\left (-3-3 \ln \left (2 \ln \left (2\right )\right )\right ) \ln \left (x \right )-10 \ln \left (2\right ) \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )-10 \ln \left (\ln \left (2\right )\right ) \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )+\ln \left (2\right ) \left (x \ln \left (x \right )-x \right )+\ln \left (\ln \left (2\right )\right ) \left (x \ln \left (x \right )-x \right )-\ln \left (2\right ) \ln \left (x \right )^{2}-\ln \left (x \right )^{2} \ln \left (\ln \left (2\right )\right )-x \ln \left (x \right )-\frac {\ln \left (2 \ln \left (2\right )\right ) \ln \left (5 x^{2}\right )^{2}}{4}\) | \(127\) |
risch | \(-2 \ln \left (2\right ) \ln \left (x \right )^{2}-2 \ln \left (x \right )^{2} \ln \left (\ln \left (2\right )\right )+\left (-5 x^{2} \ln \left (2\right )-5 x^{2} \ln \left (\ln \left (2\right )\right )+x \ln \left (2\right )+x \ln \left (\ln \left (2\right )\right )-x \right ) \ln \left (x \right )+\frac {i \left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right ) \ln \left (2\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2} \ln \left (2\right )+\pi \operatorname {csgn}\left (i x^{2}\right )^{3} \ln \left (2\right )+\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right ) \ln \left (\ln \left (2\right )\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2} \ln \left (\ln \left (2\right )\right )+\pi \operatorname {csgn}\left (i x^{2}\right )^{3} \ln \left (\ln \left (2\right )\right )-6 i \ln \left (\ln \left (2\right )\right )+2 i \ln \left (2\right ) \ln \left (5\right )-6 i \ln \left (2\right )+2 i \ln \left (5\right ) \ln \left (\ln \left (2\right )\right )-6 i\right ) \ln \left (x \right )}{2}\) | \(187\) |
int(((-ln(5*x^2)+(-10*x^2+x-2)*ln(x)-5*x^2+x+3)*ln(2*ln(2))-x*ln(x)+3-x)/x ,x,method=_RETURNVERBOSE)
-5*ln(x)*ln(2*ln(2))*x^2+ln(x)*ln(2*ln(2))*x-ln(x)*ln(5*x^2)*ln(2*ln(2))+3 *ln(x)*ln(2*ln(2))-x*ln(x)+3*ln(x)
Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {3-x-x \log (x)+\left (3+x-5 x^2+\left (-2+x-10 x^2\right ) \log (x)-\log \left (5 x^2\right )\right ) \log (\log (4))}{x} \, dx=-{\left (x - 3\right )} \log \left (x\right ) - {\left ({\left (5 \, x^{2} - x + \log \left (5\right ) - 3\right )} \log \left (x\right ) + 2 \, \log \left (x\right )^{2}\right )} \log \left (2 \, \log \left (2\right )\right ) \]
integrate(((-log(5*x^2)+(-10*x^2+x-2)*log(x)-5*x^2+x+3)*log(2*log(2))-x*lo g(x)+3-x)/x,x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (27) = 54\).
Time = 0.14 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.93 \[ \int \frac {3-x-x \log (x)+\left (3+x-5 x^2+\left (-2+x-10 x^2\right ) \log (x)-\log \left (5 x^2\right )\right ) \log (\log (4))}{x} \, dx=\left (- 5 x^{2} \log {\left (2 \right )} - 5 x^{2} \log {\left (\log {\left (2 \right )} \right )} - x + x \log {\left (\log {\left (2 \right )} \right )} + x \log {\left (2 \right )}\right ) \log {\left (x \right )} + \left (- 2 \log {\left (2 \right )} - 2 \log {\left (\log {\left (2 \right )} \right )}\right ) \log {\left (x \right )}^{2} + \left (- \log {\left (2 \right )} \log {\left (5 \right )} + 3 \log {\left (\log {\left (2 \right )} \right )} - \log {\left (5 \right )} \log {\left (\log {\left (2 \right )} \right )} + 3 \log {\left (2 \right )} + 3\right ) \log {\left (x \right )} \]
(-5*x**2*log(2) - 5*x**2*log(log(2)) - x + x*log(log(2)) + x*log(2))*log(x ) + (-2*log(2) - 2*log(log(2)))*log(x)**2 + (-log(2)*log(5) + 3*log(log(2) ) - log(5)*log(log(2)) + 3*log(2) + 3)*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (29) = 58\).
Time = 0.19 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.31 \[ \int \frac {3-x-x \log (x)+\left (3+x-5 x^2+\left (-2+x-10 x^2\right ) \log (x)-\log \left (5 x^2\right )\right ) \log (\log (4))}{x} \, dx=-\frac {5}{2} \, x^{2} \log \left (2 \, \log \left (2\right )\right ) - \frac {1}{4} \, \log \left (5 \, x^{2}\right )^{2} \log \left (2 \, \log \left (2\right )\right ) - \log \left (x\right )^{2} \log \left (2 \, \log \left (2\right )\right ) - x \log \left (x\right ) - \frac {5}{2} \, {\left (2 \, x^{2} \log \left (x\right ) - x^{2}\right )} \log \left (2 \, \log \left (2\right )\right ) + {\left (x \log \left (x\right ) - x\right )} \log \left (2 \, \log \left (2\right )\right ) + x \log \left (2 \, \log \left (2\right )\right ) + 3 \, \log \left (x\right ) \log \left (2 \, \log \left (2\right )\right ) + 3 \, \log \left (x\right ) \]
integrate(((-log(5*x^2)+(-10*x^2+x-2)*log(x)-5*x^2+x+3)*log(2*log(2))-x*lo g(x)+3-x)/x,x, algorithm=\
-5/2*x^2*log(2*log(2)) - 1/4*log(5*x^2)^2*log(2*log(2)) - log(x)^2*log(2*l og(2)) - x*log(x) - 5/2*(2*x^2*log(x) - x^2)*log(2*log(2)) + (x*log(x) - x )*log(2*log(2)) + x*log(2*log(2)) + 3*log(x)*log(2*log(2)) + 3*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (29) = 58\).
Time = 0.25 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.24 \[ \int \frac {3-x-x \log (x)+\left (3+x-5 x^2+\left (-2+x-10 x^2\right ) \log (x)-\log \left (5 x^2\right )\right ) \log (\log (4))}{x} \, dx=-2 \, {\left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} \log \left (x\right )^{2} - {\left (5 \, x^{2} {\left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} - x {\left (\log \left (2\right ) + \log \left (\log \left (2\right )\right ) - 1\right )}\right )} \log \left (x\right ) - {\left (\log \left (5\right ) \log \left (2\right ) + \log \left (5\right ) \log \left (\log \left (2\right )\right ) - 3 \, \log \left (2\right ) - 3 \, \log \left (\log \left (2\right )\right ) - 3\right )} \log \left (x\right ) \]
integrate(((-log(5*x^2)+(-10*x^2+x-2)*log(x)-5*x^2+x+3)*log(2*log(2))-x*lo g(x)+3-x)/x,x, algorithm=\
-2*(log(2) + log(log(2)))*log(x)^2 - (5*x^2*(log(2) + log(log(2))) - x*(lo g(2) + log(log(2)) - 1))*log(x) - (log(5)*log(2) + log(5)*log(log(2)) - 3* log(2) - 3*log(log(2)) - 3)*log(x)
Time = 9.62 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int \frac {3-x-x \log (x)+\left (3+x-5 x^2+\left (-2+x-10 x^2\right ) \log (x)-\log \left (5 x^2\right )\right ) \log (\log (4))}{x} \, dx=-\ln \left (x\right )\,\left (x-3\,\ln \left (2\,\ln \left (2\right )\right )+\ln \left (2\,\ln \left (2\right )\right )\,\ln \left (5\,x^2\right )+5\,x^2\,\ln \left (2\,\ln \left (2\right )\right )-x\,\ln \left (\ln \left (4\right )\right )-3\right ) \]
int(-(x + log(2*log(2))*(log(5*x^2) - x + log(x)*(10*x^2 - x + 2) + 5*x^2 - 3) + x*log(x) - 3)/x,x)