Integrand size = 159, antiderivative size = 26 \[ \int \frac {-2 x^4-2 x^4 \log (x)+\left (5 x^2+x^4+\left (5 x^2+x^4\right ) \log (x)\right ) \log \left (5+x^2\right )+\left (2 x^2+2 x^2 \log (x)+\left (-5-x^2+\left (-5-x^2\right ) \log (x)\right ) \log \left (5+x^2\right )\right ) \log (1+\log (x))+\left (-5+9 x^2+2 x^4+\left (10 x^2+2 x^4\right ) \log (x)\right ) \log \left (5+x^2\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{\left (5 x+x^3+\left (5 x+x^3\right ) \log (x)\right ) \log \left (5+x^2\right )} \, dx=\left (x^2-\log (1+\log (x))\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right ) \]
Time = 0.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {-2 x^4-2 x^4 \log (x)+\left (5 x^2+x^4+\left (5 x^2+x^4\right ) \log (x)\right ) \log \left (5+x^2\right )+\left (2 x^2+2 x^2 \log (x)+\left (-5-x^2+\left (-5-x^2\right ) \log (x)\right ) \log \left (5+x^2\right )\right ) \log (1+\log (x))+\left (-5+9 x^2+2 x^4+\left (10 x^2+2 x^4\right ) \log (x)\right ) \log \left (5+x^2\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{\left (5 x+x^3+\left (5 x+x^3\right ) \log (x)\right ) \log \left (5+x^2\right )} \, dx=\left (x^2-\log (1+\log (x))\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right ) \]
Integrate[(-2*x^4 - 2*x^4*Log[x] + (5*x^2 + x^4 + (5*x^2 + x^4)*Log[x])*Lo g[5 + x^2] + (2*x^2 + 2*x^2*Log[x] + (-5 - x^2 + (-5 - x^2)*Log[x])*Log[5 + x^2])*Log[1 + Log[x]] + (-5 + 9*x^2 + 2*x^4 + (10*x^2 + 2*x^4)*Log[x])*L og[5 + x^2]*Log[x/(2*Log[5 + x^2])])/((5*x + x^3 + (5*x + x^3)*Log[x])*Log [5 + x^2]),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.
\(\displaystyle \int \frac {-2 x^4-2 x^4 \log (x)+\left (2 x^2+2 x^2 \log (x)+\left (-x^2+\left (-x^2-5\right ) \log (x)-5\right ) \log \left (x^2+5\right )\right ) \log (\log (x)+1)+\left (x^4+5 x^2+\left (x^4+5 x^2\right ) \log (x)\right ) \log \left (x^2+5\right )+\left (2 x^4+9 x^2+\left (2 x^4+10 x^2\right ) \log (x)-5\right ) \log \left (x^2+5\right ) \log \left (\frac {x}{2 \log \left (x^2+5\right )}\right )}{\left (x^3+\left (x^3+5 x\right ) \log (x)+5 x\right ) \log \left (x^2+5\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-2 x^4-2 x^4 \log (x)+\left (2 x^2+2 x^2 \log (x)+\left (-x^2+\left (-x^2-5\right ) \log (x)-5\right ) \log \left (x^2+5\right )\right ) \log (\log (x)+1)+\left (x^4+5 x^2+\left (x^4+5 x^2\right ) \log (x)\right ) \log \left (x^2+5\right )+\left (2 x^4+9 x^2+\left (2 x^4+10 x^2\right ) \log (x)-5\right ) \log \left (x^2+5\right ) \log \left (\frac {x}{2 \log \left (x^2+5\right )}\right )}{x \left (x^2+5\right ) (\log (x)+1) \log \left (x^2+5\right )}dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {\left (-2 x^2+x^2 \log \left (x^2+5\right )+5 \log \left (x^2+5\right )\right ) \left (x^2-\log (\log (x)+1)\right )}{x \left (x^2+5\right ) \log \left (x^2+5\right )}+\frac {\left (2 x^2+2 x^2 \log (x)-1\right ) \log \left (\frac {x}{2 \log \left (x^2+5\right )}\right )}{x (\log (x)+1)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\int \frac {\log (\log (x)+1)}{\left (i \sqrt {5}-x\right ) \log \left (x^2+5\right )}dx+\int \frac {\log (\log (x)+1)}{\left (x+i \sqrt {5}\right ) \log \left (x^2+5\right )}dx-\int \frac {\log \left (\frac {x}{2 \log \left (x^2+5\right )}\right )}{x (\log (x)+1)}dx+2 \int \frac {x \log \left (\frac {x}{2 \log \left (x^2+5\right )}\right )}{\log (x)+1}dx+2 \int \frac {x \log (x) \log \left (\frac {x}{2 \log \left (x^2+5\right )}\right )}{\log (x)+1}dx-\operatorname {LogIntegral}\left (x^2+5\right )+\frac {x^2}{2}+5 \log \left (\log \left (x^2+5\right )\right )+\log (x)-(\log (x)+1) \log (\log (x)+1)\) |
Int[(-2*x^4 - 2*x^4*Log[x] + (5*x^2 + x^4 + (5*x^2 + x^4)*Log[x])*Log[5 + x^2] + (2*x^2 + 2*x^2*Log[x] + (-5 - x^2 + (-5 - x^2)*Log[x])*Log[5 + x^2] )*Log[1 + Log[x]] + (-5 + 9*x^2 + 2*x^4 + (10*x^2 + 2*x^4)*Log[x])*Log[5 + x^2]*Log[x/(2*Log[5 + x^2])])/((5*x + x^3 + (5*x + x^3)*Log[x])*Log[5 + x ^2]),x]
3.12.14.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 299, normalized size of antiderivative = 11.50
\[\left (-x^{2}+\ln \left (\ln \left (x \right )+1\right )\right ) \ln \left (\ln \left (x^{2}+5\right )\right )-\ln \left (x \right ) \ln \left (\ln \left (x \right )+1\right )+x^{2} \ln \left (x \right )+\frac {i \pi \,x^{2} \operatorname {csgn}\left (\frac {i}{\ln \left (x^{2}+5\right )}\right ) {\operatorname {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )}^{2}}{2}+\frac {i \pi \ln \left (\ln \left (x \right )+1\right ) {\operatorname {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )}^{3}}{2}-\frac {i \pi \ln \left (\ln \left (x \right )+1\right ) \operatorname {csgn}\left (i x \right ) {\operatorname {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )}^{2}}{2}+\frac {i \pi \,x^{2} \operatorname {csgn}\left (i x \right ) {\operatorname {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )}^{2}}{2}-\frac {i \pi \,x^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x^{2}+5\right )}\right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )}{2}-\frac {i \pi \ln \left (\ln \left (x \right )+1\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x^{2}+5\right )}\right ) {\operatorname {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )}^{2}}{2}-\frac {i \pi \,x^{2} {\operatorname {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )}^{3}}{2}+\frac {i \pi \ln \left (\ln \left (x \right )+1\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x^{2}+5\right )}\right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )}{2}-x^{2} \ln \left (2\right )+\ln \left (2\right ) \ln \left (\ln \left (x \right )+1\right )\]
int(((((-x^2-5)*ln(x)-x^2-5)*ln(x^2+5)+2*x^2*ln(x)+2*x^2)*ln(ln(x)+1)+((2* x^4+10*x^2)*ln(x)+2*x^4+9*x^2-5)*ln(x^2+5)*ln(1/2*x/ln(x^2+5))+((x^4+5*x^2 )*ln(x)+x^4+5*x^2)*ln(x^2+5)-2*x^4*ln(x)-2*x^4)/((x^3+5*x)*ln(x)+x^3+5*x)/ ln(x^2+5),x)
(-x^2+ln(ln(x)+1))*ln(ln(x^2+5))-ln(x)*ln(ln(x)+1)+x^2*ln(x)+1/2*I*Pi*x^2* csgn(I/ln(x^2+5))*csgn(I*x/ln(x^2+5))^2+1/2*I*Pi*ln(ln(x)+1)*csgn(I*x/ln(x ^2+5))^3-1/2*I*Pi*ln(ln(x)+1)*csgn(I*x)*csgn(I*x/ln(x^2+5))^2+1/2*I*Pi*x^2 *csgn(I*x)*csgn(I*x/ln(x^2+5))^2-1/2*I*Pi*x^2*csgn(I*x)*csgn(I/ln(x^2+5))* csgn(I*x/ln(x^2+5))-1/2*I*Pi*ln(ln(x)+1)*csgn(I/ln(x^2+5))*csgn(I*x/ln(x^2 +5))^2-1/2*I*Pi*x^2*csgn(I*x/ln(x^2+5))^3+1/2*I*Pi*ln(ln(x)+1)*csgn(I*x)*c sgn(I/ln(x^2+5))*csgn(I*x/ln(x^2+5))-x^2*ln(2)+ln(2)*ln(ln(x)+1)
Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {-2 x^4-2 x^4 \log (x)+\left (5 x^2+x^4+\left (5 x^2+x^4\right ) \log (x)\right ) \log \left (5+x^2\right )+\left (2 x^2+2 x^2 \log (x)+\left (-5-x^2+\left (-5-x^2\right ) \log (x)\right ) \log \left (5+x^2\right )\right ) \log (1+\log (x))+\left (-5+9 x^2+2 x^4+\left (10 x^2+2 x^4\right ) \log (x)\right ) \log \left (5+x^2\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{\left (5 x+x^3+\left (5 x+x^3\right ) \log (x)\right ) \log \left (5+x^2\right )} \, dx=x^{2} \log \left (\frac {x}{2 \, \log \left (x^{2} + 5\right )}\right ) - \log \left (\frac {x}{2 \, \log \left (x^{2} + 5\right )}\right ) \log \left (\log \left (x\right ) + 1\right ) \]
integrate(((((-x^2-5)*log(x)-x^2-5)*log(x^2+5)+2*x^2*log(x)+2*x^2)*log(log (x)+1)+((2*x^4+10*x^2)*log(x)+2*x^4+9*x^2-5)*log(x^2+5)*log(1/2*x/log(x^2+ 5))+((x^4+5*x^2)*log(x)+x^4+5*x^2)*log(x^2+5)-2*x^4*log(x)-2*x^4)/((x^3+5* x)*log(x)+x^3+5*x)/log(x^2+5),x, algorithm=\
Time = 1.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {-2 x^4-2 x^4 \log (x)+\left (5 x^2+x^4+\left (5 x^2+x^4\right ) \log (x)\right ) \log \left (5+x^2\right )+\left (2 x^2+2 x^2 \log (x)+\left (-5-x^2+\left (-5-x^2\right ) \log (x)\right ) \log \left (5+x^2\right )\right ) \log (1+\log (x))+\left (-5+9 x^2+2 x^4+\left (10 x^2+2 x^4\right ) \log (x)\right ) \log \left (5+x^2\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{\left (5 x+x^3+\left (5 x+x^3\right ) \log (x)\right ) \log \left (5+x^2\right )} \, dx=\left (x^{2} - \log {\left (\log {\left (x \right )} + 1 \right )}\right ) \log {\left (\frac {x}{2 \log {\left (x^{2} + 5 \right )}} \right )} \]
integrate(((((-x**2-5)*ln(x)-x**2-5)*ln(x**2+5)+2*x**2*ln(x)+2*x**2)*ln(ln (x)+1)+((2*x**4+10*x**2)*ln(x)+2*x**4+9*x**2-5)*ln(x**2+5)*ln(1/2*x/ln(x** 2+5))+((x**4+5*x**2)*ln(x)+x**4+5*x**2)*ln(x**2+5)-2*x**4*ln(x)-2*x**4)/(( x**3+5*x)*ln(x)+x**3+5*x)/ln(x**2+5),x)
Time = 0.31 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.81 \[ \int \frac {-2 x^4-2 x^4 \log (x)+\left (5 x^2+x^4+\left (5 x^2+x^4\right ) \log (x)\right ) \log \left (5+x^2\right )+\left (2 x^2+2 x^2 \log (x)+\left (-5-x^2+\left (-5-x^2\right ) \log (x)\right ) \log \left (5+x^2\right )\right ) \log (1+\log (x))+\left (-5+9 x^2+2 x^4+\left (10 x^2+2 x^4\right ) \log (x)\right ) \log \left (5+x^2\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{\left (5 x+x^3+\left (5 x+x^3\right ) \log (x)\right ) \log \left (5+x^2\right )} \, dx=-x^{2} \log \left (2\right ) + x^{2} \log \left (x\right ) + {\left (\log \left (2\right ) - \log \left (x\right )\right )} \log \left (\log \left (x\right ) + 1\right ) - {\left (x^{2} - \log \left (\log \left (x\right ) + 1\right )\right )} \log \left (\log \left (x^{2} + 5\right )\right ) \]
integrate(((((-x^2-5)*log(x)-x^2-5)*log(x^2+5)+2*x^2*log(x)+2*x^2)*log(log (x)+1)+((2*x^4+10*x^2)*log(x)+2*x^4+9*x^2-5)*log(x^2+5)*log(1/2*x/log(x^2+ 5))+((x^4+5*x^2)*log(x)+x^4+5*x^2)*log(x^2+5)-2*x^4*log(x)-2*x^4)/((x^3+5* x)*log(x)+x^3+5*x)/log(x^2+5),x, algorithm=\
-x^2*log(2) + x^2*log(x) + (log(2) - log(x))*log(log(x) + 1) - (x^2 - log( log(x) + 1))*log(log(x^2 + 5))
Time = 0.37 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.46 \[ \int \frac {-2 x^4-2 x^4 \log (x)+\left (5 x^2+x^4+\left (5 x^2+x^4\right ) \log (x)\right ) \log \left (5+x^2\right )+\left (2 x^2+2 x^2 \log (x)+\left (-5-x^2+\left (-5-x^2\right ) \log (x)\right ) \log \left (5+x^2\right )\right ) \log (1+\log (x))+\left (-5+9 x^2+2 x^4+\left (10 x^2+2 x^4\right ) \log (x)\right ) \log \left (5+x^2\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{\left (5 x+x^3+\left (5 x+x^3\right ) \log (x)\right ) \log \left (5+x^2\right )} \, dx=x^{2} \log \left (x\right ) - {\left (x^{2} - \log \left (\log \left (x\right ) + 1\right )\right )} \log \left (2 \, \log \left (x^{2} + 5\right )\right ) - \log \left (x\right ) \log \left (\log \left (x\right ) + 1\right ) \]
integrate(((((-x^2-5)*log(x)-x^2-5)*log(x^2+5)+2*x^2*log(x)+2*x^2)*log(log (x)+1)+((2*x^4+10*x^2)*log(x)+2*x^4+9*x^2-5)*log(x^2+5)*log(1/2*x/log(x^2+ 5))+((x^4+5*x^2)*log(x)+x^4+5*x^2)*log(x^2+5)-2*x^4*log(x)-2*x^4)/((x^3+5* x)*log(x)+x^3+5*x)/log(x^2+5),x, algorithm=\
Time = 10.59 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {-2 x^4-2 x^4 \log (x)+\left (5 x^2+x^4+\left (5 x^2+x^4\right ) \log (x)\right ) \log \left (5+x^2\right )+\left (2 x^2+2 x^2 \log (x)+\left (-5-x^2+\left (-5-x^2\right ) \log (x)\right ) \log \left (5+x^2\right )\right ) \log (1+\log (x))+\left (-5+9 x^2+2 x^4+\left (10 x^2+2 x^4\right ) \log (x)\right ) \log \left (5+x^2\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{\left (5 x+x^3+\left (5 x+x^3\right ) \log (x)\right ) \log \left (5+x^2\right )} \, dx=\left (\ln \left (\ln \left (x\right )+1\right )-x^2\right )\,\left (\ln \left (\ln \left (x^2+5\right )\right )+\ln \left (2\right )-\ln \left (x\right )\right ) \]
int((log(x^2 + 5)*(log(x)*(5*x^2 + x^4) + 5*x^2 + x^4) - 2*x^4*log(x) + lo g(log(x) + 1)*(2*x^2*log(x) - log(x^2 + 5)*(log(x)*(x^2 + 5) + x^2 + 5) + 2*x^2) - 2*x^4 + log(x/(2*log(x^2 + 5)))*log(x^2 + 5)*(log(x)*(10*x^2 + 2* x^4) + 9*x^2 + 2*x^4 - 5))/(log(x^2 + 5)*(5*x + log(x)*(5*x + x^3) + x^3)) ,x)