Integrand size = 100, antiderivative size = 27 \[ \int \frac {(2+2 x) \log \left (\frac {5}{4 x^3}\right )+\left (2+4 x+2 x^2\right ) \log ^2\left (\frac {5}{4 x^3}\right )+\left (-3-3 x+x \log \left (\frac {5}{4 x^3}\right )\right ) \log \left (x^2\right )}{x+\left (2 x+2 x^2\right ) \log \left (\frac {5}{4 x^3}\right )+\left (x+2 x^2+x^3\right ) \log ^2\left (\frac {5}{4 x^3}\right )} \, dx=\frac {x \log \left (x^2\right )}{x+\frac {x}{(1+x) \log \left (\frac {5}{4 x^3}\right )}} \]
Time = 0.30 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {(2+2 x) \log \left (\frac {5}{4 x^3}\right )+\left (2+4 x+2 x^2\right ) \log ^2\left (\frac {5}{4 x^3}\right )+\left (-3-3 x+x \log \left (\frac {5}{4 x^3}\right )\right ) \log \left (x^2\right )}{x+\left (2 x+2 x^2\right ) \log \left (\frac {5}{4 x^3}\right )+\left (x+2 x^2+x^3\right ) \log ^2\left (\frac {5}{4 x^3}\right )} \, dx=2 \log (x)-\frac {\log \left (x^2\right )}{1+(1+x) \log \left (\frac {5}{4 x^3}\right )} \]
Integrate[((2 + 2*x)*Log[5/(4*x^3)] + (2 + 4*x + 2*x^2)*Log[5/(4*x^3)]^2 + (-3 - 3*x + x*Log[5/(4*x^3)])*Log[x^2])/(x + (2*x + 2*x^2)*Log[5/(4*x^3)] + (x + 2*x^2 + x^3)*Log[5/(4*x^3)]^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.
\(\displaystyle \int \frac {(2 x+2) \log \left (\frac {5}{4 x^3}\right )+\left (2 x^2+4 x+2\right ) \log ^2\left (\frac {5}{4 x^3}\right )+\left (x \log \left (\frac {5}{4 x^3}\right )-3 x-3\right ) \log \left (x^2\right )}{\left (x^3+2 x^2+x\right ) \log ^2\left (\frac {5}{4 x^3}\right )+\left (2 x^2+2 x\right ) \log \left (\frac {5}{4 x^3}\right )+x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {(2 x+2) \log \left (\frac {5}{4 x^3}\right )+\left (2 x^2+4 x+2\right ) \log ^2\left (\frac {5}{4 x^3}\right )+\left (x \log \left (\frac {5}{4 x^3}\right )-3 x-3\right ) \log \left (x^2\right )}{x \left (x \log \left (\frac {5}{4 x^3}\right )+\log \left (\frac {5}{4 x^3}\right )+1\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 (x+1) \log \left (\frac {5}{4 x^3}\right )}{x \left (x \log \left (\frac {5}{4 x^3}\right )+\log \left (\frac {5}{4 x^3}\right )+1\right )}+\frac {\left (x \log \left (\frac {5}{4 x^3}\right )-3 x-3\right ) \log \left (x^2\right )}{x \left (x \log \left (\frac {5}{4 x^3}\right )+\log \left (\frac {5}{4 x^3}\right )+1\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 \int \frac {1}{x \left (x \log \left (\frac {5}{4 x^3}\right )+\log \left (\frac {5}{4 x^3}\right )+1\right )}dx-3 \int \frac {\log \left (x^2\right )}{\left (x \log \left (\frac {5}{4 x^3}\right )+\log \left (\frac {5}{4 x^3}\right )+1\right )^2}dx-3 \int \frac {\log \left (x^2\right )}{x \left (x \log \left (\frac {5}{4 x^3}\right )+\log \left (\frac {5}{4 x^3}\right )+1\right )^2}dx+\int \frac {\log \left (\frac {5}{4 x^3}\right ) \log \left (x^2\right )}{\left (x \log \left (\frac {5}{4 x^3}\right )+\log \left (\frac {5}{4 x^3}\right )+1\right )^2}dx+2 \log (x)\) |
Int[((2 + 2*x)*Log[5/(4*x^3)] + (2 + 4*x + 2*x^2)*Log[5/(4*x^3)]^2 + (-3 - 3*x + x*Log[5/(4*x^3)])*Log[x^2])/(x + (2*x + 2*x^2)*Log[5/(4*x^3)] + (x + 2*x^2 + x^3)*Log[5/(4*x^3)]^2),x]
3.18.69.3.1 Defintions of rubi rules used
Time = 2.77 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74
method | result | size |
parallelrisch | \(\frac {8 \ln \left (x^{2}\right ) \ln \left (\frac {5}{4 x^{3}}\right )+8 x \ln \left (x^{2}\right ) \ln \left (\frac {5}{4 x^{3}}\right )}{8 x \ln \left (\frac {5}{4 x^{3}}\right )+8 \ln \left (\frac {5}{4 x^{3}}\right )+8}\) | \(47\) |
risch | \(\frac {2 x \ln \left (x \right )+2 \ln \left (x \right )+\frac {2}{3}}{1+x}-\frac {4-2 i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )^{2}+2 i x \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )+2 i x \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+2 i x \pi \operatorname {csgn}\left (i x^{3}\right )^{3}+2 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )-i x \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 x \ln \left (5\right )-8 x \ln \left (2\right )+4 \ln \left (5\right )-8 \ln \left (2\right )+2 i \pi \operatorname {csgn}\left (i x^{3}\right )^{3}+2 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-2 i x \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )^{2}-2 i x \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right )^{2}-2 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right )^{2}-i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-i x \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )}{3 \left (1+x \right ) \left (2-i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )^{2}+i x \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )-2 i x \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+i x \pi \operatorname {csgn}\left (i x^{3}\right )^{3}+i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )+i x \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+i \pi \operatorname {csgn}\left (i x^{3}\right )^{3}-2 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-i x \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )^{2}-i x \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right )^{2}-i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right )^{2}+2 x \ln \left (5\right )-4 x \ln \left (2\right )-6 x \ln \left (x \right )+2 \ln \left (5\right )-4 \ln \left (2\right )-6 \ln \left (x \right )+i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+i x \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )\right )}\) | \(584\) |
int(((x*ln(5/4/x^3)-3*x-3)*ln(x^2)+(2*x^2+4*x+2)*ln(5/4/x^3)^2+(2+2*x)*ln( 5/4/x^3))/((x^3+2*x^2+x)*ln(5/4/x^3)^2+(2*x^2+2*x)*ln(5/4/x^3)+x),x,method =_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {(2+2 x) \log \left (\frac {5}{4 x^3}\right )+\left (2+4 x+2 x^2\right ) \log ^2\left (\frac {5}{4 x^3}\right )+\left (-3-3 x+x \log \left (\frac {5}{4 x^3}\right )\right ) \log \left (x^2\right )}{x+\left (2 x+2 x^2\right ) \log \left (\frac {5}{4 x^3}\right )+\left (x+2 x^2+x^3\right ) \log ^2\left (\frac {5}{4 x^3}\right )} \, dx=-\frac {2 \, {\left (x + 1\right )} \log \left (\frac {5}{4 \, x^{3}}\right )^{2} + \log \left (\frac {25}{16}\right )}{3 \, {\left ({\left (x + 1\right )} \log \left (\frac {5}{4 \, x^{3}}\right ) + 1\right )}} \]
integrate(((x*log(5/4/x^3)-3*x-3)*log(x^2)+(2*x^2+4*x+2)*log(5/4/x^3)^2+(2 +2*x)*log(5/4/x^3))/((x^3+2*x^2+x)*log(5/4/x^3)^2+(2*x^2+2*x)*log(5/4/x^3) +x),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (20) = 40\).
Time = 0.32 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.59 \[ \int \frac {(2+2 x) \log \left (\frac {5}{4 x^3}\right )+\left (2+4 x+2 x^2\right ) \log ^2\left (\frac {5}{4 x^3}\right )+\left (-3-3 x+x \log \left (\frac {5}{4 x^3}\right )\right ) \log \left (x^2\right )}{x+\left (2 x+2 x^2\right ) \log \left (\frac {5}{4 x^3}\right )+\left (x+2 x^2+x^3\right ) \log ^2\left (\frac {5}{4 x^3}\right )} \, dx=\frac {- 8 x \log {\left (2 \right )} + 4 x \log {\left (5 \right )} - 8 \log {\left (2 \right )} + 4 + 4 \log {\left (5 \right )}}{- 6 x^{2} \log {\left (5 \right )} + 12 x^{2} \log {\left (2 \right )} - 12 x \log {\left (5 \right )} - 6 x + 24 x \log {\left (2 \right )} + \left (9 x^{2} + 18 x + 9\right ) \log {\left (x^{2} \right )} - 6 \log {\left (5 \right )} - 6 + 12 \log {\left (2 \right )}} + 2 \log {\left (x \right )} + \frac {2}{3 x + 3} \]
integrate(((x*ln(5/4/x**3)-3*x-3)*ln(x**2)+(2*x**2+4*x+2)*ln(5/4/x**3)**2+ (2+2*x)*ln(5/4/x**3))/((x**3+2*x**2+x)*ln(5/4/x**3)**2+(2*x**2+2*x)*ln(5/4 /x**3)+x),x)
(-8*x*log(2) + 4*x*log(5) - 8*log(2) + 4 + 4*log(5))/(-6*x**2*log(5) + 12* x**2*log(2) - 12*x*log(5) - 6*x + 24*x*log(2) + (9*x**2 + 18*x + 9)*log(x* *2) - 6*log(5) - 6 + 12*log(2)) + 2*log(x) + 2/(3*x + 3)
Time = 0.31 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \frac {(2+2 x) \log \left (\frac {5}{4 x^3}\right )+\left (2+4 x+2 x^2\right ) \log ^2\left (\frac {5}{4 x^3}\right )+\left (-3-3 x+x \log \left (\frac {5}{4 x^3}\right )\right ) \log \left (x^2\right )}{x+\left (2 x+2 x^2\right ) \log \left (\frac {5}{4 x^3}\right )+\left (x+2 x^2+x^3\right ) \log ^2\left (\frac {5}{4 x^3}\right )} \, dx=-\frac {2 \, \log \left (x\right )}{x {\left (\log \left (5\right ) - 2 \, \log \left (2\right )\right )} - 3 \, {\left (x + 1\right )} \log \left (x\right ) + \log \left (5\right ) - 2 \, \log \left (2\right ) + 1} + 2 \, \log \left (x\right ) \]
integrate(((x*log(5/4/x^3)-3*x-3)*log(x^2)+(2*x^2+4*x+2)*log(5/4/x^3)^2+(2 +2*x)*log(5/4/x^3))/((x^3+2*x^2+x)*log(5/4/x^3)^2+(2*x^2+2*x)*log(5/4/x^3) +x),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (25) = 50\).
Time = 0.32 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.00 \[ \int \frac {(2+2 x) \log \left (\frac {5}{4 x^3}\right )+\left (2+4 x+2 x^2\right ) \log ^2\left (\frac {5}{4 x^3}\right )+\left (-3-3 x+x \log \left (\frac {5}{4 x^3}\right )\right ) \log \left (x^2\right )}{x+\left (2 x+2 x^2\right ) \log \left (\frac {5}{4 x^3}\right )+\left (x+2 x^2+x^3\right ) \log ^2\left (\frac {5}{4 x^3}\right )} \, dx=-\frac {2 \, {\left (x \log \left (5\right ) - 2 \, x \log \left (2\right ) + \log \left (5\right ) - 2 \, \log \left (2\right ) + 1\right )}}{3 \, {\left (x^{2} \log \left (5\right ) - 2 \, x^{2} \log \left (2\right ) - 3 \, x^{2} \log \left (x\right ) + 2 \, x \log \left (5\right ) - 4 \, x \log \left (2\right ) - 6 \, x \log \left (x\right ) + x + \log \left (5\right ) - 2 \, \log \left (2\right ) - 3 \, \log \left (x\right ) + 1\right )}} + \frac {2}{3 \, {\left (x + 1\right )}} + 2 \, \log \left (x\right ) \]
integrate(((x*log(5/4/x^3)-3*x-3)*log(x^2)+(2*x^2+4*x+2)*log(5/4/x^3)^2+(2 +2*x)*log(5/4/x^3))/((x^3+2*x^2+x)*log(5/4/x^3)^2+(2*x^2+2*x)*log(5/4/x^3) +x),x, algorithm=\
-2/3*(x*log(5) - 2*x*log(2) + log(5) - 2*log(2) + 1)/(x^2*log(5) - 2*x^2*l og(2) - 3*x^2*log(x) + 2*x*log(5) - 4*x*log(2) - 6*x*log(x) + x + log(5) - 2*log(2) - 3*log(x) + 1) + 2/3/(x + 1) + 2*log(x)
Timed out. \[ \int \frac {(2+2 x) \log \left (\frac {5}{4 x^3}\right )+\left (2+4 x+2 x^2\right ) \log ^2\left (\frac {5}{4 x^3}\right )+\left (-3-3 x+x \log \left (\frac {5}{4 x^3}\right )\right ) \log \left (x^2\right )}{x+\left (2 x+2 x^2\right ) \log \left (\frac {5}{4 x^3}\right )+\left (x+2 x^2+x^3\right ) \log ^2\left (\frac {5}{4 x^3}\right )} \, dx=\int \frac {\ln \left (\frac {5}{4\,x^3}\right )\,\left (2\,x+2\right )-\ln \left (x^2\right )\,\left (3\,x-x\,\ln \left (\frac {5}{4\,x^3}\right )+3\right )+{\ln \left (\frac {5}{4\,x^3}\right )}^2\,\left (2\,x^2+4\,x+2\right )}{\left (x^3+2\,x^2+x\right )\,{\ln \left (\frac {5}{4\,x^3}\right )}^2+\left (2\,x^2+2\,x\right )\,\ln \left (\frac {5}{4\,x^3}\right )+x} \,d x \]
int((log(5/(4*x^3))*(2*x + 2) - log(x^2)*(3*x - x*log(5/(4*x^3)) + 3) + lo g(5/(4*x^3))^2*(4*x + 2*x^2 + 2))/(x + log(5/(4*x^3))^2*(x + 2*x^2 + x^3) + log(5/(4*x^3))*(2*x + 2*x^2)),x)