Integrand size = 65, antiderivative size = 35 \[ \int \frac {-5+\left (10+14 x^2+5 x^3-2 x^4\right ) \log (x)+\left (-4-5 x+2 x^2\right ) \log (x) \log \left (\frac {\log (x)}{x^2}\right )}{-5 x^3 \log (x)+5 x \log (x) \log \left (\frac {\log (x)}{x^2}\right )} \, dx=-x+\frac {1}{5} \left (x^2+\log (x)\right )-\log \left (\frac {1}{5} x \left (-x^2+\log \left (\frac {\log (x)}{x^2}\right )\right )\right ) \] Output:
1/5*ln(x)+1/5*x^2-ln(1/5*x*(ln(ln(x)/x^2)-x^2))-x
Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \frac {-5+\left (10+14 x^2+5 x^3-2 x^4\right ) \log (x)+\left (-4-5 x+2 x^2\right ) \log (x) \log \left (\frac {\log (x)}{x^2}\right )}{-5 x^3 \log (x)+5 x \log (x) \log \left (\frac {\log (x)}{x^2}\right )} \, dx=-x+\frac {x^2}{5}-\frac {4 \log (x)}{5}-\log \left (x^2-\log \left (\frac {\log (x)}{x^2}\right )\right ) \] Input:
Integrate[(-5 + (10 + 14*x^2 + 5*x^3 - 2*x^4)*Log[x] + (-4 - 5*x + 2*x^2)* Log[x]*Log[Log[x]/x^2])/(-5*x^3*Log[x] + 5*x*Log[x]*Log[Log[x]/x^2]),x]
Output:
-x + x^2/5 - (4*Log[x])/5 - Log[x^2 - Log[Log[x]/x^2]]
Time = 1.14 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {7292, 27, 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.
\(\displaystyle \int \frac {\left (2 x^2-5 x-4\right ) \log \left (\frac {\log (x)}{x^2}\right ) \log (x)+\left (-2 x^4+5 x^3+14 x^2+10\right ) \log (x)-5}{5 x \log (x) \log \left (\frac {\log (x)}{x^2}\right )-5 x^3 \log (x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-\left (2 x^2-5 x-4\right ) \log \left (\frac {\log (x)}{x^2}\right ) \log (x)-\left (\left (-2 x^4+5 x^3+14 x^2+10\right ) \log (x)\right )+5}{5 x \log (x) \left (x^2-\log \left (\frac {\log (x)}{x^2}\right )\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{5} \int \frac {-\left (\left (-2 x^4+5 x^3+14 x^2+10\right ) \log (x)\right )+\left (-2 x^2+5 x+4\right ) \log \left (\frac {\log (x)}{x^2}\right ) \log (x)+5}{x \log (x) \left (x^2-\log \left (\frac {\log (x)}{x^2}\right )\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{5} \int \left (\frac {2 x^2-5 x-4}{x}-\frac {5 \left (2 \log (x) x^2+2 \log (x)-1\right )}{x \log (x) \left (x^2-\log \left (\frac {\log (x)}{x^2}\right )\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{5} \left (x^2-5 \log \left (x^2-\log \left (\frac {\log (x)}{x^2}\right )\right )-5 x-4 \log (x)\right )\) |
Input:
Int[(-5 + (10 + 14*x^2 + 5*x^3 - 2*x^4)*Log[x] + (-4 - 5*x + 2*x^2)*Log[x] *Log[Log[x]/x^2])/(-5*x^3*Log[x] + 5*x*Log[x]*Log[Log[x]/x^2]),x]
Output:
(-5*x + x^2 - 4*Log[x] - 5*Log[x^2 - Log[Log[x]/x^2]])/5
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 13.68 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86
method | result | size |
parallelrisch | \(\frac {x^{2}}{5}-x -\frac {4 \ln \left (x \right )}{5}-\ln \left (x^{2}-\ln \left (\frac {\ln \left (x \right )}{x^{2}}\right )\right )\) | \(30\) |
risch | \(\frac {x^{2}}{5}-x -\frac {4 \ln \left (x \right )}{5}-\ln \left (\ln \left (\ln \left (x \right )\right )+\frac {i \left (\pi \,\operatorname {csgn}\left (i \ln \left (x \right )\right ) \operatorname {csgn}\left (\frac {i \ln \left (x \right )}{x^{2}}\right )^{2}-\pi \,\operatorname {csgn}\left (i \ln \left (x \right )\right ) \operatorname {csgn}\left (\frac {i \ln \left (x \right )}{x^{2}}\right ) \operatorname {csgn}\left (\frac {i}{x^{2}}\right )-\pi \operatorname {csgn}\left (\frac {i \ln \left (x \right )}{x^{2}}\right )^{3}+\pi \operatorname {csgn}\left (\frac {i \ln \left (x \right )}{x^{2}}\right )^{2} \operatorname {csgn}\left (\frac {i}{x^{2}}\right )+\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+2 i x^{2}+4 i \ln \left (x \right )\right )}{2}\right )\) | \(158\) |
Input:
int(((2*x^2-5*x-4)*ln(x)*ln(ln(x)/x^2)+(-2*x^4+5*x^3+14*x^2+10)*ln(x)-5)/( 5*x*ln(x)*ln(ln(x)/x^2)-5*x^3*ln(x)),x,method=_RETURNVERBOSE)
Output:
1/5*x^2-x-4/5*ln(x)-ln(x^2-ln(ln(x)/x^2))
Time = 0.12 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \frac {-5+\left (10+14 x^2+5 x^3-2 x^4\right ) \log (x)+\left (-4-5 x+2 x^2\right ) \log (x) \log \left (\frac {\log (x)}{x^2}\right )}{-5 x^3 \log (x)+5 x \log (x) \log \left (\frac {\log (x)}{x^2}\right )} \, dx=\frac {1}{5} \, x^{2} - x - \log \left (-x^{2} + \log \left (\frac {\log \left (x\right )}{x^{2}}\right )\right ) - \frac {4}{5} \, \log \left (x\right ) \] Input:
integrate(((2*x^2-5*x-4)*log(x)*log(log(x)/x^2)+(-2*x^4+5*x^3+14*x^2+10)*l og(x)-5)/(5*x*log(x)*log(log(x)/x^2)-5*x^3*log(x)),x, algorithm="fricas")
Output:
1/5*x^2 - x - log(-x^2 + log(log(x)/x^2)) - 4/5*log(x)
Time = 0.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.74 \[ \int \frac {-5+\left (10+14 x^2+5 x^3-2 x^4\right ) \log (x)+\left (-4-5 x+2 x^2\right ) \log (x) \log \left (\frac {\log (x)}{x^2}\right )}{-5 x^3 \log (x)+5 x \log (x) \log \left (\frac {\log (x)}{x^2}\right )} \, dx=\frac {x^{2}}{5} - x - \frac {4 \log {\left (x \right )}}{5} - \log {\left (- x^{2} + \log {\left (\frac {\log {\left (x \right )}}{x^{2}} \right )} \right )} \] Input:
integrate(((2*x**2-5*x-4)*ln(x)*ln(ln(x)/x**2)+(-2*x**4+5*x**3+14*x**2+10) *ln(x)-5)/(5*x*ln(x)*ln(ln(x)/x**2)-5*x**3*ln(x)),x)
Output:
x**2/5 - x - 4*log(x)/5 - log(-x**2 + log(log(x)/x**2))
Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \frac {-5+\left (10+14 x^2+5 x^3-2 x^4\right ) \log (x)+\left (-4-5 x+2 x^2\right ) \log (x) \log \left (\frac {\log (x)}{x^2}\right )}{-5 x^3 \log (x)+5 x \log (x) \log \left (\frac {\log (x)}{x^2}\right )} \, dx=\frac {1}{5} \, x^{2} - x - \log \left (-x^{2} - 2 \, \log \left (x\right ) + \log \left (\log \left (x\right )\right )\right ) - \frac {4}{5} \, \log \left (x\right ) \] Input:
integrate(((2*x^2-5*x-4)*log(x)*log(log(x)/x^2)+(-2*x^4+5*x^3+14*x^2+10)*l og(x)-5)/(5*x*log(x)*log(log(x)/x^2)-5*x^3*log(x)),x, algorithm="maxima")
Output:
1/5*x^2 - x - log(-x^2 - 2*log(x) + log(log(x))) - 4/5*log(x)
Time = 0.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \frac {-5+\left (10+14 x^2+5 x^3-2 x^4\right ) \log (x)+\left (-4-5 x+2 x^2\right ) \log (x) \log \left (\frac {\log (x)}{x^2}\right )}{-5 x^3 \log (x)+5 x \log (x) \log \left (\frac {\log (x)}{x^2}\right )} \, dx=\frac {1}{5} \, x^{2} - x - \log \left (x^{2} + 2 \, \log \left (x\right ) - \log \left (\log \left (x\right )\right )\right ) - \frac {4}{5} \, \log \left (x\right ) \] Input:
integrate(((2*x^2-5*x-4)*log(x)*log(log(x)/x^2)+(-2*x^4+5*x^3+14*x^2+10)*l og(x)-5)/(5*x*log(x)*log(log(x)/x^2)-5*x^3*log(x)),x, algorithm="giac")
Output:
1/5*x^2 - x - log(x^2 + 2*log(x) - log(log(x))) - 4/5*log(x)
Time = 3.44 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \frac {-5+\left (10+14 x^2+5 x^3-2 x^4\right ) \log (x)+\left (-4-5 x+2 x^2\right ) \log (x) \log \left (\frac {\log (x)}{x^2}\right )}{-5 x^3 \log (x)+5 x \log (x) \log \left (\frac {\log (x)}{x^2}\right )} \, dx=\frac {x^2}{5}-\ln \left (\ln \left (\frac {\ln \left (x\right )}{x^2}\right )-x^2\right )-\frac {4\,\ln \left (x\right )}{5}-x \] Input:
int((log(log(x)/x^2)*log(x)*(5*x - 2*x^2 + 4) - log(x)*(14*x^2 + 5*x^3 - 2 *x^4 + 10) + 5)/(5*x^3*log(x) - 5*x*log(log(x)/x^2)*log(x)),x)
Output:
x^2/5 - log(log(log(x)/x^2) - x^2) - (4*log(x))/5 - x
Time = 0.17 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \frac {-5+\left (10+14 x^2+5 x^3-2 x^4\right ) \log (x)+\left (-4-5 x+2 x^2\right ) \log (x) \log \left (\frac {\log (x)}{x^2}\right )}{-5 x^3 \log (x)+5 x \log (x) \log \left (\frac {\log (x)}{x^2}\right )} \, dx=-\mathrm {log}\left (\mathrm {log}\left (\frac {\mathrm {log}\left (x \right )}{x^{2}}\right )-x^{2}\right )-\frac {4 \,\mathrm {log}\left (x \right )}{5}+\frac {x^{2}}{5}-x \] Input:
int(((2*x^2-5*x-4)*log(x)*log(log(x)/x^2)+(-2*x^4+5*x^3+14*x^2+10)*log(x)- 5)/(5*x*log(x)*log(log(x)/x^2)-5*x^3*log(x)),x)
Output:
( - 5*log(log(log(x)/x**2) - x**2) - 4*log(x) + x**2 - 5*x)/5