Integrand size = 117, antiderivative size = 25 \[ \int \frac {-160-2 x^2+\left (320-4 x^2\right ) \log \left (-\frac {x}{-80+x^2}\right )+\left (-160+2 x^2\right ) \log \left (-\frac {x}{-80+x^2}\right ) \log \left (x \log \left (-\frac {x}{-80+x^2}\right )\right )}{\left (240 x-3 x^3\right ) \log \left (-\frac {x}{-80+x^2}\right )+\left (-80 x+x^3\right ) \log \left (-\frac {x}{-80+x^2}\right ) \log \left (x \log \left (-\frac {x}{-80+x^2}\right )\right )} \, dx=\log \left (5 x^2 \left (-3+\log \left (x \log \left (\frac {x}{80-x^2}\right )\right )\right )^2\right ) \] Output:
ln(5*x^2*(ln(x*ln(x/(-x^2+80)))-3)^2)
Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {-160-2 x^2+\left (320-4 x^2\right ) \log \left (-\frac {x}{-80+x^2}\right )+\left (-160+2 x^2\right ) \log \left (-\frac {x}{-80+x^2}\right ) \log \left (x \log \left (-\frac {x}{-80+x^2}\right )\right )}{\left (240 x-3 x^3\right ) \log \left (-\frac {x}{-80+x^2}\right )+\left (-80 x+x^3\right ) \log \left (-\frac {x}{-80+x^2}\right ) \log \left (x \log \left (-\frac {x}{-80+x^2}\right )\right )} \, dx=2 \left (\log (x)+\log \left (3-\log \left (x \log \left (-\frac {x}{-80+x^2}\right )\right )\right )\right ) \] Input:
Integrate[(-160 - 2*x^2 + (320 - 4*x^2)*Log[-(x/(-80 + x^2))] + (-160 + 2* x^2)*Log[-(x/(-80 + x^2))]*Log[x*Log[-(x/(-80 + x^2))]])/((240*x - 3*x^3)* Log[-(x/(-80 + x^2))] + (-80*x + x^3)*Log[-(x/(-80 + x^2))]*Log[x*Log[-(x/ (-80 + x^2))]]),x]
Output:
2*(Log[x] + Log[3 - Log[x*Log[-(x/(-80 + x^2))]]])
Time = 0.58 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {7292, 7236}
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.
\(\displaystyle \int \frac {-2 x^2+\left (320-4 x^2\right ) \log \left (-\frac {x}{x^2-80}\right )+\left (2 x^2-160\right ) \log \left (-\frac {x}{x^2-80}\right ) \log \left (x \log \left (-\frac {x}{x^2-80}\right )\right )-160}{\left (240 x-3 x^3\right ) \log \left (-\frac {x}{x^2-80}\right )+\left (x^3-80 x\right ) \log \left (x \log \left (-\frac {x}{x^2-80}\right )\right ) \log \left (-\frac {x}{x^2-80}\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-2 x^2+\left (320-4 x^2\right ) \log \left (-\frac {x}{x^2-80}\right )+\left (2 x^2-160\right ) \log \left (-\frac {x}{x^2-80}\right ) \log \left (x \log \left (-\frac {x}{x^2-80}\right )\right )-160}{x \left (80-x^2\right ) \log \left (-\frac {x}{x^2-80}\right ) \left (3-\log \left (x \log \left (-\frac {x}{x^2-80}\right )\right )\right )}dx\) |
\(\Big \downarrow \) 7236 |
\(\displaystyle 2 \log \left (-x \left (3-\log \left (x \log \left (\frac {x}{80-x^2}\right )\right )\right )\right )\) |
Input:
Int[(-160 - 2*x^2 + (320 - 4*x^2)*Log[-(x/(-80 + x^2))] + (-160 + 2*x^2)*L og[-(x/(-80 + x^2))]*Log[x*Log[-(x/(-80 + x^2))]])/((240*x - 3*x^3)*Log[-( x/(-80 + x^2))] + (-80*x + x^3)*Log[-(x/(-80 + x^2))]*Log[x*Log[-(x/(-80 + x^2))]]),x]
Output:
2*Log[-(x*(3 - Log[x*Log[x/(80 - x^2)]]))]
Int[(u_)/((w_)*(y_)), x_Symbol] :> With[{q = DerivativeDivides[y*w, u, x]}, Simp[q*Log[RemoveContent[y*w, x]], x] /; !FalseQ[q]]
Time = 1.09 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68
method | result | size |
parallelrisch | \(2 \ln \left (\ln \left (x \ln \left (-\frac {x}{x^{2}-80}\right )\right )-3\right )+2 \ln \left (x^{2}-80\right )+2 \ln \left (-\frac {x}{x^{2}-80}\right )\) | \(42\) |
Input:
int(((2*x^2-160)*ln(-x/(x^2-80))*ln(x*ln(-x/(x^2-80)))+(-4*x^2+320)*ln(-x/ (x^2-80))-2*x^2-160)/((x^3-80*x)*ln(-x/(x^2-80))*ln(x*ln(-x/(x^2-80)))+(-3 *x^3+240*x)*ln(-x/(x^2-80))),x,method=_RETURNVERBOSE)
Output:
2*ln(ln(x*ln(-x/(x^2-80)))-3)+2*ln(x^2-80)+2*ln(-x/(x^2-80))
Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {-160-2 x^2+\left (320-4 x^2\right ) \log \left (-\frac {x}{-80+x^2}\right )+\left (-160+2 x^2\right ) \log \left (-\frac {x}{-80+x^2}\right ) \log \left (x \log \left (-\frac {x}{-80+x^2}\right )\right )}{\left (240 x-3 x^3\right ) \log \left (-\frac {x}{-80+x^2}\right )+\left (-80 x+x^3\right ) \log \left (-\frac {x}{-80+x^2}\right ) \log \left (x \log \left (-\frac {x}{-80+x^2}\right )\right )} \, dx=2 \, \log \left (x\right ) + 2 \, \log \left (\log \left (x \log \left (-\frac {x}{x^{2} - 80}\right )\right ) - 3\right ) \] Input:
integrate(((2*x^2-160)*log(-x/(x^2-80))*log(x*log(-x/(x^2-80)))+(-4*x^2+32 0)*log(-x/(x^2-80))-2*x^2-160)/((x^3-80*x)*log(-x/(x^2-80))*log(x*log(-x/( x^2-80)))+(-3*x^3+240*x)*log(-x/(x^2-80))),x, algorithm="fricas")
Output:
2*log(x) + 2*log(log(x*log(-x/(x^2 - 80))) - 3)
Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {-160-2 x^2+\left (320-4 x^2\right ) \log \left (-\frac {x}{-80+x^2}\right )+\left (-160+2 x^2\right ) \log \left (-\frac {x}{-80+x^2}\right ) \log \left (x \log \left (-\frac {x}{-80+x^2}\right )\right )}{\left (240 x-3 x^3\right ) \log \left (-\frac {x}{-80+x^2}\right )+\left (-80 x+x^3\right ) \log \left (-\frac {x}{-80+x^2}\right ) \log \left (x \log \left (-\frac {x}{-80+x^2}\right )\right )} \, dx=2 \log {\left (x \right )} + 2 \log {\left (\log {\left (x \log {\left (- \frac {x}{x^{2} - 80} \right )} \right )} - 3 \right )} \] Input:
integrate(((2*x**2-160)*ln(-x/(x**2-80))*ln(x*ln(-x/(x**2-80)))+(-4*x**2+3 20)*ln(-x/(x**2-80))-2*x**2-160)/((x**3-80*x)*ln(-x/(x**2-80))*ln(x*ln(-x/ (x**2-80)))+(-3*x**3+240*x)*ln(-x/(x**2-80))),x)
Output:
2*log(x) + 2*log(log(x*log(-x/(x**2 - 80))) - 3)
Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {-160-2 x^2+\left (320-4 x^2\right ) \log \left (-\frac {x}{-80+x^2}\right )+\left (-160+2 x^2\right ) \log \left (-\frac {x}{-80+x^2}\right ) \log \left (x \log \left (-\frac {x}{-80+x^2}\right )\right )}{\left (240 x-3 x^3\right ) \log \left (-\frac {x}{-80+x^2}\right )+\left (-80 x+x^3\right ) \log \left (-\frac {x}{-80+x^2}\right ) \log \left (x \log \left (-\frac {x}{-80+x^2}\right )\right )} \, dx=2 \, \log \left (x\right ) + 2 \, \log \left (\log \left (x\right ) + \log \left (-\log \left (-x^{2} + 80\right ) + \log \left (x\right )\right ) - 3\right ) \] Input:
integrate(((2*x^2-160)*log(-x/(x^2-80))*log(x*log(-x/(x^2-80)))+(-4*x^2+32 0)*log(-x/(x^2-80))-2*x^2-160)/((x^3-80*x)*log(-x/(x^2-80))*log(x*log(-x/( x^2-80)))+(-3*x^3+240*x)*log(-x/(x^2-80))),x, algorithm="maxima")
Output:
2*log(x) + 2*log(log(x) + log(-log(-x^2 + 80) + log(x)) - 3)
\[ \int \frac {-160-2 x^2+\left (320-4 x^2\right ) \log \left (-\frac {x}{-80+x^2}\right )+\left (-160+2 x^2\right ) \log \left (-\frac {x}{-80+x^2}\right ) \log \left (x \log \left (-\frac {x}{-80+x^2}\right )\right )}{\left (240 x-3 x^3\right ) \log \left (-\frac {x}{-80+x^2}\right )+\left (-80 x+x^3\right ) \log \left (-\frac {x}{-80+x^2}\right ) \log \left (x \log \left (-\frac {x}{-80+x^2}\right )\right )} \, dx=\int { \frac {2 \, {\left ({\left (x^{2} - 80\right )} \log \left (x \log \left (-\frac {x}{x^{2} - 80}\right )\right ) \log \left (-\frac {x}{x^{2} - 80}\right ) - x^{2} - 2 \, {\left (x^{2} - 80\right )} \log \left (-\frac {x}{x^{2} - 80}\right ) - 80\right )}}{{\left (x^{3} - 80 \, x\right )} \log \left (x \log \left (-\frac {x}{x^{2} - 80}\right )\right ) \log \left (-\frac {x}{x^{2} - 80}\right ) - 3 \, {\left (x^{3} - 80 \, x\right )} \log \left (-\frac {x}{x^{2} - 80}\right )} \,d x } \] Input:
integrate(((2*x^2-160)*log(-x/(x^2-80))*log(x*log(-x/(x^2-80)))+(-4*x^2+32 0)*log(-x/(x^2-80))-2*x^2-160)/((x^3-80*x)*log(-x/(x^2-80))*log(x*log(-x/( x^2-80)))+(-3*x^3+240*x)*log(-x/(x^2-80))),x, algorithm="giac")
Output:
integrate(2*((x^2 - 80)*log(x*log(-x/(x^2 - 80)))*log(-x/(x^2 - 80)) - x^2 - 2*(x^2 - 80)*log(-x/(x^2 - 80)) - 80)/((x^3 - 80*x)*log(x*log(-x/(x^2 - 80)))*log(-x/(x^2 - 80)) - 3*(x^3 - 80*x)*log(-x/(x^2 - 80))), x)
Time = 3.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {-160-2 x^2+\left (320-4 x^2\right ) \log \left (-\frac {x}{-80+x^2}\right )+\left (-160+2 x^2\right ) \log \left (-\frac {x}{-80+x^2}\right ) \log \left (x \log \left (-\frac {x}{-80+x^2}\right )\right )}{\left (240 x-3 x^3\right ) \log \left (-\frac {x}{-80+x^2}\right )+\left (-80 x+x^3\right ) \log \left (-\frac {x}{-80+x^2}\right ) \log \left (x \log \left (-\frac {x}{-80+x^2}\right )\right )} \, dx=2\,\ln \left (x\right )+2\,\ln \left (\ln \left (x\,\ln \left (-\frac {x}{x^2-80}\right )\right )-3\right ) \] Input:
int(-(log(-x/(x^2 - 80))*(4*x^2 - 320) + 2*x^2 - log(x*log(-x/(x^2 - 80))) *log(-x/(x^2 - 80))*(2*x^2 - 160) + 160)/(log(-x/(x^2 - 80))*(240*x - 3*x^ 3) - log(x*log(-x/(x^2 - 80)))*log(-x/(x^2 - 80))*(80*x - x^3)),x)
Output:
2*log(x) + 2*log(log(x*log(-x/(x^2 - 80))) - 3)
Time = 0.17 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {-160-2 x^2+\left (320-4 x^2\right ) \log \left (-\frac {x}{-80+x^2}\right )+\left (-160+2 x^2\right ) \log \left (-\frac {x}{-80+x^2}\right ) \log \left (x \log \left (-\frac {x}{-80+x^2}\right )\right )}{\left (240 x-3 x^3\right ) \log \left (-\frac {x}{-80+x^2}\right )+\left (-80 x+x^3\right ) \log \left (-\frac {x}{-80+x^2}\right ) \log \left (x \log \left (-\frac {x}{-80+x^2}\right )\right )} \, dx=2 \,\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (-\frac {x}{x^{2}-80}\right ) x \right )-3\right )+2 \,\mathrm {log}\left (x \right ) \] Input:
int(((2*x^2-160)*log(-x/(x^2-80))*log(x*log(-x/(x^2-80)))+(-4*x^2+320)*log (-x/(x^2-80))-2*x^2-160)/((x^3-80*x)*log(-x/(x^2-80))*log(x*log(-x/(x^2-80 )))+(-3*x^3+240*x)*log(-x/(x^2-80))),x)
Output:
2*(log(log(log(( - x)/(x**2 - 80))*x) - 3) + log(x))