Integrand size = 85, antiderivative size = 25 \[ \int \frac {22500 x-7800 x^2+100 x^3+\left (22500 x-7800 x^2+100 x^3\right ) \log (2)+\left (-22500+37500 x-15100 x^2+100 x^3+\left (-22500+37500 x-15100 x^2+100 x^3\right ) \log (2)\right ) \log (-1+x)}{-5625+5775 x-151 x^2+x^3} \, dx=\frac {4 (3-x) x (1+\log (2)) \log (-1+x)}{3-\frac {x}{25}} \] Output:
4*(3-x)*x*ln(-1+x)*(1+ln(2))/(3-1/25*x)
Leaf count is larger than twice the leaf count of optimal. \(53\) vs. \(2(25)=50\).
Time = 0.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.12 \[ \int \frac {22500 x-7800 x^2+100 x^3+\left (22500 x-7800 x^2+100 x^3\right ) \log (2)+\left (-22500+37500 x-15100 x^2+100 x^3+\left (-22500+37500 x-15100 x^2+100 x^3\right ) \log (2)\right ) \log (-1+x)}{-5625+5775 x-151 x^2+x^3} \, dx=\frac {100}{37} (1+\log (2)) \left (5400 \text {arctanh}\left (\frac {1}{37} (-38+x)\right )+\log (1-x)+2700 \log (75-x)+\frac {199800 \log (-1+x)}{-75+x}+37 (-1+x) \log (-1+x)\right ) \] Input:
Integrate[(22500*x - 7800*x^2 + 100*x^3 + (22500*x - 7800*x^2 + 100*x^3)*L og[2] + (-22500 + 37500*x - 15100*x^2 + 100*x^3 + (-22500 + 37500*x - 1510 0*x^2 + 100*x^3)*Log[2])*Log[-1 + x])/(-5625 + 5775*x - 151*x^2 + x^3),x]
Output:
(100*(1 + Log[2])*(5400*ArcTanh[(-38 + x)/37] + Log[1 - x] + 2700*Log[75 - x] + (199800*Log[-1 + x])/(-75 + x) + 37*(-1 + x)*Log[-1 + x]))/37
Time = 0.84 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.80, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {2463, 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.
\(\displaystyle \int \frac {100 x^3-7800 x^2+\left (100 x^3-15100 x^2+\left (100 x^3-15100 x^2+37500 x-22500\right ) \log (2)+37500 x-22500\right ) \log (x-1)+\left (100 x^3-7800 x^2+22500 x\right ) \log (2)+22500 x}{x^3-151 x^2+5775 x-5625} \, dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (-\frac {100 x^3-7800 x^2+\left (100 x^3-15100 x^2+\left (100 x^3-15100 x^2+37500 x-22500\right ) \log (2)+37500 x-22500\right ) \log (x-1)+\left (100 x^3-7800 x^2+22500 x\right ) \log (2)+22500 x}{5476 (x-75)}+\frac {100 x^3-7800 x^2+\left (100 x^3-15100 x^2+\left (100 x^3-15100 x^2+37500 x-22500\right ) \log (2)+37500 x-22500\right ) \log (x-1)+\left (100 x^3-7800 x^2+22500 x\right ) \log (2)+22500 x}{5476 (x-1)}+\frac {100 x^3-7800 x^2+\left (100 x^3-15100 x^2+\left (100 x^3-15100 x^2+37500 x-22500\right ) \log (2)+37500 x-22500\right ) \log (x-1)+\left (100 x^3-7800 x^2+22500 x\right ) \log (2)+22500 x}{74 (x-75)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 7300 (1+\log (2)) \log (1-x)-100 (1-x) (1+\log (2)) \log (x-1)-\frac {540000 (1+\log (2)) \log (x-1)}{75-x}\) |
Input:
Int[(22500*x - 7800*x^2 + 100*x^3 + (22500*x - 7800*x^2 + 100*x^3)*Log[2] + (-22500 + 37500*x - 15100*x^2 + 100*x^3 + (-22500 + 37500*x - 15100*x^2 + 100*x^3)*Log[2])*Log[-1 + x])/(-5625 + 5775*x - 151*x^2 + x^3),x]
Output:
7300*(1 + Log[2])*Log[1 - x] - 100*(1 - x)*(1 + Log[2])*Log[-1 + x] - (540 000*(1 + Log[2])*Log[-1 + x])/(75 - x)
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 0.40 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36
method | result | size |
norman | \(\frac {\left (100+100 \ln \left (2\right )\right ) x^{2} \ln \left (-1+x \right )+\left (-300-300 \ln \left (2\right )\right ) x \ln \left (-1+x \right )}{x -75}\) | \(34\) |
parallelrisch | \(\frac {100 \ln \left (-1+x \right ) \ln \left (2\right ) x^{2}-300 \ln \left (-1+x \right ) \ln \left (2\right ) x +100 \ln \left (-1+x \right ) x^{2}-300 \ln \left (-1+x \right ) x}{x -75}\) | \(44\) |
risch | \(\frac {100 \left (x^{2} \ln \left (2\right )-75 x \ln \left (2\right )+x^{2}+5400 \ln \left (2\right )-75 x +5400\right ) \ln \left (-1+x \right )}{x -75}+7200 \ln \left (-1+x \right ) \ln \left (2\right )+7200 \ln \left (-1+x \right )\) | \(50\) |
derivativedivides | \(100 \ln \left (2\right ) \left (\ln \left (-1+x \right ) \left (-1+x \right )+\frac {2700 \ln \left (-1+x \right ) \left (-1+x \right )}{37 \left (x -75\right )}+\frac {\ln \left (-1+x \right )}{37}\right )+100 \ln \left (-1+x \right ) \left (-1+x \right )+\frac {270000 \ln \left (-1+x \right ) \left (-1+x \right )}{37 \left (x -75\right )}+\frac {100 \ln \left (-1+x \right )}{37}\) | \(64\) |
default | \(100 \ln \left (2\right ) \left (\ln \left (-1+x \right ) \left (-1+x \right )+\frac {2700 \ln \left (-1+x \right ) \left (-1+x \right )}{37 \left (x -75\right )}+\frac {\ln \left (-1+x \right )}{37}\right )+100 \ln \left (-1+x \right ) \left (-1+x \right )+\frac {270000 \ln \left (-1+x \right ) \left (-1+x \right )}{37 \left (x -75\right )}+\frac {100 \ln \left (-1+x \right )}{37}\) | \(64\) |
parts | \(100 \left (1+\ln \left (2\right )\right ) \left (x +\frac {\ln \left (-1+x \right )}{37}+\frac {2700 \ln \left (x -75\right )}{37}\right )+100 \ln \left (-1+x \right ) \left (-1+x \right )-100 x +100+100 \ln \left (2\right ) \left (\ln \left (-1+x \right ) \left (-1+x \right )-x +1\right )+100 \left (-5400-5400 \ln \left (2\right )\right ) \left (\frac {\ln \left (x -75\right )}{74}-\frac {\ln \left (-1+x \right ) \left (-1+x \right )}{74 \left (x -75\right )}\right )\) | \(81\) |
orering | \(\frac {\left (x^{5}-227 x^{4}+1197 x^{3}+62775 x^{2}-128250 x +64800\right ) \left (\left (\left (100 x^{3}-15100 x^{2}+37500 x -22500\right ) \ln \left (2\right )+100 x^{3}-15100 x^{2}+37500 x -22500\right ) \ln \left (-1+x \right )+\left (100 x^{3}-7800 x^{2}+22500 x \right ) \ln \left (2\right )+100 x^{3}-7800 x^{2}+22500 x \right )}{\left (x^{4}-299 x^{3}+1125 x^{2}-2025 x +1350\right ) \left (x^{3}-151 x^{2}+5775 x -5625\right )}-\frac {\left (x^{4}-897 x^{2}+2250 x -1350\right ) \left (x -75\right ) \left (-1+x \right ) \left (\frac {\left (\left (300 x^{2}-30200 x +37500\right ) \ln \left (2\right )+300 x^{2}-30200 x +37500\right ) \ln \left (-1+x \right )+\frac {\left (100 x^{3}-15100 x^{2}+37500 x -22500\right ) \ln \left (2\right )+100 x^{3}-15100 x^{2}+37500 x -22500}{-1+x}+\left (300 x^{2}-15600 x +22500\right ) \ln \left (2\right )+300 x^{2}-15600 x +22500}{x^{3}-151 x^{2}+5775 x -5625}-\frac {\left (\left (\left (100 x^{3}-15100 x^{2}+37500 x -22500\right ) \ln \left (2\right )+100 x^{3}-15100 x^{2}+37500 x -22500\right ) \ln \left (-1+x \right )+\left (100 x^{3}-7800 x^{2}+22500 x \right ) \ln \left (2\right )+100 x^{3}-7800 x^{2}+22500 x \right ) \left (3 x^{2}-302 x +5775\right )}{\left (x^{3}-151 x^{2}+5775 x -5625\right )^{2}}\right )}{x^{4}-299 x^{3}+1125 x^{2}-2025 x +1350}\) | \(374\) |
Input:
int((((100*x^3-15100*x^2+37500*x-22500)*ln(2)+100*x^3-15100*x^2+37500*x-22 500)*ln(-1+x)+(100*x^3-7800*x^2+22500*x)*ln(2)+100*x^3-7800*x^2+22500*x)/( x^3-151*x^2+5775*x-5625),x,method=_RETURNVERBOSE)
Output:
((100+100*ln(2))*x^2*ln(-1+x)+(-300-300*ln(2))*x*ln(-1+x))/(x-75)
Time = 0.11 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {22500 x-7800 x^2+100 x^3+\left (22500 x-7800 x^2+100 x^3\right ) \log (2)+\left (-22500+37500 x-15100 x^2+100 x^3+\left (-22500+37500 x-15100 x^2+100 x^3\right ) \log (2)\right ) \log (-1+x)}{-5625+5775 x-151 x^2+x^3} \, dx=\frac {100 \, {\left (x^{2} + {\left (x^{2} - 3 \, x\right )} \log \left (2\right ) - 3 \, x\right )} \log \left (x - 1\right )}{x - 75} \] Input:
integrate((((100*x^3-15100*x^2+37500*x-22500)*log(2)+100*x^3-15100*x^2+375 00*x-22500)*log(-1+x)+(100*x^3-7800*x^2+22500*x)*log(2)+100*x^3-7800*x^2+2 2500*x)/(x^3-151*x^2+5775*x-5625),x, algorithm="fricas")
Output:
100*(x^2 + (x^2 - 3*x)*log(2) - 3*x)*log(x - 1)/(x - 75)
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (20) = 40\).
Time = 0.13 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.96 \[ \int \frac {22500 x-7800 x^2+100 x^3+\left (22500 x-7800 x^2+100 x^3\right ) \log (2)+\left (-22500+37500 x-15100 x^2+100 x^3+\left (-22500+37500 x-15100 x^2+100 x^3\right ) \log (2)\right ) \log (-1+x)}{-5625+5775 x-151 x^2+x^3} \, dx=\left (7200 \log {\left (2 \right )} + 7200\right ) \log {\left (x - 1 \right )} + \frac {\left (100 x^{2} \log {\left (2 \right )} + 100 x^{2} - 7500 x - 7500 x \log {\left (2 \right )} + 540000 \log {\left (2 \right )} + 540000\right ) \log {\left (x - 1 \right )}}{x - 75} \] Input:
integrate((((100*x**3-15100*x**2+37500*x-22500)*ln(2)+100*x**3-15100*x**2+ 37500*x-22500)*ln(-1+x)+(100*x**3-7800*x**2+22500*x)*ln(2)+100*x**3-7800*x **2+22500*x)/(x**3-151*x**2+5775*x-5625),x)
Output:
(7200*log(2) + 7200)*log(x - 1) + (100*x**2*log(2) + 100*x**2 - 7500*x - 7 500*x*log(2) + 540000*log(2) + 540000)*log(x - 1)/(x - 75)
Time = 0.15 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {22500 x-7800 x^2+100 x^3+\left (22500 x-7800 x^2+100 x^3\right ) \log (2)+\left (-22500+37500 x-15100 x^2+100 x^3+\left (-22500+37500 x-15100 x^2+100 x^3\right ) \log (2)\right ) \log (-1+x)}{-5625+5775 x-151 x^2+x^3} \, dx=\frac {100 \, {\left (x^{2} {\left (\log \left (2\right ) + 1\right )} - 3 \, x {\left (\log \left (2\right ) + 1\right )}\right )} \log \left (x - 1\right )}{x - 75} \] Input:
integrate((((100*x^3-15100*x^2+37500*x-22500)*log(2)+100*x^3-15100*x^2+375 00*x-22500)*log(-1+x)+(100*x^3-7800*x^2+22500*x)*log(2)+100*x^3-7800*x^2+2 2500*x)/(x^3-151*x^2+5775*x-5625),x, algorithm="maxima")
Output:
100*(x^2*(log(2) + 1) - 3*x*(log(2) + 1))*log(x - 1)/(x - 75)
Time = 0.12 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \frac {22500 x-7800 x^2+100 x^3+\left (22500 x-7800 x^2+100 x^3\right ) \log (2)+\left (-22500+37500 x-15100 x^2+100 x^3+\left (-22500+37500 x-15100 x^2+100 x^3\right ) \log (2)\right ) \log (-1+x)}{-5625+5775 x-151 x^2+x^3} \, dx=100 \, {\left (x {\left (\log \left (2\right ) + 1\right )} + \frac {5400 \, {\left (\log \left (2\right ) + 1\right )}}{x - 75}\right )} \log \left (x - 1\right ) + 7200 \, {\left (\log \left (2\right ) + 1\right )} \log \left (x - 1\right ) \] Input:
integrate((((100*x^3-15100*x^2+37500*x-22500)*log(2)+100*x^3-15100*x^2+375 00*x-22500)*log(-1+x)+(100*x^3-7800*x^2+22500*x)*log(2)+100*x^3-7800*x^2+2 2500*x)/(x^3-151*x^2+5775*x-5625),x, algorithm="giac")
Output:
100*(x*(log(2) + 1) + 5400*(log(2) + 1)/(x - 75))*log(x - 1) + 7200*(log(2 ) + 1)*log(x - 1)
Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {22500 x-7800 x^2+100 x^3+\left (22500 x-7800 x^2+100 x^3\right ) \log (2)+\left (-22500+37500 x-15100 x^2+100 x^3+\left (-22500+37500 x-15100 x^2+100 x^3\right ) \log (2)\right ) \log (-1+x)}{-5625+5775 x-151 x^2+x^3} \, dx=\frac {100\,x\,\ln \left (x-1\right )\,\left (\ln \left (2\right )+1\right )\,\left (x-3\right )}{x-75} \] Input:
int((22500*x + log(2)*(22500*x - 7800*x^2 + 100*x^3) + log(x - 1)*(37500*x + log(2)*(37500*x - 15100*x^2 + 100*x^3 - 22500) - 15100*x^2 + 100*x^3 - 22500) - 7800*x^2 + 100*x^3)/(5775*x - 151*x^2 + x^3 - 5625),x)
Output:
(100*x*log(x - 1)*(log(2) + 1)*(x - 3))/(x - 75)
Time = 0.15 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {22500 x-7800 x^2+100 x^3+\left (22500 x-7800 x^2+100 x^3\right ) \log (2)+\left (-22500+37500 x-15100 x^2+100 x^3+\left (-22500+37500 x-15100 x^2+100 x^3\right ) \log (2)\right ) \log (-1+x)}{-5625+5775 x-151 x^2+x^3} \, dx=\frac {100 \,\mathrm {log}\left (x -1\right ) x \left (\mathrm {log}\left (2\right ) x -3 \,\mathrm {log}\left (2\right )+x -3\right )}{x -75} \] Input:
int((((100*x^3-15100*x^2+37500*x-22500)*log(2)+100*x^3-15100*x^2+37500*x-2 2500)*log(-1+x)+(100*x^3-7800*x^2+22500*x)*log(2)+100*x^3-7800*x^2+22500*x )/(x^3-151*x^2+5775*x-5625),x)
Output:
(100*log(x - 1)*x*(log(2)*x - 3*log(2) + x - 3))/(x - 75)