Integrand size = 88, antiderivative size = 23 \[ \int \frac {125-335 x-210 x^2+454 x^3-193 x^4+33 x^5-2 x^6+\left (125-75 x-110 x^2+74 x^3-15 x^4+x^5\right ) \log \left (1-x^2\right )}{125-75 x-110 x^2+74 x^3-15 x^4+x^5} \, dx=x \left (1-x-\frac {x}{(-5+x)^2}+\log \left (1-x^2\right )\right ) \] Output:
x*(1-x+ln(-x^2+1)-x/(-5+x)^2)
Leaf count is larger than twice the leaf count of optimal. \(47\) vs. \(2(23)=46\).
Time = 0.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.04 \[ \int \frac {125-335 x-210 x^2+454 x^3-193 x^4+33 x^5-2 x^6+\left (125-75 x-110 x^2+74 x^3-15 x^4+x^5\right ) \log \left (1-x^2\right )}{125-75 x-110 x^2+74 x^3-15 x^4+x^5} \, dx=-\frac {25}{(-5+x)^2}-\frac {10}{-5+x}+x-x^2+2 \text {arctanh}(x)+\log (1-x)-\log (1+x)+x \log \left (1-x^2\right ) \] Input:
Integrate[(125 - 335*x - 210*x^2 + 454*x^3 - 193*x^4 + 33*x^5 - 2*x^6 + (1 25 - 75*x - 110*x^2 + 74*x^3 - 15*x^4 + x^5)*Log[1 - x^2])/(125 - 75*x - 1 10*x^2 + 74*x^3 - 15*x^4 + x^5),x]
Output:
-25/(-5 + x)^2 - 10/(-5 + x) + x - x^2 + 2*ArcTanh[x] + Log[1 - x] - Log[1 + x] + x*Log[1 - x^2]
Leaf count is larger than twice the leaf count of optimal. \(171\) vs. \(2(23)=46\).
Time = 2.33 (sec) , antiderivative size = 171, normalized size of antiderivative = 7.43, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, 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.
\(\displaystyle \int \frac {-2 x^6+33 x^5-193 x^4+454 x^3-210 x^2+\left (x^5-15 x^4+74 x^3-110 x^2-75 x+125\right ) \log \left (1-x^2\right )-335 x+125}{x^5-15 x^4+74 x^3-110 x^2-75 x+125} \, dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {19 \left (-2 x^6+33 x^5-193 x^4+454 x^3-210 x^2+\left (x^5-15 x^4+74 x^3-110 x^2-75 x+125\right ) \log \left (1-x^2\right )-335 x+125\right )}{3456 (x-5)}-\frac {-2 x^6+33 x^5-193 x^4+454 x^3-210 x^2+\left (x^5-15 x^4+74 x^3-110 x^2-75 x+125\right ) \log \left (1-x^2\right )-335 x+125}{128 (x-1)}+\frac {-2 x^6+33 x^5-193 x^4+454 x^3-210 x^2+\left (x^5-15 x^4+74 x^3-110 x^2-75 x+125\right ) \log \left (1-x^2\right )-335 x+125}{432 (x+1)}-\frac {5 \left (-2 x^6+33 x^5-193 x^4+454 x^3-210 x^2+\left (x^5-15 x^4+74 x^3-110 x^2-75 x+125\right ) \log \left (1-x^2\right )-335 x+125\right )}{288 (x-5)^2}+\frac {-2 x^6+33 x^5-193 x^4+454 x^3-210 x^2+\left (x^5-15 x^4+74 x^3-110 x^2-75 x+125\right ) \log \left (1-x^2\right )-335 x+125}{24 (x-5)^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \text {arctanh}(x)-x^2+\frac {67 x^2}{48 (5-x)^2}+\frac {5 (5-x)^4 \log \left (1-x^2\right )}{1152}-\frac {19}{432} (5-x)^3 \log \left (1-x^2\right )-\frac {37}{288} x \log \left (1-x^2\right )-\frac {5}{576} \left (1-x^2\right ) \log \left (1-x^2\right )+\frac {9655 \log \left (1-x^2\right )}{3456}-\frac {5 x^4 \log \left (1-x^2\right )}{1152}+\frac {37}{864} x^3 \log \left (1-x^2\right )+x+\frac {575}{24 (5-x)}-\frac {2875}{48 (5-x)^2}+\log (1-x)-\log (x+1)\) |
Input:
Int[(125 - 335*x - 210*x^2 + 454*x^3 - 193*x^4 + 33*x^5 - 2*x^6 + (125 - 7 5*x - 110*x^2 + 74*x^3 - 15*x^4 + x^5)*Log[1 - x^2])/(125 - 75*x - 110*x^2 + 74*x^3 - 15*x^4 + x^5),x]
Output:
-2875/(48*(5 - x)^2) + 575/(24*(5 - x)) + x - x^2 + (67*x^2)/(48*(5 - x)^2 ) + 2*ArcTanh[x] + Log[1 - x] - Log[1 + x] + (9655*Log[1 - x^2])/3456 - (1 9*(5 - x)^3*Log[1 - x^2])/432 + (5*(5 - x)^4*Log[1 - x^2])/1152 - (37*x*Lo g[1 - x^2])/288 + (37*x^3*Log[1 - x^2])/864 - (5*x^4*Log[1 - x^2])/1152 - (5*(1 - x^2)*Log[1 - x^2])/576
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 = 1.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39
method | result | size |
default | \(-x^{2}+x -\frac {25}{\left (-5+x \right )^{2}}-\frac {10}{-5+x}+\ln \left (-x^{2}+1\right ) x\) | \(32\) |
parts | \(-x^{2}+x -\frac {25}{\left (-5+x \right )^{2}}-\frac {10}{-5+x}+\ln \left (-x^{2}+1\right ) x\) | \(32\) |
risch | \(\ln \left (-x^{2}+1\right ) x -\frac {x^{4}-11 x^{3}+35 x^{2}-15 x -25}{x^{2}-10 x +25}\) | \(42\) |
norman | \(\frac {-335 x +\ln \left (-x^{2}+1\right ) x^{3}-10 \ln \left (-x^{2}+1\right ) x^{2}+25 \ln \left (-x^{2}+1\right ) x +11 x^{3}-x^{4}+900}{\left (-5+x \right )^{2}}\) | \(58\) |
parallelrisch | \(\frac {850-x^{4}+\ln \left (-x^{2}+1\right ) x^{3}+11 x^{3}-10 \ln \left (-x^{2}+1\right ) x^{2}-2 x^{2}+25 \ln \left (-x^{2}+1\right ) x -315 x}{x^{2}-10 x +25}\) | \(68\) |
orering | \(\frac {x \left (\left (x^{5}-15 x^{4}+74 x^{3}-110 x^{2}-75 x +125\right ) \ln \left (-x^{2}+1\right )-2 x^{6}+33 x^{5}-193 x^{4}+454 x^{3}-210 x^{2}-335 x +125\right )}{x^{5}-15 x^{4}+74 x^{3}-110 x^{2}-75 x +125}-\frac {\left (x^{7}-17 x^{6}+1280 x^{4}-7446 x^{3}+11460 x^{2}+7725 x -12875\right ) \left (-1+x \right ) \left (-5+x \right ) \left (1+x \right ) \left (\frac {\left (5 x^{4}-60 x^{3}+222 x^{2}-220 x -75\right ) \ln \left (-x^{2}+1\right )-\frac {2 \left (x^{5}-15 x^{4}+74 x^{3}-110 x^{2}-75 x +125\right ) x}{-x^{2}+1}-12 x^{5}+165 x^{4}-772 x^{3}+1362 x^{2}-420 x -335}{x^{5}-15 x^{4}+74 x^{3}-110 x^{2}-75 x +125}-\frac {\left (\left (x^{5}-15 x^{4}+74 x^{3}-110 x^{2}-75 x +125\right ) \ln \left (-x^{2}+1\right )-2 x^{6}+33 x^{5}-193 x^{4}+454 x^{3}-210 x^{2}-335 x +125\right ) \left (5 x^{4}-60 x^{3}+222 x^{2}-220 x -75\right )}{\left (x^{5}-15 x^{4}+74 x^{3}-110 x^{2}-75 x +125\right )^{2}}\right )}{2 \left (x^{8}-21 x^{7}+168 x^{6}-597 x^{5}+791 x^{4}+785 x^{3}-2650 x^{2}+1385 x +650\right )}\) | \(395\) |
Input:
int(((x^5-15*x^4+74*x^3-110*x^2-75*x+125)*ln(-x^2+1)-2*x^6+33*x^5-193*x^4+ 454*x^3-210*x^2-335*x+125)/(x^5-15*x^4+74*x^3-110*x^2-75*x+125),x,method=_ RETURNVERBOSE)
Output:
-x^2+x-25/(-5+x)^2-10/(-5+x)+ln(-x^2+1)*x
Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (23) = 46\).
Time = 0.09 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.26 \[ \int \frac {125-335 x-210 x^2+454 x^3-193 x^4+33 x^5-2 x^6+\left (125-75 x-110 x^2+74 x^3-15 x^4+x^5\right ) \log \left (1-x^2\right )}{125-75 x-110 x^2+74 x^3-15 x^4+x^5} \, dx=-\frac {x^{4} - 11 \, x^{3} + 35 \, x^{2} - {\left (x^{3} - 10 \, x^{2} + 25 \, x\right )} \log \left (-x^{2} + 1\right ) - 15 \, x - 25}{x^{2} - 10 \, x + 25} \] Input:
integrate(((x^5-15*x^4+74*x^3-110*x^2-75*x+125)*log(-x^2+1)-2*x^6+33*x^5-1 93*x^4+454*x^3-210*x^2-335*x+125)/(x^5-15*x^4+74*x^3-110*x^2-75*x+125),x, algorithm="fricas")
Output:
-(x^4 - 11*x^3 + 35*x^2 - (x^3 - 10*x^2 + 25*x)*log(-x^2 + 1) - 15*x - 25) /(x^2 - 10*x + 25)
Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {125-335 x-210 x^2+454 x^3-193 x^4+33 x^5-2 x^6+\left (125-75 x-110 x^2+74 x^3-15 x^4+x^5\right ) \log \left (1-x^2\right )}{125-75 x-110 x^2+74 x^3-15 x^4+x^5} \, dx=- x^{2} + x \log {\left (1 - x^{2} \right )} + x - \frac {10 x - 25}{x^{2} - 10 x + 25} \] Input:
integrate(((x**5-15*x**4+74*x**3-110*x**2-75*x+125)*ln(-x**2+1)-2*x**6+33* x**5-193*x**4+454*x**3-210*x**2-335*x+125)/(x**5-15*x**4+74*x**3-110*x**2- 75*x+125),x)
Output:
-x**2 + x*log(1 - x**2) + x - (10*x - 25)/(x**2 - 10*x + 25)
Leaf count of result is larger than twice the leaf count of optimal. 152 vs. \(2 (23) = 46\).
Time = 0.08 (sec) , antiderivative size = 152, normalized size of antiderivative = 6.61 \[ \int \frac {125-335 x-210 x^2+454 x^3-193 x^4+33 x^5-2 x^6+\left (125-75 x-110 x^2+74 x^3-15 x^4+x^5\right ) \log \left (1-x^2\right )}{125-75 x-110 x^2+74 x^3-15 x^4+x^5} \, dx=-x^{2} + {\left (x + 1\right )} \log \left (x + 1\right ) + {\left (x - 1\right )} \log \left (-x + 1\right ) + x + \frac {3125 \, {\left (47 \, x - 205\right )}}{144 \, {\left (x^{2} - 10 \, x + 25\right )}} + \frac {24125 \, {\left (23 \, x - 85\right )}}{288 \, {\left (x^{2} - 10 \, x + 25\right )}} - \frac {335 \, {\left (13 \, x - 95\right )}}{288 \, {\left (x^{2} - 10 \, x + 25\right )}} - \frac {5675 \, {\left (11 \, x - 25\right )}}{144 \, {\left (x^{2} - 10 \, x + 25\right )}} - \frac {34375 \, {\left (7 \, x - 29\right )}}{96 \, {\left (x^{2} - 10 \, x + 25\right )}} + \frac {125 \, {\left (5 \, x - 31\right )}}{288 \, {\left (x^{2} - 10 \, x + 25\right )}} - \frac {175 \, {\left (x - 35\right )}}{48 \, {\left (x^{2} - 10 \, x + 25\right )}} - \log \left (x + 1\right ) + \log \left (x - 1\right ) \] Input:
integrate(((x^5-15*x^4+74*x^3-110*x^2-75*x+125)*log(-x^2+1)-2*x^6+33*x^5-1 93*x^4+454*x^3-210*x^2-335*x+125)/(x^5-15*x^4+74*x^3-110*x^2-75*x+125),x, algorithm="maxima")
Output:
-x^2 + (x + 1)*log(x + 1) + (x - 1)*log(-x + 1) + x + 3125/144*(47*x - 205 )/(x^2 - 10*x + 25) + 24125/288*(23*x - 85)/(x^2 - 10*x + 25) - 335/288*(1 3*x - 95)/(x^2 - 10*x + 25) - 5675/144*(11*x - 25)/(x^2 - 10*x + 25) - 343 75/96*(7*x - 29)/(x^2 - 10*x + 25) + 125/288*(5*x - 31)/(x^2 - 10*x + 25) - 175/48*(x - 35)/(x^2 - 10*x + 25) - log(x + 1) + log(x - 1)
Time = 0.12 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48 \[ \int \frac {125-335 x-210 x^2+454 x^3-193 x^4+33 x^5-2 x^6+\left (125-75 x-110 x^2+74 x^3-15 x^4+x^5\right ) \log \left (1-x^2\right )}{125-75 x-110 x^2+74 x^3-15 x^4+x^5} \, dx=-x^{2} + x \log \left (-x^{2} + 1\right ) + x - \frac {5 \, {\left (2 \, x - 5\right )}}{x^{2} - 10 \, x + 25} \] Input:
integrate(((x^5-15*x^4+74*x^3-110*x^2-75*x+125)*log(-x^2+1)-2*x^6+33*x^5-1 93*x^4+454*x^3-210*x^2-335*x+125)/(x^5-15*x^4+74*x^3-110*x^2-75*x+125),x, algorithm="giac")
Output:
-x^2 + x*log(-x^2 + 1) + x - 5*(2*x - 5)/(x^2 - 10*x + 25)
Time = 3.36 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48 \[ \int \frac {125-335 x-210 x^2+454 x^3-193 x^4+33 x^5-2 x^6+\left (125-75 x-110 x^2+74 x^3-15 x^4+x^5\right ) \log \left (1-x^2\right )}{125-75 x-110 x^2+74 x^3-15 x^4+x^5} \, dx=x-\frac {10\,x-25}{x^2-10\,x+25}-x^2+x\,\ln \left (1-x^2\right ) \] Input:
int((335*x + log(1 - x^2)*(75*x + 110*x^2 - 74*x^3 + 15*x^4 - x^5 - 125) + 210*x^2 - 454*x^3 + 193*x^4 - 33*x^5 + 2*x^6 - 125)/(75*x + 110*x^2 - 74* x^3 + 15*x^4 - x^5 - 125),x)
Output:
x - (10*x - 25)/(x^2 - 10*x + 25) - x^2 + x*log(1 - x^2)
Time = 0.15 (sec) , antiderivative size = 124, normalized size of antiderivative = 5.39 \[ \int \frac {125-335 x-210 x^2+454 x^3-193 x^4+33 x^5-2 x^6+\left (125-75 x-110 x^2+74 x^3-15 x^4+x^5\right ) \log \left (1-x^2\right )}{125-75 x-110 x^2+74 x^3-15 x^4+x^5} \, dx=\frac {2 \,\mathrm {log}\left (-x^{2}+1\right ) x^{3}-18 \,\mathrm {log}\left (-x^{2}+1\right ) x^{2}+30 \,\mathrm {log}\left (-x^{2}+1\right ) x +50 \,\mathrm {log}\left (-x^{2}+1\right )-2 \,\mathrm {log}\left (x -1\right ) x^{2}+20 \,\mathrm {log}\left (x -1\right ) x -50 \,\mathrm {log}\left (x -1\right )-2 \,\mathrm {log}\left (x +1\right ) x^{2}+20 \,\mathrm {log}\left (x +1\right ) x -50 \,\mathrm {log}\left (x +1\right )-2 x^{4}+22 x^{3}-57 x^{2}-100 x +375}{2 x^{2}-20 x +50} \] Input:
int(((x^5-15*x^4+74*x^3-110*x^2-75*x+125)*log(-x^2+1)-2*x^6+33*x^5-193*x^4 +454*x^3-210*x^2-335*x+125)/(x^5-15*x^4+74*x^3-110*x^2-75*x+125),x)
Output:
(2*log( - x**2 + 1)*x**3 - 18*log( - x**2 + 1)*x**2 + 30*log( - x**2 + 1)* x + 50*log( - x**2 + 1) - 2*log(x - 1)*x**2 + 20*log(x - 1)*x - 50*log(x - 1) - 2*log(x + 1)*x**2 + 20*log(x + 1)*x - 50*log(x + 1) - 2*x**4 + 22*x* *3 - 57*x**2 - 100*x + 375)/(2*(x**2 - 10*x + 25))