Integrand size = 132, antiderivative size = 25 \[ \int \frac {9-21 x+6 x^2+(3-6 x) \log (4)+(-9+9 x+(-3+3 x) \log (4)) \log \left (-x+x^2\right )+\left (-9+15 x-7 x^2+x^3+\left (-6+8 x-2 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right ) \log ^2\left (-x+x^2\right )}{\left (-9+15 x-7 x^2+x^3+\left (-6+8 x-2 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right ) \log ^2\left (-x+x^2\right )} \, dx=-1+x+\frac {3 x}{(3-x+\log (4)) \log \left (-x+x^2\right )} \] Output:
3*x/ln(x^2-x)/(2*ln(2)+3-x)+x-1
\[ \int \frac {9-21 x+6 x^2+(3-6 x) \log (4)+(-9+9 x+(-3+3 x) \log (4)) \log \left (-x+x^2\right )+\left (-9+15 x-7 x^2+x^3+\left (-6+8 x-2 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right ) \log ^2\left (-x+x^2\right )}{\left (-9+15 x-7 x^2+x^3+\left (-6+8 x-2 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right ) \log ^2\left (-x+x^2\right )} \, dx=\int \frac {9-21 x+6 x^2+(3-6 x) \log (4)+(-9+9 x+(-3+3 x) \log (4)) \log \left (-x+x^2\right )+\left (-9+15 x-7 x^2+x^3+\left (-6+8 x-2 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right ) \log ^2\left (-x+x^2\right )}{\left (-9+15 x-7 x^2+x^3+\left (-6+8 x-2 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right ) \log ^2\left (-x+x^2\right )} \, dx \] Input:
Integrate[(9 - 21*x + 6*x^2 + (3 - 6*x)*Log[4] + (-9 + 9*x + (-3 + 3*x)*Lo g[4])*Log[-x + x^2] + (-9 + 15*x - 7*x^2 + x^3 + (-6 + 8*x - 2*x^2)*Log[4] + (-1 + x)*Log[4]^2)*Log[-x + x^2]^2)/((-9 + 15*x - 7*x^2 + x^3 + (-6 + 8 *x - 2*x^2)*Log[4] + (-1 + x)*Log[4]^2)*Log[-x + x^2]^2),x]
Output:
Integrate[(9 - 21*x + 6*x^2 + (3 - 6*x)*Log[4] + (-9 + 9*x + (-3 + 3*x)*Lo g[4])*Log[-x + x^2] + (-9 + 15*x - 7*x^2 + x^3 + (-6 + 8*x - 2*x^2)*Log[4] + (-1 + x)*Log[4]^2)*Log[-x + x^2]^2)/((-9 + 15*x - 7*x^2 + x^3 + (-6 + 8 *x - 2*x^2)*Log[4] + (-1 + x)*Log[4]^2)*Log[-x + x^2]^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.
| \(\displaystyle \int \frac {6 x^2+(9 x+(3 x-3) \log (4)-9) \log \left (x^2-x\right )+\left (x^3-7 x^2+\left (-2 x^2+8 x-6\right ) \log (4)+15 x+(x-1) \log ^2(4)-9\right ) \log ^2\left (x^2-x\right )-21 x+(3-6 x) \log (4)+9}{\left (x^3-7 x^2+\left (-2 x^2+8 x-6\right ) \log (4)+15 x+(x-1) \log ^2(4)-9\right ) \log ^2\left (x^2-x\right )} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {6 x^2+(9 x+(3 x-3) \log (4)-9) \log \left (x^2-x\right )+\left (x^3-7 x^2+\left (-2 x^2+8 x-6\right ) \log (4)+15 x+(x-1) \log ^2(4)-9\right ) \log ^2\left (x^2-x\right )-21 x+(3-6 x) \log (4)+9}{(2+\log (4)) (-x+3+\log (4))^2 \log ^2\left (x^2-x\right )}+\frac {6 x^2+(9 x+(3 x-3) \log (4)-9) \log \left (x^2-x\right )+\left (x^3-7 x^2+\left (-2 x^2+8 x-6\right ) \log (4)+15 x+(x-1) \log ^2(4)-9\right ) \log ^2\left (x^2-x\right )-21 x+(3-6 x) \log (4)+9}{(x-1) (2+\log (4))^2 \log ^2\left (x^2-x\right )}-\frac {6 x^2+(9 x+(3 x-3) \log (4)-9) \log \left (x^2-x\right )+\left (x^3-7 x^2+\left (-2 x^2+8 x-6\right ) \log (4)+15 x+(x-1) \log ^2(4)-9\right ) \log ^2\left (x^2-x\right )-21 x+(3-6 x) \log (4)+9}{(2+\log (4))^2 (x-3-\log (4)) \log ^2\left (x^2-x\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6 \int \frac {x^2}{(x-\log (4)-3)^2 \log ^2((x-1) x)}dx}{2+\log (4)}-\frac {3 \int \frac {2 x-1}{\log ^2((x-1) x)}dx}{(2+\log (4))^2}+\frac {9 \int \frac {1}{(x-\log (4)-3)^2 \log ^2((x-1) x)}dx}{2+\log (4)}-\frac {21 \int \frac {x}{(x-\log (4)-3)^2 \log ^2((x-1) x)}dx}{2+\log (4)}-\frac {3 \log (4) \int \frac {2 x-1}{(x-\log (4)-3)^2 \log ^2((x-1) x)}dx}{2+\log (4)}+\frac {3 \int \frac {(2 x-1) (x-\log (4)-3)}{(x-1) \log ^2((x-1) x)}dx}{(2+\log (4))^2}+\frac {(9+\log (64)) \int \frac {1}{\log ((x-1) x)}dx}{(2+\log (4))^2}+\frac {3 (3+\log (4)) \int \frac {x-1}{(x-\log (4)-3)^2 \log ((x-1) x)}dx}{2+\log (4)}-\frac {3 (3+\log (4)) \int \frac {x-1}{(x-\log (4)-3) \log ((x-1) x)}dx}{(2+\log (4))^2}-\frac {x^3}{3 (2+\log (4))^2}+\frac {x^2 (4+\log (4))}{2 (2+\log (4))^2}+\frac {x^2}{2 (2+\log (4))}-\frac {x (3+\log (4))}{(2+\log (4))^2}-\frac {x}{2+\log (4)}-\frac {(-x+3+\log (4))^3}{3 (2+\log (4))^2}\) |
Input:
Int[(9 - 21*x + 6*x^2 + (3 - 6*x)*Log[4] + (-9 + 9*x + (-3 + 3*x)*Log[4])* Log[-x + x^2] + (-9 + 15*x - 7*x^2 + x^3 + (-6 + 8*x - 2*x^2)*Log[4] + (-1 + x)*Log[4]^2)*Log[-x + x^2]^2)/((-9 + 15*x - 7*x^2 + x^3 + (-6 + 8*x - 2 *x^2)*Log[4] + (-1 + x)*Log[4]^2)*Log[-x + x^2]^2),x]
Output:
$Aborted
Time = 2.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08
| method | result | size |
| risch | \(x +\frac {3 x}{\ln \left (x^{2}-x \right ) \left (2 \ln \left (2\right )+3-x \right )}\) | \(27\) |
| norman | \(\frac {\left (4 \ln \left (2\right )^{2}+12 \ln \left (2\right )+9\right ) \ln \left (x^{2}-x \right )+3 x -\ln \left (x^{2}-x \right ) x^{2}}{\left (2 \ln \left (2\right )+3-x \right ) \ln \left (x^{2}-x \right )}\) | \(61\) |
| parallelrisch | \(\frac {16 \ln \left (2\right )^{3} \ln \left (x^{2}-x \right )-4 \ln \left (2\right ) \ln \left (x^{2}-x \right ) x^{2}+92 \ln \left (2\right )^{2} \ln \left (x^{2}-x \right )-8 \ln \left (2\right ) \ln \left (x^{2}-x \right ) x -7 \ln \left (x^{2}-x \right ) x^{2}+12 x \ln \left (2\right )+172 \ln \left (2\right ) \ln \left (x^{2}-x \right )-14 \ln \left (x^{2}-x \right ) x +21 x +105 \ln \left (x^{2}-x \right )}{\left (2 \ln \left (2\right )+3-x \right ) \ln \left (x^{2}-x \right ) \left (4 \ln \left (2\right )+7\right )}\) | \(142\) |
Input:
int(((4*(x-1)*ln(2)^2+2*(-2*x^2+8*x-6)*ln(2)+x^3-7*x^2+15*x-9)*ln(x^2-x)^2 +(2*(-3+3*x)*ln(2)+9*x-9)*ln(x^2-x)+2*(-6*x+3)*ln(2)+6*x^2-21*x+9)/(4*(x-1 )*ln(2)^2+2*(-2*x^2+8*x-6)*ln(2)+x^3-7*x^2+15*x-9)/ln(x^2-x)^2,x,method=_R ETURNVERBOSE)
Output:
x+3*x/ln(x^2-x)/(2*ln(2)+3-x)
Time = 0.08 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.80 \[ \int \frac {9-21 x+6 x^2+(3-6 x) \log (4)+(-9+9 x+(-3+3 x) \log (4)) \log \left (-x+x^2\right )+\left (-9+15 x-7 x^2+x^3+\left (-6+8 x-2 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right ) \log ^2\left (-x+x^2\right )}{\left (-9+15 x-7 x^2+x^3+\left (-6+8 x-2 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right ) \log ^2\left (-x+x^2\right )} \, dx=\frac {{\left (x^{2} - 2 \, x \log \left (2\right ) - 3 \, x\right )} \log \left (x^{2} - x\right ) - 3 \, x}{{\left (x - 2 \, \log \left (2\right ) - 3\right )} \log \left (x^{2} - x\right )} \] Input:
integrate(((4*(-1+x)*log(2)^2+2*(-2*x^2+8*x-6)*log(2)+x^3-7*x^2+15*x-9)*lo g(x^2-x)^2+(2*(-3+3*x)*log(2)+9*x-9)*log(x^2-x)+2*(-6*x+3)*log(2)+6*x^2-21 *x+9)/(4*(-1+x)*log(2)^2+2*(-2*x^2+8*x-6)*log(2)+x^3-7*x^2+15*x-9)/log(x^2 -x)^2,x, algorithm="fricas")
Output:
((x^2 - 2*x*log(2) - 3*x)*log(x^2 - x) - 3*x)/((x - 2*log(2) - 3)*log(x^2 - x))
Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {9-21 x+6 x^2+(3-6 x) \log (4)+(-9+9 x+(-3+3 x) \log (4)) \log \left (-x+x^2\right )+\left (-9+15 x-7 x^2+x^3+\left (-6+8 x-2 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right ) \log ^2\left (-x+x^2\right )}{\left (-9+15 x-7 x^2+x^3+\left (-6+8 x-2 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right ) \log ^2\left (-x+x^2\right )} \, dx=x - \frac {3 x}{\left (x - 3 - 2 \log {\left (2 \right )}\right ) \log {\left (x^{2} - x \right )}} \] Input:
integrate(((4*(-1+x)*ln(2)**2+2*(-2*x**2+8*x-6)*ln(2)+x**3-7*x**2+15*x-9)* ln(x**2-x)**2+(2*(-3+3*x)*ln(2)+9*x-9)*ln(x**2-x)+2*(-6*x+3)*ln(2)+6*x**2- 21*x+9)/(4*(-1+x)*ln(2)**2+2*(-2*x**2+8*x-6)*ln(2)+x**3-7*x**2+15*x-9)/ln( x**2-x)**2,x)
Output:
x - 3*x/((x - 3 - 2*log(2))*log(x**2 - x))
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (25) = 50\).
Time = 0.18 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.56 \[ \int \frac {9-21 x+6 x^2+(3-6 x) \log (4)+(-9+9 x+(-3+3 x) \log (4)) \log \left (-x+x^2\right )+\left (-9+15 x-7 x^2+x^3+\left (-6+8 x-2 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right ) \log ^2\left (-x+x^2\right )}{\left (-9+15 x-7 x^2+x^3+\left (-6+8 x-2 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right ) \log ^2\left (-x+x^2\right )} \, dx=\frac {{\left (x^{2} - x {\left (2 \, \log \left (2\right ) + 3\right )}\right )} \log \left (x - 1\right ) + {\left (x^{2} - x {\left (2 \, \log \left (2\right ) + 3\right )}\right )} \log \left (x\right ) - 3 \, x}{{\left (x - 2 \, \log \left (2\right ) - 3\right )} \log \left (x - 1\right ) + {\left (x - 2 \, \log \left (2\right ) - 3\right )} \log \left (x\right )} \] Input:
integrate(((4*(-1+x)*log(2)^2+2*(-2*x^2+8*x-6)*log(2)+x^3-7*x^2+15*x-9)*lo g(x^2-x)^2+(2*(-3+3*x)*log(2)+9*x-9)*log(x^2-x)+2*(-6*x+3)*log(2)+6*x^2-21 *x+9)/(4*(-1+x)*log(2)^2+2*(-2*x^2+8*x-6)*log(2)+x^3-7*x^2+15*x-9)/log(x^2 -x)^2,x, algorithm="maxima")
Output:
((x^2 - x*(2*log(2) + 3))*log(x - 1) + (x^2 - x*(2*log(2) + 3))*log(x) - 3 *x)/((x - 2*log(2) - 3)*log(x - 1) + (x - 2*log(2) - 3)*log(x))
Time = 0.17 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.60 \[ \int \frac {9-21 x+6 x^2+(3-6 x) \log (4)+(-9+9 x+(-3+3 x) \log (4)) \log \left (-x+x^2\right )+\left (-9+15 x-7 x^2+x^3+\left (-6+8 x-2 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right ) \log ^2\left (-x+x^2\right )}{\left (-9+15 x-7 x^2+x^3+\left (-6+8 x-2 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right ) \log ^2\left (-x+x^2\right )} \, dx=x - \frac {3 \, x}{x \log \left (x^{2} - x\right ) - 2 \, \log \left (2\right ) \log \left (x^{2} - x\right ) - 3 \, \log \left (x^{2} - x\right )} \] Input:
integrate(((4*(-1+x)*log(2)^2+2*(-2*x^2+8*x-6)*log(2)+x^3-7*x^2+15*x-9)*lo g(x^2-x)^2+(2*(-3+3*x)*log(2)+9*x-9)*log(x^2-x)+2*(-6*x+3)*log(2)+6*x^2-21 *x+9)/(4*(-1+x)*log(2)^2+2*(-2*x^2+8*x-6)*log(2)+x^3-7*x^2+15*x-9)/log(x^2 -x)^2,x, algorithm="giac")
Output:
x - 3*x/(x*log(x^2 - x) - 2*log(2)*log(x^2 - x) - 3*log(x^2 - x))
Time = 1.05 (sec) , antiderivative size = 712, normalized size of antiderivative = 28.48 \[ \int \frac {9-21 x+6 x^2+(3-6 x) \log (4)+(-9+9 x+(-3+3 x) \log (4)) \log \left (-x+x^2\right )+\left (-9+15 x-7 x^2+x^3+\left (-6+8 x-2 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right ) \log ^2\left (-x+x^2\right )}{\left (-9+15 x-7 x^2+x^3+\left (-6+8 x-2 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right ) \log ^2\left (-x+x^2\right )} \, dx=\text {Too large to display} \] Input:
int((log(x^2 - x)*(9*x + 2*log(2)*(3*x - 3) - 9) - 2*log(2)*(6*x - 3) - 21 *x + log(x^2 - x)^2*(15*x + 4*log(2)^2*(x - 1) - 2*log(2)*(2*x^2 - 8*x + 6 ) - 7*x^2 + x^3 - 9) + 6*x^2 + 9)/(log(x^2 - x)^2*(15*x + 4*log(2)^2*(x - 1) - 2*log(2)*(2*x^2 - 8*x + 6) - 7*x^2 + x^3 - 9)),x)
Output:
x + ((3*x)/(2*log(2) - x + 3) - (x*log(x^2 - x)*(log(64) + 9)*(x - 1))/((2 *x - 1)*(12*log(2) - 6*x - 4*x*log(2) + 4*log(2)^2 + x^2 + 9)))/log(x^2 - x) - ((1386*log(2) + 1017*log(4) + 72*log(8) - 404*log(512) - 6*log(2)*log (4) + 300*log(4)*log(8) - 232*log(4)*log(512) - 444*log(2)*log(4)^2 - 1248 *log(2)^2*log(4) - 96*log(2)*log(4)^3 + 248*log(4)^2*log(8) + 76*log(4)^3* log(8) + 8*log(4)^4*log(8) - 32*log(4)^2*log(512) - 1356*log(2)^2 + 936*lo g(4)^2 + 306*log(4)^3 + 36*log(4)^4 - 408*log(2)^2*log(4)^2 - 48*log(2)^2* log(4)^3)/(2*(150*log(4) + 60*log(4)^2 + 8*log(4)^3 + 125)) - (x*(552*log( 2) + 2169*log(4) + 204*log(8) - 228*log(512) - 810*log(2)*log(4) + 464*log (4)*log(8) - 72*log(4)*log(512) - 612*log(2)*log(4)^2 - 1008*log(2)^2*log( 4) - 96*log(2)*log(4)^3 + 312*log(4)^2*log(8) + 84*log(4)^3*log(8) + 8*log (4)^4*log(8) - 1092*log(2)^2 + 1296*log(4)^2 + 342*log(4)^3 + 36*log(4)^4 - 360*log(2)^2*log(4)^2 - 48*log(2)^2*log(4)^3 + 1125))/(150*log(4) + 60*l og(4)^2 + 8*log(4)^3 + 125) + (x^2*(432*log(2) + 1404*log(4) + 214*log(8) - 48*log(512) + 72*log(2)*log(4) + 288*log(4)*log(8) + 120*log(4)^2*log(8) + 16*log(4)^3*log(8) - 72*log(2)^2 + 540*log(4)^2 + 72*log(4)^3 + 1125))/ (150*log(4) + 60*log(4)^2 + 8*log(4)^3 + 125))/(6*log(4) - x*(14*log(4) + 2*log(4)^2 + 24) + x^2*(4*log(4) + 13) + log(4)^2 - 2*x^3 + 9) - (atan(((4 *x - (1300*log(4) + 720*log(4)^2 + 176*log(4)^3 + 16*log(4)^4 + 875)/(150* log(4) + 60*log(4)^2 + 8*log(4)^3 + 125))*(150*log(4) + 60*log(4)^2 + 8...
Time = 0.32 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.32 \[ \int \frac {9-21 x+6 x^2+(3-6 x) \log (4)+(-9+9 x+(-3+3 x) \log (4)) \log \left (-x+x^2\right )+\left (-9+15 x-7 x^2+x^3+\left (-6+8 x-2 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right ) \log ^2\left (-x+x^2\right )}{\left (-9+15 x-7 x^2+x^3+\left (-6+8 x-2 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right ) \log ^2\left (-x+x^2\right )} \, dx=\frac {x \left (2 \,\mathrm {log}\left (x^{2}-x \right ) \mathrm {log}\left (2\right )-\mathrm {log}\left (x^{2}-x \right ) x +3 \,\mathrm {log}\left (x^{2}-x \right )+3\right )}{\mathrm {log}\left (x^{2}-x \right ) \left (2 \,\mathrm {log}\left (2\right )-x +3\right )} \] Input:
int(((4*(-1+x)*log(2)^2+2*(-2*x^2+8*x-6)*log(2)+x^3-7*x^2+15*x-9)*log(x^2- x)^2+(2*(-3+3*x)*log(2)+9*x-9)*log(x^2-x)+2*(-6*x+3)*log(2)+6*x^2-21*x+9)/ (4*(-1+x)*log(2)^2+2*(-2*x^2+8*x-6)*log(2)+x^3-7*x^2+15*x-9)/log(x^2-x)^2, x)
Output:
(x*(2*log(x**2 - x)*log(2) - log(x**2 - x)*x + 3*log(x**2 - x) + 3))/(log( x**2 - x)*(2*log(2) - x + 3))