Integrand size = 131, antiderivative size = 18 \[ \int \frac {-81+x-7776 x^5+96 x^6+64 x^5 \log (-81+x)+\left (-12960 x^4+160 x^5\right ) \log ^2(-81+x)+96 x^4 \log ^3(-81+x)+\left (-7776 x^3+96 x^4\right ) \log ^4(-81+x)+48 x^3 \log ^5(-81+x)+\left (-1944 x^2+24 x^3\right ) \log ^6(-81+x)+8 x^2 \log ^7(-81+x)+\left (-162 x+2 x^2\right ) \log ^8(-81+x)}{-81+x} \, dx=x+x^2 \left (2 x+\log ^2(-81+x)\right )^4 \]
Leaf count is larger than twice the leaf count of optimal. \(51\) vs. \(2(18)=36\).
Time = 0.13 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.83 \[ \int \frac {-81+x-7776 x^5+96 x^6+64 x^5 \log (-81+x)+\left (-12960 x^4+160 x^5\right ) \log ^2(-81+x)+96 x^4 \log ^3(-81+x)+\left (-7776 x^3+96 x^4\right ) \log ^4(-81+x)+48 x^3 \log ^5(-81+x)+\left (-1944 x^2+24 x^3\right ) \log ^6(-81+x)+8 x^2 \log ^7(-81+x)+\left (-162 x+2 x^2\right ) \log ^8(-81+x)}{-81+x} \, dx=-4518872583777+x+16 x^6+32 x^5 \log ^2(-81+x)+24 x^4 \log ^4(-81+x)+8 x^3 \log ^6(-81+x)+x^2 \log ^8(-81+x) \]
Integrate[(-81 + x - 7776*x^5 + 96*x^6 + 64*x^5*Log[-81 + x] + (-12960*x^4 + 160*x^5)*Log[-81 + x]^2 + 96*x^4*Log[-81 + x]^3 + (-7776*x^3 + 96*x^4)* Log[-81 + x]^4 + 48*x^3*Log[-81 + x]^5 + (-1944*x^2 + 24*x^3)*Log[-81 + x] ^6 + 8*x^2*Log[-81 + x]^7 + (-162*x + 2*x^2)*Log[-81 + x]^8)/(-81 + x),x]
-4518872583777 + x + 16*x^6 + 32*x^5*Log[-81 + x]^2 + 24*x^4*Log[-81 + x]^ 4 + 8*x^3*Log[-81 + x]^6 + x^2*Log[-81 + x]^8
Leaf count is larger than twice the leaf count of optimal. \(170\) vs. \(2(18)=36\).
Time = 2.09 (sec) , antiderivative size = 170, normalized size of antiderivative = 9.44, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.015, Rules used = {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.
\(\displaystyle \int \frac {96 x^6-7776 x^5+64 x^5 \log (x-81)+96 x^4 \log ^3(x-81)+48 x^3 \log ^5(x-81)+\left (2 x^2-162 x\right ) \log ^8(x-81)+8 x^2 \log ^7(x-81)+\left (160 x^5-12960 x^4\right ) \log ^2(x-81)+\left (96 x^4-7776 x^3\right ) \log ^4(x-81)+\left (24 x^3-1944 x^2\right ) \log ^6(x-81)+x-81}{x-81} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (96 x^5+\frac {64 x^5 \log (x-81)}{x-81}+\frac {96 x^4 \log ^3(x-81)}{x-81}+160 x^4 \log ^2(x-81)+\frac {48 x^3 \log ^5(x-81)}{x-81}+96 x^3 \log ^4(x-81)+\frac {8 x^2 \log ^7(x-81)}{x-81}+24 x^2 \log ^6(x-81)+2 x \log ^8(x-81)+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 16 x^6+32 x^5 \log ^2(x-81)+x+(81-x)^2 \log ^8(x-81)-162 (81-x) \log ^8(x-81)+6561 \log ^8(x-81)-8 (81-x)^3 \log ^6(x-81)+1944 (81-x)^2 \log ^6(x-81)-157464 (81-x) \log ^6(x-81)+4251528 \log ^6(x-81)+24 (81-x)^4 \log ^4(x-81)-7776 (81-x)^3 \log ^4(x-81)+944784 (81-x)^2 \log ^4(x-81)-51018336 (81-x) \log ^4(x-81)+1033121304 \log ^4(x-81)\) |
Int[(-81 + x - 7776*x^5 + 96*x^6 + 64*x^5*Log[-81 + x] + (-12960*x^4 + 160 *x^5)*Log[-81 + x]^2 + 96*x^4*Log[-81 + x]^3 + (-7776*x^3 + 96*x^4)*Log[-8 1 + x]^4 + 48*x^3*Log[-81 + x]^5 + (-1944*x^2 + 24*x^3)*Log[-81 + x]^6 + 8 *x^2*Log[-81 + x]^7 + (-162*x + 2*x^2)*Log[-81 + x]^8)/(-81 + x),x]
x + 16*x^6 + 32*x^5*Log[-81 + x]^2 + 1033121304*Log[-81 + x]^4 - 51018336* (81 - x)*Log[-81 + x]^4 + 944784*(81 - x)^2*Log[-81 + x]^4 - 7776*(81 - x) ^3*Log[-81 + x]^4 + 24*(81 - x)^4*Log[-81 + x]^4 + 4251528*Log[-81 + x]^6 - 157464*(81 - x)*Log[-81 + x]^6 + 1944*(81 - x)^2*Log[-81 + x]^6 - 8*(81 - x)^3*Log[-81 + x]^6 + 6561*Log[-81 + x]^8 - 162*(81 - x)*Log[-81 + x]^8 + (81 - x)^2*Log[-81 + x]^8
3.30.72.3.1 Defintions of rubi rules used
Leaf count of result is larger than twice the leaf count of optimal. \(50\) vs. \(2(18)=36\).
Time = 0.48 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.83
method | result | size |
risch | \(\ln \left (x -81\right )^{8} x^{2}+8 \ln \left (x -81\right )^{6} x^{3}+24 \ln \left (x -81\right )^{4} x^{4}+32 \ln \left (x -81\right )^{2} x^{5}+16 x^{6}+x\) | \(51\) |
parallelrisch | \(\ln \left (x -81\right )^{8} x^{2}+8 \ln \left (x -81\right )^{6} x^{3}+24 \ln \left (x -81\right )^{4} x^{4}+32 \ln \left (x -81\right )^{2} x^{5}+16 x^{6}+x +\frac {81}{2}\) | \(52\) |
parts | \(x +111577100832 \ln \left (x -81\right )^{2}+16 x^{6}+12960 \ln \left (x -81\right )^{2} \left (x -81\right )^{4}+170061120 \ln \left (x -81\right )^{2} \left (x -81\right )^{2}+6887475360 \ln \left (x -81\right )^{2} \left (x -81\right )+51018336 \ln \left (x -81\right )^{4} \left (x -81\right )+2099520 \ln \left (x -81\right )^{2} \left (x -81\right )^{3}+1944 \ln \left (x -81\right )^{6} \left (x -81\right )^{2}+24 \ln \left (x -81\right )^{4} \left (x -81\right )^{4}+157464 \ln \left (x -81\right )^{6} \left (x -81\right )+\ln \left (x -81\right )^{8} \left (x -81\right )^{2}+162 \ln \left (x -81\right )^{8} \left (x -81\right )+8 \ln \left (x -81\right )^{6} \left (x -81\right )^{3}+6561 \ln \left (x -81\right )^{8}+4251528 \ln \left (x -81\right )^{6}+1033121304 \ln \left (x -81\right )^{4}+7776 \ln \left (x -81\right )^{4} \left (x -81\right )^{3}+32 \ln \left (x -81\right )^{2} \left (x -81\right )^{5}+944784 \ln \left (x -81\right )^{4} \left (x -81\right )^{2}\) | \(213\) |
derivativedivides | \(-27113235502257+334731302497 x +111577100832 \ln \left (x -81\right )^{2}+12960 \ln \left (x -81\right )^{2} \left (x -81\right )^{4}+170061120 \ln \left (x -81\right )^{2} \left (x -81\right )^{2}+6887475360 \ln \left (x -81\right )^{2} \left (x -81\right )+51018336 \ln \left (x -81\right )^{4} \left (x -81\right )+2099520 \ln \left (x -81\right )^{2} \left (x -81\right )^{3}+1944 \ln \left (x -81\right )^{6} \left (x -81\right )^{2}+24 \ln \left (x -81\right )^{4} \left (x -81\right )^{4}+157464 \ln \left (x -81\right )^{6} \left (x -81\right )+\ln \left (x -81\right )^{8} \left (x -81\right )^{2}+162 \ln \left (x -81\right )^{8} \left (x -81\right )+8 \ln \left (x -81\right )^{6} \left (x -81\right )^{3}+10331213040 \left (x -81\right )^{2}+16 \left (x -81\right )^{6}+7776 \left (x -81\right )^{5}+1574640 \left (x -81\right )^{4}+170061120 \left (x -81\right )^{3}+6561 \ln \left (x -81\right )^{8}+4251528 \ln \left (x -81\right )^{6}+1033121304 \ln \left (x -81\right )^{4}+7776 \ln \left (x -81\right )^{4} \left (x -81\right )^{3}+32 \ln \left (x -81\right )^{2} \left (x -81\right )^{5}+944784 \ln \left (x -81\right )^{4} \left (x -81\right )^{2}\) | \(246\) |
default | \(-27113235502257+334731302497 x +111577100832 \ln \left (x -81\right )^{2}+12960 \ln \left (x -81\right )^{2} \left (x -81\right )^{4}+170061120 \ln \left (x -81\right )^{2} \left (x -81\right )^{2}+6887475360 \ln \left (x -81\right )^{2} \left (x -81\right )+51018336 \ln \left (x -81\right )^{4} \left (x -81\right )+2099520 \ln \left (x -81\right )^{2} \left (x -81\right )^{3}+1944 \ln \left (x -81\right )^{6} \left (x -81\right )^{2}+24 \ln \left (x -81\right )^{4} \left (x -81\right )^{4}+157464 \ln \left (x -81\right )^{6} \left (x -81\right )+\ln \left (x -81\right )^{8} \left (x -81\right )^{2}+162 \ln \left (x -81\right )^{8} \left (x -81\right )+8 \ln \left (x -81\right )^{6} \left (x -81\right )^{3}+10331213040 \left (x -81\right )^{2}+16 \left (x -81\right )^{6}+7776 \left (x -81\right )^{5}+1574640 \left (x -81\right )^{4}+170061120 \left (x -81\right )^{3}+6561 \ln \left (x -81\right )^{8}+4251528 \ln \left (x -81\right )^{6}+1033121304 \ln \left (x -81\right )^{4}+7776 \ln \left (x -81\right )^{4} \left (x -81\right )^{3}+32 \ln \left (x -81\right )^{2} \left (x -81\right )^{5}+944784 \ln \left (x -81\right )^{4} \left (x -81\right )^{2}\) | \(246\) |
int(((2*x^2-162*x)*ln(x-81)^8+8*x^2*ln(x-81)^7+(24*x^3-1944*x^2)*ln(x-81)^ 6+48*x^3*ln(x-81)^5+(96*x^4-7776*x^3)*ln(x-81)^4+96*x^4*ln(x-81)^3+(160*x^ 5-12960*x^4)*ln(x-81)^2+64*x^5*ln(x-81)+96*x^6-7776*x^5+x-81)/(x-81),x,met hod=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (18) = 36\).
Time = 0.25 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.78 \[ \int \frac {-81+x-7776 x^5+96 x^6+64 x^5 \log (-81+x)+\left (-12960 x^4+160 x^5\right ) \log ^2(-81+x)+96 x^4 \log ^3(-81+x)+\left (-7776 x^3+96 x^4\right ) \log ^4(-81+x)+48 x^3 \log ^5(-81+x)+\left (-1944 x^2+24 x^3\right ) \log ^6(-81+x)+8 x^2 \log ^7(-81+x)+\left (-162 x+2 x^2\right ) \log ^8(-81+x)}{-81+x} \, dx=x^{2} \log \left (x - 81\right )^{8} + 8 \, x^{3} \log \left (x - 81\right )^{6} + 24 \, x^{4} \log \left (x - 81\right )^{4} + 32 \, x^{5} \log \left (x - 81\right )^{2} + 16 \, x^{6} + x \]
integrate(((2*x^2-162*x)*log(x-81)^8+8*x^2*log(x-81)^7+(24*x^3-1944*x^2)*l og(x-81)^6+48*x^3*log(x-81)^5+(96*x^4-7776*x^3)*log(x-81)^4+96*x^4*log(x-8 1)^3+(160*x^5-12960*x^4)*log(x-81)^2+64*x^5*log(x-81)+96*x^6-7776*x^5+x-81 )/(x-81),x, algorithm=\
x^2*log(x - 81)^8 + 8*x^3*log(x - 81)^6 + 24*x^4*log(x - 81)^4 + 32*x^5*lo g(x - 81)^2 + 16*x^6 + x
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (15) = 30\).
Time = 0.10 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.83 \[ \int \frac {-81+x-7776 x^5+96 x^6+64 x^5 \log (-81+x)+\left (-12960 x^4+160 x^5\right ) \log ^2(-81+x)+96 x^4 \log ^3(-81+x)+\left (-7776 x^3+96 x^4\right ) \log ^4(-81+x)+48 x^3 \log ^5(-81+x)+\left (-1944 x^2+24 x^3\right ) \log ^6(-81+x)+8 x^2 \log ^7(-81+x)+\left (-162 x+2 x^2\right ) \log ^8(-81+x)}{-81+x} \, dx=16 x^{6} + 32 x^{5} \log {\left (x - 81 \right )}^{2} + 24 x^{4} \log {\left (x - 81 \right )}^{4} + 8 x^{3} \log {\left (x - 81 \right )}^{6} + x^{2} \log {\left (x - 81 \right )}^{8} + x \]
integrate(((2*x**2-162*x)*ln(x-81)**8+8*x**2*ln(x-81)**7+(24*x**3-1944*x** 2)*ln(x-81)**6+48*x**3*ln(x-81)**5+(96*x**4-7776*x**3)*ln(x-81)**4+96*x**4 *ln(x-81)**3+(160*x**5-12960*x**4)*ln(x-81)**2+64*x**5*ln(x-81)+96*x**6-77 76*x**5+x-81)/(x-81),x)
16*x**6 + 32*x**5*log(x - 81)**2 + 24*x**4*log(x - 81)**4 + 8*x**3*log(x - 81)**6 + x**2*log(x - 81)**8 + x
Leaf count of result is larger than twice the leaf count of optimal. 1044 vs. \(2 (18) = 36\).
Time = 0.23 (sec) , antiderivative size = 1044, normalized size of antiderivative = 58.00 \[ \int \frac {-81+x-7776 x^5+96 x^6+64 x^5 \log (-81+x)+\left (-12960 x^4+160 x^5\right ) \log ^2(-81+x)+96 x^4 \log ^3(-81+x)+\left (-7776 x^3+96 x^4\right ) \log ^4(-81+x)+48 x^3 \log ^5(-81+x)+\left (-1944 x^2+24 x^3\right ) \log ^6(-81+x)+8 x^2 \log ^7(-81+x)+\left (-162 x+2 x^2\right ) \log ^8(-81+x)}{-81+x} \, dx=\text {Too large to display} \]
integrate(((2*x^2-162*x)*log(x-81)^8+8*x^2*log(x-81)^7+(24*x^3-1944*x^2)*l og(x-81)^6+48*x^3*log(x-81)^5+(96*x^4-7776*x^3)*log(x-81)^4+96*x^4*log(x-8 1)^3+(160*x^5-12960*x^4)*log(x-81)^2+64*x^5*log(x-81)+96*x^6-7776*x^5+x-81 )/(x-81),x, algorithm=\
6561*log(x - 81)^8 + 32/25*(25*log(x - 81)^2 - 10*log(x - 81) + 2)*(x - 81 )^5 + 16*x^6 + 4251528*log(x - 81)^6 + 3/4*(32*log(x - 81)^4 - 32*log(x - 81)^3 + 24*log(x - 81)^2 - 12*log(x - 81) + 3)*(x - 81)^4 + 3/4*(32*log(x - 81)^3 - 24*log(x - 81)^2 + 12*log(x - 81) - 3)*(x - 81)^4 + 1620*(8*log( x - 81)^2 - 4*log(x - 81) + 1)*(x - 81)^4 - 64/25*x^5 + 8/81*(81*log(x - 8 1)^6 - 162*log(x - 81)^5 + 270*log(x - 81)^4 - 360*log(x - 81)^3 + 360*log (x - 81)^2 - 240*log(x - 81) + 80)*(x - 81)^3 + 16/81*(81*log(x - 81)^5 - 135*log(x - 81)^4 + 180*log(x - 81)^3 - 180*log(x - 81)^2 + 120*log(x - 81 ) - 40)*(x - 81)^3 + 288*(27*log(x - 81)^4 - 36*log(x - 81)^3 + 36*log(x - 81)^2 - 24*log(x - 81) + 8)*(x - 81)^3 + 1152*(9*log(x - 81)^3 - 9*log(x - 81)^2 + 6*log(x - 81) - 2)*(x - 81)^3 + 233280*(9*log(x - 81)^2 - 6*log( x - 81) + 2)*(x - 81)^3 - 2916/5*x^4 + 1033121304*log(x - 81)^4 + 1/2*(2*l og(x - 81)^8 - 8*log(x - 81)^7 + 28*log(x - 81)^6 - 84*log(x - 81)^5 + 210 *log(x - 81)^4 - 420*log(x - 81)^3 + 630*log(x - 81)^2 - 630*log(x - 81) + 315)*(x - 81)^2 + 1/2*(8*log(x - 81)^7 - 28*log(x - 81)^6 + 84*log(x - 81 )^5 - 210*log(x - 81)^4 + 420*log(x - 81)^3 - 630*log(x - 81)^2 + 630*log( x - 81) - 315)*(x - 81)^2 + 486*(4*log(x - 81)^6 - 12*log(x - 81)^5 + 30*l og(x - 81)^4 - 60*log(x - 81)^3 + 90*log(x - 81)^2 - 90*log(x - 81) + 45)* (x - 81)^2 + 1458*(4*log(x - 81)^5 - 10*log(x - 81)^4 + 20*log(x - 81)^3 - 30*log(x - 81)^2 + 30*log(x - 81) - 15)*(x - 81)^2 + 472392*(2*log(x -...
Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (18) = 36\).
Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.78 \[ \int \frac {-81+x-7776 x^5+96 x^6+64 x^5 \log (-81+x)+\left (-12960 x^4+160 x^5\right ) \log ^2(-81+x)+96 x^4 \log ^3(-81+x)+\left (-7776 x^3+96 x^4\right ) \log ^4(-81+x)+48 x^3 \log ^5(-81+x)+\left (-1944 x^2+24 x^3\right ) \log ^6(-81+x)+8 x^2 \log ^7(-81+x)+\left (-162 x+2 x^2\right ) \log ^8(-81+x)}{-81+x} \, dx=x^{2} \log \left (x - 81\right )^{8} + 8 \, x^{3} \log \left (x - 81\right )^{6} + 24 \, x^{4} \log \left (x - 81\right )^{4} + 32 \, x^{5} \log \left (x - 81\right )^{2} + 16 \, x^{6} + x \]
integrate(((2*x^2-162*x)*log(x-81)^8+8*x^2*log(x-81)^7+(24*x^3-1944*x^2)*l og(x-81)^6+48*x^3*log(x-81)^5+(96*x^4-7776*x^3)*log(x-81)^4+96*x^4*log(x-8 1)^3+(160*x^5-12960*x^4)*log(x-81)^2+64*x^5*log(x-81)+96*x^6-7776*x^5+x-81 )/(x-81),x, algorithm=\
x^2*log(x - 81)^8 + 8*x^3*log(x - 81)^6 + 24*x^4*log(x - 81)^4 + 32*x^5*lo g(x - 81)^2 + 16*x^6 + x
Time = 9.24 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.78 \[ \int \frac {-81+x-7776 x^5+96 x^6+64 x^5 \log (-81+x)+\left (-12960 x^4+160 x^5\right ) \log ^2(-81+x)+96 x^4 \log ^3(-81+x)+\left (-7776 x^3+96 x^4\right ) \log ^4(-81+x)+48 x^3 \log ^5(-81+x)+\left (-1944 x^2+24 x^3\right ) \log ^6(-81+x)+8 x^2 \log ^7(-81+x)+\left (-162 x+2 x^2\right ) \log ^8(-81+x)}{-81+x} \, dx=16\,x^6+32\,x^5\,{\ln \left (x-81\right )}^2+24\,x^4\,{\ln \left (x-81\right )}^4+8\,x^3\,{\ln \left (x-81\right )}^6+x^2\,{\ln \left (x-81\right )}^8+x \]
int((x - log(x - 81)^8*(162*x - 2*x^2) + 64*x^5*log(x - 81) - log(x - 81)^ 6*(1944*x^2 - 24*x^3) - log(x - 81)^4*(7776*x^3 - 96*x^4) - log(x - 81)^2* (12960*x^4 - 160*x^5) - 7776*x^5 + 96*x^6 + 96*x^4*log(x - 81)^3 + 48*x^3* log(x - 81)^5 + 8*x^2*log(x - 81)^7 - 81)/(x - 81),x)