Integrand size = 83, antiderivative size = 27 \[ \int \frac {-45+54 x+7 x^2+x^3+\left (-18+18 x+6 x^2\right ) \log (3)+(-9+9 x) \log ^2(3)+(-9+9 x) \log (x)}{54 x+6 x^2+x^3+\left (18 x+6 x^2\right ) \log (3)+9 x \log ^2(3)+9 x \log (x)} \, dx=x+\log \left (\frac {5+\left (-1-\frac {x}{3}-\log (3)\right )^2+\log (x)}{26 x}\right ) \] Output:
ln(1/26*(ln(x)+(-1-ln(3)-1/3*x)^2+5)/x)+x
\[ \int \frac {-45+54 x+7 x^2+x^3+\left (-18+18 x+6 x^2\right ) \log (3)+(-9+9 x) \log ^2(3)+(-9+9 x) \log (x)}{54 x+6 x^2+x^3+\left (18 x+6 x^2\right ) \log (3)+9 x \log ^2(3)+9 x \log (x)} \, dx=\int \frac {-45+54 x+7 x^2+x^3+\left (-18+18 x+6 x^2\right ) \log (3)+(-9+9 x) \log ^2(3)+(-9+9 x) \log (x)}{54 x+6 x^2+x^3+\left (18 x+6 x^2\right ) \log (3)+9 x \log ^2(3)+9 x \log (x)} \, dx \] Input:
Integrate[(-45 + 54*x + 7*x^2 + x^3 + (-18 + 18*x + 6*x^2)*Log[3] + (-9 + 9*x)*Log[3]^2 + (-9 + 9*x)*Log[x])/(54*x + 6*x^2 + x^3 + (18*x + 6*x^2)*Lo g[3] + 9*x*Log[3]^2 + 9*x*Log[x]),x]
Output:
Integrate[(-45 + 54*x + 7*x^2 + x^3 + (-18 + 18*x + 6*x^2)*Log[3] + (-9 + 9*x)*Log[3]^2 + (-9 + 9*x)*Log[x])/(54*x + 6*x^2 + x^3 + (18*x + 6*x^2)*Lo g[3] + 9*x*Log[3]^2 + 9*x*Log[x]), x]
Time = 0.53 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {6, 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.
\(\displaystyle \int \frac {x^3+7 x^2+\left (6 x^2+18 x-18\right ) \log (3)+54 x+(9 x-9) \log ^2(3)+(9 x-9) \log (x)-45}{x^3+6 x^2+\left (6 x^2+18 x\right ) \log (3)+54 x+9 x \log ^2(3)+9 x \log (x)} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {x^3+7 x^2+\left (6 x^2+18 x-18\right ) \log (3)+54 x+(9 x-9) \log ^2(3)+(9 x-9) \log (x)-45}{x^3+6 x^2+\left (6 x^2+18 x\right ) \log (3)+x \left (54+9 \log ^2(3)\right )+9 x \log (x)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 x^2+6 x (1+\log (3))+9}{x \left (x^2+6 x (1+\log (3))+9 \log (x)+54 \left (1+\frac {1}{6} \left (\log ^2(3)+\log (9)\right )\right )\right )}+\frac {x-1}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \log \left (x^2+6 x (1+\log (3))+9 \log (x)+9 \left (6+\log ^2(3)+\log (9)\right )\right )+x-\log (x)\) |
Input:
Int[(-45 + 54*x + 7*x^2 + x^3 + (-18 + 18*x + 6*x^2)*Log[3] + (-9 + 9*x)*L og[3]^2 + (-9 + 9*x)*Log[x])/(54*x + 6*x^2 + x^3 + (18*x + 6*x^2)*Log[3] + 9*x*Log[3]^2 + 9*x*Log[x]),x]
Output:
x - Log[x] + Log[x^2 + 6*x*(1 + Log[3]) + 9*(6 + Log[3]^2 + Log[9]) + 9*Lo g[x]]
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Time = 0.32 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22
method | result | size |
risch | \(x -\ln \left (x \right )+\ln \left (\ln \left (3\right )^{2}+\frac {2 x \ln \left (3\right )}{3}+\frac {x^{2}}{9}+2 \ln \left (3\right )+\frac {2 x}{3}+\ln \left (x \right )+6\right )\) | \(33\) |
default | \(-\ln \left (x \right )+x +\ln \left (9 \ln \left (3\right )^{2}+6 x \ln \left (3\right )+x^{2}+18 \ln \left (3\right )+9 \ln \left (x \right )+6 x +54\right )\) | \(35\) |
norman | \(-\ln \left (x \right )+x +\ln \left (9 \ln \left (3\right )^{2}+6 x \ln \left (3\right )+x^{2}+18 \ln \left (3\right )+9 \ln \left (x \right )+6 x +54\right )\) | \(35\) |
parallelrisch | \(-\ln \left (x \right )+x +\ln \left (9 \ln \left (3\right )^{2}+6 x \ln \left (3\right )+x^{2}+18 \ln \left (3\right )+9 \ln \left (x \right )+6 x +54\right )\) | \(35\) |
Input:
int(((9*x-9)*ln(x)+(9*x-9)*ln(3)^2+(6*x^2+18*x-18)*ln(3)+x^3+7*x^2+54*x-45 )/(9*x*ln(x)+9*x*ln(3)^2+(6*x^2+18*x)*ln(3)+x^3+6*x^2+54*x),x,method=_RETU RNVERBOSE)
Output:
x-ln(x)+ln(ln(3)^2+2/3*x*ln(3)+1/9*x^2+2*ln(3)+2/3*x+ln(x)+6)
Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {-45+54 x+7 x^2+x^3+\left (-18+18 x+6 x^2\right ) \log (3)+(-9+9 x) \log ^2(3)+(-9+9 x) \log (x)}{54 x+6 x^2+x^3+\left (18 x+6 x^2\right ) \log (3)+9 x \log ^2(3)+9 x \log (x)} \, dx=x + \log \left (x^{2} + 6 \, {\left (x + 3\right )} \log \left (3\right ) + 9 \, \log \left (3\right )^{2} + 6 \, x + 9 \, \log \left (x\right ) + 54\right ) - \log \left (x\right ) \] Input:
integrate(((9*x-9)*log(x)+(9*x-9)*log(3)^2+(6*x^2+18*x-18)*log(3)+x^3+7*x^ 2+54*x-45)/(9*x*log(x)+9*x*log(3)^2+(6*x^2+18*x)*log(3)+x^3+6*x^2+54*x),x, algorithm="fricas")
Output:
x + log(x^2 + 6*(x + 3)*log(3) + 9*log(3)^2 + 6*x + 9*log(x) + 54) - log(x )
Time = 0.11 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {-45+54 x+7 x^2+x^3+\left (-18+18 x+6 x^2\right ) \log (3)+(-9+9 x) \log ^2(3)+(-9+9 x) \log (x)}{54 x+6 x^2+x^3+\left (18 x+6 x^2\right ) \log (3)+9 x \log ^2(3)+9 x \log (x)} \, dx=x - \log {\left (x \right )} + \log {\left (\frac {x^{2}}{9} + \frac {2 x}{3} + \frac {2 x \log {\left (3 \right )}}{3} + \log {\left (x \right )} + \log {\left (3 \right )}^{2} + 2 \log {\left (3 \right )} + 6 \right )} \] Input:
integrate(((9*x-9)*ln(x)+(9*x-9)*ln(3)**2+(6*x**2+18*x-18)*ln(3)+x**3+7*x* *2+54*x-45)/(9*x*ln(x)+9*x*ln(3)**2+(6*x**2+18*x)*ln(3)+x**3+6*x**2+54*x), x)
Output:
x - log(x) + log(x**2/9 + 2*x/3 + 2*x*log(3)/3 + log(x) + log(3)**2 + 2*lo g(3) + 6)
Time = 0.16 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {-45+54 x+7 x^2+x^3+\left (-18+18 x+6 x^2\right ) \log (3)+(-9+9 x) \log ^2(3)+(-9+9 x) \log (x)}{54 x+6 x^2+x^3+\left (18 x+6 x^2\right ) \log (3)+9 x \log ^2(3)+9 x \log (x)} \, dx=x + \log \left (\frac {1}{9} \, x^{2} + \frac {2}{3} \, x {\left (\log \left (3\right ) + 1\right )} + \log \left (3\right )^{2} + 2 \, \log \left (3\right ) + \log \left (x\right ) + 6\right ) - \log \left (x\right ) \] Input:
integrate(((9*x-9)*log(x)+(9*x-9)*log(3)^2+(6*x^2+18*x-18)*log(3)+x^3+7*x^ 2+54*x-45)/(9*x*log(x)+9*x*log(3)^2+(6*x^2+18*x)*log(3)+x^3+6*x^2+54*x),x, algorithm="maxima")
Output:
x + log(1/9*x^2 + 2/3*x*(log(3) + 1) + log(3)^2 + 2*log(3) + log(x) + 6) - log(x)
Time = 0.11 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {-45+54 x+7 x^2+x^3+\left (-18+18 x+6 x^2\right ) \log (3)+(-9+9 x) \log ^2(3)+(-9+9 x) \log (x)}{54 x+6 x^2+x^3+\left (18 x+6 x^2\right ) \log (3)+9 x \log ^2(3)+9 x \log (x)} \, dx=x + \log \left (x^{2} + 6 \, x \log \left (3\right ) + 9 \, \log \left (3\right )^{2} + 6 \, x + 18 \, \log \left (3\right ) + 9 \, \log \left (x\right ) + 54\right ) - \log \left (x\right ) \] Input:
integrate(((9*x-9)*log(x)+(9*x-9)*log(3)^2+(6*x^2+18*x-18)*log(3)+x^3+7*x^ 2+54*x-45)/(9*x*log(x)+9*x*log(3)^2+(6*x^2+18*x)*log(3)+x^3+6*x^2+54*x),x, algorithm="giac")
Output:
x + log(x^2 + 6*x*log(3) + 9*log(3)^2 + 6*x + 18*log(3) + 9*log(x) + 54) - log(x)
Time = 4.07 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {-45+54 x+7 x^2+x^3+\left (-18+18 x+6 x^2\right ) \log (3)+(-9+9 x) \log ^2(3)+(-9+9 x) \log (x)}{54 x+6 x^2+x^3+\left (18 x+6 x^2\right ) \log (3)+9 x \log ^2(3)+9 x \log (x)} \, dx=x+\ln \left (\frac {2\,x}{3}+\ln \left (9\,x\right )+\frac {2\,x\,\ln \left (3\right )}{3}+{\ln \left (3\right )}^2+\frac {x^2}{9}+6\right )-\ln \left (x\right ) \] Input:
int((54*x + log(3)*(18*x + 6*x^2 - 18) + log(3)^2*(9*x - 9) + log(x)*(9*x - 9) + 7*x^2 + x^3 - 45)/(54*x + log(3)*(18*x + 6*x^2) + 9*x*log(3)^2 + 9* x*log(x) + 6*x^2 + x^3),x)
Output:
x + log((2*x)/3 + log(9*x) + (2*x*log(3))/3 + log(3)^2 + x^2/9 + 6) - log( x)
Time = 0.17 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {-45+54 x+7 x^2+x^3+\left (-18+18 x+6 x^2\right ) \log (3)+(-9+9 x) \log ^2(3)+(-9+9 x) \log (x)}{54 x+6 x^2+x^3+\left (18 x+6 x^2\right ) \log (3)+9 x \log ^2(3)+9 x \log (x)} \, dx=\mathrm {log}\left (9 \,\mathrm {log}\left (x \right )+9 \mathrm {log}\left (3\right )^{2}+6 \,\mathrm {log}\left (3\right ) x +18 \,\mathrm {log}\left (3\right )+x^{2}+6 x +54\right )-\mathrm {log}\left (x \right )+x \] Input:
int(((9*x-9)*log(x)+(9*x-9)*log(3)^2+(6*x^2+18*x-18)*log(3)+x^3+7*x^2+54*x -45)/(9*x*log(x)+9*x*log(3)^2+(6*x^2+18*x)*log(3)+x^3+6*x^2+54*x),x)
Output:
log(9*log(x) + 9*log(3)**2 + 6*log(3)*x + 18*log(3) + x**2 + 6*x + 54) - l og(x) + x