Integrand size = 49, antiderivative size = 24 \[ \int \frac {-18+33 x^3-15 x^6+\left (1-2 x^3+x^6\right ) \log (6)-9 x^3 \log (x)}{\left (x-2 x^4+x^7\right ) \log (6)} \, dx=\log (x)-\frac {3 \left (5+\frac {x}{x-x^4}\right ) \log (x)}{\log (6)} \] Output:
ln(x)-3*(x/(-x^4+x)+5)/ln(6)*ln(x)
Time = 0.14 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-18+33 x^3-15 x^6+\left (1-2 x^3+x^6\right ) \log (6)-9 x^3 \log (x)}{\left (x-2 x^4+x^7\right ) \log (6)} \, dx=\frac {\frac {3 \log (x)}{-1+x^3}+(-15+\log (6)) \log (x)}{\log (6)} \] Input:
Integrate[(-18 + 33*x^3 - 15*x^6 + (1 - 2*x^3 + x^6)*Log[6] - 9*x^3*Log[x] )/((x - 2*x^4 + x^7)*Log[6]),x]
Output:
((3*Log[x])/(-1 + x^3) + (-15 + Log[6])*Log[x])/Log[6]
Time = 0.58 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {27, 25, 2026, 1380, 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.
\(\displaystyle \int \frac {-15 x^6+33 x^3-9 x^3 \log (x)+\left (x^6-2 x^3+1\right ) \log (6)-18}{\left (x^7-2 x^4+x\right ) \log (6)} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int -\frac {15 x^6+9 \log (x) x^3-33 x^3-\left (x^6-2 x^3+1\right ) \log (6)+18}{x^7-2 x^4+x}dx}{\log (6)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {15 x^6+9 \log (x) x^3-33 x^3-\left (x^6-2 x^3+1\right ) \log (6)+18}{x^7-2 x^4+x}dx}{\log (6)}\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle -\frac {\int \frac {15 x^6+9 \log (x) x^3-33 x^3-\left (x^6-2 x^3+1\right ) \log (6)+18}{x \left (x^6-2 x^3+1\right )}dx}{\log (6)}\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle -\frac {\int \frac {15 x^6+9 \log (x) x^3-33 x^3-\left (x^6-2 x^3+1\right ) \log (6)+18}{x \left (1-x^3\right )^2}dx}{\log (6)}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {\int \left (\frac {15 x^5}{\left (x^3-1\right )^2}+\frac {9 \log (x) x^2}{\left (x^3-1\right )^2}-\frac {33 x^2}{\left (x^3-1\right )^2}+\frac {18}{\left (x^3-1\right )^2 x}-\frac {\log (6)}{x}\right )dx}{\log (6)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\frac {3 x^3 \log (x)}{1-x^3}-\log (6) \log (x)+18 \log (x)}{\log (6)}\) |
Input:
Int[(-18 + 33*x^3 - 15*x^6 + (1 - 2*x^3 + x^6)*Log[6] - 9*x^3*Log[x])/((x - 2*x^4 + x^7)*Log[6]),x]
Output:
-((18*Log[x] + (3*x^3*Log[x])/(1 - x^3) - Log[6]*Log[x])/Log[6])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Time = 0.36 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62
method | result | size |
parallelrisch | \(\frac {\ln \left (x \right ) x^{3} \ln \left (6\right )-15 x^{3} \ln \left (x \right )-\ln \left (6\right ) \ln \left (x \right )+18 \ln \left (x \right )}{\ln \left (6\right ) \left (x^{3}-1\right )}\) | \(39\) |
norman | \(\frac {\frac {\left (18-\ln \left (6\right )\right ) \ln \left (x \right )}{\ln \left (6\right )}-\frac {\left (-\ln \left (6\right )+15\right ) x^{3} \ln \left (x \right )}{\ln \left (6\right )}}{x^{3}-1}\) | \(40\) |
risch | \(\frac {3 \ln \left (x \right )}{\left (\ln \left (2\right )+\ln \left (3\right )\right ) \left (x^{3}-1\right )}+\frac {\ln \left (x \right ) \ln \left (3\right )}{\ln \left (2\right )+\ln \left (3\right )}+\frac {\ln \left (x \right ) \ln \left (2\right )}{\ln \left (2\right )+\ln \left (3\right )}-\frac {15 \ln \left (x \right )}{\ln \left (2\right )+\ln \left (3\right )}\) | \(55\) |
default | \(\frac {\ln \left (6\right ) \ln \left (x \right )-\frac {i \sqrt {3}\, \ln \left (x \right ) \ln \left (\frac {i \sqrt {3}-2 x -1}{-1+i \sqrt {3}}\right )}{3}+\frac {i \sqrt {3}\, \ln \left (x \right ) \ln \left (\frac {i \sqrt {3}+2 x +1}{1+i \sqrt {3}}\right )}{3}+\frac {\ln \left (x \right ) x}{-1+x}+\frac {\ln \left (x \right ) \left (i \ln \left (\frac {i \sqrt {3}-2 x -1}{-1+i \sqrt {3}}\right ) \sqrt {3}\, x^{2}-i \ln \left (\frac {i \sqrt {3}+2 x +1}{1+i \sqrt {3}}\right ) \sqrt {3}\, x^{2}+i \ln \left (\frac {i \sqrt {3}-2 x -1}{-1+i \sqrt {3}}\right ) \sqrt {3}\, x -i \ln \left (\frac {i \sqrt {3}+2 x +1}{1+i \sqrt {3}}\right ) \sqrt {3}\, x +i \sqrt {3}\, \ln \left (\frac {i \sqrt {3}-2 x -1}{-1+i \sqrt {3}}\right )-i \sqrt {3}\, \ln \left (\frac {i \sqrt {3}+2 x +1}{1+i \sqrt {3}}\right )+6 x^{2}+3 x \right )}{3 x^{2}+3 x +3}-18 \ln \left (x \right )}{\ln \left (6\right )}\) | \(290\) |
parts | \(\frac {\ln \left (-1+x \right )+\ln \left (x^{2}+x +1\right )+\left (\ln \left (6\right )-18\right ) \ln \left (x \right )}{\ln \left (6\right )}-\frac {9 \left (\frac {i \sqrt {3}\, \ln \left (x \right ) \ln \left (\frac {i \sqrt {3}-2 x -1}{-1+i \sqrt {3}}\right )}{27}-\frac {i \sqrt {3}\, \ln \left (x \right ) \ln \left (\frac {i \sqrt {3}+2 x +1}{1+i \sqrt {3}}\right )}{27}+\frac {\ln \left (-1+x \right )}{9}-\frac {\ln \left (x \right ) x}{9 \left (-1+x \right )}+\frac {\ln \left (x^{2}+x +1\right )}{9}-\frac {\ln \left (x \right ) \left (i \ln \left (\frac {i \sqrt {3}-2 x -1}{-1+i \sqrt {3}}\right ) \sqrt {3}\, x^{2}-i \ln \left (\frac {i \sqrt {3}+2 x +1}{1+i \sqrt {3}}\right ) \sqrt {3}\, x^{2}+i \ln \left (\frac {i \sqrt {3}-2 x -1}{-1+i \sqrt {3}}\right ) \sqrt {3}\, x -i \ln \left (\frac {i \sqrt {3}+2 x +1}{1+i \sqrt {3}}\right ) \sqrt {3}\, x +i \sqrt {3}\, \ln \left (\frac {i \sqrt {3}-2 x -1}{-1+i \sqrt {3}}\right )-i \sqrt {3}\, \ln \left (\frac {i \sqrt {3}+2 x +1}{1+i \sqrt {3}}\right )+6 x^{2}+3 x \right )}{27 \left (x^{2}+x +1\right )}\right )}{\ln \left (6\right )}\) | \(323\) |
Input:
int((-9*x^3*ln(x)+(x^6-2*x^3+1)*ln(6)-15*x^6+33*x^3-18)/(x^7-2*x^4+x)/ln(6 ),x,method=_RETURNVERBOSE)
Output:
1/ln(6)*(ln(x)*x^3*ln(6)-15*x^3*ln(x)-ln(6)*ln(x)+18*ln(x))/(x^3-1)
Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {-18+33 x^3-15 x^6+\left (1-2 x^3+x^6\right ) \log (6)-9 x^3 \log (x)}{\left (x-2 x^4+x^7\right ) \log (6)} \, dx=-\frac {{\left (15 \, x^{3} - {\left (x^{3} - 1\right )} \log \left (6\right ) - 18\right )} \log \left (x\right )}{{\left (x^{3} - 1\right )} \log \left (6\right )} \] Input:
integrate((-9*x^3*log(x)+(x^6-2*x^3+1)*log(6)-15*x^6+33*x^3-18)/(x^7-2*x^4 +x)/log(6),x, algorithm="fricas")
Output:
-(15*x^3 - (x^3 - 1)*log(6) - 18)*log(x)/((x^3 - 1)*log(6))
Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {-18+33 x^3-15 x^6+\left (1-2 x^3+x^6\right ) \log (6)-9 x^3 \log (x)}{\left (x-2 x^4+x^7\right ) \log (6)} \, dx=\frac {\left (-15 + \log {\left (6 \right )}\right ) \log {\left (x \right )}}{\log {\left (6 \right )}} + \frac {3 \log {\left (x \right )}}{x^{3} \log {\left (6 \right )} - \log {\left (6 \right )}} \] Input:
integrate((-9*x**3*ln(x)+(x**6-2*x**3+1)*ln(6)-15*x**6+33*x**3-18)/(x**7-2 *x**4+x)/ln(6),x)
Output:
(-15 + log(6))*log(x)/log(6) + 3*log(x)/(x**3*log(6) - log(6))
Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (24) = 48\).
Time = 0.12 (sec) , antiderivative size = 114, normalized size of antiderivative = 4.75 \[ \int \frac {-18+33 x^3-15 x^6+\left (1-2 x^3+x^6\right ) \log (6)-9 x^3 \log (x)}{\left (x-2 x^4+x^7\right ) \log (6)} \, dx=-\frac {{\left (\frac {1}{x^{3} - 1} + \log \left (x^{2} + x + 1\right ) + \log \left (x - 1\right ) - 3 \, \log \left (x\right )\right )} \log \left (6\right ) + {\left (\frac {1}{x^{3} - 1} - \log \left (x^{2} + x + 1\right ) - \log \left (x - 1\right )\right )} \log \left (6\right ) - \frac {2 \, \log \left (6\right )}{x^{3} - 1} - \frac {9 \, \log \left (x\right )}{x^{3} - 1} + 3 \, \log \left (x^{3} - 1\right ) - 3 \, \log \left (x^{3}\right ) - 3 \, \log \left (x^{2} + x + 1\right ) - 3 \, \log \left (x - 1\right ) + 54 \, \log \left (x\right )}{3 \, \log \left (6\right )} \] Input:
integrate((-9*x^3*log(x)+(x^6-2*x^3+1)*log(6)-15*x^6+33*x^3-18)/(x^7-2*x^4 +x)/log(6),x, algorithm="maxima")
Output:
-1/3*((1/(x^3 - 1) + log(x^2 + x + 1) + log(x - 1) - 3*log(x))*log(6) + (1 /(x^3 - 1) - log(x^2 + x + 1) - log(x - 1))*log(6) - 2*log(6)/(x^3 - 1) - 9*log(x)/(x^3 - 1) + 3*log(x^3 - 1) - 3*log(x^3) - 3*log(x^2 + x + 1) - 3* log(x - 1) + 54*log(x))/log(6)
Time = 0.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-18+33 x^3-15 x^6+\left (1-2 x^3+x^6\right ) \log (6)-9 x^3 \log (x)}{\left (x-2 x^4+x^7\right ) \log (6)} \, dx=\frac {{\left (\log \left (6\right ) - 15\right )} \log \left (x\right ) + \frac {3 \, \log \left (x\right )}{x^{3} - 1}}{\log \left (6\right )} \] Input:
integrate((-9*x^3*log(x)+(x^6-2*x^3+1)*log(6)-15*x^6+33*x^3-18)/(x^7-2*x^4 +x)/log(6),x, algorithm="giac")
Output:
((log(6) - 15)*log(x) + 3*log(x)/(x^3 - 1))/log(6)
Time = 3.47 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {-18+33 x^3-15 x^6+\left (1-2 x^3+x^6\right ) \log (6)-9 x^3 \log (x)}{\left (x-2 x^4+x^7\right ) \log (6)} \, dx=-\frac {\ln \left (x\right )\,\left (\ln \left (6\right )-x^3\,\ln \left (6\right )+15\,x^3-18\right )}{\ln \left (6\right )\,\left (x^3-1\right )} \] Input:
int(-(9*x^3*log(x) - log(6)*(x^6 - 2*x^3 + 1) - 33*x^3 + 15*x^6 + 18)/(log (6)*(x - 2*x^4 + x^7)),x)
Output:
-(log(x)*(log(6) - x^3*log(6) + 15*x^3 - 18))/(log(6)*(x^3 - 1))
Time = 0.16 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {-18+33 x^3-15 x^6+\left (1-2 x^3+x^6\right ) \log (6)-9 x^3 \log (x)}{\left (x-2 x^4+x^7\right ) \log (6)} \, dx=\frac {\mathrm {log}\left (x \right ) \left (\mathrm {log}\left (6\right ) x^{3}-\mathrm {log}\left (6\right )-15 x^{3}+18\right )}{\mathrm {log}\left (6\right ) \left (x^{3}-1\right )} \] Input:
int((-9*x^3*log(x)+(x^6-2*x^3+1)*log(6)-15*x^6+33*x^3-18)/(x^7-2*x^4+x)/lo g(6),x)
Output:
(log(x)*(log(6)*x**3 - log(6) - 15*x**3 + 18))/(log(6)*(x**3 - 1))