Integrand size = 119, antiderivative size = 27 \[ \int \frac {3-3 x^2-3 x^3+\left (4+4 x^2+8 x^3\right ) \log \left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )+\left (1-x^2-x^3\right ) \log ^2\left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )}{-3+3 x^2+3 x^3} \, dx=-x-\frac {1}{3} x \log ^2\left (\frac {\log (3)}{\left (-\frac {1}{x}+x+x^2\right )^2}\right ) \]
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.85 (sec) , antiderivative size = 223, normalized size of antiderivative = 8.26 \[ \int \frac {3-3 x^2-3 x^3+\left (4+4 x^2+8 x^3\right ) \log \left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )+\left (1-x^2-x^3\right ) \log ^2\left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )}{-3+3 x^2+3 x^3} \, dx=\frac {x \left (3+\log ^2\left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )\right ) \left (3+2 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,1\right ] \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,2\right ]^2+2 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,1\right ]^2 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,3\right ]+2 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,2\right ] \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,3\right ]^2\right )}{3 \left (\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,2\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,3\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,2\right ]-\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,3\right ]\right )} \]
Integrate[(3 - 3*x^2 - 3*x^3 + (4 + 4*x^2 + 8*x^3)*Log[(x^2*Log[3])/(1 - 2 *x^2 - 2*x^3 + x^4 + 2*x^5 + x^6)] + (1 - x^2 - x^3)*Log[(x^2*Log[3])/(1 - 2*x^2 - 2*x^3 + x^4 + 2*x^5 + x^6)]^2)/(-3 + 3*x^2 + 3*x^3),x]
(x*(3 + Log[(x^2*Log[3])/(-1 + x^2 + x^3)^2]^2)*(3 + 2*Root[-1 + #1^2 + #1 ^3 & , 1, 0]*Root[-1 + #1^2 + #1^3 & , 2, 0]^2 + 2*Root[-1 + #1^2 + #1^3 & , 1, 0]^2*Root[-1 + #1^2 + #1^3 & , 3, 0] + 2*Root[-1 + #1^2 + #1^3 & , 2 , 0]*Root[-1 + #1^2 + #1^3 & , 3, 0]^2))/(3*(Root[-1 + #1^2 + #1^3 & , 1, 0] - Root[-1 + #1^2 + #1^3 & , 2, 0])*(Root[-1 + #1^2 + #1^3 & , 1, 0] - R oot[-1 + #1^2 + #1^3 & , 3, 0])*(Root[-1 + #1^2 + #1^3 & , 2, 0] - Root[-1 + #1^2 + #1^3 & , 3, 0]))
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.
\(\displaystyle \int \frac {-3 x^3-3 x^2+\left (-x^3-x^2+1\right ) \log ^2\left (\frac {x^2 \log (3)}{x^6+2 x^5+x^4-2 x^3-2 x^2+1}\right )+\left (8 x^3+4 x^2+4\right ) \log \left (\frac {x^2 \log (3)}{x^6+2 x^5+x^4-2 x^3-2 x^2+1}\right )+3}{3 x^3+3 x^2-3} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \left (-\frac {1}{3} \log ^2\left (\frac {x^2 \log (3)}{\left (x^3+x^2-1\right )^2}\right )+\frac {4 \left (2 x^3+x^2+1\right ) \log \left (\frac {x^2 \log (3)}{\left (x^3+x^2-1\right )^2}\right )}{3 \left (x^3+x^2-1\right )}-1\right )dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \left (-\frac {1}{3} \log ^2\left (\frac {x^2 \log (3)}{\left (x^3+x^2-1\right )^2}\right )+\frac {4 \left (2 x^3+x^2+1\right ) \log \left (\frac {x^2 \log (3)}{\left (x^3+x^2-1\right )^2}\right )}{3 \left (x^3+x^2-1\right )}-1\right )dx\) |
Int[(3 - 3*x^2 - 3*x^3 + (4 + 4*x^2 + 8*x^3)*Log[(x^2*Log[3])/(1 - 2*x^2 - 2*x^3 + x^4 + 2*x^5 + x^6)] + (1 - x^2 - x^3)*Log[(x^2*Log[3])/(1 - 2*x^2 - 2*x^3 + x^4 + 2*x^5 + x^6)]^2)/(-3 + 3*x^2 + 3*x^3),x]
3.27.14.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.42 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56
method | result | size |
norman | \(-x -\frac {x \ln \left (\frac {x^{2} \ln \left (3\right )}{x^{6}+2 x^{5}+x^{4}-2 x^{3}-2 x^{2}+1}\right )^{2}}{3}\) | \(42\) |
risch | \(-x -\frac {x \ln \left (\frac {x^{2} \ln \left (3\right )}{x^{6}+2 x^{5}+x^{4}-2 x^{3}-2 x^{2}+1}\right )^{2}}{3}\) | \(42\) |
parallelrisch | \(-\frac {x \ln \left (\frac {x^{2} \ln \left (3\right )}{x^{6}+2 x^{5}+x^{4}-2 x^{3}-2 x^{2}+1}\right )^{2}}{3}+2-x\) | \(43\) |
int(((-x^3-x^2+1)*ln(x^2*ln(3)/(x^6+2*x^5+x^4-2*x^3-2*x^2+1))^2+(8*x^3+4*x ^2+4)*ln(x^2*ln(3)/(x^6+2*x^5+x^4-2*x^3-2*x^2+1))-3*x^3-3*x^2+3)/(3*x^3+3* x^2-3),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {3-3 x^2-3 x^3+\left (4+4 x^2+8 x^3\right ) \log \left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )+\left (1-x^2-x^3\right ) \log ^2\left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )}{-3+3 x^2+3 x^3} \, dx=-\frac {1}{3} \, x \log \left (\frac {x^{2} \log \left (3\right )}{x^{6} + 2 \, x^{5} + x^{4} - 2 \, x^{3} - 2 \, x^{2} + 1}\right )^{2} - x \]
integrate(((-x^3-x^2+1)*log(x^2*log(3)/(x^6+2*x^5+x^4-2*x^3-2*x^2+1))^2+(8 *x^3+4*x^2+4)*log(x^2*log(3)/(x^6+2*x^5+x^4-2*x^3-2*x^2+1))-3*x^3-3*x^2+3) /(3*x^3+3*x^2-3),x, algorithm=\
Time = 0.11 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {3-3 x^2-3 x^3+\left (4+4 x^2+8 x^3\right ) \log \left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )+\left (1-x^2-x^3\right ) \log ^2\left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )}{-3+3 x^2+3 x^3} \, dx=- \frac {x \log {\left (\frac {x^{2} \log {\left (3 \right )}}{x^{6} + 2 x^{5} + x^{4} - 2 x^{3} - 2 x^{2} + 1} \right )}^{2}}{3} - x \]
integrate(((-x**3-x**2+1)*ln(x**2*ln(3)/(x**6+2*x**5+x**4-2*x**3-2*x**2+1) )**2+(8*x**3+4*x**2+4)*ln(x**2*ln(3)/(x**6+2*x**5+x**4-2*x**3-2*x**2+1))-3 *x**3-3*x**2+3)/(3*x**3+3*x**2-3),x)
Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (25) = 50\).
Time = 0.28 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.30 \[ \int \frac {3-3 x^2-3 x^3+\left (4+4 x^2+8 x^3\right ) \log \left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )+\left (1-x^2-x^3\right ) \log ^2\left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )}{-3+3 x^2+3 x^3} \, dx=-\frac {4}{3} \, x \log \left (x^{3} + x^{2} - 1\right )^{2} - \frac {4}{3} \, x \log \left (x\right )^{2} - \frac {4}{3} \, x \log \left (x\right ) \log \left (\log \left (3\right )\right ) - \frac {1}{3} \, {\left (\log \left (\log \left (3\right )\right )^{2} + 3\right )} x + \frac {4}{3} \, {\left (2 \, x \log \left (x\right ) + x \log \left (\log \left (3\right )\right )\right )} \log \left (x^{3} + x^{2} - 1\right ) \]
integrate(((-x^3-x^2+1)*log(x^2*log(3)/(x^6+2*x^5+x^4-2*x^3-2*x^2+1))^2+(8 *x^3+4*x^2+4)*log(x^2*log(3)/(x^6+2*x^5+x^4-2*x^3-2*x^2+1))-3*x^3-3*x^2+3) /(3*x^3+3*x^2-3),x, algorithm=\
-4/3*x*log(x^3 + x^2 - 1)^2 - 4/3*x*log(x)^2 - 4/3*x*log(x)*log(log(3)) - 1/3*(log(log(3))^2 + 3)*x + 4/3*(2*x*log(x) + x*log(log(3)))*log(x^3 + x^2 - 1)
Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (25) = 50\).
Time = 0.91 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.93 \[ \int \frac {3-3 x^2-3 x^3+\left (4+4 x^2+8 x^3\right ) \log \left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )+\left (1-x^2-x^3\right ) \log ^2\left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )}{-3+3 x^2+3 x^3} \, dx=-\frac {1}{3} \, x \log \left (x^{6} + 2 \, x^{5} + x^{4} - 2 \, x^{3} - 2 \, x^{2} + 1\right )^{2} + \frac {2}{3} \, x \log \left (x^{6} + 2 \, x^{5} + x^{4} - 2 \, x^{3} - 2 \, x^{2} + 1\right ) \log \left (x^{2} \log \left (3\right )\right ) - \frac {1}{3} \, x \log \left (x^{2} \log \left (3\right )\right )^{2} - x \]
integrate(((-x^3-x^2+1)*log(x^2*log(3)/(x^6+2*x^5+x^4-2*x^3-2*x^2+1))^2+(8 *x^3+4*x^2+4)*log(x^2*log(3)/(x^6+2*x^5+x^4-2*x^3-2*x^2+1))-3*x^3-3*x^2+3) /(3*x^3+3*x^2-3),x, algorithm=\
-1/3*x*log(x^6 + 2*x^5 + x^4 - 2*x^3 - 2*x^2 + 1)^2 + 2/3*x*log(x^6 + 2*x^ 5 + x^4 - 2*x^3 - 2*x^2 + 1)*log(x^2*log(3)) - 1/3*x*log(x^2*log(3))^2 - x
Time = 8.82 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {3-3 x^2-3 x^3+\left (4+4 x^2+8 x^3\right ) \log \left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )+\left (1-x^2-x^3\right ) \log ^2\left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )}{-3+3 x^2+3 x^3} \, dx=-\frac {x\,\left ({\ln \left (\frac {x^2\,\ln \left (3\right )}{x^6+2\,x^5+x^4-2\,x^3-2\,x^2+1}\right )}^2+3\right )}{3} \]