Integrand size = 149, antiderivative size = 31 \[ \int \frac {\left (-12+24 x-12 x^2-12 x^3-2 x^4\right ) \log \left (e^{2 x} x^2\right )+\left (-12+18 x+22 x^2-20 x^3-14 x^4-2 x^5\right ) \log \left (\frac {3 x-3 x^2-x^3}{-2+3 x+x^2}\right )}{\left (6 x-15 x^2+4 x^3+6 x^4+x^5\right ) \log \left (e^{2 x} x^2\right ) \log \left (\frac {3 x-3 x^2-x^3}{-2+3 x+x^2}\right )} \, dx=\log \left (\frac {1}{\log \left (e^{2 x} x^2\right ) \log ^2\left (-x+\frac {1}{3-\frac {2}{x}+x}\right )}\right ) \]
Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.39 \[ \int \frac {\left (-12+24 x-12 x^2-12 x^3-2 x^4\right ) \log \left (e^{2 x} x^2\right )+\left (-12+18 x+22 x^2-20 x^3-14 x^4-2 x^5\right ) \log \left (\frac {3 x-3 x^2-x^3}{-2+3 x+x^2}\right )}{\left (6 x-15 x^2+4 x^3+6 x^4+x^5\right ) \log \left (e^{2 x} x^2\right ) \log \left (\frac {3 x-3 x^2-x^3}{-2+3 x+x^2}\right )} \, dx=2 \left (-\frac {1}{2} \log \left (\log \left (e^{2 x} x^2\right )\right )-\log \left (\log \left (-\frac {x \left (-3+3 x+x^2\right )}{-2+3 x+x^2}\right )\right )\right ) \]
Integrate[((-12 + 24*x - 12*x^2 - 12*x^3 - 2*x^4)*Log[E^(2*x)*x^2] + (-12 + 18*x + 22*x^2 - 20*x^3 - 14*x^4 - 2*x^5)*Log[(3*x - 3*x^2 - x^3)/(-2 + 3 *x + x^2)])/((6*x - 15*x^2 + 4*x^3 + 6*x^4 + x^5)*Log[E^(2*x)*x^2]*Log[(3* x - 3*x^2 - x^3)/(-2 + 3*x + x^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.
| \(\displaystyle \int \frac {\left (-2 x^4-12 x^3-12 x^2+24 x-12\right ) \log \left (e^{2 x} x^2\right )+\left (-2 x^5-14 x^4-20 x^3+22 x^2+18 x-12\right ) \log \left (\frac {-x^3-3 x^2+3 x}{x^2+3 x-2}\right )}{\left (x^5+6 x^4+4 x^3-15 x^2+6 x\right ) \log \left (e^{2 x} x^2\right ) \log \left (\frac {-x^3-3 x^2+3 x}{x^2+3 x-2}\right )} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\left (-2 x^4-12 x^3-12 x^2+24 x-12\right ) \log \left (e^{2 x} x^2\right )+\left (-2 x^5-14 x^4-20 x^3+22 x^2+18 x-12\right ) \log \left (\frac {-x^3-3 x^2+3 x}{x^2+3 x-2}\right )}{x \left (x^4+6 x^3+4 x^2-15 x+6\right ) \log \left (e^{2 x} x^2\right ) \log \left (\frac {-x^3-3 x^2+3 x}{x^2+3 x-2}\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {\left (-2 x^4-12 x^3-12 x^2+24 x-12\right ) \log \left (e^{2 x} x^2\right )+\left (-2 x^5-14 x^4-20 x^3+22 x^2+18 x-12\right ) \log \left (\frac {-x^3-3 x^2+3 x}{x^2+3 x-2}\right )}{x \left (-x^2-3 x+2\right ) \log \left (e^{2 x} x^2\right ) \log \left (\frac {-x^3-3 x^2+3 x}{x^2+3 x-2}\right )}+\frac {\left (-2 x^4-12 x^3-12 x^2+24 x-12\right ) \log \left (e^{2 x} x^2\right )+\left (-2 x^5-14 x^4-20 x^3+22 x^2+18 x-12\right ) \log \left (\frac {-x^3-3 x^2+3 x}{x^2+3 x-2}\right )}{x \left (x^2+3 x-3\right ) \log \left (e^{2 x} x^2\right ) \log \left (\frac {-x^3-3 x^2+3 x}{x^2+3 x-2}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \int \frac {1}{\log \left (e^{2 x} x^2\right )}dx-2 \int \frac {1}{x \log \left (e^{2 x} x^2\right )}dx-\frac {12 \int \frac {1}{\left (-2 x+\sqrt {17}-3\right ) \log \left (-\frac {x \left (x^2+3 x-3\right )}{x^2+3 x-2}\right )}dx}{\sqrt {17}}+4 \sqrt {\frac {3}{7}} \int \frac {1}{\left (-2 x+\sqrt {21}-3\right ) \log \left (-\frac {x \left (x^2+3 x-3\right )}{x^2+3 x-2}\right )}dx-2 \int \frac {1}{x \log \left (-\frac {x \left (x^2+3 x-3\right )}{x^2+3 x-2}\right )}dx+\frac {4}{17} \left (17-3 \sqrt {17}\right ) \int \frac {1}{\left (2 x-\sqrt {17}+3\right ) \log \left (-\frac {x \left (x^2+3 x-3\right )}{x^2+3 x-2}\right )}dx+\frac {4}{17} \left (17+3 \sqrt {17}\right ) \int \frac {1}{\left (2 x+\sqrt {17}+3\right ) \log \left (-\frac {x \left (x^2+3 x-3\right )}{x^2+3 x-2}\right )}dx-\frac {12 \int \frac {1}{\left (2 x+\sqrt {17}+3\right ) \log \left (-\frac {x \left (x^2+3 x-3\right )}{x^2+3 x-2}\right )}dx}{\sqrt {17}}-\frac {4}{7} \left (7-\sqrt {21}\right ) \int \frac {1}{\left (2 x-\sqrt {21}+3\right ) \log \left (-\frac {x \left (x^2+3 x-3\right )}{x^2+3 x-2}\right )}dx-\frac {4}{7} \left (7+\sqrt {21}\right ) \int \frac {1}{\left (2 x+\sqrt {21}+3\right ) \log \left (-\frac {x \left (x^2+3 x-3\right )}{x^2+3 x-2}\right )}dx+4 \sqrt {\frac {3}{7}} \int \frac {1}{\left (2 x+\sqrt {21}+3\right ) \log \left (-\frac {x \left (x^2+3 x-3\right )}{x^2+3 x-2}\right )}dx\) |
Int[((-12 + 24*x - 12*x^2 - 12*x^3 - 2*x^4)*Log[E^(2*x)*x^2] + (-12 + 18*x + 22*x^2 - 20*x^3 - 14*x^4 - 2*x^5)*Log[(3*x - 3*x^2 - x^3)/(-2 + 3*x + x ^2)])/((6*x - 15*x^2 + 4*x^3 + 6*x^4 + x^5)*Log[E^(2*x)*x^2]*Log[(3*x - 3* x^2 - x^3)/(-2 + 3*x + x^2)]),x]
3.9.72.3.1 Defintions of rubi rules used
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])
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 98.82 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.29
| method | result | size |
| default | \(-\ln \left (\ln \left ({\mathrm e}^{2 x} x^{2}\right )\right )-2 \ln \left (\ln \left (\frac {x \left (-x^{2}-3 x +3\right )}{x^{2}+3 x -2}\right )\right )\) | \(40\) |
| parts | \(-\ln \left (\ln \left ({\mathrm e}^{2 x} x^{2}\right )\right )-2 \ln \left (\ln \left (\frac {x \left (-x^{2}-3 x +3\right )}{x^{2}+3 x -2}\right )\right )\) | \(40\) |
| parallelrisch | \(-\ln \left (\ln \left ({\mathrm e}^{2 x} x^{2}\right )\right )-2 \ln \left (\ln \left (\frac {-x^{3}-3 x^{2}+3 x}{x^{2}+3 x -2}\right )\right )\) | \(43\) |
| risch | \(-2 \ln \left (\ln \left (x^{2}+3 x -2\right )+\frac {i \left (\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i \left (x^{2}+3 x -3\right )}{x^{2}+3 x -2}\right ) \operatorname {csgn}\left (\frac {i x \left (x^{2}+3 x -3\right )}{x^{2}+3 x -2}\right )-\pi \,\operatorname {csgn}\left (i x \right ) {\operatorname {csgn}\left (\frac {i x \left (x^{2}+3 x -3\right )}{x^{2}+3 x -2}\right )}^{2}+\pi \,\operatorname {csgn}\left (i \left (x^{2}+3 x -3\right )\right ) \operatorname {csgn}\left (\frac {i}{x^{2}+3 x -2}\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}+3 x -3\right )}{x^{2}+3 x -2}\right )-\pi \,\operatorname {csgn}\left (i \left (x^{2}+3 x -3\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+3 x -3\right )}{x^{2}+3 x -2}\right )}^{2}-\pi \,\operatorname {csgn}\left (\frac {i}{x^{2}+3 x -2}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+3 x -3\right )}{x^{2}+3 x -2}\right )}^{2}+\pi {\operatorname {csgn}\left (\frac {i \left (x^{2}+3 x -3\right )}{x^{2}+3 x -2}\right )}^{3}-\pi \,\operatorname {csgn}\left (\frac {i \left (x^{2}+3 x -3\right )}{x^{2}+3 x -2}\right ) {\operatorname {csgn}\left (\frac {i x \left (x^{2}+3 x -3\right )}{x^{2}+3 x -2}\right )}^{2}-\pi {\operatorname {csgn}\left (\frac {i x \left (x^{2}+3 x -3\right )}{x^{2}+3 x -2}\right )}^{3}+2 \pi {\operatorname {csgn}\left (\frac {i x \left (x^{2}+3 x -3\right )}{x^{2}+3 x -2}\right )}^{2}+2 i \ln \left (x \right )+2 i \ln \left (x^{2}+3 x -3\right )-2 \pi \right )}{2}\right )-\ln \left (\ln \left ({\mathrm e}^{x}\right )-\frac {i \left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+\pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{2 x} x^{2}\right )-\pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{2 x} x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )^{3}-2 \pi \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{x}\right )-\pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{2 x} x^{2}\right )^{2}+\pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{x}\right )^{2}+\pi \operatorname {csgn}\left (i {\mathrm e}^{2 x} x^{2}\right )^{3}+4 i \ln \left (x \right )\right )}{4}\right )\) | \(585\) |
int(((-2*x^4-12*x^3-12*x^2+24*x-12)*ln(exp(x)^2*x^2)+(-2*x^5-14*x^4-20*x^3 +22*x^2+18*x-12)*ln((-x^3-3*x^2+3*x)/(x^2+3*x-2)))/(x^5+6*x^4+4*x^3-15*x^2 +6*x)/ln((-x^3-3*x^2+3*x)/(x^2+3*x-2))/ln(exp(x)^2*x^2),x,method=_RETURNVE RBOSE)
Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \[ \int \frac {\left (-12+24 x-12 x^2-12 x^3-2 x^4\right ) \log \left (e^{2 x} x^2\right )+\left (-12+18 x+22 x^2-20 x^3-14 x^4-2 x^5\right ) \log \left (\frac {3 x-3 x^2-x^3}{-2+3 x+x^2}\right )}{\left (6 x-15 x^2+4 x^3+6 x^4+x^5\right ) \log \left (e^{2 x} x^2\right ) \log \left (\frac {3 x-3 x^2-x^3}{-2+3 x+x^2}\right )} \, dx=-\log \left (\log \left (x^{2} e^{\left (2 \, x\right )}\right )\right ) - 2 \, \log \left (\log \left (-\frac {x^{3} + 3 \, x^{2} - 3 \, x}{x^{2} + 3 \, x - 2}\right )\right ) \]
integrate(((-2*x^4-12*x^3-12*x^2+24*x-12)*log(exp(x)^2*x^2)+(-2*x^5-14*x^4 -20*x^3+22*x^2+18*x-12)*log((-x^3-3*x^2+3*x)/(x^2+3*x-2)))/(x^5+6*x^4+4*x^ 3-15*x^2+6*x)/log((-x^3-3*x^2+3*x)/(x^2+3*x-2))/log(exp(x)^2*x^2),x, algor ithm=\
Time = 0.46 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int \frac {\left (-12+24 x-12 x^2-12 x^3-2 x^4\right ) \log \left (e^{2 x} x^2\right )+\left (-12+18 x+22 x^2-20 x^3-14 x^4-2 x^5\right ) \log \left (\frac {3 x-3 x^2-x^3}{-2+3 x+x^2}\right )}{\left (6 x-15 x^2+4 x^3+6 x^4+x^5\right ) \log \left (e^{2 x} x^2\right ) \log \left (\frac {3 x-3 x^2-x^3}{-2+3 x+x^2}\right )} \, dx=- \log {\left (\log {\left (x^{2} e^{2 x} \right )} \right )} - 2 \log {\left (\log {\left (\frac {- x^{3} - 3 x^{2} + 3 x}{x^{2} + 3 x - 2} \right )} \right )} \]
integrate(((-2*x**4-12*x**3-12*x**2+24*x-12)*ln(exp(x)**2*x**2)+(-2*x**5-1 4*x**4-20*x**3+22*x**2+18*x-12)*ln((-x**3-3*x**2+3*x)/(x**2+3*x-2)))/(x**5 +6*x**4+4*x**3-15*x**2+6*x)/ln((-x**3-3*x**2+3*x)/(x**2+3*x-2))/ln(exp(x)* *2*x**2),x)
Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {\left (-12+24 x-12 x^2-12 x^3-2 x^4\right ) \log \left (e^{2 x} x^2\right )+\left (-12+18 x+22 x^2-20 x^3-14 x^4-2 x^5\right ) \log \left (\frac {3 x-3 x^2-x^3}{-2+3 x+x^2}\right )}{\left (6 x-15 x^2+4 x^3+6 x^4+x^5\right ) \log \left (e^{2 x} x^2\right ) \log \left (\frac {3 x-3 x^2-x^3}{-2+3 x+x^2}\right )} \, dx=-\log \left (x + \log \left (x\right )\right ) - 2 \, \log \left (-\log \left (x^{2} + 3 \, x - 2\right ) + \log \left (-x^{2} - 3 \, x + 3\right ) + \log \left (x\right )\right ) \]
integrate(((-2*x^4-12*x^3-12*x^2+24*x-12)*log(exp(x)^2*x^2)+(-2*x^5-14*x^4 -20*x^3+22*x^2+18*x-12)*log((-x^3-3*x^2+3*x)/(x^2+3*x-2)))/(x^5+6*x^4+4*x^ 3-15*x^2+6*x)/log((-x^3-3*x^2+3*x)/(x^2+3*x-2))/log(exp(x)^2*x^2),x, algor ithm=\
Time = 0.43 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {\left (-12+24 x-12 x^2-12 x^3-2 x^4\right ) \log \left (e^{2 x} x^2\right )+\left (-12+18 x+22 x^2-20 x^3-14 x^4-2 x^5\right ) \log \left (\frac {3 x-3 x^2-x^3}{-2+3 x+x^2}\right )}{\left (6 x-15 x^2+4 x^3+6 x^4+x^5\right ) \log \left (e^{2 x} x^2\right ) \log \left (\frac {3 x-3 x^2-x^3}{-2+3 x+x^2}\right )} \, dx=-\log \left (x + \log \left (x\right )\right ) - 2 \, \log \left (-\log \left (x^{2} + 3 \, x - 2\right ) + \log \left (-x^{2} - 3 \, x + 3\right ) + \log \left (x\right )\right ) \]
integrate(((-2*x^4-12*x^3-12*x^2+24*x-12)*log(exp(x)^2*x^2)+(-2*x^5-14*x^4 -20*x^3+22*x^2+18*x-12)*log((-x^3-3*x^2+3*x)/(x^2+3*x-2)))/(x^5+6*x^4+4*x^ 3-15*x^2+6*x)/log((-x^3-3*x^2+3*x)/(x^2+3*x-2))/log(exp(x)^2*x^2),x, algor ithm=\
Time = 10.60 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.29 \[ \int \frac {\left (-12+24 x-12 x^2-12 x^3-2 x^4\right ) \log \left (e^{2 x} x^2\right )+\left (-12+18 x+22 x^2-20 x^3-14 x^4-2 x^5\right ) \log \left (\frac {3 x-3 x^2-x^3}{-2+3 x+x^2}\right )}{\left (6 x-15 x^2+4 x^3+6 x^4+x^5\right ) \log \left (e^{2 x} x^2\right ) \log \left (\frac {3 x-3 x^2-x^3}{-2+3 x+x^2}\right )} \, dx=-\ln \left (2\,x+\ln \left (x^2\right )\right )-2\,\ln \left (\ln \left (-\frac {x^3+3\,x^2-3\,x}{x^2+3\,x-2}\right )\right ) \]