Integrand size = 101, antiderivative size = 27 \[ \int \frac {3456 x^3+6912 x^4-8 x^5-2 x^6+\left (1728 x^2+864 x^3-1728 x^4\right ) \log (x)+\left (288 x-720 x^2-864 x^3\right ) \log ^2(x)+\left (16-184 x-144 x^2\right ) \log ^3(x)+(-12-8 x) \log ^4(x)}{4 x^4+4 x^5+x^6} \, dx=\frac {2 \left (2-x \left (-1+x-\left (6+\frac {\log (x)}{x}\right )^4\right )\right )}{2+x} \] Output:
2/(2+x)*(2-x*(x-1-(6+ln(x)/x)^4))
Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.89 \[ \int \frac {3456 x^3+6912 x^4-8 x^5-2 x^6+\left (1728 x^2+864 x^3-1728 x^4\right ) \log (x)+\left (288 x-720 x^2-864 x^3\right ) \log ^2(x)+\left (16-184 x-144 x^2\right ) \log ^3(x)+(-12-8 x) \log ^4(x)}{4 x^4+4 x^5+x^6} \, dx=\frac {2 \left (-x^3 \left (2596+2 x+x^2\right )+864 x^3 \log (x)+216 x^2 \log ^2(x)+24 x \log ^3(x)+\log ^4(x)\right )}{x^3 (2+x)} \] Input:
Integrate[(3456*x^3 + 6912*x^4 - 8*x^5 - 2*x^6 + (1728*x^2 + 864*x^3 - 172 8*x^4)*Log[x] + (288*x - 720*x^2 - 864*x^3)*Log[x]^2 + (16 - 184*x - 144*x ^2)*Log[x]^3 + (-12 - 8*x)*Log[x]^4)/(4*x^4 + 4*x^5 + x^6),x]
Output:
(2*(-(x^3*(2596 + 2*x + x^2)) + 864*x^3*Log[x] + 216*x^2*Log[x]^2 + 24*x*L og[x]^3 + Log[x]^4))/(x^3*(2 + x))
Leaf count is larger than twice the leaf count of optimal. \(140\) vs. \(2(27)=54\).
Time = 1.55 (sec) , antiderivative size = 140, normalized size of antiderivative = 5.19, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2026, 2007, 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.
\(\displaystyle \int \frac {-2 x^6-8 x^5+6912 x^4+3456 x^3+\left (-144 x^2-184 x+16\right ) \log ^3(x)+\left (-864 x^3-720 x^2+288 x\right ) \log ^2(x)+\left (-1728 x^4+864 x^3+1728 x^2\right ) \log (x)+(-8 x-12) \log ^4(x)}{x^6+4 x^5+4 x^4} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {-2 x^6-8 x^5+6912 x^4+3456 x^3+\left (-144 x^2-184 x+16\right ) \log ^3(x)+\left (-864 x^3-720 x^2+288 x\right ) \log ^2(x)+\left (-1728 x^4+864 x^3+1728 x^2\right ) \log (x)+(-8 x-12) \log ^4(x)}{x^4 \left (x^2+4 x+4\right )}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {-2 x^6-8 x^5+6912 x^4+3456 x^3+\left (-144 x^2-184 x+16\right ) \log ^3(x)+\left (-864 x^3-720 x^2+288 x\right ) \log ^2(x)+\left (-1728 x^4+864 x^3+1728 x^2\right ) \log (x)+(-8 x-12) \log ^4(x)}{x^4 (x+2)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {4 (2 x+3) \log ^4(x)}{x^4 (x+2)^2}-\frac {2 x^2}{(x+2)^2}-\frac {864 \left (2 x^2-x-2\right ) \log (x)}{x^2 (x+2)^2}-\frac {8 \left (18 x^2+23 x-2\right ) \log ^3(x)}{x^4 (x+2)^2}-\frac {144 \left (6 x^2+5 x-2\right ) \log ^2(x)}{x^3 (x+2)^2}-\frac {8 x}{(x+2)^2}+\frac {3456}{x (x+2)^2}+\frac {6912}{(x+2)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\log ^4(x)}{x^3}-\frac {\log ^4(x)}{2 x^2}+\frac {24 \log ^3(x)}{x^2}-2 x-\frac {5192}{x+2}+\frac {\log ^4(x)}{4 x}+\frac {x \log ^4(x)}{8 (x+2)}-\frac {\log ^4(x)}{8}-\frac {12 \log ^3(x)}{x}-\frac {6 x \log ^3(x)}{x+2}+6 \log ^3(x)+\frac {216 \log ^2(x)}{x}+\frac {108 x \log ^2(x)}{x+2}-108 \log ^2(x)-\frac {864 x \log (x)}{x+2}+864 \log (x)\) |
Input:
Int[(3456*x^3 + 6912*x^4 - 8*x^5 - 2*x^6 + (1728*x^2 + 864*x^3 - 1728*x^4) *Log[x] + (288*x - 720*x^2 - 864*x^3)*Log[x]^2 + (16 - 184*x - 144*x^2)*Lo g[x]^3 + (-12 - 8*x)*Log[x]^4)/(4*x^4 + 4*x^5 + x^6),x]
Output:
-2*x - 5192/(2 + x) + 864*Log[x] - (864*x*Log[x])/(2 + x) - 108*Log[x]^2 + (216*Log[x]^2)/x + (108*x*Log[x]^2)/(2 + x) + 6*Log[x]^3 + (24*Log[x]^3)/ x^2 - (12*Log[x]^3)/x - (6*x*Log[x]^3)/(2 + x) - Log[x]^4/8 + Log[x]^4/x^3 - Log[x]^4/(2*x^2) + Log[x]^4/(4*x) + (x*Log[x]^4)/(8*(2 + x))
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
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 = 19.30 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.85
method | result | size |
parallelrisch | \(\frac {-2 x^{5}+1728 x^{3} \ln \left (x \right )+432 x^{2} \ln \left (x \right )^{2}+48 x \ln \left (x \right )^{3}+2 \ln \left (x \right )^{4}-5184 x^{3}}{x^{3} \left (2+x \right )}\) | \(50\) |
risch | \(\frac {2 \ln \left (x \right )^{4}}{x^{3} \left (2+x \right )}+\frac {48 \ln \left (x \right )^{3}}{x^{2} \left (2+x \right )}+\frac {432 \ln \left (x \right )^{2}}{x \left (2+x \right )}+\frac {1728 \ln \left (x \right )}{2+x}-\frac {2 \left (x^{2}+2 x +2596\right )}{2+x}\) | \(68\) |
orering | \(\text {Expression too large to display}\) | \(5372\) |
Input:
int(((-8*x-12)*ln(x)^4+(-144*x^2-184*x+16)*ln(x)^3+(-864*x^3-720*x^2+288*x )*ln(x)^2+(-1728*x^4+864*x^3+1728*x^2)*ln(x)-2*x^6-8*x^5+6912*x^4+3456*x^3 )/(x^6+4*x^5+4*x^4),x,method=_RETURNVERBOSE)
Output:
1/x^3*(-2*x^5+1728*x^3*ln(x)+432*x^2*ln(x)^2+48*x*ln(x)^3+2*ln(x)^4-5184*x ^3)/(2+x)
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (26) = 52\).
Time = 0.09 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.07 \[ \int \frac {3456 x^3+6912 x^4-8 x^5-2 x^6+\left (1728 x^2+864 x^3-1728 x^4\right ) \log (x)+\left (288 x-720 x^2-864 x^3\right ) \log ^2(x)+\left (16-184 x-144 x^2\right ) \log ^3(x)+(-12-8 x) \log ^4(x)}{4 x^4+4 x^5+x^6} \, dx=-\frac {2 \, {\left (x^{5} + 2 \, x^{4} - 864 \, x^{3} \log \left (x\right ) - 216 \, x^{2} \log \left (x\right )^{2} - 24 \, x \log \left (x\right )^{3} - \log \left (x\right )^{4} + 2596 \, x^{3}\right )}}{x^{4} + 2 \, x^{3}} \] Input:
integrate(((-8*x-12)*log(x)^4+(-144*x^2-184*x+16)*log(x)^3+(-864*x^3-720*x ^2+288*x)*log(x)^2+(-1728*x^4+864*x^3+1728*x^2)*log(x)-2*x^6-8*x^5+6912*x^ 4+3456*x^3)/(x^6+4*x^5+4*x^4),x, algorithm="fricas")
Output:
-2*(x^5 + 2*x^4 - 864*x^3*log(x) - 216*x^2*log(x)^2 - 24*x*log(x)^3 - log( x)^4 + 2596*x^3)/(x^4 + 2*x^3)
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (19) = 38\).
Time = 0.16 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.22 \[ \int \frac {3456 x^3+6912 x^4-8 x^5-2 x^6+\left (1728 x^2+864 x^3-1728 x^4\right ) \log (x)+\left (288 x-720 x^2-864 x^3\right ) \log ^2(x)+\left (16-184 x-144 x^2\right ) \log ^3(x)+(-12-8 x) \log ^4(x)}{4 x^4+4 x^5+x^6} \, dx=- 2 x + \frac {2 \log {\left (x \right )}^{4}}{x^{4} + 2 x^{3}} + \frac {48 \log {\left (x \right )}^{3}}{x^{3} + 2 x^{2}} + \frac {432 \log {\left (x \right )}^{2}}{x^{2} + 2 x} + \frac {1728 \log {\left (x \right )}}{x + 2} - \frac {5192}{x + 2} \] Input:
integrate(((-8*x-12)*ln(x)**4+(-144*x**2-184*x+16)*ln(x)**3+(-864*x**3-720 *x**2+288*x)*ln(x)**2+(-1728*x**4+864*x**3+1728*x**2)*ln(x)-2*x**6-8*x**5+ 6912*x**4+3456*x**3)/(x**6+4*x**5+4*x**4),x)
Output:
-2*x + 2*log(x)**4/(x**4 + 2*x**3) + 48*log(x)**3/(x**3 + 2*x**2) + 432*lo g(x)**2/(x**2 + 2*x) + 1728*log(x)/(x + 2) - 5192/(x + 2)
Time = 0.07 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.93 \[ \int \frac {3456 x^3+6912 x^4-8 x^5-2 x^6+\left (1728 x^2+864 x^3-1728 x^4\right ) \log (x)+\left (288 x-720 x^2-864 x^3\right ) \log ^2(x)+\left (16-184 x-144 x^2\right ) \log ^3(x)+(-12-8 x) \log ^4(x)}{4 x^4+4 x^5+x^6} \, dx=-2 \, x + \frac {2 \, {\left (864 \, x^{3} \log \left (x\right ) + 216 \, x^{2} \log \left (x\right )^{2} + 24 \, x \log \left (x\right )^{3} + \log \left (x\right )^{4}\right )}}{x^{4} + 2 \, x^{3}} - \frac {5192}{x + 2} \] Input:
integrate(((-8*x-12)*log(x)^4+(-144*x^2-184*x+16)*log(x)^3+(-864*x^3-720*x ^2+288*x)*log(x)^2+(-1728*x^4+864*x^3+1728*x^2)*log(x)-2*x^6-8*x^5+6912*x^ 4+3456*x^3)/(x^6+4*x^5+4*x^4),x, algorithm="maxima")
Output:
-2*x + 2*(864*x^3*log(x) + 216*x^2*log(x)^2 + 24*x*log(x)^3 + log(x)^4)/(x ^4 + 2*x^3) - 5192/(x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (26) = 52\).
Time = 0.12 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.04 \[ \int \frac {3456 x^3+6912 x^4-8 x^5-2 x^6+\left (1728 x^2+864 x^3-1728 x^4\right ) \log (x)+\left (288 x-720 x^2-864 x^3\right ) \log ^2(x)+\left (16-184 x-144 x^2\right ) \log ^3(x)+(-12-8 x) \log ^4(x)}{4 x^4+4 x^5+x^6} \, dx=-\frac {1}{4} \, {\left (\frac {1}{x + 2} - \frac {x^{2} - 2 \, x + 4}{x^{3}}\right )} \log \left (x\right )^{4} + 12 \, {\left (\frac {1}{x + 2} - \frac {x - 2}{x^{2}}\right )} \log \left (x\right )^{3} - 216 \, {\left (\frac {1}{x + 2} - \frac {1}{x}\right )} \log \left (x\right )^{2} - 2 \, x + \frac {1728 \, \log \left (x\right )}{x + 2} - \frac {5192}{x + 2} \] Input:
integrate(((-8*x-12)*log(x)^4+(-144*x^2-184*x+16)*log(x)^3+(-864*x^3-720*x ^2+288*x)*log(x)^2+(-1728*x^4+864*x^3+1728*x^2)*log(x)-2*x^6-8*x^5+6912*x^ 4+3456*x^3)/(x^6+4*x^5+4*x^4),x, algorithm="giac")
Output:
-1/4*(1/(x + 2) - (x^2 - 2*x + 4)/x^3)*log(x)^4 + 12*(1/(x + 2) - (x - 2)/ x^2)*log(x)^3 - 216*(1/(x + 2) - 1/x)*log(x)^2 - 2*x + 1728*log(x)/(x + 2) - 5192/(x + 2)
Time = 4.64 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {3456 x^3+6912 x^4-8 x^5-2 x^6+\left (1728 x^2+864 x^3-1728 x^4\right ) \log (x)+\left (288 x-720 x^2-864 x^3\right ) \log ^2(x)+\left (16-184 x-144 x^2\right ) \log ^3(x)+(-12-8 x) \log ^4(x)}{4 x^4+4 x^5+x^6} \, dx=\frac {2\,\left (-x^5+1296\,x^4+864\,x^3\,\ln \left (x\right )+216\,x^2\,{\ln \left (x\right )}^2+24\,x\,{\ln \left (x\right )}^3+{\ln \left (x\right )}^4\right )}{x^3\,\left (x+2\right )} \] Input:
int(-(log(x)^3*(184*x + 144*x^2 - 16) + log(x)^2*(720*x^2 - 288*x + 864*x^ 3) - log(x)*(1728*x^2 + 864*x^3 - 1728*x^4) - 3456*x^3 - 6912*x^4 + 8*x^5 + 2*x^6 + log(x)^4*(8*x + 12))/(4*x^4 + 4*x^5 + x^6),x)
Output:
(2*(24*x*log(x)^3 + 864*x^3*log(x) + log(x)^4 + 216*x^2*log(x)^2 + 1296*x^ 4 - x^5))/(x^3*(x + 2))
Time = 0.16 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {3456 x^3+6912 x^4-8 x^5-2 x^6+\left (1728 x^2+864 x^3-1728 x^4\right ) \log (x)+\left (288 x-720 x^2-864 x^3\right ) \log ^2(x)+\left (16-184 x-144 x^2\right ) \log ^3(x)+(-12-8 x) \log ^4(x)}{4 x^4+4 x^5+x^6} \, dx=\frac {2 \mathrm {log}\left (x \right )^{4}+48 \mathrm {log}\left (x \right )^{3} x +432 \mathrm {log}\left (x \right )^{2} x^{2}+1728 \,\mathrm {log}\left (x \right ) x^{3}-2 x^{5}+2592 x^{4}}{x^{3} \left (x +2\right )} \] Input:
int(((-8*x-12)*log(x)^4+(-144*x^2-184*x+16)*log(x)^3+(-864*x^3-720*x^2+288 *x)*log(x)^2+(-1728*x^4+864*x^3+1728*x^2)*log(x)-2*x^6-8*x^5+6912*x^4+3456 *x^3)/(x^6+4*x^5+4*x^4),x)
Output:
(2*(log(x)**4 + 24*log(x)**3*x + 216*log(x)**2*x**2 + 864*log(x)*x**3 - x* *5 + 1296*x**4))/(x**3*(x + 2))