Integrand size = 72, antiderivative size = 30 \begin {dmath*} \int \frac {-192 x^2-208 x^3-56 x^4+\left (-384 x-576 x^2-304 x^3-56 x^4\right ) \log \left (\frac {3}{2+x}\right )}{\left (1728+3024 x+1764 x^2+343 x^3\right ) \log ^3\left (\frac {3}{2+x}\right )} \, dx=4 \left (5-\frac {x^2}{\left (6+\frac {x}{2+x}\right )^2 \log ^2\left (\frac {3}{2+x}\right )}\right ) \end {dmath*}
Time = 0.36 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \begin {dmath*} \int \frac {-192 x^2-208 x^3-56 x^4+\left (-384 x-576 x^2-304 x^3-56 x^4\right ) \log \left (\frac {3}{2+x}\right )}{\left (1728+3024 x+1764 x^2+343 x^3\right ) \log ^3\left (\frac {3}{2+x}\right )} \, dx=-\frac {4 x^2 (2+x)^2}{(12+7 x)^2 \log ^2\left (\frac {3}{2+x}\right )} \end {dmath*}
Integrate[(-192*x^2 - 208*x^3 - 56*x^4 + (-384*x - 576*x^2 - 304*x^3 - 56* x^4)*Log[3/(2 + x)])/((1728 + 3024*x + 1764*x^2 + 343*x^3)*Log[3/(2 + x)]^ 3),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.
\(\displaystyle \int \frac {-56 x^4-208 x^3-192 x^2+\left (-56 x^4-304 x^3-576 x^2-384 x\right ) \log \left (\frac {3}{x+2}\right )}{\left (343 x^3+1764 x^2+3024 x+1728\right ) \log ^3\left (\frac {3}{x+2}\right )} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {-56 x^4-208 x^3-192 x^2+\left (-56 x^4-304 x^3-576 x^2-384 x\right ) \log \left (\frac {3}{x+2}\right )}{(7 x+12)^3 \log ^3\left (\frac {3}{x+2}\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {8 (x+2) x^2}{(7 x+12)^2 \log ^3\left (\frac {3}{x+2}\right )}-\frac {8 (x+2) \left (7 x^2+24 x+24\right ) x}{(7 x+12)^3 \log ^2\left (\frac {3}{x+2}\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2304}{343} \int \frac {1}{(7 x+12)^2 \log ^3\left (\frac {3}{x+2}\right )}dx-\frac {768}{343} \int \frac {1}{(7 x+12) \log ^3\left (\frac {3}{x+2}\right )}dx+\frac {4608}{343} \int \frac {1}{(7 x+12)^3 \log ^2\left (\frac {3}{x+2}\right )}dx+\frac {1920}{343} \int \frac {1}{(7 x+12)^2 \log ^2\left (\frac {3}{x+2}\right )}dx-\frac {4 x (x+2)}{49 \log ^2\left (\frac {3}{x+2}\right )}+\frac {40 (x+2)}{343 \log ^2\left (\frac {3}{x+2}\right )}\) |
Int[(-192*x^2 - 208*x^3 - 56*x^4 + (-384*x - 576*x^2 - 304*x^3 - 56*x^4)*L og[3/(2 + x)])/((1728 + 3024*x + 1764*x^2 + 343*x^3)*Log[3/(2 + x)]^3),x]
3.8.46.3.1 Defintions of rubi rules used
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]
Time = 38.87 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10
method | result | size |
risch | \(-\frac {4 x^{2} \left (2+x \right )^{2}}{\left (49 x^{2}+168 x +144\right ) \ln \left (\frac {3}{2+x}\right )^{2}}\) | \(33\) |
norman | \(\frac {-4 x^{4}-16 x^{3}-16 x^{2}}{\left (7 x +12\right )^{2} \ln \left (\frac {3}{2+x}\right )^{2}}\) | \(35\) |
parallelrisch | \(\frac {-196 x^{4}-784 x^{3}-784 x^{2}}{49 \ln \left (\frac {3}{2+x}\right )^{2} \left (49 x^{2}+168 x +144\right )}\) | \(41\) |
derivativedivides | \(-\frac {4 \left (2+x \right )^{2}}{49 \ln \left (\frac {3}{2+x}\right )^{2}}+\frac {\frac {192}{343}+\frac {96 x}{343}}{\ln \left (\frac {3}{2+x}\right )^{2}}-\frac {3456}{343 \ln \left (\frac {3}{2+x}\right )^{3} \left (\frac {6}{2+x}-21\right )}-\frac {1152 \left (\frac {3 \ln \left (\frac {3}{2+x}\right )}{2+x}+21 \ln \left (\frac {3}{2+x}\right )-\frac {18}{2+x}+63\right )}{343 \ln \left (\frac {3}{2+x}\right )^{3} \left (\frac {6}{2+x}-21\right )^{2}}\) | \(114\) |
default | \(-\frac {4 \left (2+x \right )^{2}}{49 \ln \left (\frac {3}{2+x}\right )^{2}}+\frac {\frac {192}{343}+\frac {96 x}{343}}{\ln \left (\frac {3}{2+x}\right )^{2}}-\frac {3456}{343 \ln \left (\frac {3}{2+x}\right )^{3} \left (\frac {6}{2+x}-21\right )}-\frac {1152 \left (\frac {3 \ln \left (\frac {3}{2+x}\right )}{2+x}+21 \ln \left (\frac {3}{2+x}\right )-\frac {18}{2+x}+63\right )}{343 \ln \left (\frac {3}{2+x}\right )^{3} \left (\frac {6}{2+x}-21\right )^{2}}\) | \(114\) |
int(((-56*x^4-304*x^3-576*x^2-384*x)*ln(3/(2+x))-56*x^4-208*x^3-192*x^2)/( 343*x^3+1764*x^2+3024*x+1728)/ln(3/(2+x))^3,x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27 \begin {dmath*} \int \frac {-192 x^2-208 x^3-56 x^4+\left (-384 x-576 x^2-304 x^3-56 x^4\right ) \log \left (\frac {3}{2+x}\right )}{\left (1728+3024 x+1764 x^2+343 x^3\right ) \log ^3\left (\frac {3}{2+x}\right )} \, dx=-\frac {4 \, {\left (x^{4} + 4 \, x^{3} + 4 \, x^{2}\right )}}{{\left (49 \, x^{2} + 168 \, x + 144\right )} \log \left (\frac {3}{x + 2}\right )^{2}} \end {dmath*}
integrate(((-56*x^4-304*x^3-576*x^2-384*x)*log(3/(2+x))-56*x^4-208*x^3-192 *x^2)/(343*x^3+1764*x^2+3024*x+1728)/log(3/(2+x))^3,x, algorithm=\
Time = 0.11 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \begin {dmath*} \int \frac {-192 x^2-208 x^3-56 x^4+\left (-384 x-576 x^2-304 x^3-56 x^4\right ) \log \left (\frac {3}{2+x}\right )}{\left (1728+3024 x+1764 x^2+343 x^3\right ) \log ^3\left (\frac {3}{2+x}\right )} \, dx=\frac {- 4 x^{4} - 16 x^{3} - 16 x^{2}}{\left (49 x^{2} + 168 x + 144\right ) \log {\left (\frac {3}{x + 2} \right )}^{2}} \end {dmath*}
integrate(((-56*x**4-304*x**3-576*x**2-384*x)*ln(3/(2+x))-56*x**4-208*x**3 -192*x**2)/(343*x**3+1764*x**2+3024*x+1728)/ln(3/(2+x))**3,x)
Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (28) = 56\).
Time = 0.33 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.70 \begin {dmath*} \int \frac {-192 x^2-208 x^3-56 x^4+\left (-384 x-576 x^2-304 x^3-56 x^4\right ) \log \left (\frac {3}{2+x}\right )}{\left (1728+3024 x+1764 x^2+343 x^3\right ) \log ^3\left (\frac {3}{2+x}\right )} \, dx=-\frac {4 \, {\left (x^{4} + 4 \, x^{3} + 4 \, x^{2}\right )}}{49 \, x^{2} \log \left (3\right )^{2} + 168 \, x \log \left (3\right )^{2} + {\left (49 \, x^{2} + 168 \, x + 144\right )} \log \left (x + 2\right )^{2} + 144 \, \log \left (3\right )^{2} - 2 \, {\left (49 \, x^{2} \log \left (3\right ) + 168 \, x \log \left (3\right ) + 144 \, \log \left (3\right )\right )} \log \left (x + 2\right )} \end {dmath*}
integrate(((-56*x^4-304*x^3-576*x^2-384*x)*log(3/(2+x))-56*x^4-208*x^3-192 *x^2)/(343*x^3+1764*x^2+3024*x+1728)/log(3/(2+x))^3,x, algorithm=\
-4*(x^4 + 4*x^3 + 4*x^2)/(49*x^2*log(3)^2 + 168*x*log(3)^2 + (49*x^2 + 168 *x + 144)*log(x + 2)^2 + 144*log(3)^2 - 2*(49*x^2*log(3) + 168*x*log(3) + 144*log(3))*log(x + 2))
Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (28) = 56\).
Time = 0.29 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.40 \begin {dmath*} \int \frac {-192 x^2-208 x^3-56 x^4+\left (-384 x-576 x^2-304 x^3-56 x^4\right ) \log \left (\frac {3}{2+x}\right )}{\left (1728+3024 x+1764 x^2+343 x^3\right ) \log ^3\left (\frac {3}{2+x}\right )} \, dx=\frac {4 \, {\left (\frac {4}{x + 2} - \frac {4}{{\left (x + 2\right )}^{2}} - 1\right )}}{\frac {49 \, \log \left (\frac {3}{x + 2}\right )^{2}}{{\left (x + 2\right )}^{2}} - \frac {28 \, \log \left (\frac {3}{x + 2}\right )^{2}}{{\left (x + 2\right )}^{3}} + \frac {4 \, \log \left (\frac {3}{x + 2}\right )^{2}}{{\left (x + 2\right )}^{4}}} \end {dmath*}
integrate(((-56*x^4-304*x^3-576*x^2-384*x)*log(3/(2+x))-56*x^4-208*x^3-192 *x^2)/(343*x^3+1764*x^2+3024*x+1728)/log(3/(2+x))^3,x, algorithm=\
4*(4/(x + 2) - 4/(x + 2)^2 - 1)/(49*log(3/(x + 2))^2/(x + 2)^2 - 28*log(3/ (x + 2))^2/(x + 2)^3 + 4*log(3/(x + 2))^2/(x + 2)^4)
Time = 14.32 (sec) , antiderivative size = 200, normalized size of antiderivative = 6.67 \begin {dmath*} \int \frac {-192 x^2-208 x^3-56 x^4+\left (-384 x-576 x^2-304 x^3-56 x^4\right ) \log \left (\frac {3}{2+x}\right )}{\left (1728+3024 x+1764 x^2+343 x^3\right ) \log ^3\left (\frac {3}{2+x}\right )} \, dx=\frac {\frac {4\,\left (x+2\right )\,\left (7\,x^4+38\,x^3+72\,x^2+48\,x\right )}{{\left (7\,x+12\right )}^3}+\frac {8\,\ln \left (\frac {3}{x+2}\right )\,\left (x+2\right )\,\left (49\,x^5+392\,x^4+1248\,x^3+1992\,x^2+1632\,x+576\right )}{{\left (7\,x+12\right )}^4}}{\ln \left (\frac {3}{x+2}\right )}-\frac {\frac {4\,x^2\,{\left (x+2\right )}^2}{{\left (7\,x+12\right )}^2}+\frac {4\,x\,\ln \left (\frac {3}{x+2}\right )\,\left (x+2\right )\,\left (7\,x^3+38\,x^2+72\,x+48\right )}{{\left (7\,x+12\right )}^3}}{{\ln \left (\frac {3}{x+2}\right )}^2}-\frac {176\,x}{343}-\frac {8\,x^2}{49}-\frac {\frac {960\,x^3}{16807}+\frac {44928\,x^2}{117649}+\frac {694272\,x}{823543}+\frac {506880}{823543}}{x^4+\frac {48\,x^3}{7}+\frac {864\,x^2}{49}+\frac {6912\,x}{343}+\frac {20736}{2401}} \end {dmath*}
int(-(log(3/(x + 2))*(384*x + 576*x^2 + 304*x^3 + 56*x^4) + 192*x^2 + 208* x^3 + 56*x^4)/(log(3/(x + 2))^3*(3024*x + 1764*x^2 + 343*x^3 + 1728)),x)
((4*(x + 2)*(48*x + 72*x^2 + 38*x^3 + 7*x^4))/(7*x + 12)^3 + (8*log(3/(x + 2))*(x + 2)*(1632*x + 1992*x^2 + 1248*x^3 + 392*x^4 + 49*x^5 + 576))/(7*x + 12)^4)/log(3/(x + 2)) - ((4*x^2*(x + 2)^2)/(7*x + 12)^2 + (4*x*log(3/(x + 2))*(x + 2)*(72*x + 38*x^2 + 7*x^3 + 48))/(7*x + 12)^3)/log(3/(x + 2))^ 2 - (176*x)/343 - (8*x^2)/49 - ((694272*x)/823543 + (44928*x^2)/117649 + ( 960*x^3)/16807 + 506880/823543)/((6912*x)/343 + (864*x^2)/49 + (48*x^3)/7 + x^4 + 20736/2401)