Integrand size = 78, antiderivative size = 26 \[ \int \frac {-9-6 x-x^2-192 x^3+192 x^4-8 x^5-20 x^6+\left (48 x^3-48 x^4+2 x^5+5 x^6\right ) \log (4)}{-36-24 x-4 x^2+\left (9+6 x+x^2\right ) \log (4)} \, dx=-4+\frac {(-2+x)^2 x^4}{3+x}+\frac {x}{4-\log (4)} \]
Leaf count is larger than twice the leaf count of optimal. \(58\) vs. \(2(26)=52\).
Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.23 \[ \int \frac {-9-6 x-x^2-192 x^3+192 x^4-8 x^5-20 x^6+\left (48 x^3-48 x^4+2 x^5+5 x^6\right ) \log (4)}{-36-24 x-4 x^2+\left (9+6 x+x^2\right ) \log (4)} \, dx=-\frac {42129+x^2-4 x^4 (-4+\log (4))+4 x^5 (-4+\log (4))-x^6 (-4+\log (4))-10530 \log (4)-6 x (-2341+585 \log (4))}{(3+x) (-4+\log (4))} \]
Integrate[(-9 - 6*x - x^2 - 192*x^3 + 192*x^4 - 8*x^5 - 20*x^6 + (48*x^3 - 48*x^4 + 2*x^5 + 5*x^6)*Log[4])/(-36 - 24*x - 4*x^2 + (9 + 6*x + x^2)*Log [4]),x]
-((42129 + x^2 - 4*x^4*(-4 + Log[4]) + 4*x^5*(-4 + Log[4]) - x^6*(-4 + Log [4]) - 10530*Log[4] - 6*x*(-2341 + 585*Log[4]))/((3 + x)*(-4 + Log[4])))
Time = 0.35 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {2007, 2389, 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.
\(\displaystyle \int \frac {-20 x^6-8 x^5+192 x^4-192 x^3-x^2+\left (5 x^6+2 x^5-48 x^4+48 x^3\right ) \log (4)-6 x-9}{-4 x^2+\left (x^2+6 x+9\right ) \log (4)-24 x-36} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {-20 x^6-8 x^5+192 x^4-192 x^3-x^2+\left (5 x^6+2 x^5-48 x^4+48 x^3\right ) \log (4)-6 x-9}{\left (i x \sqrt {4-\log (4)}+3 i \sqrt {4-\log (4)}\right )^2}dx\) |
\(\Big \downarrow \) 2389 |
\(\displaystyle \int \left (5 x^4-28 x^3+75 x^2-150 x-\frac {2025}{(x+3)^2}+\frac {225 \log (4)-901}{\log (4)-4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^5-7 x^4+25 x^3-75 x^2+\frac {2025}{x+3}+\frac {x (901-225 \log (4))}{4-\log (4)}\) |
Int[(-9 - 6*x - x^2 - 192*x^3 + 192*x^4 - 8*x^5 - 20*x^6 + (48*x^3 - 48*x^ 4 + 2*x^5 + 5*x^6)*Log[4])/(-36 - 24*x - 4*x^2 + (9 + 6*x + x^2)*Log[4]),x ]
3.2.60.3.1 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]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Time = 0.77 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.54
method | result | size |
norman | \(\frac {x^{6}+4 x^{4}-4 x^{5}-\frac {x^{2}}{2 \left (\ln \left (2\right )-2\right )}+\frac {9}{2 \left (\ln \left (2\right )-2\right )}}{3+x}\) | \(40\) |
parallelrisch | \(\frac {2 x^{6} \ln \left (2\right )-8 x^{5} \ln \left (2\right )-4 x^{6}+8 x^{4} \ln \left (2\right )+16 x^{5}-16 x^{4}-x^{2}+9}{2 \left (\ln \left (2\right )-2\right ) \left (3+x \right )}\) | \(57\) |
gosper | \(\frac {2 x^{6} \ln \left (2\right )-8 x^{5} \ln \left (2\right )-4 x^{6}+8 x^{4} \ln \left (2\right )+16 x^{5}-16 x^{4}-x^{2}+9}{2 x \ln \left (2\right )+6 \ln \left (2\right )-4 x -12}\) | \(61\) |
default | \(\frac {2 x^{5} \ln \left (2\right )-14 x^{4} \ln \left (2\right )-4 x^{5}+50 x^{3} \ln \left (2\right )+28 x^{4}-150 x^{2} \ln \left (2\right )-100 x^{3}+450 x \ln \left (2\right )+300 x^{2}-901 x -\frac {-4050 \ln \left (2\right )+8100}{3+x}}{2 \ln \left (2\right )-4}\) | \(79\) |
risch | \(\frac {2 \ln \left (2\right ) x^{5}}{2 \ln \left (2\right )-4}-\frac {14 \ln \left (2\right ) x^{4}}{2 \ln \left (2\right )-4}-\frac {4 x^{5}}{2 \ln \left (2\right )-4}+\frac {50 \ln \left (2\right ) x^{3}}{2 \ln \left (2\right )-4}+\frac {28 x^{4}}{2 \ln \left (2\right )-4}-\frac {150 \ln \left (2\right ) x^{2}}{2 \ln \left (2\right )-4}-\frac {100 x^{3}}{2 \ln \left (2\right )-4}+\frac {450 x \ln \left (2\right )}{2 \ln \left (2\right )-4}+\frac {300 x^{2}}{2 \ln \left (2\right )-4}-\frac {901 x}{2 \ln \left (2\right )-4}+\frac {4050 \ln \left (2\right )^{2}}{\left (2 \ln \left (2\right )-4\right ) \left (x \ln \left (2\right )+3 \ln \left (2\right )-2 x -6\right )}-\frac {16200 \ln \left (2\right )}{\left (2 \ln \left (2\right )-4\right ) \left (x \ln \left (2\right )+3 \ln \left (2\right )-2 x -6\right )}+\frac {16200}{\left (2 \ln \left (2\right )-4\right ) \left (x \ln \left (2\right )+3 \ln \left (2\right )-2 x -6\right )}\) | \(219\) |
meijerg | \(-\frac {x}{\left (2 \ln \left (2\right )-4\right ) \left (1+\frac {x}{3}\right )}-\frac {3 \left (\frac {\left (6+x \right ) x}{3 x +9}-2 \ln \left (1+\frac {x}{3}\right )\right )}{2 \ln \left (2\right )-4}-\frac {6 \left (-\frac {x}{3 \left (1+\frac {x}{3}\right )}+\ln \left (1+\frac {x}{3}\right )\right )}{2 \ln \left (2\right )-4}+\frac {2187 \left (\frac {10 \ln \left (2\right )}{9}-\frac {20}{9}\right ) \left (\frac {x \left (\frac {14}{243} x^{5}-\frac {7}{27} x^{4}+\frac {35}{27} x^{3}-\frac {70}{9} x^{2}+70 x +420\right )}{210+70 x}-6 \ln \left (1+\frac {x}{3}\right )\right )}{2 \ln \left (2\right )-4}+\frac {729 \left (\frac {4 \ln \left (2\right )}{9}-\frac {8}{9}\right ) \left (-\frac {x \left (-\frac {1}{27} x^{4}+\frac {5}{27} x^{3}-\frac {10}{9} x^{2}+10 x +60\right )}{36 \left (1+\frac {x}{3}\right )}+5 \ln \left (1+\frac {x}{3}\right )\right )}{2 \ln \left (2\right )-4}+\frac {243 \left (-\frac {32 \ln \left (2\right )}{3}+\frac {64}{3}\right ) \left (\frac {x \left (\frac {5}{27} x^{3}-\frac {10}{9} x^{2}+10 x +60\right )}{15 x +45}-4 \ln \left (1+\frac {x}{3}\right )\right )}{2 \ln \left (2\right )-4}+\frac {81 \left (\frac {32 \ln \left (2\right )}{3}-\frac {64}{3}\right ) \left (-\frac {x \left (-\frac {2}{9} x^{2}+2 x +12\right )}{12 \left (1+\frac {x}{3}\right )}+3 \ln \left (1+\frac {x}{3}\right )\right )}{2 \ln \left (2\right )-4}\) | \(289\) |
int((2*(5*x^6+2*x^5-48*x^4+48*x^3)*ln(2)-20*x^6-8*x^5+192*x^4-192*x^3-x^2- 6*x-9)/(2*(x^2+6*x+9)*ln(2)-4*x^2-24*x-36),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (25) = 50\).
Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.31 \[ \int \frac {-9-6 x-x^2-192 x^3+192 x^4-8 x^5-20 x^6+\left (48 x^3-48 x^4+2 x^5+5 x^6\right ) \log (4)}{-36-24 x-4 x^2+\left (9+6 x+x^2\right ) \log (4)} \, dx=-\frac {4 \, x^{6} - 16 \, x^{5} + 16 \, x^{4} + x^{2} - 2 \, {\left (x^{6} - 4 \, x^{5} + 4 \, x^{4} + 675 \, x + 2025\right )} \log \left (2\right ) + 2703 \, x + 8100}{2 \, {\left ({\left (x + 3\right )} \log \left (2\right ) - 2 \, x - 6\right )}} \]
integrate((2*(5*x^6+2*x^5-48*x^4+48*x^3)*log(2)-20*x^6-8*x^5+192*x^4-192*x ^3-x^2-6*x-9)/(2*(x^2+6*x+9)*log(2)-4*x^2-24*x-36),x, algorithm=\
-1/2*(4*x^6 - 16*x^5 + 16*x^4 + x^2 - 2*(x^6 - 4*x^5 + 4*x^4 + 675*x + 202 5)*log(2) + 2703*x + 8100)/((x + 3)*log(2) - 2*x - 6)
Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (20) = 40\).
Time = 0.37 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69 \[ \int \frac {-9-6 x-x^2-192 x^3+192 x^4-8 x^5-20 x^6+\left (48 x^3-48 x^4+2 x^5+5 x^6\right ) \log (4)}{-36-24 x-4 x^2+\left (9+6 x+x^2\right ) \log (4)} \, dx=x^{5} - 7 x^{4} + 25 x^{3} - 75 x^{2} + x \left (\frac {450 \log {\left (2 \right )}}{-4 + 2 \log {\left (2 \right )}} - \frac {901}{-4 + 2 \log {\left (2 \right )}}\right ) + \frac {2025}{x + 3} \]
integrate((2*(5*x**6+2*x**5-48*x**4+48*x**3)*ln(2)-20*x**6-8*x**5+192*x**4 -192*x**3-x**2-6*x-9)/(2*(x**2+6*x+9)*ln(2)-4*x**2-24*x-36),x)
x**5 - 7*x**4 + 25*x**3 - 75*x**2 + x*(450*log(2)/(-4 + 2*log(2)) - 901/(- 4 + 2*log(2))) + 2025/(x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (25) = 50\).
Time = 0.19 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.35 \[ \int \frac {-9-6 x-x^2-192 x^3+192 x^4-8 x^5-20 x^6+\left (48 x^3-48 x^4+2 x^5+5 x^6\right ) \log (4)}{-36-24 x-4 x^2+\left (9+6 x+x^2\right ) \log (4)} \, dx=\frac {2 \, x^{5} {\left (\log \left (2\right ) - 2\right )} - 14 \, x^{4} {\left (\log \left (2\right ) - 2\right )} + 50 \, x^{3} {\left (\log \left (2\right ) - 2\right )} - 150 \, x^{2} {\left (\log \left (2\right ) - 2\right )} + x {\left (450 \, \log \left (2\right ) - 901\right )}}{2 \, {\left (\log \left (2\right ) - 2\right )}} + \frac {2025}{x + 3} \]
integrate((2*(5*x^6+2*x^5-48*x^4+48*x^3)*log(2)-20*x^6-8*x^5+192*x^4-192*x ^3-x^2-6*x-9)/(2*(x^2+6*x+9)*log(2)-4*x^2-24*x-36),x, algorithm=\
1/2*(2*x^5*(log(2) - 2) - 14*x^4*(log(2) - 2) + 50*x^3*(log(2) - 2) - 150* x^2*(log(2) - 2) + x*(450*log(2) - 901))/(log(2) - 2) + 2025/(x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (25) = 50\).
Time = 0.25 (sec) , antiderivative size = 269, normalized size of antiderivative = 10.35 \[ \int \frac {-9-6 x-x^2-192 x^3+192 x^4-8 x^5-20 x^6+\left (48 x^3-48 x^4+2 x^5+5 x^6\right ) \log (4)}{-36-24 x-4 x^2+\left (9+6 x+x^2\right ) \log (4)} \, dx=\frac {2 \, x^{5} \log \left (2\right )^{5} - 20 \, x^{5} \log \left (2\right )^{4} - 14 \, x^{4} \log \left (2\right )^{5} + 80 \, x^{5} \log \left (2\right )^{3} + 140 \, x^{4} \log \left (2\right )^{4} + 50 \, x^{3} \log \left (2\right )^{5} - 160 \, x^{5} \log \left (2\right )^{2} - 560 \, x^{4} \log \left (2\right )^{3} - 500 \, x^{3} \log \left (2\right )^{4} - 150 \, x^{2} \log \left (2\right )^{5} + 160 \, x^{5} \log \left (2\right ) + 1120 \, x^{4} \log \left (2\right )^{2} + 2000 \, x^{3} \log \left (2\right )^{3} + 1500 \, x^{2} \log \left (2\right )^{4} + 450 \, x \log \left (2\right )^{5} - 64 \, x^{5} - 1120 \, x^{4} \log \left (2\right ) - 4000 \, x^{3} \log \left (2\right )^{2} - 6000 \, x^{2} \log \left (2\right )^{3} - 4501 \, x \log \left (2\right )^{4} + 448 \, x^{4} + 4000 \, x^{3} \log \left (2\right ) + 12000 \, x^{2} \log \left (2\right )^{2} + 18008 \, x \log \left (2\right )^{3} - 1600 \, x^{3} - 12000 \, x^{2} \log \left (2\right ) - 36024 \, x \log \left (2\right )^{2} + 4800 \, x^{2} + 36032 \, x \log \left (2\right ) - 14416 \, x}{2 \, {\left (\log \left (2\right )^{5} - 10 \, \log \left (2\right )^{4} + 40 \, \log \left (2\right )^{3} - 80 \, \log \left (2\right )^{2} + 80 \, \log \left (2\right ) - 32\right )}} + \frac {2025}{x + 3} \]
integrate((2*(5*x^6+2*x^5-48*x^4+48*x^3)*log(2)-20*x^6-8*x^5+192*x^4-192*x ^3-x^2-6*x-9)/(2*(x^2+6*x+9)*log(2)-4*x^2-24*x-36),x, algorithm=\
1/2*(2*x^5*log(2)^5 - 20*x^5*log(2)^4 - 14*x^4*log(2)^5 + 80*x^5*log(2)^3 + 140*x^4*log(2)^4 + 50*x^3*log(2)^5 - 160*x^5*log(2)^2 - 560*x^4*log(2)^3 - 500*x^3*log(2)^4 - 150*x^2*log(2)^5 + 160*x^5*log(2) + 1120*x^4*log(2)^ 2 + 2000*x^3*log(2)^3 + 1500*x^2*log(2)^4 + 450*x*log(2)^5 - 64*x^5 - 1120 *x^4*log(2) - 4000*x^3*log(2)^2 - 6000*x^2*log(2)^3 - 4501*x*log(2)^4 + 44 8*x^4 + 4000*x^3*log(2) + 12000*x^2*log(2)^2 + 18008*x*log(2)^3 - 1600*x^3 - 12000*x^2*log(2) - 36024*x*log(2)^2 + 4800*x^2 + 36032*x*log(2) - 14416 *x)/(log(2)^5 - 10*log(2)^4 + 40*log(2)^3 - 80*log(2)^2 + 80*log(2) - 32) + 2025/(x + 3)
Timed out. \[ \int \frac {-9-6 x-x^2-192 x^3+192 x^4-8 x^5-20 x^6+\left (48 x^3-48 x^4+2 x^5+5 x^6\right ) \log (4)}{-36-24 x-4 x^2+\left (9+6 x+x^2\right ) \log (4)} \, dx=\int \frac {6\,x-2\,\ln \left (2\right )\,\left (5\,x^6+2\,x^5-48\,x^4+48\,x^3\right )+x^2+192\,x^3-192\,x^4+8\,x^5+20\,x^6+9}{24\,x+4\,x^2-2\,\ln \left (2\right )\,\left (x^2+6\,x+9\right )+36} \,d x \]
int((6*x - 2*log(2)*(48*x^3 - 48*x^4 + 2*x^5 + 5*x^6) + x^2 + 192*x^3 - 19 2*x^4 + 8*x^5 + 20*x^6 + 9)/(24*x + 4*x^2 - 2*log(2)*(6*x + x^2 + 9) + 36) ,x)