Integrand size = 45, antiderivative size = 26 \[ \int \frac {-9-486 x-6237 x^2+(-6-324 x) \log (2)}{9+462 x+5929 x^2+(12+308 x) \log (2)+4 \log ^2(2)} \, dx=81-\frac {x}{1-\frac {2 (2 x-\log (2))}{3+81 x}} \] Output:
81-x/(1-(2*x-ln(2))/(81/2*x+3/2))
Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42 \[ \int \frac {-9-486 x-6237 x^2+(-6-324 x) \log (2)}{9+462 x+5929 x^2+(12+308 x) \log (2)+4 \log ^2(2)} \, dx=-\frac {3 \left (255+160083 x^2+247 \log (4)+54 \log ^2(4)+4158 x (3+\log (4))\right )}{5929 (3+77 x+\log (4))} \] Input:
Integrate[(-9 - 486*x - 6237*x^2 + (-6 - 324*x)*Log[2])/(9 + 462*x + 5929* x^2 + (12 + 308*x)*Log[2] + 4*Log[2]^2),x]
Output:
(-3*(255 + 160083*x^2 + 247*Log[4] + 54*Log[4]^2 + 4158*x*(3 + Log[4])))/( 5929*(3 + 77*x + Log[4]))
Time = 8.95 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.35, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {2007, 2081, 1107, 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.
\(\displaystyle \int \frac {-6237 x^2-486 x+(-324 x-6) \log (2)-9}{5929 x^2+462 x+(308 x+12) \log (2)+9+4 \log ^2(2)} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {-6237 x^2-486 x+(-324 x-6) \log (2)-9}{(77 x+3+\log (4))^2}dx\) |
\(\Big \downarrow \) 2081 |
\(\displaystyle \int \frac {-6237 x^2-162 x (3+\log (4))-3 (3+2 \log (2))}{(77 x+3+\log (4))^2}dx\) |
\(\Big \downarrow \) 1107 |
\(\displaystyle \int \left (\frac {3 \left (12+27 \log ^2(4)-154 \log (2)+162 \log (4)\right )}{77 (77 x+3+\log (4))^2}-\frac {81}{77}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {81 x}{77}-\frac {3 \left (12+27 \log ^2(4)-154 \log (2)+162 \log (4)\right )}{5929 (77 x+3+\log (4))}\) |
Input:
Int[(-9 - 486*x - 6237*x^2 + (-6 - 324*x)*Log[2])/(9 + 462*x + 5929*x^2 + (12 + 308*x)*Log[2] + 4*Log[2]^2),x]
Output:
(-81*x)/77 - (3*(12 - 154*Log[2] + 162*Log[4] + 27*Log[4]^2))/(5929*(3 + 7 7*x + Log[4]))
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; F reeQ[{a, b, c, d, e, m}, x] && EqQ[2*c*d - b*e, 0] && IGtQ[p, 0] && !(EqQ[ m, 3] && NeQ[p, 1])
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[(u_)^(m_.)*(v_)^(p_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*ExpandToSum [v, x]^p, x] /; FreeQ[{m, p}, x] && LinearQ[u, x] && QuadraticQ[v, x] && ! (LinearMatchQ[u, x] && QuadraticMatchQ[v, x])
Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92
method | result | size |
norman | \(\frac {-81 x^{2}+\frac {6 \ln \left (2\right )}{77}+\frac {9}{77}}{2 \ln \left (2\right )+77 x +3}\) | \(24\) |
gosper | \(\frac {-81 x^{2}+\frac {6 \ln \left (2\right )}{77}+\frac {9}{77}}{2 \ln \left (2\right )+77 x +3}\) | \(25\) |
parallelrisch | \(\frac {9-6237 x^{2}+6 \ln \left (2\right )}{154 \ln \left (2\right )+5929 x +231}\) | \(25\) |
default | \(-\frac {81 x}{77}-\frac {3 \left (\frac {108 \ln \left (2\right )^{2}}{5929}+\frac {170 \ln \left (2\right )}{5929}+\frac {12}{5929}\right )}{2 \ln \left (2\right )+77 x +3}\) | \(30\) |
risch | \(-\frac {81 x}{77}-\frac {162 \ln \left (2\right )^{2}}{5929 \left (\ln \left (2\right )+\frac {77 x}{2}+\frac {3}{2}\right )}-\frac {255 \ln \left (2\right )}{5929 \left (\ln \left (2\right )+\frac {77 x}{2}+\frac {3}{2}\right )}-\frac {18}{5929 \left (\ln \left (2\right )+\frac {77 x}{2}+\frac {3}{2}\right )}\) | \(44\) |
meijerg | \(-\frac {9 x}{77 \left (\frac {2 \ln \left (2\right )}{77}+\frac {3}{77}\right ) \left (2 \ln \left (2\right )+3\right ) \left (1+\frac {77 x}{2 \ln \left (2\right )+3}\right )}-\frac {81 \left (2 \ln \left (2\right )+3\right )^{2} \left (\frac {77 x \left (\frac {231 x}{2 \ln \left (2\right )+3}+6\right )}{3 \left (2 \ln \left (2\right )+3\right ) \left (1+\frac {77 x}{2 \ln \left (2\right )+3}\right )}-2 \ln \left (1+\frac {77 x}{2 \ln \left (2\right )+3}\right )\right )}{456533 \left (\frac {2 \ln \left (2\right )}{77}+\frac {3}{77}\right )}+\left (-\frac {324 \ln \left (2\right )}{5929}-\frac {486}{5929}\right ) \left (-\frac {77 x}{\left (2 \ln \left (2\right )+3\right ) \left (1+\frac {77 x}{2 \ln \left (2\right )+3}\right )}+\ln \left (1+\frac {77 x}{2 \ln \left (2\right )+3}\right )\right )-\frac {6 \ln \left (2\right ) x}{77 \left (\frac {2 \ln \left (2\right )}{77}+\frac {3}{77}\right ) \left (2 \ln \left (2\right )+3\right ) \left (1+\frac {77 x}{2 \ln \left (2\right )+3}\right )}\) | \(194\) |
Input:
int(((-324*x-6)*ln(2)-6237*x^2-486*x-9)/(4*ln(2)^2+(308*x+12)*ln(2)+5929*x ^2+462*x+9),x,method=_RETURNVERBOSE)
Output:
(-81*x^2+6/77*ln(2)+9/77)/(2*ln(2)+77*x+3)
Time = 0.11 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.46 \[ \int \frac {-9-486 x-6237 x^2+(-6-324 x) \log (2)}{9+462 x+5929 x^2+(12+308 x) \log (2)+4 \log ^2(2)} \, dx=-\frac {3 \, {\left (160083 \, x^{2} + 2 \, {\left (2079 \, x + 85\right )} \log \left (2\right ) + 108 \, \log \left (2\right )^{2} + 6237 \, x + 12\right )}}{5929 \, {\left (77 \, x + 2 \, \log \left (2\right ) + 3\right )}} \] Input:
integrate(((-324*x-6)*log(2)-6237*x^2-486*x-9)/(4*log(2)^2+(308*x+12)*log( 2)+5929*x^2+462*x+9),x, algorithm="fricas")
Output:
-3/5929*(160083*x^2 + 2*(2079*x + 85)*log(2) + 108*log(2)^2 + 6237*x + 12) /(77*x + 2*log(2) + 3)
Time = 0.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {-9-486 x-6237 x^2+(-6-324 x) \log (2)}{9+462 x+5929 x^2+(12+308 x) \log (2)+4 \log ^2(2)} \, dx=- \frac {81 x}{77} - \frac {36 + 324 \log {\left (2 \right )}^{2} + 510 \log {\left (2 \right )}}{456533 x + 11858 \log {\left (2 \right )} + 17787} \] Input:
integrate(((-324*x-6)*ln(2)-6237*x**2-486*x-9)/(4*ln(2)**2+(308*x+12)*ln(2 )+5929*x**2+462*x+9),x)
Output:
-81*x/77 - (36 + 324*log(2)**2 + 510*log(2))/(456533*x + 11858*log(2) + 17 787)
Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {-9-486 x-6237 x^2+(-6-324 x) \log (2)}{9+462 x+5929 x^2+(12+308 x) \log (2)+4 \log ^2(2)} \, dx=-\frac {81}{77} \, x - \frac {6 \, {\left (54 \, \log \left (2\right )^{2} + 85 \, \log \left (2\right ) + 6\right )}}{5929 \, {\left (77 \, x + 2 \, \log \left (2\right ) + 3\right )}} \] Input:
integrate(((-324*x-6)*log(2)-6237*x^2-486*x-9)/(4*log(2)^2+(308*x+12)*log( 2)+5929*x^2+462*x+9),x, algorithm="maxima")
Output:
-81/77*x - 6/5929*(54*log(2)^2 + 85*log(2) + 6)/(77*x + 2*log(2) + 3)
Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {-9-486 x-6237 x^2+(-6-324 x) \log (2)}{9+462 x+5929 x^2+(12+308 x) \log (2)+4 \log ^2(2)} \, dx=-\frac {81}{77} \, x - \frac {6 \, {\left (54 \, \log \left (2\right )^{2} + 85 \, \log \left (2\right ) + 6\right )}}{5929 \, {\left (77 \, x + 2 \, \log \left (2\right ) + 3\right )}} \] Input:
integrate(((-324*x-6)*log(2)-6237*x^2-486*x-9)/(4*log(2)^2+(308*x+12)*log( 2)+5929*x^2+462*x+9),x, algorithm="giac")
Output:
-81/77*x - 6/5929*(54*log(2)^2 + 85*log(2) + 6)/(77*x + 2*log(2) + 3)
Time = 0.12 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {-9-486 x-6237 x^2+(-6-324 x) \log (2)}{9+462 x+5929 x^2+(12+308 x) \log (2)+4 \log ^2(2)} \, dx=-\frac {81\,x}{77}-\frac {\frac {510\,\ln \left (2\right )}{5929}+\frac {324\,{\ln \left (2\right )}^2}{5929}+\frac {36}{5929}}{77\,x+\ln \left (4\right )+3} \] Input:
int(-(486*x + log(2)*(324*x + 6) + 6237*x^2 + 9)/(462*x + log(2)*(308*x + 12) + 4*log(2)^2 + 5929*x^2 + 9),x)
Output:
- (81*x)/77 - ((510*log(2))/5929 + (324*log(2)^2)/5929 + 36/5929)/(77*x + log(4) + 3)
Time = 0.17 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73 \[ \int \frac {-9-486 x-6237 x^2+(-6-324 x) \log (2)}{9+462 x+5929 x^2+(12+308 x) \log (2)+4 \log ^2(2)} \, dx=\frac {3 x \left (-27 x -1\right )}{2 \,\mathrm {log}\left (2\right )+77 x +3} \] Input:
int(((-324*x-6)*log(2)-6237*x^2-486*x-9)/(4*log(2)^2+(308*x+12)*log(2)+592 9*x^2+462*x+9),x)
Output:
(3*x*( - 27*x - 1))/(2*log(2) + 77*x + 3)