Integrand size = 66, antiderivative size = 21 \[ \int \frac {36 x^2+\left (12-36 x+12 x^2\right ) \log (4)+\left (8-6 x+x^2\right ) \log ^2(4)}{36 x^2+\left (-36 x+12 x^2\right ) \log (4)+\left (9-6 x+x^2\right ) \log ^2(4)} \, dx=x-\frac {2-x}{-3+x+\frac {6 x}{\log (4)}} \] Output:
x-(2-x)/(x-3+3*x/ln(2))
Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33 \[ \int \frac {36 x^2+\left (12-36 x+12 x^2\right ) \log (4)+\left (8-6 x+x^2\right ) \log ^2(4)}{36 x^2+\left (-36 x+12 x^2\right ) \log (4)+\left (9-6 x+x^2\right ) \log ^2(4)} \, dx=x+\frac {(-12+\log (4)) \log (4)}{(6+\log (4)) (-3 \log (4)+x (6+\log (4)))} \] Input:
Integrate[(36*x^2 + (12 - 36*x + 12*x^2)*Log[4] + (8 - 6*x + x^2)*Log[4]^2 )/(36*x^2 + (-36*x + 12*x^2)*Log[4] + (9 - 6*x + x^2)*Log[4]^2),x]
Output:
x + ((-12 + Log[4])*Log[4])/((6 + Log[4])*(-3*Log[4] + x*(6 + Log[4])))
Time = 0.33 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.81, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2083, 1294, 25, 27, 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.
\(\displaystyle \int \frac {36 x^2+\left (x^2-6 x+8\right ) \log ^2(4)+\left (12 x^2-36 x+12\right ) \log (4)}{36 x^2+\left (x^2-6 x+9\right ) \log ^2(4)+\left (12 x^2-36 x\right ) \log (4)} \, dx\) |
\(\Big \downarrow \) 2083 |
\(\displaystyle \int \frac {x^2 (6+\log (4))^2-6 x \log (4) (6+\log (4))+4 \log (4) (3+\log (16))}{x^2 (6+\log (4))^2-6 x \log (4) (6+\log (4))+9 \log ^2(4)}dx\) |
\(\Big \downarrow \) 1294 |
\(\displaystyle (6+\log (4))^2 \int -\frac {-(6+\log (4))^2 x^2+6 \log (4) (6+\log (4)) x-4 \log (4) (3+\log (16))}{(6+\log (4))^2 (3 \log (4)-x (6+\log (4)))^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -(6+\log (4))^2 \int \frac {-(6+\log (4))^2 x^2+6 \log (4) (6+\log (4)) x-4 \log (4) (3+\log (16))}{(6+\log (4))^2 (x (6+\log (4))-\log (64))^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\int \frac {-(6+\log (4))^2 x^2+6 \log (4) (6+\log (4)) x-4 \log (4) (3+\log (16))}{(x (6+\log (4))-\log (64))^2}dx\) |
\(\Big \downarrow \) 1107 |
\(\displaystyle -\int \left (\frac {-\log ^2(64)-2 \log (4) (6-\log (1024))}{(x (6+\log (4))-\log (64))^2}-1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x-\frac {\log ^2(64)+2 \log (4) (6-\log (1024))}{(6+\log (4)) (x (6+\log (4))-\log (64))}\) |
Input:
Int[(36*x^2 + (12 - 36*x + 12*x^2)*Log[4] + (8 - 6*x + x^2)*Log[4]^2)/(36* x^2 + (-36*x + 12*x^2)*Log[4] + (9 - 6*x + x^2)*Log[4]^2),x]
Output:
x - (Log[64]^2 + 2*Log[4]*(6 - Log[1024]))/((6 + Log[4])*(x*(6 + Log[4]) - Log[64]))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.)*((d_.) + (e_.)*(x_) + (f_.)*(x _)^2)^(q_.), x_Symbol] :> Simp[1/c^p Int[(b/2 + c*x)^(2*p)*(d + e*x + f*x ^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Int[(u_)^(p_.)*(v_)^(q_.), x_Symbol] :> Int[ExpandToSum[u, x]^p*ExpandToSum [v, x]^q, x] /; FreeQ[{p, q}, x] && QuadraticQ[{u, v}, x] && !QuadraticMat chQ[{u, v}, x]
Time = 0.31 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.43
method | result | size |
default | \(x +\frac {\ln \left (2\right ) \left (\ln \left (2\right )-6\right )}{\left (3+\ln \left (2\right )\right ) \left (x \ln \left (2\right )-3 \ln \left (2\right )+3 x \right )}\) | \(30\) |
gosper | \(\frac {x \left (3 x \ln \left (2\right )-8 \ln \left (2\right )+9 x -6\right )}{3 x \ln \left (2\right )-9 \ln \left (2\right )+9 x}\) | \(32\) |
norman | \(\frac {\left (3+\ln \left (2\right )\right ) x^{2}+\left (-\frac {8 \ln \left (2\right )}{3}-2\right ) x}{x \ln \left (2\right )-3 \ln \left (2\right )+3 x}\) | \(33\) |
parallelrisch | \(\frac {3 x^{2} \ln \left (2\right )^{2}-8 x \ln \left (2\right )^{2}+9 x^{2} \ln \left (2\right )-6 x \ln \left (2\right )}{3 \ln \left (2\right ) \left (x \ln \left (2\right )-3 \ln \left (2\right )+3 x \right )}\) | \(50\) |
risch | \(x +\frac {\ln \left (2\right )^{2}}{\left (3+\ln \left (2\right )\right ) \left (x \ln \left (2\right )-3 \ln \left (2\right )+3 x \right )}-\frac {6 \ln \left (2\right )}{\left (3+\ln \left (2\right )\right ) \left (x \ln \left (2\right )-3 \ln \left (2\right )+3 x \right )}\) | \(52\) |
meijerg | \(-\frac {3 \ln \left (2\right ) \left (-\frac {x \left (3+\ln \left (2\right )\right ) \left (-\frac {x \left (3+\ln \left (2\right )\right )}{\ln \left (2\right )}+6\right )}{9 \ln \left (2\right ) \left (1-\frac {x \left (3+\ln \left (2\right )\right )}{3 \ln \left (2\right )}\right )}-2 \ln \left (1-\frac {x \left (3+\ln \left (2\right )\right )}{3 \ln \left (2\right )}\right )\right )}{3+\ln \left (2\right )}+\frac {9 \left (-\frac {8 \ln \left (2\right )^{2}}{3}-8 \ln \left (2\right )\right ) \left (\frac {x \left (3+\ln \left (2\right )\right )}{3 \ln \left (2\right ) \left (1-\frac {x \left (3+\ln \left (2\right )\right )}{3 \ln \left (2\right )}\right )}+\ln \left (1-\frac {x \left (3+\ln \left (2\right )\right )}{3 \ln \left (2\right )}\right )\right )}{4 \ln \left (2\right )^{2}+24 \ln \left (2\right )+36}+\frac {32 \left (3+\ln \left (2\right )\right )^{2} x}{9 \left (4 \ln \left (2\right )^{2}+24 \ln \left (2\right )+36\right ) \left (1-\frac {x \left (3+\ln \left (2\right )\right )}{3 \ln \left (2\right )}\right )}+\frac {8 \left (3+\ln \left (2\right )\right )^{2} x}{3 \ln \left (2\right ) \left (4 \ln \left (2\right )^{2}+24 \ln \left (2\right )+36\right ) \left (1-\frac {x \left (3+\ln \left (2\right )\right )}{3 \ln \left (2\right )}\right )}\) | \(216\) |
Input:
int((4*(x^2-6*x+8)*ln(2)^2+2*(12*x^2-36*x+12)*ln(2)+36*x^2)/(4*(x^2-6*x+9) *ln(2)^2+2*(12*x^2-36*x)*ln(2)+36*x^2),x,method=_RETURNVERBOSE)
Output:
x+ln(2)*(ln(2)-6)/(3+ln(2))/(x*ln(2)-3*ln(2)+3*x)
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (18) = 36\).
Time = 0.11 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.71 \[ \int \frac {36 x^2+\left (12-36 x+12 x^2\right ) \log (4)+\left (8-6 x+x^2\right ) \log ^2(4)}{36 x^2+\left (-36 x+12 x^2\right ) \log (4)+\left (9-6 x+x^2\right ) \log ^2(4)} \, dx=\frac {{\left (x^{2} - 3 \, x + 1\right )} \log \left (2\right )^{2} + 9 \, x^{2} + 3 \, {\left (2 \, x^{2} - 3 \, x - 2\right )} \log \left (2\right )}{{\left (x - 3\right )} \log \left (2\right )^{2} + 3 \, {\left (2 \, x - 3\right )} \log \left (2\right ) + 9 \, x} \] Input:
integrate((4*(x^2-6*x+8)*log(2)^2+2*(12*x^2-36*x+12)*log(2)+36*x^2)/(4*(x^ 2-6*x+9)*log(2)^2+2*(12*x^2-36*x)*log(2)+36*x^2),x, algorithm="fricas")
Output:
((x^2 - 3*x + 1)*log(2)^2 + 9*x^2 + 3*(2*x^2 - 3*x - 2)*log(2))/((x - 3)*l og(2)^2 + 3*(2*x - 3)*log(2) + 9*x)
Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (14) = 28\).
Time = 0.18 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.71 \[ \int \frac {36 x^2+\left (12-36 x+12 x^2\right ) \log (4)+\left (8-6 x+x^2\right ) \log ^2(4)}{36 x^2+\left (-36 x+12 x^2\right ) \log (4)+\left (9-6 x+x^2\right ) \log ^2(4)} \, dx=x + \frac {- 6 \log {\left (2 \right )} + \log {\left (2 \right )}^{2}}{x \left (\log {\left (2 \right )}^{2} + 6 \log {\left (2 \right )} + 9\right ) - 9 \log {\left (2 \right )} - 3 \log {\left (2 \right )}^{2}} \] Input:
integrate((4*(x**2-6*x+8)*ln(2)**2+2*(12*x**2-36*x+12)*ln(2)+36*x**2)/(4*( x**2-6*x+9)*ln(2)**2+2*(12*x**2-36*x)*ln(2)+36*x**2),x)
Output:
x + (-6*log(2) + log(2)**2)/(x*(log(2)**2 + 6*log(2) + 9) - 9*log(2) - 3*l og(2)**2)
Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (18) = 36\).
Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.76 \[ \int \frac {36 x^2+\left (12-36 x+12 x^2\right ) \log (4)+\left (8-6 x+x^2\right ) \log ^2(4)}{36 x^2+\left (-36 x+12 x^2\right ) \log (4)+\left (9-6 x+x^2\right ) \log ^2(4)} \, dx=x + \frac {\log \left (2\right )^{2} - 6 \, \log \left (2\right )}{{\left (\log \left (2\right )^{2} + 6 \, \log \left (2\right ) + 9\right )} x - 3 \, \log \left (2\right )^{2} - 9 \, \log \left (2\right )} \] Input:
integrate((4*(x^2-6*x+8)*log(2)^2+2*(12*x^2-36*x+12)*log(2)+36*x^2)/(4*(x^ 2-6*x+9)*log(2)^2+2*(12*x^2-36*x)*log(2)+36*x^2),x, algorithm="maxima")
Output:
x + (log(2)^2 - 6*log(2))/((log(2)^2 + 6*log(2) + 9)*x - 3*log(2)^2 - 9*lo g(2))
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (18) = 36\).
Time = 0.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.81 \[ \int \frac {36 x^2+\left (12-36 x+12 x^2\right ) \log (4)+\left (8-6 x+x^2\right ) \log ^2(4)}{36 x^2+\left (-36 x+12 x^2\right ) \log (4)+\left (9-6 x+x^2\right ) \log ^2(4)} \, dx=\frac {x \log \left (2\right )^{2} + 6 \, x \log \left (2\right ) + 9 \, x}{\log \left (2\right )^{2} + 6 \, \log \left (2\right ) + 9} + \frac {\log \left (2\right )^{2} - 6 \, \log \left (2\right )}{{\left (x \log \left (2\right ) + 3 \, x - 3 \, \log \left (2\right )\right )} {\left (\log \left (2\right ) + 3\right )}} \] Input:
integrate((4*(x^2-6*x+8)*log(2)^2+2*(12*x^2-36*x+12)*log(2)+36*x^2)/(4*(x^ 2-6*x+9)*log(2)^2+2*(12*x^2-36*x)*log(2)+36*x^2),x, algorithm="giac")
Output:
(x*log(2)^2 + 6*x*log(2) + 9*x)/(log(2)^2 + 6*log(2) + 9) + (log(2)^2 - 6* log(2))/((x*log(2) + 3*x - 3*log(2))*(log(2) + 3))
Time = 4.01 (sec) , antiderivative size = 112, normalized size of antiderivative = 5.33 \[ \int \frac {36 x^2+\left (12-36 x+12 x^2\right ) \log (4)+\left (8-6 x+x^2\right ) \log ^2(4)}{36 x^2+\left (-36 x+12 x^2\right ) \log (4)+\left (9-6 x+x^2\right ) \log ^2(4)} \, dx=x-\frac {\mathrm {atan}\left (\frac {\frac {\left (18\,\ln \left (2\right )+6\,{\ln \left (2\right )}^2\right )\,\left (\ln \left (64\right )-{\ln \left (2\right )}^2\right )}{6\,\ln \left (2\right )\,\sqrt {\ln \left (64\right )-6\,\ln \left (2\right )}}-\frac {x\,\left (\ln \left (64\right )-{\ln \left (2\right )}^2\right )\,\left (2\,\ln \left (64\right )+2\,{\ln \left (2\right )}^2+18\right )}{6\,\ln \left (2\right )\,\sqrt {\ln \left (64\right )-6\,\ln \left (2\right )}}}{\ln \left (64\right )-{\ln \left (2\right )}^2}\right )\,\left (\ln \left (64\right )-{\ln \left (2\right )}^2\right )}{3\,\ln \left (2\right )\,\sqrt {\ln \left (64\right )-6\,\ln \left (2\right )}} \] Input:
int((2*log(2)*(12*x^2 - 36*x + 12) + 4*log(2)^2*(x^2 - 6*x + 8) + 36*x^2)/ (4*log(2)^2*(x^2 - 6*x + 9) - 2*log(2)*(36*x - 12*x^2) + 36*x^2),x)
Output:
x - (atan((((18*log(2) + 6*log(2)^2)*(log(64) - log(2)^2))/(6*log(2)*(log( 64) - 6*log(2))^(1/2)) - (x*(log(64) - log(2)^2)*(2*log(64) + 2*log(2)^2 + 18))/(6*log(2)*(log(64) - 6*log(2))^(1/2)))/(log(64) - log(2)^2))*(log(64 ) - log(2)^2))/(3*log(2)*(log(64) - 6*log(2))^(1/2))
Time = 0.17 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.48 \[ \int \frac {36 x^2+\left (12-36 x+12 x^2\right ) \log (4)+\left (8-6 x+x^2\right ) \log ^2(4)}{36 x^2+\left (-36 x+12 x^2\right ) \log (4)+\left (9-6 x+x^2\right ) \log ^2(4)} \, dx=\frac {x \left (3 \,\mathrm {log}\left (2\right ) x -8 \,\mathrm {log}\left (2\right )+9 x -6\right )}{3 \,\mathrm {log}\left (2\right ) x -9 \,\mathrm {log}\left (2\right )+9 x} \] Input:
int((4*(x^2-6*x+8)*log(2)^2+2*(12*x^2-36*x+12)*log(2)+36*x^2)/(4*(x^2-6*x+ 9)*log(2)^2+2*(12*x^2-36*x)*log(2)+36*x^2),x)
Output:
(x*(3*log(2)*x - 8*log(2) + 9*x - 6))/(3*(log(2)*x - 3*log(2) + 3*x))