Integrand size = 109, antiderivative size = 32 \[ \int \frac {612+840 x-519 x^2+12 x^3+\left (-420+336 x-6 x^2\right ) \log (4)}{2000-1600 x-680 x^2+800 x^3-35 x^4-100 x^5+20 x^6+\left (1000 x-800 x^2-90 x^3+200 x^4-40 x^5\right ) \log (4)+\left (125 x^2-100 x^3+20 x^4\right ) \log ^2(4)} \, dx=\frac {3 \left (5+\frac {3-x}{5}\right )}{(-5+2 x) (-4+x (x-\log (4)))} \] Output:
(28/5-1/5*x)/(1/3*x*(x-2*ln(2))-4/3)/(-5+2*x)
Leaf count is larger than twice the leaf count of optimal. \(132\) vs. \(2(32)=64\).
Time = 0.07 (sec) , antiderivative size = 132, normalized size of antiderivative = 4.12 \[ \int \frac {612+840 x-519 x^2+12 x^3+\left (-420+336 x-6 x^2\right ) \log (4)}{2000-1600 x-680 x^2+800 x^3-35 x^4-100 x^5+20 x^6+\left (1000 x-800 x^2-90 x^3+200 x^4-40 x^5\right ) \log (4)+\left (125 x^2-100 x^3+20 x^4\right ) \log ^2(4)} \, dx=\frac {3 \left (x \left (-2240 \log ^4(4)+72 (-18+107 \log (16))+2 \log ^3(4) (2007+535 \log (16))+48 \log (4) (-261+589 \log (16))-\log ^2(4) (58225+1917 \log (16))\right )+4 \left (9072-5640 \log ^3(4)+700 \log ^4(4)+39280 \log (16)-5 \log (4) (19744+1129 \log (16))+\log ^2(4) (23057+2190 \log (16))\right )\right )}{5 (-5+2 x) (9-10 \log (4))^2 \left (-4+x^2-x \log (4)\right ) \left (16+\log ^2(4)\right )} \] Input:
Integrate[(612 + 840*x - 519*x^2 + 12*x^3 + (-420 + 336*x - 6*x^2)*Log[4]) /(2000 - 1600*x - 680*x^2 + 800*x^3 - 35*x^4 - 100*x^5 + 20*x^6 + (1000*x - 800*x^2 - 90*x^3 + 200*x^4 - 40*x^5)*Log[4] + (125*x^2 - 100*x^3 + 20*x^ 4)*Log[4]^2),x]
Output:
(3*(x*(-2240*Log[4]^4 + 72*(-18 + 107*Log[16]) + 2*Log[4]^3*(2007 + 535*Lo g[16]) + 48*Log[4]*(-261 + 589*Log[16]) - Log[4]^2*(58225 + 1917*Log[16])) + 4*(9072 - 5640*Log[4]^3 + 700*Log[4]^4 + 39280*Log[16] - 5*Log[4]*(1974 4 + 1129*Log[16]) + Log[4]^2*(23057 + 2190*Log[16]))))/(5*(-5 + 2*x)*(9 - 10*Log[4])^2*(-4 + x^2 - x*Log[4])*(16 + Log[4]^2))
Time = 0.47 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.78, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {2462, 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.
\(\displaystyle \int \frac {12 x^3-519 x^2+\left (-6 x^2+336 x-420\right ) \log (4)+840 x+612}{20 x^6-100 x^5-35 x^4+800 x^3-680 x^2+\left (20 x^4-100 x^3+125 x^2\right ) \log ^2(4)+\left (-40 x^5+200 x^4-90 x^3-800 x^2+1000 x\right ) \log (4)-1600 x+2000} \, dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (\frac {3 \left (x (264-61 \log (4))+4 \left (102+14 \log ^2(4)-33 \log (4)\right )\right )}{5 (9-10 \log (4)) \left (-x^2+x \log (4)+4\right )^2}+\frac {153}{5 (10 \log (4)-9) \left (-x^2+x \log (4)+4\right )}+\frac {612}{5 (2 x-5)^2 (10 \log (4)-9)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 (51 x+4 (33-14 \log (4)))}{5 (9-10 \log (4)) \left (-x^2+x \log (4)+4\right )}-\frac {306}{5 (5-2 x) (9-10 \log (4))}\) |
Input:
Int[(612 + 840*x - 519*x^2 + 12*x^3 + (-420 + 336*x - 6*x^2)*Log[4])/(2000 - 1600*x - 680*x^2 + 800*x^3 - 35*x^4 - 100*x^5 + 20*x^6 + (1000*x - 800* x^2 - 90*x^3 + 200*x^4 - 40*x^5)*Log[4] + (125*x^2 - 100*x^3 + 20*x^4)*Log [4]^2),x]
Output:
-306/(5*(5 - 2*x)*(9 - 10*Log[4])) + (3*(51*x + 4*(33 - 14*Log[4])))/(5*(9 - 10*Log[4])*(4 - x^2 + x*Log[4]))
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 0.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88
method | result | size |
norman | \(\frac {\frac {3 x}{5}-\frac {84}{5}}{\left (-5+2 x \right ) \left (2 x \ln \left (2\right )-x^{2}+4\right )}\) | \(28\) |
gosper | \(\frac {\frac {3 x}{5}-\frac {84}{5}}{4 x^{2} \ln \left (2\right )-2 x^{3}-10 x \ln \left (2\right )+5 x^{2}+8 x -20}\) | \(35\) |
risch | \(\frac {\frac {3 x}{20}-\frac {21}{5}}{x^{2} \ln \left (2\right )-\frac {x^{3}}{2}-\frac {5 x \ln \left (2\right )}{2}+\frac {5 x^{2}}{4}+2 x -5}\) | \(35\) |
parallelrisch | \(-\frac {168-6 x}{10 \left (4 x^{2} \ln \left (2\right )-2 x^{3}-10 x \ln \left (2\right )+5 x^{2}+8 x -20\right )}\) | \(37\) |
default | \(-\frac {3 \left (\frac {51 x}{2}-56 \ln \left (2\right )+66\right )}{5 \left (20 \ln \left (2\right )-9\right ) \left (x \ln \left (2\right )-\frac {x^{2}}{2}+2\right )}-\frac {306}{5 \left (20 \ln \left (2\right )-9\right ) \left (-5+2 x \right )}\) | \(51\) |
Input:
int((2*(-6*x^2+336*x-420)*ln(2)+12*x^3-519*x^2+840*x+612)/(4*(20*x^4-100*x ^3+125*x^2)*ln(2)^2+2*(-40*x^5+200*x^4-90*x^3-800*x^2+1000*x)*ln(2)+20*x^6 -100*x^5-35*x^4+800*x^3-680*x^2-1600*x+2000),x,method=_RETURNVERBOSE)
Output:
(3/5*x-84/5)/(-5+2*x)/(2*x*ln(2)-x^2+4)
Time = 0.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {612+840 x-519 x^2+12 x^3+\left (-420+336 x-6 x^2\right ) \log (4)}{2000-1600 x-680 x^2+800 x^3-35 x^4-100 x^5+20 x^6+\left (1000 x-800 x^2-90 x^3+200 x^4-40 x^5\right ) \log (4)+\left (125 x^2-100 x^3+20 x^4\right ) \log ^2(4)} \, dx=-\frac {3 \, {\left (x - 28\right )}}{5 \, {\left (2 \, x^{3} - 5 \, x^{2} - 2 \, {\left (2 \, x^{2} - 5 \, x\right )} \log \left (2\right ) - 8 \, x + 20\right )}} \] Input:
integrate((2*(-6*x^2+336*x-420)*log(2)+12*x^3-519*x^2+840*x+612)/(4*(20*x^ 4-100*x^3+125*x^2)*log(2)^2+2*(-40*x^5+200*x^4-90*x^3-800*x^2+1000*x)*log( 2)+20*x^6-100*x^5-35*x^4+800*x^3-680*x^2-1600*x+2000),x, algorithm="fricas ")
Output:
-3/5*(x - 28)/(2*x^3 - 5*x^2 - 2*(2*x^2 - 5*x)*log(2) - 8*x + 20)
Time = 1.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {612+840 x-519 x^2+12 x^3+\left (-420+336 x-6 x^2\right ) \log (4)}{2000-1600 x-680 x^2+800 x^3-35 x^4-100 x^5+20 x^6+\left (1000 x-800 x^2-90 x^3+200 x^4-40 x^5\right ) \log (4)+\left (125 x^2-100 x^3+20 x^4\right ) \log ^2(4)} \, dx=\frac {84 - 3 x}{10 x^{3} + x^{2} \left (-25 - 20 \log {\left (2 \right )}\right ) + x \left (-40 + 50 \log {\left (2 \right )}\right ) + 100} \] Input:
integrate((2*(-6*x**2+336*x-420)*ln(2)+12*x**3-519*x**2+840*x+612)/(4*(20* x**4-100*x**3+125*x**2)*ln(2)**2+2*(-40*x**5+200*x**4-90*x**3-800*x**2+100 0*x)*ln(2)+20*x**6-100*x**5-35*x**4+800*x**3-680*x**2-1600*x+2000),x)
Output:
(84 - 3*x)/(10*x**3 + x**2*(-25 - 20*log(2)) + x*(-40 + 50*log(2)) + 100)
Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {612+840 x-519 x^2+12 x^3+\left (-420+336 x-6 x^2\right ) \log (4)}{2000-1600 x-680 x^2+800 x^3-35 x^4-100 x^5+20 x^6+\left (1000 x-800 x^2-90 x^3+200 x^4-40 x^5\right ) \log (4)+\left (125 x^2-100 x^3+20 x^4\right ) \log ^2(4)} \, dx=-\frac {3 \, {\left (x - 28\right )}}{5 \, {\left (2 \, x^{3} - x^{2} {\left (4 \, \log \left (2\right ) + 5\right )} + 2 \, x {\left (5 \, \log \left (2\right ) - 4\right )} + 20\right )}} \] Input:
integrate((2*(-6*x^2+336*x-420)*log(2)+12*x^3-519*x^2+840*x+612)/(4*(20*x^ 4-100*x^3+125*x^2)*log(2)^2+2*(-40*x^5+200*x^4-90*x^3-800*x^2+1000*x)*log( 2)+20*x^6-100*x^5-35*x^4+800*x^3-680*x^2-1600*x+2000),x, algorithm="maxima ")
Output:
-3/5*(x - 28)/(2*x^3 - x^2*(4*log(2) + 5) + 2*x*(5*log(2) - 4) + 20)
Time = 0.13 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {612+840 x-519 x^2+12 x^3+\left (-420+336 x-6 x^2\right ) \log (4)}{2000-1600 x-680 x^2+800 x^3-35 x^4-100 x^5+20 x^6+\left (1000 x-800 x^2-90 x^3+200 x^4-40 x^5\right ) \log (4)+\left (125 x^2-100 x^3+20 x^4\right ) \log ^2(4)} \, dx=-\frac {3 \, {\left (x - 28\right )}}{5 \, {\left (2 \, x^{3} - 4 \, x^{2} \log \left (2\right ) - 5 \, x^{2} + 10 \, x \log \left (2\right ) - 8 \, x + 20\right )}} \] Input:
integrate((2*(-6*x^2+336*x-420)*log(2)+12*x^3-519*x^2+840*x+612)/(4*(20*x^ 4-100*x^3+125*x^2)*log(2)^2+2*(-40*x^5+200*x^4-90*x^3-800*x^2+1000*x)*log( 2)+20*x^6-100*x^5-35*x^4+800*x^3-680*x^2-1600*x+2000),x, algorithm="giac")
Output:
-3/5*(x - 28)/(2*x^3 - 4*x^2*log(2) - 5*x^2 + 10*x*log(2) - 8*x + 20)
Time = 0.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {612+840 x-519 x^2+12 x^3+\left (-420+336 x-6 x^2\right ) \log (4)}{2000-1600 x-680 x^2+800 x^3-35 x^4-100 x^5+20 x^6+\left (1000 x-800 x^2-90 x^3+200 x^4-40 x^5\right ) \log (4)+\left (125 x^2-100 x^3+20 x^4\right ) \log ^2(4)} \, dx=\frac {3\,\left (x-28\right )}{5\,\left (2\,x-5\right )\,\left (-x^2+2\,\ln \left (2\right )\,x+4\right )} \] Input:
int(-(840*x - 2*log(2)*(6*x^2 - 336*x + 420) - 519*x^2 + 12*x^3 + 612)/(16 00*x - 4*log(2)^2*(125*x^2 - 100*x^3 + 20*x^4) + 680*x^2 - 800*x^3 + 35*x^ 4 + 100*x^5 - 20*x^6 + 2*log(2)*(800*x^2 - 1000*x + 90*x^3 - 200*x^4 + 40* x^5) - 2000),x)
Output:
(3*(x - 28))/(5*(2*x - 5)*(2*x*log(2) - x^2 + 4))
Time = 0.21 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {612+840 x-519 x^2+12 x^3+\left (-420+336 x-6 x^2\right ) \log (4)}{2000-1600 x-680 x^2+800 x^3-35 x^4-100 x^5+20 x^6+\left (1000 x-800 x^2-90 x^3+200 x^4-40 x^5\right ) \log (4)+\left (125 x^2-100 x^3+20 x^4\right ) \log ^2(4)} \, dx=\frac {3 x -84}{20 \,\mathrm {log}\left (2\right ) x^{2}-50 \,\mathrm {log}\left (2\right ) x -10 x^{3}+25 x^{2}+40 x -100} \] Input:
int((2*(-6*x^2+336*x-420)*log(2)+12*x^3-519*x^2+840*x+612)/(4*(20*x^4-100* x^3+125*x^2)*log(2)^2+2*(-40*x^5+200*x^4-90*x^3-800*x^2+1000*x)*log(2)+20* x^6-100*x^5-35*x^4+800*x^3-680*x^2-1600*x+2000),x)
Output:
(3*(x - 28))/(5*(4*log(2)*x**2 - 10*log(2)*x - 2*x**3 + 5*x**2 + 8*x - 20) )