Integrand size = 109, antiderivative size = 22 \[ \int \frac {-16+6 x-163840 x^3+69632 x^4-5120 x^5-1920 x^6+440 x^7-35 x^8+x^9+\left (-131072 x^3+16384 x^4+20480 x^5-7680 x^6+1120 x^7-76 x^8+2 x^9\right ) \log (x)}{-32768 x^3+20480 x^4-5120 x^5+640 x^6-40 x^7+x^8} \, dx=-2-\frac {1}{(8-x)^4 x^2}+x+x (4+x) \log (x) \] Output:
ln(x)*(4+x)*x+x-2-1/x^2/(8-x)^4
Leaf count is larger than twice the leaf count of optimal. \(48\) vs. \(2(22)=44\).
Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.18 \[ \int \frac {-16+6 x-163840 x^3+69632 x^4-5120 x^5-1920 x^6+440 x^7-35 x^8+x^9+\left (-131072 x^3+16384 x^4+20480 x^5-7680 x^6+1120 x^7-76 x^8+2 x^9\right ) \log (x)}{-32768 x^3+20480 x^4-5120 x^5+640 x^6-40 x^7+x^8} \, dx=\frac {-1+4096 x^3-2048 x^4+384 x^5-32 x^6+x^7+(-8+x)^4 x^3 (4+x) \log (x)}{(-8+x)^4 x^2} \] Input:
Integrate[(-16 + 6*x - 163840*x^3 + 69632*x^4 - 5120*x^5 - 1920*x^6 + 440* x^7 - 35*x^8 + x^9 + (-131072*x^3 + 16384*x^4 + 20480*x^5 - 7680*x^6 + 112 0*x^7 - 76*x^8 + 2*x^9)*Log[x])/(-32768*x^3 + 20480*x^4 - 5120*x^5 + 640*x ^6 - 40*x^7 + x^8),x]
Output:
(-1 + 4096*x^3 - 2048*x^4 + 384*x^5 - 32*x^6 + x^7 + (-8 + x)^4*x^3*(4 + x )*Log[x])/((-8 + x)^4*x^2)
Leaf count is larger than twice the leaf count of optimal. \(84\) vs. \(2(22)=44\).
Time = 0.84 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.82, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {2026, 2007, 7293, 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.
\(\displaystyle \int \frac {x^9-35 x^8+440 x^7-1920 x^6-5120 x^5+69632 x^4-163840 x^3+\left (2 x^9-76 x^8+1120 x^7-7680 x^6+20480 x^5+16384 x^4-131072 x^3\right ) \log (x)+6 x-16}{x^8-40 x^7+640 x^6-5120 x^5+20480 x^4-32768 x^3} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {x^9-35 x^8+440 x^7-1920 x^6-5120 x^5+69632 x^4-163840 x^3+\left (2 x^9-76 x^8+1120 x^7-7680 x^6+20480 x^5+16384 x^4-131072 x^3\right ) \log (x)+6 x-16}{x^3 \left (x^5-40 x^4+640 x^3-5120 x^2+20480 x-32768\right )}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {x^9-35 x^8+440 x^7-1920 x^6-5120 x^5+69632 x^4-163840 x^3+\left (2 x^9-76 x^8+1120 x^7-7680 x^6+20480 x^5+16384 x^4-131072 x^3\right ) \log (x)+6 x-16}{(x-8)^5 x^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x^6}{(x-8)^5}-\frac {35 x^5}{(x-8)^5}+\frac {440 x^4}{(x-8)^5}-\frac {1920 x^3}{(x-8)^5}-\frac {16}{(x-8)^5 x^3}-\frac {5120 x^2}{(x-8)^5}+\frac {6}{(x-8)^5 x^2}+\frac {69632 x}{(x-8)^5}-\frac {163840}{(x-8)^5}+2 (x+2) \log (x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {60 x^4}{(8-x)^4}-\frac {1}{4096 x^2}+x+\frac {15728639}{8192 (8-x)}-\frac {94371843}{4096 (8-x)^2}+\frac {31457279}{256 (8-x)^3}-\frac {15728641}{64 (8-x)^4}-\frac {1}{8192 x}+(x+2)^2 \log (x)-4 \log (x)\) |
Input:
Int[(-16 + 6*x - 163840*x^3 + 69632*x^4 - 5120*x^5 - 1920*x^6 + 440*x^7 - 35*x^8 + x^9 + (-131072*x^3 + 16384*x^4 + 20480*x^5 - 7680*x^6 + 1120*x^7 - 76*x^8 + 2*x^9)*Log[x])/(-32768*x^3 + 20480*x^4 - 5120*x^5 + 640*x^6 - 4 0*x^7 + x^8),x]
Output:
-15728641/(64*(8 - x)^4) + 31457279/(256*(8 - x)^3) - 94371843/(4096*(8 - x)^2) + 15728639/(8192*(8 - x)) - 1/(4096*x^2) - 1/(8192*x) + x + (60*x^4) /(8 - x)^4 - 4*Log[x] + (2 + x)^2*Log[x]
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[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Leaf count of result is larger than twice the leaf count of optimal. \(51\) vs. \(2(22)=44\).
Time = 1.66 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.36
method | result | size |
default | \(x^{2} \ln \left (x \right )+4 x \ln \left (x \right )+x -\frac {1}{4096 x^{2}}-\frac {1}{8192 x}-\frac {1}{64 \left (-8+x \right )^{4}}+\frac {1}{256 \left (-8+x \right )^{3}}-\frac {3}{4096 \left (-8+x \right )^{2}}+\frac {1}{-65536+8192 x}\) | \(52\) |
parts | \(x^{2} \ln \left (x \right )+4 x \ln \left (x \right )+x -\frac {1}{4096 x^{2}}-\frac {1}{8192 x}-\frac {1}{64 \left (-8+x \right )^{4}}+\frac {1}{256 \left (-8+x \right )^{3}}-\frac {3}{4096 \left (-8+x \right )^{2}}+\frac {1}{-65536+8192 x}\) | \(52\) |
risch | \(\left (x^{2}+4 x \right ) \ln \left (x \right )+\frac {x^{7}-32 x^{6}+384 x^{5}-2048 x^{4}+4096 x^{3}-1}{x^{2} \left (x^{4}-32 x^{3}+384 x^{2}-2048 x +4096\right )}\) | \(61\) |
norman | \(\frac {-1+x^{7}+x^{8} \ln \left (x \right )-61440 x^{3}-640 x^{5}+10240 x^{4}+131072 x^{2}-4096 x^{4} \ln \left (x \right )-512 x^{5} \ln \left (x \right )+256 x^{6} \ln \left (x \right )+16384 x^{3} \ln \left (x \right )-28 x^{7} \ln \left (x \right )}{x^{2} \left (-8+x \right )^{4}}\) | \(76\) |
parallelrisch | \(\frac {-6144+6144 x^{8} \ln \left (x \right )-3145728 x^{5} \ln \left (x \right )+1572864 x^{6} \ln \left (x \right )-172032 x^{7} \ln \left (x \right )-25165824 x^{4} \ln \left (x \right )+6144 x^{7}+100663296 x^{3} \ln \left (x \right )-163840 x^{6}+134217728 x^{2}-41943040 x^{3}+1310720 x^{5}}{6144 x^{2} \left (x^{4}-32 x^{3}+384 x^{2}-2048 x +4096\right )}\) | \(95\) |
orering | \(\frac {x \left (3 x^{11}-130 x^{10}+1848 x^{9}-304640 x^{7}+3956736 x^{6}-24084480 x^{5}+74448896 x^{4}-94371880 x^{3}+24 x^{2}+640 x -1536\right ) \left (\left (2 x^{9}-76 x^{8}+1120 x^{7}-7680 x^{6}+20480 x^{5}+16384 x^{4}-131072 x^{3}\right ) \ln \left (x \right )+x^{9}-35 x^{8}+440 x^{7}-1920 x^{6}-5120 x^{5}+69632 x^{4}-163840 x^{3}+6 x -16\right )}{2 \left (2 x^{11}-91 x^{10}+1688 x^{9}-16064 x^{8}+79360 x^{7}-167936 x^{6}+32768 x^{5}-262144 x^{4}+2097104 x^{3}+204 x^{2}-64 x -768\right ) \left (x^{8}-40 x^{7}+640 x^{6}-5120 x^{5}+20480 x^{4}-32768 x^{3}\right )}-\frac {x^{2} \left (x^{10}-33 x^{9}+13008 x^{7}-215680 x^{6}+1472512 x^{5}-3751936 x^{4}-3604480 x^{3}+24641544 x^{2}-4 x -96\right ) \left (-8+x \right ) \left (\frac {\left (18 x^{8}-608 x^{7}+7840 x^{6}-46080 x^{5}+102400 x^{4}+65536 x^{3}-393216 x^{2}\right ) \ln \left (x \right )+\frac {2 x^{9}-76 x^{8}+1120 x^{7}-7680 x^{6}+20480 x^{5}+16384 x^{4}-131072 x^{3}}{x}+9 x^{8}-280 x^{7}+3080 x^{6}-11520 x^{5}-25600 x^{4}+278528 x^{3}-491520 x^{2}+6}{x^{8}-40 x^{7}+640 x^{6}-5120 x^{5}+20480 x^{4}-32768 x^{3}}-\frac {\left (\left (2 x^{9}-76 x^{8}+1120 x^{7}-7680 x^{6}+20480 x^{5}+16384 x^{4}-131072 x^{3}\right ) \ln \left (x \right )+x^{9}-35 x^{8}+440 x^{7}-1920 x^{6}-5120 x^{5}+69632 x^{4}-163840 x^{3}+6 x -16\right ) \left (8 x^{7}-280 x^{6}+3840 x^{5}-25600 x^{4}+81920 x^{3}-98304 x^{2}\right )}{\left (x^{8}-40 x^{7}+640 x^{6}-5120 x^{5}+20480 x^{4}-32768 x^{3}\right )^{2}}\right )}{2 \left (2 x^{11}-91 x^{10}+1688 x^{9}-16064 x^{8}+79360 x^{7}-167936 x^{6}+32768 x^{5}-262144 x^{4}+2097104 x^{3}+204 x^{2}-64 x -768\right )}\) | \(618\) |
Input:
int(((2*x^9-76*x^8+1120*x^7-7680*x^6+20480*x^5+16384*x^4-131072*x^3)*ln(x) +x^9-35*x^8+440*x^7-1920*x^6-5120*x^5+69632*x^4-163840*x^3+6*x-16)/(x^8-40 *x^7+640*x^6-5120*x^5+20480*x^4-32768*x^3),x,method=_RETURNVERBOSE)
Output:
x^2*ln(x)+4*x*ln(x)+x-1/4096/x^2-1/8192/x-1/64/(-8+x)^4+1/256/(-8+x)^3-3/4 096/(-8+x)^2+1/8192/(-8+x)
Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (20) = 40\).
Time = 0.09 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.82 \[ \int \frac {-16+6 x-163840 x^3+69632 x^4-5120 x^5-1920 x^6+440 x^7-35 x^8+x^9+\left (-131072 x^3+16384 x^4+20480 x^5-7680 x^6+1120 x^7-76 x^8+2 x^9\right ) \log (x)}{-32768 x^3+20480 x^4-5120 x^5+640 x^6-40 x^7+x^8} \, dx=\frac {x^{7} - 32 \, x^{6} + 384 \, x^{5} - 2048 \, x^{4} + 4096 \, x^{3} + {\left (x^{8} - 28 \, x^{7} + 256 \, x^{6} - 512 \, x^{5} - 4096 \, x^{4} + 16384 \, x^{3}\right )} \log \left (x\right ) - 1}{x^{6} - 32 \, x^{5} + 384 \, x^{4} - 2048 \, x^{3} + 4096 \, x^{2}} \] Input:
integrate(((2*x^9-76*x^8+1120*x^7-7680*x^6+20480*x^5+16384*x^4-131072*x^3) *log(x)+x^9-35*x^8+440*x^7-1920*x^6-5120*x^5+69632*x^4-163840*x^3+6*x-16)/ (x^8-40*x^7+640*x^6-5120*x^5+20480*x^4-32768*x^3),x, algorithm="fricas")
Output:
(x^7 - 32*x^6 + 384*x^5 - 2048*x^4 + 4096*x^3 + (x^8 - 28*x^7 + 256*x^6 - 512*x^5 - 4096*x^4 + 16384*x^3)*log(x) - 1)/(x^6 - 32*x^5 + 384*x^4 - 2048 *x^3 + 4096*x^2)
Time = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.64 \[ \int \frac {-16+6 x-163840 x^3+69632 x^4-5120 x^5-1920 x^6+440 x^7-35 x^8+x^9+\left (-131072 x^3+16384 x^4+20480 x^5-7680 x^6+1120 x^7-76 x^8+2 x^9\right ) \log (x)}{-32768 x^3+20480 x^4-5120 x^5+640 x^6-40 x^7+x^8} \, dx=x + \left (x^{2} + 4 x\right ) \log {\left (x \right )} - \frac {1}{x^{6} - 32 x^{5} + 384 x^{4} - 2048 x^{3} + 4096 x^{2}} \] Input:
integrate(((2*x**9-76*x**8+1120*x**7-7680*x**6+20480*x**5+16384*x**4-13107 2*x**3)*ln(x)+x**9-35*x**8+440*x**7-1920*x**6-5120*x**5+69632*x**4-163840* x**3+6*x-16)/(x**8-40*x**7+640*x**6-5120*x**5+20480*x**4-32768*x**3),x)
Output:
x + (x**2 + 4*x)*log(x) - 1/(x**6 - 32*x**5 + 384*x**4 - 2048*x**3 + 4096* x**2)
Leaf count of result is larger than twice the leaf count of optimal. 628 vs. \(2 (20) = 40\).
Time = 0.09 (sec) , antiderivative size = 628, normalized size of antiderivative = 28.55 \[ \int \frac {-16+6 x-163840 x^3+69632 x^4-5120 x^5-1920 x^6+440 x^7-35 x^8+x^9+\left (-131072 x^3+16384 x^4+20480 x^5-7680 x^6+1120 x^7-76 x^8+2 x^9\right ) \log (x)}{-32768 x^3+20480 x^4-5120 x^5+640 x^6-40 x^7+x^8} \, dx =\text {Too large to display} \] Input:
integrate(((2*x^9-76*x^8+1120*x^7-7680*x^6+20480*x^5+16384*x^4-131072*x^3) *log(x)+x^9-35*x^8+440*x^7-1920*x^6-5120*x^5+69632*x^4-163840*x^3+6*x-16)/ (x^8-40*x^7+640*x^6-5120*x^5+20480*x^4-32768*x^3),x, algorithm="maxima")
Output:
1/2*x^2 + 5*x + 7680*(x^3 - 12*x^2 + 64*x - 128)*log(x)/(x^4 - 32*x^3 + 38 4*x^2 - 2048*x + 4096) - 10240/3*(3*x^2 - 16*x + 32)*log(x)/(x^4 - 32*x^3 + 384*x^2 - 2048*x + 4096) - 16384/3*(x - 2)*log(x)/(x^4 - 32*x^3 + 384*x^ 2 - 2048*x + 4096) - 1/16384*(15*x^5 - 420*x^4 + 4160*x^3 - 16000*x^2 + 12 288*x + 16384)/(x^6 - 32*x^5 + 384*x^4 - 2048*x^3 + 4096*x^2) + 1/16384*(1 5*x^4 - 420*x^3 + 4160*x^2 - 16000*x + 12288)/(x^5 - 32*x^4 + 384*x^3 - 20 48*x^2 + 4096*x) - 1/6*(3*x^6 - 72*x^5 + 384*x^4 + 36864*x^3 - 737280*x^2 - 6*(x^6 - 28*x^5)*log(x) + 4980736*x - 11534336)/(x^4 - 32*x^3 + 384*x^2 - 2048*x + 4096) + 4480/3*(15*x^3 - 300*x^2 + 2080*x - 4928)/(x^4 - 32*x^3 + 384*x^2 - 2048*x + 4096) - 2048*(5*x^3 - 105*x^2 + 752*x - 1824)/(x^4 - 32*x^3 + 384*x^2 - 2048*x + 4096) - 14080/3*(3*x^3 - 54*x^2 + 352*x - 800 )/(x^4 - 32*x^3 + 384*x^2 - 2048*x + 4096) + 1920*(x^3 - 12*x^2 + 64*x - 1 28)/(x^4 - 32*x^3 + 384*x^2 - 2048*x + 4096) + 640*(9*x^2 - 108*x + 352)/( x^3 - 24*x^2 + 192*x - 512) + 2560/3*(3*x^2 - 16*x + 32)/(x^4 - 32*x^3 + 3 84*x^2 - 2048*x + 4096) + 64/3*(3*x^2 - 60*x + 352)/(x^3 - 24*x^2 + 192*x - 512) - 640/3*(x^2 + 4*x - 32)/(x^3 - 24*x^2 + 192*x - 512) + 64/3*(x^2 - 20*x + 32)/(x^3 - 24*x^2 + 192*x - 512) - 69632/3*(x - 2)/(x^4 - 32*x^3 + 384*x^2 - 2048*x + 4096) + 32768*log(x)/(x^4 - 32*x^3 + 384*x^2 - 2048*x + 4096) + 40960/(x^4 - 32*x^3 + 384*x^2 - 2048*x + 4096) + 256*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (20) = 40\).
Time = 0.12 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.50 \[ \int \frac {-16+6 x-163840 x^3+69632 x^4-5120 x^5-1920 x^6+440 x^7-35 x^8+x^9+\left (-131072 x^3+16384 x^4+20480 x^5-7680 x^6+1120 x^7-76 x^8+2 x^9\right ) \log (x)}{-32768 x^3+20480 x^4-5120 x^5+640 x^6-40 x^7+x^8} \, dx={\left (x^{2} + 4 \, x\right )} \log \left (x\right ) + x + \frac {x^{3} - 30 \, x^{2} + 320 \, x - 1280}{8192 \, {\left (x^{4} - 32 \, x^{3} + 384 \, x^{2} - 2048 \, x + 4096\right )}} - \frac {x + 2}{8192 \, x^{2}} \] Input:
integrate(((2*x^9-76*x^8+1120*x^7-7680*x^6+20480*x^5+16384*x^4-131072*x^3) *log(x)+x^9-35*x^8+440*x^7-1920*x^6-5120*x^5+69632*x^4-163840*x^3+6*x-16)/ (x^8-40*x^7+640*x^6-5120*x^5+20480*x^4-32768*x^3),x, algorithm="giac")
Output:
(x^2 + 4*x)*log(x) + x + 1/8192*(x^3 - 30*x^2 + 320*x - 1280)/(x^4 - 32*x^ 3 + 384*x^2 - 2048*x + 4096) - 1/8192*(x + 2)/x^2
Time = 2.71 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.82 \[ \int \frac {-16+6 x-163840 x^3+69632 x^4-5120 x^5-1920 x^6+440 x^7-35 x^8+x^9+\left (-131072 x^3+16384 x^4+20480 x^5-7680 x^6+1120 x^7-76 x^8+2 x^9\right ) \log (x)}{-32768 x^3+20480 x^4-5120 x^5+640 x^6-40 x^7+x^8} \, dx=x+\ln \left (x\right )\,\left (x^2+4\,x\right )-\frac {1}{x^6-32\,x^5+384\,x^4-2048\,x^3+4096\,x^2} \] Input:
int(-(6*x + log(x)*(16384*x^4 - 131072*x^3 + 20480*x^5 - 7680*x^6 + 1120*x ^7 - 76*x^8 + 2*x^9) - 163840*x^3 + 69632*x^4 - 5120*x^5 - 1920*x^6 + 440* x^7 - 35*x^8 + x^9 - 16)/(32768*x^3 - 20480*x^4 + 5120*x^5 - 640*x^6 + 40* x^7 - x^8),x)
Output:
x + log(x)*(4*x + x^2) - 1/(4096*x^2 - 2048*x^3 + 384*x^4 - 32*x^5 + x^6)
Time = 0.17 (sec) , antiderivative size = 95, normalized size of antiderivative = 4.32 \[ \int \frac {-16+6 x-163840 x^3+69632 x^4-5120 x^5-1920 x^6+440 x^7-35 x^8+x^9+\left (-131072 x^3+16384 x^4+20480 x^5-7680 x^6+1120 x^7-76 x^8+2 x^9\right ) \log (x)}{-32768 x^3+20480 x^4-5120 x^5+640 x^6-40 x^7+x^8} \, dx=\frac {\mathrm {log}\left (x \right ) x^{8}-28 \,\mathrm {log}\left (x \right ) x^{7}+256 \,\mathrm {log}\left (x \right ) x^{6}-512 \,\mathrm {log}\left (x \right ) x^{5}-4096 \,\mathrm {log}\left (x \right ) x^{4}+16384 \,\mathrm {log}\left (x \right ) x^{3}+x^{7}-4 x^{6}-512 x^{5}+8704 x^{4}-53248 x^{3}+114688 x^{2}-1}{x^{2} \left (x^{4}-32 x^{3}+384 x^{2}-2048 x +4096\right )} \] Input:
int(((2*x^9-76*x^8+1120*x^7-7680*x^6+20480*x^5+16384*x^4-131072*x^3)*log(x )+x^9-35*x^8+440*x^7-1920*x^6-5120*x^5+69632*x^4-163840*x^3+6*x-16)/(x^8-4 0*x^7+640*x^6-5120*x^5+20480*x^4-32768*x^3),x)
Output:
(log(x)*x**8 - 28*log(x)*x**7 + 256*log(x)*x**6 - 512*log(x)*x**5 - 4096*l og(x)*x**4 + 16384*log(x)*x**3 + x**7 - 4*x**6 - 512*x**5 + 8704*x**4 - 53 248*x**3 + 114688*x**2 - 1)/(x**2*(x**4 - 32*x**3 + 384*x**2 - 2048*x + 40 96))