Integrand size = 173, antiderivative size = 30 \[ \int \frac {-65536 x^2-163840 x^3-204800 x^4-163840 x^5-92160 x^6-37888 x^7-11520 x^8-2560 x^9-400 x^{10}-40 x^{11}-2 x^{12}+\left (32768+146440 x+265716 x^2+287097 x^3+210432 x^4+111344 x^5+43704 x^6+12806 x^7+2760 x^8+420 x^9+41 x^{10}+2 x^{11}\right ) \log (5)}{\left (32768 x+81920 x^2+102400 x^3+81920 x^4+46080 x^5+18944 x^6+5760 x^7+1280 x^8+200 x^9+20 x^{10}+x^{11}\right ) \log (5)} \, dx=x+x \left (1-\frac {1}{\left (4+(2+x)^2\right )^2}\right )^2-\frac {x^2}{\log (5)}+\log (x) \] Output:
ln(x)+(1-1/((2+x)^2+4)^2)^2*x+x-x^2/ln(5)
Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.50 \[ \int \frac {-65536 x^2-163840 x^3-204800 x^4-163840 x^5-92160 x^6-37888 x^7-11520 x^8-2560 x^9-400 x^{10}-40 x^{11}-2 x^{12}+\left (32768+146440 x+265716 x^2+287097 x^3+210432 x^4+111344 x^5+43704 x^6+12806 x^7+2760 x^8+420 x^9+41 x^{10}+2 x^{11}\right ) \log (5)}{\left (32768 x+81920 x^2+102400 x^3+81920 x^4+46080 x^5+18944 x^6+5760 x^7+1280 x^8+200 x^9+20 x^{10}+x^{11}\right ) \log (5)} \, dx=\frac {x \left (-x+\left (\frac {1}{\left (8+4 x+x^2\right )^4}-\frac {2}{\left (8+4 x+x^2\right )^2}\right ) \log (5)+\log (25)\right )+\log (5) \log (x)}{\log (5)} \] Input:
Integrate[(-65536*x^2 - 163840*x^3 - 204800*x^4 - 163840*x^5 - 92160*x^6 - 37888*x^7 - 11520*x^8 - 2560*x^9 - 400*x^10 - 40*x^11 - 2*x^12 + (32768 + 146440*x + 265716*x^2 + 287097*x^3 + 210432*x^4 + 111344*x^5 + 43704*x^6 + 12806*x^7 + 2760*x^8 + 420*x^9 + 41*x^10 + 2*x^11)*Log[5])/((32768*x + 8 1920*x^2 + 102400*x^3 + 81920*x^4 + 46080*x^5 + 18944*x^6 + 5760*x^7 + 128 0*x^8 + 200*x^9 + 20*x^10 + x^11)*Log[5]),x]
Output:
(x*(-x + ((8 + 4*x + x^2)^(-4) - 2/(8 + 4*x + x^2)^2)*Log[5] + Log[25]) + Log[5]*Log[x])/Log[5]
Time = 1.51 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.70, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {27, 25, 2026, 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.
| \(\displaystyle \int \frac {-2 x^{12}-40 x^{11}-400 x^{10}-2560 x^9-11520 x^8-37888 x^7-92160 x^6-163840 x^5-204800 x^4-163840 x^3-65536 x^2+\left (2 x^{11}+41 x^{10}+420 x^9+2760 x^8+12806 x^7+43704 x^6+111344 x^5+210432 x^4+287097 x^3+265716 x^2+146440 x+32768\right ) \log (5)}{\left (x^{11}+20 x^{10}+200 x^9+1280 x^8+5760 x^7+18944 x^6+46080 x^5+81920 x^4+102400 x^3+81920 x^2+32768 x\right ) \log (5)} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {2 x^{12}+40 x^{11}+400 x^{10}+2560 x^9+11520 x^8+37888 x^7+92160 x^6+163840 x^5+204800 x^4+163840 x^3+65536 x^2-\left (2 x^{11}+41 x^{10}+420 x^9+2760 x^8+12806 x^7+43704 x^6+111344 x^5+210432 x^4+287097 x^3+265716 x^2+146440 x+32768\right ) \log (5)}{x^{11}+20 x^{10}+200 x^9+1280 x^8+5760 x^7+18944 x^6+46080 x^5+81920 x^4+102400 x^3+81920 x^2+32768 x}dx}{\log (5)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {2 x^{12}+40 x^{11}+400 x^{10}+2560 x^9+11520 x^8+37888 x^7+92160 x^6+163840 x^5+204800 x^4+163840 x^3+65536 x^2-\left (2 x^{11}+41 x^{10}+420 x^9+2760 x^8+12806 x^7+43704 x^6+111344 x^5+210432 x^4+287097 x^3+265716 x^2+146440 x+32768\right ) \log (5)}{x^{11}+20 x^{10}+200 x^9+1280 x^8+5760 x^7+18944 x^6+46080 x^5+81920 x^4+102400 x^3+81920 x^2+32768 x}dx}{\log (5)}\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle -\frac {\int \frac {2 x^{12}+40 x^{11}+400 x^{10}+2560 x^9+11520 x^8+37888 x^7+92160 x^6+163840 x^5+204800 x^4+163840 x^3+65536 x^2-\left (2 x^{11}+41 x^{10}+420 x^9+2760 x^8+12806 x^7+43704 x^6+111344 x^5+210432 x^4+287097 x^3+265716 x^2+146440 x+32768\right ) \log (5)}{x \left (x^{10}+20 x^9+200 x^8+1280 x^7+5760 x^6+18944 x^5+46080 x^4+81920 x^3+102400 x^2+81920 x+32768\right )}dx}{\log (5)}\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle -\frac {\int \left (2 x-\frac {6 \log (5)}{\left (x^2+4 x+8\right )^2}+\frac {16 (x+4) \log (5)}{\left (x^2+4 x+8\right )^3}+\frac {7 \log (5)}{\left (x^2+4 x+8\right )^4}-\frac {16 (x+4) \log (5)}{\left (x^2+4 x+8\right )^5}-2 \log (5)-\frac {\log (5)}{x}\right )dx}{\log (5)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {x^2+\frac {2 x \log (5)}{\left (x^2+4 x+8\right )^2}-\frac {x \log (5)}{\left (x^2+4 x+8\right )^4}-2 x \log (5)-\log (5) \log (x)}{\log (5)}\) |
Input:
Int[(-65536*x^2 - 163840*x^3 - 204800*x^4 - 163840*x^5 - 92160*x^6 - 37888 *x^7 - 11520*x^8 - 2560*x^9 - 400*x^10 - 40*x^11 - 2*x^12 + (32768 + 14644 0*x + 265716*x^2 + 287097*x^3 + 210432*x^4 + 111344*x^5 + 43704*x^6 + 1280 6*x^7 + 2760*x^8 + 420*x^9 + 41*x^10 + 2*x^11)*Log[5])/((32768*x + 81920*x ^2 + 102400*x^3 + 81920*x^4 + 46080*x^5 + 18944*x^6 + 5760*x^7 + 1280*x^8 + 200*x^9 + 20*x^10 + x^11)*Log[5]),x]
Output:
-((x^2 - 2*x*Log[5] - (x*Log[5])/(8 + 4*x + x^2)^4 + (2*x*Log[5])/(8 + 4*x + x^2)^2 - Log[5]*Log[x])/Log[5])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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 = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.97
| method | result | size |
| default | \(\frac {-x^{2}+2 x \ln \left (5\right )+\ln \left (5\right ) \ln \left (x \right )+\frac {\ln \left (5\right ) \left (-2 x^{5}-16 x^{4}-64 x^{3}-128 x^{2}-127 x \right )}{\left (x^{2}+4 x +8\right )^{4}}}{\ln \left (5\right )}\) | \(59\) |
| risch | \(2 x -\frac {x^{2}}{\ln \left (5\right )}+\frac {-2 x^{5} \ln \left (5\right )-16 x^{4} \ln \left (5\right )-64 x^{3} \ln \left (5\right )-128 x^{2} \ln \left (5\right )-127 x \ln \left (5\right )}{\ln \left (5\right ) \left (x^{8}+16 x^{7}+128 x^{6}+640 x^{5}+2176 x^{4}+5120 x^{3}+8192 x^{2}+8192 x +4096\right )}+\ln \left (x \right )\) | \(95\) |
| norman | \(\frac {-\frac {128 \left (2 \ln \left (5\right )-11\right ) x^{7}}{\ln \left (5\right )}-\frac {128 \left (22 \ln \left (5\right )-111\right ) x^{6}}{\ln \left (5\right )}-\frac {10 \left (1613 \ln \left (5\right )-7680\right ) x^{5}}{\ln \left (5\right )}-\frac {16 \left (3713 \ln \left (5\right )-16896\right ) x^{4}}{\ln \left (5\right )}-\frac {64 \left (2305 \ln \left (5\right )-10112\right ) x^{3}}{\ln \left (5\right )}-\frac {2176 \left (113 \ln \left (5\right )-480\right ) x^{2}}{\ln \left (5\right )}-\frac {\left (254079 \ln \left (5\right )-1048576\right ) x}{\ln \left (5\right )}-\frac {x^{10}}{\ln \left (5\right )}+\frac {2 \left (-8+\ln \left (5\right )\right ) x^{9}}{\ln \left (5\right )}-\frac {131072 \left (\ln \left (5\right )-4\right )}{\ln \left (5\right )}}{\left (x^{2}+4 x +8\right )^{4}}+\ln \left (x \right )\) | \(151\) |
| parallelrisch | \(\frac {524288+1048576 x -16130 x^{5} \ln \left (5\right )-2816 x^{6} \ln \left (5\right )-254079 x \ln \left (5\right )-147520 x^{3} \ln \left (5\right )+8192 x \ln \left (5\right ) \ln \left (x \right )+8192 x^{2} \ln \left (5\right ) \ln \left (x \right )-59408 x^{4} \ln \left (5\right )+4096 \ln \left (5\right ) \ln \left (x \right )-245888 x^{2} \ln \left (5\right )-131072 \ln \left (5\right )+1408 x^{7}-x^{10}-16 x^{9}+270336 x^{4}+647168 x^{3}+1044480 x^{2}+14208 x^{6}+76800 x^{5}-256 \ln \left (5\right ) x^{7}+2 \ln \left (5\right ) x^{9}+\ln \left (5\right ) \ln \left (x \right ) x^{8}+2176 \ln \left (x \right ) \ln \left (5\right ) x^{4}+5120 \ln \left (x \right ) \ln \left (5\right ) x^{3}+16 \ln \left (5\right ) \ln \left (x \right ) x^{7}+128 \ln \left (5\right ) \ln \left (x \right ) x^{6}+640 \ln \left (5\right ) \ln \left (x \right ) x^{5}}{\ln \left (5\right ) \left (x^{8}+16 x^{7}+128 x^{6}+640 x^{5}+2176 x^{4}+5120 x^{3}+8192 x^{2}+8192 x +4096\right )}\) | \(224\) |
Input:
int(((2*x^11+41*x^10+420*x^9+2760*x^8+12806*x^7+43704*x^6+111344*x^5+21043 2*x^4+287097*x^3+265716*x^2+146440*x+32768)*ln(5)-2*x^12-40*x^11-400*x^10- 2560*x^9-11520*x^8-37888*x^7-92160*x^6-163840*x^5-204800*x^4-163840*x^3-65 536*x^2)/(x^11+20*x^10+200*x^9+1280*x^8+5760*x^7+18944*x^6+46080*x^5+81920 *x^4+102400*x^3+81920*x^2+32768*x)/ln(5),x,method=_RETURNVERBOSE)
Output:
1/ln(5)*(-x^2+2*x*ln(5)+ln(5)*ln(x)+ln(5)*(-2*x^5-16*x^4-64*x^3-128*x^2-12 7*x)/(x^2+4*x+8)^4)
Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (28) = 56\).
Time = 0.07 (sec) , antiderivative size = 182, normalized size of antiderivative = 6.07 \[ \int \frac {-65536 x^2-163840 x^3-204800 x^4-163840 x^5-92160 x^6-37888 x^7-11520 x^8-2560 x^9-400 x^{10}-40 x^{11}-2 x^{12}+\left (32768+146440 x+265716 x^2+287097 x^3+210432 x^4+111344 x^5+43704 x^6+12806 x^7+2760 x^8+420 x^9+41 x^{10}+2 x^{11}\right ) \log (5)}{\left (32768 x+81920 x^2+102400 x^3+81920 x^4+46080 x^5+18944 x^6+5760 x^7+1280 x^8+200 x^9+20 x^{10}+x^{11}\right ) \log (5)} \, dx=-\frac {x^{10} + 16 \, x^{9} + 128 \, x^{8} + 640 \, x^{7} + 2176 \, x^{6} + 5120 \, x^{5} + 8192 \, x^{4} + 8192 \, x^{3} - {\left (x^{8} + 16 \, x^{7} + 128 \, x^{6} + 640 \, x^{5} + 2176 \, x^{4} + 5120 \, x^{3} + 8192 \, x^{2} + 8192 \, x + 4096\right )} \log \left (5\right ) \log \left (x\right ) + 4096 \, x^{2} - {\left (2 \, x^{9} + 32 \, x^{8} + 256 \, x^{7} + 1280 \, x^{6} + 4350 \, x^{5} + 10224 \, x^{4} + 16320 \, x^{3} + 16256 \, x^{2} + 8065 \, x\right )} \log \left (5\right )}{{\left (x^{8} + 16 \, x^{7} + 128 \, x^{6} + 640 \, x^{5} + 2176 \, x^{4} + 5120 \, x^{3} + 8192 \, x^{2} + 8192 \, x + 4096\right )} \log \left (5\right )} \] Input:
integrate(((2*x^11+41*x^10+420*x^9+2760*x^8+12806*x^7+43704*x^6+111344*x^5 +210432*x^4+287097*x^3+265716*x^2+146440*x+32768)*log(5)-2*x^12-40*x^11-40 0*x^10-2560*x^9-11520*x^8-37888*x^7-92160*x^6-163840*x^5-204800*x^4-163840 *x^3-65536*x^2)/(x^11+20*x^10+200*x^9+1280*x^8+5760*x^7+18944*x^6+46080*x^ 5+81920*x^4+102400*x^3+81920*x^2+32768*x)/log(5),x, algorithm="fricas")
Output:
-(x^10 + 16*x^9 + 128*x^8 + 640*x^7 + 2176*x^6 + 5120*x^5 + 8192*x^4 + 819 2*x^3 - (x^8 + 16*x^7 + 128*x^6 + 640*x^5 + 2176*x^4 + 5120*x^3 + 8192*x^2 + 8192*x + 4096)*log(5)*log(x) + 4096*x^2 - (2*x^9 + 32*x^8 + 256*x^7 + 1 280*x^6 + 4350*x^5 + 10224*x^4 + 16320*x^3 + 16256*x^2 + 8065*x)*log(5))/( (x^8 + 16*x^7 + 128*x^6 + 640*x^5 + 2176*x^4 + 5120*x^3 + 8192*x^2 + 8192* x + 4096)*log(5))
Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (26) = 52\).
Time = 0.61 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.50 \[ \int \frac {-65536 x^2-163840 x^3-204800 x^4-163840 x^5-92160 x^6-37888 x^7-11520 x^8-2560 x^9-400 x^{10}-40 x^{11}-2 x^{12}+\left (32768+146440 x+265716 x^2+287097 x^3+210432 x^4+111344 x^5+43704 x^6+12806 x^7+2760 x^8+420 x^9+41 x^{10}+2 x^{11}\right ) \log (5)}{\left (32768 x+81920 x^2+102400 x^3+81920 x^4+46080 x^5+18944 x^6+5760 x^7+1280 x^8+200 x^9+20 x^{10}+x^{11}\right ) \log (5)} \, dx=- \frac {x^{2}}{\log {\left (5 \right )}} + 2 x - \frac {2 x^{5} + 16 x^{4} + 64 x^{3} + 128 x^{2} + 127 x}{x^{8} + 16 x^{7} + 128 x^{6} + 640 x^{5} + 2176 x^{4} + 5120 x^{3} + 8192 x^{2} + 8192 x + 4096} + \log {\left (x \right )} \] Input:
integrate(((2*x**11+41*x**10+420*x**9+2760*x**8+12806*x**7+43704*x**6+1113 44*x**5+210432*x**4+287097*x**3+265716*x**2+146440*x+32768)*ln(5)-2*x**12- 40*x**11-400*x**10-2560*x**9-11520*x**8-37888*x**7-92160*x**6-163840*x**5- 204800*x**4-163840*x**3-65536*x**2)/(x**11+20*x**10+200*x**9+1280*x**8+576 0*x**7+18944*x**6+46080*x**5+81920*x**4+102400*x**3+81920*x**2+32768*x)/ln (5),x)
Output:
-x**2/log(5) + 2*x - (2*x**5 + 16*x**4 + 64*x**3 + 128*x**2 + 127*x)/(x**8 + 16*x**7 + 128*x**6 + 640*x**5 + 2176*x**4 + 5120*x**3 + 8192*x**2 + 819 2*x + 4096) + log(x)
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (28) = 56\).
Time = 0.03 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.20 \[ \int \frac {-65536 x^2-163840 x^3-204800 x^4-163840 x^5-92160 x^6-37888 x^7-11520 x^8-2560 x^9-400 x^{10}-40 x^{11}-2 x^{12}+\left (32768+146440 x+265716 x^2+287097 x^3+210432 x^4+111344 x^5+43704 x^6+12806 x^7+2760 x^8+420 x^9+41 x^{10}+2 x^{11}\right ) \log (5)}{\left (32768 x+81920 x^2+102400 x^3+81920 x^4+46080 x^5+18944 x^6+5760 x^7+1280 x^8+200 x^9+20 x^{10}+x^{11}\right ) \log (5)} \, dx=-\frac {x^{2} - 2 \, x \log \left (5\right ) - \log \left (5\right ) \log \left (x\right ) + \frac {2 \, x^{5} \log \left (5\right ) + 16 \, x^{4} \log \left (5\right ) + 64 \, x^{3} \log \left (5\right ) + 128 \, x^{2} \log \left (5\right ) + 127 \, x \log \left (5\right )}{x^{8} + 16 \, x^{7} + 128 \, x^{6} + 640 \, x^{5} + 2176 \, x^{4} + 5120 \, x^{3} + 8192 \, x^{2} + 8192 \, x + 4096}}{\log \left (5\right )} \] Input:
integrate(((2*x^11+41*x^10+420*x^9+2760*x^8+12806*x^7+43704*x^6+111344*x^5 +210432*x^4+287097*x^3+265716*x^2+146440*x+32768)*log(5)-2*x^12-40*x^11-40 0*x^10-2560*x^9-11520*x^8-37888*x^7-92160*x^6-163840*x^5-204800*x^4-163840 *x^3-65536*x^2)/(x^11+20*x^10+200*x^9+1280*x^8+5760*x^7+18944*x^6+46080*x^ 5+81920*x^4+102400*x^3+81920*x^2+32768*x)/log(5),x, algorithm="maxima")
Output:
-(x^2 - 2*x*log(5) - log(5)*log(x) + (2*x^5*log(5) + 16*x^4*log(5) + 64*x^ 3*log(5) + 128*x^2*log(5) + 127*x*log(5))/(x^8 + 16*x^7 + 128*x^6 + 640*x^ 5 + 2176*x^4 + 5120*x^3 + 8192*x^2 + 8192*x + 4096))/log(5)
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (28) = 56\).
Time = 0.12 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.23 \[ \int \frac {-65536 x^2-163840 x^3-204800 x^4-163840 x^5-92160 x^6-37888 x^7-11520 x^8-2560 x^9-400 x^{10}-40 x^{11}-2 x^{12}+\left (32768+146440 x+265716 x^2+287097 x^3+210432 x^4+111344 x^5+43704 x^6+12806 x^7+2760 x^8+420 x^9+41 x^{10}+2 x^{11}\right ) \log (5)}{\left (32768 x+81920 x^2+102400 x^3+81920 x^4+46080 x^5+18944 x^6+5760 x^7+1280 x^8+200 x^9+20 x^{10}+x^{11}\right ) \log (5)} \, dx=-\frac {x^{2} - 2 \, x \log \left (5\right ) - \log \left (5\right ) \log \left ({\left | x \right |}\right ) + \frac {2 \, x^{5} \log \left (5\right ) + 16 \, x^{4} \log \left (5\right ) + 64 \, x^{3} \log \left (5\right ) + 128 \, x^{2} \log \left (5\right ) + 127 \, x \log \left (5\right )}{{\left (x^{2} + 4 \, x + 8\right )}^{4}}}{\log \left (5\right )} \] Input:
integrate(((2*x^11+41*x^10+420*x^9+2760*x^8+12806*x^7+43704*x^6+111344*x^5 +210432*x^4+287097*x^3+265716*x^2+146440*x+32768)*log(5)-2*x^12-40*x^11-40 0*x^10-2560*x^9-11520*x^8-37888*x^7-92160*x^6-163840*x^5-204800*x^4-163840 *x^3-65536*x^2)/(x^11+20*x^10+200*x^9+1280*x^8+5760*x^7+18944*x^6+46080*x^ 5+81920*x^4+102400*x^3+81920*x^2+32768*x)/log(5),x, algorithm="giac")
Output:
-(x^2 - 2*x*log(5) - log(5)*log(abs(x)) + (2*x^5*log(5) + 16*x^4*log(5) + 64*x^3*log(5) + 128*x^2*log(5) + 127*x*log(5))/(x^2 + 4*x + 8)^4)/log(5)
Time = 0.37 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.20 \[ \int \frac {-65536 x^2-163840 x^3-204800 x^4-163840 x^5-92160 x^6-37888 x^7-11520 x^8-2560 x^9-400 x^{10}-40 x^{11}-2 x^{12}+\left (32768+146440 x+265716 x^2+287097 x^3+210432 x^4+111344 x^5+43704 x^6+12806 x^7+2760 x^8+420 x^9+41 x^{10}+2 x^{11}\right ) \log (5)}{\left (32768 x+81920 x^2+102400 x^3+81920 x^4+46080 x^5+18944 x^6+5760 x^7+1280 x^8+200 x^9+20 x^{10}+x^{11}\right ) \log (5)} \, dx=\ln \left (x\right )-\frac {x^2}{\ln \left (5\right )}-\frac {2\,x^5+16\,x^4+64\,x^3+128\,x^2+127\,x}{x^8+16\,x^7+128\,x^6+640\,x^5+2176\,x^4+5120\,x^3+8192\,x^2+8192\,x+4096}+x\,\left (\frac {40}{\ln \left (5\right )}+\frac {\ln \left (25\right )-40}{\ln \left (5\right )}\right ) \] Input:
int(-(65536*x^2 - log(5)*(146440*x + 265716*x^2 + 287097*x^3 + 210432*x^4 + 111344*x^5 + 43704*x^6 + 12806*x^7 + 2760*x^8 + 420*x^9 + 41*x^10 + 2*x^ 11 + 32768) + 163840*x^3 + 204800*x^4 + 163840*x^5 + 92160*x^6 + 37888*x^7 + 11520*x^8 + 2560*x^9 + 400*x^10 + 40*x^11 + 2*x^12)/(log(5)*(32768*x + 81920*x^2 + 102400*x^3 + 81920*x^4 + 46080*x^5 + 18944*x^6 + 5760*x^7 + 12 80*x^8 + 200*x^9 + 20*x^10 + x^11)),x)
Output:
log(x) - x^2/log(5) - (127*x + 128*x^2 + 64*x^3 + 16*x^4 + 2*x^5)/(8192*x + 8192*x^2 + 5120*x^3 + 2176*x^4 + 640*x^5 + 128*x^6 + 16*x^7 + x^8 + 4096 ) + x*(40/log(5) + (log(25) - 40)/log(5))
Time = 0.21 (sec) , antiderivative size = 223, normalized size of antiderivative = 7.43 \[ \int \frac {-65536 x^2-163840 x^3-204800 x^4-163840 x^5-92160 x^6-37888 x^7-11520 x^8-2560 x^9-400 x^{10}-40 x^{11}-2 x^{12}+\left (32768+146440 x+265716 x^2+287097 x^3+210432 x^4+111344 x^5+43704 x^6+12806 x^7+2760 x^8+420 x^9+41 x^{10}+2 x^{11}\right ) \log (5)}{\left (32768 x+81920 x^2+102400 x^3+81920 x^4+46080 x^5+18944 x^6+5760 x^7+1280 x^8+200 x^9+20 x^{10}+x^{11}\right ) \log (5)} \, dx=\frac {163840+327680 x +16 \,\mathrm {log}\left (5\right ) x^{8}-768 \,\mathrm {log}\left (5\right ) x^{6}-5890 \,\mathrm {log}\left (5\right ) x^{5}-24592 \,\mathrm {log}\left (5\right ) x^{4}-65600 \,\mathrm {log}\left (5\right ) x^{3}+4096 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (5\right )+2 \,\mathrm {log}\left (5\right ) x^{9}-x^{10}-16 x^{9}-114816 \,\mathrm {log}\left (5\right ) x^{2}-123007 \,\mathrm {log}\left (5\right ) x +323584 x^{2}+20480 x^{5}+196608 x^{3}+\mathrm {log}\left (x \right ) \mathrm {log}\left (5\right ) x^{8}-88 x^{8}+2944 x^{6}+8192 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (5\right ) x +78848 x^{4}+16 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (5\right ) x^{7}+128 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (5\right ) x^{6}+640 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (5\right ) x^{5}+2176 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (5\right ) x^{4}+5120 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (5\right ) x^{3}+8192 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (5\right ) x^{2}-65536 \,\mathrm {log}\left (5\right )}{\mathrm {log}\left (5\right ) \left (x^{8}+16 x^{7}+128 x^{6}+640 x^{5}+2176 x^{4}+5120 x^{3}+8192 x^{2}+8192 x +4096\right )} \] Input:
int(((2*x^11+41*x^10+420*x^9+2760*x^8+12806*x^7+43704*x^6+111344*x^5+21043 2*x^4+287097*x^3+265716*x^2+146440*x+32768)*log(5)-2*x^12-40*x^11-400*x^10 -2560*x^9-11520*x^8-37888*x^7-92160*x^6-163840*x^5-204800*x^4-163840*x^3-6 5536*x^2)/(x^11+20*x^10+200*x^9+1280*x^8+5760*x^7+18944*x^6+46080*x^5+8192 0*x^4+102400*x^3+81920*x^2+32768*x)/log(5),x)
Output:
(log(x)*log(5)*x**8 + 16*log(x)*log(5)*x**7 + 128*log(x)*log(5)*x**6 + 640 *log(x)*log(5)*x**5 + 2176*log(x)*log(5)*x**4 + 5120*log(x)*log(5)*x**3 + 8192*log(x)*log(5)*x**2 + 8192*log(x)*log(5)*x + 4096*log(x)*log(5) + 2*lo g(5)*x**9 + 16*log(5)*x**8 - 768*log(5)*x**6 - 5890*log(5)*x**5 - 24592*lo g(5)*x**4 - 65600*log(5)*x**3 - 114816*log(5)*x**2 - 123007*log(5)*x - 655 36*log(5) - x**10 - 16*x**9 - 88*x**8 + 2944*x**6 + 20480*x**5 + 78848*x** 4 + 196608*x**3 + 323584*x**2 + 327680*x + 163840)/(log(5)*(x**8 + 16*x**7 + 128*x**6 + 640*x**5 + 2176*x**4 + 5120*x**3 + 8192*x**2 + 8192*x + 4096 ))