Integrand size = 177, antiderivative size = 38 \[ \int \frac {32768-8192 x^2+1536 x^3-128 x^4-128 x^5+16 x^6-2 x^7+\frac {e^{25+\log ^2(2)} \left (2048-1536 x+384 x^2-64 x^3\right )}{1024}}{-16384 x^3-4096 x^4+3072 x^5+768 x^6-192 x^7-48 x^8+4 x^9+x^{10}+\frac {e^{50+2 \log ^2(2)} \left (-64 x^3+48 x^4-12 x^5+x^6\right )}{1048576}+\frac {e^{25+\log ^2(2)} \left (-2048 x^3+512 x^4+256 x^5-64 x^6-8 x^7+2 x^8\right )}{1024}} \, dx=\frac {\left (\frac {4}{4-x}+\frac {4-x}{x}\right )^2}{e^{(-5+\log (2))^2}+(4+x)^2} \]
Time = 0.34 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {32768-8192 x^2+1536 x^3-128 x^4-128 x^5+16 x^6-2 x^7+\frac {e^{25+\log ^2(2)} \left (2048-1536 x+384 x^2-64 x^3\right )}{1024}}{-16384 x^3-4096 x^4+3072 x^5+768 x^6-192 x^7-48 x^8+4 x^9+x^{10}+\frac {e^{50+2 \log ^2(2)} \left (-64 x^3+48 x^4-12 x^5+x^6\right )}{1048576}+\frac {e^{25+\log ^2(2)} \left (-2048 x^3+512 x^4+256 x^5-64 x^6-8 x^7+2 x^8\right )}{1024}} \, dx=\frac {1024 \left (16-4 x+x^2\right )^2}{(-4+x)^2 x^2 \left (e^{25+\log ^2(2)}+1024 (4+x)^2\right )} \]
Integrate[(32768 - 8192*x^2 + 1536*x^3 - 128*x^4 - 128*x^5 + 16*x^6 - 2*x^ 7 + (E^(25 + Log[2]^2)*(2048 - 1536*x + 384*x^2 - 64*x^3))/1024)/(-16384*x ^3 - 4096*x^4 + 3072*x^5 + 768*x^6 - 192*x^7 - 48*x^8 + 4*x^9 + x^10 + (E^ (50 + 2*Log[2]^2)*(-64*x^3 + 48*x^4 - 12*x^5 + x^6))/1048576 + (E^(25 + Lo g[2]^2)*(-2048*x^3 + 512*x^4 + 256*x^5 - 64*x^6 - 8*x^7 + 2*x^8))/1024),x]
Leaf count is larger than twice the leaf count of optimal. \(194\) vs. \(2(38)=76\).
Time = 1.34 (sec) , antiderivative size = 194, normalized size of antiderivative = 5.11, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {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.
| \(\displaystyle \int \frac {-2 x^7+16 x^6-128 x^5-128 x^4+1536 x^3-8192 x^2+\frac {\left (-64 x^3+384 x^2-1536 x+2048\right ) e^{25+\log ^2(2)}}{1024}+32768}{x^{10}+4 x^9-48 x^8-192 x^7+768 x^6+3072 x^5-4096 x^4-16384 x^3+\frac {\left (x^6-12 x^5+48 x^4-64 x^3\right ) e^{50+2 \log ^2(2)}}{1048576}+\frac {\left (2 x^8-8 x^7-64 x^6+256 x^5+512 x^4-2048 x^3\right ) e^{25+\log ^2(2)}}{1024}} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {-2 x^7+16 x^6-128 x^5-128 x^4+1536 x^3-8192 x^2+\frac {\left (-64 x^3+384 x^2-1536 x+2048\right ) e^{25+\log ^2(2)}}{1024}+32768}{x^3 \left (x^7+4 x^6-\frac {1}{512} x^5 \left (24576-e^{25+\log ^2(2)}\right )-\frac {1}{128} x^4 \left (24576+e^{25+\log ^2(2)}\right )+\frac {x^3 \left (16384-e^{25+\log ^2(2)}\right ) \left (49152-e^{25+\log ^2(2)}\right )}{1048576}+\frac {x^2 \left (805306368+65536 e^{25+\log ^2(2)}-3 e^{50+2 \log ^2(2)}\right )}{262144}-\frac {x \left (16384-3 e^{25+\log ^2(2)}\right ) \left (16384+e^{25+\log ^2(2)}\right )}{65536}-\frac {\left (16384+e^{25+\log ^2(2)}\right )^2}{16384}\right )}dx\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (-\frac {32768}{x^3 \left (16384+e^{25+\log ^2(2)}\right )}+\frac {2097152 \left (-\left (x \left (2594073385365405696+171523813933056 e^{25+\log ^2(2)}+e^{4 \left (25+\log ^2(2)\right )}+7516192768 e^{50+2 \log ^2(2)}+131072 e^{75+3 \log ^2(2)}\right )\right )-4 \left (2594073385365405696+13194139533312 e^{25+\log ^2(2)}+e^{4 \left (25+\log ^2(2)\right )}+1073741824 e^{50+2 \log ^2(2)}+32768 e^{75+3 \log ^2(2)}\right )\right )}{\left (16384+e^{25+\log ^2(2)}\right )^2 \left (65536+e^{25+\log ^2(2)}\right )^2 \left (1024 x^2+8192 x+16384+e^{25+\log ^2(2)}\right )^2}-\frac {412316860416 \left (1610612736+65536 e^{25+\log ^2(2)}+e^{50+2 \log ^2(2)}\right )}{\left (16384+e^{25+\log ^2(2)}\right )^2 \left (65536+e^{25+\log ^2(2)}\right )^2 \left (1024 x^2+8192 x+16384+e^{25+\log ^2(2)}\right )}+\frac {134217728}{x^2 \left (16384+e^{25+\log ^2(2)}\right )^2}+\frac {268435456}{(x-4)^2 \left (65536+e^{25+\log ^2(2)}\right )^2}-\frac {32768}{(x-4)^3 \left (65536+e^{25+\log ^2(2)}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1024 \left (402653184 x \left (1610612736+65536 e^{25+\log ^2(2)}+e^{50+2 \log ^2(2)}\right )+5188146770730811392+131072 e^{75+3 \log ^2(2)}+9126805504 e^{50+2 \log ^2(2)}+e^{4 \left (25+\log ^2(2)\right )}+277076930199552 e^{25+\log ^2(2)}\right )}{\left (16384+e^{25+\log ^2(2)}\right )^2 \left (65536+e^{25+\log ^2(2)}\right )^2 \left (1024 x^2+8192 x+16384+e^{25+\log ^2(2)}\right )}+\frac {16384}{x^2 \left (16384+e^{25+\log ^2(2)}\right )}+\frac {268435456}{(4-x) \left (65536+e^{25+\log ^2(2)}\right )^2}-\frac {134217728}{x \left (16384+e^{25+\log ^2(2)}\right )^2}+\frac {16384}{(4-x)^2 \left (65536+e^{25+\log ^2(2)}\right )}\) |
Int[(32768 - 8192*x^2 + 1536*x^3 - 128*x^4 - 128*x^5 + 16*x^6 - 2*x^7 + (E ^(25 + Log[2]^2)*(2048 - 1536*x + 384*x^2 - 64*x^3))/1024)/(-16384*x^3 - 4 096*x^4 + 3072*x^5 + 768*x^6 - 192*x^7 - 48*x^8 + 4*x^9 + x^10 + (E^(50 + 2*Log[2]^2)*(-64*x^3 + 48*x^4 - 12*x^5 + x^6))/1048576 + (E^(25 + Log[2]^2 )*(-2048*x^3 + 512*x^4 + 256*x^5 - 64*x^6 - 8*x^7 + 2*x^8))/1024),x]
16384/((65536 + E^(25 + Log[2]^2))*(4 - x)^2) + 268435456/((65536 + E^(25 + Log[2]^2))^2*(4 - x)) + 16384/((16384 + E^(25 + Log[2]^2))*x^2) - 134217 728/((16384 + E^(25 + Log[2]^2))^2*x) + (1024*(5188146770730811392 + 27707 6930199552*E^(25 + Log[2]^2) + E^(4*(25 + Log[2]^2)) + 9126805504*E^(50 + 2*Log[2]^2) + 131072*E^(75 + 3*Log[2]^2) + 402653184*(1610612736 + 65536*E ^(25 + Log[2]^2) + E^(50 + 2*Log[2]^2))*x))/((16384 + E^(25 + Log[2]^2))^2 *(65536 + E^(25 + Log[2]^2))^2*(16384 + E^(25 + Log[2]^2) + 8192*x + 1024* x^2))
3.3.27.3.1 Defintions of rubi rules used
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 = 8.93 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.29
| method | result | size |
| norman | \(\frac {1024 x^{4}-8192 x^{3}+49152 x^{2}-131072 x +262144}{x^{2} \left (x -4\right )^{2} \left (1024 x^{2}+{\mathrm e}^{25+\ln \left (2\right )^{2}}+8192 x +16384\right )}\) | \(49\) |
| gosper | \(\frac {\left (x^{2}-4 x +16\right )^{2}}{x^{2} \left (x^{4}+{\mathrm e}^{\ln \left (2\right )^{2}-10 \ln \left (2\right )+25} x^{2}-8 \,{\mathrm e}^{\ln \left (2\right )^{2}-10 \ln \left (2\right )+25} x -32 x^{2}+16 \,{\mathrm e}^{\ln \left (2\right )^{2}-10 \ln \left (2\right )+25}+256\right )}\) | \(69\) |
| risch | \(\frac {1024 x^{4}-8192 x^{3}+49152 x^{2}-131072 x +262144}{x^{2} \left ({\mathrm e}^{25+\ln \left (2\right )^{2}} x^{2}+1024 x^{4}-8 \,{\mathrm e}^{25+\ln \left (2\right )^{2}} x +16 \,{\mathrm e}^{25+\ln \left (2\right )^{2}}-32768 x^{2}+262144\right )}\) | \(69\) |
| parallelrisch | \(\frac {x^{4}-8 x^{3}+48 x^{2}-128 x +256}{x^{2} \left (x^{4}+{\mathrm e}^{\ln \left (2\right )^{2}-10 \ln \left (2\right )+25} x^{2}-8 \,{\mathrm e}^{\ln \left (2\right )^{2}-10 \ln \left (2\right )+25} x -32 x^{2}+16 \,{\mathrm e}^{\ln \left (2\right )^{2}-10 \ln \left (2\right )+25}+256\right )}\) | \(77\) |
int(((-64*x^3+384*x^2-1536*x+2048)*exp(ln(2)^2-10*ln(2)+25)-2*x^7+16*x^6-1 28*x^5-128*x^4+1536*x^3-8192*x^2+32768)/((x^6-12*x^5+48*x^4-64*x^3)*exp(ln (2)^2-10*ln(2)+25)^2+(2*x^8-8*x^7-64*x^6+256*x^5+512*x^4-2048*x^3)*exp(ln( 2)^2-10*ln(2)+25)+x^10+4*x^9-48*x^8-192*x^7+768*x^6+3072*x^5-4096*x^4-1638 4*x^3),x,method=_RETURNVERBOSE)
(1024*x^4-8192*x^3+49152*x^2-131072*x+262144)/x^2/(x-4)^2/(1024*x^2+exp(25 +ln(2)^2)+8192*x+16384)
Time = 0.26 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.61 \[ \int \frac {32768-8192 x^2+1536 x^3-128 x^4-128 x^5+16 x^6-2 x^7+\frac {e^{25+\log ^2(2)} \left (2048-1536 x+384 x^2-64 x^3\right )}{1024}}{-16384 x^3-4096 x^4+3072 x^5+768 x^6-192 x^7-48 x^8+4 x^9+x^{10}+\frac {e^{50+2 \log ^2(2)} \left (-64 x^3+48 x^4-12 x^5+x^6\right )}{1048576}+\frac {e^{25+\log ^2(2)} \left (-2048 x^3+512 x^4+256 x^5-64 x^6-8 x^7+2 x^8\right )}{1024}} \, dx=\frac {x^{4} - 8 \, x^{3} + 48 \, x^{2} - 128 \, x + 256}{x^{6} - 32 \, x^{4} + 256 \, x^{2} + {\left (x^{4} - 8 \, x^{3} + 16 \, x^{2}\right )} e^{\left (\log \left (2\right )^{2} - 10 \, \log \left (2\right ) + 25\right )}} \]
integrate(((-64*x^3+384*x^2-1536*x+2048)*exp(log(2)^2-10*log(2)+25)-2*x^7+ 16*x^6-128*x^5-128*x^4+1536*x^3-8192*x^2+32768)/((x^6-12*x^5+48*x^4-64*x^3 )*exp(log(2)^2-10*log(2)+25)^2+(2*x^8-8*x^7-64*x^6+256*x^5+512*x^4-2048*x^ 3)*exp(log(2)^2-10*log(2)+25)+x^10+4*x^9-48*x^8-192*x^7+768*x^6+3072*x^5-4 096*x^4-16384*x^3),x, algorithm=\
(x^4 - 8*x^3 + 48*x^2 - 128*x + 256)/(x^6 - 32*x^4 + 256*x^2 + (x^4 - 8*x^ 3 + 16*x^2)*e^(log(2)^2 - 10*log(2) + 25))
Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (24) = 48\).
Time = 7.05 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.92 \[ \int \frac {32768-8192 x^2+1536 x^3-128 x^4-128 x^5+16 x^6-2 x^7+\frac {e^{25+\log ^2(2)} \left (2048-1536 x+384 x^2-64 x^3\right )}{1024}}{-16384 x^3-4096 x^4+3072 x^5+768 x^6-192 x^7-48 x^8+4 x^9+x^{10}+\frac {e^{50+2 \log ^2(2)} \left (-64 x^3+48 x^4-12 x^5+x^6\right )}{1048576}+\frac {e^{25+\log ^2(2)} \left (-2048 x^3+512 x^4+256 x^5-64 x^6-8 x^7+2 x^8\right )}{1024}} \, dx=- \frac {- 1024 x^{4} + 8192 x^{3} - 49152 x^{2} + 131072 x - 262144}{1024 x^{6} + x^{4} \left (-32768 + e^{25} e^{\log {\left (2 \right )}^{2}}\right ) - 8 x^{3} e^{25} e^{\log {\left (2 \right )}^{2}} + x^{2} \cdot \left (262144 + 16 e^{25} e^{\log {\left (2 \right )}^{2}}\right )} \]
integrate(((-64*x**3+384*x**2-1536*x+2048)*exp(ln(2)**2-10*ln(2)+25)-2*x** 7+16*x**6-128*x**5-128*x**4+1536*x**3-8192*x**2+32768)/((x**6-12*x**5+48*x **4-64*x**3)*exp(ln(2)**2-10*ln(2)+25)**2+(2*x**8-8*x**7-64*x**6+256*x**5+ 512*x**4-2048*x**3)*exp(ln(2)**2-10*ln(2)+25)+x**10+4*x**9-48*x**8-192*x** 7+768*x**6+3072*x**5-4096*x**4-16384*x**3),x)
-(-1024*x**4 + 8192*x**3 - 49152*x**2 + 131072*x - 262144)/(1024*x**6 + x* *4*(-32768 + exp(25)*exp(log(2)**2)) - 8*x**3*exp(25)*exp(log(2)**2) + x** 2*(262144 + 16*exp(25)*exp(log(2)**2)))
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (33) = 66\).
Time = 0.76 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.76 \[ \int \frac {32768-8192 x^2+1536 x^3-128 x^4-128 x^5+16 x^6-2 x^7+\frac {e^{25+\log ^2(2)} \left (2048-1536 x+384 x^2-64 x^3\right )}{1024}}{-16384 x^3-4096 x^4+3072 x^5+768 x^6-192 x^7-48 x^8+4 x^9+x^{10}+\frac {e^{50+2 \log ^2(2)} \left (-64 x^3+48 x^4-12 x^5+x^6\right )}{1048576}+\frac {e^{25+\log ^2(2)} \left (-2048 x^3+512 x^4+256 x^5-64 x^6-8 x^7+2 x^8\right )}{1024}} \, dx=\frac {1024 \, {\left (x^{4} - 8 \, x^{3} + 48 \, x^{2} - 128 \, x + 256\right )}}{1024 \, x^{6} + x^{4} {\left (e^{\left (\log \left (2\right )^{2} + 25\right )} - 32768\right )} - 8 \, x^{3} e^{\left (\log \left (2\right )^{2} + 25\right )} + 16 \, x^{2} {\left (e^{\left (\log \left (2\right )^{2} + 25\right )} + 16384\right )}} \]
integrate(((-64*x^3+384*x^2-1536*x+2048)*exp(log(2)^2-10*log(2)+25)-2*x^7+ 16*x^6-128*x^5-128*x^4+1536*x^3-8192*x^2+32768)/((x^6-12*x^5+48*x^4-64*x^3 )*exp(log(2)^2-10*log(2)+25)^2+(2*x^8-8*x^7-64*x^6+256*x^5+512*x^4-2048*x^ 3)*exp(log(2)^2-10*log(2)+25)+x^10+4*x^9-48*x^8-192*x^7+768*x^6+3072*x^5-4 096*x^4-16384*x^3),x, algorithm=\
1024*(x^4 - 8*x^3 + 48*x^2 - 128*x + 256)/(1024*x^6 + x^4*(e^(log(2)^2 + 2 5) - 32768) - 8*x^3*e^(log(2)^2 + 25) + 16*x^2*(e^(log(2)^2 + 25) + 16384) )
Leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (33) = 66\).
Time = 0.28 (sec) , antiderivative size = 435, normalized size of antiderivative = 11.45 \[ \int \frac {32768-8192 x^2+1536 x^3-128 x^4-128 x^5+16 x^6-2 x^7+\frac {e^{25+\log ^2(2)} \left (2048-1536 x+384 x^2-64 x^3\right )}{1024}}{-16384 x^3-4096 x^4+3072 x^5+768 x^6-192 x^7-48 x^8+4 x^9+x^{10}+\frac {e^{50+2 \log ^2(2)} \left (-64 x^3+48 x^4-12 x^5+x^6\right )}{1048576}+\frac {e^{25+\log ^2(2)} \left (-2048 x^3+512 x^4+256 x^5-64 x^6-8 x^7+2 x^8\right )}{1024}} \, dx =\text {Too large to display} \]
integrate(((-64*x^3+384*x^2-1536*x+2048)*exp(log(2)^2-10*log(2)+25)-2*x^7+ 16*x^6-128*x^5-128*x^4+1536*x^3-8192*x^2+32768)/((x^6-12*x^5+48*x^4-64*x^3 )*exp(log(2)^2-10*log(2)+25)^2+(2*x^8-8*x^7-64*x^6+256*x^5+512*x^4-2048*x^ 3)*exp(log(2)^2-10*log(2)+25)+x^10+4*x^9-48*x^8-192*x^7+768*x^6+3072*x^5-4 096*x^4-16384*x^3),x, algorithm=\
(384*x*e^(2*log(2)^2 - 20*log(2) + 50) + 24576*x*e^(log(2)^2 - 10*log(2) + 25) + 589824*x + e^(4*log(2)^2 - 40*log(2) + 100) + 128*e^(3*log(2)^2 - 3 0*log(2) + 75) + 8704*e^(2*log(2)^2 - 20*log(2) + 50) + 258048*e^(log(2)^2 - 10*log(2) + 25) + 4718592)/((x^2 + 8*x + e^(log(2)^2 - 10*log(2) + 25) + 16)*(e^(4*log(2)^2 - 40*log(2) + 100) + 160*e^(3*log(2)^2 - 30*log(2) + 75) + 8448*e^(2*log(2)^2 - 20*log(2) + 50) + 163840*e^(log(2)^2 - 10*log(2 ) + 25) + 1048576)) - 32*(12*x^3*e^(2*log(2)^2 - 20*log(2) + 50) + 768*x^3 *e^(log(2)^2 - 10*log(2) + 25) + 18432*x^3 - x^2*e^(3*log(2)^2 - 30*log(2) + 75) - 184*x^2*e^(2*log(2)^2 - 20*log(2) + 50) - 9344*x^2*e^(log(2)^2 - 10*log(2) + 25) - 180224*x^2 + 4*x*e^(3*log(2)^2 - 30*log(2) + 75) + 640*x *e^(2*log(2)^2 - 20*log(2) + 50) + 32768*x*e^(log(2)^2 - 10*log(2) + 25) + 524288*x - 8*e^(3*log(2)^2 - 30*log(2) + 75) - 1152*e^(2*log(2)^2 - 20*lo g(2) + 50) - 49152*e^(log(2)^2 - 10*log(2) + 25) - 524288)/((x^2 - 4*x)^2* (e^(4*log(2)^2 - 40*log(2) + 100) + 160*e^(3*log(2)^2 - 30*log(2) + 75) + 8448*e^(2*log(2)^2 - 20*log(2) + 50) + 163840*e^(log(2)^2 - 10*log(2) + 25 ) + 1048576))
Time = 9.88 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.82 \[ \int \frac {32768-8192 x^2+1536 x^3-128 x^4-128 x^5+16 x^6-2 x^7+\frac {e^{25+\log ^2(2)} \left (2048-1536 x+384 x^2-64 x^3\right )}{1024}}{-16384 x^3-4096 x^4+3072 x^5+768 x^6-192 x^7-48 x^8+4 x^9+x^{10}+\frac {e^{50+2 \log ^2(2)} \left (-64 x^3+48 x^4-12 x^5+x^6\right )}{1048576}+\frac {e^{25+\log ^2(2)} \left (-2048 x^3+512 x^4+256 x^5-64 x^6-8 x^7+2 x^8\right )}{1024}} \, dx=\frac {1024\,x^4-8192\,x^3+49152\,x^2-131072\,x+262144}{1024\,x^6+\left ({\mathrm {e}}^{{\ln \left (2\right )}^2+25}-32768\right )\,x^4-8\,{\mathrm {e}}^{{\ln \left (2\right )}^2+25}\,x^3+\left (16\,{\mathrm {e}}^{{\ln \left (2\right )}^2+25}+262144\right )\,x^2} \]
int((exp(log(2)^2 - 10*log(2) + 25)*(1536*x - 384*x^2 + 64*x^3 - 2048) + 8 192*x^2 - 1536*x^3 + 128*x^4 + 128*x^5 - 16*x^6 + 2*x^7 - 32768)/(exp(2*lo g(2)^2 - 20*log(2) + 50)*(64*x^3 - 48*x^4 + 12*x^5 - x^6) + exp(log(2)^2 - 10*log(2) + 25)*(2048*x^3 - 512*x^4 - 256*x^5 + 64*x^6 + 8*x^7 - 2*x^8) + 16384*x^3 + 4096*x^4 - 3072*x^5 - 768*x^6 + 192*x^7 + 48*x^8 - 4*x^9 - x^ 10),x)