Integrand size = 146, antiderivative size = 26 \[ \int \frac {-e^{2 x}-4096 x^5-65536 x^8+e^x \left (-4 x^2-512 x^4\right )+2 e^{2 x} \left (2 e^{2 x} x^2+1024 e^x x^6+131072 x^{10}\right )}{1024 x^6+65536 x^9+262144 x^{10}+e^{2 x} \left (x+4 x^2\right )+e^x \left (4 x^2+512 x^5+2048 x^6\right )+2 e^{2 x} \left (e^{2 x} x^2+512 e^x x^6+65536 x^{10}\right )} \, dx=\log \left (4+2 e^{2 x}+\frac {1}{x}+\frac {4}{e^x+256 x^4}\right ) \] Output:
ln(exp(ln(2)+2*x)+1/x+4/(exp(x)+256*x^4)+4)
Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(26)=52\).
Time = 10.08 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.27 \[ \int \frac {-e^{2 x}-4096 x^5-65536 x^8+e^x \left (-4 x^2-512 x^4\right )+2 e^{2 x} \left (2 e^{2 x} x^2+1024 e^x x^6+131072 x^{10}\right )}{1024 x^6+65536 x^9+262144 x^{10}+e^{2 x} \left (x+4 x^2\right )+e^x \left (4 x^2+512 x^5+2048 x^6\right )+2 e^{2 x} \left (e^{2 x} x^2+512 e^x x^6+65536 x^{10}\right )} \, dx=-\log (x)-\log \left (e^x+256 x^4\right )+\log \left (e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5\right ) \] Input:
Integrate[(-E^(2*x) - 4096*x^5 - 65536*x^8 + E^x*(-4*x^2 - 512*x^4) + 2*E^ (2*x)*(2*E^(2*x)*x^2 + 1024*E^x*x^6 + 131072*x^10))/(1024*x^6 + 65536*x^9 + 262144*x^10 + E^(2*x)*(x + 4*x^2) + E^x*(4*x^2 + 512*x^5 + 2048*x^6) + 2 *E^(2*x)*(E^(2*x)*x^2 + 512*E^x*x^6 + 65536*x^10)),x]
Output:
-Log[x] - Log[E^x + 256*x^4] + Log[E^x + 4*x + 4*E^x*x + 2*E^(3*x)*x + 256 *x^4 + 1024*x^5 + 512*E^(2*x)*x^5]
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.
| \(\displaystyle \int \frac {-65536 x^8-4096 x^5+e^x \left (-512 x^4-4 x^2\right )+2 e^{2 x} \left (131072 x^{10}+1024 e^x x^6+2 e^{2 x} x^2\right )-e^{2 x}}{262144 x^{10}+65536 x^9+1024 x^6+e^{2 x} \left (4 x^2+x\right )+2 e^{2 x} \left (65536 x^{10}+512 e^x x^6+e^{2 x} x^2\right )+e^x \left (2048 x^6+512 x^5+4 x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {256 (x-4) x^3}{256 x^4+e^x}-\frac {512 e^{2 x} x^6+3072 x^6-2048 e^{2 x} x^5-3328 x^5-768 x^4+8 e^x x^2+12 x^2+2 e^x x+e^x}{\left (512 e^{2 x} x^5+1024 x^5+256 x^4+4 e^x x+2 e^{3 x} x+4 x+e^x\right ) x}+2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 256 \int \frac {x^4}{256 x^4+e^x}dx-2 \int \frac {e^x}{512 e^{2 x} x^5+1024 x^5+256 x^4+4 e^x x+2 e^{3 x} x+4 x+e^x}dx-\int \frac {e^x}{x \left (512 e^{2 x} x^5+1024 x^5+256 x^4+4 e^x x+2 e^{3 x} x+4 x+e^x\right )}dx-12 \int \frac {x}{512 e^{2 x} x^5+1024 x^5+256 x^4+4 e^x x+2 e^{3 x} x+4 x+e^x}dx-8 \int \frac {e^x x}{512 e^{2 x} x^5+1024 x^5+256 x^4+4 e^x x+2 e^{3 x} x+4 x+e^x}dx+3328 \int \frac {x^4}{512 e^{2 x} x^5+1024 x^5+256 x^4+4 e^x x+2 e^{3 x} x+4 x+e^x}dx+2048 \int \frac {e^{2 x} x^4}{512 e^{2 x} x^5+1024 x^5+256 x^4+4 e^x x+2 e^{3 x} x+4 x+e^x}dx-3072 \int \frac {x^5}{512 e^{2 x} x^5+1024 x^5+256 x^4+4 e^x x+2 e^{3 x} x+4 x+e^x}dx-512 \int \frac {e^{2 x} x^5}{512 e^{2 x} x^5+1024 x^5+256 x^4+4 e^x x+2 e^{3 x} x+4 x+e^x}dx-1024 \int \frac {x^3}{256 x^4+e^x}dx+768 \int \frac {x^3}{512 e^{2 x} x^5+1024 x^5+256 x^4+4 e^x x+2 e^{3 x} x+4 x+e^x}dx+2 x\) |
Input:
Int[(-E^(2*x) - 4096*x^5 - 65536*x^8 + E^x*(-4*x^2 - 512*x^4) + 2*E^(2*x)* (2*E^(2*x)*x^2 + 1024*E^x*x^6 + 131072*x^10))/(1024*x^6 + 65536*x^9 + 2621 44*x^10 + E^(2*x)*(x + 4*x^2) + E^x*(4*x^2 + 512*x^5 + 2048*x^6) + 2*E^(2* x)*(E^(2*x)*x^2 + 512*E^x*x^6 + 65536*x^10)),x]
Output:
$Aborted
Time = 3.10 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.96
| method | result | size |
| risch | \(-\ln \left ({\mathrm e}^{x}+256 x^{4}\right )+\ln \left ({\mathrm e}^{3 x}+256 \,{\mathrm e}^{2 x} x^{4}+\frac {\left (1+4 x \right ) {\mathrm e}^{x}}{2 x}+512 x^{4}+128 x^{3}+2\right )\) | \(51\) |
| parallelrisch | \(-\ln \left (x \right )-\ln \left (x^{4}+\frac {{\mathrm e}^{x}}{256}\right )+\ln \left (x^{5} {\mathrm e}^{\ln \left (2\right )+2 x}+4 x^{5}+x^{4}+\frac {x \,{\mathrm e}^{x} {\mathrm e}^{\ln \left (2\right )+2 x}}{256}+\frac {{\mathrm e}^{x} x}{64}+\frac {x}{64}+\frac {{\mathrm e}^{x}}{256}\right )\) | \(62\) |
Input:
int(((2*exp(x)^2*x^2+1024*x^6*exp(x)+131072*x^10)*exp(ln(2)+2*x)-exp(x)^2+ (-512*x^4-4*x^2)*exp(x)-65536*x^8-4096*x^5)/((exp(x)^2*x^2+512*x^6*exp(x)+ 65536*x^10)*exp(ln(2)+2*x)+(4*x^2+x)*exp(x)^2+(2048*x^6+512*x^5+4*x^2)*exp (x)+262144*x^10+65536*x^9+1024*x^6),x,method=_RETURNVERBOSE)
Output:
-ln(exp(x)+256*x^4)+ln(exp(3*x)+256*exp(2*x)*x^4+1/2*(1+4*x)/x*exp(x)+512* x^4+128*x^3+2)
Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (25) = 50\).
Time = 0.11 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.12 \[ \int \frac {-e^{2 x}-4096 x^5-65536 x^8+e^x \left (-4 x^2-512 x^4\right )+2 e^{2 x} \left (2 e^{2 x} x^2+1024 e^x x^6+131072 x^{10}\right )}{1024 x^6+65536 x^9+262144 x^{10}+e^{2 x} \left (x+4 x^2\right )+e^x \left (4 x^2+512 x^5+2048 x^6\right )+2 e^{2 x} \left (e^{2 x} x^2+512 e^x x^6+65536 x^{10}\right )} \, dx=-\log \left (256 \, x^{4} + e^{x}\right ) + \log \left (\frac {512 \, x^{5} e^{\left (2 \, x\right )} + 1024 \, x^{5} + 256 \, x^{4} + 2 \, x e^{\left (3 \, x\right )} + {\left (4 \, x + 1\right )} e^{x} + 4 \, x}{x}\right ) \] Input:
integrate(((2*exp(x)^2*x^2+1024*x^6*exp(x)+131072*x^10)*exp(log(2)+2*x)-ex p(x)^2+(-512*x^4-4*x^2)*exp(x)-65536*x^8-4096*x^5)/((exp(x)^2*x^2+512*x^6* exp(x)+65536*x^10)*exp(log(2)+2*x)+(4*x^2+x)*exp(x)^2+(2048*x^6+512*x^5+4* x^2)*exp(x)+262144*x^10+65536*x^9+1024*x^6),x, algorithm="fricas")
Output:
-log(256*x^4 + e^x) + log((512*x^5*e^(2*x) + 1024*x^5 + 256*x^4 + 2*x*e^(3 *x) + (4*x + 1)*e^x + 4*x)/x)
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (22) = 44\).
Time = 0.74 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.88 \[ \int \frac {-e^{2 x}-4096 x^5-65536 x^8+e^x \left (-4 x^2-512 x^4\right )+2 e^{2 x} \left (2 e^{2 x} x^2+1024 e^x x^6+131072 x^{10}\right )}{1024 x^6+65536 x^9+262144 x^{10}+e^{2 x} \left (x+4 x^2\right )+e^x \left (4 x^2+512 x^5+2048 x^6\right )+2 e^{2 x} \left (e^{2 x} x^2+512 e^x x^6+65536 x^{10}\right )} \, dx=- \log {\left (256 x^{4} + e^{x} \right )} + \log {\left (256 x^{4} e^{2 x} + 512 x^{4} + 128 x^{3} + e^{3 x} + 2 + \frac {\left (4 x + 1\right ) e^{x}}{2 x} \right )} \] Input:
integrate(((2*exp(x)**2*x**2+1024*x**6*exp(x)+131072*x**10)*exp(ln(2)+2*x) -exp(x)**2+(-512*x**4-4*x**2)*exp(x)-65536*x**8-4096*x**5)/((exp(x)**2*x** 2+512*x**6*exp(x)+65536*x**10)*exp(ln(2)+2*x)+(4*x**2+x)*exp(x)**2+(2048*x **6+512*x**5+4*x**2)*exp(x)+262144*x**10+65536*x**9+1024*x**6),x)
Output:
-log(256*x**4 + exp(x)) + log(256*x**4*exp(2*x) + 512*x**4 + 128*x**3 + ex p(3*x) + 2 + (4*x + 1)*exp(x)/(2*x))
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (25) = 50\).
Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.15 \[ \int \frac {-e^{2 x}-4096 x^5-65536 x^8+e^x \left (-4 x^2-512 x^4\right )+2 e^{2 x} \left (2 e^{2 x} x^2+1024 e^x x^6+131072 x^{10}\right )}{1024 x^6+65536 x^9+262144 x^{10}+e^{2 x} \left (x+4 x^2\right )+e^x \left (4 x^2+512 x^5+2048 x^6\right )+2 e^{2 x} \left (e^{2 x} x^2+512 e^x x^6+65536 x^{10}\right )} \, dx=-\log \left (256 \, x^{4} + e^{x}\right ) + \log \left (\frac {512 \, x^{5} e^{\left (2 \, x\right )} + 1024 \, x^{5} + 256 \, x^{4} + 2 \, x e^{\left (3 \, x\right )} + {\left (4 \, x + 1\right )} e^{x} + 4 \, x}{2 \, x}\right ) \] Input:
integrate(((2*exp(x)^2*x^2+1024*x^6*exp(x)+131072*x^10)*exp(log(2)+2*x)-ex p(x)^2+(-512*x^4-4*x^2)*exp(x)-65536*x^8-4096*x^5)/((exp(x)^2*x^2+512*x^6* exp(x)+65536*x^10)*exp(log(2)+2*x)+(4*x^2+x)*exp(x)^2+(2048*x^6+512*x^5+4* x^2)*exp(x)+262144*x^10+65536*x^9+1024*x^6),x, algorithm="maxima")
Output:
-log(256*x^4 + e^x) + log(1/2*(512*x^5*e^(2*x) + 1024*x^5 + 256*x^4 + 2*x* e^(3*x) + (4*x + 1)*e^x + 4*x)/x)
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (25) = 50\).
Time = 1.33 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.08 \[ \int \frac {-e^{2 x}-4096 x^5-65536 x^8+e^x \left (-4 x^2-512 x^4\right )+2 e^{2 x} \left (2 e^{2 x} x^2+1024 e^x x^6+131072 x^{10}\right )}{1024 x^6+65536 x^9+262144 x^{10}+e^{2 x} \left (x+4 x^2\right )+e^x \left (4 x^2+512 x^5+2048 x^6\right )+2 e^{2 x} \left (e^{2 x} x^2+512 e^x x^6+65536 x^{10}\right )} \, dx=\log \left (512 \, x^{5} e^{\left (2 \, x\right )} + 1024 \, x^{5} + 256 \, x^{4} + 2 \, x e^{\left (3 \, x\right )} + 4 \, x e^{x} + 4 \, x + e^{x}\right ) - \log \left (256 \, x^{4} + e^{x}\right ) - \log \left (x\right ) \] Input:
integrate(((2*exp(x)^2*x^2+1024*x^6*exp(x)+131072*x^10)*exp(log(2)+2*x)-ex p(x)^2+(-512*x^4-4*x^2)*exp(x)-65536*x^8-4096*x^5)/((exp(x)^2*x^2+512*x^6* exp(x)+65536*x^10)*exp(log(2)+2*x)+(4*x^2+x)*exp(x)^2+(2048*x^6+512*x^5+4* x^2)*exp(x)+262144*x^10+65536*x^9+1024*x^6),x, algorithm="giac")
Output:
log(512*x^5*e^(2*x) + 1024*x^5 + 256*x^4 + 2*x*e^(3*x) + 4*x*e^x + 4*x + e ^x) - log(256*x^4 + e^x) - log(x)
Timed out. \[ \int \frac {-e^{2 x}-4096 x^5-65536 x^8+e^x \left (-4 x^2-512 x^4\right )+2 e^{2 x} \left (2 e^{2 x} x^2+1024 e^x x^6+131072 x^{10}\right )}{1024 x^6+65536 x^9+262144 x^{10}+e^{2 x} \left (x+4 x^2\right )+e^x \left (4 x^2+512 x^5+2048 x^6\right )+2 e^{2 x} \left (e^{2 x} x^2+512 e^x x^6+65536 x^{10}\right )} \, dx=\int -\frac {{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (512\,x^4+4\,x^2\right )-{\mathrm {e}}^{2\,x+\ln \left (2\right )}\,\left (1024\,x^6\,{\mathrm {e}}^x+2\,x^2\,{\mathrm {e}}^{2\,x}+131072\,x^{10}\right )+4096\,x^5+65536\,x^8}{{\mathrm {e}}^x\,\left (2048\,x^6+512\,x^5+4\,x^2\right )+{\mathrm {e}}^{2\,x+\ln \left (2\right )}\,\left (512\,x^6\,{\mathrm {e}}^x+x^2\,{\mathrm {e}}^{2\,x}+65536\,x^{10}\right )+{\mathrm {e}}^{2\,x}\,\left (4\,x^2+x\right )+1024\,x^6+65536\,x^9+262144\,x^{10}} \,d x \] Input:
int(-(exp(2*x) + exp(x)*(4*x^2 + 512*x^4) - exp(2*x + log(2))*(1024*x^6*ex p(x) + 2*x^2*exp(2*x) + 131072*x^10) + 4096*x^5 + 65536*x^8)/(exp(x)*(4*x^ 2 + 512*x^5 + 2048*x^6) + exp(2*x + log(2))*(512*x^6*exp(x) + x^2*exp(2*x) + 65536*x^10) + exp(2*x)*(x + 4*x^2) + 1024*x^6 + 65536*x^9 + 262144*x^10 ),x)
Output:
int(-(exp(2*x) + exp(x)*(4*x^2 + 512*x^4) - exp(2*x + log(2))*(1024*x^6*ex p(x) + 2*x^2*exp(2*x) + 131072*x^10) + 4096*x^5 + 65536*x^8)/(exp(x)*(4*x^ 2 + 512*x^5 + 2048*x^6) + exp(2*x + log(2))*(512*x^6*exp(x) + x^2*exp(2*x) + 65536*x^10) + exp(2*x)*(x + 4*x^2) + 1024*x^6 + 65536*x^9 + 262144*x^10 ), x)
\[ \int \frac {-e^{2 x}-4096 x^5-65536 x^8+e^x \left (-4 x^2-512 x^4\right )+2 e^{2 x} \left (2 e^{2 x} x^2+1024 e^x x^6+131072 x^{10}\right )}{1024 x^6+65536 x^9+262144 x^{10}+e^{2 x} \left (x+4 x^2\right )+e^x \left (4 x^2+512 x^5+2048 x^6\right )+2 e^{2 x} \left (e^{2 x} x^2+512 e^x x^6+65536 x^{10}\right )} \, dx=-\left (\int \frac {e^{2 x}}{2 e^{4 x} x^{2}+1024 e^{3 x} x^{6}+131072 e^{2 x} x^{10}+4 e^{2 x} x^{2}+e^{2 x} x +2048 e^{x} x^{6}+512 e^{x} x^{5}+4 e^{x} x^{2}+262144 x^{10}+65536 x^{9}+1024 x^{6}}d x \right )-65536 \left (\int \frac {x^{7}}{2 e^{4 x} x +1024 e^{3 x} x^{5}+131072 e^{2 x} x^{9}+4 e^{2 x} x +e^{2 x}+2048 e^{x} x^{5}+512 e^{x} x^{4}+4 e^{x} x +262144 x^{9}+65536 x^{8}+1024 x^{5}}d x \right )-4096 \left (\int \frac {x^{4}}{2 e^{4 x} x +1024 e^{3 x} x^{5}+131072 e^{2 x} x^{9}+4 e^{2 x} x +e^{2 x}+2048 e^{x} x^{5}+512 e^{x} x^{4}+4 e^{x} x +262144 x^{9}+65536 x^{8}+1024 x^{5}}d x \right )+4 \left (\int \frac {e^{4 x} x}{2 e^{4 x} x +1024 e^{3 x} x^{5}+131072 e^{2 x} x^{9}+4 e^{2 x} x +e^{2 x}+2048 e^{x} x^{5}+512 e^{x} x^{4}+4 e^{x} x +262144 x^{9}+65536 x^{8}+1024 x^{5}}d x \right )+2048 \left (\int \frac {e^{3 x} x^{5}}{2 e^{4 x} x +1024 e^{3 x} x^{5}+131072 e^{2 x} x^{9}+4 e^{2 x} x +e^{2 x}+2048 e^{x} x^{5}+512 e^{x} x^{4}+4 e^{x} x +262144 x^{9}+65536 x^{8}+1024 x^{5}}d x \right )+262144 \left (\int \frac {e^{2 x} x^{9}}{2 e^{4 x} x +1024 e^{3 x} x^{5}+131072 e^{2 x} x^{9}+4 e^{2 x} x +e^{2 x}+2048 e^{x} x^{5}+512 e^{x} x^{4}+4 e^{x} x +262144 x^{9}+65536 x^{8}+1024 x^{5}}d x \right )-512 \left (\int \frac {e^{x} x^{3}}{2 e^{4 x} x +1024 e^{3 x} x^{5}+131072 e^{2 x} x^{9}+4 e^{2 x} x +e^{2 x}+2048 e^{x} x^{5}+512 e^{x} x^{4}+4 e^{x} x +262144 x^{9}+65536 x^{8}+1024 x^{5}}d x \right )-4 \left (\int \frac {e^{x} x}{2 e^{4 x} x +1024 e^{3 x} x^{5}+131072 e^{2 x} x^{9}+4 e^{2 x} x +e^{2 x}+2048 e^{x} x^{5}+512 e^{x} x^{4}+4 e^{x} x +262144 x^{9}+65536 x^{8}+1024 x^{5}}d x \right ) \] Input:
int(((2*exp(x)^2*x^2+1024*x^6*exp(x)+131072*x^10)*exp(log(2)+2*x)-exp(x)^2 +(-512*x^4-4*x^2)*exp(x)-65536*x^8-4096*x^5)/((exp(x)^2*x^2+512*x^6*exp(x) +65536*x^10)*exp(log(2)+2*x)+(4*x^2+x)*exp(x)^2+(2048*x^6+512*x^5+4*x^2)*e xp(x)+262144*x^10+65536*x^9+1024*x^6),x)
Output:
- int(e**(2*x)/(2*e**(4*x)*x**2 + 1024*e**(3*x)*x**6 + 131072*e**(2*x)*x* *10 + 4*e**(2*x)*x**2 + e**(2*x)*x + 2048*e**x*x**6 + 512*e**x*x**5 + 4*e* *x*x**2 + 262144*x**10 + 65536*x**9 + 1024*x**6),x) - 65536*int(x**7/(2*e* *(4*x)*x + 1024*e**(3*x)*x**5 + 131072*e**(2*x)*x**9 + 4*e**(2*x)*x + e**( 2*x) + 2048*e**x*x**5 + 512*e**x*x**4 + 4*e**x*x + 262144*x**9 + 65536*x** 8 + 1024*x**5),x) - 4096*int(x**4/(2*e**(4*x)*x + 1024*e**(3*x)*x**5 + 131 072*e**(2*x)*x**9 + 4*e**(2*x)*x + e**(2*x) + 2048*e**x*x**5 + 512*e**x*x* *4 + 4*e**x*x + 262144*x**9 + 65536*x**8 + 1024*x**5),x) + 4*int((e**(4*x) *x)/(2*e**(4*x)*x + 1024*e**(3*x)*x**5 + 131072*e**(2*x)*x**9 + 4*e**(2*x) *x + e**(2*x) + 2048*e**x*x**5 + 512*e**x*x**4 + 4*e**x*x + 262144*x**9 + 65536*x**8 + 1024*x**5),x) + 2048*int((e**(3*x)*x**5)/(2*e**(4*x)*x + 1024 *e**(3*x)*x**5 + 131072*e**(2*x)*x**9 + 4*e**(2*x)*x + e**(2*x) + 2048*e** x*x**5 + 512*e**x*x**4 + 4*e**x*x + 262144*x**9 + 65536*x**8 + 1024*x**5), x) + 262144*int((e**(2*x)*x**9)/(2*e**(4*x)*x + 1024*e**(3*x)*x**5 + 13107 2*e**(2*x)*x**9 + 4*e**(2*x)*x + e**(2*x) + 2048*e**x*x**5 + 512*e**x*x**4 + 4*e**x*x + 262144*x**9 + 65536*x**8 + 1024*x**5),x) - 512*int((e**x*x** 3)/(2*e**(4*x)*x + 1024*e**(3*x)*x**5 + 131072*e**(2*x)*x**9 + 4*e**(2*x)* x + e**(2*x) + 2048*e**x*x**5 + 512*e**x*x**4 + 4*e**x*x + 262144*x**9 + 6 5536*x**8 + 1024*x**5),x) - 4*int((e**x*x)/(2*e**(4*x)*x + 1024*e**(3*x)*x **5 + 131072*e**(2*x)*x**9 + 4*e**(2*x)*x + e**(2*x) + 2048*e**x*x**5 +...