Integrand size = 78, antiderivative size = 25 \[ \int \frac {256+16 e-64 x+24 x^2-2 x^3}{\left (-4096+2944 x-704 x^2+52 x^3+x^4+e (-16+4 x)\right ) \log ^2\left (\frac {1024+4 e-480 x+56 x^2+x^3}{64-32 x+4 x^2}\right )} \, dx=\frac {2}{\log \left (16+\frac {x}{4}+\frac {e+4 x}{(4-x)^2}\right )} \] Output:
2/ln((exp(1)+4*x)/(4-x)^2+1/4*x+16)
Time = 0.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {256+16 e-64 x+24 x^2-2 x^3}{\left (-4096+2944 x-704 x^2+52 x^3+x^4+e (-16+4 x)\right ) \log ^2\left (\frac {1024+4 e-480 x+56 x^2+x^3}{64-32 x+4 x^2}\right )} \, dx=\frac {2}{\log \left (\frac {1024+4 e-480 x+56 x^2+x^3}{4 (-4+x)^2}\right )} \] Input:
Integrate[(256 + 16*E - 64*x + 24*x^2 - 2*x^3)/((-4096 + 2944*x - 704*x^2 + 52*x^3 + x^4 + E*(-16 + 4*x))*Log[(1024 + 4*E - 480*x + 56*x^2 + x^3)/(6 4 - 32*x + 4*x^2)]^2),x]
Output:
2/Log[(1024 + 4*E - 480*x + 56*x^2 + x^3)/(4*(-4 + x)^2)]
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.
\(\displaystyle \int \frac {-2 x^3+24 x^2-64 x+16 e+256}{\left (x^4+52 x^3-704 x^2+2944 x+e (4 x-16)-4096\right ) \log ^2\left (\frac {x^3+56 x^2-480 x+4 e+1024}{4 x^2-32 x+64}\right )} \, dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {-2 x^3+24 x^2-64 x+16 e+256}{4 (16+e) (x-4) \log ^2\left (\frac {x^3+56 x^2-480 x+4 e+1024}{4 x^2-32 x+64}\right )}+\frac {\left (-x^2-60 x+240\right ) \left (-2 x^3+24 x^2-64 x+16 e+256\right )}{4 (16+e) \left (x^3+56 x^2-480 x+4 (256+e)\right ) \log ^2\left (\frac {x^3+56 x^2-480 x+4 e+1024}{4 x^2-32 x+64}\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -4 \int \frac {1}{(4-x) \log ^2\left (\frac {x^3+56 x^2-480 x+4 (256+e)}{4 (x-4)^2}\right )}dx+224 \int \frac {x}{\left (-x^3-56 x^2+480 x-4 (256+e)\right ) \log ^2\left (\frac {x^3+56 x^2-480 x+4 (256+e)}{4 (x-4)^2}\right )}dx+6 \int \frac {x^2}{\left (-x^3-56 x^2+480 x-4 (256+e)\right ) \log ^2\left (\frac {x^3+56 x^2-480 x+4 (256+e)}{4 (x-4)^2}\right )}dx+960 \int \frac {1}{\left (x^3+56 x^2-480 x+4 (256+e)\right ) \log ^2\left (\frac {x^3+56 x^2-480 x+4 (256+e)}{4 (x-4)^2}\right )}dx\) |
Input:
Int[(256 + 16*E - 64*x + 24*x^2 - 2*x^3)/((-4096 + 2944*x - 704*x^2 + 52*x ^3 + x^4 + E*(-16 + 4*x))*Log[(1024 + 4*E - 480*x + 56*x^2 + x^3)/(64 - 32 *x + 4*x^2)]^2),x]
Output:
$Aborted
Time = 0.62 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40
method | result | size |
parallelrisch | \(\frac {2}{\ln \left (\frac {4 \,{\mathrm e}+x^{3}+56 x^{2}-480 x +1024}{4 x^{2}-32 x +64}\right )}\) | \(35\) |
norman | \(\frac {2}{\ln \left (\frac {4 \,{\mathrm e}+x^{3}+56 x^{2}-480 x +1024}{4 x^{2}-32 x +64}\right )}\) | \(36\) |
risch | \(\frac {2}{\ln \left (\frac {4 \,{\mathrm e}+x^{3}+56 x^{2}-480 x +1024}{4 x^{2}-32 x +64}\right )}\) | \(36\) |
default | \(-\frac {2}{2 \ln \left (2\right )-\ln \left (\frac {4 \,{\mathrm e}+x^{3}+56 x^{2}-480 x +1024}{x^{2}-8 x +16}\right )}\) | \(41\) |
Input:
int((16*exp(1)-2*x^3+24*x^2-64*x+256)/((4*x-16)*exp(1)+x^4+52*x^3-704*x^2+ 2944*x-4096)/ln((4*exp(1)+x^3+56*x^2-480*x+1024)/(4*x^2-32*x+64))^2,x,meth od=_RETURNVERBOSE)
Output:
2/ln(1/4*(4*exp(1)+x^3+56*x^2-480*x+1024)/(x^2-8*x+16))
Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {256+16 e-64 x+24 x^2-2 x^3}{\left (-4096+2944 x-704 x^2+52 x^3+x^4+e (-16+4 x)\right ) \log ^2\left (\frac {1024+4 e-480 x+56 x^2+x^3}{64-32 x+4 x^2}\right )} \, dx=\frac {2}{\log \left (\frac {x^{3} + 56 \, x^{2} - 480 \, x + 4 \, e + 1024}{4 \, {\left (x^{2} - 8 \, x + 16\right )}}\right )} \] Input:
integrate((16*exp(1)-2*x^3+24*x^2-64*x+256)/((4*x-16)*exp(1)+x^4+52*x^3-70 4*x^2+2944*x-4096)/log((4*exp(1)+x^3+56*x^2-480*x+1024)/(4*x^2-32*x+64))^2 ,x, algorithm="fricas")
Output:
2/log(1/4*(x^3 + 56*x^2 - 480*x + 4*e + 1024)/(x^2 - 8*x + 16))
Time = 0.12 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {256+16 e-64 x+24 x^2-2 x^3}{\left (-4096+2944 x-704 x^2+52 x^3+x^4+e (-16+4 x)\right ) \log ^2\left (\frac {1024+4 e-480 x+56 x^2+x^3}{64-32 x+4 x^2}\right )} \, dx=\frac {2}{\log {\left (\frac {x^{3} + 56 x^{2} - 480 x + 4 e + 1024}{4 x^{2} - 32 x + 64} \right )}} \] Input:
integrate((16*exp(1)-2*x**3+24*x**2-64*x+256)/((4*x-16)*exp(1)+x**4+52*x** 3-704*x**2+2944*x-4096)/ln((4*exp(1)+x**3+56*x**2-480*x+1024)/(4*x**2-32*x +64))**2,x)
Output:
2/log((x**3 + 56*x**2 - 480*x + 4*E + 1024)/(4*x**2 - 32*x + 64))
Time = 0.16 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \frac {256+16 e-64 x+24 x^2-2 x^3}{\left (-4096+2944 x-704 x^2+52 x^3+x^4+e (-16+4 x)\right ) \log ^2\left (\frac {1024+4 e-480 x+56 x^2+x^3}{64-32 x+4 x^2}\right )} \, dx=-\frac {2}{2 \, \log \left (2\right ) - \log \left (x^{3} + 56 \, x^{2} - 480 \, x + 4 \, e + 1024\right ) + 2 \, \log \left (x - 4\right )} \] Input:
integrate((16*exp(1)-2*x^3+24*x^2-64*x+256)/((4*x-16)*exp(1)+x^4+52*x^3-70 4*x^2+2944*x-4096)/log((4*exp(1)+x^3+56*x^2-480*x+1024)/(4*x^2-32*x+64))^2 ,x, algorithm="maxima")
Output:
-2/(2*log(2) - log(x^3 + 56*x^2 - 480*x + 4*e + 1024) + 2*log(x - 4))
Time = 0.16 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {256+16 e-64 x+24 x^2-2 x^3}{\left (-4096+2944 x-704 x^2+52 x^3+x^4+e (-16+4 x)\right ) \log ^2\left (\frac {1024+4 e-480 x+56 x^2+x^3}{64-32 x+4 x^2}\right )} \, dx=\frac {2}{\log \left (\frac {x^{3} + 56 \, x^{2} - 480 \, x + 4 \, e + 1024}{4 \, {\left (x^{2} - 8 \, x + 16\right )}}\right )} \] Input:
integrate((16*exp(1)-2*x^3+24*x^2-64*x+256)/((4*x-16)*exp(1)+x^4+52*x^3-70 4*x^2+2944*x-4096)/log((4*exp(1)+x^3+56*x^2-480*x+1024)/(4*x^2-32*x+64))^2 ,x, algorithm="giac")
Output:
2/log(1/4*(x^3 + 56*x^2 - 480*x + 4*e + 1024)/(x^2 - 8*x + 16))
Time = 6.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \frac {256+16 e-64 x+24 x^2-2 x^3}{\left (-4096+2944 x-704 x^2+52 x^3+x^4+e (-16+4 x)\right ) \log ^2\left (\frac {1024+4 e-480 x+56 x^2+x^3}{64-32 x+4 x^2}\right )} \, dx=\frac {2}{\ln \left (\frac {x^3+56\,x^2-480\,x+4\,\mathrm {e}+1024}{4\,x^2-32\,x+64}\right )} \] Input:
int((16*exp(1) - 64*x + 24*x^2 - 2*x^3 + 256)/(log((4*exp(1) - 480*x + 56* x^2 + x^3 + 1024)/(4*x^2 - 32*x + 64))^2*(2944*x - 704*x^2 + 52*x^3 + x^4 + exp(1)*(4*x - 16) - 4096)),x)
Output:
2/log((4*exp(1) - 480*x + 56*x^2 + x^3 + 1024)/(4*x^2 - 32*x + 64))
Time = 0.18 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {256+16 e-64 x+24 x^2-2 x^3}{\left (-4096+2944 x-704 x^2+52 x^3+x^4+e (-16+4 x)\right ) \log ^2\left (\frac {1024+4 e-480 x+56 x^2+x^3}{64-32 x+4 x^2}\right )} \, dx=\frac {2}{\mathrm {log}\left (\frac {x^{3}+56 x^{2}+4 e -480 x +1024}{4 x^{2}-32 x +64}\right )} \] Input:
int((16*exp(1)-2*x^3+24*x^2-64*x+256)/((4*x-16)*exp(1)+x^4+52*x^3-704*x^2+ 2944*x-4096)/log((4*exp(1)+x^3+56*x^2-480*x+1024)/(4*x^2-32*x+64))^2,x)
Output:
2/log((4*e + x**3 + 56*x**2 - 480*x + 1024)/(4*x**2 - 32*x + 64))