Integrand size = 100, antiderivative size = 25 \[ \int \frac {4^{-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}} \left (12 x-512 x^2+1024 x^4+\left (-256-48 x+1024 x^2-1024 x^4\right ) \log \left (\frac {1}{4} \left (16+3 x-64 x^2+64 x^4\right )\right )\right )}{16 x^5+3 x^6-64 x^7+64 x^9} \, dx=4 \left (x+\frac {1}{4} \left (-x+\left (4-8 x^2\right )^2\right )\right )^{\frac {1}{x^4}} \] Output:
4*exp(ln(1/4*(-8*x^2+4)^2+3/4*x)/x^4)
Time = 0.35 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {4^{-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}} \left (12 x-512 x^2+1024 x^4+\left (-256-48 x+1024 x^2-1024 x^4\right ) \log \left (\frac {1}{4} \left (16+3 x-64 x^2+64 x^4\right )\right )\right )}{16 x^5+3 x^6-64 x^7+64 x^9} \, dx=4 \left (4+\frac {3 x}{4}-16 x^2+16 x^4\right )^{\frac {1}{x^4}} \] Input:
Integrate[((16 + 3*x - 64*x^2 + 64*x^4)^x^(-4)*(12*x - 512*x^2 + 1024*x^4 + (-256 - 48*x + 1024*x^2 - 1024*x^4)*Log[(16 + 3*x - 64*x^2 + 64*x^4)/4]) )/(4^x^(-4)*(16*x^5 + 3*x^6 - 64*x^7 + 64*x^9)),x]
Output:
4*(4 + (3*x)/4 - 16*x^2 + 16*x^4)^x^(-4)
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.
\(\displaystyle \int \frac {4^{-\frac {1}{x^4}} \left (64 x^4-64 x^2+3 x+16\right )^{\frac {1}{x^4}} \left (1024 x^4-512 x^2+\left (-1024 x^4+1024 x^2-48 x-256\right ) \log \left (\frac {1}{4} \left (64 x^4-64 x^2+3 x+16\right )\right )+12 x\right )}{64 x^9-64 x^7+3 x^6+16 x^5} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {4^{-\frac {1}{x^4}} \left (64 x^4-64 x^2+3 x+16\right )^{\frac {1}{x^4}-1} \left (1024 x^4-512 x^2+\left (-1024 x^4+1024 x^2-48 x-256\right ) \log \left (\frac {1}{4} \left (64 x^4-64 x^2+3 x+16\right )\right )+12 x\right )}{x^5}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {4^{1-\frac {1}{x^4}} \left (256 x^3-128 x+3\right ) \left (64 x^4-64 x^2+3 x+16\right )^{\frac {1}{x^4}-1}}{x^4}-\frac {4^{2-\frac {1}{x^4}} \left (64 x^4-64 x^2+3 x+16\right )^{\frac {1}{x^4}} \log \left (16 x^4-16 x^2+\frac {3 x}{4}+4\right )}{x^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \int \frac {4^{1-\frac {1}{x^4}} \left (64 x^4-64 x^2+3 x+16\right )^{\frac {1}{x^4}-1}}{x^4}dx+\int \frac {4^{5-\frac {1}{x^4}} \left (64 x^4-64 x^2+3 x+16\right )^{\frac {1}{x^4}-1}}{x}dx+3 \int \frac {\int \frac {4^{2-\frac {1}{x^4}} \left (64 x^4-64 x^2+3 x+16\right )^{\frac {1}{x^4}}}{x^5}dx}{64 x^4-64 x^2+3 x+16}dx-128 \int \frac {x \int \frac {4^{2-\frac {1}{x^4}} \left (64 x^4-64 x^2+3 x+16\right )^{\frac {1}{x^4}}}{x^5}dx}{64 x^4-64 x^2+3 x+16}dx-\log \left (16 x^4-16 x^2+\frac {3 x}{4}+4\right ) \int \frac {4^{2-\frac {1}{x^4}} \left (64 x^4-64 x^2+3 x+16\right )^{\frac {1}{x^4}}}{x^5}dx-\int \frac {2^{9-\frac {2}{x^4}} \left (64 x^4-64 x^2+3 x+16\right )^{\frac {1}{x^4}-1}}{x^3}dx+256 \int \frac {x^3 \int \frac {4^{2-\frac {1}{x^4}} \left (64 x^4-64 x^2+3 x+16\right )^{\frac {1}{x^4}}}{x^5}dx}{64 x^4-64 x^2+3 x+16}dx\) |
Input:
Int[((16 + 3*x - 64*x^2 + 64*x^4)^x^(-4)*(12*x - 512*x^2 + 1024*x^4 + (-25 6 - 48*x + 1024*x^2 - 1024*x^4)*Log[(16 + 3*x - 64*x^2 + 64*x^4)/4]))/(4^x ^(-4)*(16*x^5 + 3*x^6 - 64*x^7 + 64*x^9)),x]
Output:
$Aborted
Time = 22.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88
method | result | size |
risch | \(4 \left (16 x^{4}-16 x^{2}+\frac {3}{4} x +4\right )^{\frac {1}{x^{4}}}\) | \(22\) |
parallelrisch | \(4 \,{\mathrm e}^{\frac {\ln \left (16 x^{4}-16 x^{2}+\frac {3}{4} x +4\right )}{x^{4}}}\) | \(24\) |
Input:
int(((-1024*x^4+1024*x^2-48*x-256)*ln(16*x^4-16*x^2+3/4*x+4)+1024*x^4-512* x^2+12*x)*exp(ln(16*x^4-16*x^2+3/4*x+4)/x^4)/(64*x^9-64*x^7+3*x^6+16*x^5), x,method=_RETURNVERBOSE)
Output:
4*(16*x^4-16*x^2+3/4*x+4)^(1/x^4)
Time = 0.10 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {4^{-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}} \left (12 x-512 x^2+1024 x^4+\left (-256-48 x+1024 x^2-1024 x^4\right ) \log \left (\frac {1}{4} \left (16+3 x-64 x^2+64 x^4\right )\right )\right )}{16 x^5+3 x^6-64 x^7+64 x^9} \, dx=4 \, {\left (16 \, x^{4} - 16 \, x^{2} + \frac {3}{4} \, x + 4\right )}^{\left (\frac {1}{x^{4}}\right )} \] Input:
integrate(((-1024*x^4+1024*x^2-48*x-256)*log(16*x^4-16*x^2+3/4*x+4)+1024*x ^4-512*x^2+12*x)*exp(log(16*x^4-16*x^2+3/4*x+4)/x^4)/(64*x^9-64*x^7+3*x^6+ 16*x^5),x, algorithm="fricas")
Output:
4*(16*x^4 - 16*x^2 + 3/4*x + 4)^(x^(-4))
Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {4^{-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}} \left (12 x-512 x^2+1024 x^4+\left (-256-48 x+1024 x^2-1024 x^4\right ) \log \left (\frac {1}{4} \left (16+3 x-64 x^2+64 x^4\right )\right )\right )}{16 x^5+3 x^6-64 x^7+64 x^9} \, dx=4 e^{\frac {\log {\left (16 x^{4} - 16 x^{2} + \frac {3 x}{4} + 4 \right )}}{x^{4}}} \] Input:
integrate(((-1024*x**4+1024*x**2-48*x-256)*ln(16*x**4-16*x**2+3/4*x+4)+102 4*x**4-512*x**2+12*x)*exp(ln(16*x**4-16*x**2+3/4*x+4)/x**4)/(64*x**9-64*x* *7+3*x**6+16*x**5),x)
Output:
4*exp(log(16*x**4 - 16*x**2 + 3*x/4 + 4)/x**4)
Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {4^{-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}} \left (12 x-512 x^2+1024 x^4+\left (-256-48 x+1024 x^2-1024 x^4\right ) \log \left (\frac {1}{4} \left (16+3 x-64 x^2+64 x^4\right )\right )\right )}{16 x^5+3 x^6-64 x^7+64 x^9} \, dx=4 \, e^{\left (-\frac {2 \, \log \left (2\right )}{x^{4}} + \frac {\log \left (64 \, x^{4} - 64 \, x^{2} + 3 \, x + 16\right )}{x^{4}}\right )} \] Input:
integrate(((-1024*x^4+1024*x^2-48*x-256)*log(16*x^4-16*x^2+3/4*x+4)+1024*x ^4-512*x^2+12*x)*exp(log(16*x^4-16*x^2+3/4*x+4)/x^4)/(64*x^9-64*x^7+3*x^6+ 16*x^5),x, algorithm="maxima")
Output:
4*e^(-2*log(2)/x^4 + log(64*x^4 - 64*x^2 + 3*x + 16)/x^4)
\[ \int \frac {4^{-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}} \left (12 x-512 x^2+1024 x^4+\left (-256-48 x+1024 x^2-1024 x^4\right ) \log \left (\frac {1}{4} \left (16+3 x-64 x^2+64 x^4\right )\right )\right )}{16 x^5+3 x^6-64 x^7+64 x^9} \, dx=\int { \frac {4 \, {\left (256 \, x^{4} - 128 \, x^{2} - 4 \, {\left (64 \, x^{4} - 64 \, x^{2} + 3 \, x + 16\right )} \log \left (16 \, x^{4} - 16 \, x^{2} + \frac {3}{4} \, x + 4\right ) + 3 \, x\right )} {\left (16 \, x^{4} - 16 \, x^{2} + \frac {3}{4} \, x + 4\right )}^{\left (\frac {1}{x^{4}}\right )}}{64 \, x^{9} - 64 \, x^{7} + 3 \, x^{6} + 16 \, x^{5}} \,d x } \] Input:
integrate(((-1024*x^4+1024*x^2-48*x-256)*log(16*x^4-16*x^2+3/4*x+4)+1024*x ^4-512*x^2+12*x)*exp(log(16*x^4-16*x^2+3/4*x+4)/x^4)/(64*x^9-64*x^7+3*x^6+ 16*x^5),x, algorithm="giac")
Output:
integrate(4*(256*x^4 - 128*x^2 - 4*(64*x^4 - 64*x^2 + 3*x + 16)*log(16*x^4 - 16*x^2 + 3/4*x + 4) + 3*x)*(16*x^4 - 16*x^2 + 3/4*x + 4)^(x^(-4))/(64*x ^9 - 64*x^7 + 3*x^6 + 16*x^5), x)
Time = 3.07 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {4^{-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}} \left (12 x-512 x^2+1024 x^4+\left (-256-48 x+1024 x^2-1024 x^4\right ) \log \left (\frac {1}{4} \left (16+3 x-64 x^2+64 x^4\right )\right )\right )}{16 x^5+3 x^6-64 x^7+64 x^9} \, dx=4\,{\left (16\,x^4-16\,x^2+\frac {3\,x}{4}+4\right )}^{\frac {1}{x^4}} \] Input:
int((exp(log((3*x)/4 - 16*x^2 + 16*x^4 + 4)/x^4)*(12*x - log((3*x)/4 - 16* x^2 + 16*x^4 + 4)*(48*x - 1024*x^2 + 1024*x^4 + 256) - 512*x^2 + 1024*x^4) )/(16*x^5 + 3*x^6 - 64*x^7 + 64*x^9),x)
Output:
4*((3*x)/4 - 16*x^2 + 16*x^4 + 4)^(1/x^4)
Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {4^{-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}} \left (12 x-512 x^2+1024 x^4+\left (-256-48 x+1024 x^2-1024 x^4\right ) \log \left (\frac {1}{4} \left (16+3 x-64 x^2+64 x^4\right )\right )\right )}{16 x^5+3 x^6-64 x^7+64 x^9} \, dx=4 e^{\frac {\mathrm {log}\left (16 x^{4}-16 x^{2}+\frac {3}{4} x +4\right )}{x^{4}}} \] Input:
int(((-1024*x^4+1024*x^2-48*x-256)*log(16*x^4-16*x^2+3/4*x+4)+1024*x^4-512 *x^2+12*x)*exp(log(16*x^4-16*x^2+3/4*x+4)/x^4)/(64*x^9-64*x^7+3*x^6+16*x^5 ),x)
Output:
4*e**(log((64*x**4 - 64*x**2 + 3*x + 16)/4)/x**4)