\(\int \frac {4^{-\frac {1}{x^4}} (16+3 x-64 x^2+64 x^4)^{\frac {1}{x^4}} (12 x-512 x^2+1024 x^4+(-256-48 x+1024 x^2-1024 x^4) \log (\frac {1}{4} (16+3 x-64 x^2+64 x^4)))}{16 x^5+3 x^6-64 x^7+64 x^9} \, dx\) [7017]

Optimal result

Integrand size = 100, antiderivative size = 25 \[ \int \frac {4^{-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}} \left (12 x-512 x^2+1024 x^4+\left (-256-48 x+1024 x^2-1024 x^4\right ) \log \left (\frac {1}{4} \left (16+3 x-64 x^2+64 x^4\right )\right )\right )}{16 x^5+3 x^6-64 x^7+64 x^9} \, dx=4 \left (x+\frac {1}{4} \left (-x+\left (4-8 x^2\right )^2\right )\right )^{\frac {1}{x^4}} \]

[Out]

4*exp(ln(1/4*(-8*x^2+4)^2+3/4*x)/x^4)

Rubi [F]

\[ \int \frac {4^{-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}} \left (12 x-512 x^2+1024 x^4+\left (-256-48 x+1024 x^2-1024 x^4\right ) \log \left (\frac {1}{4} \left (16+3 x-64 x^2+64 x^4\right )\right )\right )}{16 x^5+3 x^6-64 x^7+64 x^9} \, dx=\int \frac {4^{-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}} \left (12 x-512 x^2+1024 x^4+\left (-256-48 x+1024 x^2-1024 x^4\right ) \log \left (\frac {1}{4} \left (16+3 x-64 x^2+64 x^4\right )\right )\right )}{16 x^5+3 x^6-64 x^7+64 x^9} \, dx \]

[In]

Int[((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]

[Out]

3*Defer[Int][(4 + (3*x)/4 - 16*x^2 + 16*x^4)^(-1 + x^(-4))/x^4, x] - 128*Defer[Int][(4 + (3*x)/4 - 16*x^2 + 16
*x^4)^(-1 + x^(-4))/x^3, x] + 256*Defer[Int][(4 + (3*x)/4 - 16*x^2 + 16*x^4)^(-1 + x^(-4))/x, x] - Log[4 + (3*
x)/4 - 16*x^2 + 16*x^4]*Defer[Int][(4^(2 - x^(-4))*(16 + 3*x - 64*x^2 + 64*x^4)^x^(-4))/x^5, x] + 3*Defer[Int]
[Defer[Int][(4^(2 - x^(-4))*(16 + 3*x - 64*x^2 + 64*x^4)^x^(-4))/x^5, x]/(16 + 3*x - 64*x^2 + 64*x^4), x] - 12
8*Defer[Int][(x*Defer[Int][(4^(2 - x^(-4))*(16 + 3*x - 64*x^2 + 64*x^4)^x^(-4))/x^5, x])/(16 + 3*x - 64*x^2 +
64*x^4), x] + 256*Defer[Int][(x^3*Defer[Int][(4^(2 - x^(-4))*(16 + 3*x - 64*x^2 + 64*x^4)^x^(-4))/x^5, x])/(16
+ 3*x - 64*x^2 + 64*x^4), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {4^{-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{-1+\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 )}{x^5} \, dx \\ & = \int \left (\frac {4^{1-\frac {1}{x^4}} \left (3-128 x+256 x^3\right ) \left (16+3 x-64 x^2+64 x^4\right )^{-1+\frac {1}{x^4}}}{x^4}-\frac {4^{2-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}} \log \left (4+\frac {3 x}{4}-16 x^2+16 x^4\right )}{x^5}\right ) \, dx \\ & = \int \frac {4^{1-\frac {1}{x^4}} \left (3-128 x+256 x^3\right ) \left (16+3 x-64 x^2+64 x^4\right )^{-1+\frac {1}{x^4}}}{x^4} \, dx-\int \frac {4^{2-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}} \log \left (4+\frac {3 x}{4}-16 x^2+16 x^4\right )}{x^5} \, dx \\ & = -\left (\log \left (4+\frac {3 x}{4}-16 x^2+16 x^4\right ) \int \frac {4^{2-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}}}{x^5} \, dx\right )+\int \frac {\left (3-128 x+256 x^3\right ) \left (4+\frac {3 x}{4}-16 x^2+16 x^4\right )^{-1+\frac {1}{x^4}}}{x^4} \, dx+\int \frac {\left (3-128 x+256 x^3\right ) \int \frac {4^{2-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}}}{x^5} \, dx}{16+3 x-64 x^2+64 x^4} \, dx \\ & = -\left (\log \left (4+\frac {3 x}{4}-16 x^2+16 x^4\right ) \int \frac {4^{2-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}}}{x^5} \, dx\right )+\int \left (\frac {3 \left (4+\frac {3 x}{4}-16 x^2+16 x^4\right )^{-1+\frac {1}{x^4}}}{x^4}-\frac {128 \left (4+\frac {3 x}{4}-16 x^2+16 x^4\right )^{-1+\frac {1}{x^4}}}{x^3}+\frac {256 \left (4+\frac {3 x}{4}-16 x^2+16 x^4\right )^{-1+\frac {1}{x^4}}}{x}\right ) \, dx+\int \left (\frac {3 \int \frac {4^{2-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}}}{x^5} \, dx}{16+3 x-64 x^2+64 x^4}-\frac {128 x \int \frac {4^{2-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}}}{x^5} \, dx}{16+3 x-64 x^2+64 x^4}+\frac {256 x^3 \int \frac {4^{2-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}}}{x^5} \, dx}{16+3 x-64 x^2+64 x^4}\right ) \, dx \\ & = 3 \int \frac {\left (4+\frac {3 x}{4}-16 x^2+16 x^4\right )^{-1+\frac {1}{x^4}}}{x^4} \, dx+3 \int \frac {\int \frac {4^{2-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}}}{x^5} \, dx}{16+3 x-64 x^2+64 x^4} \, dx-128 \int \frac {\left (4+\frac {3 x}{4}-16 x^2+16 x^4\right )^{-1+\frac {1}{x^4}}}{x^3} \, dx-128 \int \frac {x \int \frac {4^{2-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}}}{x^5} \, dx}{16+3 x-64 x^2+64 x^4} \, dx+256 \int \frac {\left (4+\frac {3 x}{4}-16 x^2+16 x^4\right )^{-1+\frac {1}{x^4}}}{x} \, dx+256 \int \frac {x^3 \int \frac {4^{2-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}}}{x^5} \, dx}{16+3 x-64 x^2+64 x^4} \, dx-\log \left (4+\frac {3 x}{4}-16 x^2+16 x^4\right ) \int \frac {4^{2-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}}}{x^5} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {4^{-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}} \left (12 x-512 x^2+1024 x^4+\left (-256-48 x+1024 x^2-1024 x^4\right ) \log \left (\frac {1}{4} \left (16+3 x-64 x^2+64 x^4\right )\right )\right )}{16 x^5+3 x^6-64 x^7+64 x^9} \, dx=4 \left (4+\frac {3 x}{4}-16 x^2+16 x^4\right )^{\frac {1}{x^4}} \]

[In]

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]

[Out]

4*(4 + (3*x)/4 - 16*x^2 + 16*x^4)^x^(-4)

Maple [A] (verified)

Time = 11.52 (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	\(4 \,{\mathrm e}^{\frac {\ln \left (16 x^{4}-16 x^{2}+\frac {3}{4} x +4\right )}{x^{4}}}\)	\(24\)

[In]

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)

[Out]

4*(16*x^4-16*x^2+3/4*x+4)^(1/x^4)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {4^{-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}} \left (12 x-512 x^2+1024 x^4+\left (-256-48 x+1024 x^2-1024 x^4\right ) \log \left (\frac {1}{4} \left (16+3 x-64 x^2+64 x^4\right )\right )\right )}{16 x^5+3 x^6-64 x^7+64 x^9} \, dx=4 \, {\left (16 \, x^{4} - 16 \, x^{2} + \frac {3}{4} \, x + 4\right )}^{\left (\frac {1}{x^{4}}\right )} \]

[In]

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")

[Out]

4*(16*x^4 - 16*x^2 + 3/4*x + 4)^(x^(-4))

Sympy [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {4^{-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}} \left (12 x-512 x^2+1024 x^4+\left (-256-48 x+1024 x^2-1024 x^4\right ) \log \left (\frac {1}{4} \left (16+3 x-64 x^2+64 x^4\right )\right )\right )}{16 x^5+3 x^6-64 x^7+64 x^9} \, dx=4 e^{\frac {\log {\left (16 x^{4} - 16 x^{2} + \frac {3 x}{4} + 4 \right )}}{x^{4}}} \]

[In]

integrate(((-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)

[Out]

4*exp(log(16*x**4 - 16*x**2 + 3*x/4 + 4)/x**4)

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {4^{-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}} \left (12 x-512 x^2+1024 x^4+\left (-256-48 x+1024 x^2-1024 x^4\right ) \log \left (\frac {1}{4} \left (16+3 x-64 x^2+64 x^4\right )\right )\right )}{16 x^5+3 x^6-64 x^7+64 x^9} \, dx=4 \, e^{\left (-\frac {2 \, \log \left (2\right )}{x^{4}} + \frac {\log \left (64 \, x^{4} - 64 \, x^{2} + 3 \, x + 16\right )}{x^{4}}\right )} \]

[In]

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")

[Out]

4*e^(-2*log(2)/x^4 + log(64*x^4 - 64*x^2 + 3*x + 16)/x^4)

Giac [F]

\[ \int \frac {4^{-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}} \left (12 x-512 x^2+1024 x^4+\left (-256-48 x+1024 x^2-1024 x^4\right ) \log \left (\frac {1}{4} \left (16+3 x-64 x^2+64 x^4\right )\right )\right )}{16 x^5+3 x^6-64 x^7+64 x^9} \, dx=\int { \frac {4 \, {\left (256 \, x^{4} - 128 \, x^{2} - 4 \, {\left (64 \, x^{4} - 64 \, x^{2} + 3 \, x + 16\right )} \log \left (16 \, x^{4} - 16 \, x^{2} + \frac {3}{4} \, x + 4\right ) + 3 \, x\right )} {\left (16 \, x^{4} - 16 \, x^{2} + \frac {3}{4} \, x + 4\right )}^{\left (\frac {1}{x^{4}}\right )}}{64 \, x^{9} - 64 \, x^{7} + 3 \, x^{6} + 16 \, x^{5}} \,d x } \]

[In]

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")

[Out]

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)

Mupad [B] (verification not implemented)

Time = 18.55 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {4^{-\frac {1}{x^4}} \left (16+3 x-64 x^2+64 x^4\right )^{\frac {1}{x^4}} \left (12 x-512 x^2+1024 x^4+\left (-256-48 x+1024 x^2-1024 x^4\right ) \log \left (\frac {1}{4} \left (16+3 x-64 x^2+64 x^4\right )\right )\right )}{16 x^5+3 x^6-64 x^7+64 x^9} \, dx=4\,{\left (16\,x^4-16\,x^2+\frac {3\,x}{4}+4\right )}^{\frac {1}{x^4}} \]

[In]

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)

[Out]

4*((3*x)/4 - 16*x^2 + 16*x^4 + 4)^(1/x^4)