\(\int \frac {4^{\frac {1}{x}} (\frac {1}{1-4 x+6 x^2-4 x^3+x^4})^{\frac {1}{x}} (-4 x-3 x^2-x^3+(1-x^2) \log (\frac {4}{1-4 x+6 x^2-4 x^3+x^4}))}{-2 x^2-2 x^3+2 x^4+2 x^5} \, dx\) [348]

Optimal result

Integrand size = 98, antiderivative size = 26 \[ \int \frac {4^{\frac {1}{x}} \left (\frac {1}{1-4 x+6 x^2-4 x^3+x^4}\right )^{\frac {1}{x}} \left (-4 x-3 x^2-x^3+\left (1-x^2\right ) \log \left (\frac {4}{1-4 x+6 x^2-4 x^3+x^4}\right )\right )}{-2 x^2-2 x^3+2 x^4+2 x^5} \, dx=\frac {2^{-1+\frac {2}{x}} \left (\frac {1}{(1-x)^4}\right )^{\frac {1}{x}}}{1+x} \] Output:

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

Mathematica [F]

\[ \int \frac {4^{\frac {1}{x}} \left (\frac {1}{1-4 x+6 x^2-4 x^3+x^4}\right )^{\frac {1}{x}} \left (-4 x-3 x^2-x^3+\left (1-x^2\right ) \log \left (\frac {4}{1-4 x+6 x^2-4 x^3+x^4}\right )\right )}{-2 x^2-2 x^3+2 x^4+2 x^5} \, dx=\int \frac {4^{\frac {1}{x}} \left (\frac {1}{1-4 x+6 x^2-4 x^3+x^4}\right )^{\frac {1}{x}} \left (-4 x-3 x^2-x^3+\left (1-x^2\right ) \log \left (\frac {4}{1-4 x+6 x^2-4 x^3+x^4}\right )\right )}{-2 x^2-2 x^3+2 x^4+2 x^5} \, dx \] Input:

Integrate[(4^x^(-1)*((1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(-1))^x^(-1)*(-4*x - 
3*x^2 - x^3 + (1 - x^2)*Log[4/(1 - 4*x + 6*x^2 - 4*x^3 + x^4)]))/(-2*x^2 - 
 2*x^3 + 2*x^4 + 2*x^5),x]
 

Output:

Integrate[(4^x^(-1)*((1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(-1))^x^(-1)*(-4*x - 
3*x^2 - x^3 + (1 - x^2)*Log[4/(1 - 4*x + 6*x^2 - 4*x^3 + x^4)]))/(-2*x^2 - 
 2*x^3 + 2*x^4 + 2*x^5), x]
 

Rubi [F]

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.

\(\displaystyle \int \frac {4^{\frac {1}{x}} \left (\frac {1}{x^4-4 x^3+6 x^2-4 x+1}\right )^{\frac {1}{x}} \left (-x^3-3 x^2+\left (1-x^2\right ) \log \left (\frac {4}{x^4-4 x^3+6 x^2-4 x+1}\right )-4 x\right )}{2 x^5+2 x^4-2 x^3-2 x^2} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {4^{\frac {1}{x}} \left (\frac {1}{x^4-4 x^3+6 x^2-4 x+1}\right )^{\frac {1}{x}} \left (-x^3-3 x^2+\left (1-x^2\right ) \log \left (\frac {4}{x^4-4 x^3+6 x^2-4 x+1}\right )-4 x\right )}{x^2 \left (2 x^3+2 x^2-2 x-2\right )}dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (\frac {4^{\frac {1}{x}-1} \left (\frac {1}{x^4-4 x^3+6 x^2-4 x+1}\right )^{\frac {1}{x}} \left (-x^3-3 x^2+\left (1-x^2\right ) \log \left (\frac {4}{x^4-4 x^3+6 x^2-4 x+1}\right )-4 x\right )}{x^2 \left (x^2-1\right )}-\frac {4^{\frac {1}{x}-1} \left (\frac {1}{x^4-4 x^3+6 x^2-4 x+1}\right )^{\frac {1}{x}} \left (-x^3-3 x^2+\left (1-x^2\right ) \log \left (\frac {4}{x^4-4 x^3+6 x^2-4 x+1}\right )-4 x\right )}{x^2 (x+1)^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -8 \int \frac {\int \frac {4^{\frac {1}{x}-1} \left (\frac {1}{(x-1)^4}\right )^{\frac {1}{x}}}{x^2}dx}{x-1}dx-2 \log \left (\frac {4}{(1-x)^4}\right ) \int \frac {4^{\frac {1}{x}-1} \left (\frac {1}{(x-1)^4}\right )^{\frac {1}{x}}}{x^2}dx+\int \frac {4^{\frac {1}{x}-1} \left (\frac {1}{(x-1)^4}\right )^{\frac {1}{x}}}{-x-1}dx-\int \frac {4^{\frac {1}{x}} \left (\frac {1}{(x-1)^4}\right )^{\frac {1}{x}}}{x-1}dx+2 \int \frac {4^{\frac {1}{x}} \left (\frac {1}{(x-1)^4}\right )^{\frac {1}{x}}}{x}dx-\int \frac {2^{\frac {2}{x}-1} \left (\frac {1}{(x-1)^4}\right )^{\frac {1}{x}}}{(x+1)^2}dx-3 \int \frac {4^{\frac {1}{x}-1} \left (\frac {1}{(x-1)^4}\right )^{\frac {1}{x}}}{x+1}dx+4 \int \frac {\int \frac {2^{\frac {2}{x}-1} \left (\frac {1}{(x-1)^4}\right )^{\frac {1}{x}}}{x}dx}{x-1}dx-4 \int \frac {\int \frac {2^{\frac {2}{x}-1} \left (\frac {1}{(x-1)^4}\right )^{\frac {1}{x}}}{x+1}dx}{x-1}dx+\log \left (\frac {4}{(1-x)^4}\right ) \int \frac {2^{\frac {2}{x}-1} \left (\frac {1}{(x-1)^4}\right )^{\frac {1}{x}}}{x}dx-\log \left (\frac {4}{(1-x)^4}\right ) \int \frac {2^{\frac {2}{x}-1} \left (\frac {1}{(x-1)^4}\right )^{\frac {1}{x}}}{x+1}dx\)

Input:

Int[(4^x^(-1)*((1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(-1))^x^(-1)*(-4*x - 3*x^2 
- x^3 + (1 - x^2)*Log[4/(1 - 4*x + 6*x^2 - 4*x^3 + x^4)]))/(-2*x^2 - 2*x^3 
 + 2*x^4 + 2*x^5),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.39 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31

method	result	size
risch	\(\frac {\left (\frac {4}{x^{4}-4 x^{3}+6 x^{2}-4 x +1}\right )^{\frac {1}{x}}}{2+2 x}\)	\(34\)
parallelrisch	\(\frac {{\mathrm e}^{\frac {\ln \left (\frac {4}{x^{4}-4 x^{3}+6 x^{2}-4 x +1}\right )}{x}}}{2+2 x}\)	\(36\)

Input:

int(((-x^2+1)*ln(4/(x^4-4*x^3+6*x^2-4*x+1))-x^3-3*x^2-4*x)*exp(ln(4/(x^4-4 
*x^3+6*x^2-4*x+1))/x)/(2*x^5+2*x^4-2*x^3-2*x^2),x,method=_RETURNVERBOSE)
 

Output:

1/2/(1+x)*(4/(x^4-4*x^3+6*x^2-4*x+1))^(1/x)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27 \[ \int \frac {4^{\frac {1}{x}} \left (\frac {1}{1-4 x+6 x^2-4 x^3+x^4}\right )^{\frac {1}{x}} \left (-4 x-3 x^2-x^3+\left (1-x^2\right ) \log \left (\frac {4}{1-4 x+6 x^2-4 x^3+x^4}\right )\right )}{-2 x^2-2 x^3+2 x^4+2 x^5} \, dx=\frac {\left (\frac {4}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right )^{\left (\frac {1}{x}\right )}}{2 \, {\left (x + 1\right )}} \] Input:

integrate(((-x^2+1)*log(4/(x^4-4*x^3+6*x^2-4*x+1))-x^3-3*x^2-4*x)*exp(log( 
4/(x^4-4*x^3+6*x^2-4*x+1))/x)/(2*x^5+2*x^4-2*x^3-2*x^2),x, algorithm="fric 
as")
 

Output:

1/2*(4/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1))^(1/x)/(x + 1)
 

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {4^{\frac {1}{x}} \left (\frac {1}{1-4 x+6 x^2-4 x^3+x^4}\right )^{\frac {1}{x}} \left (-4 x-3 x^2-x^3+\left (1-x^2\right ) \log \left (\frac {4}{1-4 x+6 x^2-4 x^3+x^4}\right )\right )}{-2 x^2-2 x^3+2 x^4+2 x^5} \, dx=\frac {e^{\frac {\log {\left (\frac {4}{x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1} \right )}}{x}}}{2 x + 2} \] Input:

integrate(((-x**2+1)*ln(4/(x**4-4*x**3+6*x**2-4*x+1))-x**3-3*x**2-4*x)*exp 
(ln(4/(x**4-4*x**3+6*x**2-4*x+1))/x)/(2*x**5+2*x**4-2*x**3-2*x**2),x)
 

Output:

exp(log(4/(x**4 - 4*x**3 + 6*x**2 - 4*x + 1))/x)/(2*x + 2)
 

Maxima [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {4^{\frac {1}{x}} \left (\frac {1}{1-4 x+6 x^2-4 x^3+x^4}\right )^{\frac {1}{x}} \left (-4 x-3 x^2-x^3+\left (1-x^2\right ) \log \left (\frac {4}{1-4 x+6 x^2-4 x^3+x^4}\right )\right )}{-2 x^2-2 x^3+2 x^4+2 x^5} \, dx=\frac {e^{\left (\frac {2 \, \log \left (2\right )}{x} - \frac {4 \, \log \left (x - 1\right )}{x}\right )}}{2 \, {\left (x + 1\right )}} \] Input:

integrate(((-x^2+1)*log(4/(x^4-4*x^3+6*x^2-4*x+1))-x^3-3*x^2-4*x)*exp(log( 
4/(x^4-4*x^3+6*x^2-4*x+1))/x)/(2*x^5+2*x^4-2*x^3-2*x^2),x, algorithm="maxi 
ma")
 

Output:

1/2*e^(2*log(2)/x - 4*log(x - 1)/x)/(x + 1)
 

Giac [F]

\[ \int \frac {4^{\frac {1}{x}} \left (\frac {1}{1-4 x+6 x^2-4 x^3+x^4}\right )^{\frac {1}{x}} \left (-4 x-3 x^2-x^3+\left (1-x^2\right ) \log \left (\frac {4}{1-4 x+6 x^2-4 x^3+x^4}\right )\right )}{-2 x^2-2 x^3+2 x^4+2 x^5} \, dx=\int { -\frac {{\left (x^{3} + 3 \, x^{2} + {\left (x^{2} - 1\right )} \log \left (\frac {4}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right ) + 4 \, x\right )} \left (\frac {4}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right )^{\left (\frac {1}{x}\right )}}{2 \, {\left (x^{5} + x^{4} - x^{3} - x^{2}\right )}} \,d x } \] Input:

integrate(((-x^2+1)*log(4/(x^4-4*x^3+6*x^2-4*x+1))-x^3-3*x^2-4*x)*exp(log( 
4/(x^4-4*x^3+6*x^2-4*x+1))/x)/(2*x^5+2*x^4-2*x^3-2*x^2),x, algorithm="giac 
")
 

Output:

integrate(-1/2*(x^3 + 3*x^2 + (x^2 - 1)*log(4/(x^4 - 4*x^3 + 6*x^2 - 4*x + 
 1)) + 4*x)*(4/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1))^(1/x)/(x^5 + x^4 - x^3 - x 
^2), x)
 

Mupad [B] (verification not implemented)

Time = 0.72 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {4^{\frac {1}{x}} \left (\frac {1}{1-4 x+6 x^2-4 x^3+x^4}\right )^{\frac {1}{x}} \left (-4 x-3 x^2-x^3+\left (1-x^2\right ) \log \left (\frac {4}{1-4 x+6 x^2-4 x^3+x^4}\right )\right )}{-2 x^2-2 x^3+2 x^4+2 x^5} \, dx=\frac {{\left (\frac {4}{x^4-4\,x^3+6\,x^2-4\,x+1}\right )}^{1/x}}{2\,\left (x+1\right )} \] Input:

int((exp(log(4/(6*x^2 - 4*x - 4*x^3 + x^4 + 1))/x)*(4*x + log(4/(6*x^2 - 4 
*x - 4*x^3 + x^4 + 1))*(x^2 - 1) + 3*x^2 + x^3))/(2*x^2 + 2*x^3 - 2*x^4 - 
2*x^5),x)
 

Output:

(4/(6*x^2 - 4*x - 4*x^3 + x^4 + 1))^(1/x)/(2*(x + 1))
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42 \[ \int \frac {4^{\frac {1}{x}} \left (\frac {1}{1-4 x+6 x^2-4 x^3+x^4}\right )^{\frac {1}{x}} \left (-4 x-3 x^2-x^3+\left (1-x^2\right ) \log \left (\frac {4}{1-4 x+6 x^2-4 x^3+x^4}\right )\right )}{-2 x^2-2 x^3+2 x^4+2 x^5} \, dx=\frac {e^{\frac {\mathrm {log}\left (\frac {4}{x^{4}-4 x^{3}+6 x^{2}-4 x +1}\right )}{x}}}{2 x +2} \] Input:

int(((-x^2+1)*log(4/(x^4-4*x^3+6*x^2-4*x+1))-x^3-3*x^2-4*x)*exp(log(4/(x^4 
-4*x^3+6*x^2-4*x+1))/x)/(2*x^5+2*x^4-2*x^3-2*x^2),x)
 

Output:

e**(log(4/(x**4 - 4*x**3 + 6*x**2 - 4*x + 1))/x)/(2*(x + 1))