\(\int \frac {(\frac {x+4 \log (\frac {(-4+4 x) \log (4)}{5 x})}{\log (\frac {(-4+4 x) \log (4)}{5 x})})^{\frac {1}{\log (x^2)}} (-x \log (x^2)+(-x+x^2) \log (x^2) \log (\frac {(-4+4 x) \log (4)}{5 x})+((2 x-2 x^2) \log (\frac {(-4+4 x) \log (4)}{5 x})+(8-8 x) \log ^2(\frac {(-4+4 x) \log (4)}{5 x})) \log (\frac {x+4 \log (\frac {(-4+4 x) \log (4)}{5 x})}{\log (\frac {(-4+4 x) \log (4)}{5 x})}))}{(-x^2+x^3) \log ^2(x^2) \log (\frac {(-4+4 x) \log (4)}{5 x})+(-4 x+4 x^2) \log ^2(x^2) \log ^2(\frac {(-4+4 x) \log (4)}{5 x})} \, dx\) [1049]

Optimal result

Integrand size = 235, antiderivative size = 26 \[ \int \frac {\left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )^{\frac {1}{\log \left (x^2\right )}} \left (-x \log \left (x^2\right )+\left (-x+x^2\right ) \log \left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (\left (2 x-2 x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+(8-8 x) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )\right ) \log \left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )\right )}{\left (-x^2+x^3\right ) \log ^2\left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (-4 x+4 x^2\right ) \log ^2\left (x^2\right ) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )} \, dx=\left (4+\frac {x}{\log \left (\frac {4 (-1+x) \log (4)}{5 x}\right )}\right )^{\frac {1}{\log \left (x^2\right )}} \]

[Out]

exp(ln(4+x/ln(8/5*(-1+x)/x*ln(2)))/ln(x^2))

Rubi [F]

\[ \int \frac {\left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )^{\frac {1}{\log \left (x^2\right )}} \left (-x \log \left (x^2\right )+\left (-x+x^2\right ) \log \left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (\left (2 x-2 x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+(8-8 x) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )\right ) \log \left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )\right )}{\left (-x^2+x^3\right ) \log ^2\left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (-4 x+4 x^2\right ) \log ^2\left (x^2\right ) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )} \, dx=\int \frac {\left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )^{\frac {1}{\log \left (x^2\right )}} \left (-x \log \left (x^2\right )+\left (-x+x^2\right ) \log \left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (\left (2 x-2 x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+(8-8 x) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )\right ) \log \left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )\right )}{\left (-x^2+x^3\right ) \log ^2\left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (-4 x+4 x^2\right ) \log ^2\left (x^2\right ) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )} \, dx \]

[In]

Int[(((x + 4*Log[((-4 + 4*x)*Log[4])/(5*x)])/Log[((-4 + 4*x)*Log[4])/(5*x)])^Log[x^2]^(-1)*(-(x*Log[x^2]) + (-
x + x^2)*Log[x^2]*Log[((-4 + 4*x)*Log[4])/(5*x)] + ((2*x - 2*x^2)*Log[((-4 + 4*x)*Log[4])/(5*x)] + (8 - 8*x)*L
og[((-4 + 4*x)*Log[4])/(5*x)]^2)*Log[(x + 4*Log[((-4 + 4*x)*Log[4])/(5*x)])/Log[((-4 + 4*x)*Log[4])/(5*x)]]))/
((-x^2 + x^3)*Log[x^2]^2*Log[((-4 + 4*x)*Log[4])/(5*x)] + (-4*x + 4*x^2)*Log[x^2]^2*Log[((-4 + 4*x)*Log[4])/(5
*x)]^2),x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )^{\frac {1}{\log \left (x^2\right )}} \left (-x \log \left (x^2\right )+\left (-x+x^2\right ) \log \left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (\left (2 x-2 x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+(8-8 x) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )\right ) \log \left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )\right )}{\left (-x^2+x^3\right ) \log ^2\left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (-4 x+4 x^2\right ) \log ^2\left (x^2\right ) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )} \, dx=\left (4+\frac {x}{\log \left (\frac {4 (-1+x) \log (4)}{5 x}\right )}\right )^{\frac {1}{\log \left (x^2\right )}} \]

[In]

Integrate[(((x + 4*Log[((-4 + 4*x)*Log[4])/(5*x)])/Log[((-4 + 4*x)*Log[4])/(5*x)])^Log[x^2]^(-1)*(-(x*Log[x^2]
) + (-x + x^2)*Log[x^2]*Log[((-4 + 4*x)*Log[4])/(5*x)] + ((2*x - 2*x^2)*Log[((-4 + 4*x)*Log[4])/(5*x)] + (8 -
8*x)*Log[((-4 + 4*x)*Log[4])/(5*x)]^2)*Log[(x + 4*Log[((-4 + 4*x)*Log[4])/(5*x)])/Log[((-4 + 4*x)*Log[4])/(5*x
)]]))/((-x^2 + x^3)*Log[x^2]^2*Log[((-4 + 4*x)*Log[4])/(5*x)] + (-4*x + 4*x^2)*Log[x^2]^2*Log[((-4 + 4*x)*Log[
4])/(5*x)]^2),x]

[Out]

(4 + x/Log[(4*(-1 + x)*Log[4])/(5*x)])^Log[x^2]^(-1)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.20 (sec) , antiderivative size = 1267, normalized size of antiderivative = 48.73

\[\text {Expression too large to display}\]

[In]

int((((-8*x+8)*ln(2/5*(-4+4*x)*ln(2)/x)^2+(-2*x^2+2*x)*ln(2/5*(-4+4*x)*ln(2)/x))*ln((4*ln(2/5*(-4+4*x)*ln(2)/x
)+x)/ln(2/5*(-4+4*x)*ln(2)/x))+(x^2-x)*ln(x^2)*ln(2/5*(-4+4*x)*ln(2)/x)-x*ln(x^2))*exp(ln((4*ln(2/5*(-4+4*x)*l
n(2)/x)+x)/ln(2/5*(-4+4*x)*ln(2)/x))/ln(x^2))/((4*x^2-4*x)*ln(x^2)^2*ln(2/5*(-4+4*x)*ln(2)/x)^2+(x^3-x^2)*ln(x
^2)^2*ln(2/5*(-4+4*x)*ln(2)/x)),x)

[Out]

16^(-1/2/(I*Pi*csgn(I*x^2)-I*Pi*csgn(I*x)-2*ln(x)))*(-2*I*ln(x)+2*I*ln(-1+x)-Pi*csgn(I/x)*csgn(I*(-1+x)/x)^2-P
i*csgn(I*(-1+x))*csgn(I*(-1+x)/x)^2+Pi*csgn(I*(-1+x)/x)^3+Pi*csgn(I/x)*csgn(I*(-1+x))*csgn(I*(-1+x)/x)+2*I*(3*
ln(2)-ln(5)+ln(ln(2))))^(1/(I*Pi*csgn(I*x^2)-I*Pi*csgn(I*x)-2*ln(x)))*(1/2*I*x-2*I*ln(x)+2*I*ln(-1+x)-Pi*csgn(
I/x)*csgn(I*(-1+x)/x)^2-Pi*csgn(I*(-1+x))*csgn(I*(-1+x)/x)^2+Pi*csgn(I*(-1+x)/x)^3+Pi*csgn(I/x)*csgn(I*(-1+x))
*csgn(I*(-1+x)/x)+2*I*(3*ln(2)-ln(5)+ln(ln(2))))^(-1/(I*Pi*csgn(I*x^2)-I*Pi*csgn(I*x)-2*ln(x)))*exp(I*Pi*csgn(
I*(-1/2*I*x+2*I*ln(x)-2*I*ln(-1+x)+Pi*csgn(I/x)*csgn(I*(-1+x)/x)^2+Pi*csgn(I*(-1+x))*csgn(I*(-1+x)/x)^2-Pi*csg
n(I*(-1+x)/x)^3-Pi*csgn(I/x)*csgn(I*(-1+x))*csgn(I*(-1+x)/x)-2*I*ln(8/5*ln(2)))/(Pi*csgn(I/x)*csgn(I*(-1+x)/x)
^2-Pi*csgn(I/x)*csgn(I*(-1+x))*csgn(I*(-1+x)/x)-Pi*csgn(I*(-1+x)/x)^3+Pi*csgn(I*(-1+x))*csgn(I*(-1+x)/x)^2+2*I
*ln(x)-2*I*ln(8/5*ln(2))-2*I*ln(-1+x)))*(-csgn(I*(-1/2*I*x+2*I*ln(x)-2*I*ln(-1+x)+Pi*csgn(I/x)*csgn(I*(-1+x)/x
)^2+Pi*csgn(I*(-1+x))*csgn(I*(-1+x)/x)^2-Pi*csgn(I*(-1+x)/x)^3-Pi*csgn(I/x)*csgn(I*(-1+x))*csgn(I*(-1+x)/x)-2*
I*ln(8/5*ln(2)))/(Pi*csgn(I/x)*csgn(I*(-1+x)/x)^2-Pi*csgn(I/x)*csgn(I*(-1+x))*csgn(I*(-1+x)/x)-Pi*csgn(I*(-1+x
)/x)^3+Pi*csgn(I*(-1+x))*csgn(I*(-1+x)/x)^2+2*I*ln(x)-2*I*ln(8/5*ln(2))-2*I*ln(-1+x)))+csgn(-1/2*x+2*ln(x)-2*l
n(-1+x)-I*Pi*csgn(I/x)*csgn(I*(-1+x)/x)^2-I*Pi*csgn(I*(-1+x))*csgn(I*(-1+x)/x)^2+I*Pi*csgn(I*(-1+x)/x)^3+I*Pi*
csgn(I/x)*csgn(I*(-1+x))*csgn(I*(-1+x)/x)-2*ln(8/5*ln(2))))*(csgn(I*(-1/2*I*x+2*I*ln(x)-2*I*ln(-1+x)+Pi*csgn(I
/x)*csgn(I*(-1+x)/x)^2+Pi*csgn(I*(-1+x))*csgn(I*(-1+x)/x)^2-Pi*csgn(I*(-1+x)/x)^3-Pi*csgn(I/x)*csgn(I*(-1+x))*
csgn(I*(-1+x)/x)-2*I*ln(8/5*ln(2)))/(Pi*csgn(I/x)*csgn(I*(-1+x)/x)^2-Pi*csgn(I/x)*csgn(I*(-1+x))*csgn(I*(-1+x)
/x)-Pi*csgn(I*(-1+x)/x)^3+Pi*csgn(I*(-1+x))*csgn(I*(-1+x)/x)^2+2*I*ln(x)-2*I*ln(8/5*ln(2))-2*I*ln(-1+x)))+csgn
(I/(Pi*csgn(I/x)*csgn(I*(-1+x)/x)^2-Pi*csgn(I/x)*csgn(I*(-1+x))*csgn(I*(-1+x)/x)-Pi*csgn(I*(-1+x)/x)^3+Pi*csgn
(I*(-1+x))*csgn(I*(-1+x)/x)^2+2*I*ln(x)-2*I*ln(8/5*ln(2))-2*I*ln(-1+x))))/(-I*Pi*csgn(I*x^2)^3+2*I*Pi*csgn(I*x
^2)^2*csgn(I*x)-I*Pi*csgn(I*x^2)*csgn(I*x)^2+4*ln(x)))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {\left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )^{\frac {1}{\log \left (x^2\right )}} \left (-x \log \left (x^2\right )+\left (-x+x^2\right ) \log \left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (\left (2 x-2 x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+(8-8 x) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )\right ) \log \left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )\right )}{\left (-x^2+x^3\right ) \log ^2\left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (-4 x+4 x^2\right ) \log ^2\left (x^2\right ) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )} \, dx=\left (\frac {x + 4 \, \log \left (\frac {8 \, {\left (x - 1\right )} \log \left (2\right )}{5 \, x}\right )}{\log \left (\frac {8 \, {\left (x - 1\right )} \log \left (2\right )}{5 \, x}\right )}\right )^{\left (\frac {1}{\log \left (x^{2}\right )}\right )} \]

[In]

integrate((((-8*x+8)*log(2/5*(-4+4*x)*log(2)/x)^2+(-2*x^2+2*x)*log(2/5*(-4+4*x)*log(2)/x))*log((4*log(2/5*(-4+
4*x)*log(2)/x)+x)/log(2/5*(-4+4*x)*log(2)/x))+(x^2-x)*log(x^2)*log(2/5*(-4+4*x)*log(2)/x)-x*log(x^2))*exp(log(
(4*log(2/5*(-4+4*x)*log(2)/x)+x)/log(2/5*(-4+4*x)*log(2)/x))/log(x^2))/((4*x^2-4*x)*log(x^2)^2*log(2/5*(-4+4*x
)*log(2)/x)^2+(x^3-x^2)*log(x^2)^2*log(2/5*(-4+4*x)*log(2)/x)),x, algorithm="fricas")

[Out]

((x + 4*log(8/5*(x - 1)*log(2)/x))/log(8/5*(x - 1)*log(2)/x))^(1/log(x^2))

Sympy [F(-2)]

Exception generated. \[ \int \frac {\left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )^{\frac {1}{\log \left (x^2\right )}} \left (-x \log \left (x^2\right )+\left (-x+x^2\right ) \log \left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (\left (2 x-2 x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+(8-8 x) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )\right ) \log \left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )\right )}{\left (-x^2+x^3\right ) \log ^2\left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (-4 x+4 x^2\right ) \log ^2\left (x^2\right ) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((((-8*x+8)*ln(2/5*(-4+4*x)*ln(2)/x)**2+(-2*x**2+2*x)*ln(2/5*(-4+4*x)*ln(2)/x))*ln((4*ln(2/5*(-4+4*x)
*ln(2)/x)+x)/ln(2/5*(-4+4*x)*ln(2)/x))+(x**2-x)*ln(x**2)*ln(2/5*(-4+4*x)*ln(2)/x)-x*ln(x**2))*exp(ln((4*ln(2/5
*(-4+4*x)*ln(2)/x)+x)/ln(2/5*(-4+4*x)*ln(2)/x))/ln(x**2))/((4*x**2-4*x)*ln(x**2)**2*ln(2/5*(-4+4*x)*ln(2)/x)**
2+(x**3-x**2)*ln(x**2)**2*ln(2/5*(-4+4*x)*ln(2)/x)),x)

[Out]

Exception raised: TypeError >> '>' not supported between instances of 'Poly' and 'int'

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (24) = 48\).

Time = 0.67 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.35 \[ \int \frac {\left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )^{\frac {1}{\log \left (x^2\right )}} \left (-x \log \left (x^2\right )+\left (-x+x^2\right ) \log \left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (\left (2 x-2 x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+(8-8 x) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )\right ) \log \left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )\right )}{\left (-x^2+x^3\right ) \log ^2\left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (-4 x+4 x^2\right ) \log ^2\left (x^2\right ) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )} \, dx=e^{\left (\frac {\log \left (x - 4 \, \log \left (5\right ) + 12 \, \log \left (2\right ) + 4 \, \log \left (x - 1\right ) - 4 \, \log \left (x\right ) + 4 \, \log \left (\log \left (2\right )\right )\right )}{2 \, \log \left (x\right )} - \frac {\log \left (-\log \left (5\right ) + 3 \, \log \left (2\right ) + \log \left (x - 1\right ) - \log \left (x\right ) + \log \left (\log \left (2\right )\right )\right )}{2 \, \log \left (x\right )}\right )} \]

[In]

integrate((((-8*x+8)*log(2/5*(-4+4*x)*log(2)/x)^2+(-2*x^2+2*x)*log(2/5*(-4+4*x)*log(2)/x))*log((4*log(2/5*(-4+
4*x)*log(2)/x)+x)/log(2/5*(-4+4*x)*log(2)/x))+(x^2-x)*log(x^2)*log(2/5*(-4+4*x)*log(2)/x)-x*log(x^2))*exp(log(
(4*log(2/5*(-4+4*x)*log(2)/x)+x)/log(2/5*(-4+4*x)*log(2)/x))/log(x^2))/((4*x^2-4*x)*log(x^2)^2*log(2/5*(-4+4*x
)*log(2)/x)^2+(x^3-x^2)*log(x^2)^2*log(2/5*(-4+4*x)*log(2)/x)),x, algorithm="maxima")

[Out]

e^(1/2*log(x - 4*log(5) + 12*log(2) + 4*log(x - 1) - 4*log(x) + 4*log(log(2)))/log(x) - 1/2*log(-log(5) + 3*lo
g(2) + log(x - 1) - log(x) + log(log(2)))/log(x))

Giac [F]

\[ \int \frac {\left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )^{\frac {1}{\log \left (x^2\right )}} \left (-x \log \left (x^2\right )+\left (-x+x^2\right ) \log \left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (\left (2 x-2 x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+(8-8 x) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )\right ) \log \left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )\right )}{\left (-x^2+x^3\right ) \log ^2\left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (-4 x+4 x^2\right ) \log ^2\left (x^2\right ) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )} \, dx=\int { \frac {{\left ({\left (x^{2} - x\right )} \log \left (x^{2}\right ) \log \left (\frac {8 \, {\left (x - 1\right )} \log \left (2\right )}{5 \, x}\right ) - x \log \left (x^{2}\right ) - 2 \, {\left (4 \, {\left (x - 1\right )} \log \left (\frac {8 \, {\left (x - 1\right )} \log \left (2\right )}{5 \, x}\right )^{2} + {\left (x^{2} - x\right )} \log \left (\frac {8 \, {\left (x - 1\right )} \log \left (2\right )}{5 \, x}\right )\right )} \log \left (\frac {x + 4 \, \log \left (\frac {8 \, {\left (x - 1\right )} \log \left (2\right )}{5 \, x}\right )}{\log \left (\frac {8 \, {\left (x - 1\right )} \log \left (2\right )}{5 \, x}\right )}\right )\right )} \left (\frac {x + 4 \, \log \left (\frac {8 \, {\left (x - 1\right )} \log \left (2\right )}{5 \, x}\right )}{\log \left (\frac {8 \, {\left (x - 1\right )} \log \left (2\right )}{5 \, x}\right )}\right )^{\left (\frac {1}{\log \left (x^{2}\right )}\right )}}{4 \, {\left (x^{2} - x\right )} \log \left (x^{2}\right )^{2} \log \left (\frac {8 \, {\left (x - 1\right )} \log \left (2\right )}{5 \, x}\right )^{2} + {\left (x^{3} - x^{2}\right )} \log \left (x^{2}\right )^{2} \log \left (\frac {8 \, {\left (x - 1\right )} \log \left (2\right )}{5 \, x}\right )} \,d x } \]

[In]

integrate((((-8*x+8)*log(2/5*(-4+4*x)*log(2)/x)^2+(-2*x^2+2*x)*log(2/5*(-4+4*x)*log(2)/x))*log((4*log(2/5*(-4+
4*x)*log(2)/x)+x)/log(2/5*(-4+4*x)*log(2)/x))+(x^2-x)*log(x^2)*log(2/5*(-4+4*x)*log(2)/x)-x*log(x^2))*exp(log(
(4*log(2/5*(-4+4*x)*log(2)/x)+x)/log(2/5*(-4+4*x)*log(2)/x))/log(x^2))/((4*x^2-4*x)*log(x^2)^2*log(2/5*(-4+4*x
)*log(2)/x)^2+(x^3-x^2)*log(x^2)^2*log(2/5*(-4+4*x)*log(2)/x)),x, algorithm="giac")

[Out]

integrate(((x^2 - x)*log(x^2)*log(8/5*(x - 1)*log(2)/x) - x*log(x^2) - 2*(4*(x - 1)*log(8/5*(x - 1)*log(2)/x)^
2 + (x^2 - x)*log(8/5*(x - 1)*log(2)/x))*log((x + 4*log(8/5*(x - 1)*log(2)/x))/log(8/5*(x - 1)*log(2)/x)))*((x
+ 4*log(8/5*(x - 1)*log(2)/x))/log(8/5*(x - 1)*log(2)/x))^(1/log(x^2))/(4*(x^2 - x)*log(x^2)^2*log(8/5*(x - 1
)*log(2)/x)^2 + (x^3 - x^2)*log(x^2)^2*log(8/5*(x - 1)*log(2)/x)), x)

Mupad [B] (verification not implemented)

Time = 9.77 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {\left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )^{\frac {1}{\log \left (x^2\right )}} \left (-x \log \left (x^2\right )+\left (-x+x^2\right ) \log \left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (\left (2 x-2 x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+(8-8 x) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )\right ) \log \left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )\right )}{\left (-x^2+x^3\right ) \log ^2\left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (-4 x+4 x^2\right ) \log ^2\left (x^2\right ) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )} \, dx={\left (\frac {x}{\ln \left (-\frac {8\,\ln \left (2\right )-8\,x\,\ln \left (2\right )}{5\,x}\right )}+4\right )}^{\frac {1}{\ln \left (x^2\right )}} \]

[In]

int((exp(log((x + 4*log((2*log(2)*(4*x - 4))/(5*x)))/log((2*log(2)*(4*x - 4))/(5*x)))/log(x^2))*(x*log(x^2) -
log((x + 4*log((2*log(2)*(4*x - 4))/(5*x)))/log((2*log(2)*(4*x - 4))/(5*x)))*(log((2*log(2)*(4*x - 4))/(5*x))*
(2*x - 2*x^2) - log((2*log(2)*(4*x - 4))/(5*x))^2*(8*x - 8)) + log(x^2)*log((2*log(2)*(4*x - 4))/(5*x))*(x - x
^2)))/(log(x^2)^2*log((2*log(2)*(4*x - 4))/(5*x))^2*(4*x - 4*x^2) + log(x^2)^2*log((2*log(2)*(4*x - 4))/(5*x))
*(x^2 - x^3)),x)

[Out]

(x/log(-(8*log(2) - 8*x*log(2))/(5*x)) + 4)^(1/log(x^2))