\(\int \frac {\log (-1+4 x+4 \sqrt {(-1+x) x})}{x} \, dx\) [105]

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int \frac {\log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{x} \, dx=\text {Int}\left (\frac {\log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{x},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{x} \, dx=\int \frac {\log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{x} \, dx \]

[In]

Int[Log[-1 + 4*x + 4*Sqrt[(-1 + x)*x]]/x,x]

[Out]

Defer[Int][Log[-1 + 4*x + 4*Sqrt[-x + x^2]]/x, x]

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

Mathematica [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{x} \, dx=\int \frac {\log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{x} \, dx \]

[In]

Integrate[Log[-1 + 4*x + 4*Sqrt[(-1 + x)*x]]/x,x]

[Out]

Integrate[Log[-1 + 4*x + 4*Sqrt[(-1 + x)*x]]/x, x]

Maple [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90

\[\int \frac {\ln \left (-1+4 x +4 \sqrt {\left (-1+x \right ) x}\right )}{x}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{x} \, dx=\int { \frac {\log \left (4 \, x + 4 \, \sqrt {{\left (x - 1\right )} x} - 1\right )}{x} \,d x } \]

[In]

integrate(log(-1+4*x+4*((-1+x)*x)^(1/2))/x,x, algorithm="fricas")

[Out]

integral(log(4*x + 4*sqrt(x^2 - x) - 1)/x, x)

Sympy [N/A]

Not integrable

Time = 44.31 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {\log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{x} \, dx=\int \frac {\log {\left (4 x + 4 \sqrt {x^{2} - x} - 1 \right )}}{x}\, dx \]

[In]

integrate(ln(-1+4*x+4*((-1+x)*x)**(1/2))/x,x)

[Out]

Integral(log(4*x + 4*sqrt(x**2 - x) - 1)/x, x)

Maxima [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{x} \, dx=\int { \frac {\log \left (4 \, x + 4 \, \sqrt {{\left (x - 1\right )} x} - 1\right )}{x} \,d x } \]

[In]

integrate(log(-1+4*x+4*((-1+x)*x)^(1/2))/x,x, algorithm="maxima")

[Out]

integrate(log(4*x + 4*sqrt((x - 1)*x) - 1)/x, x)

Giac [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{x} \, dx=\int { \frac {\log \left (4 \, x + 4 \, \sqrt {{\left (x - 1\right )} x} - 1\right )}{x} \,d x } \]

[In]

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

[Out]

integrate(log(4*x + 4*sqrt((x - 1)*x) - 1)/x, x)

Mupad [N/A]

Not integrable

Time = 1.42 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{x} \, dx=\int \frac {\ln \left (4\,x+4\,\sqrt {x\,\left (x-1\right )}-1\right )}{x} \,d x \]

[In]

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

[Out]

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