### 3.105 $$\int \frac{\log (-1+4 x+4 \sqrt{(-1+x) x})}{x} \, dx$$

Optimal. Leaf size=25 $\text{CannotIntegrate}\left (\frac{\log \left (4 \sqrt{x^2-x}+4 x-1\right )}{x},x\right )$

[Out]

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

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Rubi [A]  time = 0.0763766, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0., Rules used = {} $\int \frac{\log \left (-1+4 x+4 \sqrt{(-1+x) x}\right )}{x} \, dx$

Veriﬁcation is Not applicable to the result.

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

Mathematica [A]  time = 0.453661, size = 0, normalized size = 0. $\int \frac{\log \left (-1+4 x+4 \sqrt{(-1+x) x}\right )}{x} \, dx$

Veriﬁcation is Not applicable to the result.

[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]

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Maple [A]  time = 0.005, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}\ln \left ( -1+4\,x+4\,\sqrt{ \left ( -1+x \right ) x} \right ) }\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (4 \, x + 4 \, \sqrt{{\left (x - 1\right )} x} - 1\right )}{x}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (4 \, x + 4 \, \sqrt{x^{2} - x} - 1\right )}{x}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (4 \, x + 4 \, \sqrt{{\left (x - 1\right )} x} - 1\right )}{x}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)