\(\int \log (-1+\sqrt {\frac {1+x}{x}}) \, dx\) [297]

Optimal result

Integrand size = 14, antiderivative size = 50 \[ \int \log \left (-1+\sqrt {\frac {1+x}{x}}\right ) \, dx=-\frac {1}{2 \left (1-\sqrt {1+\frac {1}{x}}\right )}-\frac {1}{2} \text {arctanh}\left (\sqrt {1+\frac {1}{x}}\right )+x \log \left (-1+\sqrt {\frac {1+x}{x}}\right ) \]

[Out]

-1/2*arctanh((1+1/x)^(1/2))+x*ln(-1+((1+x)/x)^(1/2))-1/2/(1-(1+1/x)^(1/2))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2628, 46, 213} \[ \int \log \left (-1+\sqrt {\frac {1+x}{x}}\right ) \, dx=-\frac {1}{2} \text {arctanh}\left (\sqrt {\frac {1}{x}+1}\right )-\frac {1}{2 \left (1-\sqrt {\frac {1}{x}+1}\right )}+x \log \left (\sqrt {\frac {x+1}{x}}-1\right ) \]

[In]

Int[Log[-1 + Sqrt[(1 + x)/x]],x]

[Out]

-1/2*1/(1 - Sqrt[1 + x^(-1)]) - ArcTanh[Sqrt[1 + x^(-1)]]/2 + x*Log[-1 + Sqrt[(1 + x)/x]]

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 2628

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[x*(D[u, x]/u), x], x] /; InverseFunctionFr
eeQ[u, x]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.06 \[ \int \log \left (-1+\sqrt {\frac {1+x}{x}}\right ) \, dx=\frac {1}{2} \left (1+\sqrt {1+\frac {1}{x}}\right ) x+x \log \left (-1+\sqrt {1+\frac {1}{x}}\right )-\frac {1}{4} \log \left (1+\left (2+2 \sqrt {1+\frac {1}{x}}\right ) x\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.60

method	result	size
default	\(x \ln \left (-1+\sqrt {\frac {x +1}{x}}\right )-\frac {-2 \sqrt {\frac {x +1}{x}}\, x^{2}+\ln \left (\frac {1}{2}+x +\sqrt {x^{2}+x}\right ) \sqrt {x \left (x +1\right )}-2 \sqrt {x \left (x +1\right )}\, \sqrt {x^{2}+x}}{4 \sqrt {\frac {x +1}{x}}\, x}\)	\(80\)
parts	\(x \ln \left (-1+\sqrt {\frac {x +1}{x}}\right )-\frac {-2 \sqrt {\frac {x +1}{x}}\, x^{2}+\ln \left (\frac {1}{2}+x +\sqrt {x^{2}+x}\right ) \sqrt {x \left (x +1\right )}-2 \sqrt {x \left (x +1\right )}\, \sqrt {x^{2}+x}}{4 \sqrt {\frac {x +1}{x}}\, x}\)	\(80\)

[In]

int(ln(-1+((x+1)/x)^(1/2)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.98 \[ \int \log \left (-1+\sqrt {\frac {1+x}{x}}\right ) \, dx=\frac {1}{4} \, {\left (4 \, x + 1\right )} \log \left (\sqrt {\frac {x + 1}{x}} - 1\right ) + \frac {1}{2} \, x \sqrt {\frac {x + 1}{x}} + \frac {1}{2} \, x - \frac {1}{4} \, \log \left (\sqrt {\frac {x + 1}{x}} + 1\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 47.55 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.06 \[ \int \log \left (-1+\sqrt {\frac {1+x}{x}}\right ) \, dx=x \log {\left (\sqrt {\frac {x + 1}{x}} - 1 \right )} + \frac {\log {\left (\sqrt {1 + \frac {1}{x}} - 1 \right )}}{4} - \frac {\log {\left (\sqrt {1 + \frac {1}{x}} + 1 \right )}}{4} + \frac {1}{2 \left (\sqrt {1 + \frac {1}{x}} - 1\right )} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.36 \[ \int \log \left (-1+\sqrt {\frac {1+x}{x}}\right ) \, dx=\frac {\log \left (\sqrt {\frac {x + 1}{x}} - 1\right )}{\frac {x + 1}{x} - 1} + \frac {1}{2 \, {\left (\sqrt {\frac {x + 1}{x}} - 1\right )}} - \frac {1}{4} \, \log \left (\sqrt {\frac {x + 1}{x}} + 1\right ) + \frac {1}{4} \, \log \left (\sqrt {\frac {x + 1}{x}} - 1\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.06 \[ \int \log \left (-1+\sqrt {\frac {1+x}{x}}\right ) \, dx=x \log \left (\sqrt {\frac {x + 1}{x}} - 1\right ) + \frac {1}{2} \, x + \frac {\log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} + x} - 1 \right |}\right )}{4 \, \mathrm {sgn}\left (x\right )} + \frac {\sqrt {x^{2} + x}}{2 \, \mathrm {sgn}\left (x\right )} \]

[In]

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

[Out]

x*log(sqrt((x + 1)/x) - 1) + 1/2*x + 1/4*log(abs(-2*x + 2*sqrt(x^2 + x) - 1))/sgn(x) + 1/2*sqrt(x^2 + x)/sgn(x
)

Mupad [B] (verification not implemented)

Time = 1.57 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.76 \[ \int \log \left (-1+\sqrt {\frac {1+x}{x}}\right ) \, dx=\frac {x}{2}-\frac {\mathrm {atanh}\left (\sqrt {\frac {1}{x}+1}\right )}{2}+x\,\ln \left (\sqrt {\frac {x+1}{x}}-1\right )+\frac {x\,\sqrt {\frac {1}{x}+1}}{2} \]

[In]

int(log(((x + 1)/x)^(1/2) - 1),x)

[Out]

x/2 - atanh((1/x + 1)^(1/2))/2 + x*log(((x + 1)/x)^(1/2) - 1) + (x*(1/x + 1)^(1/2))/2