3.294 \(\int \log (2+\sqrt{\frac{1+x}{x}}) \, dx\)

Optimal. Leaf size=67 \[ -\frac{1}{6} \log \left (1-\sqrt{\frac{1}{x}+1}\right )+\frac{1}{2} \log \left (\sqrt{\frac{1}{x}+1}+1\right )-\frac{1}{3} \log \left (\sqrt{\frac{1}{x}+1}+2\right )+x \log \left (\sqrt{\frac{x+1}{x}}+2\right ) \]

[Out]

-Log[1 - Sqrt[1 + x^(-1)]]/6 + Log[1 + Sqrt[1 + x^(-1)]]/2 - Log[2 + Sqrt[1 + x^(-1)]]/3 + x*Log[2 + Sqrt[(1 +
x)/x]]

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Rubi [A]  time = 0.0742242, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2548, 12, 2058} \[ -\frac{1}{6} \log \left (1-\sqrt{\frac{1}{x}+1}\right )+\frac{1}{2} \log \left (\sqrt{\frac{1}{x}+1}+1\right )-\frac{1}{3} \log \left (\sqrt{\frac{1}{x}+1}+2\right )+x \log \left (\sqrt{\frac{x+1}{x}}+2\right ) \]

Antiderivative was successfully verified.

[In]

Int[Log[2 + Sqrt[(1 + x)/x]],x]

[Out]

-Log[1 - Sqrt[1 + x^(-1)]]/6 + Log[1 + Sqrt[1 + x^(-1)]]/2 - Log[2 + Sqrt[1 + x^(-1)]]/3 + x*Log[2 + Sqrt[(1 +
x)/x]]

Rule 2548

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2058

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P]}, Int[ExpandIntegrand[u^p, x], x] /;  !SumQ[NonfreeFactors[u,
x]]] /; PolyQ[P, x] && ILtQ[p, 0]

Rubi steps

\begin{align*} \int \log \left (2+\sqrt{\frac{1+x}{x}}\right ) \, dx &=x \log \left (2+\sqrt{\frac{1+x}{x}}\right )-\int \frac{1}{2 \left (-1-x-2 x \sqrt{\frac{1+x}{x}}\right )} \, dx\\ &=x \log \left (2+\sqrt{\frac{1+x}{x}}\right )-\frac{1}{2} \int \frac{1}{-1-x-2 x \sqrt{\frac{1+x}{x}}} \, dx\\ &=x \log \left (2+\sqrt{\frac{1+x}{x}}\right )+\operatorname{Subst}\left (\int \frac{1}{2+x-2 x^2-x^3} \, dx,x,\sqrt{\frac{1+x}{x}}\right )\\ &=x \log \left (2+\sqrt{\frac{1+x}{x}}\right )+\operatorname{Subst}\left (\int \left (-\frac{1}{6 (-1+x)}+\frac{1}{2 (1+x)}-\frac{1}{3 (2+x)}\right ) \, dx,x,\sqrt{\frac{1+x}{x}}\right )\\ &=-\frac{1}{6} \log \left (1-\sqrt{1+\frac{1}{x}}\right )+\frac{1}{2} \log \left (1+\sqrt{1+\frac{1}{x}}\right )-\frac{1}{3} \log \left (2+\sqrt{1+\frac{1}{x}}\right )+x \log \left (2+\sqrt{\frac{1+x}{x}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0305202, size = 53, normalized size = 0.79 \[ x \log \left (\sqrt{\frac{1}{x}+1}+2\right )+\frac{1}{3} \tanh ^{-1}\left (\frac{1}{3} \left (2 \sqrt{\frac{1}{x}+1}+1\right )\right )-\tanh ^{-1}\left (2 \sqrt{\frac{1}{x}+1}+3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[2 + Sqrt[(1 + x)/x]],x]

[Out]

ArcTanh[(1 + 2*Sqrt[1 + x^(-1)])/3]/3 - ArcTanh[3 + 2*Sqrt[1 + x^(-1)]] + x*Log[2 + Sqrt[1 + x^(-1)]]

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Maple [A]  time = 0.046, size = 107, normalized size = 1.6 \begin{align*} x\ln \left ( 2+\sqrt{{\frac{1+x}{x}}} \right ) -{\frac{1}{18\,x} \left ( \sqrt{9}\ln \left ({\frac{1}{9\,x-3} \left ( 4\,\sqrt{9}\sqrt{{x}^{2}+x}+15\,x+3 \right ) } \right ) \sqrt{x \left ( 1+x \right ) }+3\,\sqrt{{\frac{1+x}{x}}}x\ln \left ( -3\,x+1 \right ) -6\,\ln \left ( 1/2+x+\sqrt{{x}^{2}+x} \right ) \sqrt{x \left ( 1+x \right ) } \right ){\frac{1}{\sqrt{{\frac{1+x}{x}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(2+((1+x)/x)^(1/2)),x)

[Out]

x*ln(2+((1+x)/x)^(1/2))-1/18/((1+x)/x)^(1/2)/x*(9^(1/2)*ln(1/3*(4*9^(1/2)*(x^2+x)^(1/2)+15*x+3)/(3*x-1))*(x*(1
+x))^(1/2)+3*((1+x)/x)^(1/2)*x*ln(-3*x+1)-6*ln(1/2+x+(x^2+x)^(1/2))*(x*(1+x))^(1/2))

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Maxima [A]  time = 1.01949, size = 90, normalized size = 1.34 \begin{align*} \frac{\log \left (\sqrt{\frac{x + 1}{x}} + 2\right )}{\frac{x + 1}{x} - 1} - \frac{1}{3} \, \log \left (\sqrt{\frac{x + 1}{x}} + 2\right ) + \frac{1}{2} \, \log \left (\sqrt{\frac{x + 1}{x}} + 1\right ) - \frac{1}{6} \, \log \left (\sqrt{\frac{x + 1}{x}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sqrt((x + 1)/x) + 2)/((x + 1)/x - 1) - 1/3*log(sqrt((x + 1)/x) + 2) + 1/2*log(sqrt((x + 1)/x) + 1) - 1/6*l
og(sqrt((x + 1)/x) - 1)

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Fricas [A]  time = 2.25612, size = 138, normalized size = 2.06 \begin{align*} \frac{1}{3} \,{\left (3 \, x - 1\right )} \log \left (\sqrt{\frac{x + 1}{x}} + 2\right ) + \frac{1}{2} \, \log \left (\sqrt{\frac{x + 1}{x}} + 1\right ) - \frac{1}{6} \, \log \left (\sqrt{\frac{x + 1}{x}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 161.722, size = 53, normalized size = 0.79 \begin{align*} x \log{\left (\sqrt{\frac{x + 1}{x}} + 2 \right )} - \frac{\log{\left (\sqrt{1 + \frac{1}{x}} - 1 \right )}}{6} + \frac{\log{\left (\sqrt{1 + \frac{1}{x}} + 1 \right )}}{2} - \frac{\log{\left (\sqrt{1 + \frac{1}{x}} + 2 \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.3405, size = 157, normalized size = 2.34 \begin{align*} x \log \left (\sqrt{\frac{x + 1}{x}} + 2\right ) - \frac{1}{6} \,{\left (\frac{\log \left ({\left | -2 \,{\left (x - \sqrt{x^{2} + x}\right )} \mathrm{sgn}\left (x\right ) + x - \sqrt{x^{2} + x} + 1 \right |}\right )}{\mathrm{sgn}\left (x\right )} - \frac{\log \left ({\left | -2 \,{\left (x - \sqrt{x^{2} + x}\right )} \mathrm{sgn}\left (x\right ) - x + \sqrt{x^{2} + x} - 1 \right |}\right )}{\mathrm{sgn}\left (x\right )} + 2 \, \log \left ({\left | -2 \, x + 2 \, \sqrt{x^{2} + x} - 1 \right |}\right )\right )} \mathrm{sgn}\left (x\right ) - \frac{1}{6} \, \log \left ({\left | 3 \, x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(sqrt((x + 1)/x) + 2) - 1/6*(log(abs(-2*(x - sqrt(x^2 + x))*sgn(x) + x - sqrt(x^2 + x) + 1))/sgn(x) - log
(abs(-2*(x - sqrt(x^2 + x))*sgn(x) - x + sqrt(x^2 + x) - 1))/sgn(x) + 2*log(abs(-2*x + 2*sqrt(x^2 + x) - 1)))*
sgn(x) - 1/6*log(abs(3*x - 1))