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

Optimal. Leaf size=50 \[ -\frac{1}{2 \left (\sqrt{\frac{1}{x}+1}+1\right )}+x \log \left (\sqrt{\frac{x+1}{x}}+1\right )+\frac{1}{2} \tanh ^{-1}\left (\sqrt{\frac{x+1}{x}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.056997, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2548, 12, 44, 207} \[ -\frac{1}{2 \left (\sqrt{\frac{1}{x}+1}+1\right )}+x \log \left (\sqrt{\frac{x+1}{x}}+1\right )+\frac{1}{2} \tanh ^{-1}\left (\sqrt{\frac{x+1}{x}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Log[1 + Sqrt[(1 + x)/x]],x]

[Out]

-1/(2*(1 + Sqrt[1 + x^(-1)])) + ArcTanh[Sqrt[(1 + x)/x]]/2 + x*Log[1 + 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 44

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] && L
tQ[m + n + 2, 0])

Rule 207

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

Rubi steps

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

Mathematica [A]  time = 0.0373296, size = 53, normalized size = 1.06 \[ \frac{1}{4} \left (-2 \sqrt{\frac{1}{x}+1} x+2 x+4 x \log \left (\sqrt{\frac{1}{x}+1}+1\right )+\log \left (\left (2 \sqrt{\frac{1}{x}+1}+2\right ) x+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[1 + Sqrt[(1 + x)/x]],x]

[Out]

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

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \int \ln \left ( 1+\sqrt{{\frac{1+x}{x}}} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.03464, size = 92, normalized size = 1.84 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 2.18177, size = 139, normalized size = 2.78 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 160.093, size = 53, normalized size = 1.06 \begin{align*} 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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

undef