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]

-1/6*ln(1-(1+1/x)^(1/2))+1/2*ln(1+(1+1/x)^(1/2))-1/3*ln(2+(1+1/x)^(1/2))+x*ln(2+((1+x)/x)^(1/2))

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Rubi [A]  time = 0.07, antiderivative size = 67, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.03, 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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fricas [A]  time = 0.45, size = 48, normalized size = 0.72 \[ \frac {1}{3} \, {\left (3 \, x - 1\right )} \log \left (\sqrt {\frac {x + 1}{x}} + 2\right ) + \frac {1}{2} \, \log \left (\sqrt {\frac {x + 1}{x}} + 1\right ) - \frac {1}{6} \, \log \left (\sqrt {\frac {x + 1}{x}} - 1\right ) \]

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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giac [A]  time = 0.25, size = 88, normalized size = 1.31 \[ x \log \left (\sqrt {\frac {x + 1}{x}} + 2\right ) - \frac {\log \left ({\left | -x + \sqrt {x^{2} + x} + 1 \right |}\right )}{6 \, \mathrm {sgn}\relax (x)} - \frac {\log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} + x} - 1 \right |}\right )}{3 \, \mathrm {sgn}\relax (x)} + \frac {\log \left ({\left | -3 \, x + 3 \, \sqrt {x^{2} + x} - 1 \right |}\right )}{6 \, \mathrm {sgn}\relax (x)} - \frac {1}{6} \, \log \left ({\left | 3 \, x - 1 \right |}\right ) \]

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

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maple [B]  time = 0.14, size = 108, normalized size = 1.61 \[ x \ln \left (2+\sqrt {\frac {x +1}{x}}\right )+\frac {-3 \sqrt {\frac {x +1}{x}}\, x \ln \left (-3 x +1\right )-\sqrt {9}\, \sqrt {\left (x +1\right ) x}\, \ln \left (\frac {15 x +4 \sqrt {9}\, \sqrt {x^{2}+x}+3}{9 x -3}\right )+6 \sqrt {\left (x +1\right ) x}\, \ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{18 \sqrt {\frac {x +1}{x}}\, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.56, size = 67, normalized size = 1.00 \[ \frac {\log \left (\sqrt {\frac {x + 1}{x}} + 2\right )}{\frac {x + 1}{x} - 1} - \frac {1}{3} \, \log \left (\sqrt {\frac {x + 1}{x}} + 2\right ) + \frac {1}{2} \, \log \left (\sqrt {\frac {x + 1}{x}} + 1\right ) - \frac {1}{6} \, \log \left (\sqrt {\frac {x + 1}{x}} - 1\right ) \]

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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mupad [B]  time = 0.64, size = 63, normalized size = 0.94 \[ \frac {\ln \left (-5\,\sqrt {\frac {x+1}{x}}-5\right )}{2}-\frac {\ln \left (\frac {\sqrt {\frac {x+1}{x}}}{9}-\frac {1}{9}\right )}{6}-\frac {\ln \left (-\frac {5\,\sqrt {\frac {x+1}{x}}}{9}-\frac {10}{9}\right )}{3}+x\,\ln \left (\sqrt {\frac {x+1}{x}}+2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(((x + 1)/x)^(1/2) + 2),x)

[Out]

log(- 5*((x + 1)/x)^(1/2) - 5)/2 - log(((x + 1)/x)^(1/2)/9 - 1/9)/6 - log(- (5*((x + 1)/x)^(1/2))/9 - 10/9)/3
+ x*log(((x + 1)/x)^(1/2) + 2)

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sympy [A]  time = 40.98, size = 53, normalized size = 0.79 \[ x \log {\left (\sqrt {\frac {x + 1}{x}} + 2 \right )} - \frac {\log {\left (\sqrt {1 + \frac {1}{x}} - 1 \right )}}{6} + \frac {\log {\left (\sqrt {1 + \frac {1}{x}} + 1 \right )}}{2} - \frac {\log {\left (\sqrt {1 + \frac {1}{x}} + 2 \right )}}{3} \]

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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