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

Optimal. Leaf size=24 \[ 2 \tanh ^{-1}\left (\sqrt{\frac{1}{x}+1}\right )-2 \sqrt{\frac{1}{x}+1} \]

[Out]

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

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Rubi [A]  time = 0.0170135, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1973, 266, 50, 63, 207} \[ 2 \tanh ^{-1}\left (\sqrt{\frac{1}{x}+1}\right )-2 \sqrt{\frac{1}{x}+1} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[(1 + x)/x]/x,x]

[Out]

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

Rule 1973

Int[(u_)^(p_.)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[(c*x)^m*ExpandToSum[u, x]^p, x] /; FreeQ[{c, m, p}, x] &&
BinomialQ[u, x] &&  !BinomialMatchQ[u, x]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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

Mathematica [A]  time = 0.0055733, size = 24, normalized size = 1. \[ 2 \tanh ^{-1}\left (\sqrt{\frac{1}{x}+1}\right )-2 \sqrt{\frac{1}{x}+1} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[(1 + x)/x]/x,x]

[Out]

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

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Maple [B]  time = 0.004, size = 60, normalized size = 2.5 \begin{align*} -{\frac{1}{x}\sqrt{{\frac{1+x}{x}}} \left ( 2\, \left ({x}^{2}+x \right ) ^{3/2}-2\,{x}^{2}\sqrt{{x}^{2}+x}-\ln \left ({\frac{1}{2}}+x+\sqrt{{x}^{2}+x} \right ){x}^{2} \right ){\frac{1}{\sqrt{x \left ( 1+x \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.14745, size = 51, normalized size = 2.12 \begin{align*} -2 \, \sqrt{\frac{x + 1}{x}} + \log \left (\sqrt{\frac{x + 1}{x}} + 1\right ) - \log \left (\sqrt{\frac{x + 1}{x}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.47816, size = 100, normalized size = 4.17 \begin{align*} -2 \, \sqrt{\frac{x + 1}{x}} + \log \left (\sqrt{\frac{x + 1}{x}} + 1\right ) - \log \left (\sqrt{\frac{x + 1}{x}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 3.05388, size = 32, normalized size = 1.33 \begin{align*} - 2 \sqrt{1 + \frac{1}{x}} - \log{\left (\sqrt{1 + \frac{1}{x}} - 1 \right )} + \log{\left (\sqrt{1 + \frac{1}{x}} + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.32662, size = 51, normalized size = 2.12 \begin{align*} -\log \left ({\left | -2 \, x + 2 \, \sqrt{x^{2} + x} - 1 \right |}\right ) \mathrm{sgn}\left (x\right ) + \frac{2 \, \mathrm{sgn}\left (x\right )}{x - \sqrt{x^{2} + x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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