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

Optimal. Leaf size=10 \[ -2 \tanh ^{-1}\left (\sqrt{x+1}\right ) \]

[Out]

-2*ArcTanh[Sqrt[1 + x]]

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Rubi [A]  time = 0.0030611, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {63, 207} \[ -2 \tanh ^{-1}\left (\sqrt{x+1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[1 + x]),x]

[Out]

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

Mathematica [A]  time = 0.0019041, size = 10, normalized size = 1. \[ -2 \tanh ^{-1}\left (\sqrt{x+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[1 + x]),x]

[Out]

-2*ArcTanh[Sqrt[1 + x]]

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Maple [A]  time = 0.003, size = 9, normalized size = 0.9 \begin{align*} -2\,{\it Artanh} \left ( \sqrt{1+x} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 0.936395, size = 26, normalized size = 2.6 \begin{align*} -\log \left (\sqrt{x + 1} + 1\right ) + \log \left (\sqrt{x + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(sqrt(x + 1) + 1) + log(sqrt(x + 1) - 1)

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Fricas [B]  time = 1.87211, size = 62, normalized size = 6.2 \begin{align*} -\log \left (\sqrt{x + 1} + 1\right ) + \log \left (\sqrt{x + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(sqrt(x + 1) + 1) + log(sqrt(x + 1) - 1)

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Sympy [A]  time = 0.48158, size = 26, normalized size = 2.6 \begin{align*} \begin{cases} - 2 \operatorname{acoth}{\left (\sqrt{x + 1} \right )} & \text{for}\: \left |{x + 1}\right | > 1 \\- 2 \operatorname{atanh}{\left (\sqrt{x + 1} \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*acoth(sqrt(x + 1)), Abs(x + 1) > 1), (-2*atanh(sqrt(x + 1)), True))

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Giac [B]  time = 1.05432, size = 27, normalized size = 2.7 \begin{align*} -\log \left (\sqrt{x + 1} + 1\right ) + \log \left ({\left | \sqrt{x + 1} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(sqrt(x + 1) + 1) + log(abs(sqrt(x + 1) - 1))