3.144 \(\int \frac{1}{\sqrt{\tanh ^{-1}(\tanh (a+b x))}} \, dx\)

Optimal. Leaf size=16 \[ \frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{b} \]

[Out]

(2*Sqrt[ArcTanh[Tanh[a + b*x]]])/b

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Rubi [A]  time = 0.0044057, antiderivative size = 16, 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 = {2157, 30} \[ \frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{b} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[ArcTanh[Tanh[a + b*x]]],x]

[Out]

(2*Sqrt[ArcTanh[Tanh[a + b*x]]])/b

Rule 2157

Int[(u_)^(m_.), x_Symbol] :> With[{c = Simplify[D[u, x]]}, Dist[1/c, Subst[Int[x^m, x], x, u], x]] /; FreeQ[m,
x] && PiecewiseLinearQ[u, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.006343, size = 16, normalized size = 1. \[ \frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[ArcTanh[Tanh[a + b*x]]],x]

[Out]

(2*Sqrt[ArcTanh[Tanh[a + b*x]]])/b

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Maple [A]  time = 0.032, size = 15, normalized size = 0.9 \begin{align*} 2\,{\frac{\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }}{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/arctanh(tanh(b*x+a))^(1/2),x)

[Out]

2*arctanh(tanh(b*x+a))^(1/2)/b

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Maxima [A]  time = 1.70043, size = 16, normalized size = 1. \begin{align*} \frac{2 \, \sqrt{b x + a}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arctanh(tanh(b*x+a))^(1/2),x, algorithm="maxima")

[Out]

2*sqrt(b*x + a)/b

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Fricas [A]  time = 2.05399, size = 26, normalized size = 1.62 \begin{align*} \frac{2 \, \sqrt{b x + a}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arctanh(tanh(b*x+a))^(1/2),x, algorithm="fricas")

[Out]

2*sqrt(b*x + a)/b

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Sympy [A]  time = 18.5199, size = 24, normalized size = 1.5 \begin{align*} \begin{cases} \frac{2 \sqrt{\operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}}{b} & \text{for}\: b \neq 0 \\\frac{x}{\sqrt{\operatorname{atanh}{\left (\tanh{\left (a \right )} \right )}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/atanh(tanh(b*x+a))**(1/2),x)

[Out]

Piecewise((2*sqrt(atanh(tanh(a + b*x)))/b, Ne(b, 0)), (x/sqrt(atanh(tanh(a))), True))

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Giac [A]  time = 1.16846, size = 16, normalized size = 1. \begin{align*} \frac{2 \, \sqrt{b x + a}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arctanh(tanh(b*x+a))^(1/2),x, algorithm="giac")

[Out]

2*sqrt(b*x + a)/b