3.153 \(\int \frac{1}{\tanh ^{-1}(\tanh (a+b x))^{3/2}} \, dx\)

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

[Out]

-2/(b*Sqrt[ArcTanh[Tanh[a + b*x]]])

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Rubi [A]  time = 0.0048986, 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} \[ -\frac{2}{b \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[Tanh[a + b*x]]^(-3/2),x]

[Out]

-2/(b*Sqrt[ArcTanh[Tanh[a + b*x]]])

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}{\tanh ^{-1}(\tanh (a+b x))^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^{3/2}} \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{b}\\ &=-\frac{2}{b \sqrt{\tanh ^{-1}(\tanh (a+b x))}}\\ \end{align*}

Mathematica [A]  time = 0.0067302, size = 16, normalized size = 1. \[ -\frac{2}{b \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[Tanh[a + b*x]]^(-3/2),x]

[Out]

-2/(b*Sqrt[ArcTanh[Tanh[a + b*x]]])

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Maple [A]  time = 0.027, size = 15, normalized size = 0.9 \begin{align*} -2\,{\frac{1}{b\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/arctanh(tanh(b*x+a))^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.7012, 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))^(3/2),x, algorithm="maxima")

[Out]

-2/(sqrt(b*x + a)*b)

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Fricas [A]  time = 2.00522, size = 43, normalized size = 2.69 \begin{align*} -\frac{2 \, \sqrt{b x + a}}{b^{2} x + a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arctanh(tanh(b*x+a))^(3/2),x, algorithm="fricas")

[Out]

-2*sqrt(b*x + a)/(b^2*x + a*b)

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Sympy [A]  time = 35.1407, size = 26, normalized size = 1.62 \begin{align*} \begin{cases} - \frac{2}{b \sqrt{\operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}} & \text{for}\: b \neq 0 \\\frac{x}{\operatorname{atanh}^{\frac{3}{2}}{\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))**(3/2),x)

[Out]

Piecewise((-2/(b*sqrt(atanh(tanh(a + b*x)))), Ne(b, 0)), (x/atanh(tanh(a))**(3/2), True))

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Giac [A]  time = 1.1385, 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))^(3/2),x, algorithm="giac")

[Out]

-2/(sqrt(b*x + a)*b)