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

Optimal. Leaf size=33 \[ \frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]

[Out]

(2*Sqrt[ArcTanh[Tanh[a + b*x]]])/(Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]]))

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Rubi [A]  time = 0.0133372, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2167} \[ \frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(3/2)*Sqrt[ArcTanh[Tanh[a + b*x]]]),x]

[Out]

(2*Sqrt[ArcTanh[Tanh[a + b*x]]])/(Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]]))

Rule 2167

Int[(u_)^(m_)*(v_)^(n_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, -Simp[(u^(m + 1)*v^
(n + 1))/((m + 1)*(b*u - a*v)), x] /; NeQ[b*u - a*v, 0]] /; FreeQ[{m, n}, x] && PiecewiseLinearQ[u, v, x] && E
qQ[m + n + 2, 0] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0392659, size = 32, normalized size = 0.97 \[ -\frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{x} \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(3/2)*Sqrt[ArcTanh[Tanh[a + b*x]]]),x]

[Out]

(-2*Sqrt[ArcTanh[Tanh[a + b*x]]])/(Sqrt[x]*(-(b*x) + ArcTanh[Tanh[a + b*x]]))

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Maple [A]  time = 0.17, size = 29, normalized size = 0.9 \begin{align*} -2\,{\frac{\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }}{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx \right ) \sqrt{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.47852, size = 20, normalized size = 0.61 \begin{align*} -\frac{2 \, \sqrt{b x + a}}{a \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(b*x + a)/(a*sqrt(x))

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Fricas [A]  time = 2.06163, size = 41, normalized size = 1.24 \begin{align*} -\frac{2 \, \sqrt{b x + a}}{a \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(b*x + a)/(a*sqrt(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{\frac{3}{2}} \sqrt{\operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(3/2)/atanh(tanh(b*x+a))**(1/2),x)

[Out]

Integral(1/(x**(3/2)*sqrt(atanh(tanh(a + b*x)))), x)

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Giac [A]  time = 1.14597, size = 41, normalized size = 1.24 \begin{align*} \frac{4 \, \sqrt{b}}{{\left (\sqrt{b} \sqrt{x} - \sqrt{b x + a}\right )}^{2} - a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*sqrt(b)/((sqrt(b)*sqrt(x) - sqrt(b*x + a))^2 - a)