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

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

[Out]

-1/(2*b*ArcTanh[Tanh[a + b*x]]^2)

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Rubi [A]  time = 0.004478, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2157, 30} \[ -\frac{1}{2 b \tanh ^{-1}(\tanh (a+b x))^2} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[Tanh[a + b*x]]^(-3),x]

[Out]

-1/(2*b*ArcTanh[Tanh[a + b*x]]^2)

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

Mathematica [A]  time = 0.006647, size = 16, normalized size = 1. \[ -\frac{1}{2 b \tanh ^{-1}(\tanh (a+b x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(2*b*ArcTanh[Tanh[a + b*x]]^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.47506, size = 16, normalized size = 1. \begin{align*} -\frac{1}{2 \,{\left (b x + a\right )}^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/((b*x + a)^2*b)

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Fricas [A]  time = 1.49182, size = 49, normalized size = 3.06 \begin{align*} -\frac{1}{2 \,{\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(b^3*x^2 + 2*a*b^2*x + a^2*b)

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Sympy [A]  time = 15.3124, size = 24, normalized size = 1.5 \begin{align*} \begin{cases} - \frac{1}{2 b \operatorname{atanh}^{2}{\left (\tanh{\left (a + b x \right )} \right )}} & \text{for}\: b \neq 0 \\\frac{x}{\operatorname{atanh}^{3}{\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,x)

[Out]

Piecewise((-1/(2*b*atanh(tanh(a + b*x))**2), Ne(b, 0)), (x/atanh(tanh(a))**3, True))

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Giac [A]  time = 1.12099, size = 16, normalized size = 1. \begin{align*} -\frac{1}{2 \,{\left (b x + a\right )}^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/((b*x + a)^2*b)