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

Optimal. Leaf size=14 \[ -\frac{1}{b \tanh ^{-1}(\tanh (a+b x))} \]

[Out]

-(1/(b*ArcTanh[Tanh[a + b*x]]))

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Rubi [A]  time = 0.0048739, antiderivative size = 14, 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} \[ -\frac{1}{b \tanh ^{-1}(\tanh (a+b x))} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[Tanh[a + b*x]]^(-2),x]

[Out]

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

Mathematica [A]  time = 0.0057617, size = 14, normalized size = 1. \[ -\frac{1}{b \tanh ^{-1}(\tanh (a+b x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(b*ArcTanh[Tanh[a + b*x]]))

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Maple [A]  time = 0.027, size = 15, normalized size = 1.1 \begin{align*} -{\frac{1}{b{\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))^2,x)

[Out]

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

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Maxima [A]  time = 1.47179, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{{\left (b x + a\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/((b*x + a)*b)

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Fricas [A]  time = 1.44121, size = 24, normalized size = 1.71 \begin{align*} -\frac{1}{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))^2,x, algorithm="fricas")

[Out]

-1/(b^2*x + a*b)

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Sympy [A]  time = 14.972, size = 20, normalized size = 1.43 \begin{align*} \begin{cases} - \frac{1}{b \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}} & \text{for}\: b \neq 0 \\\frac{x}{\operatorname{atanh}^{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))**2,x)

[Out]

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

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Giac [A]  time = 1.14276, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{{\left (b x + a\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/((b*x + a)*b)