3.270 \(\int \tanh ^{-1}(\tanh (a+b x))^n \, dx\)

Optimal. Leaf size=20 \[ \frac{\tanh ^{-1}(\tanh (a+b x))^{n+1}}{b (n+1)} \]

[Out]

ArcTanh[Tanh[a + b*x]]^(1 + n)/(b*(1 + n))

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Rubi [A]  time = 0.0067712, antiderivative size = 20, 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} \[ \frac{\tanh ^{-1}(\tanh (a+b x))^{n+1}}{b (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[Tanh[a + b*x]]^n,x]

[Out]

ArcTanh[Tanh[a + b*x]]^(1 + n)/(b*(1 + n))

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

Mathematica [A]  time = 0.0147276, size = 20, normalized size = 1. \[ \frac{\tanh ^{-1}(\tanh (a+b x))^{n+1}}{b (n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[Tanh[a + b*x]]^n,x]

[Out]

ArcTanh[Tanh[a + b*x]]^(1 + n)/(b*(1 + n))

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Maple [A]  time = 0.027, size = 21, normalized size = 1.1 \begin{align*}{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{1+n}}{b \left ( 1+n \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(tanh(b*x+a))^n,x)

[Out]

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

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Maxima [A]  time = 1.69616, size = 28, normalized size = 1.4 \begin{align*} \frac{{\left (b x + a\right )}{\left (b x + a\right )}^{n}}{b{\left (n + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x + a)*(b*x + a)^n/(b*(n + 1))

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Fricas [A]  time = 2.33867, size = 104, normalized size = 5.2 \begin{align*} \frac{{\left (b x + a\right )} \cosh \left (n \log \left (b x + a\right )\right ) +{\left (b x + a\right )} \sinh \left (n \log \left (b x + a\right )\right )}{b n + b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*x + a)*cosh(n*log(b*x + a)) + (b*x + a)*sinh(n*log(b*x + a)))/(b*n + b)

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Sympy [A]  time = 1.46915, size = 51, normalized size = 2.55 \begin{align*} \begin{cases} \frac{x}{\operatorname{atanh}{\left (\tanh{\left (a \right )} \right )}} & \text{for}\: b = 0 \wedge n = -1 \\x \operatorname{atanh}^{n}{\left (\tanh{\left (a \right )} \right )} & \text{for}\: b = 0 \\\frac{\log{\left (\operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )} \right )}}{b} & \text{for}\: n = -1 \\\frac{\operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )} \operatorname{atanh}^{n}{\left (\tanh{\left (a + b x \right )} \right )}}{b n + b} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(tanh(b*x+a))**n,x)

[Out]

Piecewise((x/atanh(tanh(a)), Eq(b, 0) & Eq(n, -1)), (x*atanh(tanh(a))**n, Eq(b, 0)), (log(atanh(tanh(a + b*x))
)/b, Eq(n, -1)), (atanh(tanh(a + b*x))*atanh(tanh(a + b*x))**n/(b*n + b), True))

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Giac [A]  time = 1.16034, size = 38, normalized size = 1.9 \begin{align*} \frac{{\left (b x + a\right )}^{n} b x +{\left (b x + a\right )}^{n} a}{b n + b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*x + a)^n*b*x + (b*x + a)^n*a)/(b*n + b)