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

Optimal. Leaf size=79 \[ \frac{x^m \left (\frac{b x}{b x-\coth ^{-1}(\tanh (a+b x))}\right )^{-m} \coth ^{-1}(\tanh (a+b x))^{n+1} \text{Hypergeometric2F1}\left (-m,n+1,n+2,-\frac{\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{b (n+1)} \]

[Out]

(x^m*ArcCoth[Tanh[a + b*x]]^(1 + n)*Hypergeometric2F1[-m, 1 + n, 2 + n, -(ArcCoth[Tanh[a + b*x]]/(b*x - ArcCot
h[Tanh[a + b*x]]))])/(b*(1 + n)*((b*x)/(b*x - ArcCoth[Tanh[a + b*x]]))^m)

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Rubi [A]  time = 0.0494153, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2173} \[ \frac{x^m \left (\frac{b x}{b x-\coth ^{-1}(\tanh (a+b x))}\right )^{-m} \coth ^{-1}(\tanh (a+b x))^{n+1} \, _2F_1\left (-m,n+1;n+2;-\frac{\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{b (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[x^m*ArcCoth[Tanh[a + b*x]]^n,x]

[Out]

(x^m*ArcCoth[Tanh[a + b*x]]^(1 + n)*Hypergeometric2F1[-m, 1 + n, 2 + n, -(ArcCoth[Tanh[a + b*x]]/(b*x - ArcCot
h[Tanh[a + b*x]]))])/(b*(1 + n)*((b*x)/(b*x - ArcCoth[Tanh[a + b*x]]))^m)

Rule 2173

Int[(u_)^(m_)*(v_)^(n_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(u^m*v^(n + 1)
*Hypergeometric2F1[-m, n + 1, n + 2, -((a*v)/(b*u - a*v))])/(b*(n + 1)*((b*u)/(b*u - a*v))^m), x] /; NeQ[b*u -
a*v, 0]] /; PiecewiseLinearQ[u, v, x] &&  !IntegerQ[m] &&  !IntegerQ[n]

Rubi steps

\begin{align*} \int x^m \coth ^{-1}(\tanh (a+b x))^n \, dx &=\frac{x^m \left (\frac{b x}{b x-\coth ^{-1}(\tanh (a+b x))}\right )^{-m} \coth ^{-1}(\tanh (a+b x))^{1+n} \, _2F_1\left (-m,1+n;2+n;-\frac{\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{b (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.132399, size = 71, normalized size = 0.9 \[ \frac{x^{m+1} \coth ^{-1}(\tanh (a+b x))^n \left (\frac{b x}{\coth ^{-1}(\tanh (a+b x))-b x}+1\right )^{-n} \text{Hypergeometric2F1}\left (m+1,-n,m+2,-\frac{b x}{\coth ^{-1}(\tanh (a+b x))-b x}\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*ArcCoth[Tanh[a + b*x]]^n,x]

[Out]

(x^(1 + m)*ArcCoth[Tanh[a + b*x]]^n*Hypergeometric2F1[1 + m, -n, 2 + m, -((b*x)/(-(b*x) + ArcCoth[Tanh[a + b*x
]]))])/((1 + m)*(1 + (b*x)/(-(b*x) + ArcCoth[Tanh[a + b*x]]))^n)

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Maple [F]  time = 2.566, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ({\rm arccoth} \left (\tanh \left ( bx+a \right ) \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m*arccoth(tanh(b*x + a))^n, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{m} \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^m*arccoth(tanh(b*x + a))^n, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \operatorname{acoth}^{n}{\left (\tanh{\left (a + b x \right )} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*acoth(tanh(b*x+a))**n,x)

[Out]

Integral(x**m*acoth(tanh(a + b*x))**n, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m*arccoth(tanh(b*x + a))^n, x)