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

Optimal. Leaf size=64 \[ \frac{\coth ^{-1}(\tanh (a+b x))^{n+1} \text{Hypergeometric2F1}\left (1,n+1,n+2,-\frac{\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{(n+1) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0208313, antiderivative size = 64, 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 = {2164} \[ \frac{\coth ^{-1}(\tanh (a+b x))^{n+1} \, _2F_1\left (1,n+1;n+2;-\frac{\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{(n+1) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )} \]

Antiderivative was successfully verified.

[In]

Int[ArcCoth[Tanh[a + b*x]]^n/x,x]

[Out]

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

Rule 2164

Int[(v_)^(n_)/(u_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(v^(n + 1)*Hypergeo
metric2F1[1, n + 1, n + 2, -((a*v)/(b*u - a*v))])/((n + 1)*(b*u - a*v)), x] /; NeQ[b*u - a*v, 0]] /; Piecewise
LinearQ[u, v, x] &&  !IntegerQ[n]

Rubi steps

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

Mathematica [A]  time = 0.0811981, size = 60, normalized size = 0.94 \[ \frac{\coth ^{-1}(\tanh (a+b x))^n \left (\frac{\coth ^{-1}(\tanh (a+b x))}{b x}\right )^{-n} \text{Hypergeometric2F1}\left (-n,-n,1-n,1-\frac{\coth ^{-1}(\tanh (a+b x))}{b x}\right )}{n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.855, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm arccoth} \left (\tanh \left ( bx+a \right ) \right ) \right ) ^{n}}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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