\(\int x^m \coth ^{-1}(\tanh (a+b x))^n \, dx\) [184]

Optimal result

Integrand size = 13, antiderivative size = 79 \[ \int x^m \coth ^{-1}(\tanh (a+b x))^n \, dx=\frac {x^m \left (\frac {b x}{b x-\coth ^{-1}(\tanh (a+b x))}\right )^{-m} \coth ^{-1}(\tanh (a+b x))^{1+n} \operatorname {Hypergeometric2F1}\left (-m,1+n,2+n,-\frac {\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{b (1+n)} \]

[Out]

x^m*arccoth(tanh(b*x+a))^(1+n)*hypergeom([-m, 1+n],[2+n],-arccoth(tanh(b*x+a))/(b*x-arccoth(tanh(b*x+a))))/b/(
1+n)/((b*x/(b*x-arccoth(tanh(b*x+a))))^m)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2204} \[ \int x^m \coth ^{-1}(\tanh (a+b x))^n \, dx=\frac {x^m \left (\frac {b x}{b x-\coth ^{-1}(\tanh (a+b x))}\right )^{-m} \coth ^{-1}(\tanh (a+b x))^{n+1} \operatorname {Hypergeometric2F1}\left (-m,n+1,n+2,-\frac {\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{b (n+1)} \]

[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 2204

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

Rubi steps \begin{align*} \text {integral}& = \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} \operatorname {Hypergeometric2F1}\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] (verified)

Time = 0.10 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.90 \[ \int x^m \coth ^{-1}(\tanh (a+b x))^n \, dx=\frac {x^{1+m} \coth ^{-1}(\tanh (a+b x))^n \left (1+\frac {b x}{-b x+\coth ^{-1}(\tanh (a+b x))}\right )^{-n} \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {b x}{-b x+\coth ^{-1}(\tanh (a+b x))}\right )}{1+m} \]

[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)

Maple [F]

\[\int x^{m} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int x^m \coth ^{-1}(\tanh (a+b x))^n \, dx=\int { x^{m} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{n} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int x^m \coth ^{-1}(\tanh (a+b x))^n \, dx=\int x^{m} \operatorname {acoth}^{n}{\left (\tanh {\left (a + b x \right )} \right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int x^m \coth ^{-1}(\tanh (a+b x))^n \, dx=\int { x^{m} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{n} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int x^m \coth ^{-1}(\tanh (a+b x))^n \, dx=\int { x^{m} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{n} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int x^m \coth ^{-1}(\tanh (a+b x))^n \, dx=\int x^m\,{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^n \,d x \]

[In]

int(x^m*acoth(tanh(a + b*x))^n,x)

[Out]

int(x^m*acoth(tanh(a + b*x))^n, x)