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

Optimal. Leaf size=79 \[ \frac {x^m \left (\frac {b x}{b x-\coth ^{-1}(\tanh (a+b x))}\right )^{-m} \coth ^{-1}(\tanh (a+b x))^{n+1} \, _2F_1\left (-m,n+1;n+2;-\frac {\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{b (n+1)} \]

[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)

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Rubi [A]  time = 0.05, antiderivative size = 79, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.14, size = 71, normalized size = 0.90 \[ \frac {x^{m+1} \coth ^{-1}(\tanh (a+b x))^n \left (\frac {b x}{\coth ^{-1}(\tanh (a+b x))-b x}+1\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac {b x}{\coth ^{-1}(\tanh (a+b x))-b x}\right )}{m+1} \]

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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fricas [F]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{n}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{n}\,{d x} \]

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)

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maple [F]  time = 14.58, size = 0, normalized size = 0.00 \[ \int x^{m} \mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )^{n}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{n}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^m\,{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^n \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*acoth(tanh(a + b*x))^n,x)

[Out]

int(x^m*acoth(tanh(a + b*x))^n, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \operatorname {acoth}^{n}{\left (\tanh {\left (a + b x \right )} \right )}\, dx \]

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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