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

Optimal. Leaf size=64 \[ \frac {\coth ^{-1}(\tanh (a+b x))^{n+1} \, _2F_1\left (1,n+1;n+2;-\frac {\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{(n+1) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 64, 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 = {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*}

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Mathematica [A]  time = 0.09, size = 60, normalized size = 0.94 \[ \frac {\coth ^{-1}(\tanh (a+b x))^n \left (\frac {\coth ^{-1}(\tanh (a+b x))}{b x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;1-\frac {\coth ^{-1}(\tanh (a+b x))}{b x}\right )}{n} \]

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)

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fricas [F]  time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{n}}{x}, x\right ) \]

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)

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

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)

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maple [F]  time = 0.68, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )^{n}}{x}\, dx \]

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)

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

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^n}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acoth(tanh(a + b*x))^n/x,x)

[Out]

int(acoth(tanh(a + b*x))^n/x, x)

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

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)

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