3.192 \(\int \frac{\coth ^{-1}(\tanh (a+b x))^n}{x^3} \, dx\)

Optimal. Leaf size=101 \[ \frac{b^2 n \coth ^{-1}(\tanh (a+b x))^{n-1} \text{Hypergeometric2F1}\left (1,n-1,n,-\frac{\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}-\frac{\coth ^{-1}(\tanh (a+b x))^n}{2 x^2}-\frac{b n \coth ^{-1}(\tanh (a+b x))^{n-1}}{2 x} \]

[Out]

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

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Rubi [A]  time = 0.0692208, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2168, 2164} \[ \frac{b^2 n \coth ^{-1}(\tanh (a+b x))^{n-1} \, _2F_1\left (1,n-1;n;-\frac{\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}-\frac{\coth ^{-1}(\tanh (a+b x))^n}{2 x^2}-\frac{b n \coth ^{-1}(\tanh (a+b x))^{n-1}}{2 x} \]

Antiderivative was successfully verified.

[In]

Int[ArcCoth[Tanh[a + b*x]]^n/x^3,x]

[Out]

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

Rule 2168

Int[(u_)^(m_)*(v_)^(n_.), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(u^(m + 1)*v^
n)/(a*(m + 1)), x] - Dist[(b*n)/(a*(m + 1)), Int[u^(m + 1)*v^(n - 1), x], x] /; NeQ[b*u - a*v, 0]] /; FreeQ[{m
, n}, x] && PiecewiseLinearQ[u, v, x] && NeQ[m, -1] && ((LtQ[m, -1] && GtQ[n, 0] &&  !(ILtQ[m + n, -2] && (Fra
ctionQ[m] || GeQ[2*n + m + 1, 0]))) || (IGtQ[n, 0] && IGtQ[m, 0] && LeQ[n, m]) || (IGtQ[n, 0] &&  !IntegerQ[m]
) || (ILtQ[m, 0] &&  !IntegerQ[n]))

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^3} \, dx &=-\frac{\coth ^{-1}(\tanh (a+b x))^n}{2 x^2}+\frac{1}{2} (b n) \int \frac{\coth ^{-1}(\tanh (a+b x))^{-1+n}}{x^2} \, dx\\ &=-\frac{b n \coth ^{-1}(\tanh (a+b x))^{-1+n}}{2 x}-\frac{\coth ^{-1}(\tanh (a+b x))^n}{2 x^2}-\frac{1}{2} \left (b^2 (1-n) n\right ) \int \frac{\coth ^{-1}(\tanh (a+b x))^{-2+n}}{x} \, dx\\ &=-\frac{b n \coth ^{-1}(\tanh (a+b x))^{-1+n}}{2 x}-\frac{\coth ^{-1}(\tanh (a+b x))^n}{2 x^2}+\frac{b^2 n \coth ^{-1}(\tanh (a+b x))^{-1+n} \, _2F_1\left (1,-1+n;n;-\frac{\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}\\ \end{align*}

Mathematica [A]  time = 0.0411354, size = 67, normalized size = 0.66 \[ \frac{\coth ^{-1}(\tanh (a+b x))^n \left (\frac{\coth ^{-1}(\tanh (a+b x))}{b x}\right )^{-n} \text{Hypergeometric2F1}\left (2-n,-n,3-n,1-\frac{\coth ^{-1}(\tanh (a+b x))}{b x}\right )}{(n-2) x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCoth[Tanh[a + b*x]]^n/x^3,x]

[Out]

(ArcCoth[Tanh[a + b*x]]^n*Hypergeometric2F1[2 - n, -n, 3 - n, 1 - ArcCoth[Tanh[a + b*x]]/(b*x)])/((-2 + n)*x^2
*(ArcCoth[Tanh[a + b*x]]/(b*x))^n)

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Maple [F]  time = 2.029, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm arccoth} \left (\tanh \left ( bx+a \right ) \right ) \right ) ^{n}}{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccoth(tanh(b*x+a))^n/x^3,x)

[Out]

int(arccoth(tanh(b*x+a))^n/x^3,x)

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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^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(tanh(b*x+a))^n/x^3,x, algorithm="maxima")

[Out]

integrate(arccoth(tanh(b*x + a))^n/x^3, x)

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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^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(tanh(b*x+a))^n/x^3,x, algorithm="fricas")

[Out]

integral(arccoth(tanh(b*x + a))^n/x^3, x)

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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^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acoth(tanh(b*x+a))**n/x**3,x)

[Out]

Integral(acoth(tanh(a + b*x))**n/x**3, x)

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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^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(tanh(b*x+a))^n/x^3,x, algorithm="giac")

[Out]

integrate(arccoth(tanh(b*x + a))^n/x^3, x)