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

Optimal. Leaf size=53 \[ -\frac{x^{m+1} \text{Hypergeometric2F1}\left (1,m+1,m+2,\frac{b x}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{(m+1) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )} \]

[Out]

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

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Rubi [A]  time = 0.0284971, antiderivative size = 53, 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} \[ -\frac{x^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b x}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{(m+1) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^m/ArcCoth[Tanh[a + b*x]],x]

[Out]

-((x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (b*x)/(b*x - ArcCoth[Tanh[a + b*x]])])/((1 + m)*(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{x^m}{\coth ^{-1}(\tanh (a+b x))} \, dx &=-\frac{x^{1+m} \, _2F_1\left (1,1+m;2+m;\frac{b x}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{(1+m) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}\\ \end{align*}

Mathematica [A]  time = 0.0842978, size = 51, normalized size = 0.96 \[ \frac{x^{m+1} \text{Hypergeometric2F1}\left (1,m+1,m+2,-\frac{b x}{\coth ^{-1}(\tanh (a+b x))-b x}\right )}{(m+1) \left (\coth ^{-1}(\tanh (a+b x))-b x\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.346, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{{\rm arccoth} \left (\tanh \left ( bx+a \right ) \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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