3.65 \(\int \coth ^{-1}(a+b x) \, dx\)

Optimal. Leaf size=35 \[ \frac{\log \left (1-(a+b x)^2\right )}{2 b}+\frac{(a+b x) \coth ^{-1}(a+b x)}{b} \]

[Out]

((a + b*x)*ArcCoth[a + b*x])/b + Log[1 - (a + b*x)^2]/(2*b)

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Rubi [A]  time = 0.0159401, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6104, 5911, 260} \[ \frac{\log \left (1-(a+b x)^2\right )}{2 b}+\frac{(a+b x) \coth ^{-1}(a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[ArcCoth[a + b*x],x]

[Out]

((a + b*x)*ArcCoth[a + b*x])/b + Log[1 - (a + b*x)^2]/(2*b)

Rule 6104

Int[((a_.) + ArcCoth[(c_) + (d_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Dist[1/d, Subst[Int[(a + b*ArcCoth[x])^p, x
], x, c + d*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]

Rule 5911

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCoth[c*x])^p, x] - Dist[b*c*p, In
t[(x*(a + b*ArcCoth[c*x])^(p - 1))/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 0]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \coth ^{-1}(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \coth ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac{(a+b x) \coth ^{-1}(a+b x)}{b}-\frac{\operatorname{Subst}\left (\int \frac{x}{1-x^2} \, dx,x,a+b x\right )}{b}\\ &=\frac{(a+b x) \coth ^{-1}(a+b x)}{b}+\frac{\log \left (1-(a+b x)^2\right )}{2 b}\\ \end{align*}

Mathematica [A]  time = 0.0145124, size = 43, normalized size = 1.23 \[ \frac{(a+1) \log (a+b x+1)-(a-1) \log (-a-b x+1)}{2 b}+x \coth ^{-1}(a+b x) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCoth[a + b*x],x]

[Out]

x*ArcCoth[a + b*x] + (-((-1 + a)*Log[1 - a - b*x]) + (1 + a)*Log[1 + a + b*x])/(2*b)

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Maple [A]  time = 0.03, size = 36, normalized size = 1. \begin{align*} x{\rm arccoth} \left (bx+a\right )+{\frac{{\rm arccoth} \left (bx+a\right )a}{b}}+{\frac{\ln \left ( \left ( bx+a \right ) ^{2}-1 \right ) }{2\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccoth(b*x+a),x)

[Out]

x*arccoth(b*x+a)+1/b*arccoth(b*x+a)*a+1/2/b*ln((b*x+a)^2-1)

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Maxima [A]  time = 0.973925, size = 42, normalized size = 1.2 \begin{align*} \frac{2 \,{\left (b x + a\right )} \operatorname{arcoth}\left (b x + a\right ) + \log \left (-{\left (b x + a\right )}^{2} + 1\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(b*x+a),x, algorithm="maxima")

[Out]

1/2*(2*(b*x + a)*arccoth(b*x + a) + log(-(b*x + a)^2 + 1))/b

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Fricas [A]  time = 1.54404, size = 135, normalized size = 3.86 \begin{align*} \frac{b x \log \left (\frac{b x + a + 1}{b x + a - 1}\right ) +{\left (a + 1\right )} \log \left (b x + a + 1\right ) -{\left (a - 1\right )} \log \left (b x + a - 1\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(b*x+a),x, algorithm="fricas")

[Out]

1/2*(b*x*log((b*x + a + 1)/(b*x + a - 1)) + (a + 1)*log(b*x + a + 1) - (a - 1)*log(b*x + a - 1))/b

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Sympy [A]  time = 0.8191, size = 41, normalized size = 1.17 \begin{align*} \begin{cases} \frac{a \operatorname{acoth}{\left (a + b x \right )}}{b} + x \operatorname{acoth}{\left (a + b x \right )} + \frac{\log{\left (a + b x + 1 \right )}}{b} - \frac{\operatorname{acoth}{\left (a + b x \right )}}{b} & \text{for}\: b \neq 0 \\x \operatorname{acoth}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acoth(b*x+a),x)

[Out]

Piecewise((a*acoth(a + b*x)/b + x*acoth(a + b*x) + log(a + b*x + 1)/b - acoth(a + b*x)/b, Ne(b, 0)), (x*acoth(
a), True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(b*x+a),x, algorithm="giac")

[Out]

integrate(arccoth(b*x + a), x)