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

Optimal. Leaf size=12 \[ \frac {\log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b} \]

[Out]

ln(arccoth(tanh(b*x+a)))/b

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Rubi [A]  time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2157, 29} \[ \frac {\log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[ArcCoth[Tanh[a + b*x]]^(-1),x]

[Out]

Log[ArcCoth[Tanh[a + b*x]]]/b

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2157

Int[(u_)^(m_.), x_Symbol] :> With[{c = Simplify[D[u, x]]}, Dist[1/c, Subst[Int[x^m, x], x, u], x]] /; FreeQ[m,
x] && PiecewiseLinearQ[u, x]

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 12, normalized size = 1.00 \[ \frac {\log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCoth[Tanh[a + b*x]]^(-1),x]

[Out]

Log[ArcCoth[Tanh[a + b*x]]]/b

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fricas [B]  time = 0.68, size = 28, normalized size = 2.33 \[ \frac {\log \left (4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arccoth(tanh(b*x+a)),x, algorithm="fricas")

[Out]

1/2*log(4*b^2*x^2 + 8*a*b*x + pi^2 + 4*a^2)/b

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arccoth(tanh(b*x+a)),x, algorithm="giac")

[Out]

integrate(1/arccoth(tanh(b*x + a)), x)

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maple [A]  time = 0.07, size = 13, normalized size = 1.08 \[ \frac {\ln \left (\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/arccoth(tanh(b*x+a)),x)

[Out]

ln(arccoth(tanh(b*x+a)))/b

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maxima [C]  time = 0.41, size = 16, normalized size = 1.33 \[ \frac {\log \left (-\frac {1}{2} i \, \pi - b x - a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arccoth(tanh(b*x+a)),x, algorithm="maxima")

[Out]

log(-1/2*I*pi - b*x - a)/b

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mupad [B]  time = 1.18, size = 12, normalized size = 1.00 \[ \frac {\ln \left (\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/acoth(tanh(a + b*x)),x)

[Out]

log(acoth(tanh(a + b*x)))/b

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/acoth(tanh(b*x+a)),x)

[Out]

Exception raised: TypeError

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