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3.141 \(\int \frac{\coth ^{-1}(\tanh (a+b x))^2}{x} \, dx\)

Optimal. Leaf size=49 \[ -b x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )+\frac{1}{2} \coth ^{-1}(\tanh (a+b x))^2+\log (x) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \]

[Out]

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

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Rubi [A]  time = 0.0259427, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2159, 2158, 29} \[ -b x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )+\frac{1}{2} \coth ^{-1}(\tanh (a+b x))^2+\log (x) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \]

Antiderivative was successfully verified.

[In]

Int[ArcCoth[Tanh[a + b*x]]^2/x,x]

[Out]

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

Rule 2159

Int[(v_)^(n_)/(u_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[v^n/(a*n), x] - Dis
t[(b*u - a*v)/a, Int[v^(n - 1)/u, x], x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x] && GtQ[n, 0] && Ne
Q[n, 1]

Rule 2158

Int[(v_)/(u_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(b*x)/a, x] - Dist[(b*u
- a*v)/a, Int[1/u, x], x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x]

Rule 29

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

Rubi steps

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

Mathematica [A]  time = 0.0427117, size = 53, normalized size = 1.08 \[ \frac{1}{2} (a+b x)^2-(a+b x) \left (-2 \coth ^{-1}(\tanh (a+b x))+a+2 b x\right )+\log (b x) \left (\coth ^{-1}(\tanh (a+b x))-b x\right )^2 \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCoth[Tanh[a + b*x]]^2/x,x]

[Out]

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

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Maple [C]  time = 0.284, size = 3774, normalized size = 77. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccoth(tanh(b*x+a))^2/x,x)

[Out]

-1/8*ln(x)*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))^2*csgn(I*exp(2*b*x+2*a)
/(exp(2*b*x+2*a)+1))+I*Pi*b*x*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2-1/2*I*ln(exp(b*x+a))*ln(x)*Pi*csgn(I
*exp(2*b*x+2*a))^3-1/2*I*Pi*x*b*csgn(I*exp(2*b*x+2*a))^3-1/2*I*Pi*x*b*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1)
)^3-1/8*ln(x)*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))^4*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1
))-1/2*I*ln(exp(b*x+a))*ln(x)*Pi*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3+1/8*ln(x)*Pi^2*csgn(I/(exp(2*b*x+
2*a)+1))^2*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3+1/2*I*ln(x)*Pi*x*b*csgn(I*exp(2*
b*x+2*a)/(exp(2*b*x+2*a)+1))^3+1/2*I*Pi*x*b*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2
-1/2*I*ln(x)*Pi*x*b*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2-1/2*I*ln(exp(b*x+a))*ln
(x)*Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))+b^2*x^2*ln(
x)+2*b*ln(exp(b*x+a))*x-1/2*Pi^2*ln(x)*csgn(I/(exp(2*b*x+2*a)+1))^3+1/2*Pi^2*ln(x)*csgn(I/(exp(2*b*x+2*a)+1))^
2-1/2*I*Pi*x*b*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))+1/2
*ln(x)*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^3*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2-1/16*ln(x)*Pi^2*csgn(I*ex
p(b*x+a))^4*csgn(I*exp(2*b*x+2*a))^2+1/4*ln(x)*Pi^2*csgn(I*exp(b*x+a))^3*csgn(I*exp(2*b*x+2*a))^3-1/4*ln(x)*Pi
^2*csgn(I/(exp(2*b*x+2*a)+1))^3*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))+1/4*ln(x)*Pi^2*csgn(I/(exp(2*b*x+2
*a)+1))^3*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2-1/16*ln(x)*Pi^2*csgn(I*exp(2*b*x+
2*a))^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^4+1/8*ln(x)*Pi^2*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a
)/(exp(2*b*x+2*a)+1))^5-1/4*ln(x)*Pi^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3*csgn(I/(exp(2*b*x+2*a)+1))^
3+1/4*ln(x)*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2-I*ln(exp(b*x+a))*ln(x)
*Pi*csgn(I/(exp(2*b*x+2*a)+1))^3+I*ln(exp(b*x+a))*ln(x)*Pi*csgn(I/(exp(2*b*x+2*a)+1))^2-I*Pi*x*b*csgn(I/(exp(2
*b*x+2*a)+1))^3+1/8*ln(x)*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))^3*csgn(I*exp(2*b*x+2*a)/(exp(
2*b*x+2*a)+1))^2-1/16*ln(x)*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^2*csgn(I*exp(2*b*x+2*a))^2*csgn(I*exp(2*b*x+2*a)/(
exp(2*b*x+2*a)+1))^2+I*Pi*x*b*csgn(I/(exp(2*b*x+2*a)+1))^2-1/4*ln(x)*Pi^2*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+
2*a))^3*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2-3/2*b^2*x^2+1/8*ln(x)*Pi^2*csgn(I*exp(b*x+a))^2*csgn(I*exp
(2*b*x+2*a))^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+1/8*ln(x)*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*ex
p(2*b*x+2*a))^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3-1/4*ln(x)*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*e
xp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))-1/4*ln(x)*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^4*csgn(I*ex
p(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))+1/4*ln(x)*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^3*csgn(I*exp
(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))-1/2*ln(x)*Pi^2*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a)
)^2*csgn(I/(exp(2*b*x+2*a)+1))^2-1/4*ln(x)*Pi^2*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1
))^2*csgn(I/(exp(2*b*x+2*a)+1))^2+1/4*ln(x)*Pi^2*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))*csgn(I/(exp(2*b*x
+2*a)+1))^2+1/2*ln(x)*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^5-1/4*ln(x)*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^4-1/4*ln(x)*
Pi^2*csgn(I*exp(2*b*x+2*a))^3-1/16*ln(x)*Pi^2*csgn(I*exp(2*b*x+2*a))^6-1/4*ln(x)*Pi^2*csgn(I/(exp(2*b*x+2*a)+1
))^6-1/4*ln(x)*Pi^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3-I*ln(exp(b*x+a))*ln(x)*Pi-I*Pi*x*b+1/4*ln(x)*P
i^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3*csgn(I/(exp(2*b*x+2*a)+1))^2-1/16*ln(x)*Pi^2*csgn(I*exp(2*b*x+
2*a)/(exp(2*b*x+2*a)+1))^6-3/8*ln(x)*Pi^2*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))^4+1/4*ln(x)*Pi^2*csgn(I*
exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^5+1/8*ln(x)*Pi^2*csgn(I*exp(2*b*x+2*a))^4*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+
2*a)+1))^2-1/8*ln(x)*Pi^2*csgn(I*exp(2*b*x+2*a))^3*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3-1/16*ln(x)*Pi^2
*csgn(I/(exp(2*b*x+2*a)+1))^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^4+1/4*ln(x)*Pi^2*csgn(I/(exp(2*b*x+2*a
)+1))^4*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2-1/4*ln(x)*Pi^2*csgn(I*exp(2*b*x+2*a))^3*csgn(I/(exp(2*b*x+
2*a)+1))^3-1/4*Pi^2*ln(x)*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))+1/2*Pi^2*ln(x)*csgn(I*exp(b*x+a))*csgn(I
*exp(2*b*x+2*a))^2+1/4*Pi^2*ln(x)*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2-1/4*ln(x)
*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^3*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+1/4*ln(x)*Pi^2*csgn(I/(exp(2*b*
x+2*a)+1))^2*csgn(I*exp(2*b*x+2*a))^3+1/2*I*ln(x)*Pi*x*b*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))*csg
n(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))-2*b*ln(exp(b*x+a))*ln(x)*x+I*ln(x)*Pi*x*b+ln(x)*ln(exp(b*x+a))^2+1/8*ln
(x)*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^5-1/8*ln(x)*Pi^2*csgn(I*exp(b*x+
a))^2*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3+1/4*ln(x)*Pi^2*csgn(I*exp(b*x+a))*csg
n(I*exp(2*b*x+2*a))^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3-1/4*ln(x)*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))*cs
gn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^4-I*ln(x)*Pi*x*b*csgn(I/(exp(2*b*x+2*a)+1))^2-1
/4*Pi^2*ln(x)+I*ln(x)*Pi*x*b*csgn(I/(exp(2*b*x+2*a)+1))^3+1/4*ln(x)*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp
(b*x+a))*csgn(I*exp(2*b*x+2*a))^3*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))-1/4*ln(x)*Pi^2*csgn(I/(exp(2*b*x+2
*a)+1))*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+1/2*I*ln(x)*Pi
*x*b*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))-I*ln(x)*Pi*x*b*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2+1/
2*I*ln(exp(b*x+a))*ln(x)*Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2-1/2*I*ln(ex
p(b*x+a))*ln(x)*Pi*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))+1/2*I*ln(x)*Pi*x*b*csgn(I*exp(2*b*x+2*a))^3+1/2
*I*ln(exp(b*x+a))*ln(x)*Pi*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2-1/2*I*Pi*x*b*csg
n(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))+1/8*ln(x)*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(b*x+a))^2*csgn(I
*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+I*ln(exp(b*x+a))*ln(x)*Pi*csgn(I*exp(b*x+a))*csgn
(I*exp(2*b*x+2*a))^2+1/2*I*Pi*b*x*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2-1/2*I
*ln(x)*Pi*x*b*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2

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Maxima [C]  time = 2.47337, size = 51, normalized size = 1.04 \begin{align*} \frac{1}{2} \, b^{2} x^{2} + \frac{1}{8} \,{\left (-8 i \, \pi b + 16 \, a b\right )} x - \frac{1}{4} \,{\left (\pi ^{2} + 4 i \, \pi a - 4 \, a^{2}\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(tanh(b*x+a))^2/x,x, algorithm="maxima")

[Out]

1/2*b^2*x^2 + 1/8*(-8*I*pi*b + 16*a*b)*x - 1/4*(pi^2 + 4*I*pi*a - 4*a^2)*log(x)

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Fricas [A]  time = 1.65362, size = 69, normalized size = 1.41 \begin{align*} \frac{1}{2} \, b^{2} x^{2} + 2 \, a b x - \frac{1}{4} \,{\left (\pi ^{2} - 4 \, a^{2}\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(tanh(b*x+a))^2/x,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acoth}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acoth(tanh(b*x+a))**2/x,x)

[Out]

Integral(acoth(tanh(a + b*x))**2/x, 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 )^{2}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(tanh(b*x+a))^2/x,x, algorithm="giac")

[Out]

integrate(arccoth(tanh(b*x + a))^2/x, x)

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