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3.145 \(\int \frac {\coth ^{-1}(\tanh (a+b x))^2}{x^5} \, dx\)

Optimal. Leaf size=64 \[ \frac {\coth ^{-1}(\tanh (a+b x))^3}{4 x^4 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}+\frac {b \coth ^{-1}(\tanh (a+b x))^3}{12 x^3 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2} \]

[Out]

1/12*b*arccoth(tanh(b*x+a))^3/x^3/(b*x-arccoth(tanh(b*x+a)))^2+1/4*arccoth(tanh(b*x+a))^3/x^4/(b*x-arccoth(tan
h(b*x+a)))

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Rubi [A]  time = 0.04, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2171, 2167} \[ \frac {\coth ^{-1}(\tanh (a+b x))^3}{4 x^4 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}+\frac {b \coth ^{-1}(\tanh (a+b x))^3}{12 x^3 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[ArcCoth[Tanh[a + b*x]]^2/x^5,x]

[Out]

(b*ArcCoth[Tanh[a + b*x]]^3)/(12*x^3*(b*x - ArcCoth[Tanh[a + b*x]])^2) + ArcCoth[Tanh[a + b*x]]^3/(4*x^4*(b*x
- ArcCoth[Tanh[a + b*x]]))

Rule 2167

Int[(u_)^(m_)*(v_)^(n_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, -Simp[(u^(m + 1)*v^
(n + 1))/((m + 1)*(b*u - a*v)), x] /; NeQ[b*u - a*v, 0]] /; FreeQ[{m, n}, x] && PiecewiseLinearQ[u, v, x] && E
qQ[m + n + 2, 0] && NeQ[m, -1]

Rule 2171

Int[(u_)^(m_)*(v_)^(n_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, -Simp[(u^(m + 1)*v^
(n + 1))/((m + 1)*(b*u - a*v)), x] + Dist[(b*(m + n + 2))/((m + 1)*(b*u - a*v)), Int[u^(m + 1)*v^n, x], x] /;
NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x] && NeQ[m + n + 2, 0] && LtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\coth ^{-1}(\tanh (a+b x))^2}{x^5} \, dx &=\frac {\coth ^{-1}(\tanh (a+b x))^3}{4 x^4 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}+\frac {b \int \frac {\coth ^{-1}(\tanh (a+b x))^2}{x^4} \, dx}{4 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}\\ &=\frac {b \coth ^{-1}(\tanh (a+b x))^3}{12 x^3 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}+\frac {\coth ^{-1}(\tanh (a+b x))^3}{4 x^4 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 37, normalized size = 0.58 \[ -\frac {2 b x \coth ^{-1}(\tanh (a+b x))+3 \coth ^{-1}(\tanh (a+b x))^2+b^2 x^2}{12 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCoth[Tanh[a + b*x]]^2/x^5,x]

[Out]

-1/12*(b^2*x^2 + 2*b*x*ArcCoth[Tanh[a + b*x]] + 3*ArcCoth[Tanh[a + b*x]]^2)/x^4

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fricas [A]  time = 0.61, size = 29, normalized size = 0.45 \[ -\frac {24 \, b^{2} x^{2} + 32 \, a b x - 3 \, \pi ^{2} + 12 \, a^{2}}{48 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(tanh(b*x+a))^2/x^5,x, algorithm="fricas")

[Out]

-1/48*(24*b^2*x^2 + 32*a*b*x - 3*pi^2 + 12*a^2)/x^4

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{2}}{x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(tanh(b*x+a))^2/x^5,x, algorithm="giac")

[Out]

integrate(arccoth(tanh(b*x + a))^2/x^5, x)

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maple [C]  time = 0.39, size = 3217, normalized size = 50.27 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccoth(tanh(b*x+a))^2/x^5,x)

[Out]

-1/4/x^4*ln(exp(b*x+a))^2-1/24*(4*b*x-3*I*Pi*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))+6*I*Pi*csgn(I*exp(b*x
+a))*csgn(I*exp(2*b*x+2*a))^2-6*I*Pi-3*I*Pi*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3-6*I*Pi*csgn(I/(exp(2*b
*x+2*a)+1))^3+3*I*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))^2-3*I*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))+6*I*Pi*csgn(I/(exp(2*b*x+2*a)
+1))^2+3*I*Pi*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2-3*I*Pi*csgn(I*exp(2*b*x+2*a))
^3)/x^4*ln(exp(b*x+a))-1/192*(-12*Pi^2+16*b^2*x^2+12*Pi^2*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^5+24*Pi^2*
csgn(I/(exp(2*b*x+2*a)+1))^5+12*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))^2-3*Pi^2
*csgn(I*exp(b*x+a))^4*csgn(I*exp(2*b*x+2*a))^2+12*Pi^2*csgn(I*exp(b*x+a))^3*csgn(I*exp(2*b*x+2*a))^3-12*csgn(I
*exp(2*b*x+2*a))^3*Pi^2-12*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^6+24*csgn(I/(exp(2*b*x+2*a)+1))^2*Pi^2-12*csgn(I*ex
p(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3*Pi^2-6*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))+12*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))-3*Pi^2*csgn(I*exp(2*b*x+2*a))^6-3*Pi^2*csgn(I*exp(2*b*x+2*a)/(exp(
2*b*x+2*a)+1))^6+16*I*Pi*b*x*csgn(I/(exp(2*b*x+2*a)+1))^2-16*I*Pi*b*x*csgn(I/(exp(2*b*x+2*a)+1))^3+24*Pi^2*csg
n(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2+12*Pi^2*csgn(I*exp(2*b*x+2*a))^3*csgn(I/(exp(2*b*x+2*a)+1))^2+12*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+6*Pi^2*csgn(I*exp(2*b*x+2*a))*csgn(I*e
xp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^5-12*Pi^2*csgn(I*exp(2*b*x+2*a))^3*csgn(I/(exp(2*b*x+2*a)+1))^3-3*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))^4-16*I*Pi*x*b+6*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))^5-12*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^4-12*Pi^2*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))^3+12*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))^2+12*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+24*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))^3+6*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))*c
sgn(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+12*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))-8*I*Pi*b*x*c
sgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))+16*I*Pi*b*x*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2+8*I*Pi*b*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+8*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))^2-12*Pi^2*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))^4-8*I*Pi*x*b*csgn(I*exp(2*b*x+2*a))^3-8*I*Pi*x*b*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x
+2*a)+1))^3-12*Pi^2*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))-18*Pi^2*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+
2*a))^4-6*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))^3-8*I*Pi*b*x*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))-3*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+6*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))^2-12*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))^3-12*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))-6*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+12*Pi^2*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2*csgn(I*e
xp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3-12*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^4*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b
*x+2*a)/(exp(2*b*x+2*a)+1))-24*csgn(I/(exp(2*b*x+2*a)+1))^3*Pi^2+12*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))^3-3*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+6*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+6*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))^2*csgn
(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3-12*Pi^2*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))+12*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
-12*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))^2-24*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+6*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))^2-12*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+6*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^2*csgn(I*exp(2*b*x+2*a))*cs
gn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3-6*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))^4*csgn(I*ex
p(2*b*x+2*a)/(exp(2*b*x+2*a)+1))-12*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)/x^4

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maxima [A]  time = 0.45, size = 36, normalized size = 0.56 \[ -\frac {b^{2}}{12 \, x^{2}} - \frac {b \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )}{6 \, x^{3}} - \frac {\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{2}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(tanh(b*x+a))^2/x^5,x, algorithm="maxima")

[Out]

-1/12*b^2/x^2 - 1/6*b*arccoth(tanh(b*x + a))/x^3 - 1/4*arccoth(tanh(b*x + a))^2/x^4

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mupad [B]  time = 1.17, size = 36, normalized size = 0.56 \[ -\frac {{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{4\,x^4}-\frac {b^2}{12\,x^2}-\frac {b\,\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{6\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acoth(tanh(a + b*x))^2/x^5,x)

[Out]

- acoth(tanh(a + b*x))^2/(4*x^4) - b^2/(12*x^2) - (b*acoth(tanh(a + b*x)))/(6*x^3)

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sympy [A]  time = 1.29, size = 39, normalized size = 0.61 \[ - \frac {b^{2}}{12 x^{2}} - \frac {b \operatorname {acoth}{\left (\tanh {\left (a + b x \right )} \right )}}{6 x^{3}} - \frac {\operatorname {acoth}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acoth(tanh(b*x+a))**2/x**5,x)

[Out]

-b**2/(12*x**2) - b*acoth(tanh(a + b*x))/(6*x**3) - acoth(tanh(a + b*x))**2/(4*x**4)

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