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]
(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]]))
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Rubi [A] time = 0.0351377, antiderivative size = 64, normalized size of antiderivative = 1.,
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 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]
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]
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*}
Mathematica [A] time = 0.0306651, 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]
-(b^2*x^2 + 2*b*x*ArcCoth[Tanh[a + b*x]] + 3*ArcCoth[Tanh[a + b*x]]^2)/(12*x^4)
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Maple [C] time = 0.362, size = 3217, normalized size = 50.3 \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^5,x)
[Out]
-1/4/x^4*ln(exp(b*x+a))^2-1/24*(4*b*x+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(b*x+a))^2*csgn(I*exp(2*b*x+2*a))-6*I*Pi-3*I*Pi*csgn(I*exp(2*b*x+2*a)/(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))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))+6*I*Pi*csg
n(I/(exp(2*b*x+2*a)+1))^2-3*I*Pi*csgn(I*exp(2*b*x+2*a))^3+6*I*Pi*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2+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))^2-6*I*Pi*csgn(I/(exp(2*b*x+2*a)+1))
^3)/x^4*ln(exp(b*x+a))-1/192*(-18*Pi^2*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))^4-12*Pi^2-12*Pi^2*csgn(I*ex
p(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3-16*I*Pi*x*b*csgn(I/(exp(2*b*x+2*a)+1))^3-3*Pi^2*csgn(I*exp(2*b*x+2*a))^6-8*
I*Pi*b*x*csgn(I*exp(2*b*x+2*a))^3-8*I*Pi*b*x*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3+16*I*Pi*x*b*csgn(I/(e
xp(2*b*x+2*a)+1))^2-12*Pi^2*csgn(I*exp(2*b*x+2*a))^3+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+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))^2-12*Pi^2*csg
n(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))^3*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)/(exp(2*b*x+2*a)+1))^4-8*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))-12*Pi^2*csgn(I/
(exp(2*b*x+2*a)+1))^6+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))^2-6*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+16*b^2*x^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))^3*csgn(I/(exp(2*b*x+2*a)+1))^2-1
2*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+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*x*b*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))+24*Pi^2*csgn(I*exp(b*x+a))*csgn(I*exp(
2*b*x+2*a))^2+24*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^2-24*Pi^2*csgn(I/(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))^5+6*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))^5-3*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+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))^3*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)/(exp(2*b*x+2*a)+1))^3*csgn(I/(exp(2*b*x+2*a)+1))^2-12*Pi^2*csgn(I*exp(b
*x+a))^2*csgn(I*exp(2*b*x+2*a))-16*I*Pi*x*b+12*Pi^2*csgn(I*exp(b*x+a))^3*csgn(I*exp(2*b*x+2*a))^3+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+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(b*x+a))^4*csgn(I*exp(2*b*x+2*a))^
2+24*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^5-12*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^4+6*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))^3-6*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))+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(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))^2-24*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^2*csgn(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)+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))-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))+24*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+12*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^3*csgn(I*exp(2*b*x+2*a))*csg
n(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2-3*Pi^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^6-12*Pi^2*csgn(I/(ex
p(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*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))^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3-6*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(2*b*x+2*a)+1))^2*csgn(I*exp(b*x+a))^2*csgn(I*exp(2
*b*x+2*a))-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))
^4-3*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(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))^2*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*e
xp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))-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)))/x^4
________________________________________________________________________________________
Maxima [A] time = 1.40713, size = 49, normalized size = 0.77 \begin{align*} -\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}} \end{align*}
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
________________________________________________________________________________________
Fricas [A] time = 1.61147, size = 72, normalized size = 1.12 \begin{align*} -\frac{24 \, b^{2} x^{2} + 32 \, a b x - 3 \, \pi ^{2} + 12 \, a^{2}}{48 \, x^{4}} \end{align*}
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
________________________________________________________________________________________
Sympy [A] time = 1.92498, size = 39, normalized size = 0.61 \begin{align*} - \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}} \end{align*}
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)
________________________________________________________________________________________
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^{5}}\,{d x} \end{align*}
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)