3.144 \(\int \frac{\coth ^{-1}(\tanh (a+b x))^2}{x^4} \, dx\)
Optimal. Leaf size=31 \[ \frac{\coth ^{-1}(\tanh (a+b x))^3}{3 x^3 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )} \]
[Out]
ArcCoth[Tanh[a + b*x]]^3/(3*x^3*(b*x - ArcCoth[Tanh[a + b*x]]))
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Rubi [A] time = 0.0135441, antiderivative size = 31, normalized size of antiderivative = 1.,
number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used =
{2167} \[ \frac{\coth ^{-1}(\tanh (a+b x))^3}{3 x^3 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )} \]
Antiderivative was successfully verified.
[In]
Int[ArcCoth[Tanh[a + b*x]]^2/x^4,x]
[Out]
ArcCoth[Tanh[a + b*x]]^3/(3*x^3*(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]
Rubi steps
\begin{align*} \int \frac{\coth ^{-1}(\tanh (a+b x))^2}{x^4} \, dx &=\frac{\coth ^{-1}(\tanh (a+b x))^3}{3 x^3 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}\\ \end{align*}
Mathematica [A] time = 0.0474149, size = 34, normalized size = 1.1 \[ -\frac{b x \coth ^{-1}(\tanh (a+b x))+\coth ^{-1}(\tanh (a+b x))^2+b^2 x^2}{3 x^3} \]
Antiderivative was successfully verified.
[In]
Integrate[ArcCoth[Tanh[a + b*x]]^2/x^4,x]
[Out]
-(b^2*x^2 + b*x*ArcCoth[Tanh[a + b*x]] + ArcCoth[Tanh[a + b*x]]^2)/(3*x^3)
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Maple [C] time = 0.357, size = 3217, normalized size = 103.8 \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^4,x)
[Out]
-1/3/x^3*ln(exp(b*x+a))^2-1/6*(2*b*x+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-I
*Pi*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))-2*I*Pi+2*I*Pi*csgn(I/(exp(2*b*x+2*a)+1))^2-I*Pi*csgn(I*exp(2*b
*x+2*a)/(exp(2*b*x+2*a)+1))^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)/(ex
p(2*b*x+2*a)+1))-I*Pi*csgn(I*exp(2*b*x+2*a))^3+2*I*Pi*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2+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))^2-2*I*Pi*csgn(I/(exp(2*b*x+2*a)+1))^3)/x^3*ln(ex
p(b*x+a))-1/48*(-4*I*Pi*x*b*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))^4-4*Pi^2-4*Pi^2*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))^6-4*Pi^2*
csgn(I*exp(2*b*x+2*a))^3+4*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+4*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))^2-4*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))^3-4*Pi^2*csgn(I*exp(2*b*x+2*a))^3*csgn(I/(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))^4-4*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))+4*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+8*I*Pi*b*x*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2+4*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))^2-4*I*Pi*x*b*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3
-4*I*Pi*b*x*csgn(I*exp(2*b*x+2*a))^3-4*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^6+2*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))^2-2*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+4*Pi^2*csgn(I*exp(2*b*x+2*a))^3*csgn(I/(exp(2*b*x+2*a)+1))^2-4*Pi^2*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*csgn(I/(exp(2*b*x+2*a)+1))^2+8*I*Pi*x*b*csgn(I/(exp(2*b*x+2*a)+1))^2-
8*I*Pi*x*b*csgn(I/(exp(2*b*x+2*a)+1))^3+8*Pi^2*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2+8*Pi^2*csgn(I/(exp(
2*b*x+2*a)+1))^2-8*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^3+4*Pi^2*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^5+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))^5-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+2*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-
4*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+4*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))^2-4*Pi^2*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))-8*I*Pi*x*
b+4*Pi^2*csgn(I*exp(b*x+a))^3*csgn(I*exp(2*b*x+2*a))^3+4*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+4*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))-Pi^2*csgn(I*exp(b*x+a))^4*csgn(I*exp(2*b*x+2*a))^2+8*Pi^2*csgn(I/(exp(2*b*x+2*a)+1))^5-4*Pi^2*cs
gn(I/(exp(2*b*x+2*a)+1))^4+2*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)/(e
xp(2*b*x+2*a)+1))^3-2*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))+2*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+2*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))^3-4*P
i^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))^2-8*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))^2-4*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))-4*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))+8*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+4*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-Pi^2*csgn(I*ex
p(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^6-4*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))+2*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+4*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))^3-2*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+4*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))-4*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-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+2*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))^2-4*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+4*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))-2*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^3
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Maxima [A] time = 1.40024, size = 49, normalized size = 1.58 \begin{align*} -\frac{b^{2}}{3 \, x} - \frac{b \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )}{3 \, x^{2}} - \frac{\operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{2}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arccoth(tanh(b*x+a))^2/x^4,x, algorithm="maxima")
[Out]
-1/3*b^2/x - 1/3*b*arccoth(tanh(b*x + a))/x^2 - 1/3*arccoth(tanh(b*x + a))^2/x^3
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Fricas [A] time = 1.58129, size = 68, normalized size = 2.19 \begin{align*} -\frac{12 \, b^{2} x^{2} + 12 \, a b x - \pi ^{2} + 4 \, a^{2}}{12 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arccoth(tanh(b*x+a))^2/x^4,x, algorithm="fricas")
[Out]
-1/12*(12*b^2*x^2 + 12*a*b*x - pi^2 + 4*a^2)/x^3
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Sympy [A] time = 1.18208, size = 37, normalized size = 1.19 \begin{align*} - \frac{b^{2}}{3 x} - \frac{b \operatorname{acoth}{\left (\tanh{\left (a + b x \right )} \right )}}{3 x^{2}} - \frac{\operatorname{acoth}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{3 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(acoth(tanh(b*x+a))**2/x**4,x)
[Out]
-b**2/(3*x) - b*acoth(tanh(a + b*x))/(3*x**2) - acoth(tanh(a + b*x))**2/(3*x**3)
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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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arccoth(tanh(b*x+a))^2/x^4,x, algorithm="giac")
[Out]
integrate(arccoth(tanh(b*x + a))^2/x^4, x)