3.139 \(\int x \coth ^{-1}(\tanh (a+b x))^2 \, dx\)
Optimal. Leaf size=34 \[ \frac{x \coth ^{-1}(\tanh (a+b x))^3}{3 b}-\frac{\coth ^{-1}(\tanh (a+b x))^4}{12 b^2} \]
[Out]
(x*ArcCoth[Tanh[a + b*x]]^3)/(3*b) - ArcCoth[Tanh[a + b*x]]^4/(12*b^2)
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Rubi [A] time = 0.0253549, antiderivative size = 34, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used =
{2168, 2157, 30} \[ \frac{x \coth ^{-1}(\tanh (a+b x))^3}{3 b}-\frac{\coth ^{-1}(\tanh (a+b x))^4}{12 b^2} \]
Antiderivative was successfully verified.
[In]
Int[x*ArcCoth[Tanh[a + b*x]]^2,x]
[Out]
(x*ArcCoth[Tanh[a + b*x]]^3)/(3*b) - ArcCoth[Tanh[a + b*x]]^4/(12*b^2)
Rule 2168
Int[(u_)^(m_)*(v_)^(n_.), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(u^(m + 1)*v^
n)/(a*(m + 1)), x] - Dist[(b*n)/(a*(m + 1)), Int[u^(m + 1)*v^(n - 1), x], x] /; NeQ[b*u - a*v, 0]] /; FreeQ[{m
, n}, x] && PiecewiseLinearQ[u, v, x] && NeQ[m, -1] && ((LtQ[m, -1] && GtQ[n, 0] && !(ILtQ[m + n, -2] && (Fra
ctionQ[m] || GeQ[2*n + m + 1, 0]))) || (IGtQ[n, 0] && IGtQ[m, 0] && LeQ[n, m]) || (IGtQ[n, 0] && !IntegerQ[m]
) || (ILtQ[m, 0] && !IntegerQ[n]))
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]
Rule 30
Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]
Rubi steps
\begin{align*} \int x \coth ^{-1}(\tanh (a+b x))^2 \, dx &=\frac{x \coth ^{-1}(\tanh (a+b x))^3}{3 b}-\frac{\int \coth ^{-1}(\tanh (a+b x))^3 \, dx}{3 b}\\ &=\frac{x \coth ^{-1}(\tanh (a+b x))^3}{3 b}-\frac{\operatorname{Subst}\left (\int x^3 \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{3 b^2}\\ &=\frac{x \coth ^{-1}(\tanh (a+b x))^3}{3 b}-\frac{\coth ^{-1}(\tanh (a+b x))^4}{12 b^2}\\ \end{align*}
Mathematica [B] time = 0.0822737, size = 74, normalized size = 2.18 \[ \frac{(a+b x) \left (4 \left (2 a^2+a b x-b^2 x^2\right ) \coth ^{-1}(\tanh (a+b x))-(3 a-b x) (a+b x)^2-6 (a-b x) \coth ^{-1}(\tanh (a+b x))^2\right )}{12 b^2} \]
Antiderivative was successfully verified.
[In]
Integrate[x*ArcCoth[Tanh[a + b*x]]^2,x]
[Out]
((a + b*x)*(-((3*a - b*x)*(a + b*x)^2) + 4*(2*a^2 + a*b*x - b^2*x^2)*ArcCoth[Tanh[a + b*x]] - 6*(a - b*x)*ArcC
oth[Tanh[a + b*x]]^2))/(12*b^2)
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Maple [C] time = 0.368, size = 3418, normalized size = 100.5 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*arccoth(tanh(b*x+a))^2,x)
[Out]
-1/4*Pi^2*x^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))^2-1/8*Pi^2*x^2*csgn(I/(ex
p(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))^2+1/4*Pi^2*x^2*csgn(I/(exp
(2*b*x+2*a)+1))^5+1/16*Pi^2*x^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+1/16*Pi^2*x^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*Pi^2*x^2+
1/2*x^2*ln(exp(b*x+a))^2+1/16*Pi^2*x^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*Pi^2*x^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))^3-1/8*Pi^2*x^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
-1/8*Pi^2*x^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+1/8*Pi^2*x^2*csgn(I/(ex
p(2*b*x+2*a)+1))^2*csgn(I*exp(2*b*x+2*a))^3+1/8*Pi^2*x^2*csgn(I/(exp(2*b*x+2*a)+1))^2*csgn(I*exp(2*b*x+2*a)/(e
xp(2*b*x+2*a)+1))^3-1/8*Pi^2*x^2*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))+1/4*Pi^2*x^2*csgn(I*exp(b*x+a))*c
sgn(I*exp(2*b*x+2*a))^2+1/8*Pi^2*x^2*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2-1/32*P
i^2*x^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))^4-1/32*Pi^2*x^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+1/16*Pi^2*x^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+1/8*Pi^2*x^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))^2-3
/16*Pi^2*x^2*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))^4+1/8*Pi^2*x^2*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*
a))^5+1/16*Pi^2*x^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))^2-1/16*Pi^2*x^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))^3-1/8*Pi^2*x^2*csgn(I*exp(2*b*x+2*a))^3*csgn(I/(ex
p(2*b*x+2*a)+1))^3+1/8*Pi^2*x^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))^2-1/32*
Pi^2*x^2*csgn(I*exp(b*x+a))^4*csgn(I*exp(2*b*x+2*a))^2+1/8*Pi^2*x^2*csgn(I*exp(b*x+a))^3*csgn(I*exp(2*b*x+2*a)
)^3-1/32*Pi^2*x^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+1/16*Pi^2*x^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))^3-1/16*Pi^2*x^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))-1/4*Pi^2*x^2*csgn(I/(exp(2*b*x+2*a)+1))^3-1/8*Pi^2*x^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3-1/32*Pi
^2*x^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^6-1/16*Pi^2*x^2*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))*c
sgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3-1/8*Pi^2*x^2*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))*csgn(I/(ex
p(2*b*x+2*a)+1))^3+1/12*I*Pi*b*x^3*csgn(I*exp(2*b*x+2*a))^3-1/16*Pi^2*x^2*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*ex
p(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))+1/16*Pi^2*x^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))^2-1/8*Pi^2*x^2
*csgn(I/(exp(2*b*x+2*a)+1))^6-1/8*Pi^2*x^2*csgn(I*exp(2*b*x+2*a))^3-1/32*Pi^2*x^2*csgn(I*exp(2*b*x+2*a))^6-1/8
*Pi^2*x^2*csgn(I/(exp(2*b*x+2*a)+1))^4+1/4*Pi^2*x^2*csgn(I/(exp(2*b*x+2*a)+1))^2+(-1/3*b*x^3+1/4*I*Pi*x^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))^2-1/4*I*Pi*x^2*csgn(I/(exp(2*b*x+2*a)+1))*csg
n(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))+1/4*I*Pi*x^2*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2
*b*x+2*a)/(exp(2*b*x+2*a)+1))^2-1/2*I*Pi*x^2*csgn(I/(exp(2*b*x+2*a)+1))^3-1/4*I*Pi*x^2*csgn(I*exp(2*b*x+2*a)/(
exp(2*b*x+2*a)+1))^3+1/2*I*Pi*x^2*csgn(I/(exp(2*b*x+2*a)+1))^2+1/2*I*Pi*x^2*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*
x+2*a))^2-1/4*I*Pi*x^2*csgn(I*exp(2*b*x+2*a))^3-1/2*I*Pi*x^2-1/4*I*Pi*x^2*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*
x+2*a)))*ln(exp(b*x+a))+1/12*I*Pi*b*x^3*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3+1/8*Pi^2*x^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))+1/12*x^4*b^2+1/8*Pi^2*x^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+1/8*Pi^2*x^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))+1/6*I*Pi*b*x^3*csgn(I/(exp(2*b*x+2*a)+1))
^3+1/4*Pi^2*x^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))^3-1/8*Pi^2*x^2*csgn(I*e
xp(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))^2-1/8*Pi^2*x^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))+1/6*I*Pi*b*x^3+1/12*I*Pi*b*x^3*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))-1/8*Pi^2*x^2*csgn(I/(e
xp(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+1/8*Pi^2*x^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))^3-1/6*I*Pi*b*x^3*csgn(I/(exp(2*b*x+2
*a)+1))^2-1/8*Pi^2*x^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))-1/6*I*Pi*b*x^3*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2-1/12*I*Pi*b*x^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+1/8*Pi^2*x^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))-1/8*Pi^2*x^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))^2-1/12*I*Pi*b*x^3*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+1/12*I*Pi*b*x^3*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))
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Maxima [A] time = 1.36817, size = 49, normalized size = 1.44 \begin{align*} \frac{1}{12} \, b^{2} x^{4} - \frac{1}{3} \, b x^{3} \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right ) + \frac{1}{2} \, x^{2} \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*arccoth(tanh(b*x+a))^2,x, algorithm="maxima")
[Out]
1/12*b^2*x^4 - 1/3*b*x^3*arccoth(tanh(b*x + a)) + 1/2*x^2*arccoth(tanh(b*x + a))^2
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Fricas [A] time = 1.52462, size = 70, normalized size = 2.06 \begin{align*} \frac{1}{4} \, b^{2} x^{4} + \frac{2}{3} \, a b x^{3} - \frac{1}{8} \,{\left (\pi ^{2} - 4 \, a^{2}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*arccoth(tanh(b*x+a))^2,x, algorithm="fricas")
[Out]
1/4*b^2*x^4 + 2/3*a*b*x^3 - 1/8*(pi^2 - 4*a^2)*x^2
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Sympy [A] time = 0.586048, size = 37, normalized size = 1.09 \begin{align*} \frac{b^{2} x^{4}}{12} - \frac{b x^{3} \operatorname{acoth}{\left (\tanh{\left (a + b x \right )} \right )}}{3} + \frac{x^{2} \operatorname{acoth}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*acoth(tanh(b*x+a))**2,x)
[Out]
b**2*x**4/12 - b*x**3*acoth(tanh(a + b*x))/3 + x**2*acoth(tanh(a + b*x))**2/2
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*arccoth(tanh(b*x+a))^2,x, algorithm="giac")
[Out]
integrate(x*arccoth(tanh(b*x + a))^2, x)