3.138 \(\int x^2 \coth ^{-1}(\tanh (a+b x))^2 \, dx\)
Optimal. Leaf size=42 \[ -\frac{1}{6} b x^4 \coth ^{-1}(\tanh (a+b x))+\frac{1}{3} x^3 \coth ^{-1}(\tanh (a+b x))^2+\frac{b^2 x^5}{30} \]
[Out]
(b^2*x^5)/30 - (b*x^4*ArcCoth[Tanh[a + b*x]])/6 + (x^3*ArcCoth[Tanh[a + b*x]]^2)/3
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Rubi [A] time = 0.02533, antiderivative size = 42, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used =
{2168, 30} \[ -\frac{1}{6} b x^4 \coth ^{-1}(\tanh (a+b x))+\frac{1}{3} x^3 \coth ^{-1}(\tanh (a+b x))^2+\frac{b^2 x^5}{30} \]
Antiderivative was successfully verified.
[In]
Int[x^2*ArcCoth[Tanh[a + b*x]]^2,x]
[Out]
(b^2*x^5)/30 - (b*x^4*ArcCoth[Tanh[a + b*x]])/6 + (x^3*ArcCoth[Tanh[a + b*x]]^2)/3
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 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^2 \coth ^{-1}(\tanh (a+b x))^2 \, dx &=\frac{1}{3} x^3 \coth ^{-1}(\tanh (a+b x))^2-\frac{1}{3} (2 b) \int x^3 \coth ^{-1}(\tanh (a+b x)) \, dx\\ &=-\frac{1}{6} b x^4 \coth ^{-1}(\tanh (a+b x))+\frac{1}{3} x^3 \coth ^{-1}(\tanh (a+b x))^2+\frac{1}{6} b^2 \int x^4 \, dx\\ &=\frac{b^2 x^5}{30}-\frac{1}{6} b x^4 \coth ^{-1}(\tanh (a+b x))+\frac{1}{3} x^3 \coth ^{-1}(\tanh (a+b x))^2\\ \end{align*}
Mathematica [A] time = 0.0561201, size = 37, normalized size = 0.88 \[ \frac{1}{30} x^3 \left (-5 b x \coth ^{-1}(\tanh (a+b x))+10 \coth ^{-1}(\tanh (a+b x))^2+b^2 x^2\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[x^2*ArcCoth[Tanh[a + b*x]]^2,x]
[Out]
(x^3*(b^2*x^2 - 5*b*x*ArcCoth[Tanh[a + b*x]] + 10*ArcCoth[Tanh[a + b*x]]^2))/30
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Maple [C] time = 0.372, size = 3418, normalized size = 81.4 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*arccoth(tanh(b*x+a))^2,x)
[Out]
-1/12*Pi^2*x^3*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))^2
-1/6*Pi^2*x^3*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+1/6*Pi^2*x^3*csgn(I/(ex
p(2*b*x+2*a)+1))^5-1/12*Pi^2*x^3*csgn(I/(exp(2*b*x+2*a)+1))^6+1/12*Pi^2*x^3*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*Pi^2*x^3*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2
*csgn(I/(exp(2*b*x+2*a)+1))^3-1/12*Pi^2*x^3+1/3*x^3*ln(exp(b*x+a))^2-1/12*Pi^2*x^3*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))-1/12*Pi^2*x^3*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))-1/48*Pi^2*x^3*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+(-1/6*b*x^4-1/3*I*Pi*x^3*csgn(I/(exp(2*b*x+2*
a)+1))^3-1/6*I*Pi*x^3*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))-1/6*I*Pi*x^3*csgn(I*exp(2*b*x+2*a)/(exp(2*b*
x+2*a)+1))^3+1/3*I*Pi*x^3*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2-1/6*I*Pi*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))+1/6*I*Pi*x^3*csgn(I*exp(2*b*x+2*a))*csgn(I*ex
p(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2-1/6*I*Pi*x^3*csgn(I*exp(2*b*x+2*a))^3-1/3*I*Pi*x^3+1/6*I*Pi*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+1/3*I*Pi*x^3*csgn(I/(exp(2*b*x+2*a)+1))^2)*ln(exp(
b*x+a))-1/12*I*Pi*b*x^4*csgn(I/(exp(2*b*x+2*a)+1))^2+1/12*I*Pi*b*x^4*csgn(I/(exp(2*b*x+2*a)+1))^3+1/12*Pi^2*x^
3*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+1/24*Pi^2*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))^5-1/12*Pi^2*x^3*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+1/12*Pi^2*x^3*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-
1/24*Pi^2*x^3*csgn(I*exp(2*b*x+2*a))^3*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3-1/12*Pi^2*x^3*csgn(I*exp(2*
b*x+2*a))^3*csgn(I/(exp(2*b*x+2*a)+1))^3-1/48*Pi^2*x^3*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+1/12*Pi^2*x^3*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^5+1/24*Pi^2*x^3*csgn(I*exp(2*b*x+2*a))^4
*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2-1/48*Pi^2*x^3*csgn(I*exp(b*x+a))^4*csgn(I*exp(2*b*x+2*a))^2+1/12*
Pi^2*x^3*csgn(I*exp(b*x+a))^3*csgn(I*exp(2*b*x+2*a))^3-1/8*Pi^2*x^3*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a)
)^4-1/48*Pi^2*x^3*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+1/24*Pi^2*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))^5+1/12*Pi^2*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))^2-1/12*Pi^2*x^3*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+1/12*Pi^2*x^3*csgn(I*exp(2*b*x+2*a))^3*csgn(I/(exp(2*b*x+2*a)+1))^2-1/6*Pi^2*x^3*csgn(I/(exp(2*b
*x+2*a)+1))^3-1/24*I*Pi*b*x^4*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2-1/12*Pi^2*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))^4-1/24*Pi^2*x^3*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))^3-1/12*Pi^2*x^3*csgn(I*ex
p(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3-1/12*Pi^2*x^3*csgn(I*exp(2*b*x+2*a))^3+1/6*Pi^2*x^3*csgn(I/(exp(2*b*x+2*a)+
1))^2-1/12*Pi^2*x^3*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))*csgn(I/(exp(2*b*x+2*a)+1))^3-1/12*Pi^2*x^3*csg
n(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+1/12*Pi^2*x^3*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))-1/24*Pi^2*x^3*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))+1/24*Pi^
2*x^3*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+1/12*Pi^2*x^3*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))-1/12*Pi^2*x^3*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))^2-1/48*Pi^2*x^3*csgn(I*exp(2*b*x+2*a))^6-1/48*Pi^2*x^3*csgn(I*exp
(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^6+1/30*x^5*b^2+1/12*Pi^2*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*Pi^2*x^3*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))+1/6*Pi^2*x^3*csgn(I*exp(b*x+a)
)*csgn(I*exp(2*b*x+2*a))^2+1/24*I*Pi*b*x^4*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))-1/12*Pi^2*x^3*csgn(I/(exp(2*b*x+2*a)+1))^4-1/24*I*Pi*b*x^4*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+1/24*I*Pi*b*x^4*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))-1/12*
I*Pi*b*x^4*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2+1/12*Pi^2*x^3*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))+1/24*I*Pi*b*x^4*csgn(I*exp(2*b*x+2*a))^3+1/24*Pi^2*x^3*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+1/24*Pi^2*x^3*csgn(I*exp(b*x+a))^2*cs
gn(I*exp(2*b*x+2*a))^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+1/24*I*Pi*b*x^4*csgn(I*exp(2*b*x+2*a)/(exp(
2*b*x+2*a)+1))^3+1/12*I*Pi*b*x^4-1/24*Pi^2*x^3*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))+1/24*Pi^2*x^3*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+1/24*Pi^2*x^3*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+1/12*Pi^2*x^3*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
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Maxima [A] time = 1.37034, size = 49, normalized size = 1.17 \begin{align*} \frac{1}{30} \, b^{2} x^{5} - \frac{1}{6} \, b x^{4} \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right ) + \frac{1}{3} \, x^{3} \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^2*arccoth(tanh(b*x+a))^2,x, algorithm="maxima")
[Out]
1/30*b^2*x^5 - 1/6*b*x^4*arccoth(tanh(b*x + a)) + 1/3*x^3*arccoth(tanh(b*x + a))^2
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Fricas [A] time = 1.63241, size = 72, normalized size = 1.71 \begin{align*} \frac{1}{5} \, b^{2} x^{5} + \frac{1}{2} \, a b x^{4} - \frac{1}{12} \,{\left (\pi ^{2} - 4 \, a^{2}\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*arccoth(tanh(b*x+a))^2,x, algorithm="fricas")
[Out]
1/5*b^2*x^5 + 1/2*a*b*x^4 - 1/12*(pi^2 - 4*a^2)*x^3
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Sympy [A] time = 1.88637, size = 60, normalized size = 1.43 \begin{align*} \begin{cases} \frac{x^{2} \operatorname{acoth}^{3}{\left (\tanh{\left (a + b x \right )} \right )}}{3 b} - \frac{x \operatorname{acoth}^{4}{\left (\tanh{\left (a + b x \right )} \right )}}{6 b^{2}} + \frac{\operatorname{acoth}^{5}{\left (\tanh{\left (a + b x \right )} \right )}}{30 b^{3}} & \text{for}\: b \neq 0 \\\frac{x^{3} \operatorname{acoth}^{2}{\left (\tanh{\left (a \right )} \right )}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**2*acoth(tanh(b*x+a))**2,x)
[Out]
Piecewise((x**2*acoth(tanh(a + b*x))**3/(3*b) - x*acoth(tanh(a + b*x))**4/(6*b**2) + acoth(tanh(a + b*x))**5/(
30*b**3), Ne(b, 0)), (x**3*acoth(tanh(a))**2/3, True))
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \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^2*arccoth(tanh(b*x+a))^2,x, algorithm="giac")
[Out]
integrate(x^2*arccoth(tanh(b*x + a))^2, x)