3.137 \(\int x^3 \coth ^{-1}(\tanh (a+b x))^2 \, dx\)
Optimal. Leaf size=42 \[ -\frac{1}{10} b x^5 \coth ^{-1}(\tanh (a+b x))+\frac{1}{4} x^4 \coth ^{-1}(\tanh (a+b x))^2+\frac{b^2 x^6}{60} \]
[Out]
(b^2*x^6)/60 - (b*x^5*ArcCoth[Tanh[a + b*x]])/10 + (x^4*ArcCoth[Tanh[a + b*x]]^2)/4
________________________________________________________________________________________
Rubi [A] time = 0.0260149, 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} \[ -\frac{1}{10} b x^5 \coth ^{-1}(\tanh (a+b x))+\frac{1}{4} x^4 \coth ^{-1}(\tanh (a+b x))^2+\frac{b^2 x^6}{60} \]
Antiderivative was successfully verified.
[In]
Int[x^3*ArcCoth[Tanh[a + b*x]]^2,x]
[Out]
(b^2*x^6)/60 - (b*x^5*ArcCoth[Tanh[a + b*x]])/10 + (x^4*ArcCoth[Tanh[a + b*x]]^2)/4
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^3 \coth ^{-1}(\tanh (a+b x))^2 \, dx &=\frac{1}{4} x^4 \coth ^{-1}(\tanh (a+b x))^2-\frac{1}{2} b \int x^4 \coth ^{-1}(\tanh (a+b x)) \, dx\\ &=-\frac{1}{10} b x^5 \coth ^{-1}(\tanh (a+b x))+\frac{1}{4} x^4 \coth ^{-1}(\tanh (a+b x))^2+\frac{1}{10} b^2 \int x^5 \, dx\\ &=\frac{b^2 x^6}{60}-\frac{1}{10} b x^5 \coth ^{-1}(\tanh (a+b x))+\frac{1}{4} x^4 \coth ^{-1}(\tanh (a+b x))^2\\ \end{align*}
Mathematica [A] time = 0.0336754, size = 37, normalized size = 0.88 \[ \frac{1}{60} x^4 \left (-6 b x \coth ^{-1}(\tanh (a+b x))+15 \coth ^{-1}(\tanh (a+b x))^2+b^2 x^2\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[x^3*ArcCoth[Tanh[a + b*x]]^2,x]
[Out]
(x^4*(b^2*x^2 - 6*b*x*ArcCoth[Tanh[a + b*x]] + 15*ArcCoth[Tanh[a + b*x]]^2))/60
________________________________________________________________________________________
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^3*arccoth(tanh(b*x+a))^2,x)
[Out]
1/20*I*Pi*b*x^5-1/16*Pi^2*x^4*csgn(I/(exp(2*b*x+2*a)+1))^6-1/16*Pi^2*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))-1/20*I*Pi*b*x^5*csgn(I/(exp(2*b*x+2*a)+1))^2+1/8*Pi^2*x
^4*csgn(I/(exp(2*b*x+2*a)+1))^5-1/16*Pi^2*x^4+1/4*x^4*ln(exp(b*x+a))^2+1/16*Pi^2*x^4*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))^2-1/16*Pi^2*x^4*csgn(I/(exp(2*b*x+2*a)+1))^3*csgn(I*exp(2*b*x+2*
a))^3+1/60*x^6*b^2-1/16*Pi^2*x^4*csgn(I/(exp(2*b*x+2*a)+1))^4-1/64*Pi^2*x^4*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2
*a)+1))^6-1/16*Pi^2*x^4*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))^3-1/16*Pi^2*x^4
*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))^2-1/32*Pi^2*x^4*csgn(I*exp(2*b*x+2*a))
^3*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3-1/64*Pi^2*x^4*csgn(I*exp(2*b*x+2*a))^2*csgn(I*exp(2*b*x+2*a)/(e
xp(2*b*x+2*a)+1))^4+1/32*Pi^2*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))^5-3/32*Pi^2
*x^4*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))^4+1/16*Pi^2*x^4*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^5+1
/32*Pi^2*x^4*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^4*csgn(I*exp(2*b
*x+2*a))^3*csgn(I/(exp(2*b*x+2*a)+1))^2+1/16*Pi^2*x^4*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+1/20*I*Pi*b*x^5*csgn(I/(exp(2*b*x+2*a)+1))^3-1/16*Pi^2*x^4*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))-1/16*Pi^2*x^4*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))-1/8*Pi^2*x^4*csgn(I/(exp(2*b*x+2*a)+1))^3+1/8*Pi^2*x^4*csgn(I/(exp(2*b*x+
2*a)+1))^2+1/8*Pi^2*x^4*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2+1/16*Pi^2*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))^2-1/64*Pi^2*x^4*csgn(I*exp(2*b*x+2*a))^6-1/16*Pi^2*x^4*csgn(I*exp(b*x+a))
^2*csgn(I*exp(2*b*x+2*a))-1/16*Pi^2*x^4*csgn(I*exp(2*b*x+2*a))^3+1/16*Pi^2*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))^2-1/64*Pi^2*x^4*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^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))^5-1/64*Pi^
2*x^4*csgn(I*exp(b*x+a))^4*csgn(I*exp(2*b*x+2*a))^2+1/16*Pi^2*x^4*csgn(I*exp(b*x+a))^3*csgn(I*exp(2*b*x+2*a))^
3-1/16*Pi^2*x^4*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3-1/40*I*Pi*b*x^5*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+1/40*I*Pi*b*x^5*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))-1/20*I*Pi*b*x
^5*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2-1/40*I*Pi*b*x^5*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(e
xp(2*b*x+2*a)+1))^2+1/40*I*Pi*b*x^5*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)/(e
xp(2*b*x+2*a)+1))+1/8*Pi^2*x^4*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+1/16*P
i^2*x^4*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+1/16*P
i^2*x^4*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))*csgn(I/(exp(2*b*x+2*a)+1))^2-1/8*Pi^2*x^4*csgn(I*exp(b*x+a
))*csgn(I*exp(2*b*x+2*a))^2*csgn(I/(exp(2*b*x+2*a)+1))^2-1/16*Pi^2*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*csgn(I/(exp(2*b*x+2*a)+1))^2-1/32*Pi^2*x^4*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))+1/32*Pi^2*x^4*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+1/16*Pi^2*x^4*c
sgn(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)
)-1/16*Pi^2*x^4*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))^2-1/16*Pi^2*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)/(ex
p(2*b*x+2*a)+1))^4-1/32*Pi^2*x^4*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+1/16*Pi^2*x^4*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))
^3-1/32*Pi^2*x^4*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))
+1/32*Pi^2*x^4*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
-1/64*Pi^2*x^4*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+1/32*Pi^2*x^4*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+1/32*Pi^2*x^4*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+1/16*Pi^2*x^4*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))
+(-1/10*b*x^5-1/8*I*Pi*x^4*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3-1/8*I*Pi*x^4*csgn(I*exp(2*b*x+2*a))^3+1
/4*I*Pi*x^4*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2+1/8*I*Pi*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+1/8*I*Pi*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-
1/8*I*Pi*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))-1/8*I
*Pi*x^4*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))-1/4*I*Pi*x^4*csgn(I/(exp(2*b*x+2*a)+1))^3-1/4*I*Pi*x^4+1/4
*I*Pi*x^4*csgn(I/(exp(2*b*x+2*a)+1))^2)*ln(exp(b*x+a))+1/40*I*Pi*b*x^5*csgn(I*exp(2*b*x+2*a))^3+1/40*I*Pi*b*x^
5*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3+1/32*Pi^2*x^4*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^4*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
________________________________________________________________________________________
Maxima [A] time = 1.3594, size = 49, normalized size = 1.17 \begin{align*} \frac{1}{60} \, b^{2} x^{6} - \frac{1}{10} \, b x^{5} \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right ) + \frac{1}{4} \, x^{4} \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^3*arccoth(tanh(b*x+a))^2,x, algorithm="maxima")
[Out]
1/60*b^2*x^6 - 1/10*b*x^5*arccoth(tanh(b*x + a)) + 1/4*x^4*arccoth(tanh(b*x + a))^2
________________________________________________________________________________________
Fricas [A] time = 1.57062, size = 72, normalized size = 1.71 \begin{align*} \frac{1}{6} \, b^{2} x^{6} + \frac{2}{5} \, a b x^{5} - \frac{1}{16} \,{\left (\pi ^{2} - 4 \, a^{2}\right )} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*arccoth(tanh(b*x+a))^2,x, algorithm="fricas")
[Out]
1/6*b^2*x^6 + 2/5*a*b*x^5 - 1/16*(pi^2 - 4*a^2)*x^4
________________________________________________________________________________________
Sympy [A] time = 3.67829, size = 78, normalized size = 1.86 \begin{align*} \begin{cases} \frac{x^{3} \operatorname{acoth}^{3}{\left (\tanh{\left (a + b x \right )} \right )}}{3 b} - \frac{x^{2} \operatorname{acoth}^{4}{\left (\tanh{\left (a + b x \right )} \right )}}{4 b^{2}} + \frac{x \operatorname{acoth}^{5}{\left (\tanh{\left (a + b x \right )} \right )}}{10 b^{3}} - \frac{\operatorname{acoth}^{6}{\left (\tanh{\left (a + b x \right )} \right )}}{60 b^{4}} & \text{for}\: b \neq 0 \\\frac{x^{4} \operatorname{acoth}^{2}{\left (\tanh{\left (a \right )} \right )}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**3*acoth(tanh(b*x+a))**2,x)
[Out]
Piecewise((x**3*acoth(tanh(a + b*x))**3/(3*b) - x**2*acoth(tanh(a + b*x))**4/(4*b**2) + x*acoth(tanh(a + b*x))
**5/(10*b**3) - acoth(tanh(a + b*x))**6/(60*b**4), Ne(b, 0)), (x**4*acoth(tanh(a))**2/4, True))
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \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^3*arccoth(tanh(b*x+a))^2,x, algorithm="giac")
[Out]
integrate(x^3*arccoth(tanh(b*x + a))^2, x)