3.148 \(\int x^3 \coth ^{-1}(\tanh (a+b x))^3 \, dx\)

Optimal. Leaf size=61 \[ \frac{1}{20} b^2 x^6 \coth ^{-1}(\tanh (a+b x))-\frac{3}{20} b x^5 \coth ^{-1}(\tanh (a+b x))^2+\frac{1}{4} x^4 \coth ^{-1}(\tanh (a+b x))^3-\frac{1}{140} b^3 x^7 \]

[Out]

-(b^3*x^7)/140 + (b^2*x^6*ArcCoth[Tanh[a + b*x]])/20 - (3*b*x^5*ArcCoth[Tanh[a + b*x]]^2)/20 + (x^4*ArcCoth[Ta
nh[a + b*x]]^3)/4

________________________________________________________________________________________

Rubi [A]  time = 0.0407912, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2168, 30} \[ \frac{1}{20} b^2 x^6 \coth ^{-1}(\tanh (a+b x))-\frac{3}{20} b x^5 \coth ^{-1}(\tanh (a+b x))^2+\frac{1}{4} x^4 \coth ^{-1}(\tanh (a+b x))^3-\frac{1}{140} b^3 x^7 \]

Antiderivative was successfully verified.

[In]

Int[x^3*ArcCoth[Tanh[a + b*x]]^3,x]

[Out]

-(b^3*x^7)/140 + (b^2*x^6*ArcCoth[Tanh[a + b*x]])/20 - (3*b*x^5*ArcCoth[Tanh[a + b*x]]^2)/20 + (x^4*ArcCoth[Ta
nh[a + b*x]]^3)/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))^3 \, dx &=\frac{1}{4} x^4 \coth ^{-1}(\tanh (a+b x))^3-\frac{1}{4} (3 b) \int x^4 \coth ^{-1}(\tanh (a+b x))^2 \, dx\\ &=-\frac{3}{20} b x^5 \coth ^{-1}(\tanh (a+b x))^2+\frac{1}{4} x^4 \coth ^{-1}(\tanh (a+b x))^3+\frac{1}{10} \left (3 b^2\right ) \int x^5 \coth ^{-1}(\tanh (a+b x)) \, dx\\ &=\frac{1}{20} b^2 x^6 \coth ^{-1}(\tanh (a+b x))-\frac{3}{20} b x^5 \coth ^{-1}(\tanh (a+b x))^2+\frac{1}{4} x^4 \coth ^{-1}(\tanh (a+b x))^3-\frac{1}{20} b^3 \int x^6 \, dx\\ &=-\frac{1}{140} b^3 x^7+\frac{1}{20} b^2 x^6 \coth ^{-1}(\tanh (a+b x))-\frac{3}{20} b x^5 \coth ^{-1}(\tanh (a+b x))^2+\frac{1}{4} x^4 \coth ^{-1}(\tanh (a+b x))^3\\ \end{align*}

Mathematica [A]  time = 0.0271736, size = 54, normalized size = 0.89 \[ -\frac{1}{140} x^4 \left (-7 b^2 x^2 \coth ^{-1}(\tanh (a+b x))+21 b x \coth ^{-1}(\tanh (a+b x))^2-35 \coth ^{-1}(\tanh (a+b x))^3+b^3 x^3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*ArcCoth[Tanh[a + b*x]]^3,x]

[Out]

-(x^4*(b^3*x^3 - 7*b^2*x^2*ArcCoth[Tanh[a + b*x]] + 21*b*x*ArcCoth[Tanh[a + b*x]]^2 - 35*ArcCoth[Tanh[a + b*x]
]^3))/140

________________________________________________________________________________________

Maple [C]  time = 1.096, size = 18111, normalized size = 296.9 \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))^3,x)

[Out]

result too large to display

________________________________________________________________________________________

Maxima [A]  time = 1.56117, size = 73, normalized size = 1.2 \begin{align*} -\frac{3}{20} \, b x^{5} \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{2} + \frac{1}{4} \, x^{4} \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{3} - \frac{1}{140} \,{\left (b^{2} x^{7} - 7 \, b x^{6} \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )\right )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccoth(tanh(b*x+a))^3,x, algorithm="maxima")

[Out]

-3/20*b*x^5*arccoth(tanh(b*x + a))^2 + 1/4*x^4*arccoth(tanh(b*x + a))^3 - 1/140*(b^2*x^7 - 7*b*x^6*arccoth(tan
h(b*x + a)))*b

________________________________________________________________________________________

Fricas [A]  time = 1.5684, size = 120, normalized size = 1.97 \begin{align*} \frac{1}{7} \, b^{3} x^{7} + \frac{1}{2} \, a b^{2} x^{6} - \frac{3}{20} \,{\left (\pi ^{2} b - 4 \, a^{2} b\right )} x^{5} - \frac{1}{16} \,{\left (3 \, \pi ^{2} a - 4 \, a^{3}\right )} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccoth(tanh(b*x+a))^3,x, algorithm="fricas")

[Out]

1/7*b^3*x^7 + 1/2*a*b^2*x^6 - 3/20*(pi^2*b - 4*a^2*b)*x^5 - 1/16*(3*pi^2*a - 4*a^3)*x^4

________________________________________________________________________________________

Sympy [A]  time = 3.559, size = 58, normalized size = 0.95 \begin{align*} - \frac{b^{3} x^{7}}{140} + \frac{b^{2} x^{6} \operatorname{acoth}{\left (\tanh{\left (a + b x \right )} \right )}}{20} - \frac{3 b x^{5} \operatorname{acoth}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{20} + \frac{x^{4} \operatorname{acoth}^{3}{\left (\tanh{\left (a + b x \right )} \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*acoth(tanh(b*x+a))**3,x)

[Out]

-b**3*x**7/140 + b**2*x**6*acoth(tanh(a + b*x))/20 - 3*b*x**5*acoth(tanh(a + b*x))**2/20 + x**4*acoth(tanh(a +
b*x))**3/4

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccoth(tanh(b*x+a))^3,x, algorithm="giac")

[Out]

integrate(x^3*arccoth(tanh(b*x + a))^3, x)