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

Optimal. Leaf size=34 \[ \frac{x \tanh ^{-1}(\tanh (a+b x))^4}{4 b}-\frac{\tanh ^{-1}(\tanh (a+b x))^5}{20 b^2} \]

[Out]

(x*ArcTanh[Tanh[a + b*x]]^4)/(4*b) - ArcTanh[Tanh[a + b*x]]^5/(20*b^2)

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Rubi [A]  time = 0.0139659, 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} \[ \frac{x \tanh ^{-1}(\tanh (a+b x))^4}{4 b}-\frac{\tanh ^{-1}(\tanh (a+b x))^5}{20 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x*ArcTanh[Tanh[a + b*x]]^3,x]

[Out]

(x*ArcTanh[Tanh[a + b*x]]^4)/(4*b) - ArcTanh[Tanh[a + b*x]]^5/(20*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 \tanh ^{-1}(\tanh (a+b x))^3 \, dx &=\frac{x \tanh ^{-1}(\tanh (a+b x))^4}{4 b}-\frac{\int \tanh ^{-1}(\tanh (a+b x))^4 \, dx}{4 b}\\ &=\frac{x \tanh ^{-1}(\tanh (a+b x))^4}{4 b}-\frac{\operatorname{Subst}\left (\int x^4 \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{4 b^2}\\ &=\frac{x \tanh ^{-1}(\tanh (a+b x))^4}{4 b}-\frac{\tanh ^{-1}(\tanh (a+b x))^5}{20 b^2}\\ \end{align*}

Mathematica [B]  time = 0.0727119, size = 99, normalized size = 2.91 \[ \frac{(a+b x) \left (10 \left (2 a^2+a b x-b^2 x^2\right ) \tanh ^{-1}(\tanh (a+b x))^2+(4 a-b x) (a+b x)^3-5 (3 a-b x) (a+b x)^2 \tanh ^{-1}(\tanh (a+b x))-10 (a-b x) \tanh ^{-1}(\tanh (a+b x))^3\right )}{20 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)*((4*a - b*x)*(a + b*x)^3 - 5*(3*a - b*x)*(a + b*x)^2*ArcTanh[Tanh[a + b*x]] + 10*(2*a^2 + a*b*x - b
^2*x^2)*ArcTanh[Tanh[a + b*x]]^2 - 10*(a - b*x)*ArcTanh[Tanh[a + b*x]]^3))/(20*b^2)

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Maple [A]  time = 0.037, size = 56, normalized size = 1.7 \begin{align*}{\frac{{x}^{2} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3}}{2}}-{\frac{3\,b}{2} \left ({\frac{{x}^{3} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{3}}-{\frac{2\,b}{3} \left ({\frac{{x}^{4}{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{4}}-{\frac{b{x}^{5}}{20}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arctanh(tanh(b*x+a))^3,x)

[Out]

1/2*x^2*arctanh(tanh(b*x+a))^3-3/2*b*(1/3*x^3*arctanh(tanh(b*x+a))^2-2/3*b*(1/4*x^4*arctanh(tanh(b*x+a))-1/20*
b*x^5))

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Maxima [A]  time = 1.54725, size = 73, normalized size = 2.15 \begin{align*} -\frac{1}{2} \, b x^{3} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2} + \frac{1}{2} \, x^{2} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{3} - \frac{1}{20} \,{\left (b^{2} x^{5} - 5 \, b x^{4} \operatorname{artanh}\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*arctanh(tanh(b*x+a))^3,x, algorithm="maxima")

[Out]

-1/2*b*x^3*arctanh(tanh(b*x + a))^2 + 1/2*x^2*arctanh(tanh(b*x + a))^3 - 1/20*(b^2*x^5 - 5*b*x^4*arctanh(tanh(
b*x + a)))*b

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Fricas [A]  time = 1.42563, size = 74, normalized size = 2.18 \begin{align*} \frac{1}{5} \, b^{3} x^{5} + \frac{3}{4} \, a b^{2} x^{4} + a^{2} b x^{3} + \frac{1}{2} \, a^{3} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b^3*x^5 + 3/4*a*b^2*x^4 + a^2*b*x^3 + 1/2*a^3*x^2

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Sympy [A]  time = 6.83063, size = 56, normalized size = 1.65 \begin{align*} - \frac{b^{3} x^{5}}{20} + \frac{b^{2} x^{4} \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}{4} - \frac{b x^{3} \operatorname{atanh}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{2} + \frac{x^{2} \operatorname{atanh}^{3}{\left (\tanh{\left (a + b x \right )} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b**3*x**5/20 + b**2*x**4*atanh(tanh(a + b*x))/4 - b*x**3*atanh(tanh(a + b*x))**2/2 + x**2*atanh(tanh(a + b*x)
)**3/2

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Giac [A]  time = 1.12829, size = 46, normalized size = 1.35 \begin{align*} \frac{1}{5} \, b^{3} x^{5} + \frac{3}{4} \, a b^{2} x^{4} + a^{2} b x^{3} + \frac{1}{2} \, a^{3} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b^3*x^5 + 3/4*a*b^2*x^4 + a^2*b*x^3 + 1/2*a^3*x^2