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

Optimal. Leaf size=38 \[ \frac{2 x \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 b}-\frac{4 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{15 b^2} \]

[Out]

(2*x*ArcTanh[Tanh[a + b*x]]^(3/2))/(3*b) - (4*ArcTanh[Tanh[a + b*x]]^(5/2))/(15*b^2)

________________________________________________________________________________________

Rubi [A]  time = 0.0140504, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2168, 2157, 30} \[ \frac{2 x \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 b}-\frac{4 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{15 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[ArcTanh[Tanh[a + b*x]]],x]

[Out]

(2*x*ArcTanh[Tanh[a + b*x]]^(3/2))/(3*b) - (4*ArcTanh[Tanh[a + b*x]]^(5/2))/(15*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 \sqrt{\tanh ^{-1}(\tanh (a+b x))} \, dx &=\frac{2 x \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 b}-\frac{2 \int \tanh ^{-1}(\tanh (a+b x))^{3/2} \, dx}{3 b}\\ &=\frac{2 x \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 b}-\frac{2 \operatorname{Subst}\left (\int x^{3/2} \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{3 b^2}\\ &=\frac{2 x \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 b}-\frac{4 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{15 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0491836, size = 32, normalized size = 0.84 \[ \frac{2 \left (5 b x-2 \tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^{3/2}}{15 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[ArcTanh[Tanh[a + b*x]]],x]

[Out]

(2*(5*b*x - 2*ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]^(3/2))/(15*b^2)

________________________________________________________________________________________

Maple [A]  time = 0.046, size = 42, normalized size = 1.1 \begin{align*} 2\,{\frac{1/5\, \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{5/2}+1/3\, \left ( bx-{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3/2}}{{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arctanh(tanh(b*x+a))^(1/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.76433, size = 41, normalized size = 1.08 \begin{align*} \frac{2 \,{\left (3 \, b^{2} x^{2} + a b x - 2 \, a^{2}\right )} \sqrt{b x + a}}{15 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arctanh(tanh(b*x+a))^(1/2),x, algorithm="maxima")

[Out]

2/15*(3*b^2*x^2 + a*b*x - 2*a^2)*sqrt(b*x + a)/b^2

________________________________________________________________________________________

Fricas [A]  time = 1.52821, size = 70, normalized size = 1.84 \begin{align*} \frac{2 \,{\left (3 \, b^{2} x^{2} + a b x - 2 \, a^{2}\right )} \sqrt{b x + a}}{15 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arctanh(tanh(b*x+a))^(1/2),x, algorithm="fricas")

[Out]

2/15*(3*b^2*x^2 + a*b*x - 2*a^2)*sqrt(b*x + a)/b^2

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sqrt{\operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*atanh(tanh(b*x+a))**(1/2),x)

[Out]

Integral(x*sqrt(atanh(tanh(a + b*x))), x)

________________________________________________________________________________________

Giac [A]  time = 1.12343, size = 34, normalized size = 0.89 \begin{align*} \frac{2 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )}}{15 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arctanh(tanh(b*x+a))^(1/2),x, algorithm="giac")

[Out]

2/15*(3*(b*x + a)^(5/2) - 5*(b*x + a)^(3/2)*a)/b^2