3.152 \(\int \frac{x}{\tanh ^{-1}(\tanh (a+b x))^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0148316, antiderivative size = 34, 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} \[ \frac{4 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{b^2}-\frac{2 x}{b \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \]

Antiderivative was successfully verified.

[In]

Int[x/ArcTanh[Tanh[a + b*x]]^(3/2),x]

[Out]

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

Mathematica [A]  time = 0.0572761, size = 29, normalized size = 0.85 \[ \frac{4 \tanh ^{-1}(\tanh (a+b x))-2 b x}{b^2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/ArcTanh[Tanh[a + b*x]]^(3/2),x]

[Out]

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

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Maple [A]  time = 0.037, size = 40, normalized size = 1.2 \begin{align*} 2\,{\frac{1}{{b}^{2}} \left ( \sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }-{\frac{bx-{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/arctanh(tanh(b*x+a))^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.78442, size = 41, normalized size = 1.21 \begin{align*} \frac{2 \,{\left (b^{2} x^{2} + 3 \, a b x + 2 \, a^{2}\right )}}{{\left (b x + a\right )}^{\frac{3}{2}} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/arctanh(tanh(b*x+a))^(3/2),x, algorithm="maxima")

[Out]

2*(b^2*x^2 + 3*a*b*x + 2*a^2)/((b*x + a)^(3/2)*b^2)

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Fricas [A]  time = 2.11797, size = 61, normalized size = 1.79 \begin{align*} \frac{2 \,{\left (b x + 2 \, a\right )} \sqrt{b x + a}}{b^{3} x + a b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/arctanh(tanh(b*x+a))^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 35.2198, size = 46, normalized size = 1.35 \begin{align*} \begin{cases} - \frac{2 x}{b \sqrt{\operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}} + \frac{4 \sqrt{\operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}}{b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 \operatorname{atanh}^{\frac{3}{2}}{\left (\tanh{\left (a \right )} \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/atanh(tanh(b*x+a))**(3/2),x)

[Out]

Piecewise((-2*x/(b*sqrt(atanh(tanh(a + b*x)))) + 4*sqrt(atanh(tanh(a + b*x)))/b**2, Ne(b, 0)), (x**2/(2*atanh(
tanh(a))**(3/2)), True))

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Giac [A]  time = 1.15173, size = 39, normalized size = 1.15 \begin{align*} \frac{2 \,{\left (\frac{\sqrt{b x + a}}{b} + \frac{a}{\sqrt{b x + a} b}\right )}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/arctanh(tanh(b*x+a))^(3/2),x, algorithm="giac")

[Out]

2*(sqrt(b*x + a)/b + a/(sqrt(b*x + a)*b))/b