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

Optimal. Leaf size=44 \[ 8 b \sqrt{x} \tanh ^{-1}(\tanh (a+b x))-\frac{2 \tanh ^{-1}(\tanh (a+b x))^2}{\sqrt{x}}-\frac{16}{3} b^2 x^{3/2} \]

[Out]

(-16*b^2*x^(3/2))/3 + 8*b*Sqrt[x]*ArcTanh[Tanh[a + b*x]] - (2*ArcTanh[Tanh[a + b*x]]^2)/Sqrt[x]

________________________________________________________________________________________

Rubi [A]  time = 0.0217644, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2168, 30} \[ 8 b \sqrt{x} \tanh ^{-1}(\tanh (a+b x))-\frac{2 \tanh ^{-1}(\tanh (a+b x))^2}{\sqrt{x}}-\frac{16}{3} b^2 x^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-16*b^2*x^(3/2))/3 + 8*b*Sqrt[x]*ArcTanh[Tanh[a + b*x]] - (2*ArcTanh[Tanh[a + b*x]]^2)/Sqrt[x]

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 \frac{\tanh ^{-1}(\tanh (a+b x))^2}{x^{3/2}} \, dx &=-\frac{2 \tanh ^{-1}(\tanh (a+b x))^2}{\sqrt{x}}+(4 b) \int \frac{\tanh ^{-1}(\tanh (a+b x))}{\sqrt{x}} \, dx\\ &=8 b \sqrt{x} \tanh ^{-1}(\tanh (a+b x))-\frac{2 \tanh ^{-1}(\tanh (a+b x))^2}{\sqrt{x}}-\left (8 b^2\right ) \int \sqrt{x} \, dx\\ &=-\frac{16}{3} b^2 x^{3/2}+8 b \sqrt{x} \tanh ^{-1}(\tanh (a+b x))-\frac{2 \tanh ^{-1}(\tanh (a+b x))^2}{\sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.042224, size = 40, normalized size = 0.91 \[ -\frac{2 \left (-12 b x \tanh ^{-1}(\tanh (a+b x))+3 \tanh ^{-1}(\tanh (a+b x))^2+8 b^2 x^2\right )}{3 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(8*b^2*x^2 - 12*b*x*ArcTanh[Tanh[a + b*x]] + 3*ArcTanh[Tanh[a + b*x]]^2))/(3*Sqrt[x])

________________________________________________________________________________________

Maple [A]  time = 0.04, size = 37, normalized size = 0.8 \begin{align*} -2\,{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{\sqrt{x}}}+8\,b \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \sqrt{x}-2/3\,b{x}^{3/2} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*arctanh(tanh(b*x+a))^2/x^(1/2)+8*b*(arctanh(tanh(b*x+a))*x^(1/2)-2/3*b*x^(3/2))

________________________________________________________________________________________

Maxima [A]  time = 1.01822, size = 49, normalized size = 1.11 \begin{align*} -\frac{16}{3} \, b^{2} x^{\frac{3}{2}} + 8 \, b \sqrt{x} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right ) - \frac{2 \, \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16/3*b^2*x^(3/2) + 8*b*sqrt(x)*arctanh(tanh(b*x + a)) - 2*arctanh(tanh(b*x + a))^2/sqrt(x)

________________________________________________________________________________________

Fricas [A]  time = 2.0064, size = 55, normalized size = 1.25 \begin{align*} \frac{2 \,{\left (b^{2} x^{2} + 6 \, a b x - 3 \, a^{2}\right )}}{3 \, \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(b^2*x^2 + 6*a*b*x - 3*a^2)/sqrt(x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{atanh}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{x^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.1522, size = 32, normalized size = 0.73 \begin{align*} \frac{2}{3} \, b^{2} x^{\frac{3}{2}} + 4 \, a b \sqrt{x} - \frac{2 \, a^{2}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*b^2*x^(3/2) + 4*a*b*sqrt(x) - 2*a^2/sqrt(x)