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

Optimal. Leaf size=23 \[ 4 b \sqrt{x}-\frac{2 \tanh ^{-1}(\tanh (a+b x))}{\sqrt{x}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0082348, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2168, 30} \[ 4 b \sqrt{x}-\frac{2 \tanh ^{-1}(\tanh (a+b x))}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[Tanh[a + b*x]]/x^(3/2),x]

[Out]

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

Mathematica [A]  time = 0.0196906, size = 20, normalized size = 0.87 \[ \frac{4 b x-2 \tanh ^{-1}(\tanh (a+b x))}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.034, size = 20, normalized size = 0.9 \begin{align*} -2\,{\frac{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{\sqrt{x}}}+4\,b\sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.981791, size = 26, normalized size = 1.13 \begin{align*} 4 \, b \sqrt{x} - \frac{2 \, \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*b*sqrt(x) - 2*arctanh(tanh(b*x + a))/sqrt(x)

________________________________________________________________________________________

Fricas [A]  time = 2.0859, size = 28, normalized size = 1.22 \begin{align*} \frac{2 \,{\left (b x - a\right )}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(b*x - a)/sqrt(x)

________________________________________________________________________________________

Sympy [A]  time = 1.64159, size = 22, normalized size = 0.96 \begin{align*} 4 b \sqrt{x} - \frac{2 \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*b*sqrt(x) - 2*atanh(tanh(a + b*x))/sqrt(x)

________________________________________________________________________________________

Giac [A]  time = 1.15908, size = 18, normalized size = 0.78 \begin{align*} 2 \, b \sqrt{x} - \frac{2 \, a}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b*sqrt(x) - 2*a/sqrt(x)