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

Optimal. Leaf size=91 \[ -2 \sqrt{\tanh ^{-1}(\tanh (a+b x))} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+\frac{2}{3} \tanh ^{-1}(\tanh (a+b x))^{3/2}+2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{3/2} \tan ^{-1}\left (\frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0497527, antiderivative size = 91, 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 = {2159, 2161} \[ -2 \sqrt{\tanh ^{-1}(\tanh (a+b x))} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+\frac{2}{3} \tanh ^{-1}(\tanh (a+b x))^{3/2}+2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{3/2} \tan ^{-1}\left (\frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[Tanh[a + b*x]]^(3/2)/x,x]

[Out]

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

Rule 2159

Int[(v_)^(n_)/(u_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[v^n/(a*n), x] - Dis
t[(b*u - a*v)/a, Int[v^(n - 1)/u, x], x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x] && GtQ[n, 0] && Ne
Q[n, 1]

Rule 2161

Int[1/((u_)*Sqrt[v_]), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(2*ArcTan[Sqrt[v
]/Rt[(b*u - a*v)/a, 2]])/(a*Rt[(b*u - a*v)/a, 2]), x] /; NeQ[b*u - a*v, 0] && PosQ[(b*u - a*v)/a]] /; Piecewis
eLinearQ[u, v, x]

Rubi steps

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

Mathematica [A]  time = 0.0693455, size = 80, normalized size = 0.88 \[ -\frac{2}{3} \left (-4 \tanh ^{-1}(\tanh (a+b x))^{3/2}+3 b x \sqrt{\tanh ^{-1}(\tanh (a+b x))}+3 \tanh ^{-1}\left (\frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))-b x}}\right ) \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^{3/2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.107, size = 131, normalized size = 1.4 \begin{align*}{\frac{2}{3} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{{\frac{3}{2}}}}+2\,a\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }+2\, \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx-a \right ) \sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }-2\,{\frac{{a}^{2}+2\,a \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx-a \right ) + \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx-a \right ) ^{2}}{\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx}}{\it Artanh} \left ({\frac{\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }}{\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.11955, size = 228, normalized size = 2.51 \begin{align*} \left [a^{\frac{3}{2}} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + \frac{2}{3} \,{\left (b x + 4 \, a\right )} \sqrt{b x + a}, 2 \, \sqrt{-a} a \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) + \frac{2}{3} \,{\left (b x + 4 \, a\right )} \sqrt{b x + a}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[a^(3/2)*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2/3*(b*x + 4*a)*sqrt(b*x + a), 2*sqrt(-a)*a*arctan(sqr
t(b*x + a)*sqrt(-a)/a) + 2/3*(b*x + 4*a)*sqrt(b*x + a)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15854, size = 77, normalized size = 0.85 \begin{align*} \frac{1}{3} \, \sqrt{2}{\left (\frac{3 \, \sqrt{2} a^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \sqrt{2}{\left (b x + a\right )}^{\frac{3}{2}} + 3 \, \sqrt{2} \sqrt{b x + a} a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(2)*(3*sqrt(2)*a^2*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a) + sqrt(2)*(b*x + a)^(3/2) + 3*sqrt(2)*sqrt(
b*x + a)*a)