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

Optimal. Leaf size=81 \[ -\frac{\tanh ^{-1}(\tanh (a+b x))^{3/2}}{x}+3 b \sqrt{\tanh ^{-1}(\tanh (a+b x))}-3 b \sqrt{b x-\tanh ^{-1}(\tanh (a+b x))} \tan ^{-1}\left (\frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0474029, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2168, 2159, 2161} \[ -\frac{\tanh ^{-1}(\tanh (a+b x))^{3/2}}{x}+3 b \sqrt{\tanh ^{-1}(\tanh (a+b x))}-3 b \sqrt{b x-\tanh ^{-1}(\tanh (a+b x))} \tan ^{-1}\left (\frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0350054, size = 79, normalized size = 0.98 \[ -\frac{\tanh ^{-1}(\tanh (a+b x))^{3/2}}{x}+3 b \sqrt{\tanh ^{-1}(\tanh (a+b x))}-3 b \tanh ^{-1}\left (\frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))-b x}}\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))-b x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.118, size = 85, normalized size = 1.1 \begin{align*} 2\,b \left ( \sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }+{\frac{ \left ( -1/2\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) +1/2\,bx \right ) \sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }}{bx}}-3/2\,\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 ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b*(arctanh(tanh(b*x+a))^(1/2)+(-1/2*arctanh(tanh(b*x+a))+1/2*b*x)*arctanh(tanh(b*x+a))^(1/2)/b/x-3/2*(arctan
h(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^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.21208, size = 247, normalized size = 3.05 \begin{align*} \left [\frac{3 \, \sqrt{a} b x \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (2 \, b x - a\right )} \sqrt{b x + a}}{2 \, x}, \frac{3 \, \sqrt{-a} b x \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (2 \, b x - a\right )} \sqrt{b x + a}}{x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(3*sqrt(a)*b*x*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(2*b*x - a)*sqrt(b*x + a))/x, (3*sqrt(-a)
*b*x*arctan(sqrt(b*x + a)*sqrt(-a)/a) + (2*b*x - a)*sqrt(b*x + a))/x]

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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^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15655, size = 93, normalized size = 1.15 \begin{align*} \frac{\sqrt{2}{\left (\frac{3 \, \sqrt{2} a b^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 2 \, \sqrt{2} \sqrt{b x + a} b^{2} - \frac{\sqrt{2} \sqrt{b x + a} a b}{x}\right )}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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