3.63 \(\int \frac{\tanh ^{-1}(\tanh (a+b x))^3}{x^5} \, dx\)

Optimal. Leaf size=31 \[ \frac{\tanh ^{-1}(\tanh (a+b x))^4}{4 x^4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]

[Out]

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

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Rubi [A]  time = 0.0132947, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2167} \[ \frac{\tanh ^{-1}(\tanh (a+b x))^4}{4 x^4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[Tanh[a + b*x]]^3/x^5,x]

[Out]

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

Rule 2167

Int[(u_)^(m_)*(v_)^(n_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, -Simp[(u^(m + 1)*v^
(n + 1))/((m + 1)*(b*u - a*v)), x] /; NeQ[b*u - a*v, 0]] /; FreeQ[{m, n}, x] && PiecewiseLinearQ[u, v, x] && E
qQ[m + n + 2, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))^3}{x^5} \, dx &=\frac{\tanh ^{-1}(\tanh (a+b x))^4}{4 x^4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end{align*}

Mathematica [A]  time = 0.0224993, size = 50, normalized size = 1.61 \[ -\frac{b^2 x^2 \tanh ^{-1}(\tanh (a+b x))+b x \tanh ^{-1}(\tanh (a+b x))^2+\tanh ^{-1}(\tanh (a+b x))^3+b^3 x^3}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[Tanh[a + b*x]]^3/x^5,x]

[Out]

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

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Maple [A]  time = 0.039, size = 56, normalized size = 1.8 \begin{align*} -{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3}}{4\,{x}^{4}}}+{\frac{3\,b}{4} \left ( -{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{3\,{x}^{3}}}+{\frac{2\,b}{3} \left ( -{\frac{b}{2\,x}}-{\frac{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{2\,{x}^{2}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(tanh(b*x+a))^3/x^5,x)

[Out]

-1/4*arctanh(tanh(b*x+a))^3/x^4+3/4*b*(-1/3*arctanh(tanh(b*x+a))^2/x^3+2/3*b*(-1/2*b/x-1/2*arctanh(tanh(b*x+a)
)/x^2))

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Maxima [A]  time = 1.59351, size = 72, normalized size = 2.32 \begin{align*} -\frac{1}{4} \, b{\left (\frac{b^{2}}{x} + \frac{b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )}{x^{2}}\right )} - \frac{b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{4 \, x^{3}} - \frac{\operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(tanh(b*x+a))^3/x^5,x, algorithm="maxima")

[Out]

-1/4*b*(b^2/x + b*arctanh(tanh(b*x + a))/x^2) - 1/4*b*arctanh(tanh(b*x + a))^2/x^3 - 1/4*arctanh(tanh(b*x + a)
)^3/x^4

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Fricas [A]  time = 1.42449, size = 73, normalized size = 2.35 \begin{align*} -\frac{4 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 4 \, a^{2} b x + a^{3}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(tanh(b*x+a))^3/x^5,x, algorithm="fricas")

[Out]

-1/4*(4*b^3*x^3 + 6*a*b^2*x^2 + 4*a^2*b*x + a^3)/x^4

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Sympy [B]  time = 4.61478, size = 56, normalized size = 1.81 \begin{align*} - \frac{b^{3}}{4 x} - \frac{b^{2} \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}{4 x^{2}} - \frac{b \operatorname{atanh}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{4 x^{3}} - \frac{\operatorname{atanh}^{3}{\left (\tanh{\left (a + b x \right )} \right )}}{4 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(tanh(b*x+a))**3/x**5,x)

[Out]

-b**3/(4*x) - b**2*atanh(tanh(a + b*x))/(4*x**2) - b*atanh(tanh(a + b*x))**2/(4*x**3) - atanh(tanh(a + b*x))**
3/(4*x**4)

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Giac [A]  time = 1.1343, size = 45, normalized size = 1.45 \begin{align*} -\frac{4 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 4 \, a^{2} b x + a^{3}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(tanh(b*x+a))^3/x^5,x, algorithm="giac")

[Out]

-1/4*(4*b^3*x^3 + 6*a*b^2*x^2 + 4*a^2*b*x + a^3)/x^4