3.53 \(\int \frac{\tanh ^{-1}(\tanh (a+b x))^2}{x^5} \, dx\)

Optimal. Leaf size=42 \[ -\frac{b \tanh ^{-1}(\tanh (a+b x))}{6 x^3}-\frac{\tanh ^{-1}(\tanh (a+b x))^2}{4 x^4}-\frac{b^2}{12 x^2} \]

[Out]

-b^2/(12*x^2) - (b*ArcTanh[Tanh[a + b*x]])/(6*x^3) - ArcTanh[Tanh[a + b*x]]^2/(4*x^4)

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Rubi [A]  time = 0.0303805, antiderivative size = 64, normalized size of antiderivative = 1.52, 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 = {2171, 2167} \[ \frac{\tanh ^{-1}(\tanh (a+b x))^3}{4 x^4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac{b \tanh ^{-1}(\tanh (a+b x))^3}{12 x^3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[Tanh[a + b*x]]^2/x^5,x]

[Out]

(b*ArcTanh[Tanh[a + b*x]]^3)/(12*x^3*(b*x - ArcTanh[Tanh[a + b*x]])^2) + ArcTanh[Tanh[a + b*x]]^3/(4*x^4*(b*x
- ArcTanh[Tanh[a + b*x]]))

Rule 2171

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] + Dist[(b*(m + n + 2))/((m + 1)*(b*u - a*v)), Int[u^(m + 1)*v^n, x], x] /;
NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x] && NeQ[m + n + 2, 0] && LtQ[m, -1]

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))^2}{x^5} \, dx &=\frac{\tanh ^{-1}(\tanh (a+b x))^3}{4 x^4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac{b \int \frac{\tanh ^{-1}(\tanh (a+b x))^2}{x^4} \, dx}{4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ &=\frac{b \tanh ^{-1}(\tanh (a+b x))^3}{12 x^3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}+\frac{\tanh ^{-1}(\tanh (a+b x))^3}{4 x^4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end{align*}

Mathematica [A]  time = 0.0289113, size = 37, normalized size = 0.88 \[ -\frac{2 b x \tanh ^{-1}(\tanh (a+b x))+3 \tanh ^{-1}(\tanh (a+b x))^2+b^2 x^2}{12 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[Tanh[a + b*x]]^2/x^5,x]

[Out]

-(b^2*x^2 + 2*b*x*ArcTanh[Tanh[a + b*x]] + 3*ArcTanh[Tanh[a + b*x]]^2)/(12*x^4)

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Maple [A]  time = 0.035, size = 38, normalized size = 0.9 \begin{align*} -{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{4\,{x}^{4}}}+{\frac{b}{2} \left ( -{\frac{b}{6\,{x}^{2}}}-{\frac{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{3\,{x}^{3}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(tanh(b*x+a))^2/x^5,x)

[Out]

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

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Maxima [A]  time = 1.36886, size = 49, normalized size = 1.17 \begin{align*} -\frac{b^{2}}{12 \, x^{2}} - \frac{b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )}{6 \, x^{3}} - \frac{\operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(tanh(b*x+a))^2/x^5,x, algorithm="maxima")

[Out]

-1/12*b^2/x^2 - 1/6*b*arctanh(tanh(b*x + a))/x^3 - 1/4*arctanh(tanh(b*x + a))^2/x^4

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Fricas [A]  time = 1.46397, size = 55, normalized size = 1.31 \begin{align*} -\frac{6 \, b^{2} x^{2} + 8 \, a b x + 3 \, a^{2}}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(tanh(b*x+a))^2/x^5,x, algorithm="fricas")

[Out]

-1/12*(6*b^2*x^2 + 8*a*b*x + 3*a^2)/x^4

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Sympy [A]  time = 8.01543, size = 39, normalized size = 0.93 \begin{align*} - \frac{b^{2}}{12 x^{2}} - \frac{b \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}{6 x^{3}} - \frac{\operatorname{atanh}^{2}{\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))**2/x**5,x)

[Out]

-b**2/(12*x**2) - b*atanh(tanh(a + b*x))/(6*x**3) - atanh(tanh(a + b*x))**2/(4*x**4)

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Giac [A]  time = 1.17859, size = 32, normalized size = 0.76 \begin{align*} -\frac{6 \, b^{2} x^{2} + 8 \, a b x + 3 \, a^{2}}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(tanh(b*x+a))^2/x^5,x, algorithm="giac")

[Out]

-1/12*(6*b^2*x^2 + 8*a*b*x + 3*a^2)/x^4