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

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

[Out]

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

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Rubi [A]  time = 0.0089049, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2168, 30} \[ -\frac{\tanh ^{-1}(\tanh (a+b x))}{2 x^2}-\frac{b}{2 x} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[Tanh[a + b*x]]/x^3,x]

[Out]

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

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} \, dx &=-\frac{\tanh ^{-1}(\tanh (a+b x))}{2 x^2}+\frac{1}{2} b \int \frac{1}{x^2} \, dx\\ &=-\frac{b}{2 x}-\frac{\tanh ^{-1}(\tanh (a+b x))}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0139429, size = 18, normalized size = 0.78 \[ -\frac{\tanh ^{-1}(\tanh (a+b x))+b x}{2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.033, size = 20, normalized size = 0.9 \begin{align*} -{\frac{b}{2\,x}}-{\frac{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.16409, size = 26, normalized size = 1.13 \begin{align*} -\frac{b}{2 \, x} - \frac{\operatorname{artanh}\left (\tanh \left (b x + a\right )\right )}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.66006, size = 30, normalized size = 1.3 \begin{align*} -\frac{2 \, b x + a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*b*x + a)/x^2

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Sympy [A]  time = 1.1003, size = 19, normalized size = 0.83 \begin{align*} - \frac{b}{2 x} - \frac{\operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.1388, size = 15, normalized size = 0.65 \begin{align*} -\frac{2 \, b x + a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*b*x + a)/x^2