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]

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

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Rubi [A]  time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.02, 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]

-1/4*(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)/x^4

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fricas [A]  time = 0.92, size = 33, normalized size = 1.06 \[ -\frac {4 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 4 \, a^{2} b x + a^{3}}{4 \, x^{4}} \]

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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giac [A]  time = 0.22, size = 33, normalized size = 1.06 \[ -\frac {4 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 4 \, a^{2} b x + a^{3}}{4 \, x^{4}} \]

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

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maple [A]  time = 0.14, size = 56, normalized size = 1.81 \[ -\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{3}}{4 x^{4}}+\frac {3 b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{3 x^{3}}+\frac {2 b \left (-\frac {b}{2 x}-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )}{2 x^{2}}\right )}{3}\right )}{4} \]

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 = 0.53, size = 53, normalized size = 1.71 \[ -\frac {1}{4} \, b {\left (\frac {b^{2}}{x} + \frac {b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}{x^{2}}\right )} - \frac {b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{4 \, x^{3}} - \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{4 \, x^{4}} \]

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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mupad [B]  time = 0.97, size = 48, normalized size = 1.55 \[ -\frac {b^3\,x^3+b^2\,x^2\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )+b\,x\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2+{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^3}{4\,x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(tanh(a + b*x))^3/x^5,x)

[Out]

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

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sympy [B]  time = 1.31, size = 56, normalized size = 1.81 \[ - \frac {b^{3}}{4 x} - \frac {b^{2} \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{4 x^{2}} - \frac {b \operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{4 x^{3}} - \frac {\operatorname {atanh}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}{4 x^{4}} \]

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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