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

Optimal. Leaf size=16 \[ -\frac {1}{2 b \tanh ^{-1}(\tanh (a+b x))^2} \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2157, 30} \[ -\frac {1}{2 b \tanh ^{-1}(\tanh (a+b x))^2} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[Tanh[a + b*x]]^(-3),x]

[Out]

-1/(2*b*ArcTanh[Tanh[a + b*x]]^2)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2157

Int[(u_)^(m_.), x_Symbol] :> With[{c = Simplify[D[u, x]]}, Dist[1/c, Subst[Int[x^m, x], x, u], x]] /; FreeQ[m,
x] && PiecewiseLinearQ[u, x]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \[ -\frac {1}{2 b \tanh ^{-1}(\tanh (a+b x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*1/(b*ArcTanh[Tanh[a + b*x]]^2)

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fricas [A]  time = 0.40, size = 24, normalized size = 1.50 \[ -\frac {1}{2 \, {\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arctanh(tanh(b*x+a))^3,x, algorithm="fricas")

[Out]

-1/2/(b^3*x^2 + 2*a*b^2*x + a^2*b)

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giac [A]  time = 0.17, size = 12, normalized size = 0.75 \[ -\frac {1}{2 \, {\left (b x + a\right )}^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arctanh(tanh(b*x+a))^3,x, algorithm="giac")

[Out]

-1/2/((b*x + a)^2*b)

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maple [A]  time = 0.03, size = 15, normalized size = 0.94 \[ -\frac {1}{2 b \arctanh \left (\tanh \left (b x +a \right )\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/arctanh(tanh(b*x+a))^3,x)

[Out]

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

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maxima [A]  time = 0.56, size = 12, normalized size = 0.75 \[ -\frac {1}{2 \, {\left (b x + a\right )}^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arctanh(tanh(b*x+a))^3,x, algorithm="maxima")

[Out]

-1/2/((b*x + a)^2*b)

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mupad [B]  time = 0.06, size = 14, normalized size = 0.88 \[ -\frac {1}{2\,b\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/atanh(tanh(a + b*x))^3,x)

[Out]

-1/(2*b*atanh(tanh(a + b*x))^2)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/atanh(tanh(b*x+a))**3,x)

[Out]

Exception raised: TypeError

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