3.2.7 \(\int \frac {1}{\tanh ^{-1}(\tanh (a+b x))^3} \, dx\) [107]

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 = 0.01, 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 = {2188, 30} \begin {gather*} -\frac {1}{2 b \tanh ^{-1}(\tanh (a+b x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcTanh[Tanh[a + b*x]]^(-3),x]

[Out]

-1/2*1/(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 2188

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 {\text {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 \begin {gather*} -\frac {1}{2 b \tanh ^{-1}(\tanh (a+b x))^2} \end {gather*}

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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Maple [A]
time = 0.06, size = 15, normalized size = 0.94

method	result	size
derivativedivides	\(-\frac {1}{2 b \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}\)	\(15\)
default	\(-\frac {1}{2 b \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}\)	\(15\)
risch	\(\frac {8}{b \left (-\pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}+1}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )+\pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}+1}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )^{2}-\pi \mathrm {csgn}\left (i {\mathrm e}^{b x +a}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )+2 \pi \,\mathrm {csgn}\left (i {\mathrm e}^{b x +a}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{2}-\pi \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{3}+\pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )^{2}-\pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )^{3}-4 i \ln \left ({\mathrm e}^{b x +a}\right )\right )^{2}}\)	\(273\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.47, size = 12, normalized size = 0.75 \begin {gather*} -\frac {1}{2 \, {\left (b x + a\right )}^{2} b} \end {gather*}

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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Fricas [A]
time = 0.35, size = 24, normalized size = 1.50 \begin {gather*} -\frac {1}{2 \, {\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} \end {gather*}

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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Sympy [A]
time = 25.47, size = 24, normalized size = 1.50 \begin {gather*} \begin {cases} - \frac {1}{2 b \operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}} & \text {for}\: b \neq 0 \\\frac {x}{\operatorname {atanh}^{3}{\left (\tanh {\left (a \right )} \right )}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-1/(2*b*atanh(tanh(a + b*x))**2), Ne(b, 0)), (x/atanh(tanh(a))**3, True))

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Giac [A]
time = 0.39, size = 12, normalized size = 0.75 \begin {gather*} -\frac {1}{2 \, {\left (b x + a\right )}^{2} b} \end {gather*}

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

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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