3.2.80 \(\int \frac {1}{\coth ^{-1}(\tanh (a+b x))^3} \, dx\) [180]

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

[Out]

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

________________________________________________________________________________________

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

Antiderivative was successfully verified.

[In]

Int[ArcCoth[Tanh[a + b*x]]^(-3),x]

[Out]

-1/2*1/(b*ArcCoth[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}{\coth ^{-1}(\tanh (a+b x))^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^3} \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{b}\\ &=-\frac {1}{2 b \coth ^{-1}(\tanh (a+b x))^2}\\ \end {align*}

________________________________________________________________________________________

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

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*1/(b*ArcCoth[Tanh[a + b*x]]^2)

________________________________________________________________________________________

Maple [A]
time = 0.19, size = 15, normalized size = 0.94

method	result	size
derivativedivides	\(-\frac {1}{2 b \mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )^{2}}\)	\(15\)
default	\(-\frac {1}{2 b \mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )^{2}}\)	\(15\)
risch	\(\frac {8}{b \left (\pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}+1}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}+1}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )^{2}-2 \pi \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}+1}\right )^{2}+2 \pi \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}+1}\right )^{3}+\pi \mathrm {csgn}\left (i {\mathrm e}^{b x +a}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )-2 \pi \,\mathrm {csgn}\left (i {\mathrm e}^{b x +a}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{2}+\pi \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{3}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}+1}\right )^{3}+2 \pi +4 i \ln \left ({\mathrm e}^{b x +a}\right )\right )^{2}}\)	\(318\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [C] Result contains complex when optimal does not.
time = 0.47, size = 28, normalized size = 1.75 \begin {gather*} \frac {2}{{\left (\pi ^{2} - 4 i \, \pi {\left (b x + a\right )} - 4 \, {\left (b x + a\right )}^{2}\right )} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/((pi^2 - 4*I*pi*(b*x + a) - 4*(b*x + a)^2)*b)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (14) = 28\).
time = 0.36, size = 107, normalized size = 6.69 \begin {gather*} -\frac {2 \, {\left (4 \, b^{2} x^{2} + 8 \, a b x - \pi ^{2} + 4 \, a^{2}\right )}}{16 \, b^{5} x^{4} + 64 \, a b^{4} x^{3} + \pi ^{4} b + 8 \, \pi ^{2} a^{2} b + 16 \, a^{4} b + 8 \, {\left (\pi ^{2} b^{3} + 12 \, a^{2} b^{3}\right )} x^{2} + 16 \, {\left (\pi ^{2} a b^{2} + 4 \, a^{3} b^{2}\right )} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(4*b^2*x^2 + 8*a*b*x - pi^2 + 4*a^2)/(16*b^5*x^4 + 64*a*b^4*x^3 + pi^4*b + 8*pi^2*a^2*b + 16*a^4*b + 8*(pi^
2*b^3 + 12*a^2*b^3)*x^2 + 16*(pi^2*a*b^2 + 4*a^3*b^2)*x)

________________________________________________________________________________________

Sympy [A]
time = 22.44, size = 24, normalized size = 1.50 \begin {gather*} \begin {cases} - \frac {1}{2 b \operatorname {acoth}^{2}{\left (\tanh {\left (a + b x \right )} \right )}} & \text {for}\: b \neq 0 \\\frac {x}{\operatorname {acoth}^{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/acoth(tanh(b*x+a))**3,x)

[Out]

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

________________________________________________________________________________________

Giac [C] Result contains complex when optimal does not.
time = 0.38, size = 44, normalized size = 2.75 \begin {gather*} -\frac {2 i}{4 i \, b^{3} x^{2} - 4 \, \pi b^{2} x + 8 i \, a b^{2} x - i \, \pi ^{2} b - 4 \, \pi a b + 4 i \, a^{2} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*I/(4*I*b^3*x^2 - 4*pi*b^2*x + 8*I*a*b^2*x - I*pi^2*b - 4*pi*a*b + 4*I*a^2*b)

________________________________________________________________________________________

Mupad [B]
time = 0.07, size = 14, normalized size = 0.88 \begin {gather*} -\frac {1}{2\,b\,{\mathrm {acoth}\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/acoth(tanh(a + b*x))^3,x)

[Out]

-1/(2*b*acoth(tanh(a + b*x))^2)

________________________________________________________________________________________