3.8.4 \(\int \cot ^{-1}(\cosh (x)) \coth (x) \text {csch}^3(x) \, dx\) [704]

Optimal. Leaf size=36 \[ \frac {\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{6 \sqrt {2}}+\frac {\coth (x)}{6}-\frac {1}{3} \cot ^{-1}(\cosh (x)) \text {csch}^3(x) \]

[Out]

1/6*coth(x)-1/3*arccot(cosh(x))*csch(x)^3+1/12*arctanh(1/2*2^(1/2)*tanh(x))*2^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2686, 30, 5316, 12, 464, 212} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{6 \sqrt {2}}+\frac {\coth (x)}{6}-\frac {1}{3} \text {csch}^3(x) \cot ^{-1}(\cosh (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCot[Cosh[x]]*Coth[x]*Csch[x]^3,x]

[Out]

ArcTanh[Tanh[x]/Sqrt[2]]/(6*Sqrt[2]) + Coth[x]/6 - (ArcCot[Cosh[x]]*Csch[x]^3)/3

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 464

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[c*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 2686

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 5316

Int[((a_.) + ArcCot[u_]*(b_.))*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[a + b*ArcCot[u], w, x] + Dist
[b, Int[SimplifyIntegrand[w*(D[u, x]/(1 + u^2)), x], x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x]
&& InverseFunctionFreeQ[u, x] &&  !MatchQ[v, ((c_.) + (d_.)*x)^(m_.) /; FreeQ[{c, d, m}, x]] && FalseQ[Functi
onOfLinear[v*(a + b*ArcCot[u]), x]]

Rubi steps

\begin {align*} \int \cot ^{-1}(\cosh (x)) \coth (x) \text {csch}^3(x) \, dx &=-\frac {1}{3} \cot ^{-1}(\cosh (x)) \text {csch}^3(x)+\int \frac {2 \text {csch}^2(x)}{3 (-3-\cosh (2 x))} \, dx\\ &=-\frac {1}{3} \cot ^{-1}(\cosh (x)) \text {csch}^3(x)+\frac {2}{3} \int \frac {\text {csch}^2(x)}{-3-\cosh (2 x)} \, dx\\ &=-\frac {1}{3} \cot ^{-1}(\cosh (x)) \text {csch}^3(x)-\frac {2}{3} \text {Subst}\left (\int \frac {1-x^2}{2 x^2 \left (2-x^2\right )} \, dx,x,\tanh (x)\right )\\ &=-\frac {1}{3} \cot ^{-1}(\cosh (x)) \text {csch}^3(x)-\frac {1}{3} \text {Subst}\left (\int \frac {1-x^2}{x^2 \left (2-x^2\right )} \, dx,x,\tanh (x)\right )\\ &=\frac {\coth (x)}{6}-\frac {1}{3} \cot ^{-1}(\cosh (x)) \text {csch}^3(x)+\frac {1}{6} \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\tanh (x)\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{6 \sqrt {2}}+\frac {\coth (x)}{6}-\frac {1}{3} \cot ^{-1}(\cosh (x)) \text {csch}^3(x)\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 40, normalized size = 1.11 \begin {gather*} \frac {1}{24} \left (2 \sqrt {2} \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right )+\left (-8 \cot ^{-1}(\cosh (x))-\cosh (x)+\cosh (3 x)\right ) \text {csch}^3(x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[Cosh[x]]*Coth[x]*Csch[x]^3,x]

[Out]

(2*Sqrt[2]*ArcTanh[Tanh[x]/Sqrt[2]] + (-8*ArcCot[Cosh[x]] - Cosh[x] + Cosh[3*x])*Csch[x]^3)/24

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.52, size = 854, normalized size = 23.72

method	result	size
risch	\(\text {Expression too large to display}\)	\(854\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(cosh(x))*cosh(x)/sinh(x)^4,x,method=_RETURNVERBOSE)

[Out]

4/3*I*exp(3*x)/(exp(2*x)-1)^3*ln(exp(2*x)+1+2*I*exp(x))-1/24*(-8+16*Pi*csgn(I*(exp(2*x)+1+2*I*exp(x)))*csgn(I*
exp(-x)*(exp(2*x)+1+2*I*exp(x)))^2*exp(3*x)-16*Pi*csgn(I*exp(-x)*(-exp(2*x)-1+2*I*exp(x)))^2*csgn(I*exp(-x))*e
xp(3*x)+16*Pi*csgn(I*exp(-x)*(exp(2*x)+1+2*I*exp(x)))^2*csgn(I*exp(-x))*exp(3*x)+16*Pi*csgn(I*exp(-x)*(exp(2*x
)+1+2*I*exp(x)))*csgn(exp(-x)*(exp(2*x)+1+2*I*exp(x)))^2*exp(3*x)+16*Pi*csgn(I*(-exp(2*x)-1+2*I*exp(x)))*csgn(
I*exp(-x)*(-exp(2*x)-1+2*I*exp(x)))^2*exp(3*x)-16*Pi*csgn(I*(exp(2*x)+1+2*I*exp(x)))*csgn(I*exp(-x)*(exp(2*x)+
1+2*I*exp(x)))*csgn(I*exp(-x))*exp(3*x)-16*Pi*csgn(I*exp(-x)*(-exp(2*x)-1+2*I*exp(x)))^3*exp(3*x)+16*exp(2*x)+
16*Pi*csgn(I*exp(-x)*(-exp(2*x)-1+2*I*exp(x)))*csgn(exp(-x)*(-exp(2*x)-1+2*I*exp(x)))*exp(3*x)+16*Pi*csgn(I*(-
exp(2*x)-1+2*I*exp(x)))*csgn(I*exp(-x)*(-exp(2*x)-1+2*I*exp(x)))*csgn(I*exp(-x))*exp(3*x)+16*Pi*csgn(exp(-x)*(
-exp(2*x)-1+2*I*exp(x)))^2*exp(3*x)-8*exp(4*x)+2^(1/2)*ln(exp(2*x)+(2^(1/2)-1)^2)-2^(1/2)*ln(exp(2*x)+(1+2^(1/
2))^2)-16*Pi*csgn(I*exp(-x)*(exp(2*x)+1+2*I*exp(x)))*csgn(exp(-x)*(exp(2*x)+1+2*I*exp(x)))*exp(3*x)-2^(1/2)*ln
(exp(2*x)+(2^(1/2)-1)^2)*exp(6*x)+2^(1/2)*ln(exp(2*x)+(1+2^(1/2))^2)*exp(6*x)+3*2^(1/2)*ln(exp(2*x)+(2^(1/2)-1
)^2)*exp(4*x)+32*I*exp(3*x)*ln(exp(2*x)+1-2*I*exp(x))-16*Pi*csgn(I*exp(-x)*(exp(2*x)+1+2*I*exp(x)))^3*exp(3*x)
-3*2^(1/2)*ln(exp(2*x)+(1+2^(1/2))^2)*exp(4*x)-3*2^(1/2)*ln(exp(2*x)+(2^(1/2)-1)^2)*exp(2*x)+3*2^(1/2)*ln(exp(
2*x)+(1+2^(1/2))^2)*exp(2*x)+16*Pi*csgn(I*exp(-x)*(-exp(2*x)-1+2*I*exp(x)))*csgn(exp(-x)*(-exp(2*x)-1+2*I*exp(
x)))^2*exp(3*x)-16*Pi*csgn(exp(-x)*(exp(2*x)+1+2*I*exp(x)))^3*exp(3*x)+16*Pi*csgn(exp(-x)*(-exp(2*x)-1+2*I*exp
(x)))^3*exp(3*x)+16*Pi*csgn(exp(-x)*(exp(2*x)+1+2*I*exp(x)))^2*exp(3*x))/(exp(2*x)-1)^3

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Maxima [A]
time = 2.14, size = 54, normalized size = 1.50 \begin {gather*} -\frac {1}{24} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (-2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (-2 \, x\right )} + 3}\right ) - \frac {1}{3 \, {\left (e^{\left (-2 \, x\right )} - 1\right )}} - \frac {\operatorname {arccot}\left (\cosh \left (x\right )\right )}{3 \, \sinh \left (x\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(cosh(x))*cosh(x)/sinh(x)^4,x, algorithm="maxima")

[Out]

-1/24*sqrt(2)*log(-(2*sqrt(2) - e^(-2*x) - 3)/(2*sqrt(2) + e^(-2*x) + 3)) - 1/3/(e^(-2*x) - 1) - 1/3*arccot(co
sh(x))/sinh(x)^3

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (27) = 54\).
time = 0.60, size = 423, normalized size = 11.75 \begin {gather*} \frac {8 \, \cosh \left (x\right )^{4} + 32 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + 8 \, \sinh \left (x\right )^{4} + 16 \, {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 64 \, {\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3}\right )} \arctan \left (\frac {2 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}\right ) - 16 \, \cosh \left (x\right )^{2} + {\left (\sqrt {2} \cosh \left (x\right )^{6} + 6 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sqrt {2} \sinh \left (x\right )^{6} + 3 \, {\left (5 \, \sqrt {2} \cosh \left (x\right )^{2} - \sqrt {2}\right )} \sinh \left (x\right )^{4} - 3 \, \sqrt {2} \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \sqrt {2} \cosh \left (x\right )^{3} - 3 \, \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, \sqrt {2} \cosh \left (x\right )^{4} - 6 \, \sqrt {2} \cosh \left (x\right )^{2} + \sqrt {2}\right )} \sinh \left (x\right )^{2} + 3 \, \sqrt {2} \cosh \left (x\right )^{2} + 6 \, {\left (\sqrt {2} \cosh \left (x\right )^{5} - 2 \, \sqrt {2} \cosh \left (x\right )^{3} + \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right ) - \sqrt {2}\right )} \log \left (-\frac {3 \, {\left (2 \, \sqrt {2} - 3\right )} \cosh \left (x\right )^{2} - 4 \, {\left (3 \, \sqrt {2} - 4\right )} \cosh \left (x\right ) \sinh \left (x\right ) + 3 \, {\left (2 \, \sqrt {2} - 3\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt {2} - 3}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} + 3}\right ) + 32 \, {\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 8}{24 \, {\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 3 \, {\left (5 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{4} - 3 \, \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, \cosh \left (x\right )^{4} - 6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 3 \, \cosh \left (x\right )^{2} + 6 \, {\left (\cosh \left (x\right )^{5} - 2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(cosh(x))*cosh(x)/sinh(x)^4,x, algorithm="fricas")

[Out]

1/24*(8*cosh(x)^4 + 32*cosh(x)*sinh(x)^3 + 8*sinh(x)^4 + 16*(3*cosh(x)^2 - 1)*sinh(x)^2 - 64*(cosh(x)^3 + 3*co
sh(x)^2*sinh(x) + 3*cosh(x)*sinh(x)^2 + sinh(x)^3)*arctan(2*(cosh(x) + sinh(x))/(cosh(x)^2 + 2*cosh(x)*sinh(x)
+ sinh(x)^2 + 1)) - 16*cosh(x)^2 + (sqrt(2)*cosh(x)^6 + 6*sqrt(2)*cosh(x)*sinh(x)^5 + sqrt(2)*sinh(x)^6 + 3*(
5*sqrt(2)*cosh(x)^2 - sqrt(2))*sinh(x)^4 - 3*sqrt(2)*cosh(x)^4 + 4*(5*sqrt(2)*cosh(x)^3 - 3*sqrt(2)*cosh(x))*s
inh(x)^3 + 3*(5*sqrt(2)*cosh(x)^4 - 6*sqrt(2)*cosh(x)^2 + sqrt(2))*sinh(x)^2 + 3*sqrt(2)*cosh(x)^2 + 6*(sqrt(2
)*cosh(x)^5 - 2*sqrt(2)*cosh(x)^3 + sqrt(2)*cosh(x))*sinh(x) - sqrt(2))*log(-(3*(2*sqrt(2) - 3)*cosh(x)^2 - 4*
(3*sqrt(2) - 4)*cosh(x)*sinh(x) + 3*(2*sqrt(2) - 3)*sinh(x)^2 + 2*sqrt(2) - 3)/(cosh(x)^2 + sinh(x)^2 + 3)) +
32*(cosh(x)^3 - cosh(x))*sinh(x) + 8)/(cosh(x)^6 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6 + 3*(5*cosh(x)^2 - 1)*sinh(
x)^4 - 3*cosh(x)^4 + 4*(5*cosh(x)^3 - 3*cosh(x))*sinh(x)^3 + 3*(5*cosh(x)^4 - 6*cosh(x)^2 + 1)*sinh(x)^2 + 3*c
osh(x)^2 + 6*(cosh(x)^5 - 2*cosh(x)^3 + cosh(x))*sinh(x) - 1)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (34) = 68\).
time = 76.67, size = 214, normalized size = 5.94 \begin {gather*} - \frac {\sqrt {2} \log {\left (4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \sqrt {2} \tanh {\left (\frac {x}{2} \right )} + 4 \right )}}{24} + \frac {\sqrt {2} \log {\left (4 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4 \sqrt {2} \tanh {\left (\frac {x}{2} \right )} + 4 \right )}}{24} - \frac {\tanh ^{3}{\left (\frac {x}{2} \right )} \operatorname {acot}{\left (\frac {\tanh ^{2}{\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} + \frac {1}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} \right )}}{24} + \frac {\tanh {\left (\frac {x}{2} \right )} \operatorname {acot}{\left (\frac {\tanh ^{2}{\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} + \frac {1}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} \right )}}{8} + \frac {\tanh {\left (\frac {x}{2} \right )}}{12} - \frac {\operatorname {acot}{\left (\frac {\tanh ^{2}{\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} + \frac {1}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} \right )}}{8 \tanh {\left (\frac {x}{2} \right )}} + \frac {1}{12 \tanh {\left (\frac {x}{2} \right )}} + \frac {\operatorname {acot}{\left (\frac {\tanh ^{2}{\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} + \frac {1}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} \right )}}{24 \tanh ^{3}{\left (\frac {x}{2} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(cosh(x))*cosh(x)/sinh(x)**4,x)

[Out]

-sqrt(2)*log(4*tanh(x/2)**2 - 4*sqrt(2)*tanh(x/2) + 4)/24 + sqrt(2)*log(4*tanh(x/2)**2 + 4*sqrt(2)*tanh(x/2) +
4)/24 - tanh(x/2)**3*acot(tanh(x/2)**2/(tanh(x/2)**2 - 1) + 1/(tanh(x/2)**2 - 1))/24 + tanh(x/2)*acot(tanh(x/
2)**2/(tanh(x/2)**2 - 1) + 1/(tanh(x/2)**2 - 1))/8 + tanh(x/2)/12 - acot(tanh(x/2)**2/(tanh(x/2)**2 - 1) + 1/(
tanh(x/2)**2 - 1))/(8*tanh(x/2)) + 1/(12*tanh(x/2)) + acot(tanh(x/2)**2/(tanh(x/2)**2 - 1) + 1/(tanh(x/2)**2 -
1))/(24*tanh(x/2)**3)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (27) = 54\).
time = 0.98, size = 70, normalized size = 1.94 \begin {gather*} \frac {1}{24} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (2 \, x\right )} + 3}\right ) + \frac {1}{3 \, {\left (e^{\left (2 \, x\right )} - 1\right )}} + \frac {8 \, \arctan \left (\frac {2}{e^{\left (-x\right )} + e^{x}}\right )}{3 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(cosh(x))*cosh(x)/sinh(x)^4,x, algorithm="giac")

[Out]

1/24*sqrt(2)*log(-(2*sqrt(2) - e^(2*x) - 3)/(2*sqrt(2) + e^(2*x) + 3)) + 1/3/(e^(2*x) - 1) + 8/3*arctan(2/(e^(
-x) + e^x))/(e^(-x) - e^x)^3

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Mupad [B]
time = 0.54, size = 103, normalized size = 2.86 \begin {gather*} \frac {\sqrt {2}\,\ln \left (-\frac {2\,{\mathrm {e}}^{2\,x}}{3}-\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}+4\right )}{24}\right )}{24}-\frac {\sqrt {2}\,\ln \left (\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}+4\right )}{24}-\frac {2\,{\mathrm {e}}^{2\,x}}{3}\right )}{24}+\frac {1}{3\,\left ({\mathrm {e}}^{2\,x}-1\right )}-\frac {8\,{\mathrm {e}}^{3\,x}\,\mathrm {acot}\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}{3\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((acot(cosh(x))*cosh(x))/sinh(x)^4,x)

[Out]

(2^(1/2)*log(- (2*exp(2*x))/3 - (2^(1/2)*(12*exp(2*x) + 4))/24))/24 - (2^(1/2)*log((2^(1/2)*(12*exp(2*x) + 4))
/24 - (2*exp(2*x))/3))/24 + 1/(3*(exp(2*x) - 1)) - (8*exp(3*x)*acot(exp(-x)/2 + exp(x)/2))/(3*(3*exp(2*x) - 3*
exp(4*x) + exp(6*x) - 1))

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