3.704 \(\int \cot ^{-1}(\cosh (x)) \coth (x) \text{csch}^3(x) \, dx\)
Optimal. Leaf size=36 \[ \frac{\tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )}{6 \sqrt{2}}+\frac{\coth (x)}{6}-\frac{1}{3} \text{csch}^3(x) \cot ^{-1}(\cosh (x)) \]
[Out]
ArcTanh[Tanh[x]/Sqrt[2]]/(6*Sqrt[2]) + Coth[x]/6 - (ArcCot[Cosh[x]]*Csch[x]^3)/3
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Rubi [A] time = 0.119881, antiderivative size = 36, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used =
{2606, 30, 5208, 12, 453, 206} \[ \frac{\tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )}{6 \sqrt{2}}+\frac{\coth (x)}{6}-\frac{1}{3} \text{csch}^3(x) \cot ^{-1}(\cosh (x)) \]
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 2606
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 30
Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]
Rule 5208
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]]
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 453
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 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
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} \operatorname{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} \operatorname{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} \operatorname{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*}
Mathematica [A] time = 0.167389, size = 40, normalized size = 1.11 \[ \frac{1}{24} \left (2 \sqrt{2} \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )+\text{csch}^3(x) \left (-\cosh (x)+\cosh (3 x)-8 \cot ^{-1}(\cosh (x))\right )\right ) \]
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] time = 0.462, size = 854, normalized size = 23.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(arccot(cosh(x))*cosh(x)/sinh(x)^4,x)
[Out]
4/3*I*exp(3*x)/(-1+exp(2*x))^3*ln(exp(2*x)+1+2*I*exp(x))-1/24*(-8-2^(1/2)*ln(exp(2*x)+(1+2^(1/2))^2)+2^(1/2)*l
n(exp(2*x)+(2^(1/2)-1)^2)+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(-x))*csgn(I*(exp(2*x)+1+2*I*exp(x)))*csgn(I*exp(-x)*(exp(2*x)+1+2*I*ex
p(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)-8*exp(4*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)-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*P
i*csgn(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))*csgn(I*exp(-x)*(ex
p(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)))^2*exp(3*x)+16*Pi*c
sgn(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(-x)*(
exp(2*x)+1+2*I*exp(x)))^3*exp(3*x)-16*Pi*csgn(I*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)+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)-16*Pi*csgn(exp(-x)*(exp(2*x)+1+2*I*exp(x)))^3*exp(3*x)+16*Pi*csgn(I*ex
p(-x))*csgn(I*(-exp(2*x)-1+2*I*exp(x)))*csgn(I*exp(-x)*(-exp(2*x)-1+2*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)))*exp(3*x))/(-1+exp(2*x))^3
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Maxima [B] time = 1.44779, size = 73, normalized size = 2.03 \begin{align*} -\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{align*}
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] time = 2.30467, size = 1422, normalized size = 39.5 \begin{align*} \text{result too large to display} \end{align*}
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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(acot(cosh(x))*cosh(x)/sinh(x)**4,x)
[Out]
Timed out
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Giac [B] time = 1.09246, size = 95, normalized size = 2.64 \begin{align*} \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{align*}
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