Integrand size = 10, antiderivative size = 36 \[ \int \cot ^{-1}(\cosh (x)) \coth (x) \text {csch}^3(x) \, dx=\frac {\text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{6 \sqrt {2}}+\frac {\coth (x)}{6}-\frac {1}{3} \cot ^{-1}(\cosh (x)) \text {csch}^3(x) \]
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 144, normalized size of antiderivative = 4.00 \[ \int \cot ^{-1}(\cosh (x)) \coth (x) \text {csch}^3(x) \, dx=\frac {1}{48} \text {csch}^3(x) \left (-16 \cot ^{-1}(\cosh (x))-2 \cosh (x)+2 \cosh (3 x)-3 i \sqrt {2} \arctan \left (1-i \sqrt {2} \tanh \left (\frac {x}{2}\right )\right ) \sinh (x)+3 i \sqrt {2} \arctan \left (1+i \sqrt {2} \tanh \left (\frac {x}{2}\right )\right ) \sinh (x)+i \sqrt {2} \arctan \left (1-i \sqrt {2} \tanh \left (\frac {x}{2}\right )\right ) \sinh (3 x)-i \sqrt {2} \arctan \left (1+i \sqrt {2} \tanh \left (\frac {x}{2}\right )\right ) \sinh (3 x)\right ) \]
(Csch[x]^3*(-16*ArcCot[Cosh[x]] - 2*Cosh[x] + 2*Cosh[3*x] - (3*I)*Sqrt[2]* ArcTan[1 - I*Sqrt[2]*Tanh[x/2]]*Sinh[x] + (3*I)*Sqrt[2]*ArcTan[1 + I*Sqrt[ 2]*Tanh[x/2]]*Sinh[x] + I*Sqrt[2]*ArcTan[1 - I*Sqrt[2]*Tanh[x/2]]*Sinh[3*x ] - I*Sqrt[2]*ArcTan[1 + I*Sqrt[2]*Tanh[x/2]]*Sinh[3*x]))/48
Time = 0.31 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5731, 27, 3042, 25, 4889, 27, 359, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \coth (x) \text {csch}^3(x) \cot ^{-1}(\cosh (x)) \, dx\) |
\(\Big \downarrow \) 5731 |
\(\displaystyle \int -\frac {2 \text {csch}^2(x)}{3 (\cosh (2 x)+3)}dx-\frac {1}{3} \text {csch}^3(x) \cot ^{-1}(\cosh (x))\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2}{3} \int \frac {\text {csch}^2(x)}{\cosh (2 x)+3}dx-\frac {1}{3} \text {csch}^3(x) \cot ^{-1}(\cosh (x))\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{3} \text {csch}^3(x) \cot ^{-1}(\cosh (x))-\frac {2}{3} \int -\frac {1}{(\cos (2 i x)+3) \sin (i x)^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{3} \text {csch}^3(x) \cot ^{-1}(\cosh (x))+\frac {2}{3} \int \frac {1}{(\cos (2 i x)+3) \sin (i x)^2}dx\) |
\(\Big \downarrow \) 4889 |
\(\displaystyle \frac {2}{3} \int -\frac {\coth ^2(x) \left (1-\tanh ^2(x)\right )}{2 \left (2-\tanh ^2(x)\right )}d\tanh (x)-\frac {1}{3} \text {csch}^3(x) \cot ^{-1}(\cosh (x))\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{3} \int \frac {\coth ^2(x) \left (1-\tanh ^2(x)\right )}{2-\tanh ^2(x)}d\tanh (x)-\frac {1}{3} \text {csch}^3(x) \cot ^{-1}(\cosh (x))\) |
\(\Big \downarrow \) 359 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{2} \int \frac {1}{2-\tanh ^2(x)}d\tanh (x)+\frac {\coth (x)}{2}\right )-\frac {1}{3} \text {csch}^3(x) \cot ^{-1}(\cosh (x))\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{3} \left (\frac {\text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{2 \sqrt {2}}+\frac {\coth (x)}{2}\right )-\frac {1}{3} \text {csch}^3(x) \cot ^{-1}(\cosh (x))\) |
3.8.4.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, With[{d = FreeFactors [Tan[v], x]}, Simp[d/Coefficient[v, x, 1] Subst[Int[SubstFor[1/(1 + d^2*x ^2), Tan[v]/d, u, x], x], x, Tan[v]/d], x]] /; !FalseQ[v] && FunctionOfQ[N onfreeFactors[Tan[v], x], u, x]] /; InverseFunctionFreeQ[u, x] && !MatchQ[ u, (v_.)*((c_.)*tan[w_]^(n_.)*tan[z_]^(n_.))^(p_.) /; FreeQ[{c, p}, x] && I ntegerQ[n] && LinearQ[w, x] && EqQ[z, 2*w]]
Int[((a_.) + ArcCot[u_]*(b_.))*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Simp[(a + b*ArcCot[u])*w, x] + Simp[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[FunctionOfLinear[v*(a + b*ArcCot[u]), x]]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.37 (sec) , antiderivative size = 850, normalized size of antiderivative = 23.61
\[\text {Expression too large to display}\]
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*csg n(-I*exp(2*x)+2*exp(x)-I)*csgn(I*exp(-x)*(exp(2*x)+1+2*I*exp(x)))^2*exp(3* x)-16*Pi*csgn(I*exp(2*x)+I+2*exp(x))*csgn(I*exp(-x)*(2*I*exp(x)-exp(2*x)-1 ))^2*exp(3*x)-16*Pi*csgn(I*exp(-x))*csgn(I*exp(2*x)+I+2*exp(x))*csgn(I*exp (-x)*(2*I*exp(x)-exp(2*x)-1))*exp(3*x)+16*exp(2*x)+16*Pi*csgn(I*exp(-x)*(2 *I*exp(x)-exp(2*x)-1))*csgn(exp(-x)*(2*I*exp(x)-exp(2*x)-1))*exp(3*x)-16*P i*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)-8*exp(4*x)+16*Pi*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)*(2*I*exp(x)-exp(2*x)-1))*csg n(exp(-x)*(2*I*exp(x)-exp(2*x)-1))^2*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*c sgn(I*exp(-x))*csgn(-I*exp(2*x)+2*exp(x)-I)*csgn(I*exp(-x)*(exp(2*x)+1+2*I *exp(x)))*exp(3*x)-2^(1/2)*ln(exp(2*x)+(1+2^(1/2))^2)-16*Pi*csgn(I*exp(-x) )*csgn(I*exp(-x)*(2*I*exp(x)-exp(2*x)-1))^2*exp(3*x)+2^(1/2)*ln(exp(2*x)+( 2^(1/2)-1)^2)+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)*(2*I* exp(x)-exp(2*x)-1))^3*exp(3*x)+16*Pi*csgn(exp(-x)*(2*I*exp(x)-exp(2*x)-1)) ^2*exp(3*x)-ln(exp(2*x)+(2^(1/2)-1)^2)*2^(1/2)*exp(6*x)+32*I*exp(3*x)*ln(e xp(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)+ln(exp(2*x)+(1+2^(1/2))^2)*2^(1/2)*exp(6*x)+3*ln(exp(2*x)+(2^(1/2)...
Leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (27) = 54\).
Time = 0.26 (sec) , antiderivative size = 423, normalized size of antiderivative = 11.75 \[ \int \cot ^{-1}(\cosh (x)) \coth (x) \text {csch}^3(x) \, dx=\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 )}} \]
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*cosh(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)*s inh(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))*sinh( 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*sqr t(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*co sh(x)^2 + 1)*sinh(x)^2 + 3*cosh(x)^2 + 6*(cosh(x)^5 - 2*cosh(x)^3 + cosh(x ))*sinh(x) - 1)
Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (34) = 68\).
Time = 78.09 (sec) , antiderivative size = 214, normalized size of antiderivative = 5.94 \[ \int \cot ^{-1}(\cosh (x)) \coth (x) \text {csch}^3(x) \, dx=- \frac {\sqrt {2} \log {\left (4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \sqrt {2} \tanh {\left (\frac {x}{2} \right )} + 4 \right )}}{24} + \frac {\sqrt {2} \log {\left (4 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4 \sqrt {2} \tanh {\left (\frac {x}{2} \right )} + 4 \right )}}{24} - \frac {\tanh ^{3}{\left (\frac {x}{2} \right )} \operatorname {acot}{\left (\frac {\tanh ^{2}{\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} + \frac {1}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} \right )}}{24} + \frac {\tanh {\left (\frac {x}{2} \right )} \operatorname {acot}{\left (\frac {\tanh ^{2}{\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} + \frac {1}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} \right )}}{8} + \frac {\tanh {\left (\frac {x}{2} \right )}}{12} - \frac {\operatorname {acot}{\left (\frac {\tanh ^{2}{\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} + \frac {1}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} \right )}}{8 \tanh {\left (\frac {x}{2} \right )}} + \frac {1}{12 \tanh {\left (\frac {x}{2} \right )}} + \frac {\operatorname {acot}{\left (\frac {\tanh ^{2}{\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} + \frac {1}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} \right )}}{24 \tanh ^{3}{\left (\frac {x}{2} \right )}} \]
-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(tan h(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)
Time = 0.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.50 \[ \int \cot ^{-1}(\cosh (x)) \coth (x) \text {csch}^3(x) \, dx=-\frac {1}{24} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (-2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (-2 \, x\right )} + 3}\right ) - \frac {1}{3 \, {\left (e^{\left (-2 \, x\right )} - 1\right )}} - \frac {\operatorname {arccot}\left (\cosh \left (x\right )\right )}{3 \, \sinh \left (x\right )^{3}} \]
-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(cosh(x))/sinh(x)^3
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (27) = 54\).
Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.94 \[ \int \cot ^{-1}(\cosh (x)) \coth (x) \text {csch}^3(x) \, dx=\frac {1}{24} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (2 \, x\right )} + 3}\right ) + \frac {1}{3 \, {\left (e^{\left (2 \, x\right )} - 1\right )}} + \frac {8 \, \arctan \left (\frac {2}{e^{\left (-x\right )} + e^{x}}\right )}{3 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{3}} \]
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
Time = 0.63 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.86 \[ \int \cot ^{-1}(\cosh (x)) \coth (x) \text {csch}^3(x) \, dx=\frac {\sqrt {2}\,\ln \left (-\frac {2\,{\mathrm {e}}^{2\,x}}{3}-\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}+4\right )}{24}\right )}{24}-\frac {\sqrt {2}\,\ln \left (\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}+4\right )}{24}-\frac {2\,{\mathrm {e}}^{2\,x}}{3}\right )}{24}+\frac {1}{3\,\left ({\mathrm {e}}^{2\,x}-1\right )}-\frac {8\,{\mathrm {e}}^{3\,x}\,\mathrm {acot}\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}{3\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )} \]