\(\int \frac {\coth ^2(x)}{a+b \cosh (x)} \, dx\) [184]

Optimal result

Integrand size = 13, antiderivative size = 77 \[ \int \frac {\coth ^2(x)}{a+b \cosh (x)} \, dx=\frac {2 a^2 \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} (a+b)^{3/2}}-\frac {a \coth (x)}{a^2-b^2}+\frac {b \text {csch}(x)}{a^2-b^2} \] Output:

2*a^2*arctanh((a-b)^(1/2)*tanh(1/2*x)/(a+b)^(1/2))/(a-b)^(3/2)/(a+b)^(3/2) 
-a*coth(x)/(a^2-b^2)+b*csch(x)/(a^2-b^2)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00 \[ \int \frac {\coth ^2(x)}{a+b \cosh (x)} \, dx=\frac {2 a^2 \arctan \left (\frac {(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{3/2}}-\frac {\coth \left (\frac {x}{2}\right )}{2 (a+b)}-\frac {\tanh \left (\frac {x}{2}\right )}{2 (a-b)} \] Input:

Integrate[Coth[x]^2/(a + b*Cosh[x]),x]
 

Output:

(2*a^2*ArcTan[((a - b)*Tanh[x/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) - 
Coth[x/2]/(2*(a + b)) - Tanh[x/2]/(2*(a - b))
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.14, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.923, Rules used = {3042, 25, 3206, 25, 3042, 25, 3086, 24, 3138, 221, 4254, 24}

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.

\(\displaystyle \int \frac {\coth ^2(x)}{a+b \cosh (x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -\frac {\tan \left (-\frac {\pi }{2}+i x\right )^2}{a-b \sin \left (-\frac {\pi }{2}+i x\right )}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \frac {\tan \left (i x-\frac {\pi }{2}\right )^2}{a-b \sin \left (i x-\frac {\pi }{2}\right )}dx\)

\(\Big \downarrow \) 3206

\(\displaystyle \frac {a^2 \int \frac {1}{a+b \cosh (x)}dx}{a^2-b^2}-\frac {a \int -\text {csch}^2(x)dx}{a^2-b^2}-\frac {b \int \coth (x) \text {csch}(x)dx}{a^2-b^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {a^2 \int \frac {1}{a+b \cosh (x)}dx}{a^2-b^2}+\frac {a \int \text {csch}^2(x)dx}{a^2-b^2}-\frac {b \int \coth (x) \text {csch}(x)dx}{a^2-b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a^2 \int \frac {1}{a+b \sin \left (i x+\frac {\pi }{2}\right )}dx}{a^2-b^2}+\frac {a \int -\csc (i x)^2dx}{a^2-b^2}-\frac {b \int \sec \left (i x-\frac {\pi }{2}\right ) \tan \left (i x-\frac {\pi }{2}\right )dx}{a^2-b^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {a^2 \int \frac {1}{a+b \sin \left (i x+\frac {\pi }{2}\right )}dx}{a^2-b^2}-\frac {a \int \csc (i x)^2dx}{a^2-b^2}-\frac {b \int \sec \left (i x-\frac {\pi }{2}\right ) \tan \left (i x-\frac {\pi }{2}\right )dx}{a^2-b^2}\)

\(\Big \downarrow \) 3086

\(\displaystyle \frac {a^2 \int \frac {1}{a+b \sin \left (i x+\frac {\pi }{2}\right )}dx}{a^2-b^2}-\frac {a \int \csc (i x)^2dx}{a^2-b^2}+\frac {i b \int 1d(-i \text {csch}(x))}{a^2-b^2}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {a^2 \int \frac {1}{a+b \sin \left (i x+\frac {\pi }{2}\right )}dx}{a^2-b^2}-\frac {a \int \csc (i x)^2dx}{a^2-b^2}+\frac {b \text {csch}(x)}{a^2-b^2}\)

\(\Big \downarrow \) 3138

\(\displaystyle -\frac {a \int \csc (i x)^2dx}{a^2-b^2}+\frac {2 a^2 \int \frac {1}{-\left ((a-b) \tanh ^2\left (\frac {x}{2}\right )\right )+a+b}d\tanh \left (\frac {x}{2}\right )}{a^2-b^2}+\frac {b \text {csch}(x)}{a^2-b^2}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {a \int \csc (i x)^2dx}{a^2-b^2}+\frac {2 a^2 \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b} \left (a^2-b^2\right )}+\frac {b \text {csch}(x)}{a^2-b^2}\)

\(\Big \downarrow \) 4254

\(\displaystyle -\frac {i a \int 1d(-i \coth (x))}{a^2-b^2}+\frac {2 a^2 \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b} \left (a^2-b^2\right )}+\frac {b \text {csch}(x)}{a^2-b^2}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {2 a^2 \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b} \left (a^2-b^2\right )}-\frac {a \coth (x)}{a^2-b^2}+\frac {b \text {csch}(x)}{a^2-b^2}\)

Input:

Int[Coth[x]^2/(a + b*Cosh[x]),x]
 

Output:

(2*a^2*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + 
 b]*(a^2 - b^2)) - (a*Coth[x])/(a^2 - b^2) + (b*Csch[x])/(a^2 - b^2)
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 221

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3086

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( 
n_.), x_Symbol] :> Simp[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 3138

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ 
e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d)   Subst[Int[1/(a + b + 
(a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] 
 && NeQ[a^2 - b^2, 0]
 

rule 3206

Int[((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)]), x_Symbol] :> Simp[a/(a^2 - b^2)   Int[(g*Tan[e + f*x])^p/Sin[e + f*x] 
^2, x], x] + (-Simp[b*(g/(a^2 - b^2))   Int[(g*Tan[e + f*x])^(p - 1)/Cos[e 
+ f*x], x], x] - Simp[a^2*(g^2/(a^2 - b^2))   Int[(g*Tan[e + f*x])^(p - 2)/ 
(a + b*Sin[e + f*x]), x], x]) /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2 
, 0] && IntegersQ[2*p] && GtQ[p, 1]
 

rule 4254

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1)   Subst[Int[Exp 
andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, 
 d}, x] && IGtQ[n/2, 0]
 

Maple [A] (verified)

Time = 0.82 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.01

method	result	size
default	\(-\frac {\tanh \left (\frac {x}{2}\right )}{2 \left (a -b \right )}-\frac {1}{2 \left (a +b \right ) \tanh \left (\frac {x}{2}\right )}+\frac {2 a^{2} \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\)	\(78\)
risch	\(-\frac {2 \left (-{\mathrm e}^{x} b +a \right )}{\left ({\mathrm e}^{2 x}-1\right ) \left (a^{2}-b^{2}\right )}+\frac {a^{2} \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right )}-\frac {a^{2} \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right )}\)	\(167\)

Input:

int(coth(x)^2/(a+b*cosh(x)),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (67) = 134\).

Time = 0.11 (sec) , antiderivative size = 470, normalized size of antiderivative = 6.10 \[ \int \frac {\coth ^2(x)}{a+b \cosh (x)} \, dx=\left [\frac {2 \, a^{3} - 2 \, a b^{2} + {\left (a^{2} \cosh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right ) + a^{2} \sinh \left (x\right )^{2} - a^{2}\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} - b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) + 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + b}\right ) - 2 \, {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right ) - 2 \, {\left (a^{2} b - b^{3}\right )} \sinh \left (x\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4} - {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right ) - {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )^{2}}, \frac {2 \, {\left (a^{3} - a b^{2} + {\left (a^{2} \cosh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right ) + a^{2} \sinh \left (x\right )^{2} - a^{2}\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{a^{2} - b^{2}}\right ) - {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right ) - {\left (a^{2} b - b^{3}\right )} \sinh \left (x\right )\right )}}{a^{4} - 2 \, a^{2} b^{2} + b^{4} - {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right ) - {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )^{2}}\right ] \] Input:

integrate(coth(x)^2/(a+b*cosh(x)),x, algorithm="fricas")
 

Output:

[(2*a^3 - 2*a*b^2 + (a^2*cosh(x)^2 + 2*a^2*cosh(x)*sinh(x) + a^2*sinh(x)^2 
 - a^2)*sqrt(a^2 - b^2)*log((b^2*cosh(x)^2 + b^2*sinh(x)^2 + 2*a*b*cosh(x) 
 + 2*a^2 - b^2 + 2*(b^2*cosh(x) + a*b)*sinh(x) + 2*sqrt(a^2 - b^2)*(b*cosh 
(x) + b*sinh(x) + a))/(b*cosh(x)^2 + b*sinh(x)^2 + 2*a*cosh(x) + 2*(b*cosh 
(x) + a)*sinh(x) + b)) - 2*(a^2*b - b^3)*cosh(x) - 2*(a^2*b - b^3)*sinh(x) 
)/(a^4 - 2*a^2*b^2 + b^4 - (a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2 - 2*(a^4 - 2* 
a^2*b^2 + b^4)*cosh(x)*sinh(x) - (a^4 - 2*a^2*b^2 + b^4)*sinh(x)^2), 2*(a^ 
3 - a*b^2 + (a^2*cosh(x)^2 + 2*a^2*cosh(x)*sinh(x) + a^2*sinh(x)^2 - a^2)* 
sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cosh(x) + b*sinh(x) + a)/(a^2 
 - b^2)) - (a^2*b - b^3)*cosh(x) - (a^2*b - b^3)*sinh(x))/(a^4 - 2*a^2*b^2 
 + b^4 - (a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2 - 2*(a^4 - 2*a^2*b^2 + b^4)*cos 
h(x)*sinh(x) - (a^4 - 2*a^2*b^2 + b^4)*sinh(x)^2)]
 

Sympy [F]

\[ \int \frac {\coth ^2(x)}{a+b \cosh (x)} \, dx=\int \frac {\coth ^{2}{\left (x \right )}}{a + b \cosh {\left (x \right )}}\, dx \] Input:

integrate(coth(x)**2/(a+b*cosh(x)),x)
 

Output:

Integral(coth(x)**2/(a + b*cosh(x)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\coth ^2(x)}{a+b \cosh (x)} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(coth(x)^2/(a+b*cosh(x)),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?` f 
or more de
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.99 \[ \int \frac {\coth ^2(x)}{a+b \cosh (x)} \, dx=\frac {2 \, a^{2} \arctan \left (\frac {b e^{x} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{{\left (a^{2} - b^{2}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {2 \, {\left (b e^{x} - a\right )}}{{\left (a^{2} - b^{2}\right )} {\left (e^{\left (2 \, x\right )} - 1\right )}} \] Input:

integrate(coth(x)^2/(a+b*cosh(x)),x, algorithm="giac")
 

Output:

2*a^2*arctan((b*e^x + a)/sqrt(-a^2 + b^2))/((a^2 - b^2)*sqrt(-a^2 + b^2)) 
+ 2*(b*e^x - a)/((a^2 - b^2)*(e^(2*x) - 1))
 

Mupad [B] (verification not implemented)

Time = 2.36 (sec) , antiderivative size = 337, normalized size of antiderivative = 4.38 \[ \int \frac {\coth ^2(x)}{a+b \cosh (x)} \, dx=-\frac {\frac {2\,a}{a^2-b^2}-\frac {2\,b\,{\mathrm {e}}^x}{a^2-b^2}}{{\mathrm {e}}^{2\,x}-1}-\frac {2\,\mathrm {atan}\left (\left ({\mathrm {e}}^x\,\left (\frac {2\,a^2}{b^2\,{\left (a^2-b^2\right )}^2\,\sqrt {a^4}}+\frac {2\,\left (a^3\,\sqrt {a^4}-a\,b^2\,\sqrt {a^4}\right )}{a\,b^2\,\left (a^2-b^2\right )\,\sqrt {-{\left (a^2-b^2\right )}^3}\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}}\right )-\frac {2\,\left (b^3\,\sqrt {a^4}-a^2\,b\,\sqrt {a^4}\right )}{a\,b^2\,\left (a^2-b^2\right )\,\sqrt {-{\left (a^2-b^2\right )}^3}\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}}\right )\,\left (\frac {b^3\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}}{2}-\frac {a^2\,b\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}}{2}\right )\right )\,\sqrt {a^4}}{\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}} \] Input:

int(coth(x)^2/(a + b*cosh(x)),x)
 

Output:

- ((2*a)/(a^2 - b^2) - (2*b*exp(x))/(a^2 - b^2))/(exp(2*x) - 1) - (2*atan( 
(exp(x)*((2*a^2)/(b^2*(a^2 - b^2)^2*(a^4)^(1/2)) + (2*(a^3*(a^4)^(1/2) - a 
*b^2*(a^4)^(1/2)))/(a*b^2*(a^2 - b^2)*(-(a^2 - b^2)^3)^(1/2)*(b^6 - a^6 - 
3*a^2*b^4 + 3*a^4*b^2)^(1/2))) - (2*(b^3*(a^4)^(1/2) - a^2*b*(a^4)^(1/2))) 
/(a*b^2*(a^2 - b^2)*(-(a^2 - b^2)^3)^(1/2)*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4* 
b^2)^(1/2)))*((b^3*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2))/2 - (a^2*b*( 
b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2))/2))*(a^4)^(1/2))/(b^6 - a^6 - 3* 
a^2*b^4 + 3*a^4*b^2)^(1/2)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.17 \[ \int \frac {\coth ^2(x)}{a+b \cosh (x)} \, dx=\frac {-2 e^{2 x} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a^{2}+2 \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a^{2}-2 e^{2 x} a^{3}+2 e^{2 x} a \,b^{2}+2 e^{x} a^{2} b -2 e^{x} b^{3}}{e^{2 x} a^{4}-2 e^{2 x} a^{2} b^{2}+e^{2 x} b^{4}-a^{4}+2 a^{2} b^{2}-b^{4}} \] Input:

int(coth(x)^2/(a+b*cosh(x)),x)
 

Output:

(2*( - e**(2*x)*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a**2 + b**2 
))*a**2 + sqrt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a**2 + b**2))*a** 
2 - e**(2*x)*a**3 + e**(2*x)*a*b**2 + e**x*a**2*b - e**x*b**3))/(e**(2*x)* 
a**4 - 2*e**(2*x)*a**2*b**2 + e**(2*x)*b**4 - a**4 + 2*a**2*b**2 - b**4)