\(\int \frac {\coth ^2(x)}{(a+b \sinh (x))^2} \, dx\) [241]

Optimal result

Integrand size = 13, antiderivative size = 80 \[ \int \frac {\coth ^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {2 b \text {arctanh}(\cosh (x))}{a^3}-\frac {2 \left (a^2+2 b^2\right ) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2}}-\frac {2 \coth (x)}{a^2}+\frac {\coth (x)}{a (a+b \sinh (x))} \] Output:

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

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.40 \[ \int \frac {\coth ^2(x)}{(a+b \sinh (x))^2} \, dx=-\frac {-\frac {4 \left (a^2+2 b^2\right ) \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+a \coth \left (\frac {x}{2}\right )-4 b \log \left (\cosh \left (\frac {x}{2}\right )\right )+4 b \log \left (\sinh \left (\frac {x}{2}\right )\right )+\frac {2 a b \cosh (x)}{a+b \sinh (x)}+a \tanh \left (\frac {x}{2}\right )}{2 a^3} \] Input:

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

Output:

-1/2*((-4*(a^2 + 2*b^2)*ArcTan[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2]])/Sqrt[- 
a^2 - b^2] + a*Coth[x/2] - 4*b*Log[Cosh[x/2]] + 4*b*Log[Sinh[x/2]] + (2*a* 
b*Cosh[x])/(a + b*Sinh[x]) + a*Tanh[x/2])/a^3
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.12 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.58, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.615, Rules used = {3042, 25, 3202, 25, 3042, 25, 3535, 25, 3042, 25, 3535, 3042, 26, 3480, 26, 3042, 26, 3139, 1083, 219, 4257}

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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int -\frac {1}{\tan (i x)^2 (a-i b \sin (i x))^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \frac {1}{(a-i b \sin (i x))^2 \tan (i x)^2}dx\)

\(\Big \downarrow \) 3202

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

\(\Big \downarrow \) 25

\(\displaystyle \int \frac {\left (\sinh ^2(x)+1\right ) \text {csch}^2(x)}{(a+b \sinh (x))^2}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -\frac {1-\sin (i x)^2}{\sin (i x)^2 (a-i b \sin (i x))^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \frac {1-\sin (i x)^2}{\sin (i x)^2 (a-i b \sin (i x))^2}dx\)

\(\Big \downarrow \) 3535

\(\displaystyle \frac {\coth (x)}{a (a+b \sinh (x))}-\frac {\int -\frac {\text {csch}^2(x) \left (\left (a^2+b^2\right ) \sinh ^2(x)+2 \left (a^2+b^2\right )\right )}{a+b \sinh (x)}dx}{a \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\text {csch}^2(x) \left (\left (a^2+b^2\right ) \sinh ^2(x)+2 \left (a^2+b^2\right )\right )}{a+b \sinh (x)}dx}{a \left (a^2+b^2\right )}+\frac {\coth (x)}{a (a+b \sinh (x))}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3535

\(\displaystyle \frac {\coth (x)}{a (a+b \sinh (x))}-\frac {\frac {\int \frac {\text {csch}(x) \left (2 b \left (a^2+b^2\right )-a \left (a^2+b^2\right ) \sinh (x)\right )}{a+b \sinh (x)}dx}{a}+\frac {2 \left (a^2+b^2\right ) \coth (x)}{a}}{a \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3480

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3139

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

\(\Big \downarrow \) 1083

\(\displaystyle \frac {\coth (x)}{a (a+b \sinh (x))}-\frac {\frac {2 \left (a^2+b^2\right ) \coth (x)}{a}+\frac {i \left (\frac {2 b \left (a^2+b^2\right ) \int \csc (i x)dx}{a}-\frac {4 i \left (a^2+b^2\right ) \left (a^2+2 b^2\right ) \int \frac {1}{4 \left (a^2+b^2\right )-\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )^2}d\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{a}\right )}{a}}{a \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 4257

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

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

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 219

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])
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 3042

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

rule 3139

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre 
eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d)   Subst[Int[1/(a + 2*b*e*x + a 
*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 3202

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)/tan[(e_.) + (f_.)*(x_)]^2, 
x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*((1 - Sin[e + f*x]^2)/Sin[e + f*x]^ 
2), x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0]
 

rule 3480

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ 
.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(A*b 
- a*B)/(b*c - a*d)   Int[1/(a + b*Sin[e + f*x]), x], x] + Simp[(B*c - A*d)/ 
(b*c - a*d)   Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f 
, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
 

rule 3535

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> 
Simp[(-(A*b^2 + a^2*C))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*S 
in[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + Simp[1/((m 
+ 1)*(b*c - a*d)*(a^2 - b^2))   Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin 
[e + f*x])^n*Simp[a*(m + 1)*(b*c - a*d)*(A + C) + d*(A*b^2 + a^2*C)*(m + n 
+ 2) - (c*(A*b^2 + a^2*C) + b*(m + 1)*(b*c - a*d)*(A + C))*Sin[e + f*x] - d 
*(A*b^2 + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, 
d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 
- d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) || 
 !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 
 0])))
 

rule 4257

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] 
 /; FreeQ[{c, d}, x]
 

Maple [A] (verified)

Time = 3.51 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.48

method	result	size
default	\(-\frac {\tanh \left (\frac {x}{2}\right )}{2 a^{2}}-\frac {1}{2 a^{2} \tanh \left (\frac {x}{2}\right )}-\frac {2 b \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a^{3}}-\frac {2 \left (\frac {-b^{2} \tanh \left (\frac {x}{2}\right )-a b}{\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a}-\frac {\left (a^{2}+2 b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}\right )}{a^{3}}\)	\(118\)
risch	\(\frac {2 a \,{\mathrm e}^{3 x}-4 b \,{\mathrm e}^{2 x}-6 \,{\mathrm e}^{x} a +4 b}{\left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right ) a^{2} \left ({\mathrm e}^{2 x}-1\right )}+\frac {\ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}+b^{2}}-a^{2}-b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right )}{\sqrt {a^{2}+b^{2}}\, a}+\frac {2 \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}+b^{2}}-a^{2}-b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right ) b^{2}}{\sqrt {a^{2}+b^{2}}\, a^{3}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}+b^{2}}+a^{2}+b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right )}{\sqrt {a^{2}+b^{2}}\, a}-\frac {2 \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}+b^{2}}+a^{2}+b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right ) b^{2}}{\sqrt {a^{2}+b^{2}}\, a^{3}}+\frac {2 b \ln \left ({\mathrm e}^{x}+1\right )}{a^{3}}-\frac {2 b \ln \left ({\mathrm e}^{x}-1\right )}{a^{3}}\)	\(285\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1257 vs. \(2 (76) = 152\).

Time = 0.13 (sec) , antiderivative size = 1257, normalized size of antiderivative = 15.71 \[ \int \frac {\coth ^2(x)}{(a+b \sinh (x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(4*a^3*b + 4*a*b^3 + 2*(a^4 + a^2*b^2)*cosh(x)^3 + 2*(a^4 + a^2*b^2)*sinh( 
x)^3 - 4*(a^3*b + a*b^3)*cosh(x)^2 - 2*(2*a^3*b + 2*a*b^3 - 3*(a^4 + a^2*b 
^2)*cosh(x))*sinh(x)^2 + ((a^2*b + 2*b^3)*cosh(x)^4 + (a^2*b + 2*b^3)*sinh 
(x)^4 + 2*(a^3 + 2*a*b^2)*cosh(x)^3 + 2*(a^3 + 2*a*b^2 + 2*(a^2*b + 2*b^3) 
*cosh(x))*sinh(x)^3 + a^2*b + 2*b^3 - 2*(a^2*b + 2*b^3)*cosh(x)^2 - 2*(a^2 
*b + 2*b^3 - 3*(a^2*b + 2*b^3)*cosh(x)^2 - 3*(a^3 + 2*a*b^2)*cosh(x))*sinh 
(x)^2 - 2*(a^3 + 2*a*b^2)*cosh(x) + 2*(2*(a^2*b + 2*b^3)*cosh(x)^3 - a^3 - 
 2*a*b^2 + 3*(a^3 + 2*a*b^2)*cosh(x)^2 - 2*(a^2*b + 2*b^3)*cosh(x))*sinh(x 
))*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)) - 6*(a^4 + a^2*b^2)*cosh(x) + 2*((a^2*b^2 + b^4)*cosh(x) 
^4 + (a^2*b^2 + b^4)*sinh(x)^4 + a^2*b^2 + b^4 + 2*(a^3*b + a*b^3)*cosh(x) 
^3 + 2*(a^3*b + a*b^3 + 2*(a^2*b^2 + b^4)*cosh(x))*sinh(x)^3 - 2*(a^2*b^2 
+ b^4)*cosh(x)^2 - 2*(a^2*b^2 + b^4 - 3*(a^2*b^2 + b^4)*cosh(x)^2 - 3*(a^3 
*b + a*b^3)*cosh(x))*sinh(x)^2 - 2*(a^3*b + a*b^3)*cosh(x) - 2*(a^3*b + a* 
b^3 - 2*(a^2*b^2 + b^4)*cosh(x)^3 - 3*(a^3*b + a*b^3)*cosh(x)^2 + 2*(a^2*b 
^2 + b^4)*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) + 1) - 2*((a^2*b^2 + b^4 
)*cosh(x)^4 + (a^2*b^2 + b^4)*sinh(x)^4 + a^2*b^2 + b^4 + 2*(a^3*b + a*b^3 
)*cosh(x)^3 + 2*(a^3*b + a*b^3 + 2*(a^2*b^2 + b^4)*cosh(x))*sinh(x)^3 -...
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (76) = 152\).

Time = 0.12 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.06 \[ \int \frac {\coth ^2(x)}{(a+b \sinh (x))^2} \, dx=-\frac {2 \, {\left (3 \, a e^{\left (-x\right )} - 2 \, b e^{\left (-2 \, x\right )} - a e^{\left (-3 \, x\right )} + 2 \, b\right )}}{2 \, a^{3} e^{\left (-x\right )} - 2 \, a^{2} b e^{\left (-2 \, x\right )} - 2 \, a^{3} e^{\left (-3 \, x\right )} + a^{2} b e^{\left (-4 \, x\right )} + a^{2} b} + \frac {2 \, b \log \left (e^{\left (-x\right )} + 1\right )}{a^{3}} - \frac {2 \, b \log \left (e^{\left (-x\right )} - 1\right )}{a^{3}} + \frac {{\left (a^{2} + 2 \, b^{2}\right )} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a^{3}} \] Input:

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

Output:

-2*(3*a*e^(-x) - 2*b*e^(-2*x) - a*e^(-3*x) + 2*b)/(2*a^3*e^(-x) - 2*a^2*b* 
e^(-2*x) - 2*a^3*e^(-3*x) + a^2*b*e^(-4*x) + a^2*b) + 2*b*log(e^(-x) + 1)/ 
a^3 - 2*b*log(e^(-x) - 1)/a^3 + (a^2 + 2*b^2)*log((b*e^(-x) - a - sqrt(a^2 
 + b^2))/(b*e^(-x) - a + sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a^3)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.85 \[ \int \frac {\coth ^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {2 \, b \log \left (e^{x} + 1\right )}{a^{3}} - \frac {2 \, b \log \left ({\left | e^{x} - 1 \right |}\right )}{a^{3}} + \frac {{\left (a^{2} + 2 \, b^{2}\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a^{3}} + \frac {2 \, {\left (a e^{\left (3 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} - 3 \, a e^{x} + 2 \, b\right )}}{{\left (b e^{\left (4 \, x\right )} + 2 \, a e^{\left (3 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} - 2 \, a e^{x} + b\right )} a^{2}} \] Input:

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

Output:

2*b*log(e^x + 1)/a^3 - 2*b*log(abs(e^x - 1))/a^3 + (a^2 + 2*b^2)*log(abs(2 
*b*e^x + 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*e^x + 2*a + 2*sqrt(a^2 + b^2)))/ 
(sqrt(a^2 + b^2)*a^3) + 2*(a*e^(3*x) - 2*b*e^(2*x) - 3*a*e^x + 2*b)/((b*e^ 
(4*x) + 2*a*e^(3*x) - 2*b*e^(2*x) - 2*a*e^x + b)*a^2)
 

Mupad [B] (verification not implemented)

Time = 2.61 (sec) , antiderivative size = 897, normalized size of antiderivative = 11.21 \[ \int \frac {\coth ^2(x)}{(a+b \sinh (x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

((4*(16*a^2*b^14 + 56*a^4*b^12 + 65*a^6*b^10 + 25*a^8*b^8))/(a^4*b^4*(16*b 
^9 + 56*a^2*b^7 + 65*a^4*b^5 + 25*a^6*b^3)) - (6*exp(x)*(16*a^3*b^14 + 56* 
a^5*b^12 + 65*a^7*b^10 + 25*a^9*b^8))/(a^4*b^5*(16*b^9 + 56*a^2*b^7 + 65*a 
^4*b^5 + 25*a^6*b^3)) - (4*exp(2*x)*(16*a^2*b^14 + 56*a^4*b^12 + 65*a^6*b^ 
10 + 25*a^8*b^8))/(a^4*b^4*(16*b^9 + 56*a^2*b^7 + 65*a^4*b^5 + 25*a^6*b^3) 
) + (2*exp(3*x)*(16*a^3*b^14 + 56*a^5*b^12 + 65*a^7*b^10 + 25*a^9*b^8))/(a 
^4*b^5*(16*b^9 + 56*a^2*b^7 + 65*a^4*b^5 + 25*a^6*b^3)))/(b - 2*a*exp(x) + 
 2*a*exp(3*x) - 2*b*exp(2*x) + b*exp(4*x)) - (2*b*log(64*exp(x) - 64))/a^3 
 + (2*b*log(64*exp(x) + 64))/a^3 - (log(((a^2 + 2*b^2)*((32*(a^4 + 8*b^4 + 
 12*a^2*b^2 - 12*a*b^3*exp(x) - 16*a^3*b*exp(x)))/(a^4*b^4) + ((a^2 + 2*b^ 
2)*((32*(2*a^2*b + 4*b^3 - 4*a^3*exp(x) - 7*a*b^2*exp(x)))/b^5 - (32*(a^2 
+ 2*b^2)*(a^2 + b^2)^(1/2)*(3*a^4*b + 2*a^2*b^3 - 4*a^5*exp(x) - 3*a^3*b^2 
*exp(x)))/(b^5*(a^5 + a^3*b^2)))*(a^2 + b^2)^(1/2))/(a^5 + a^3*b^2))*(a^2 
+ b^2)^(1/2))/(a^5 + a^3*b^2) - (64*(a^2 + 2*b^2)*(4*b - 7*a*exp(x)))/(a^6 
*b^3))*(a^2 + 2*b^2)*(a^2 + b^2)^(1/2))/(a^5 + a^3*b^2) + (log(- ((a^2 + 2 
*b^2)*((32*(a^4 + 8*b^4 + 12*a^2*b^2 - 12*a*b^3*exp(x) - 16*a^3*b*exp(x))) 
/(a^4*b^4) - ((a^2 + 2*b^2)*((32*(2*a^2*b + 4*b^3 - 4*a^3*exp(x) - 7*a*b^2 
*exp(x)))/b^5 + (32*(a^2 + 2*b^2)*(a^2 + b^2)^(1/2)*(3*a^4*b + 2*a^2*b^3 - 
 4*a^5*exp(x) - 3*a^3*b^2*exp(x)))/(b^5*(a^5 + a^3*b^2)))*(a^2 + b^2)^(1/2 
))/(a^5 + a^3*b^2))*(a^2 + b^2)^(1/2))/(a^5 + a^3*b^2) - (64*(a^2 + 2*b...
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 888, normalized size of antiderivative = 11.10 \[ \int \frac {\coth ^2(x)}{(a+b \sinh (x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*e**(4*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a** 
2*b*i + 4*e**(4*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b** 
2))*b**3*i + 4*e**(3*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 
+ b**2))*a**3*i + 8*e**(3*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt( 
a**2 + b**2))*a*b**2*i - 4*e**(2*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i 
)/sqrt(a**2 + b**2))*a**2*b*i - 8*e**(2*x)*sqrt(a**2 + b**2)*atan((e**x*b* 
i + a*i)/sqrt(a**2 + b**2))*b**3*i - 4*e**x*sqrt(a**2 + b**2)*atan((e**x*b 
*i + a*i)/sqrt(a**2 + b**2))*a**3*i - 8*e**x*sqrt(a**2 + b**2)*atan((e**x* 
b*i + a*i)/sqrt(a**2 + b**2))*a*b**2*i + 2*sqrt(a**2 + b**2)*atan((e**x*b* 
i + a*i)/sqrt(a**2 + b**2))*a**2*b*i + 4*sqrt(a**2 + b**2)*atan((e**x*b*i 
+ a*i)/sqrt(a**2 + b**2))*b**3*i - 2*e**(4*x)*log(e**x - 1)*a**2*b**2 - 2* 
e**(4*x)*log(e**x - 1)*b**4 + 2*e**(4*x)*log(e**x + 1)*a**2*b**2 + 2*e**(4 
*x)*log(e**x + 1)*b**4 - e**(4*x)*a**3*b - e**(4*x)*a*b**3 - 4*e**(3*x)*lo 
g(e**x - 1)*a**3*b - 4*e**(3*x)*log(e**x - 1)*a*b**3 + 4*e**(3*x)*log(e**x 
 + 1)*a**3*b + 4*e**(3*x)*log(e**x + 1)*a*b**3 + 4*e**(2*x)*log(e**x - 1)* 
a**2*b**2 + 4*e**(2*x)*log(e**x - 1)*b**4 - 4*e**(2*x)*log(e**x + 1)*a**2* 
b**2 - 4*e**(2*x)*log(e**x + 1)*b**4 - 2*e**(2*x)*a**3*b - 2*e**(2*x)*a*b* 
*3 + 4*e**x*log(e**x - 1)*a**3*b + 4*e**x*log(e**x - 1)*a*b**3 - 4*e**x*lo 
g(e**x + 1)*a**3*b - 4*e**x*log(e**x + 1)*a*b**3 - 4*e**x*a**4 - 4*e**x*a* 
*2*b**2 - 2*log(e**x - 1)*a**2*b**2 - 2*log(e**x - 1)*b**4 + 2*log(e**x...