\(\int \frac {1}{(a+b \text {csch}(c+d x))^2} \, dx\) [75]

Optimal result

Integrand size = 12, antiderivative size = 101 \[ \int \frac {1}{(a+b \text {csch}(c+d x))^2} \, dx=\frac {x}{a^2}+\frac {2 b \left (2 a^2+b^2\right ) \text {arctanh}\left (\frac {a-b \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {b^2 \coth (c+d x)}{a \left (a^2+b^2\right ) d (a+b \text {csch}(c+d x))} \] Output:

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

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.41 \[ \int \frac {1}{(a+b \text {csch}(c+d x))^2} \, dx=\frac {\text {csch}(c+d x) \left (-\frac {a b^2 \coth (c+d x)}{a^2+b^2}+(c+d x) (a+b \text {csch}(c+d x))+\frac {2 b \left (2 a^2+b^2\right ) \arctan \left (\frac {a-b \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right ) (a+b \text {csch}(c+d x))}{\left (-a^2-b^2\right )^{3/2}}\right ) (b+a \sinh (c+d x))}{a^2 d (a+b \text {csch}(c+d x))^2} \] Input:

Integrate[(a + b*Csch[c + d*x])^(-2),x]
 

Output:

(Csch[c + d*x]*(-((a*b^2*Coth[c + d*x])/(a^2 + b^2)) + (c + d*x)*(a + b*Cs 
ch[c + d*x]) + (2*b*(2*a^2 + b^2)*ArcTan[(a - b*Tanh[(c + d*x)/2])/Sqrt[-a 
^2 - b^2]]*(a + b*Csch[c + d*x]))/(-a^2 - b^2)^(3/2))*(b + a*Sinh[c + d*x] 
))/(a^2*d*(a + b*Csch[c + d*x])^2)
 

Rubi [A] (warning: unable to verify)

Time = 0.68 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.20, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {3042, 4272, 25, 3042, 4407, 26, 3042, 26, 4318, 3042, 3139, 1083, 217}

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.

\(\displaystyle \int \frac {1}{(a+b \text {csch}(c+d x))^2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(a+i b \csc (i c+i d x))^2}dx\)

\(\Big \downarrow \) 4272

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4407

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4318

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3139

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

\(\Big \downarrow \) 1083

\(\displaystyle -\frac {b^2 \coth (c+d x)}{a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))}+\frac {\frac {x \left (a^2+b^2\right )}{a}-\frac {4 i \left (2 a^2+b^2\right ) \int \frac {1}{\tanh ^2\left (\frac {1}{2} (c+d x)\right )-4 \left (\frac {a^2}{b^2}+1\right )}d\left (2 i \tanh \left (\frac {1}{2} (c+d x)\right )-\frac {2 i a}{b}\right )}{a d}}{a \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 217

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

Input:

Int[(a + b*Csch[c + d*x])^(-2),x]
 

Output:

(((a^2 + b^2)*x)/a - (2*b*(2*a^2 + b^2)*ArcTanh[(b*Tanh[(c + d*x)/2])/(2*S 
qrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d))/(a*(a^2 + b^2)) - (b^2*Coth[c + d 
*x])/(a*(a^2 + b^2)*d*(a + b*Csch[c + d*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 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[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 4272

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[b^2*Cot[ 
c + d*x]*((a + b*Csc[c + d*x])^(n + 1)/(a*d*(n + 1)*(a^2 - b^2))), x] + Sim 
p[1/(a*(n + 1)*(a^2 - b^2))   Int[(a + b*Csc[c + d*x])^(n + 1)*Simp[(a^2 - 
b^2)*(n + 1) - a*b*(n + 1)*Csc[c + d*x] + b^2*(n + 2)*Csc[c + d*x]^2, x], x 
], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integ 
erQ[2*n]
 

rule 4318

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbo 
l] :> Simp[1/b   Int[1/(1 + (a/b)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, 
f}, x] && NeQ[a^2 - b^2, 0]
 

rule 4407

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + 
 (a_)), x_Symbol] :> Simp[c*(x/a), x] - Simp[(b*c - a*d)/a   Int[Csc[e + f* 
x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c 
- a*d, 0]
 

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.74

method	result	size
derivativedivides	\(\frac {-\frac {2 b \left (\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2}}{2 a^{2}+2 b^{2}}+\frac {a b}{2 a^{2}+2 b^{2}}}{-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b}{2}+a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {b}{2}}-\frac {2 \left (2 a^{2}+b^{2}\right ) \operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (2 a^{2}+2 b^{2}\right ) \sqrt {a^{2}+b^{2}}}\right )}{a^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2}}}{d}\)	\(176\)
default	\(\frac {-\frac {2 b \left (\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2}}{2 a^{2}+2 b^{2}}+\frac {a b}{2 a^{2}+2 b^{2}}}{-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b}{2}+a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {b}{2}}-\frac {2 \left (2 a^{2}+b^{2}\right ) \operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (2 a^{2}+2 b^{2}\right ) \sqrt {a^{2}+b^{2}}}\right )}{a^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2}}}{d}\)	\(176\)
risch	\(\frac {x}{a^{2}}-\frac {2 b^{2} \left (-b \,{\mathrm e}^{d x +c}+a \right )}{d \,a^{2} \left (a^{2}+b^{2}\right ) \left (a \,{\mathrm e}^{2 d x +2 c}+2 b \,{\mathrm e}^{d x +c}-a \right )}+\frac {2 b \ln \left ({\mathrm e}^{d x +c}+\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b +a^{4}+2 a^{2} b^{2}+b^{4}}{a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d}+\frac {b^{3} \ln \left ({\mathrm e}^{d x +c}+\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b +a^{4}+2 a^{2} b^{2}+b^{4}}{a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d \,a^{2}}-\frac {2 b \ln \left ({\mathrm e}^{d x +c}+\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b -a^{4}-2 a^{2} b^{2}-b^{4}}{a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d}-\frac {b^{3} \ln \left ({\mathrm e}^{d x +c}+\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b -a^{4}-2 a^{2} b^{2}-b^{4}}{a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d \,a^{2}}\)	\(329\)

Input:

int(1/(a+csch(d*x+c)*b)^2,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 645 vs. \(2 (98) = 196\).

Time = 0.11 (sec) , antiderivative size = 645, normalized size of antiderivative = 6.39 \[ \int \frac {1}{(a+b \text {csch}(c+d x))^2} \, dx =\text {Too large to display} \] Input:

integrate(1/(a+b*csch(d*x+c))^2,x, algorithm="fricas")
 

Output:

-(2*a^3*b^2 + 2*a*b^4 - (a^5 + 2*a^3*b^2 + a*b^4)*d*x*cosh(d*x + c)^2 - (a 
^5 + 2*a^3*b^2 + a*b^4)*d*x*sinh(d*x + c)^2 + (a^5 + 2*a^3*b^2 + a*b^4)*d* 
x + (2*a^3*b + a*b^3 - (2*a^3*b + a*b^3)*cosh(d*x + c)^2 - (2*a^3*b + a*b^ 
3)*sinh(d*x + c)^2 - 2*(2*a^2*b^2 + b^4)*cosh(d*x + c) - 2*(2*a^2*b^2 + b^ 
4 + (2*a^3*b + a*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a^2 + b^2)*log((a 
^2*cosh(d*x + c)^2 + a^2*sinh(d*x + c)^2 + 2*a*b*cosh(d*x + c) + a^2 + 2*b 
^2 + 2*(a^2*cosh(d*x + c) + a*b)*sinh(d*x + c) + 2*sqrt(a^2 + b^2)*(a*cosh 
(d*x + c) + a*sinh(d*x + c) + b))/(a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 + 
 2*b*cosh(d*x + c) + 2*(a*cosh(d*x + c) + b)*sinh(d*x + c) - a)) - 2*(a^2* 
b^3 + b^5 + (a^4*b + 2*a^2*b^3 + b^5)*d*x)*cosh(d*x + c) - 2*(a^2*b^3 + b^ 
5 + (a^5 + 2*a^3*b^2 + a*b^4)*d*x*cosh(d*x + c) + (a^4*b + 2*a^2*b^3 + b^5 
)*d*x)*sinh(d*x + c))/((a^7 + 2*a^5*b^2 + a^3*b^4)*d*cosh(d*x + c)^2 + (a^ 
7 + 2*a^5*b^2 + a^3*b^4)*d*sinh(d*x + c)^2 + 2*(a^6*b + 2*a^4*b^3 + a^2*b^ 
5)*d*cosh(d*x + c) - (a^7 + 2*a^5*b^2 + a^3*b^4)*d + 2*((a^7 + 2*a^5*b^2 + 
 a^3*b^4)*d*cosh(d*x + c) + (a^6*b + 2*a^4*b^3 + a^2*b^5)*d)*sinh(d*x + c) 
)
 

Sympy [F]

\[ \int \frac {1}{(a+b \text {csch}(c+d x))^2} \, dx=\int \frac {1}{\left (a + b \operatorname {csch}{\left (c + d x \right )}\right )^{2}}\, dx \] Input:

integrate(1/(a+b*csch(d*x+c))**2,x)
 

Output:

Integral((a + b*csch(c + d*x))**(-2), x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.85 \[ \int \frac {1}{(a+b \text {csch}(c+d x))^2} \, dx=-\frac {{\left (2 \, a^{2} b + b^{3}\right )} \log \left (\frac {a e^{\left (-d x - c\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-d x - c\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{4} + a^{2} b^{2}\right )} \sqrt {a^{2} + b^{2}} d} - \frac {2 \, {\left (b^{3} e^{\left (-d x - c\right )} + a b^{2}\right )}}{{\left (a^{5} + a^{3} b^{2} + 2 \, {\left (a^{4} b + a^{2} b^{3}\right )} e^{\left (-d x - c\right )} - {\left (a^{5} + a^{3} b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} d} + \frac {d x + c}{a^{2} d} \] Input:

integrate(1/(a+b*csch(d*x+c))^2,x, algorithm="maxima")
 

Output:

-(2*a^2*b + b^3)*log((a*e^(-d*x - c) - b - sqrt(a^2 + b^2))/(a*e^(-d*x - c 
) - b + sqrt(a^2 + b^2)))/((a^4 + a^2*b^2)*sqrt(a^2 + b^2)*d) - 2*(b^3*e^( 
-d*x - c) + a*b^2)/((a^5 + a^3*b^2 + 2*(a^4*b + a^2*b^3)*e^(-d*x - c) - (a 
^5 + a^3*b^2)*e^(-2*d*x - 2*c))*d) + (d*x + c)/(a^2*d)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.59 \[ \int \frac {1}{(a+b \text {csch}(c+d x))^2} \, dx=-\frac {\frac {{\left (2 \, a^{2} b + b^{3}\right )} \log \left (\frac {{\left | 2 \, a e^{\left (d x + c\right )} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{\left (d x + c\right )} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{4} + a^{2} b^{2}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (b^{3} e^{\left (d x + c\right )} - a b^{2}\right )}}{{\left (a^{4} + a^{2} b^{2}\right )} {\left (a e^{\left (2 \, d x + 2 \, c\right )} + 2 \, b e^{\left (d x + c\right )} - a\right )}} - \frac {d x + c}{a^{2}}}{d} \] Input:

integrate(1/(a+b*csch(d*x+c))^2,x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 3.04 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.66 \[ \int \frac {1}{(a+b \text {csch}(c+d x))^2} \, dx=\frac {x}{a^2}-\frac {\frac {2\,b^2}{d\,\left (a^3+a\,b^2\right )}-\frac {2\,b^3\,{\mathrm {e}}^{c+d\,x}}{a\,d\,\left (a^3+a\,b^2\right )}}{2\,b\,{\mathrm {e}}^{c+d\,x}-a+a\,{\mathrm {e}}^{2\,c+2\,d\,x}}-\frac {b\,\ln \left (\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (2\,a^2\,b+b^3\right )}{a^3\,\left (a^2+b^2\right )}-\frac {2\,b\,\left (2\,a^2+b^2\right )\,\left (a-b\,{\mathrm {e}}^{c+d\,x}\right )}{a^3\,{\left (a^2+b^2\right )}^{3/2}}\right )\,\left (2\,a^2+b^2\right )}{a^2\,d\,{\left (a^2+b^2\right )}^{3/2}}+\frac {b\,\ln \left (\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (2\,a^2\,b+b^3\right )}{a^3\,\left (a^2+b^2\right )}+\frac {2\,b\,\left (2\,a^2+b^2\right )\,\left (a-b\,{\mathrm {e}}^{c+d\,x}\right )}{a^3\,{\left (a^2+b^2\right )}^{3/2}}\right )\,\left (2\,a^2+b^2\right )}{a^2\,d\,{\left (a^2+b^2\right )}^{3/2}} \] Input:

int(1/(a + b/sinh(c + d*x))^2,x)
 

Output:

x/a^2 - ((2*b^2)/(d*(a*b^2 + a^3)) - (2*b^3*exp(c + d*x))/(a*d*(a*b^2 + a^ 
3)))/(2*b*exp(c + d*x) - a + a*exp(2*c + 2*d*x)) - (b*log((2*exp(c + d*x)* 
(2*a^2*b + b^3))/(a^3*(a^2 + b^2)) - (2*b*(2*a^2 + b^2)*(a - b*exp(c + d*x 
)))/(a^3*(a^2 + b^2)^(3/2)))*(2*a^2 + b^2))/(a^2*d*(a^2 + b^2)^(3/2)) + (b 
*log((2*exp(c + d*x)*(2*a^2*b + b^3))/(a^3*(a^2 + b^2)) + (2*b*(2*a^2 + b^ 
2)*(a - b*exp(c + d*x)))/(a^3*(a^2 + b^2)^(3/2)))*(2*a^2 + b^2))/(a^2*d*(a 
^2 + b^2)^(3/2))
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 570, normalized size of antiderivative = 5.64 \[ \int \frac {1}{(a+b \text {csch}(c+d x))^2} \, dx=\frac {-4 e^{2 d x +2 c} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{d x +c} a i +b i}{\sqrt {a^{2}+b^{2}}}\right ) a^{3} b i -2 e^{2 d x +2 c} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{d x +c} a i +b i}{\sqrt {a^{2}+b^{2}}}\right ) a \,b^{3} i -8 e^{d x +c} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{d x +c} a i +b i}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2} i -4 e^{d x +c} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{d x +c} a i +b i}{\sqrt {a^{2}+b^{2}}}\right ) b^{4} i +4 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{d x +c} a i +b i}{\sqrt {a^{2}+b^{2}}}\right ) a^{3} b i +2 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{d x +c} a i +b i}{\sqrt {a^{2}+b^{2}}}\right ) a \,b^{3} i +e^{2 d x +2 c} a^{5} d x +2 e^{2 d x +2 c} a^{3} b^{2} d x -e^{2 d x +2 c} a^{3} b^{2}+e^{2 d x +2 c} a \,b^{4} d x -e^{2 d x +2 c} a \,b^{4}+2 e^{d x +c} a^{4} b d x +4 e^{d x +c} a^{2} b^{3} d x +2 e^{d x +c} b^{5} d x -a^{5} d x -2 a^{3} b^{2} d x -a^{3} b^{2}-a \,b^{4} d x -a \,b^{4}}{a^{2} d \left (e^{2 d x +2 c} a^{5}+2 e^{2 d x +2 c} a^{3} b^{2}+e^{2 d x +2 c} a \,b^{4}+2 e^{d x +c} a^{4} b +4 e^{d x +c} a^{2} b^{3}+2 e^{d x +c} b^{5}-a^{5}-2 a^{3} b^{2}-a \,b^{4}\right )} \] Input:

int(1/(a+b*csch(d*x+c))^2,x)
 

Output:

( - 4*e**(2*c + 2*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*a*i + b*i)/sqr 
t(a**2 + b**2))*a**3*b*i - 2*e**(2*c + 2*d*x)*sqrt(a**2 + b**2)*atan((e**( 
c + d*x)*a*i + b*i)/sqrt(a**2 + b**2))*a*b**3*i - 8*e**(c + d*x)*sqrt(a**2 
 + b**2)*atan((e**(c + d*x)*a*i + b*i)/sqrt(a**2 + b**2))*a**2*b**2*i - 4* 
e**(c + d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*a*i + b*i)/sqrt(a**2 + b 
**2))*b**4*i + 4*sqrt(a**2 + b**2)*atan((e**(c + d*x)*a*i + b*i)/sqrt(a**2 
 + b**2))*a**3*b*i + 2*sqrt(a**2 + b**2)*atan((e**(c + d*x)*a*i + b*i)/sqr 
t(a**2 + b**2))*a*b**3*i + e**(2*c + 2*d*x)*a**5*d*x + 2*e**(2*c + 2*d*x)* 
a**3*b**2*d*x - e**(2*c + 2*d*x)*a**3*b**2 + e**(2*c + 2*d*x)*a*b**4*d*x - 
 e**(2*c + 2*d*x)*a*b**4 + 2*e**(c + d*x)*a**4*b*d*x + 4*e**(c + d*x)*a**2 
*b**3*d*x + 2*e**(c + d*x)*b**5*d*x - a**5*d*x - 2*a**3*b**2*d*x - a**3*b* 
*2 - a*b**4*d*x - a*b**4)/(a**2*d*(e**(2*c + 2*d*x)*a**5 + 2*e**(2*c + 2*d 
*x)*a**3*b**2 + e**(2*c + 2*d*x)*a*b**4 + 2*e**(c + d*x)*a**4*b + 4*e**(c 
+ d*x)*a**2*b**3 + 2*e**(c + d*x)*b**5 - a**5 - 2*a**3*b**2 - a*b**4))