Integrand size = 12, antiderivative size = 101 \[ \int \frac {1}{(a+b \text {csch}(c+d x))^2} \, dx=\frac {x}{a^2}+\frac {2 b \left (2 a^2+b^2\right ) \text {arctanh}\left (\frac {a-b \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {b^2 \coth (c+d x)}{a \left (a^2+b^2\right ) d (a+b \text {csch}(c+d x))} \]
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))
Time = 0.40 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.41 \[ \int \frac {1}{(a+b \text {csch}(c+d x))^2} \, dx=\frac {\text {csch}(c+d x) \left (-\frac {a b^2 \coth (c+d x)}{a^2+b^2}+(c+d x) (a+b \text {csch}(c+d x))+\frac {2 b \left (2 a^2+b^2\right ) \arctan \left (\frac {a-b \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right ) (a+b \text {csch}(c+d x))}{\left (-a^2-b^2\right )^{3/2}}\right ) (b+a \sinh (c+d x))}{a^2 d (a+b \text {csch}(c+d x))^2} \]
(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)
Time = 0.66 (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 |
| \(\displaystyle \frac {\frac {x \left (a^2+b^2\right )}{a}-\frac {2 b \left (2 a^2+b^2\right ) \text {arctanh}\left (\frac {b \tanh \left (\frac {1}{2} (c+d x)\right )}{2 \sqrt {a^2+b^2}}\right )}{a d \sqrt {a^2+b^2}}}{a \left (a^2+b^2\right )}-\frac {b^2 \coth (c+d x)}{a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))}\) |
(((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]))
3.1.75.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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])
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]
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]
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]
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]
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]
Time = 0.34 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.74
| method | result | size |
| derivativedivides | \(\frac {\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}-\frac {2 b \left (\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2}}{2 a^{2}+2 b^{2}}+\frac {a b}{2 a^{2}+2 b^{2}}}{-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b}{2}+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a +\frac {b}{2}}-\frac {2 \left (2 a^{2}+b^{2}\right ) \operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (2 a^{2}+2 b^{2}\right ) \sqrt {a^{2}+b^{2}}}\right )}{a^{2}}}{d}\) | \(176\) |
| default | \(\frac {\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}-\frac {2 b \left (\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2}}{2 a^{2}+2 b^{2}}+\frac {a b}{2 a^{2}+2 b^{2}}}{-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b}{2}+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a +\frac {b}{2}}-\frac {2 \left (2 a^{2}+b^{2}\right ) \operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (2 a^{2}+2 b^{2}\right ) \sqrt {a^{2}+b^{2}}}\right )}{a^{2}}}{d}\) | \(176\) |
| risch | \(\frac {x}{a^{2}}-\frac {2 b^{2} \left (-{\mathrm e}^{d x +c} b +a \right )}{d \,a^{2} \left (a^{2}+b^{2}\right ) \left ({\mathrm e}^{2 d x +2 c} a +2 \,{\mathrm e}^{d x +c} b -a \right )}+\frac {2 b \ln \left ({\mathrm e}^{d x +c}+\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b +a^{4}+2 a^{2} b^{2}+b^{4}}{a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d}+\frac {b^{3} \ln \left ({\mathrm e}^{d x +c}+\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b +a^{4}+2 a^{2} b^{2}+b^{4}}{a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d \,a^{2}}-\frac {2 b \ln \left ({\mathrm e}^{d x +c}+\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b -a^{4}-2 a^{2} b^{2}-b^{4}}{a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d}-\frac {b^{3} \ln \left ({\mathrm e}^{d x +c}+\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b -a^{4}-2 a^{2} b^{2}-b^{4}}{a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d \,a^{2}}\) | \(329\) |
1/d*(1/a^2*ln(1+tanh(1/2*d*x+1/2*c))-1/a^2*ln(tanh(1/2*d*x+1/2*c)-1)-2/a^2 *b*((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+tanh(1/2*d*x+1/2*c)*a+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 ))))
Leaf count of result is larger than twice the leaf count of optimal. 645 vs. \(2 (98) = 196\).
Time = 0.28 (sec) , antiderivative size = 645, normalized size of antiderivative = 6.39 \[ \int \frac {1}{(a+b \text {csch}(c+d x))^2} \, dx=-\frac {2 \, a^{3} b^{2} + 2 \, a b^{4} - {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d x \cosh \left (d x + c\right )^{2} - {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d x \sinh \left (d x + c\right )^{2} + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d x + {\left (2 \, a^{3} b + a b^{3} - {\left (2 \, a^{3} b + a b^{3}\right )} \cosh \left (d x + c\right )^{2} - {\left (2 \, a^{3} b + a b^{3}\right )} \sinh \left (d x + c\right )^{2} - 2 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (d x + c\right ) - 2 \, {\left (2 \, a^{2} b^{2} + b^{4} + {\left (2 \, a^{3} b + a b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {a^{2} \cosh \left (d x + c\right )^{2} + a^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) + a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right ) + 2 \, \sqrt {a^{2} + b^{2}} {\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + b\right )}}{a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) + 2 \, {\left (a \cosh \left (d x + c\right ) + b\right )} \sinh \left (d x + c\right ) - a}\right ) - 2 \, {\left (a^{2} b^{3} + b^{5} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d x\right )} \cosh \left (d x + c\right ) - 2 \, {\left (a^{2} b^{3} + b^{5} + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d x \cosh \left (d x + c\right ) + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d x\right )} \sinh \left (d x + c\right )}{{\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \cosh \left (d x + c\right )^{2} + {\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \sinh \left (d x + c\right )^{2} + 2 \, {\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d \cosh \left (d x + c\right ) - {\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d + 2 \, {\left ({\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \cosh \left (d x + c\right ) + {\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d\right )} \sinh \left (d x + c\right )} \]
-(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) )
\[ \int \frac {1}{(a+b \text {csch}(c+d x))^2} \, dx=\int \frac {1}{\left (a + b \operatorname {csch}{\left (c + d x \right )}\right )^{2}}\, dx \]
Time = 0.27 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.85 \[ \int \frac {1}{(a+b \text {csch}(c+d x))^2} \, dx=-\frac {{\left (2 \, a^{2} b + b^{3}\right )} \log \left (\frac {a e^{\left (-d x - c\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-d x - c\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{4} + a^{2} b^{2}\right )} \sqrt {a^{2} + b^{2}} d} - \frac {2 \, {\left (b^{3} e^{\left (-d x - c\right )} + a b^{2}\right )}}{{\left (a^{5} + a^{3} b^{2} + 2 \, {\left (a^{4} b + a^{2} b^{3}\right )} e^{\left (-d x - c\right )} - {\left (a^{5} + a^{3} b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} d} + \frac {d x + c}{a^{2} d} \]
-(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)
Time = 0.33 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.59 \[ \int \frac {1}{(a+b \text {csch}(c+d x))^2} \, dx=-\frac {\frac {{\left (2 \, a^{2} b + b^{3}\right )} \log \left (\frac {{\left | 2 \, a e^{\left (d x + c\right )} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{\left (d x + c\right )} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{4} + a^{2} b^{2}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (b^{3} e^{\left (d x + c\right )} - a b^{2}\right )}}{{\left (a^{4} + a^{2} b^{2}\right )} {\left (a e^{\left (2 \, d x + 2 \, c\right )} + 2 \, b e^{\left (d x + c\right )} - a\right )}} - \frac {d x + c}{a^{2}}}{d} \]
-((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
Time = 2.72 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.66 \[ \int \frac {1}{(a+b \text {csch}(c+d x))^2} \, dx=\frac {x}{a^2}-\frac {\frac {2\,b^2}{d\,\left (a^3+a\,b^2\right )}-\frac {2\,b^3\,{\mathrm {e}}^{c+d\,x}}{a\,d\,\left (a^3+a\,b^2\right )}}{2\,b\,{\mathrm {e}}^{c+d\,x}-a+a\,{\mathrm {e}}^{2\,c+2\,d\,x}}-\frac {b\,\ln \left (\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (2\,a^2\,b+b^3\right )}{a^3\,\left (a^2+b^2\right )}-\frac {2\,b\,\left (2\,a^2+b^2\right )\,\left (a-b\,{\mathrm {e}}^{c+d\,x}\right )}{a^3\,{\left (a^2+b^2\right )}^{3/2}}\right )\,\left (2\,a^2+b^2\right )}{a^2\,d\,{\left (a^2+b^2\right )}^{3/2}}+\frac {b\,\ln \left (\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (2\,a^2\,b+b^3\right )}{a^3\,\left (a^2+b^2\right )}+\frac {2\,b\,\left (2\,a^2+b^2\right )\,\left (a-b\,{\mathrm {e}}^{c+d\,x}\right )}{a^3\,{\left (a^2+b^2\right )}^{3/2}}\right )\,\left (2\,a^2+b^2\right )}{a^2\,d\,{\left (a^2+b^2\right )}^{3/2}} \]
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))