\(\int \frac {\text {csch}^2(c+d x)}{(a+b \text {sech}^2(c+d x))^2} \, dx\) [38]

Optimal result

Integrand size = 23, antiderivative size = 91 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {3 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{2 (a+b)^{5/2} d}-\frac {\coth (c+d x)}{(a+b)^2 d}+\frac {b \tanh (c+d x)}{2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )} \] Output:

3/2*b^(1/2)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^(1/2))/(a+b)^(5/2)/d-coth(d* 
x+c)/(a+b)^2/d+1/2*b*tanh(d*x+c)/(a+b)^2/d/(a+b-b*tanh(d*x+c)^2)
                                                                                    
                                                                                    
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(220\) vs. \(2(91)=182\).

Time = 2.29 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.42 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^4(c+d x) \left (\frac {3 b \text {arctanh}\left (\frac {\text {sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right ) (a+2 b+a \cosh (2 (c+d x))) (\cosh (2 c)-\sinh (2 c))}{\sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}+2 (a+2 b+a \cosh (2 (c+d x))) \text {csch}(c) \text {csch}(c+d x) \sinh (d x)+b \text {sech}(2 c) \sinh (2 d x)-\frac {b (a+2 b) \tanh (2 c)}{a}\right )}{8 (a+b)^2 d \left (a+b \text {sech}^2(c+d x)\right )^2} \] Input:

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

Output:

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^4*((3*b*ArcTanh[(Sech[d*x]* 
(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt 
[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(a + 2*b + a*Cosh[2*(c + d*x)])*(C 
osh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + 2*(a 
+ 2*b + a*Cosh[2*(c + d*x)])*Csch[c]*Csch[c + d*x]*Sinh[d*x] + b*Sech[2*c] 
*Sinh[2*d*x] - (b*(a + 2*b)*Tanh[2*c])/a))/(8*(a + b)^2*d*(a + b*Sech[c + 
d*x]^2)^2)
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 25, 4620, 253, 264, 221}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4620

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

\(\Big \downarrow \) 253

\(\displaystyle \frac {\frac {3 \int \frac {\coth ^2(c+d x)}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{2 (a+b)}+\frac {\coth (c+d x)}{2 (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{d}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {\frac {3 \left (\frac {b \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a+b}-\frac {\coth (c+d x)}{a+b}\right )}{2 (a+b)}+\frac {\coth (c+d x)}{2 (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {3 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{(a+b)^{3/2}}-\frac {\coth (c+d x)}{a+b}\right )}{2 (a+b)}+\frac {\coth (c+d x)}{2 (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{d}\)

Input:

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

Output:

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

Defintions of rubi rules used

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 253

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x 
)^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 
2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m 
}, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 3042

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

rule 4620

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_ 
)]^(m_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff^(m 
+ 1)/f   Subst[Int[x^m*(ExpandToSum[a + b*(1 + ff^2*x^2)^(n/2), x]^p/(1 + f 
f^2*x^2)^(m/2 + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, 
 x] && IntegerQ[m/2] && IntegerQ[n/2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(240\) vs. \(2(79)=158\).

Time = 39.14 (sec) , antiderivative size = 241, normalized size of antiderivative = 2.65

method	result	size
derivativedivides	\(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a^{2}+2 a b +b^{2}\right )}-\frac {4 b \left (\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4}-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {3 \ln \left (-\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}-\sqrt {a +b}\right )}{16 \sqrt {b}\, \sqrt {a +b}}-\frac {3 \ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{16 \sqrt {b}\, \sqrt {a +b}}\right )}{\left (a +b \right )^{2}}-\frac {1}{2 \left (a +b \right )^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\)	\(241\)
default	\(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a^{2}+2 a b +b^{2}\right )}-\frac {4 b \left (\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4}-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {3 \ln \left (-\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}-\sqrt {a +b}\right )}{16 \sqrt {b}\, \sqrt {a +b}}-\frac {3 \ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{16 \sqrt {b}\, \sqrt {a +b}}\right )}{\left (a +b \right )^{2}}-\frac {1}{2 \left (a +b \right )^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\)	\(241\)
risch	\(-\frac {2 a^{2} {\mathrm e}^{4 d x +4 c}+a b \,{\mathrm e}^{4 d x +4 c}+2 b^{2} {\mathrm e}^{4 d x +4 c}+4 a^{2} {\mathrm e}^{2 d x +2 c}+8 a b \,{\mathrm e}^{2 d x +2 c}-2 b^{2} {\mathrm e}^{2 d x +2 c}+2 a^{2}-a b}{a d \left (a +b \right )^{2} \left ({\mathrm e}^{4 d x +4 c} a +2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right ) \left ({\mathrm e}^{2 d x +2 c}-1\right )}+\frac {3 \sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-a +2 \sqrt {b \left (a +b \right )}-2 b}{a}\right )}{4 \left (a +b \right )^{3} d}-\frac {3 \sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a +2 \sqrt {b \left (a +b \right )}+2 b}{a}\right )}{4 \left (a +b \right )^{3} d}\)	\(253\)

Input:

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

Output:

1/d*(-1/2*tanh(1/2*d*x+1/2*c)/(a^2+2*a*b+b^2)-4/(a+b)^2*b*((-1/4*tanh(1/2* 
d*x+1/2*c)^3-1/4*tanh(1/2*d*x+1/2*c))/(tanh(1/2*d*x+1/2*c)^4*a+tanh(1/2*d* 
x+1/2*c)^4*b+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)+3/16 
/b^(1/2)/(a+b)^(1/2)*ln(-(a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+ 
1/2*c)*b^(1/2)-(a+b)^(1/2))-3/16/b^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1 
/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(1/2)))-1/2/(a+b)^2/ta 
nh(1/2*d*x+1/2*c))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1065 vs. \(2 (82) = 164\).

Time = 0.36 (sec) , antiderivative size = 2407, normalized size of antiderivative = 26.45 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/4*(4*(2*a^2 + a*b + 2*b^2)*cosh(d*x + c)^4 + 16*(2*a^2 + a*b + 2*b^2)* 
cosh(d*x + c)*sinh(d*x + c)^3 + 4*(2*a^2 + a*b + 2*b^2)*sinh(d*x + c)^4 + 
8*(2*a^2 + 4*a*b - b^2)*cosh(d*x + c)^2 + 8*(3*(2*a^2 + a*b + 2*b^2)*cosh( 
d*x + c)^2 + 2*a^2 + 4*a*b - b^2)*sinh(d*x + c)^2 - 3*(a^2*cosh(d*x + c)^6 
 + 6*a^2*cosh(d*x + c)*sinh(d*x + c)^5 + a^2*sinh(d*x + c)^6 + (a^2 + 4*a* 
b)*cosh(d*x + c)^4 + (15*a^2*cosh(d*x + c)^2 + a^2 + 4*a*b)*sinh(d*x + c)^ 
4 + 4*(5*a^2*cosh(d*x + c)^3 + (a^2 + 4*a*b)*cosh(d*x + c))*sinh(d*x + c)^ 
3 - (a^2 + 4*a*b)*cosh(d*x + c)^2 + (15*a^2*cosh(d*x + c)^4 + 6*(a^2 + 4*a 
*b)*cosh(d*x + c)^2 - a^2 - 4*a*b)*sinh(d*x + c)^2 - a^2 + 2*(3*a^2*cosh(d 
*x + c)^5 + 2*(a^2 + 4*a*b)*cosh(d*x + c)^3 - (a^2 + 4*a*b)*cosh(d*x + c)) 
*sinh(d*x + c))*sqrt(b/(a + b))*log((a^2*cosh(d*x + c)^4 + 4*a^2*cosh(d*x 
+ c)*sinh(d*x + c)^3 + a^2*sinh(d*x + c)^4 + 2*(a^2 + 2*a*b)*cosh(d*x + c) 
^2 + 2*(3*a^2*cosh(d*x + c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^2 + a^2 + 8*a*b 
 + 8*b^2 + 4*(a^2*cosh(d*x + c)^3 + (a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x 
+ c) - 4*((a^2 + a*b)*cosh(d*x + c)^2 + 2*(a^2 + a*b)*cosh(d*x + c)*sinh(d 
*x + c) + (a^2 + a*b)*sinh(d*x + c)^2 + a^2 + 3*a*b + 2*b^2)*sqrt(b/(a + b 
)))/(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + 
c)^4 + 2*(a + 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 + a + 2*b)*sin 
h(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 + (a + 2*b)*cosh(d*x + c))*sinh(d*x + 
c) + a)) + 8*a^2 - 4*a*b + 16*((2*a^2 + a*b + 2*b^2)*cosh(d*x + c)^3 + ...
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (82) = 164\).

Time = 0.16 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.88 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=-\frac {3 \, b \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {{\left (a + b\right )} b} d} - \frac {2 \, a^{2} - a b + 2 \, {\left (2 \, a^{2} + 4 \, a b - b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (2 \, a^{2} + a b + 2 \, b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{{\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2} + {\left (a^{4} + 6 \, a^{3} b + 9 \, a^{2} b^{2} + 4 \, a b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - {\left (a^{4} + 6 \, a^{3} b + 9 \, a^{2} b^{2} + 4 \, a b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )} d} \] Input:

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

Output:

-3/4*b*log((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x 
 - 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^2 + 2*a*b + b^2)*sqrt((a + b)* 
b)*d) - (2*a^2 - a*b + 2*(2*a^2 + 4*a*b - b^2)*e^(-2*d*x - 2*c) + (2*a^2 + 
 a*b + 2*b^2)*e^(-4*d*x - 4*c))/((a^4 + 2*a^3*b + a^2*b^2 + (a^4 + 6*a^3*b 
 + 9*a^2*b^2 + 4*a*b^3)*e^(-2*d*x - 2*c) - (a^4 + 6*a^3*b + 9*a^2*b^2 + 4* 
a*b^3)*e^(-4*d*x - 4*c) - (a^4 + 2*a^3*b + a^2*b^2)*e^(-6*d*x - 6*c))*d)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (82) = 164\).

Time = 0.30 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.63 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {\frac {3 \, b \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {-a b - b^{2}}} - \frac {2 \, {\left (2 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + a b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 8 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{2} - a b\right )}}{{\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} {\left (a e^{\left (6 \, d x + 6 \, c\right )} + a e^{\left (4 \, d x + 4 \, c\right )} + 4 \, b e^{\left (4 \, d x + 4 \, c\right )} - a e^{\left (2 \, d x + 2 \, c\right )} - 4 \, b e^{\left (2 \, d x + 2 \, c\right )} - a\right )}}}{2 \, d} \] Input:

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

Output:

1/2*(3*b*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a*b - b^2))/((a^2 
+ 2*a*b + b^2)*sqrt(-a*b - b^2)) - 2*(2*a^2*e^(4*d*x + 4*c) + a*b*e^(4*d*x 
 + 4*c) + 2*b^2*e^(4*d*x + 4*c) + 4*a^2*e^(2*d*x + 2*c) + 8*a*b*e^(2*d*x + 
 2*c) - 2*b^2*e^(2*d*x + 2*c) + 2*a^2 - a*b)/((a^3 + 2*a^2*b + a*b^2)*(a*e 
^(6*d*x + 6*c) + a*e^(4*d*x + 4*c) + 4*b*e^(4*d*x + 4*c) - a*e^(2*d*x + 2* 
c) - 4*b*e^(2*d*x + 2*c) - a)))/d
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^4}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^2} \,d x \] Input:

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

Output:

int(cosh(c + d*x)^4/(sinh(c + d*x)^2*(b + a*cosh(c + d*x)^2)^2), x)
 

Reduce [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 1863, normalized size of antiderivative = 20.47 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(3*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) 
- a - 2*b) + e**(c + d*x)*sqrt(a))*a**2 + 12*e**(6*c + 6*d*x)*sqrt(b)*sqrt 
(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a 
))*a*b + 3*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sqrt(a 
+ b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**2 + 12*e**(6*c + 6*d*x)*sqrt(b) 
*sqrt(a + b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt 
(a))*a*b - 3*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a + b)*log(2*sqrt(b)*sqrt(a + b 
) + e**(2*c + 2*d*x)*a + a + 2*b)*a**2 - 12*e**(6*c + 6*d*x)*sqrt(b)*sqrt( 
a + b)*log(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*a*b + 3*e 
**(4*c + 4*d*x)*sqrt(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a 
- 2*b) + e**(c + d*x)*sqrt(a))*a**2 + 24*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a + 
 b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a 
*b + 48*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a 
+ b) - a - 2*b) + e**(c + d*x)*sqrt(a))*b**2 + 3*e**(4*c + 4*d*x)*sqrt(b)* 
sqrt(a + b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt( 
a))*a**2 + 24*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sqrt 
(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a*b + 48*e**(4*c + 4*d*x)*sqrt( 
b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sq 
rt(a))*b**2 - 3*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a + b)*log(2*sqrt(b)*sqrt(a 
+ b) + e**(2*c + 2*d*x)*a + a + 2*b)*a**2 - 24*e**(4*c + 4*d*x)*sqrt(b)...