\(\int \frac {\coth ^2(c+d x)}{(a+b \text {sech}^2(c+d x))^2} \, dx\) [155]

Optimal result

Integrand size = 23, antiderivative size = 121 \[ \int \frac {\coth ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {x}{a^2}-\frac {b^{3/2} (5 a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{2 a^2 (a+b)^{5/2} d}-\frac {(2 a-b) \coth (c+d x)}{2 a (a+b)^2 d}-\frac {b \coth (c+d x)}{2 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )} \] Output:

x/a^2-1/2*b^(3/2)*(5*a+2*b)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^(1/2))/a^2/( 
a+b)^(5/2)/d-1/2*(2*a-b)*coth(d*x+c)/a/(a+b)^2/d-1/2*b*coth(d*x+c)/a/(a+b) 
/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. \(268\) vs. \(2(121)=242\).

Time = 2.37 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.21 \[ \int \frac {\coth ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^4(c+d x) \left (\frac {2 x (a+2 b+a \cosh (2 (c+d x)))}{a^2}-\frac {b^2 (5 a+2 b) \text {arctanh}\left (\frac {\text {sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right ) (a+2 b+a \cosh (2 (c+d x))) (\cosh (2 c)-\sinh (2 c))}{a^2 (a+b)^{5/2} d \sqrt {b (\cosh (c)-\sinh (c))^4}}+\frac {2 (a+2 b+a \cosh (2 (c+d x))) \text {csch}(c) \text {csch}(c+d x) \sinh (d x)}{(a+b)^2 d}+\frac {b^2 \text {sech}(2 c) ((a+2 b) \sinh (2 c)-a \sinh (2 d x))}{a^2 (a+b)^2 d}\right )}{8 \left (a+b \text {sech}^2(c+d x)\right )^2} \] Input:

Integrate[Coth[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*((2*x*(a + 2*b + a*Cosh[2 
*(c + d*x)]))/a^2 - (b^2*(5*a + 2*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*(Co 
sh[c] - Sinh[c])^4])]*(a + 2*b + a*Cosh[2*(c + d*x)])*(Cosh[2*c] - Sinh[2* 
c]))/(a^2*(a + b)^(5/2)*d*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])/((a + b)^2*d) + (b^2* 
Sech[2*c]*((a + 2*b)*Sinh[2*c] - a*Sinh[2*d*x]))/(a^2*(a + b)^2*d)))/(8*(a 
 + b*Sech[c + d*x]^2)^2)
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.17, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {3042, 25, 4629, 25, 2075, 374, 25, 445, 25, 397, 219, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4629

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2075

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

\(\Big \downarrow \) 374

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 445

\(\displaystyle -\frac {\frac {b \coth (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {-\frac {\int -\frac {2 a^2+6 b a+b^2-(2 a-b) b \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{a+b}-\frac {(2 a-b) \coth (c+d x)}{a+b}}{2 a (a+b)}}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {b \coth (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {\frac {\int \frac {2 a^2+6 b a+b^2-(2 a-b) b \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{a+b}-\frac {(2 a-b) \coth (c+d x)}{a+b}}{2 a (a+b)}}{d}\)

\(\Big \downarrow \) 397

\(\displaystyle -\frac {\frac {b \coth (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {\frac {\frac {2 (a+b)^2 \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a}-\frac {b^2 (5 a+2 b) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}}{a+b}-\frac {(2 a-b) \coth (c+d x)}{a+b}}{2 a (a+b)}}{d}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\frac {b \coth (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {\frac {\frac {2 (a+b)^2 \text {arctanh}(\tanh (c+d x))}{a}-\frac {b^2 (5 a+2 b) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}}{a+b}-\frac {(2 a-b) \coth (c+d x)}{a+b}}{2 a (a+b)}}{d}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\frac {b \coth (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {\frac {\frac {2 (a+b)^2 \text {arctanh}(\tanh (c+d x))}{a}-\frac {b^{3/2} (5 a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a \sqrt {a+b}}}{a+b}-\frac {(2 a-b) \coth (c+d x)}{a+b}}{2 a (a+b)}}{d}\)

Input:

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

Output:

-((-1/2*(((2*(a + b)^2*ArcTanh[Tanh[c + d*x]])/a - (b^(3/2)*(5*a + 2*b)*Ar 
cTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a*Sqrt[a + b]))/(a + b) - ((2 
*a - b)*Coth[c + d*x])/(a + b))/(a*(a + b)) + (b*Coth[c + d*x])/(2*a*(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 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 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 374

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q 
 + 1)/(a*e*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(e*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[b*c*(m + 1) + 2*(b*c - 
a*d)*(p + 1) + d*b*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, 
c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IntBinomialQ[a, b, 
 c, d, e, m, 2, p, q, x]
 

rule 397

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ 
Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^2), x], x] - Simp[ 
(d*e - c*f)/(b*c - a*d)   Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e 
, f}, x]
 

rule 445

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ 
.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p 
+ 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) 
 Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c 
+ a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 
2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
 

rule 2075

Int[(u_)^(p_.)*(v_)^(q_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*Expa 
ndToSum[u, x]^p*ExpandToSum[v, x]^q, x] /; FreeQ[{e, m, p, q}, x] && Binomi 
alQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0] &&  ! 
BinomialMatchQ[{u, v}, x]
 

rule 3042

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

rule 4629

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*((d_.)*tan[(e_.) + (f 
_.)*(x_)])^(m_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Sim 
p[ff/f   Subst[Int[(d*ff*x)^m*((a + b*(1 + ff^2*x^2)^(n/2))^p/(1 + ff^2*x^2 
)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && Inte 
gerQ[n/2] && (IntegerQ[m/2] || EqQ[n, 2])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(287\) vs. \(2(107)=214\).

Time = 23.54 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.38

method	result	size
derivativedivides	\(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a^{2}+2 a b +b^{2}\right )}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2}}-\frac {1}{2 \left (a +b \right )^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}+\frac {2 b^{2} \left (\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a}{2}-\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {\left (5 a +2 b \right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{2}\right )}{\left (a +b \right )^{2} a^{2}}}{d}\)	\(288\)
default	\(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a^{2}+2 a b +b^{2}\right )}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2}}-\frac {1}{2 \left (a +b \right )^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}+\frac {2 b^{2} \left (\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a}{2}-\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {\left (5 a +2 b \right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{2}\right )}{\left (a +b \right )^{2} a^{2}}}{d}\)	\(288\)
risch	\(\frac {x}{a^{2}}-\frac {2 a^{3} {\mathrm e}^{4 d x +4 c}-a \,b^{2} {\mathrm e}^{4 d x +4 c}-2 b^{3} {\mathrm e}^{4 d x +4 c}+4 a^{3} {\mathrm e}^{2 d x +2 c}+8 a^{2} b \,{\mathrm e}^{2 d x +2 c}+2 b^{3} {\mathrm e}^{2 d x +2 c}+2 a^{3}+a \,b^{2}}{d \left (a +b \right )^{2} \left ({\mathrm e}^{2 d x +2 c}-1\right ) a^{2} \left ({\mathrm e}^{4 d x +4 c} a +2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )}+\frac {5 \sqrt {b \left (a +b \right )}\, b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a +2 \sqrt {b \left (a +b \right )}+2 b}{a}\right )}{4 \left (a +b \right )^{3} d a}+\frac {\sqrt {b \left (a +b \right )}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a +2 \sqrt {b \left (a +b \right )}+2 b}{a}\right )}{2 \left (a +b \right )^{3} d \,a^{2}}-\frac {5 \sqrt {b \left (a +b \right )}\, b \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-a +2 \sqrt {b \left (a +b \right )}-2 b}{a}\right )}{4 \left (a +b \right )^{3} d a}-\frac {\sqrt {b \left (a +b \right )}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-a +2 \sqrt {b \left (a +b \right )}-2 b}{a}\right )}{2 \left (a +b \right )^{3} d \,a^{2}}\)	\(379\)

Input:

int(coth(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)+1/a^2*ln(tanh(1/2*d*x+1/2*c) 
+1)-1/2/(a+b)^2/tanh(1/2*d*x+1/2*c)-1/a^2*ln(tanh(1/2*d*x+1/2*c)-1)+2*b^2/ 
(a+b)^2/a^2*((-1/2*tanh(1/2*d*x+1/2*c)^3*a-1/2*a*tanh(1/2*d*x+1/2*c))/(tan 
h(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*t 
anh(1/2*d*x+1/2*c)^2*b+a+b)+1/2*(5*a+2*b)*(-1/4/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))+ 
1/4/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)))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1674 vs. \(2 (110) = 220\).

Time = 0.37 (sec) , antiderivative size = 3624, normalized size of antiderivative = 29.95 \[ \int \frac {\coth ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

Integral(coth(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. 1070 vs. \(2 (110) = 220\).

Time = 0.24 (sec) , antiderivative size = 1070, normalized size of antiderivative = 8.84 \[ \int \frac {\coth ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/4*(2*a*b + b^2)*log(a*e^(4*d*x + 4*c) + 2*(a + 2*b)*e^(2*d*x + 2*c) + a) 
/((a^4 + 2*a^3*b + a^2*b^2)*d) - 1/4*(2*a*b + b^2)*log(2*(a + 2*b)*e^(-2*d 
*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/((a^4 + 2*a^3*b + a^2*b^2)*d) - 1/16*( 
3*a^2*b + 10*a*b^2 + 4*b^3)*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^4 + 2*a^3*b 
 + a^2*b^2)*sqrt((a + b)*b)*d) + 1/16*(3*a^2*b + 10*a*b^2 + 4*b^3)*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^4 + 2*a^3*b + a^2*b^2)*sqrt((a + b)*b)*d) - 
3/8*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) + 1/4*(2*a^3 + a^2*b + 2*a*b^2 + (2*a^3 - a^2*b - 8*a*b^2 - 8*b^3)*e^ 
(4*d*x + 4*c) + 2*(2*a^3 + 4*a^2*b + 3*a*b^2 + 4*b^3)*e^(2*d*x + 2*c))/((a 
^5 + 2*a^4*b + a^3*b^2 - (a^5 + 2*a^4*b + a^3*b^2)*e^(6*d*x + 6*c) - (a^5 
+ 6*a^4*b + 9*a^3*b^2 + 4*a^2*b^3)*e^(4*d*x + 4*c) + (a^5 + 6*a^4*b + 9*a^ 
3*b^2 + 4*a^2*b^3)*e^(2*d*x + 2*c))*d) - 1/4*(2*a^3 + a^2*b + 2*a*b^2 + 2* 
(2*a^3 + 4*a^2*b + 3*a*b^2 + 4*b^3)*e^(-2*d*x - 2*c) + (2*a^3 - a^2*b - 8* 
a*b^2 - 8*b^3)*e^(-4*d*x - 4*c))/((a^5 + 2*a^4*b + a^3*b^2 + (a^5 + 6*a^4* 
b + 9*a^3*b^2 + 4*a^2*b^3)*e^(-2*d*x - 2*c) - (a^5 + 6*a^4*b + 9*a^3*b^2 + 
 4*a^2*b^3)*e^(-4*d*x - 4*c) - (a^5 + 2*a^4*b + a^3*b^2)*e^(-6*d*x - 6*c)) 
*d) - 1/2*(2*a^2 - a*b + 2*(2*a^2 + 4*a*b - b^2)*e^(-2*d*x - 2*c) + (2*...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (110) = 220\).

Time = 0.64 (sec) , antiderivative size = 273, normalized size of antiderivative = 2.26 \[ \int \frac {\coth ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=-\frac {\frac {{\left (5 \, a b^{2} + 2 \, b^{3}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{{\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} \sqrt {-a b - b^{2}}} + \frac {2 \, {\left (2 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 2 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 8 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{3} + a b^{2}\right )}}{{\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} {\left (a e^{\left (6 \, d x + 6 \, c\right )} + a e^{\left (4 \, d x + 4 \, c\right )} + 4 \, b e^{\left (4 \, d x + 4 \, c\right )} - a e^{\left (2 \, d x + 2 \, c\right )} - 4 \, b e^{\left (2 \, d x + 2 \, c\right )} - a\right )}} - \frac {2 \, {\left (d x + c\right )}}{a^{2}}}{2 \, d} \] Input:

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

Output:

-1/2*((5*a*b^2 + 2*b^3)*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a*b 
 - b^2))/((a^4 + 2*a^3*b + a^2*b^2)*sqrt(-a*b - b^2)) + 2*(2*a^3*e^(4*d*x 
+ 4*c) - a*b^2*e^(4*d*x + 4*c) - 2*b^3*e^(4*d*x + 4*c) + 4*a^3*e^(2*d*x + 
2*c) + 8*a^2*b*e^(2*d*x + 2*c) + 2*b^3*e^(2*d*x + 2*c) + 2*a^3 + a*b^2)/(( 
a^4 + 2*a^3*b + a^2*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)) - 2*(d*x + c)/ 
a^2)/d
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 2905, normalized size of antiderivative = 24.01 \[ \int \frac {\coth ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 5*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**3*b - 22*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)*s 
qrt(a))*a**2*b**2 - 8*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a + b)*log( - sqrt(2*s 
qrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a*b**3 - 5*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**3*b - 22*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a + b)*log(s 
qrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**2*b**2 - 8 
*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 + 5*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**3*b + 22* 
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*b**2 + 8*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a + b)*lo 
g(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*a*b**3 - 5*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**3*b - 42*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* 
b**2 - 96*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**3 - 32*e**(4*c + 4*d*x)*sqr 
t(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c +...