\(\int \frac {\coth ^2(c+d x)}{(a+b \text {sech}^2(c+d x))^3} \, dx\) [166]

Optimal result

Integrand size = 23, antiderivative size = 182 \[ \int \frac {\coth ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {x}{a^3}-\frac {b^{3/2} \left (35 a^2+28 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 a^3 (a+b)^{7/2} d}-\frac {\left (8 a^2-11 a b-4 b^2\right ) \coth (c+d x)}{8 a^2 (a+b)^3 d}-\frac {b \coth (c+d x)}{4 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {b (9 a+4 b) \coth (c+d x)}{8 a^2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )} \] Output:

x/a^3-1/8*b^(3/2)*(35*a^2+28*a*b+8*b^2)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^ 
(1/2))/a^3/(a+b)^(7/2)/d-1/8*(8*a^2-11*a*b-4*b^2)*coth(d*x+c)/a^2/(a+b)^3/ 
d-1/4*b*coth(d*x+c)/a/(a+b)/d/(a+b-b*tanh(d*x+c)^2)^2-1/8*b*(9*a+4*b)*coth 
(d*x+c)/a^2/(a+b)^2/d/(a+b-b*tanh(d*x+c)^2)
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 7.73 (sec) , antiderivative size = 2083, normalized size of antiderivative = 11.45 \[ \int \frac {\coth ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Result too large to show} \] Input:

Integrate[Coth[c + d*x]^2/(a + b*Sech[c + d*x]^2)^3,x]
 

Output:

((35*a^2 + 28*a*b + 8*b^2)*(a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x] 
^6*(((I/64)*b^2*ArcTan[Sech[d*x]*(((-1/2*I)*Cosh[2*c])/(Sqrt[a + b]*Sqrt[b 
*Cosh[4*c] - b*Sinh[4*c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4* 
c] - b*Sinh[4*c]]))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*Sinh[2*c + d*x])]* 
Cosh[2*c])/(a^3*Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) - ((I/64)*b 
^2*ArcTan[Sech[d*x]*(((-1/2*I)*Cosh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - 
b*Sinh[4*c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4 
*c]]))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*Sinh[2*c + d*x])]*Sinh[2*c])/(a 
^3*Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]])))/((a + b)^3*(a + b*Sech 
[c + d*x]^2)^3) + ((a + 2*b + a*Cosh[2*c + 2*d*x])*Csch[c]*Csch[c + d*x]*S 
ech[2*c]*Sech[c + d*x]^6*(8*a^5*d*x*Cosh[d*x] + 56*a^4*b*d*x*Cosh[d*x] + 1 
84*a^3*b^2*d*x*Cosh[d*x] + 296*a^2*b^3*d*x*Cosh[d*x] + 224*a*b^4*d*x*Cosh[ 
d*x] + 64*b^5*d*x*Cosh[d*x] - 12*a^5*d*x*Cosh[3*d*x] - 68*a^4*b*d*x*Cosh[3 
*d*x] - 132*a^3*b^2*d*x*Cosh[3*d*x] - 108*a^2*b^3*d*x*Cosh[3*d*x] - 32*a*b 
^4*d*x*Cosh[3*d*x] - 8*a^5*d*x*Cosh[2*c - d*x] - 56*a^4*b*d*x*Cosh[2*c - d 
*x] - 184*a^3*b^2*d*x*Cosh[2*c - d*x] - 296*a^2*b^3*d*x*Cosh[2*c - d*x] - 
224*a*b^4*d*x*Cosh[2*c - d*x] - 64*b^5*d*x*Cosh[2*c - d*x] - 8*a^5*d*x*Cos 
h[2*c + d*x] - 56*a^4*b*d*x*Cosh[2*c + d*x] - 184*a^3*b^2*d*x*Cosh[2*c + d 
*x] - 296*a^2*b^3*d*x*Cosh[2*c + d*x] - 224*a*b^4*d*x*Cosh[2*c + d*x] - 64 
*b^5*d*x*Cosh[2*c + d*x] + 8*a^5*d*x*Cosh[4*c + d*x] + 56*a^4*b*d*x*Cos...
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.17, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {3042, 25, 4629, 25, 2075, 374, 25, 441, 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.

\(\displaystyle \int \frac {\coth ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -\frac {1}{\tan (i c+i d x)^2 \left (a+b \sec (i c+i d x)^2\right )^3}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \frac {1}{\left (b \sec (i c+i d x)^2+a\right )^3 \tan (i c+i d x)^2}dx\)

\(\Big \downarrow \) 4629

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2075

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

\(\Big \downarrow \) 374

\(\displaystyle -\frac {\frac {\int -\frac {\coth ^2(c+d x) \left (5 b \tanh ^2(c+d x)+4 a-b\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{4 a (a+b)}+\frac {b \coth (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {b \coth (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\int \frac {\coth ^2(c+d x) \left (5 b \tanh ^2(c+d x)+4 a-b\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{4 a (a+b)}}{d}\)

\(\Big \downarrow \) 441

\(\displaystyle -\frac {\frac {b \coth (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {-\frac {\int -\frac {\coth ^2(c+d x) \left (8 a^2-11 b a-4 b^2+3 b (9 a+4 b) \tanh ^2(c+d x)\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{2 a (a+b)}-\frac {b (9 a+4 b) \coth (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {b \coth (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {\int \frac {\coth ^2(c+d x) \left (8 a^2-11 b a-4 b^2+3 b (9 a+4 b) \tanh ^2(c+d x)\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{2 a (a+b)}-\frac {b (9 a+4 b) \coth (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}}{d}\)

\(\Big \downarrow \) 445

\(\displaystyle -\frac {\frac {b \coth (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {-\frac {\int -\frac {8 a^3+32 b a^2+13 b^2 a+4 b^3-b \left (8 a^2-11 b a-4 b^2\right ) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{a+b}-\frac {\left (8 a^2-11 a b-4 b^2\right ) \coth (c+d x)}{a+b}}{2 a (a+b)}-\frac {b (9 a+4 b) \coth (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {b \coth (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {\frac {\int \frac {8 a^3+32 b a^2+13 b^2 a+4 b^3-b \left (8 a^2-11 b a-4 b^2\right ) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{a+b}-\frac {\left (8 a^2-11 a b-4 b^2\right ) \coth (c+d x)}{a+b}}{2 a (a+b)}-\frac {b (9 a+4 b) \coth (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}}{d}\)

\(\Big \downarrow \) 397

\(\displaystyle -\frac {\frac {b \coth (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {\frac {\frac {8 (a+b)^3 \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a}-\frac {b^2 \left (35 a^2+28 a b+8 b^2\right ) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}}{a+b}-\frac {\left (8 a^2-11 a b-4 b^2\right ) \coth (c+d x)}{a+b}}{2 a (a+b)}-\frac {b (9 a+4 b) \coth (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}}{d}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\frac {b \coth (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {\frac {\frac {8 (a+b)^3 \text {arctanh}(\tanh (c+d x))}{a}-\frac {b^2 \left (35 a^2+28 a b+8 b^2\right ) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}}{a+b}-\frac {\left (8 a^2-11 a b-4 b^2\right ) \coth (c+d x)}{a+b}}{2 a (a+b)}-\frac {b (9 a+4 b) \coth (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}}{d}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\frac {b \coth (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {\frac {\frac {8 (a+b)^3 \text {arctanh}(\tanh (c+d x))}{a}-\frac {b^{3/2} \left (35 a^2+28 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a \sqrt {a+b}}}{a+b}-\frac {\left (8 a^2-11 a b-4 b^2\right ) \coth (c+d x)}{a+b}}{2 a (a+b)}-\frac {b (9 a+4 b) \coth (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a (a+b)}}{d}\)

Input:

Int[Coth[c + d*x]^2/(a + b*Sech[c + d*x]^2)^3,x]
 

Output:

-(((b*Coth[c + d*x])/(4*a*(a + b)*(a + b - b*Tanh[c + d*x]^2)^2) - ((((8*( 
a + b)^3*ArcTanh[Tanh[c + d*x]])/a - (b^(3/2)*(35*a^2 + 28*a*b + 8*b^2)*Ar 
cTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a*Sqrt[a + b]))/(a + b) - ((8 
*a^2 - 11*a*b - 4*b^2)*Coth[c + d*x])/(a + b))/(2*a*(a + b)) - (b*(9*a + 4 
*b)*Coth[c + d*x])/(2*a*(a + b)*(a + b - b*Tanh[c + d*x]^2)))/(4*a*(a + b) 
))/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 441

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

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. \(393\) vs. \(2(166)=332\).

Time = 142.14 (sec) , antiderivative size = 394, normalized size of antiderivative = 2.16

method	result	size
derivativedivides	\(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {1}{2 \left (a +b \right )^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 b^{2} \left (\frac {\left (-\frac {13}{8} a^{3}-\frac {17}{8} a^{2} b -\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-\frac {\left (39 a^{2}+7 a b -4 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8}-\frac {\left (39 a^{2}+7 a b -4 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8}+\left (-\frac {13}{8} a^{3}-\frac {17}{8} a^{2} b -\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {\left (35 a^{2}+28 a b +8 b^{2}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{8}\right )}{\left (a +b \right )^{3} a^{3}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3}}}{d}\)	\(394\)
default	\(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {1}{2 \left (a +b \right )^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 b^{2} \left (\frac {\left (-\frac {13}{8} a^{3}-\frac {17}{8} a^{2} b -\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-\frac {\left (39 a^{2}+7 a b -4 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8}-\frac {\left (39 a^{2}+7 a b -4 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8}+\left (-\frac {13}{8} a^{3}-\frac {17}{8} a^{2} b -\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {\left (35 a^{2}+28 a b +8 b^{2}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{8}\right )}{\left (a +b \right )^{3} a^{3}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3}}}{d}\)	\(394\)
risch	\(\frac {x}{a^{3}}-\frac {8 a^{5} {\mathrm e}^{8 d x +8 c}-13 a^{3} b^{2} {\mathrm e}^{8 d x +8 c}-36 a^{2} b^{3} {\mathrm e}^{8 d x +8 c}-16 a \,b^{4} {\mathrm e}^{8 d x +8 c}+32 a^{5} {\mathrm e}^{6 d x +6 c}+64 a^{4} b \,{\mathrm e}^{6 d x +6 c}-26 a^{3} b^{2} {\mathrm e}^{6 d x +6 c}-86 a^{2} b^{3} {\mathrm e}^{6 d x +6 c}-136 a \,b^{4} {\mathrm e}^{6 d x +6 c}-48 b^{5} {\mathrm e}^{6 d x +6 c}+48 a^{5} {\mathrm e}^{4 d x +4 c}+128 a^{4} b \,{\mathrm e}^{4 d x +4 c}+128 a^{3} b^{2} {\mathrm e}^{4 d x +4 c}+30 a^{2} b^{3} {\mathrm e}^{4 d x +4 c}+120 a \,b^{4} {\mathrm e}^{4 d x +4 c}+48 b^{5} {\mathrm e}^{4 d x +4 c}+32 a^{5} {\mathrm e}^{2 d x +2 c}+64 a^{4} b \,{\mathrm e}^{2 d x +2 c}+26 a^{3} b^{2} {\mathrm e}^{2 d x +2 c}+86 a^{2} b^{3} {\mathrm e}^{2 d x +2 c}+32 a \,b^{4} {\mathrm e}^{2 d x +2 c}+8 a^{5}+13 a^{3} b^{2}+6 a^{2} b^{3}}{4 a^{3} d \left (a +b \right )^{3} \left ({\mathrm e}^{2 d x +2 c}-1\right ) \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 )^{2}}+\frac {35 \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 )}{16 \left (a +b \right )^{4} d a}+\frac {7 \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 )}{4 \left (a +b \right )^{4} d \,a^{2}}+\frac {\sqrt {b \left (a +b \right )}\, b^{3} \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 )^{4} d \,a^{3}}-\frac {35 \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 )}{16 \left (a +b \right )^{4} d a}-\frac {7 \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 )}{4 \left (a +b \right )^{4} d \,a^{2}}-\frac {\sqrt {b \left (a +b \right )}\, b^{3} \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 )^{4} d \,a^{3}}\)	\(736\)

Input:

int(coth(d*x+c)^2/(a+b*sech(d*x+c)^2)^3,x,method=_RETURNVERBOSE)
 

Output:

1/d*(-1/2*tanh(1/2*d*x+1/2*c)/(a^3+3*a^2*b+3*a*b^2+b^3)-1/2/(a+b)^3/tanh(1 
/2*d*x+1/2*c)+2*b^2/(a+b)^3/a^3*(((-13/8*a^3-17/8*a^2*b-1/2*a*b^2)*tanh(1/ 
2*d*x+1/2*c)^7-1/8*(39*a^2+7*a*b-4*b^2)*a*tanh(1/2*d*x+1/2*c)^5-1/8*(39*a^ 
2+7*a*b-4*b^2)*a*tanh(1/2*d*x+1/2*c)^3+(-13/8*a^3-17/8*a^2*b-1/2*a*b^2)*ta 
nh(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)^2+1/8*(35*a^2+28*a*b+8* 
b^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))+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/a^3 
*ln(tanh(1/2*d*x+1/2*c)+1)-1/a^3*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. 5665 vs. \(2 (172) = 344\).

Time = 0.54 (sec) , antiderivative size = 11606, normalized size of antiderivative = 63.77 \[ \int \frac {\coth ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(coth(d*x+c)^2/(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {\coth ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {\coth ^{2}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3}}\, dx \] Input:

integrate(coth(d*x+c)**2/(a+b*sech(d*x+c)**2)**3,x)
 

Output:

Integral(coth(c + d*x)**2/(a + b*sech(c + d*x)**2)**3, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1971 vs. \(2 (172) = 344\).

Time = 0.38 (sec) , antiderivative size = 1971, normalized size of antiderivative = 10.83 \[ \int \frac {\coth ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(coth(d*x+c)^2/(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")
 

Output:

1/4*(3*a^2*b + 3*a*b^2 + b^3)*log(a*e^(4*d*x + 4*c) + 2*(a + 2*b)*e^(2*d*x 
 + 2*c) + a)/((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d) - 1/4*(3*a^2*b + 3* 
a*b^2 + b^3)*log(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/(( 
a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d) - 1/64*(15*a^3*b + 70*a^2*b^2 + 56 
*a*b^3 + 16*b^4)*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^6 + 3*a^5*b + 3*a^4*b^ 
2 + a^3*b^3)*sqrt((a + b)*b)*d) + 1/64*(15*a^3*b + 70*a^2*b^2 + 56*a*b^3 + 
 16*b^4)*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^6 + 3*a^5*b + 3*a^4*b^2 + a^ 
3*b^3)*sqrt((a + b)*b)*d) - 15/32*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^3 
 + 3*a^2*b + 3*a*b^2 + b^3)*sqrt((a + b)*b)*d) + 1/16*(8*a^5 + 9*a^4*b + 2 
8*a^3*b^2 + 12*a^2*b^3 + (8*a^5 - 9*a^4*b - 98*a^3*b^2 - 160*a^2*b^3 - 64* 
a*b^4)*e^(8*d*x + 8*c) + 2*(16*a^5 + 23*a^4*b - 77*a^3*b^2 - 246*a^2*b^3 - 
 288*a*b^4 - 96*b^5)*e^(6*d*x + 6*c) + 2*(24*a^5 + 64*a^4*b + 99*a^3*b^2 + 
 190*a^2*b^3 + 272*a*b^4 + 96*b^5)*e^(4*d*x + 4*c) + 2*(16*a^5 + 41*a^4*b 
+ 77*a^3*b^2 + 130*a^2*b^3 + 48*a*b^4)*e^(2*d*x + 2*c))/((a^8 + 3*a^7*b + 
3*a^6*b^2 + a^5*b^3 - (a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*e^(10*d*x + 10 
*c) - (3*a^8 + 17*a^7*b + 33*a^6*b^2 + 27*a^5*b^3 + 8*a^4*b^4)*e^(8*d*x + 
8*c) - 2*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (172) = 344\).

Time = 0.86 (sec) , antiderivative size = 389, normalized size of antiderivative = 2.14 \[ \int \frac {\coth ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=-\frac {\frac {{\left (35 \, a^{2} b^{2} + 28 \, a b^{3} + 8 \, b^{4}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{{\left (a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right )} \sqrt {-a b - b^{2}}} - \frac {2 \, {\left (13 \, a^{3} b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 36 \, a^{2} b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 16 \, a b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 39 \, a^{3} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 122 \, a^{2} b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 152 \, a b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 48 \, b^{5} e^{\left (4 \, d x + 4 \, c\right )} + 39 \, a^{3} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 92 \, a^{2} b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 32 \, a b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 13 \, a^{3} b^{2} + 6 \, a^{2} b^{3}\right )}}{{\left (a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}^{2}} - \frac {8 \, {\left (d x + c\right )}}{a^{3}} + \frac {16}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}}}{8 \, d} \] Input:

integrate(coth(d*x+c)^2/(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")
 

Output:

-1/8*((35*a^2*b^2 + 28*a*b^3 + 8*b^4)*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 
2*b)/sqrt(-a*b - b^2))/((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*sqrt(-a*b - 
b^2)) - 2*(13*a^3*b^2*e^(6*d*x + 6*c) + 36*a^2*b^3*e^(6*d*x + 6*c) + 16*a* 
b^4*e^(6*d*x + 6*c) + 39*a^3*b^2*e^(4*d*x + 4*c) + 122*a^2*b^3*e^(4*d*x + 
4*c) + 152*a*b^4*e^(4*d*x + 4*c) + 48*b^5*e^(4*d*x + 4*c) + 39*a^3*b^2*e^( 
2*d*x + 2*c) + 92*a^2*b^3*e^(2*d*x + 2*c) + 32*a*b^4*e^(2*d*x + 2*c) + 13* 
a^3*b^2 + 6*a^2*b^3)/((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*(a*e^(4*d*x + 
4*c) + 2*a*e^(2*d*x + 2*c) + 4*b*e^(2*d*x + 2*c) + a)^2) - 8*(d*x + c)/a^3 
 + 16/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*(e^(2*d*x + 2*c) - 1)))/d
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\coth ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6\,{\mathrm {coth}\left (c+d\,x\right )}^2}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \] Input:

int(coth(c + d*x)^2/(a + b/cosh(c + d*x)^2)^3,x)
 

Output:

int((cosh(c + d*x)^6*coth(c + d*x)^2)/(b + a*cosh(c + d*x)^2)^3, x)
 

Reduce [B] (verification not implemented)

Time = 0.62 (sec) , antiderivative size = 6415, normalized size of antiderivative = 35.25 \[ \int \frac {\coth ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:

int(coth(d*x+c)^2/(a+b*sech(d*x+c)^2)^3,x)
 

Output:

( - 105*e**(10*c + 10*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**5*b - 364*e**(10*c + 10*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**4*b**2 - 248*e**(10*c + 10*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**3 
 - 64*e**(10*c + 10*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**4 - 105*e**(10*c + 10*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**5*b - 364*e**(10*c + 10*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**4*b**2 - 248*e 
**(10*c + 10*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**3 - 64*e**(10*c + 10*d*x)*sqrt(b)*sq 
rt(a + b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a) 
)*a**2*b**4 + 105*e**(10*c + 10*d*x)*sqrt(b)*sqrt(a + b)*log(2*sqrt(b)*sqr 
t(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*a**5*b + 364*e**(10*c + 10*d*x)*s 
qrt(b)*sqrt(a + b)*log(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2* 
b)*a**4*b**2 + 248*e**(10*c + 10*d*x)*sqrt(b)*sqrt(a + b)*log(2*sqrt(b)*sq 
rt(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*a**3*b**3 + 64*e**(10*c + 10*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**4 - 315*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log( - sqrt(...