\(\int \frac {\coth ^4(c+d x)}{(a+b \text {sech}^2(c+d x))^2} \, dx\) [157]

Optimal result

Integrand size = 23, antiderivative size = 161 \[ \int \frac {\coth ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {x}{a^2}-\frac {b^{5/2} (7 a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{2 a^2 (a+b)^{7/2} d}-\frac {\left (2 a^2+6 a b-b^2\right ) \coth (c+d x)}{2 a (a+b)^3 d}-\frac {(2 a-3 b) \coth ^3(c+d x)}{6 a (a+b)^2 d}-\frac {b \coth ^3(c+d x)}{2 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )} \] Output:

x/a^2-1/2*b^(5/2)*(7*a+2*b)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^(1/2))/a^2/( 
a+b)^(7/2)/d-1/2*(2*a^2+6*a*b-b^2)*coth(d*x+c)/a/(a+b)^3/d-1/6*(2*a-3*b)*c 
oth(d*x+c)^3/a/(a+b)^2/d-1/2*b*coth(d*x+c)^3/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. \(350\) vs. \(2(161)=322\).

Time = 3.80 (sec) , antiderivative size = 350, normalized size of antiderivative = 2.17 \[ \int \frac {\coth ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^4(c+d x) \left (\frac {6 x (a+2 b+a \cosh (2 (c+d x)))}{a^2}-\frac {2 (a+2 b+a \cosh (2 (c+d x))) \coth (c) \text {csch}^2(c+d x)}{(a+b)^2 d}-\frac {3 b^3 (7 a+2 b) \text {arctanh}\left (\frac {\text {sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right ) (a+2 b+a \cosh (2 (c+d x))) (\cosh (2 c)-\sinh (2 c))}{a^2 (a+b)^{7/2} d \sqrt {b (\cosh (c)-\sinh (c))^4}}+\frac {4 (2 a+5 b) (a+2 b+a \cosh (2 (c+d x))) \text {csch}(c) \text {csch}(c+d x) \sinh (d x)}{(a+b)^3 d}+\frac {2 (a+2 b+a \cosh (2 (c+d x))) \text {csch}(c) \text {csch}^3(c+d x) \sinh (d x)}{(a+b)^2 d}+\frac {3 b^3 \text {sech}(2 c) ((a+2 b) \sinh (2 c)-a \sinh (2 d x))}{a^2 (a+b)^3 d}\right )}{24 \left (a+b \text {sech}^2(c+d x)\right )^2} \] Input:

Integrate[Coth[c + d*x]^4/(a + b*Sech[c + d*x]^2)^2,x]
 

Output:

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^4*((6*x*(a + 2*b + a*Cosh[2 
*(c + d*x)]))/a^2 - (2*(a + 2*b + a*Cosh[2*(c + d*x)])*Coth[c]*Csch[c + d* 
x]^2)/((a + b)^2*d) - (3*b^3*(7*a + 2*b)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - S 
inh[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)])*(Cosh[2*c] - Sin 
h[2*c]))/(a^2*(a + b)^(7/2)*d*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + (4*(2*a + 5 
*b)*(a + 2*b + a*Cosh[2*(c + d*x)])*Csch[c]*Csch[c + d*x]*Sinh[d*x])/((a + 
 b)^3*d) + (2*(a + 2*b + a*Cosh[2*(c + d*x)])*Csch[c]*Csch[c + d*x]^3*Sinh 
[d*x])/((a + b)^2*d) + (3*b^3*Sech[2*c]*((a + 2*b)*Sinh[2*c] - a*Sinh[2*d* 
x]))/(a^2*(a + b)^3*d)))/(24*(a + b*Sech[c + d*x]^2)^2)
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.13, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {3042, 4629, 2075, 374, 25, 445, 27, 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.

\(\displaystyle \int \frac {\coth ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\tan (i c+i d x)^4 \left (a+b \sec (i c+i d x)^2\right )^2}dx\)

\(\Big \downarrow \) 4629

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

\(\Big \downarrow \) 2075

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

\(\Big \downarrow \) 374

\(\displaystyle \frac {-\frac {\int -\frac {\coth ^4(c+d x) \left (5 b \tanh ^2(c+d x)+2 a-3 b\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{2 a (a+b)}-\frac {b \coth ^3(c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {\coth ^4(c+d x) \left (5 b \tanh ^2(c+d x)+2 a-3 b\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{2 a (a+b)}-\frac {b \coth ^3(c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{d}\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {\frac {-\frac {\int -\frac {3 \coth ^2(c+d x) \left (2 a^2+6 b a-b^2-(2 a-3 b) b \tanh ^2(c+d x)\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{3 (a+b)}-\frac {(2 a-3 b) \coth ^3(c+d x)}{3 (a+b)}}{2 a (a+b)}-\frac {b \coth ^3(c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {\coth ^2(c+d x) \left (2 a^2+6 b a-b^2-(2 a-3 b) b \tanh ^2(c+d x)\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{a+b}-\frac {(2 a-3 b) \coth ^3(c+d x)}{3 (a+b)}}{2 a (a+b)}-\frac {b \coth ^3(c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{d}\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {\frac {\frac {-\frac {\int -\frac {2 a^3+8 b a^2+12 b^2 a+b^3-b \left (2 a^2+6 b a-b^2\right ) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{a+b}-\frac {\left (2 a^2+6 a b-b^2\right ) \coth (c+d x)}{a+b}}{a+b}-\frac {(2 a-3 b) \coth ^3(c+d x)}{3 (a+b)}}{2 a (a+b)}-\frac {b \coth ^3(c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\frac {\int \frac {2 a^3+8 b a^2+12 b^2 a+b^3-b \left (2 a^2+6 b a-b^2\right ) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{a+b}-\frac {\left (2 a^2+6 a b-b^2\right ) \coth (c+d x)}{a+b}}{a+b}-\frac {(2 a-3 b) \coth ^3(c+d x)}{3 (a+b)}}{2 a (a+b)}-\frac {b \coth ^3(c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{d}\)

\(\Big \downarrow \) 397

\(\displaystyle \frac {\frac {\frac {\frac {\frac {2 (a+b)^3 \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a}-\frac {b^3 (7 a+2 b) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}}{a+b}-\frac {\left (2 a^2+6 a b-b^2\right ) \coth (c+d x)}{a+b}}{a+b}-\frac {(2 a-3 b) \coth ^3(c+d x)}{3 (a+b)}}{2 a (a+b)}-\frac {b \coth ^3(c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\frac {\frac {2 (a+b)^3 \text {arctanh}(\tanh (c+d x))}{a}-\frac {b^3 (7 a+2 b) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}}{a+b}-\frac {\left (2 a^2+6 a b-b^2\right ) \coth (c+d x)}{a+b}}{a+b}-\frac {(2 a-3 b) \coth ^3(c+d x)}{3 (a+b)}}{2 a (a+b)}-\frac {b \coth ^3(c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {\frac {\frac {2 (a+b)^3 \text {arctanh}(\tanh (c+d x))}{a}-\frac {b^{5/2} (7 a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a \sqrt {a+b}}}{a+b}-\frac {\left (2 a^2+6 a b-b^2\right ) \coth (c+d x)}{a+b}}{a+b}-\frac {(2 a-3 b) \coth ^3(c+d x)}{3 (a+b)}}{2 a (a+b)}-\frac {b \coth ^3(c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{d}\)

Input:

Int[Coth[c + d*x]^4/(a + b*Sech[c + d*x]^2)^2,x]
 

Output:

((-1/3*((2*a - 3*b)*Coth[c + d*x]^3)/(a + b) + (((2*(a + b)^3*ArcTanh[Tanh 
[c + d*x]])/a - (b^(5/2)*(7*a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[ 
a + b]])/(a*Sqrt[a + b]))/(a + b) - ((2*a^2 + 6*a*b - b^2)*Coth[c + d*x])/ 
(a + b))/(a + b))/(2*a*(a + b)) - (b*Coth[c + d*x]^3)/(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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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. \(361\) vs. \(2(145)=290\).

Time = 50.69 (sec) , antiderivative size = 362, normalized size of antiderivative = 2.25

method	result	size
derivativedivides	\(\frac {-\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a}{3}+\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+5 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+13 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 \left (a^{2}+2 a b +b^{2}\right ) \left (a +b \right )}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2}}+\frac {2 b^{3} \left (\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a}{2}-\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {\left (7 a +2 b \right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{2}\right )}{a^{2} \left (a +b \right )^{3}}-\frac {1}{24 \left (a +b \right )^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {5 a +13 b}{8 \left (a +b \right )^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}}{d}\)	\(362\)
default	\(\frac {-\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a}{3}+\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+5 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+13 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 \left (a^{2}+2 a b +b^{2}\right ) \left (a +b \right )}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2}}+\frac {2 b^{3} \left (\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a}{2}-\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {\left (7 a +2 b \right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{2}\right )}{a^{2} \left (a +b \right )^{3}}-\frac {1}{24 \left (a +b \right )^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {5 a +13 b}{8 \left (a +b \right )^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}}{d}\)	\(362\)
risch	\(\frac {x}{a^{2}}-\frac {12 a^{4} {\mathrm e}^{8 d x +8 c}+24 a^{3} b \,{\mathrm e}^{8 d x +8 c}-3 a \,b^{3} {\mathrm e}^{8 d x +8 c}-6 b^{4} {\mathrm e}^{8 d x +8 c}+12 \,{\mathrm e}^{6 d x +6 c} a^{4}+60 a^{3} b \,{\mathrm e}^{6 d x +6 c}+96 a^{2} b^{2} {\mathrm e}^{6 d x +6 c}+6 a \,b^{3} {\mathrm e}^{6 d x +6 c}+18 b^{4} {\mathrm e}^{6 d x +6 c}-4 a^{4} {\mathrm e}^{4 d x +4 c}-76 a^{3} b \,{\mathrm e}^{4 d x +4 c}-144 \,{\mathrm e}^{4 d x +4 c} a^{2} b^{2}-18 b^{4} {\mathrm e}^{4 d x +4 c}+4 a^{4} {\mathrm e}^{2 d x +2 c}+36 a^{3} b \,{\mathrm e}^{2 d x +2 c}+80 \,{\mathrm e}^{2 d x +2 c} a^{2} b^{2}-6 a \,b^{3} {\mathrm e}^{2 d x +2 c}+6 b^{4} {\mathrm e}^{2 d x +2 c}+8 a^{4}+20 a^{3} b +3 a \,b^{3}}{3 a^{2} d \left (a +b \right )^{3} \left ({\mathrm e}^{4 d x +4 c} a +2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right ) \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}+\frac {7 \sqrt {b \left (a +b \right )}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a +2 \sqrt {b \left (a +b \right )}+2 b}{a}\right )}{4 \left (a +b \right )^{4} d a}+\frac {\sqrt {b \left (a +b \right )}\, b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a +2 \sqrt {b \left (a +b \right )}+2 b}{a}\right )}{2 \left (a +b \right )^{4} d \,a^{2}}-\frac {7 \sqrt {b \left (a +b \right )}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-a +2 \sqrt {b \left (a +b \right )}-2 b}{a}\right )}{4 \left (a +b \right )^{4} d a}-\frac {\sqrt {b \left (a +b \right )}\, b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-a +2 \sqrt {b \left (a +b \right )}-2 b}{a}\right )}{2 \left (a +b \right )^{4} d \,a^{2}}\)	\(572\)

Input:

int(coth(d*x+c)^4/(a+b*sech(d*x+c)^2)^2,x,method=_RETURNVERBOSE)
 

Output:

1/d*(-1/8/(a^2+2*a*b+b^2)/(a+b)*(1/3*tanh(1/2*d*x+1/2*c)^3*a+1/3*b*tanh(1/ 
2*d*x+1/2*c)^3+5*a*tanh(1/2*d*x+1/2*c)+13*b*tanh(1/2*d*x+1/2*c))+1/a^2*ln( 
tanh(1/2*d*x+1/2*c)+1)+2*b^3/a^2/(a+b)^3*((-1/2*tanh(1/2*d*x+1/2*c)^3*a-1/ 
2*a*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)+1/2*(7*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))))-1/24/(a+b)^2/ 
tanh(1/2*d*x+1/2*c)^3-1/8*(5*a+13*b)/(a+b)^3/tanh(1/2*d*x+1/2*c)-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. 4786 vs. \(2 (148) = 296\).

Time = 0.46 (sec) , antiderivative size = 9849, normalized size of antiderivative = 61.17 \[ \int \frac {\coth ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(coth(d*x+c)^4/(a+b*sech(d*x+c)^2)^2,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

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

integrate(coth(d*x+c)**4/(a+b*sech(d*x+c)**2)**2,x)
 

Output:

Integral(coth(c + d*x)**4/(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. 2961 vs. \(2 (148) = 296\).

Time = 0.43 (sec) , antiderivative size = 2961, normalized size of antiderivative = 18.39 \[ \int \frac {\coth ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(coth(d*x+c)^4/(a+b*sech(d*x+c)^2)^2,x, algorithm="maxima")
 

Output:

1/4*(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^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d) - 1/2*b*log(a*e^(4*d* 
x + 4*c) + 2*(a + 2*b)*e^(2*d*x + 2*c) + a)/((a^3 + 3*a^2*b + 3*a*b^2 + b^ 
3)*d) - 1/4*(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^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d) + 1/2*b*log 
(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/((a^3 + 3*a^2*b + 
3*a*b^2 + b^3)*d) + 1/2*(a + 2*b)*log(e^(2*d*x + 2*c) - 1)/((a^3 + 3*a^2*b 
 + 3*a*b^2 + b^3)*d) + b*log(e^(2*d*x + 2*c) - 1)/((a^3 + 3*a^2*b + 3*a*b^ 
2 + b^3)*d) - 1/2*(a + 2*b)*log(e^(-2*d*x - 2*c) - 1)/((a^3 + 3*a^2*b + 3* 
a*b^2 + b^3)*d) - b*log(e^(-2*d*x - 2*c) - 1)/((a^3 + 3*a^2*b + 3*a*b^2 + 
b^3)*d) - 1/64*(3*a^3*b + 38*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*sqr 
t((a + b)*b)))/((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*sqrt((a + b)*b)*d) + 
 1/16*(3*a*b + 8*b^2)*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) + 1/64*(3*a^3*b + 38*a^2*b^2 + 56*a*b^3 + 1 
6*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^5 + 3*a^4*b + 3*a^3*b^2 + a^2* 
b^3)*sqrt((a + b)*b)*d) - 1/16*(3*a*b + 8*b^2)*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 +...
 

Giac [A] (verification not implemented)

Time = 1.11 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.82 \[ \int \frac {\coth ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=-\frac {\frac {3 \, {\left (7 \, a b^{3} + 2 \, b^{4}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{{\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} \sqrt {-a b - b^{2}}} - \frac {6 \, {\left (a b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, b^{4} e^{\left (2 \, d x + 2 \, c\right )} + a b^{3}\right )}}{{\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}} - \frac {6 \, {\left (d x + c\right )}}{a^{2}} + \frac {8 \, {\left (3 \, a e^{\left (4 \, d x + 4 \, c\right )} + 6 \, b e^{\left (4 \, d x + 4 \, c\right )} - 3 \, a e^{\left (2 \, d x + 2 \, c\right )} - 9 \, b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a + 5 \, b\right )}}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{6 \, d} \] Input:

integrate(coth(d*x+c)^4/(a+b*sech(d*x+c)^2)^2,x, algorithm="giac")
 

Output:

-1/6*(3*(7*a*b^3 + 2*b^4)*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a 
*b - b^2))/((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*sqrt(-a*b - b^2)) - 6*(a 
*b^3*e^(2*d*x + 2*c) + 2*b^4*e^(2*d*x + 2*c) + a*b^3)/((a^5 + 3*a^4*b + 3* 
a^3*b^2 + a^2*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)) - 6*(d*x + c)/a^2 + 8*(3*a*e^(4*d*x + 4*c) + 6*b*e^(4*d*x + 
4*c) - 3*a*e^(2*d*x + 2*c) - 9*b*e^(2*d*x + 2*c) + 2*a + 5*b)/((a^3 + 3*a^ 
2*b + 3*a*b^2 + b^3)*(e^(2*d*x + 2*c) - 1)^3))/d
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 9.58 (sec) , antiderivative size = 4961, normalized size of antiderivative = 30.81 \[ \int \frac {\coth ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 21*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**2 + 78*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**3 + 24*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*b**4 - 
21*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**2 + 78*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)*sqr 
t(a))*a**2*b**3 + 24*e**(10*c + 10*d*x)*sqrt(b)*sqrt(a + b)*log(sqrt(2*sqr 
t(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a*b**4 + 21*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**3*b**2 - 78*e**(10*c + 10*d*x)*sqrt(b)*sqrt(a + b)*log(2*s 
qrt(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*a**2*b**3 - 24*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*b**4 + 21*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log( - sqr 
t(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**3*b**2 - 162 
*e**(8*c + 8*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**3 + 288*e**(8*c + 8*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))*a*b**4 + 96*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log( - sqrt(2*s...