\(\int \coth ^5(c+d x) (a+b \text {sech}^2(c+d x))^3 \, dx\) [133]

Optimal result

Integrand size = 23, antiderivative size = 81 \[ \int \coth ^5(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=-\frac {(2 a-b) (a+b)^2 \text {csch}^2(c+d x)}{2 d}-\frac {(a+b)^3 \text {csch}^4(c+d x)}{4 d}-\frac {b^3 \log (\cosh (c+d x))}{d}+\frac {\left (a^3+b^3\right ) \log (\sinh (c+d x))}{d} \] Output:

-1/2*(2*a-b)*(a+b)^2*csch(d*x+c)^2/d-1/4*(a+b)^3*csch(d*x+c)^4/d-b^3*ln(co 
sh(d*x+c))/d+(a^3+b^3)*ln(sinh(d*x+c))/d
 

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.25 \[ \int \coth ^5(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=-\frac {2 \left (b+a \cosh ^2(c+d x)\right )^3 \left (2 (2 a-b) (a+b)^2 \text {csch}^2(c+d x)+(a+b)^3 \text {csch}^4(c+d x)+4 b^3 \log (\cosh (c+d x))-4 \left (a^3+b^3\right ) \log (\sinh (c+d x))\right )}{d (a+2 b+a \cosh (2 (c+d x)))^3} \] Input:

Integrate[Coth[c + d*x]^5*(a + b*Sech[c + d*x]^2)^3,x]
 

Output:

(-2*(b + a*Cosh[c + d*x]^2)^3*(2*(2*a - b)*(a + b)^2*Csch[c + d*x]^2 + (a 
+ b)^3*Csch[c + d*x]^4 + 4*b^3*Log[Cosh[c + d*x]] - 4*(a^3 + b^3)*Log[Sinh 
[c + d*x]]))/(d*(a + 2*b + a*Cosh[2*(c + d*x)])^3)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 26, 4626, 354, 99, 2009}

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.

\(\displaystyle \int \coth ^5(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {i \left (a+b \sec (i c+i d x)^2\right )^3}{\tan (i c+i d x)^5}dx\)

\(\Big \downarrow \) 26

\(\displaystyle i \int \frac {\left (b \sec (i c+i d x)^2+a\right )^3}{\tan (i c+i d x)^5}dx\)

\(\Big \downarrow \) 4626

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 99

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

\(\Big \downarrow \) 2009

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

Input:

Int[Coth[c + d*x]^5*(a + b*Sech[c + d*x]^2)^3,x]
 

Output:

-1/2*((a + b)^3/(2*(1 - Cosh[c + d*x]^2)^2) - ((2*a - b)*(a + b)^2)/(1 - C 
osh[c + d*x]^2) + b^3*Log[Cosh[c + d*x]^2] - (a^3 + b^3)*Log[1 - Cosh[c + 
d*x]^2])/d
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | 
| (GtQ[m, 0] && GeQ[n, -1]))
 

rule 354

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 4626

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

Maple [A] (verified)

Time = 87.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.47

method	result	size
derivativedivides	\(\frac {a^{3} \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\coth \left (d x +c \right )^{2}}{2}-\frac {\coth \left (d x +c \right )^{4}}{4}\right )+3 a^{2} b \left (-\frac {\cosh \left (d x +c \right )^{2}}{2 \sinh \left (d x +c \right )^{4}}+\frac {1}{4 \sinh \left (d x +c \right )^{4}}\right )-\frac {3 a \,b^{2}}{4 \sinh \left (d x +c \right )^{4}}+b^{3} \left (-\frac {1}{4 \sinh \left (d x +c \right )^{4}}+\frac {1}{2 \sinh \left (d x +c \right )^{2}}+\ln \left (\tanh \left (d x +c \right )\right )\right )}{d}\)	\(119\)
default	\(\frac {a^{3} \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\coth \left (d x +c \right )^{2}}{2}-\frac {\coth \left (d x +c \right )^{4}}{4}\right )+3 a^{2} b \left (-\frac {\cosh \left (d x +c \right )^{2}}{2 \sinh \left (d x +c \right )^{4}}+\frac {1}{4 \sinh \left (d x +c \right )^{4}}\right )-\frac {3 a \,b^{2}}{4 \sinh \left (d x +c \right )^{4}}+b^{3} \left (-\frac {1}{4 \sinh \left (d x +c \right )^{4}}+\frac {1}{2 \sinh \left (d x +c \right )^{2}}+\ln \left (\tanh \left (d x +c \right )\right )\right )}{d}\)	\(119\)
risch	\(-a^{3} x -\frac {2 a^{3} c}{d}-\frac {2 \,{\mathrm e}^{2 d x +2 c} \left (2 a^{3} {\mathrm e}^{4 d x +4 c}+3 a^{2} b \,{\mathrm e}^{4 d x +4 c}-b^{3} {\mathrm e}^{4 d x +4 c}-2 a^{3} {\mathrm e}^{2 d x +2 c}+6 a \,b^{2} {\mathrm e}^{2 d x +2 c}+4 b^{3} {\mathrm e}^{2 d x +2 c}+2 a^{3}+3 a^{2} b -b^{3}\right )}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{4}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) a^{3}}{d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) b^{3}}{d}-\frac {b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+1\right )}{d}\)	\(205\)

Input:

int(coth(d*x+c)^5*(a+b*sech(d*x+c)^2)^3,x,method=_RETURNVERBOSE)
 

Output:

1/d*(a^3*(ln(sinh(d*x+c))-1/2*coth(d*x+c)^2-1/4*coth(d*x+c)^4)+3*a^2*b*(-1 
/2/sinh(d*x+c)^4*cosh(d*x+c)^2+1/4/sinh(d*x+c)^4)-3/4*a*b^2/sinh(d*x+c)^4+ 
b^3*(-1/4/sinh(d*x+c)^4+1/2/sinh(d*x+c)^2+ln(tanh(d*x+c))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1830 vs. \(2 (77) = 154\).

Time = 0.30 (sec) , antiderivative size = 1830, normalized size of antiderivative = 22.59 \[ \int \coth ^5(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\text {Too large to display} \] Input:

integrate(coth(d*x+c)^5*(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")
 

Output:

-(a^3*d*x*cosh(d*x + c)^8 + 8*a^3*d*x*cosh(d*x + c)*sinh(d*x + c)^7 + a^3* 
d*x*sinh(d*x + c)^8 - 2*(2*a^3*d*x - 2*a^3 - 3*a^2*b + b^3)*cosh(d*x + c)^ 
6 + 2*(14*a^3*d*x*cosh(d*x + c)^2 - 2*a^3*d*x + 2*a^3 + 3*a^2*b - b^3)*sin 
h(d*x + c)^6 + 4*(14*a^3*d*x*cosh(d*x + c)^3 - 3*(2*a^3*d*x - 2*a^3 - 3*a^ 
2*b + b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + a^3*d*x + 2*(3*a^3*d*x - 2*a^3 
 + 6*a*b^2 + 4*b^3)*cosh(d*x + c)^4 + 2*(35*a^3*d*x*cosh(d*x + c)^4 + 3*a^ 
3*d*x - 2*a^3 + 6*a*b^2 + 4*b^3 - 15*(2*a^3*d*x - 2*a^3 - 3*a^2*b + b^3)*c 
osh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*a^3*d*x*cosh(d*x + c)^5 - 5*(2*a^3* 
d*x - 2*a^3 - 3*a^2*b + b^3)*cosh(d*x + c)^3 + (3*a^3*d*x - 2*a^3 + 6*a*b^ 
2 + 4*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - 2*(2*a^3*d*x - 2*a^3 - 3*a^2*b 
 + b^3)*cosh(d*x + c)^2 + 2*(14*a^3*d*x*cosh(d*x + c)^6 - 2*a^3*d*x - 15*( 
2*a^3*d*x - 2*a^3 - 3*a^2*b + b^3)*cosh(d*x + c)^4 + 2*a^3 + 3*a^2*b - b^3 
 + 6*(3*a^3*d*x - 2*a^3 + 6*a*b^2 + 4*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^ 
2 + (b^3*cosh(d*x + c)^8 + 8*b^3*cosh(d*x + c)*sinh(d*x + c)^7 + b^3*sinh( 
d*x + c)^8 - 4*b^3*cosh(d*x + c)^6 + 6*b^3*cosh(d*x + c)^4 + 4*(7*b^3*cosh 
(d*x + c)^2 - b^3)*sinh(d*x + c)^6 + 8*(7*b^3*cosh(d*x + c)^3 - 3*b^3*cosh 
(d*x + c))*sinh(d*x + c)^5 - 4*b^3*cosh(d*x + c)^2 + 2*(35*b^3*cosh(d*x + 
c)^4 - 30*b^3*cosh(d*x + c)^2 + 3*b^3)*sinh(d*x + c)^4 + 8*(7*b^3*cosh(d*x 
 + c)^5 - 10*b^3*cosh(d*x + c)^3 + 3*b^3*cosh(d*x + c))*sinh(d*x + c)^3 + 
b^3 + 4*(7*b^3*cosh(d*x + c)^6 - 15*b^3*cosh(d*x + c)^4 + 9*b^3*cosh(d*...
 

Sympy [F(-1)]

Timed out. \[ \int \coth ^5(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\text {Timed out} \] Input:

integrate(coth(d*x+c)**5*(a+b*sech(d*x+c)**2)**3,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (77) = 154\).

Time = 0.13 (sec) , antiderivative size = 422, normalized size of antiderivative = 5.21 \[ \int \coth ^5(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=a^{3} {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {4 \, {\left (e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} + b^{3} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} - \frac {2 \, {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 4 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} + 6 \, a^{2} b {\left (\frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}} + \frac {e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} - \frac {12 \, a b^{2}}{d {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{4}} \] Input:

integrate(coth(d*x+c)^5*(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

a^3*(x + c/d + log(e^(-d*x - c) + 1)/d + log(e^(-d*x - c) - 1)/d + 4*(e^(- 
2*d*x - 2*c) - e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c))/(d*(4*e^(-2*d*x - 2*c) 
 - 6*e^(-4*d*x - 4*c) + 4*e^(-6*d*x - 6*c) - e^(-8*d*x - 8*c) - 1))) + b^3 
*(log(e^(-d*x - c) + 1)/d + log(e^(-d*x - c) - 1)/d - log(e^(-2*d*x - 2*c) 
 + 1)/d - 2*(e^(-2*d*x - 2*c) - 4*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c))/(d* 
(4*e^(-2*d*x - 2*c) - 6*e^(-4*d*x - 4*c) + 4*e^(-6*d*x - 6*c) - e^(-8*d*x 
- 8*c) - 1))) + 6*a^2*b*(e^(-2*d*x - 2*c)/(d*(4*e^(-2*d*x - 2*c) - 6*e^(-4 
*d*x - 4*c) + 4*e^(-6*d*x - 6*c) - e^(-8*d*x - 8*c) - 1)) + e^(-6*d*x - 6* 
c)/(d*(4*e^(-2*d*x - 2*c) - 6*e^(-4*d*x - 4*c) + 4*e^(-6*d*x - 6*c) - e^(- 
8*d*x - 8*c) - 1))) - 12*a*b^2/(d*(e^(d*x + c) - e^(-d*x - c))^4)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (77) = 154\).

Time = 0.22 (sec) , antiderivative size = 227, normalized size of antiderivative = 2.80 \[ \int \coth ^5(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=-\frac {2 \, b^{3} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} + 2\right ) - 2 \, {\left (a^{3} + b^{3}\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} - 2\right ) + \frac {3 \, a^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} + 3 \, b^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} + 4 \, a^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 24 \, a^{2} b {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} - 20 \, b^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} - 4 \, a^{3} + 48 \, a b^{2} + 44 \, b^{3}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} - 2\right )}^{2}}}{4 \, d} \] Input:

integrate(coth(d*x+c)^5*(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")
 

Output:

-1/4*(2*b^3*log(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c) + 2) - 2*(a^3 + b^3)*lo 
g(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c) - 2) + (3*a^3*(e^(2*d*x + 2*c) + e^(- 
2*d*x - 2*c))^2 + 3*b^3*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c))^2 + 4*a^3*(e^ 
(2*d*x + 2*c) + e^(-2*d*x - 2*c)) + 24*a^2*b*(e^(2*d*x + 2*c) + e^(-2*d*x 
- 2*c)) - 20*b^3*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c)) - 4*a^3 + 48*a*b^2 + 
 44*b^3)/(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c) - 2)^2)/d
 

Mupad [B] (verification not implemented)

Time = 2.51 (sec) , antiderivative size = 384, normalized size of antiderivative = 4.74 \[ \int \coth ^5(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=-a^3\,x-\frac {2\,\left (4\,a^3+9\,a^2\,b+6\,a\,b^2+b^3\right )}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {\ln \left ({\mathrm {e}}^{4\,c+4\,d\,x}-1\right )\,\left (b^3\,d-d\,\left (a^3+b^3\right )\right )}{2\,d^2}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\left (a^6\,\sqrt {-d^2}+4\,b^6\,\sqrt {-d^2}+4\,a^3\,b^3\,\sqrt {-d^2}\right )}{a^3\,d\,\sqrt {a^6+4\,a^3\,b^3+4\,b^6}+2\,b^3\,d\,\sqrt {a^6+4\,a^3\,b^3+4\,b^6}}\right )\,\sqrt {a^6+4\,a^3\,b^3+4\,b^6}}{\sqrt {-d^2}}-\frac {2\,\left (2\,a^3+3\,a^2\,b-b^3\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {8\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}{d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {4\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}{d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )} \] Input:

int(coth(c + d*x)^5*(a + b/cosh(c + d*x)^2)^3,x)
 

Output:

- a^3*x - (2*(6*a*b^2 + 9*a^2*b + 4*a^3 + b^3))/(d*(exp(4*c + 4*d*x) - 2*e 
xp(2*c + 2*d*x) + 1)) - (log(exp(4*c + 4*d*x) - 1)*(b^3*d - d*(a^3 + b^3)) 
)/(2*d^2) - (atan((exp(2*c)*exp(2*d*x)*(a^6*(-d^2)^(1/2) + 4*b^6*(-d^2)^(1 
/2) + 4*a^3*b^3*(-d^2)^(1/2)))/(a^3*d*(a^6 + 4*b^6 + 4*a^3*b^3)^(1/2) + 2* 
b^3*d*(a^6 + 4*b^6 + 4*a^3*b^3)^(1/2)))*(a^6 + 4*b^6 + 4*a^3*b^3)^(1/2))/( 
-d^2)^(1/2) - (2*(3*a^2*b + 2*a^3 - b^3))/(d*(exp(2*c + 2*d*x) - 1)) - (8* 
(3*a*b^2 + 3*a^2*b + a^3 + b^3))/(d*(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d* 
x) + exp(6*c + 6*d*x) - 1)) - (4*(3*a*b^2 + 3*a^2*b + a^3 + b^3))/(d*(6*ex 
p(4*c + 4*d*x) - 4*exp(2*c + 2*d*x) - 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x 
) + 1))
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 842, normalized size of antiderivative = 10.40 \[ \int \coth ^5(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {2 e^{8 d x +8 c} \mathrm {log}\left (e^{d x +c}-1\right ) a^{3}+2 e^{8 d x +8 c} \mathrm {log}\left (e^{d x +c}-1\right ) b^{3}+2 e^{8 d x +8 c} \mathrm {log}\left (e^{d x +c}+1\right ) a^{3}+2 e^{8 d x +8 c} \mathrm {log}\left (e^{d x +c}+1\right ) b^{3}-8 e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}-1\right ) a^{3}-8 e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}-1\right ) b^{3}-8 e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}+1\right ) a^{3}-8 e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}+1\right ) b^{3}+12 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}-1\right ) a^{3}+12 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}-1\right ) b^{3}+12 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}+1\right ) a^{3}+12 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}+1\right ) b^{3}-8 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}-1\right ) a^{3}-8 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}-1\right ) b^{3}-8 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}+1\right ) a^{3}-8 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}+1\right ) b^{3}-24 e^{4 d x +4 c} a \,b^{2}+b^{3}-2 \,\mathrm {log}\left (e^{2 d x +2 c}+1\right ) b^{3}-3 e^{8 d x +8 c} a^{2} b -18 e^{4 d x +4 c} a^{2} b -10 e^{4 d x +4 c} b^{3}-2 e^{8 d x +8 c} a^{3} d x +8 e^{6 d x +6 c} a^{3} d x -2 a^{3}-2 e^{8 d x +8 c} \mathrm {log}\left (e^{2 d x +2 c}+1\right ) b^{3}+8 e^{6 d x +6 c} \mathrm {log}\left (e^{2 d x +2 c}+1\right ) b^{3}-12 e^{4 d x +4 c} \mathrm {log}\left (e^{2 d x +2 c}+1\right ) b^{3}+8 e^{2 d x +2 c} \mathrm {log}\left (e^{2 d x +2 c}+1\right ) b^{3}-3 a^{2} b -2 e^{8 d x +8 c} a^{3}+e^{8 d x +8 c} b^{3}-4 e^{4 d x +4 c} a^{3}+2 \,\mathrm {log}\left (e^{d x +c}-1\right ) a^{3}+2 \,\mathrm {log}\left (e^{d x +c}-1\right ) b^{3}+2 \,\mathrm {log}\left (e^{d x +c}+1\right ) a^{3}+2 \,\mathrm {log}\left (e^{d x +c}+1\right ) b^{3}-12 e^{4 d x +4 c} a^{3} d x -2 a^{3} d x +8 e^{2 d x +2 c} a^{3} d x}{2 d \left (e^{8 d x +8 c}-4 e^{6 d x +6 c}+6 e^{4 d x +4 c}-4 e^{2 d x +2 c}+1\right )} \] Input:

int(coth(d*x+c)^5*(a+b*sech(d*x+c)^2)^3,x)
 

Output:

( - 2*e**(8*c + 8*d*x)*log(e**(2*c + 2*d*x) + 1)*b**3 + 2*e**(8*c + 8*d*x) 
*log(e**(c + d*x) - 1)*a**3 + 2*e**(8*c + 8*d*x)*log(e**(c + d*x) - 1)*b** 
3 + 2*e**(8*c + 8*d*x)*log(e**(c + d*x) + 1)*a**3 + 2*e**(8*c + 8*d*x)*log 
(e**(c + d*x) + 1)*b**3 - 2*e**(8*c + 8*d*x)*a**3*d*x - 2*e**(8*c + 8*d*x) 
*a**3 - 3*e**(8*c + 8*d*x)*a**2*b + e**(8*c + 8*d*x)*b**3 + 8*e**(6*c + 6* 
d*x)*log(e**(2*c + 2*d*x) + 1)*b**3 - 8*e**(6*c + 6*d*x)*log(e**(c + d*x) 
- 1)*a**3 - 8*e**(6*c + 6*d*x)*log(e**(c + d*x) - 1)*b**3 - 8*e**(6*c + 6* 
d*x)*log(e**(c + d*x) + 1)*a**3 - 8*e**(6*c + 6*d*x)*log(e**(c + d*x) + 1) 
*b**3 + 8*e**(6*c + 6*d*x)*a**3*d*x - 12*e**(4*c + 4*d*x)*log(e**(2*c + 2* 
d*x) + 1)*b**3 + 12*e**(4*c + 4*d*x)*log(e**(c + d*x) - 1)*a**3 + 12*e**(4 
*c + 4*d*x)*log(e**(c + d*x) - 1)*b**3 + 12*e**(4*c + 4*d*x)*log(e**(c + d 
*x) + 1)*a**3 + 12*e**(4*c + 4*d*x)*log(e**(c + d*x) + 1)*b**3 - 12*e**(4* 
c + 4*d*x)*a**3*d*x - 4*e**(4*c + 4*d*x)*a**3 - 18*e**(4*c + 4*d*x)*a**2*b 
 - 24*e**(4*c + 4*d*x)*a*b**2 - 10*e**(4*c + 4*d*x)*b**3 + 8*e**(2*c + 2*d 
*x)*log(e**(2*c + 2*d*x) + 1)*b**3 - 8*e**(2*c + 2*d*x)*log(e**(c + d*x) - 
 1)*a**3 - 8*e**(2*c + 2*d*x)*log(e**(c + d*x) - 1)*b**3 - 8*e**(2*c + 2*d 
*x)*log(e**(c + d*x) + 1)*a**3 - 8*e**(2*c + 2*d*x)*log(e**(c + d*x) + 1)* 
b**3 + 8*e**(2*c + 2*d*x)*a**3*d*x - 2*log(e**(2*c + 2*d*x) + 1)*b**3 + 2* 
log(e**(c + d*x) - 1)*a**3 + 2*log(e**(c + d*x) - 1)*b**3 + 2*log(e**(c + 
d*x) + 1)*a**3 + 2*log(e**(c + d*x) + 1)*b**3 - 2*a**3*d*x - 2*a**3 - 3...