\(\int \coth (c+d x) (a+b \text {sech}^2(c+d x))^2 \, dx\) [117]

Optimal result

Integrand size = 21, antiderivative size = 53 \[ \int \coth (c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=-\frac {b (2 a+b) \log (\cosh (c+d x))}{d}+\frac {(a+b)^2 \log (\sinh (c+d x))}{d}+\frac {b^2 \text {sech}^2(c+d x)}{2 d} \] Output:

-b*(2*a+b)*ln(cosh(d*x+c))/d+(a+b)^2*ln(sinh(d*x+c))/d+1/2*b^2*sech(d*x+c) 
^2/d
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.58 \[ \int \coth (c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\frac {2 \left (b^2+2 \cosh ^2(c+d x) \left (-b (2 a+b) \log (\cosh (c+d x))+(a+b)^2 \log (\sinh (c+d x))\right )\right ) (a \cosh (c+d x)+b \text {sech}(c+d x))^2}{d (a+2 b+a \cosh (2 (c+d x)))^2} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.29 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4626

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 99

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/2*(b*(2*a + b)*Log[Cosh[c + d*x]^2] - (a + b)^2*Log[1 - Cosh[c + d*x]^2 
] - b^2*Sech[c + d*x])/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 = 6.62 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.94

method	result	size
derivativedivides	\(\frac {a^{2} \ln \left (\sinh \left (d x +c \right )\right )+2 a b \ln \left (\tanh \left (d x +c \right )\right )+b^{2} \left (\frac {1}{2 \cosh \left (d x +c \right )^{2}}+\ln \left (\tanh \left (d x +c \right )\right )\right )}{d}\)	\(50\)
default	\(\frac {a^{2} \ln \left (\sinh \left (d x +c \right )\right )+2 a b \ln \left (\tanh \left (d x +c \right )\right )+b^{2} \left (\frac {1}{2 \cosh \left (d x +c \right )^{2}}+\ln \left (\tanh \left (d x +c \right )\right )\right )}{d}\)	\(50\)
risch	\(-a^{2} x -\frac {2 a^{2} c}{d}+\frac {2 \,{\mathrm e}^{2 d x +2 c} b^{2}}{d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) a^{2}}{d}+\frac {2 a b \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) b^{2}}{d}-\frac {2 b \ln \left ({\mathrm e}^{2 d x +2 c}+1\right ) a}{d}-\frac {b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+1\right )}{d}\)	\(143\)

Input:

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

Output:

1/d*(a^2*ln(sinh(d*x+c))+2*a*b*ln(tanh(d*x+c))+b^2*(1/2/cosh(d*x+c)^2+ln(t 
anh(d*x+c))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 665 vs. \(2 (51) = 102\).

Time = 0.36 (sec) , antiderivative size = 665, normalized size of antiderivative = 12.55 \[ \int \coth (c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

-(a^2*d*x*cosh(d*x + c)^4 + 4*a^2*d*x*cosh(d*x + c)*sinh(d*x + c)^3 + a^2* 
d*x*sinh(d*x + c)^4 + a^2*d*x + 2*(a^2*d*x - b^2)*cosh(d*x + c)^2 + 2*(3*a 
^2*d*x*cosh(d*x + c)^2 + a^2*d*x - b^2)*sinh(d*x + c)^2 + ((2*a*b + b^2)*c 
osh(d*x + c)^4 + 4*(2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a*b + 
b^2)*sinh(d*x + c)^4 + 2*(2*a*b + b^2)*cosh(d*x + c)^2 + 2*(3*(2*a*b + b^2 
)*cosh(d*x + c)^2 + 2*a*b + b^2)*sinh(d*x + c)^2 + 2*a*b + b^2 + 4*((2*a*b 
 + b^2)*cosh(d*x + c)^3 + (2*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c))*log( 
2*cosh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) - ((a^2 + 2*a*b + b^2)*co 
sh(d*x + c)^4 + 4*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 
 + 2*a*b + b^2)*sinh(d*x + c)^4 + 2*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + 
2*(3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 + 2*a*b + b^2)*sinh(d*x + c 
)^2 + a^2 + 2*a*b + b^2 + 4*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + (a^2 + 
2*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c))*log(2*sinh(d*x + c)/(cosh(d*x + 
 c) - sinh(d*x + c))) + 4*(a^2*d*x*cosh(d*x + c)^3 + (a^2*d*x - b^2)*cosh( 
d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^4 + 4*d*cosh(d*x + c)*sinh(d*x + 
 c)^3 + d*sinh(d*x + c)^4 + 2*d*cosh(d*x + c)^2 + 2*(3*d*cosh(d*x + c)^2 + 
 d)*sinh(d*x + c)^2 + 4*(d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c 
) + d)
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (51) = 102\).

Time = 0.14 (sec) , antiderivative size = 161, normalized size of antiderivative = 3.04 \[ \int \coth (c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=b^{2} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} + 2 \, a b {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d}\right )} + \frac {a^{2} \log \left (\sinh \left (d x + c\right )\right )}{d} \] Input:

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

Output:

b^2*(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)/(d*(2*e^(-2*d*x - 2*c) + e^(-4*d*x - 4*c) 
+ 1))) + 2*a*b*(log(e^(-d*x - c) + 1)/d + log(e^(-d*x - c) - 1)/d - log(e^ 
(-2*d*x - 2*c) + 1)/d) + a^2*log(sinh(d*x + c))/d
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (51) = 102\).

Time = 0.15 (sec) , antiderivative size = 150, normalized size of antiderivative = 2.83 \[ \int \coth (c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=-\frac {{\left (2 \, a b + b^{2}\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} + 2\right ) - {\left (a^{2} + 2 \, a b + b^{2}\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} - 2\right ) - \frac {2 \, a b {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + b^{2} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 4 \, a b + 6 \, b^{2}}{e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} + 2}}{2 \, d} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 308, normalized size of antiderivative = 5.81 \[ \int \coth (c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\frac {2\,b^2}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-a^2\,x+\frac {a^2\,\ln \left ({\mathrm {e}}^{4\,c+4\,d\,x}-1\right )}{2\,d}-\frac {2\,b^2}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\left (a^4\,\sqrt {-d^2}+4\,b^4\,\sqrt {-d^2}+16\,a\,b^3\,\sqrt {-d^2}+8\,a^3\,b\,\sqrt {-d^2}+20\,a^2\,b^2\,\sqrt {-d^2}\right )}{a^2\,d\,\sqrt {a^4+8\,a^3\,b+20\,a^2\,b^2+16\,a\,b^3+4\,b^4}+2\,b^2\,d\,\sqrt {a^4+8\,a^3\,b+20\,a^2\,b^2+16\,a\,b^3+4\,b^4}+4\,a\,b\,d\,\sqrt {a^4+8\,a^3\,b+20\,a^2\,b^2+16\,a\,b^3+4\,b^4}}\right )\,\sqrt {a^4+8\,a^3\,b+20\,a^2\,b^2+16\,a\,b^3+4\,b^4}}{\sqrt {-d^2}} \] Input:

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

Output:

(2*b^2)/(d*(exp(2*c + 2*d*x) + 1)) - a^2*x + (a^2*log(exp(4*c + 4*d*x) - 1 
))/(2*d) - (2*b^2)/(d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) - (atan 
((exp(2*c)*exp(2*d*x)*(a^4*(-d^2)^(1/2) + 4*b^4*(-d^2)^(1/2) + 16*a*b^3*(- 
d^2)^(1/2) + 8*a^3*b*(-d^2)^(1/2) + 20*a^2*b^2*(-d^2)^(1/2)))/(a^2*d*(16*a 
*b^3 + 8*a^3*b + a^4 + 4*b^4 + 20*a^2*b^2)^(1/2) + 2*b^2*d*(16*a*b^3 + 8*a 
^3*b + a^4 + 4*b^4 + 20*a^2*b^2)^(1/2) + 4*a*b*d*(16*a*b^3 + 8*a^3*b + a^4 
 + 4*b^4 + 20*a^2*b^2)^(1/2)))*(16*a*b^3 + 8*a^3*b + a^4 + 4*b^4 + 20*a^2* 
b^2)^(1/2))/(-d^2)^(1/2)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 613, normalized size of antiderivative = 11.57 \[ \int \coth (c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\frac {-2 e^{2 d x +2 c} \mathrm {log}\left (e^{2 d x +2 c}+1\right ) b^{2}-2 \,\mathrm {log}\left (e^{2 d x +2 c}+1\right ) a b -2 e^{2 d x +2 c} a^{2} d x +2 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}-1\right ) a b +2 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}+1\right ) a b +4 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}-1\right ) a b +4 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}+1\right ) a b -2 e^{4 d x +4 c} \mathrm {log}\left (e^{2 d x +2 c}+1\right ) a b -4 e^{2 d x +2 c} \mathrm {log}\left (e^{2 d x +2 c}+1\right ) a b -e^{4 d x +4 c} \mathrm {log}\left (e^{2 d x +2 c}+1\right ) b^{2}-\mathrm {log}\left (e^{2 d x +2 c}+1\right ) b^{2}-e^{4 d x +4 c} a^{2} d x -a^{2} d x -b^{2}+e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}-1\right ) a^{2}+e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}-1\right ) b^{2}+e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}+1\right ) a^{2}+e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}+1\right ) b^{2}+2 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}-1\right ) a^{2}+2 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}-1\right ) b^{2}+2 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}+1\right ) a^{2}+2 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}+1\right ) b^{2}+2 \,\mathrm {log}\left (e^{d x +c}-1\right ) a b +2 \,\mathrm {log}\left (e^{d x +c}+1\right ) a b +\mathrm {log}\left (e^{d x +c}-1\right ) a^{2}+\mathrm {log}\left (e^{d x +c}-1\right ) b^{2}+\mathrm {log}\left (e^{d x +c}+1\right ) a^{2}+\mathrm {log}\left (e^{d x +c}+1\right ) b^{2}-e^{4 d x +4 c} b^{2}}{d \left (e^{4 d x +4 c}+2 e^{2 d x +2 c}+1\right )} \] Input:

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

Output:

( - 2*e**(4*c + 4*d*x)*log(e**(2*c + 2*d*x) + 1)*a*b - e**(4*c + 4*d*x)*lo 
g(e**(2*c + 2*d*x) + 1)*b**2 + e**(4*c + 4*d*x)*log(e**(c + d*x) - 1)*a**2 
 + 2*e**(4*c + 4*d*x)*log(e**(c + d*x) - 1)*a*b + e**(4*c + 4*d*x)*log(e** 
(c + d*x) - 1)*b**2 + e**(4*c + 4*d*x)*log(e**(c + d*x) + 1)*a**2 + 2*e**( 
4*c + 4*d*x)*log(e**(c + d*x) + 1)*a*b + e**(4*c + 4*d*x)*log(e**(c + d*x) 
 + 1)*b**2 - e**(4*c + 4*d*x)*a**2*d*x - e**(4*c + 4*d*x)*b**2 - 4*e**(2*c 
 + 2*d*x)*log(e**(2*c + 2*d*x) + 1)*a*b - 2*e**(2*c + 2*d*x)*log(e**(2*c + 
 2*d*x) + 1)*b**2 + 2*e**(2*c + 2*d*x)*log(e**(c + d*x) - 1)*a**2 + 4*e**( 
2*c + 2*d*x)*log(e**(c + d*x) - 1)*a*b + 2*e**(2*c + 2*d*x)*log(e**(c + d* 
x) - 1)*b**2 + 2*e**(2*c + 2*d*x)*log(e**(c + d*x) + 1)*a**2 + 4*e**(2*c + 
 2*d*x)*log(e**(c + d*x) + 1)*a*b + 2*e**(2*c + 2*d*x)*log(e**(c + d*x) + 
1)*b**2 - 2*e**(2*c + 2*d*x)*a**2*d*x - 2*log(e**(2*c + 2*d*x) + 1)*a*b - 
log(e**(2*c + 2*d*x) + 1)*b**2 + log(e**(c + d*x) - 1)*a**2 + 2*log(e**(c 
+ d*x) - 1)*a*b + log(e**(c + d*x) - 1)*b**2 + log(e**(c + d*x) + 1)*a**2 
+ 2*log(e**(c + d*x) + 1)*a*b + log(e**(c + d*x) + 1)*b**2 - a**2*d*x - b* 
*2)/(d*(e**(4*c + 4*d*x) + 2*e**(2*c + 2*d*x) + 1))