\(\int \frac {\text {sech}^2(x)}{(a+b \sinh (x))^2} \, dx\) [205]

Optimal result

Integrand size = 13, antiderivative size = 93 \[ \int \frac {\text {sech}^2(x)}{(a+b \sinh (x))^2} \, dx=-\frac {6 a b^2 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac {b \text {sech}(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {\text {sech}(x) \left (3 a b+\left (a^2-2 b^2\right ) \sinh (x)\right )}{\left (a^2+b^2\right )^2} \] Output:

-6*a*b^2*arctanh((b-a*tanh(1/2*x))/(a^2+b^2)^(1/2))/(a^2+b^2)^(5/2)-b*sech 
(x)/(a^2+b^2)/(a+b*sinh(x))+sech(x)*(3*a*b+(a^2-2*b^2)*sinh(x))/(a^2+b^2)^ 
2
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.01 \[ \int \frac {\text {sech}^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {\frac {6 a b^2 \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+2 a b \text {sech}(x)-\frac {b^3 \cosh (x)}{a+b \sinh (x)}+a^2 \tanh (x)-b^2 \tanh (x)}{\left (a^2+b^2\right )^2} \] Input:

Integrate[Sech[x]^2/(a + b*Sinh[x])^2,x]
 

Output:

((6*a*b^2*ArcTan[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] + 2 
*a*b*Sech[x] - (b^3*Cosh[x])/(a + b*Sinh[x]) + a^2*Tanh[x] - b^2*Tanh[x])/ 
(a^2 + b^2)^2
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.17, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {3042, 3173, 25, 3042, 3345, 27, 3042, 3139, 1083, 219}

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.

\(\displaystyle \int \frac {\text {sech}^2(x)}{(a+b \sinh (x))^2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\cos (i x)^2 (a-i b \sin (i x))^2}dx\)

\(\Big \downarrow \) 3173

\(\displaystyle -\frac {\int -\frac {\text {sech}^2(x) (a-2 b \sinh (x))}{a+b \sinh (x)}dx}{a^2+b^2}-\frac {b \text {sech}(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\text {sech}^2(x) (a-2 b \sinh (x))}{a+b \sinh (x)}dx}{a^2+b^2}-\frac {b \text {sech}(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b \text {sech}(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {\int \frac {a+2 i b \sin (i x)}{\cos (i x)^2 (a-i b \sin (i x))}dx}{a^2+b^2}\)

\(\Big \downarrow \) 3345

\(\displaystyle \frac {\frac {\text {sech}(x) \left (\left (a^2-2 b^2\right ) \sinh (x)+3 a b\right )}{a^2+b^2}-\frac {\int -\frac {3 a b^2}{a+b \sinh (x)}dx}{a^2+b^2}}{a^2+b^2}-\frac {b \text {sech}(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b \text {sech}(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {\frac {\text {sech}(x) \left (\left (a^2-2 b^2\right ) \sinh (x)+3 a b\right )}{a^2+b^2}+\frac {3 a b^2 \int \frac {1}{a-i b \sin (i x)}dx}{a^2+b^2}}{a^2+b^2}\)

\(\Big \downarrow \) 3139

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

\(\Big \downarrow \) 1083

\(\displaystyle \frac {\frac {\text {sech}(x) \left (\left (a^2-2 b^2\right ) \sinh (x)+3 a b\right )}{a^2+b^2}-\frac {12 a b^2 \int \frac {1}{4 \left (a^2+b^2\right )-\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )^2}d\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{a^2+b^2}}{a^2+b^2}-\frac {b \text {sech}(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\text {sech}(x) \left (\left (a^2-2 b^2\right ) \sinh (x)+3 a b\right )}{a^2+b^2}-\frac {6 a b^2 \text {arctanh}\left (\frac {2 b-2 a \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}}{a^2+b^2}-\frac {b \text {sech}(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}\)

Input:

Int[Sech[x]^2/(a + b*Sinh[x])^2,x]
 

Output:

-((b*Sech[x])/((a^2 + b^2)*(a + b*Sinh[x]))) + ((-6*a*b^2*ArcTanh[(2*b - 2 
*a*Tanh[x/2])/(2*Sqrt[a^2 + b^2])])/(a^2 + b^2)^(3/2) + (Sech[x]*(3*a*b + 
(a^2 - 2*b^2)*Sinh[x]))/(a^2 + b^2))/(a^2 + b^2)
 

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 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 3042

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

rule 3139

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre 
eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d)   Subst[Int[1/(a + 2*b*e*x + a 
*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ 
[a^2 - b^2, 0]
 

rule 3173

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + 
f*x])^(m + 1)/(f*g*(a^2 - b^2)*(m + 1))), x] + Simp[1/((a^2 - b^2)*(m + 1)) 
   Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)*(a*(m + 1) - b*(m + 
p + 2)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b 
^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*p]
 

rule 3345

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(g*Co 
s[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c - a*d - (a*c - b*d)* 
Sin[e + f*x])/(f*g*(a^2 - b^2)*(p + 1))), x] + Simp[1/(g^2*(a^2 - b^2)*(p + 
 1))   Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m*Simp[c*(a^2*(p + 
 2) - b^2*(m + p + 2)) + a*b*d*m + b*(a*c - b*d)*(m + p + 3)*Sin[e + f*x], 
x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && Lt 
Q[p, -1] && IntegerQ[2*m]
 

Maple [A] (verified)

Time = 42.93 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.48

method	result	size
default	\(-\frac {2 b^{2} \left (\frac {-\frac {b^{2} \tanh \left (\frac {x}{2}\right )}{a}-b}{\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a}-\frac {3 a \,\operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {2 \left (\left (-a^{2}+b^{2}\right ) \tanh \left (\frac {x}{2}\right )-2 a b \right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )}\)	\(138\)
risch	\(-\frac {2 \left (-3 a \,b^{2} {\mathrm e}^{3 x}-3 a^{2} b \,{\mathrm e}^{2 x}+2 a^{3} {\mathrm e}^{x}-a \,b^{2} {\mathrm e}^{x}-a^{2} b +2 b^{3}\right )}{\left (a^{2}+b^{2}\right )^{2} \left ({\mathrm e}^{2 x}+1\right ) \left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right )}+\frac {3 b^{2} a \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} a -a^{6}-3 a^{4} b^{2}-3 a^{2} b^{4}-b^{6}}{b \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}-\frac {3 b^{2} a \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} a +a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}{b \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\)	\(221\)

Input:

int(sech(x)^2/(a+b*sinh(x))^2,x,method=_RETURNVERBOSE)
 

Output:

-2/(a^2+b^2)^2*b^2*((-1/a*b^2*tanh(1/2*x)-b)/(tanh(1/2*x)^2*a-2*b*tanh(1/2 
*x)-a)-3*a/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tanh(1/2*x)-2*b)/(a^2+b^2)^(1/ 
2)))-2/(a^4+2*a^2*b^2+b^4)*((-a^2+b^2)*tanh(1/2*x)-2*a*b)/(1+tanh(1/2*x)^2 
)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 802 vs. \(2 (89) = 178\).

Time = 0.10 (sec) , antiderivative size = 802, normalized size of antiderivative = 8.62 \[ \int \frac {\text {sech}^2(x)}{(a+b \sinh (x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

-(2*a^4*b - 2*a^2*b^3 - 4*b^5 + 6*(a^3*b^2 + a*b^4)*cosh(x)^3 + 6*(a^3*b^2 
 + a*b^4)*sinh(x)^3 + 6*(a^4*b + a^2*b^3)*cosh(x)^2 + 6*(a^4*b + a^2*b^3 + 
 3*(a^3*b^2 + a*b^4)*cosh(x))*sinh(x)^2 + 3*(a*b^3*cosh(x)^4 + a*b^3*sinh( 
x)^4 + 2*a^2*b^2*cosh(x)^3 + 2*a^2*b^2*cosh(x) - a*b^3 + 2*(2*a*b^3*cosh(x 
) + a^2*b^2)*sinh(x)^3 + 6*(a*b^3*cosh(x)^2 + a^2*b^2*cosh(x))*sinh(x)^2 + 
 2*(2*a*b^3*cosh(x)^3 + 3*a^2*b^2*cosh(x)^2 + a^2*b^2)*sinh(x))*sqrt(a^2 + 
 b^2)*log((b^2*cosh(x)^2 + b^2*sinh(x)^2 + 2*a*b*cosh(x) + 2*a^2 + b^2 + 2 
*(b^2*cosh(x) + a*b)*sinh(x) - 2*sqrt(a^2 + b^2)*(b*cosh(x) + b*sinh(x) + 
a))/(b*cosh(x)^2 + b*sinh(x)^2 + 2*a*cosh(x) + 2*(b*cosh(x) + a)*sinh(x) - 
 b)) - 2*(2*a^5 + a^3*b^2 - a*b^4)*cosh(x) - 2*(2*a^5 + a^3*b^2 - a*b^4 - 
9*(a^3*b^2 + a*b^4)*cosh(x)^2 - 6*(a^4*b + a^2*b^3)*cosh(x))*sinh(x))/(a^6 
*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7 - (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*c 
osh(x)^4 - (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*sinh(x)^4 - 2*(a^7 + 3*a^ 
5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x)^3 - 2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a* 
b^6 + 2*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cosh(x))*sinh(x)^3 - 6*((a^6 
*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cosh(x)^2 + (a^7 + 3*a^5*b^2 + 3*a^3*b^4 
 + a*b^6)*cosh(x))*sinh(x)^2 - 2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cos 
h(x) - 2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6 + 2*(a^6*b + 3*a^4*b^3 + 3*a 
^2*b^5 + b^7)*cosh(x)^3 + 3*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x)^ 
2)*sinh(x))
 

Sympy [F]

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

integrate(sech(x)**2/(a+b*sinh(x))**2,x)
 

Output:

Integral(sech(x)**2/(a + b*sinh(x))**2, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (89) = 178\).

Time = 0.13 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.31 \[ \int \frac {\text {sech}^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {3 \, a b^{2} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (3 \, a^{2} b e^{\left (-2 \, x\right )} - 3 \, a b^{2} e^{\left (-3 \, x\right )} + a^{2} b - 2 \, b^{3} + {\left (2 \, a^{3} - a b^{2}\right )} e^{\left (-x\right )}\right )}}{a^{4} b + 2 \, a^{2} b^{3} + b^{5} + 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} e^{\left (-x\right )} + 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} e^{\left (-3 \, x\right )} - {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} e^{\left (-4 \, x\right )}} \] Input:

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

Output:

3*a*b^2*log((b*e^(-x) - a - sqrt(a^2 + b^2))/(b*e^(-x) - a + sqrt(a^2 + b^ 
2)))/((a^4 + 2*a^2*b^2 + b^4)*sqrt(a^2 + b^2)) + 2*(3*a^2*b*e^(-2*x) - 3*a 
*b^2*e^(-3*x) + a^2*b - 2*b^3 + (2*a^3 - a*b^2)*e^(-x))/(a^4*b + 2*a^2*b^3 
 + b^5 + 2*(a^5 + 2*a^3*b^2 + a*b^4)*e^(-x) + 2*(a^5 + 2*a^3*b^2 + a*b^4)* 
e^(-3*x) - (a^4*b + 2*a^2*b^3 + b^5)*e^(-4*x))
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.80 \[ \int \frac {\text {sech}^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {3 \, a b^{2} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (3 \, a b^{2} e^{\left (3 \, x\right )} + 3 \, a^{2} b e^{\left (2 \, x\right )} - 2 \, a^{3} e^{x} + a b^{2} e^{x} + a^{2} b - 2 \, b^{3}\right )}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (b e^{\left (4 \, x\right )} + 2 \, a e^{\left (3 \, x\right )} + 2 \, a e^{x} - b\right )}} \] Input:

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

Output:

3*a*b^2*log(abs(2*b*e^x + 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*e^x + 2*a + 2*s 
qrt(a^2 + b^2)))/((a^4 + 2*a^2*b^2 + b^4)*sqrt(a^2 + b^2)) + 2*(3*a*b^2*e^ 
(3*x) + 3*a^2*b*e^(2*x) - 2*a^3*e^x + a*b^2*e^x + a^2*b - 2*b^3)/((a^4 + 2 
*a^2*b^2 + b^4)*(b*e^(4*x) + 2*a*e^(3*x) + 2*a*e^x - b))
 

Mupad [B] (verification not implemented)

Time = 1.83 (sec) , antiderivative size = 302, normalized size of antiderivative = 3.25 \[ \int \frac {\text {sech}^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {\frac {6\,a^4\,b^4\,{\mathrm {e}}^{2\,x}}{\left (a^3+a\,b^2\right )\,\left (a^3\,b^3+a\,b^5\right )}-\frac {2\,\left (2\,a^2\,b^6-a^4\,b^4\right )}{\left (a^3+a\,b^2\right )\,\left (a^3\,b^3+a\,b^5\right )}+\frac {6\,a^3\,b^5\,{\mathrm {e}}^{3\,x}}{\left (a^3+a\,b^2\right )\,\left (a^3\,b^3+a\,b^5\right )}+\frac {2\,a\,{\mathrm {e}}^x\,\left (a^2\,b^6-2\,a^4\,b^4\right )}{b\,\left (a^3+a\,b^2\right )\,\left (a^3\,b^3+a\,b^5\right )}}{2\,a\,{\mathrm {e}}^x-b+2\,a\,{\mathrm {e}}^{3\,x}+b\,{\mathrm {e}}^{4\,x}}-\frac {3\,a\,b^2\,\ln \left (-\frac {6\,a\,b\,{\mathrm {e}}^x}{{\left (a^2+b^2\right )}^2}-\frac {6\,a\,b\,\left (b-a\,{\mathrm {e}}^x\right )}{{\left (a^2+b^2\right )}^{5/2}}\right )}{{\left (a^2+b^2\right )}^{5/2}}+\frac {3\,a\,b^2\,\ln \left (\frac {6\,a\,b\,\left (b-a\,{\mathrm {e}}^x\right )}{{\left (a^2+b^2\right )}^{5/2}}-\frac {6\,a\,b\,{\mathrm {e}}^x}{{\left (a^2+b^2\right )}^2}\right )}{{\left (a^2+b^2\right )}^{5/2}} \] Input:

int(1/(cosh(x)^2*(a + b*sinh(x))^2),x)
 

Output:

((6*a^4*b^4*exp(2*x))/((a*b^2 + a^3)*(a*b^5 + a^3*b^3)) - (2*(2*a^2*b^6 - 
a^4*b^4))/((a*b^2 + a^3)*(a*b^5 + a^3*b^3)) + (6*a^3*b^5*exp(3*x))/((a*b^2 
 + a^3)*(a*b^5 + a^3*b^3)) + (2*a*exp(x)*(a^2*b^6 - 2*a^4*b^4))/(b*(a*b^2 
+ a^3)*(a*b^5 + a^3*b^3)))/(2*a*exp(x) - b + 2*a*exp(3*x) + b*exp(4*x)) - 
(3*a*b^2*log(- (6*a*b*exp(x))/(a^2 + b^2)^2 - (6*a*b*(b - a*exp(x)))/(a^2 
+ b^2)^(5/2)))/(a^2 + b^2)^(5/2) + (3*a*b^2*log((6*a*b*(b - a*exp(x)))/(a^ 
2 + b^2)^(5/2) - (6*a*b*exp(x))/(a^2 + b^2)^2))/(a^2 + b^2)^(5/2)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 421, normalized size of antiderivative = 4.53 \[ \int \frac {\text {sech}^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {6 e^{4 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a \,b^{3} i +12 e^{3 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2} i +12 e^{x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2} i -6 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a \,b^{3} i -3 e^{4 x} a^{2} b^{3}-3 e^{4 x} b^{5}+6 e^{2 x} a^{4} b +6 e^{2 x} a^{2} b^{3}-4 e^{x} a^{5}-8 e^{x} a^{3} b^{2}-4 e^{x} a \,b^{4}+2 a^{4} b +a^{2} b^{3}-b^{5}}{e^{4 x} a^{6} b +3 e^{4 x} a^{4} b^{3}+3 e^{4 x} a^{2} b^{5}+e^{4 x} b^{7}+2 e^{3 x} a^{7}+6 e^{3 x} a^{5} b^{2}+6 e^{3 x} a^{3} b^{4}+2 e^{3 x} a \,b^{6}+2 e^{x} a^{7}+6 e^{x} a^{5} b^{2}+6 e^{x} a^{3} b^{4}+2 e^{x} a \,b^{6}-a^{6} b -3 a^{4} b^{3}-3 a^{2} b^{5}-b^{7}} \] Input:

int(sech(x)^2/(a+b*sinh(x))^2,x)
 

Output:

(6*e**(4*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a*b 
**3*i + 12*e**(3*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b* 
*2))*a**2*b**2*i + 12*e**x*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a* 
*2 + b**2))*a**2*b**2*i - 6*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a 
**2 + b**2))*a*b**3*i - 3*e**(4*x)*a**2*b**3 - 3*e**(4*x)*b**5 + 6*e**(2*x 
)*a**4*b + 6*e**(2*x)*a**2*b**3 - 4*e**x*a**5 - 8*e**x*a**3*b**2 - 4*e**x* 
a*b**4 + 2*a**4*b + a**2*b**3 - b**5)/(e**(4*x)*a**6*b + 3*e**(4*x)*a**4*b 
**3 + 3*e**(4*x)*a**2*b**5 + e**(4*x)*b**7 + 2*e**(3*x)*a**7 + 6*e**(3*x)* 
a**5*b**2 + 6*e**(3*x)*a**3*b**4 + 2*e**(3*x)*a*b**6 + 2*e**x*a**7 + 6*e** 
x*a**5*b**2 + 6*e**x*a**3*b**4 + 2*e**x*a*b**6 - a**6*b - 3*a**4*b**3 - 3* 
a**2*b**5 - b**7)