\(\int \frac {\text {sech}^2(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx\) [344]

Optimal result

Integrand size = 25, antiderivative size = 160 \[ \int \frac {\text {sech}^2(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\frac {E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{(a-b) f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {b \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a (a-b) f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}} \] Output:

EllipticE(sinh(f*x+e)/(1+sinh(f*x+e)^2)^(1/2),(1-b/a)^(1/2))*sech(f*x+e)*( 
a+b*sinh(f*x+e)^2)^(1/2)/(a-b)/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/ 
2)-b*InverseJacobiAM(arctan(sinh(f*x+e)),(1-b/a)^(1/2))*sech(f*x+e)*(a+b*s 
inh(f*x+e)^2)^(1/2)/a/(a-b)/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.67 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.99 \[ \int \frac {\text {sech}^2(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\frac {2 i a \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )-2 i (a-b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} \operatorname {EllipticF}\left (i (e+f x),\frac {b}{a}\right )+\sqrt {2} (2 a-b+b \cosh (2 (e+f x))) \tanh (e+f x)}{2 (a-b) f \sqrt {2 a-b+b \cosh (2 (e+f x))}} \] Input:

Integrate[Sech[e + f*x]^2/Sqrt[a + b*Sinh[e + f*x]^2],x]
 

Output:

((2*I)*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/ 
a] - (2*I)*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e 
+ f*x), b/a] + Sqrt[2]*(2*a - b + b*Cosh[2*(e + f*x)])*Tanh[e + f*x])/(2*( 
a - b)*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.77, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 3671, 316, 27, 324, 320, 388, 313}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3671

\(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \int \frac {1}{\left (\sinh ^2(e+f x)+1\right )^{3/2} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{f}\)

\(\Big \downarrow \) 316

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 324

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

\(\Big \downarrow \) 320

\(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{(a-b) \sqrt {\sinh ^2(e+f x)+1}}-\frac {b \left (\int \frac {\sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+\frac {\sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{a \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}\right )}{a-b}\right )}{f}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{(a-b) \sqrt {\sinh ^2(e+f x)+1}}-\frac {b \left (-\frac {\int \frac {\sqrt {b \sinh ^2(e+f x)+a}}{\left (\sinh ^2(e+f x)+1\right )^{3/2}}d\sinh (e+f x)}{b}+\frac {\sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{a \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}+\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{b \sqrt {\sinh ^2(e+f x)+1}}\right )}{a-b}\right )}{f}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{(a-b) \sqrt {\sinh ^2(e+f x)+1}}-\frac {b \left (\frac {\sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{a \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}-\frac {\sqrt {a+b \sinh ^2(e+f x)} E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{b \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}+\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{b \sqrt {\sinh ^2(e+f x)+1}}\right )}{a-b}\right )}{f}\)

Input:

Int[Sech[e + f*x]^2/Sqrt[a + b*Sinh[e + f*x]^2],x]
 

Output:

(Sqrt[Cosh[e + f*x]^2]*Sech[e + f*x]*((Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f 
*x]^2])/((a - b)*Sqrt[1 + Sinh[e + f*x]^2]) - (b*((Sinh[e + f*x]*Sqrt[a + 
b*Sinh[e + f*x]^2])/(b*Sqrt[1 + Sinh[e + f*x]^2]) - (EllipticE[ArcTan[Sinh 
[e + f*x]], 1 - b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(b*Sqrt[1 + Sinh[e + f*x 
]^2]*Sqrt[(a + b*Sinh[e + f*x]^2)/(a*(1 + Sinh[e + f*x]^2))]) + (EllipticF 
[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(a*Sqrt[1 + 
Sinh[e + f*x]^2]*Sqrt[(a + b*Sinh[e + f*x]^2)/(a*(1 + Sinh[e + f*x]^2))])) 
)/(a - b)))/f
 

Defintions of rubi rules used

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 313

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim 
p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c 
+ d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ 
[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
 

rule 316

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) 
), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d))   Int[(a + b*x^2)^(p + 1)*(c + d*x 
^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x 
], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  ! 
( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, 
 p, q, x]
 

rule 320

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( 
c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre 
eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 324

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
a   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] + Simp[b   Int[x^2/(Sqr 
t[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && PosQ[d/c 
] && PosQ[b/a]
 

rule 388

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] 
 :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b   Int[Sqrt[ 
a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 3042

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

rule 3671

Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^( 
p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff*(Sqrt[ 
Cos[e + f*x]^2]/(f*Cos[e + f*x]))   Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a 
 + b*ff^2*x^2)^p, x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] 
 && IntegerQ[m/2] &&  !IntegerQ[p]
 

Maple [A] (verified)

Time = 1.45 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.83

method	result	size
default	\(-\frac {-\sqrt {-\frac {b}{a}}\, b \sinh \left (f x +e \right )^{3}+b \sqrt {\frac {a +b \sinh \left (f x +e \right )^{2}}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-\sqrt {-\frac {b}{a}}\, a \sinh \left (f x +e \right )}{\left (a -b \right ) \sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\)	\(133\)
risch	\(\text {Expression too large to display}\)	\(4969\)

Input:

int(sech(f*x+e)^2/(a+b*sinh(f*x+e)^2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-(-(-b/a)^(1/2)*b*sinh(f*x+e)^3+b*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+ 
e)^2)^(1/2)*EllipticE(sinh(f*x+e)*(-b/a)^(1/2),(1/b*a)^(1/2))-(-b/a)^(1/2) 
*a*sinh(f*x+e))/(a-b)/(-b/a)^(1/2)/cosh(f*x+e)/(a+b*sinh(f*x+e)^2)^(1/2)/f
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 576 vs. \(2 (168) = 336\).

Time = 0.10 (sec) , antiderivative size = 576, normalized size of antiderivative = 3.60 \[ \int \frac {\text {sech}^2(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=-\frac {{\left ({\left (2 \, a - b\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (2 \, a - b\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + {\left (2 \, a - b\right )} \sinh \left (f x + e\right )^{2} - 2 \, {\left (b \cosh \left (f x + e\right )^{2} + 2 \, b \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + b \sinh \left (f x + e\right )^{2} + b\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}} + 2 \, a - b\right )} \sqrt {b} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}} E(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}} {\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} - 8 \, a b + b^{2} + 4 \, {\left (2 \, a b - b^{2}\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}}{b^{2}}) - 2 \, {\left ({\left (2 \, a - b\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (2 \, a - b\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + {\left (2 \, a - b\right )} \sinh \left (f x + e\right )^{2} + 2 \, a - b\right )} \sqrt {b} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}} F(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}} {\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} - 8 \, a b + b^{2} + 4 \, {\left (2 \, a b - b^{2}\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}}{b^{2}}) - \sqrt {2} {\left (b \cosh \left (f x + e\right ) + b \sinh \left (f x + e\right )\right )} \sqrt {\frac {b \cosh \left (f x + e\right )^{2} + b \sinh \left (f x + e\right )^{2} + 2 \, a - b}{\cosh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2}}}}{{\left (a b - b^{2}\right )} f \cosh \left (f x + e\right )^{2} + 2 \, {\left (a b - b^{2}\right )} f \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + {\left (a b - b^{2}\right )} f \sinh \left (f x + e\right )^{2} + {\left (a b - b^{2}\right )} f} \] Input:

integrate(sech(f*x+e)^2/(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="fricas")
 

Output:

-(((2*a - b)*cosh(f*x + e)^2 + 2*(2*a - b)*cosh(f*x + e)*sinh(f*x + e) + ( 
2*a - b)*sinh(f*x + e)^2 - 2*(b*cosh(f*x + e)^2 + 2*b*cosh(f*x + e)*sinh(f 
*x + e) + b*sinh(f*x + e)^2 + b)*sqrt((a^2 - a*b)/b^2) + 2*a - b)*sqrt(b)* 
sqrt((2*b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*elliptic_e(arcsin(sqrt((2*b* 
sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*(cosh(f*x + e) + sinh(f*x + e))), (8*a 
^2 - 8*a*b + b^2 + 4*(2*a*b - b^2)*sqrt((a^2 - a*b)/b^2))/b^2) - 2*((2*a - 
 b)*cosh(f*x + e)^2 + 2*(2*a - b)*cosh(f*x + e)*sinh(f*x + e) + (2*a - b)* 
sinh(f*x + e)^2 + 2*a - b)*sqrt(b)*sqrt((2*b*sqrt((a^2 - a*b)/b^2) - 2*a + 
 b)/b)*elliptic_f(arcsin(sqrt((2*b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*(co 
sh(f*x + e) + sinh(f*x + e))), (8*a^2 - 8*a*b + b^2 + 4*(2*a*b - b^2)*sqrt 
((a^2 - a*b)/b^2))/b^2) - sqrt(2)*(b*cosh(f*x + e) + b*sinh(f*x + e))*sqrt 
((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*co 
sh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2)))/((a*b - b^2)*f*cosh(f*x + e 
)^2 + 2*(a*b - b^2)*f*cosh(f*x + e)*sinh(f*x + e) + (a*b - b^2)*f*sinh(f*x 
 + e)^2 + (a*b - b^2)*f)
 

Sympy [F]

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

integrate(sech(f*x+e)**2/(a+b*sinh(f*x+e)**2)**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate(sech(f*x+e)^2/(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="maxima")
 

Output:

integrate(sech(f*x + e)^2/sqrt(b*sinh(f*x + e)^2 + a), x)
 

Giac [F]

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

integrate(sech(f*x+e)^2/(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="giac")
 

Output:

integrate(sech(f*x + e)^2/sqrt(b*sinh(f*x + e)^2 + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {sech}^2(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\int \frac {1}{{\mathrm {cosh}\left (e+f\,x\right )}^2\,\sqrt {b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a}} \,d x \] Input:

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

Output:

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

Reduce [F]

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

int(sech(f*x+e)^2/(a+b*sinh(f*x+e)^2)^(1/2),x)
 

Output:

int((sqrt(sinh(e + f*x)**2*b + a)*sech(e + f*x)**2)/(sinh(e + f*x)**2*b + 
a),x)