Integrand size = 25, antiderivative size = 219 \[ \int \frac {\text {sech}^4(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\frac {2 (a-2 b) E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 (a-b)^2 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {(a-3 b) b \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a (a-b)^2 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {\text {sech}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 (a-b) f} \]
2/3*(a-2*b)*(1/(1+sinh(f*x+e)^2))^(1/2)*(1+sinh(f*x+e)^2)^(1/2)*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)^2/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)-1/3*( a-3*b)*b*(1/(1+sinh(f*x+e)^2))^(1/2)*(1+sinh(f*x+e)^2)^(1/2)*EllipticF(sin h(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/(a-b)^2/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)+1/3*se ch(f*x+e)^2*(a+b*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)/(a-b)/f
Result contains complex when optimal does not.
Time = 1.86 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00 \[ \int \frac {\text {sech}^4(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\frac {4 i a (a-2 b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )-2 i \left (2 a^2-5 a b+3 b^2\right ) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} \operatorname {EllipticF}\left (i (e+f x),\frac {b}{a}\right )+\frac {\left (8 a^2-15 a b+4 b^2+\left (4 a^2-6 a b-2 b^2\right ) \cosh (2 (e+f x))+(a-2 b) b \cosh (4 (e+f x))\right ) \text {sech}^2(e+f x) \tanh (e+f x)}{\sqrt {2}}}{6 (a-b)^2 f \sqrt {2 a-b+b \cosh (2 (e+f x))}} \]
((4*I)*a*(a - 2*b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - (2*I)*(2*a^2 - 5*a*b + 3*b^2)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + ((8*a^2 - 15*a*b + 4*b^2 + (4*a^2 - 6*a*b - 2*b^2)*Cosh[2*(e + f*x)] + (a - 2*b)*b*Cosh[4*(e + f*x)])*Sech[ e + f*x]^2*Tanh[e + f*x])/Sqrt[2])/(6*(a - b)^2*f*Sqrt[2*a - b + b*Cosh[2* (e + f*x)]])
Time = 0.40 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.23, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3042, 3671, 316, 25, 400, 313, 320}
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.
| \(\displaystyle \int \frac {\text {sech}^4(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (i e+i f x)^4 \sqrt {a-b \sin (i e+i f x)^2}}dx\) |
| \(\Big \downarrow \) 3671 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \int \frac {1}{\left (\sinh ^2(e+f x)+1\right )^{5/2} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{f}\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 (a-b) \left (\sinh ^2(e+f x)+1\right )^{3/2}}-\frac {\int -\frac {b \sinh ^2(e+f x)+2 a-3 b}{\left (\sinh ^2(e+f x)+1\right )^{3/2} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{3 (a-b)}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\int \frac {b \sinh ^2(e+f x)+2 a-3 b}{\left (\sinh ^2(e+f x)+1\right )^{3/2} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{3 (a-b)}+\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 (a-b) \left (\sinh ^2(e+f x)+1\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\frac {2 (a-2 b) \int \frac {\sqrt {b \sinh ^2(e+f x)+a}}{\left (\sinh ^2(e+f x)+1\right )^{3/2}}d\sinh (e+f x)}{a-b}-\frac {b (a-3 b) \int \frac {1}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a-b}}{3 (a-b)}+\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 (a-b) \left (\sinh ^2(e+f x)+1\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\frac {2 (a-2 b) \sqrt {a+b \sinh ^2(e+f x)} E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{(a-b) \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}-\frac {b (a-3 b) \int \frac {1}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a-b}}{3 (a-b)}+\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 (a-b) \left (\sinh ^2(e+f x)+1\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\frac {2 (a-2 b) \sqrt {a+b \sinh ^2(e+f x)} E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{(a-b) \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}-\frac {b (a-3 b) \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{a (a-b) \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}}{3 (a-b)}+\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 (a-b) \left (\sinh ^2(e+f x)+1\right )^{3/2}}\right )}{f}\) |
(Sqrt[Cosh[e + f*x]^2]*Sech[e + f*x]*((Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f *x]^2])/(3*(a - b)*(1 + Sinh[e + f*x]^2)^(3/2)) + ((2*(a - 2*b)*EllipticE[ ArcTan[Sinh[e + f*x]], 1 - b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/((a - b)*Sqrt [1 + Sinh[e + f*x]^2]*Sqrt[(a + b*Sinh[e + f*x]^2)/(a*(1 + Sinh[e + f*x]^2 ))]) - ((a - 3*b)*b*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sqrt[a + b*S inh[e + f*x]^2])/(a*(a - b)*Sqrt[1 + Sinh[e + f*x]^2]*Sqrt[(a + b*Sinh[e + f*x]^2)/(a*(1 + Sinh[e + f*x]^2))]))/(3*(a - b))))/f
3.4.81.3.1 Defintions of rubi rules used
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]
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]
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]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^ (3/2)), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(Sqrt[a + b*x^2]* Sqrt[c + d*x^2]), x], x] - Simp[(d*e - c*f)/(b*c - a*d) Int[Sqrt[a + b*x^ 2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] & & PosQ[d/c]
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]
Time = 1.61 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.57
| method | result | size |
| default | \(\frac {\sqrt {\left (a +b \sinh \left (f x +e \right )^{2}\right ) \cosh \left (f x +e \right )^{2}}\, \left (2 \cosh \left (f x +e \right )^{4} \sqrt {-\frac {b}{a}}\, b \left (a -2 b \right ) \sinh \left (f x +e \right )+\cosh \left (f x +e \right )^{2} \sqrt {-\frac {b}{a}}\, \left (2 a^{2}-5 a b +3 b^{2}\right ) \sinh \left (f x +e \right )+\sqrt {-\frac {b}{a}}\, \left (a^{2}-2 a b +b^{2}\right ) \sinh \left (f x +e \right )+\sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, b \left (a \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-b \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-2 \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a +4 b \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )\right ) \cosh \left (f x +e \right )^{2}\right )}{3 \cosh \left (f x +e \right )^{3} \sqrt {-\frac {b}{a}}\, \sqrt {b \cosh \left (f x +e \right )^{4}+\left (a -b \right ) \cosh \left (f x +e \right )^{2}}\, \left (a^{2}-2 a b +b^{2}\right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\) | \(343\) |
| risch | \(\text {Expression too large to display}\) | \(122046\) |
1/3*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)/cosh(f*x+e)^3/(-b/a)^(1/2)/( b*cosh(f*x+e)^4+(a-b)*cosh(f*x+e)^2)^(1/2)/(a^2-2*a*b+b^2)*(2*cosh(f*x+e)^ 4*(-b/a)^(1/2)*b*(a-2*b)*sinh(f*x+e)+cosh(f*x+e)^2*(-b/a)^(1/2)*(2*a^2-5*a *b+3*b^2)*sinh(f*x+e)+(-b/a)^(1/2)*(a^2-2*a*b+b^2)*sinh(f*x+e)+(cosh(f*x+e )^2)^(1/2)*(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*b*(a*EllipticF(sinh(f*x+e)*(- b/a)^(1/2),(a/b)^(1/2))-b*EllipticF(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))- 2*EllipticE(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*a+4*b*EllipticE(sinh(f*x +e)*(-b/a)^(1/2),(a/b)^(1/2)))*cosh(f*x+e)^2)/(a+b*sinh(f*x+e)^2)^(1/2)/f
Leaf count of result is larger than twice the leaf count of optimal. 2443 vs. \(2 (231) = 462\).
Time = 0.13 (sec) , antiderivative size = 2443, normalized size of antiderivative = 11.16 \[ \int \frac {\text {sech}^4(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\text {Too large to display} \]
-2/3*(((2*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e)^6 + 6*(2*a^2 - 5*a*b + 2*b^2) *cosh(f*x + e)*sinh(f*x + e)^5 + (2*a^2 - 5*a*b + 2*b^2)*sinh(f*x + e)^6 + 3*(2*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e)^4 + 3*(5*(2*a^2 - 5*a*b + 2*b^2)* cosh(f*x + e)^2 + 2*a^2 - 5*a*b + 2*b^2)*sinh(f*x + e)^4 + 4*(5*(2*a^2 - 5 *a*b + 2*b^2)*cosh(f*x + e)^3 + 3*(2*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e))*s inh(f*x + e)^3 + 3*(2*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e)^2 + 3*(5*(2*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e)^4 + 6*(2*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e)^ 2 + 2*a^2 - 5*a*b + 2*b^2)*sinh(f*x + e)^2 + 2*a^2 - 5*a*b + 2*b^2 + 6*((2 *a^2 - 5*a*b + 2*b^2)*cosh(f*x + e)^5 + 2*(2*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e)^3 + (2*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e))*sinh(f*x + e) - 2*((a*b - 2*b^2)*cosh(f*x + e)^6 + 6*(a*b - 2*b^2)*cosh(f*x + e)*sinh(f*x + e)^5 + (a*b - 2*b^2)*sinh(f*x + e)^6 + 3*(a*b - 2*b^2)*cosh(f*x + e)^4 + 3*(5*(a* b - 2*b^2)*cosh(f*x + e)^2 + a*b - 2*b^2)*sinh(f*x + e)^4 + 4*(5*(a*b - 2* b^2)*cosh(f*x + e)^3 + 3*(a*b - 2*b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 3* (a*b - 2*b^2)*cosh(f*x + e)^2 + 3*(5*(a*b - 2*b^2)*cosh(f*x + e)^4 + 6*(a* b - 2*b^2)*cosh(f*x + e)^2 + a*b - 2*b^2)*sinh(f*x + e)^2 + a*b - 2*b^2 + 6*((a*b - 2*b^2)*cosh(f*x + e)^5 + 2*(a*b - 2*b^2)*cosh(f*x + e)^3 + (a*b - 2*b^2)*cosh(f*x + e))*sinh(f*x + e))*sqrt((a^2 - a*b)/b^2))*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^...
\[ \int \frac {\text {sech}^4(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\int \frac {\operatorname {sech}^{4}{\left (e + f x \right )}}{\sqrt {a + b \sinh ^{2}{\left (e + f x \right )}}}\, dx \]
\[ \int \frac {\text {sech}^4(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\int { \frac {\operatorname {sech}\left (f x + e\right )^{4}}{\sqrt {b \sinh \left (f x + e\right )^{2} + a}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1142 vs. \(2 (231) = 462\).
Time = 1.45 (sec) , antiderivative size = 1142, normalized size of antiderivative = 5.21 \[ \int \frac {\text {sech}^4(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\text {Too large to display} \]
1/3*(3*(a - 2*b)*arctan(-1/2*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b) + sqrt(b))/sqrt(a - b))/((a*e^(4*e) - b*e^(4*e))*sqrt(a - b)) - 2*(3*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b)) ^5*a - 6*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^5*b + 15*(sqrt(b)*e^(2*f*x + 2*e) - sqr t(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^4*a* sqrt(b) - 30*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2* f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^4*b^(3/2) + 32*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^3*a^2 - 130*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a* e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^3*a*b + 68*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^3*b^2 - 96*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a* e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^2*a^2*sqrt(b) + 30*(sqrt(b)*e^ (2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f* x + 2*e) + b))^2*a*b^(3/2) + 36*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^2*b^(5/2) - 48*( sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2 *b*e^(2*f*x + 2*e) + b))*a^3 - 96*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(...
Timed out. \[ \int \frac {\text {sech}^4(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\int \frac {1}{{\mathrm {cosh}\left (e+f\,x\right )}^4\,\sqrt {b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a}} \,d x \]