∫ sech5(e + fx)∘ ____________ a + b sinh2(e + fx) dx [356]

Optimal. Leaf size=151 \[ \frac {a (3 a-4 b) \text {ArcTan}\left (\frac {\sqrt {a-b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{8 (a-b)^{3/2} f}+\frac {(3 a-4 b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{8 (a-b) f}+\frac {\text {sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh (e+f x)}{4 (a-b) f} \]

1/8*a*(3*a-4*b)*arctan(sinh(f*x+e)*(a-b)^(1/2)/(a+b*sinh(f*x+e)^2)^(1/2))/(a-b)^(3/2)/f+1/4*sech(f*x+e)^3*(a+b

*sinh(f*x+e)^2)^(3/2)*tanh(f*x+e)/(a-b)/f+1/8*(3*a-4*b)*sech(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)/(a-b

________________________________________________________________________________________

Rubi [A]
time = 0.10, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3269, 390, 386, 385, 209} \begin {gather*} \frac {a (3 a-4 b) \text {ArcTan}\left (\frac {\sqrt {a-b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{8 f (a-b)^{3/2}}+\frac {\tanh (e+f x) \text {sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{4 f (a-b)}+\frac {(3 a-4 b) \tanh (e+f x) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{8 f (a-b)} \end {gather*}

(a*(3*a - 4*b)*ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/(8*(a - b)^(3/2)*f) + ((3*a -

4*b)*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(8*(a - b)*f) + (Sech[e + f*x]^3*(a + b*Sinh[e +

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;

FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]

, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Simp[(-x)*(a + b*x^n)^(p + 1)*(

(c + d*x^n)^q/(a*n*(p + 1))), x] - Dist[c*(q/(a*(p + 1))), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1), x], x]

/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0] && GtQ[q, 0] && NeQ[p, -1]

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*

((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a

*d)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && Eq

Q[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] || !LtQ[q, -1]) && NeQ[p, -1]

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free

Factors[Sin[e + f*x], x]}, Dist[ff/f, 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 - 1)/2]

\begin {align*} \int \text {sech}^5(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx &=\frac {\text {Subst}\left (\int \frac {\sqrt {a+b x^2}}{\left (1+x^2\right )^3} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=\frac {\text {sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh (e+f x)}{4 (a-b) f}+\frac {(3 a-4 b) \text {Subst}\left (\int \frac {\sqrt {a+b x^2}}{\left (1+x^2\right )^2} \, dx,x,\sinh (e+f x)\right )}{4 (a-b) f}\\ &=\frac {(3 a-4 b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{8 (a-b) f}+\frac {\text {sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh (e+f x)}{4 (a-b) f}+\frac {(a (3 a-4 b)) \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{8 (a-b) f}\\ &=\frac {(3 a-4 b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{8 (a-b) f}+\frac {\text {sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh (e+f x)}{4 (a-b) f}+\frac {(a (3 a-4 b)) \text {Subst}\left (\int \frac {1}{1-(-a+b) x^2} \, dx,x,\frac {\sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{8 (a-b) f}\\ &=\frac {a (3 a-4 b) \tan ^{-1}\left (\frac {\sqrt {a-b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{8 (a-b)^{3/2} f}+\frac {(3 a-4 b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{8 (a-b) f}+\frac {\text {sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh (e+f x)}{4 (a-b) f}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 10.14, size = 684, normalized size = 4.53 \begin {gather*} -\frac {\text {sech}^3(e+f x) \left (1+\frac {b \sinh ^2(e+f x)}{a}\right ) \tanh (e+f x) \left (-15 a \text {ArcSin}\left (\sqrt {\frac {(a-b) \tanh ^2(e+f x)}{a}}\right )-10 b \text {ArcSin}\left (\sqrt {\frac {(a-b) \tanh ^2(e+f x)}{a}}\right ) \sinh ^2(e+f x)-30 a \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}} \left (\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{3/2}-20 b \sinh ^2(e+f x) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}} \left (\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{3/2}-32 a \, _2F_1\left (2,4;\frac {7}{2};\frac {(a-b) \tanh ^2(e+f x)}{a}\right ) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}} \left (\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{5/2}-32 b \, _2F_1\left (2,4;\frac {7}{2};\frac {(a-b) \tanh ^2(e+f x)}{a}\right ) \sinh ^2(e+f x) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}} \left (\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{5/2}+32 a \, _2F_1\left (2,4;\frac {7}{2};\frac {(a-b) \tanh ^2(e+f x)}{a}\right ) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}} \left (\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{7/2}+32 b \, _2F_1\left (2,4;\frac {7}{2};\frac {(a-b) \tanh ^2(e+f x)}{a}\right ) \sinh ^2(e+f x) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}} \left (\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{7/2}+15 a \sqrt {\frac {(a-b) \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right ) \tanh ^2(e+f x)}{a^2}}+10 b \sinh ^2(e+f x) \sqrt {\frac {(a-b) \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right ) \tanh ^2(e+f x)}{a^2}}\right )}{40 f \sqrt {a+b \sinh ^2(e+f x)} \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}} \left (\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{3/2}} \end {gather*}

-1/40*(Sech[e + f*x]^3*(1 + (b*Sinh[e + f*x]^2)/a)*Tanh[e + f*x]*(-15*a*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/

a]] - 10*b*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]]*Sinh[e + f*x]^2 - 30*a*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[

e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(3/2) - 20*b*Sinh[e + f*x]^2*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e

+ f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(3/2) - 32*a*Hypergeometric2F1[2, 4, 7/2, ((a - b)*Tanh[e + f*x]^

2)/a]*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(5/2) - 32*b*Hypergeomet

ric2F1[2, 4, 7/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sinh[e + f*x]^2*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/

a]*(((a - b)*Tanh[e + f*x]^2)/a)^(5/2) + 32*a*Hypergeometric2F1[2, 4, 7/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sqrt[(

Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(7/2) + 32*b*Hypergeometric2F1[2, 4,

7/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sinh[e + f*x]^2*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)

*Tanh[e + f*x]^2)/a)^(7/2) + 15*a*Sqrt[((a - b)*Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)*Tanh[e + f*x]^2)/a^2]

+ 10*b*Sinh[e + f*x]^2*Sqrt[((a - b)*Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)*Tanh[e + f*x]^2)/a^2]))/(f*Sqrt[a

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

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 60.87, size = 35, normalized size = 0.23


method	result	size

default	\(\frac {\mathit {`\,int/indef0`\,}\left (\frac {\sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}{\cosh \left (f x +e \right )^{6}}, \sinh \left (f x +e \right )\right )}{f}\)	\(35\)
risch	\(\text {Expression too large to display}\)	\(155502204\)

int(sech(f*x+e)^5*(a+b*sinh(f*x+e)^2)^(1/2),x,method=_RETURNVERBOSE)

`int/indef0`(1/cosh(f*x+e)^6*(a+b*sinh(f*x+e)^2)^(1/2),sinh(f*x+e))/f

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

integrate(sech(f*x+e)^5*(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="maxima")

integrate(sqrt(b*sinh(f*x + e)^2 + a)*sech(f*x + e)^5, x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1805 vs. \(2 (135) = 270\).
time = 0.55, size = 3727, normalized size = 24.68 \begin {gather*} \text {Too large to display} \end {gather*}

integrate(sech(f*x+e)^5*(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="fricas")

[-1/16*(((3*a^2 - 4*a*b)*cosh(f*x + e)^8 + 8*(3*a^2 - 4*a*b)*cosh(f*x + e)*sinh(f*x + e)^7 + (3*a^2 - 4*a*b)*s

inh(f*x + e)^8 + 4*(3*a^2 - 4*a*b)*cosh(f*x + e)^6 + 4*(7*(3*a^2 - 4*a*b)*cosh(f*x + e)^2 + 3*a^2 - 4*a*b)*sin

h(f*x + e)^6 + 8*(7*(3*a^2 - 4*a*b)*cosh(f*x + e)^3 + 3*(3*a^2 - 4*a*b)*cosh(f*x + e))*sinh(f*x + e)^5 + 6*(3*

a^2 - 4*a*b)*cosh(f*x + e)^4 + 2*(35*(3*a^2 - 4*a*b)*cosh(f*x + e)^4 + 30*(3*a^2 - 4*a*b)*cosh(f*x + e)^2 + 9*

a^2 - 12*a*b)*sinh(f*x + e)^4 + 8*(7*(3*a^2 - 4*a*b)*cosh(f*x + e)^5 + 10*(3*a^2 - 4*a*b)*cosh(f*x + e)^3 + 3*

(3*a^2 - 4*a*b)*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(3*a^2 - 4*a*b)*cosh(f*x + e)^2 + 4*(7*(3*a^2 - 4*a*b)*cosh

(f*x + e)^6 + 15*(3*a^2 - 4*a*b)*cosh(f*x + e)^4 + 9*(3*a^2 - 4*a*b)*cosh(f*x + e)^2 + 3*a^2 - 4*a*b)*sinh(f*x

+ e)^2 + 3*a^2 - 4*a*b + 8*((3*a^2 - 4*a*b)*cosh(f*x + e)^7 + 3*(3*a^2 - 4*a*b)*cosh(f*x + e)^5 + 3*(3*a^2 -

4*a*b)*cosh(f*x + e)^3 + (3*a^2 - 4*a*b)*cosh(f*x + e))*sinh(f*x + e))*sqrt(-a + b)*log(((a - 2*b)*cosh(f*x +

e)^4 + 4*(a - 2*b)*cosh(f*x + e)*sinh(f*x + e)^3 + (a - 2*b)*sinh(f*x + e)^4 - 2*(3*a - 2*b)*cosh(f*x + e)^2 +

2*(3*(a - 2*b)*cosh(f*x + e)^2 - 3*a + 2*b)*sinh(f*x + e)^2 - 2*sqrt(2)*(cosh(f*x + e)^2 + 2*cosh(f*x + e)*si

nh(f*x + e) + sinh(f*x + e)^2 - 1)*sqrt(-a + b)*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f

*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2)) + 4*((a - 2*b)*cosh(f*x + e)^3 - (3*a - 2*b)*cos

h(f*x + e))*sinh(f*x + e) + a - 2*b)/(cosh(f*x + e)^4 + 4*cosh(f*x + e)*sinh(f*x + e)^3 + sinh(f*x + e)^4 + 2*

(3*cosh(f*x + e)^2 + 1)*sinh(f*x + e)^2 + 2*cosh(f*x + e)^2 + 4*(cosh(f*x + e)^3 + cosh(f*x + e))*sinh(f*x + e

) + 1)) - 2*sqrt(2)*((3*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e)^6 + 6*(3*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e)*sinh(f*

x + e)^5 + (3*a^2 - 5*a*b + 2*b^2)*sinh(f*x + e)^6 + (11*a^2 - 21*a*b + 10*b^2)*cosh(f*x + e)^4 + (15*(3*a^2 -

5*a*b + 2*b^2)*cosh(f*x + e)^2 + 11*a^2 - 21*a*b + 10*b^2)*sinh(f*x + e)^4 + 4*(5*(3*a^2 - 5*a*b + 2*b^2)*cos

h(f*x + e)^3 + (11*a^2 - 21*a*b + 10*b^2)*cosh(f*x + e))*sinh(f*x + e)^3 - (11*a^2 - 21*a*b + 10*b^2)*cosh(f*x

+ e)^2 + (15*(3*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e)^4 + 6*(11*a^2 - 21*a*b + 10*b^2)*cosh(f*x + e)^2 - 11*a^2

+ 21*a*b - 10*b^2)*sinh(f*x + e)^2 - 3*a^2 + 5*a*b - 2*b^2 + 2*(3*(3*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e)^5 + 2*

(11*a^2 - 21*a*b + 10*b^2)*cosh(f*x + e)^3 - (11*a^2 - 21*a*b + 10*b^2)*cosh(f*x + e))*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*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e

)^2)))/((a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^8 + 8*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)*sinh(f*x + e)^7 + (a^2 -

2*a*b + b^2)*f*sinh(f*x + e)^8 + 4*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^6 + 4*(7*(a^2 - 2*a*b + b^2)*f*cosh(f*

x + e)^2 + (a^2 - 2*a*b + b^2)*f)*sinh(f*x + e)^6 + 6*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^4 + 8*(7*(a^2 - 2*a*

b + b^2)*f*cosh(f*x + e)^3 + 3*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(35*(a^2 - 2*a*b + b^2

)*f*cosh(f*x + e)^4 + 30*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^2 + 3*(a^2 - 2*a*b + b^2)*f)*sinh(f*x + e)^4 + 4*

(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^2 + 8*(7*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^5 + 10*(a^2 - 2*a*b + b^2)*f*

cosh(f*x + e)^3 + 3*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(7*(a^2 - 2*a*b + b^2)*f*cosh(f*x

+ e)^6 + 15*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^4 + 9*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^2 + (a^2 - 2*a*b +

b^2)*f)*sinh(f*x + e)^2 + (a^2 - 2*a*b + b^2)*f + 8*((a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^7 + 3*(a^2 - 2*a*b +

b^2)*f*cosh(f*x + e)^5 + 3*(a^2 - 2*a*b + b^2)*f*cosh(f*x + e)^3 + (a^2 - 2*a*b + b^2)*f*cosh(f*x + e))*sinh(f

*x + e)), 1/8*(((3*a^2 - 4*a*b)*cosh(f*x + e)^8 + 8*(3*a^2 - 4*a*b)*cosh(f*x + e)*sinh(f*x + e)^7 + (3*a^2 - 4

*a*b)*sinh(f*x + e)^8 + 4*(3*a^2 - 4*a*b)*cosh(f*x + e)^6 + 4*(7*(3*a^2 - 4*a*b)*cosh(f*x + e)^2 + 3*a^2 - 4*a

*b)*sinh(f*x + e)^6 + 8*(7*(3*a^2 - 4*a*b)*cosh(f*x + e)^3 + 3*(3*a^2 - 4*a*b)*cosh(f*x + e))*sinh(f*x + e)^5

+ 6*(3*a^2 - 4*a*b)*cosh(f*x + e)^4 + 2*(35*(3*a^2 - 4*a*b)*cosh(f*x + e)^4 + 30*(3*a^2 - 4*a*b)*cosh(f*x + e)

^2 + 9*a^2 - 12*a*b)*sinh(f*x + e)^4 + 8*(7*(3*a^2 - 4*a*b)*cosh(f*x + e)^5 + 10*(3*a^2 - 4*a*b)*cosh(f*x + e)

^3 + 3*(3*a^2 - 4*a*b)*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(3*a^2 - 4*a*b)*cosh(f*x + e)^2 + 4*(7*(3*a^2 - 4*a*

b)*cosh(f*x + e)^6 + 15*(3*a^2 - 4*a*b)*cosh(f*x + e)^4 + 9*(3*a^2 - 4*a*b)*cosh(f*x + e)^2 + 3*a^2 - 4*a*b)*s

inh(f*x + e)^2 + 3*a^2 - 4*a*b + 8*((3*a^2 - 4*a*b)*cosh(f*x + e)^7 + 3*(3*a^2 - 4*a*b)*cosh(f*x + e)^5 + 3*(3

*a^2 - 4*a*b)*cosh(f*x + e)^3 + (3*a^2 - 4*a*b)*cosh(f*x + e))*sinh(f*x + e))*sqrt(a - b)*arctan(sqrt(2)*(cosh

(f*x + e)^2 + 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2 - 1)*sqrt(a - b)*sqrt((b*cosh(f*x + e)^2 + b*sin

h(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2))/(b*cosh(f*x + e)^

4 + 4*b*cosh(f*x + e)*sinh(f*x + e)^3 + b*sinh(f*x + e)^4 + 2*(2*a - b)*cosh(f*x + e)^2 + 2*(3*b*cosh(f*x + e)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a + b \sinh ^{2}{\left (e + f x \right )}} \operatorname {sech}^{5}{\left (e + f x \right )}\, dx \end {gather*}

Integral(sqrt(a + b*sinh(e + f*x)**2)*sech(e + f*x)**5, x)

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

integrate(sech(f*x+e)^5*(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="giac")

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a}}{{\mathrm {cosh}\left (e+f\,x\right )}^5} \,d x \end {gather*}

________________________________________________________________________________________

3.4.56 \(\int \text {sech}^5(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx\) [356]