∫ sech3(e + fx)∘ ____________ a + b sinh2(e + fx) dx [355]

Optimal. Leaf size=86 \[ \frac {a \text {ArcTan}\left (\frac {\sqrt {a-b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{2 \sqrt {a-b} f}+\frac {\text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{2 f} \]

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

________________________________________________________________________________________

Rubi [A]
time = 0.06, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3269, 386, 385, 209} \begin {gather*} \frac {a \text {ArcTan}\left (\frac {\sqrt {a-b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{2 f \sqrt {a-b}}+\frac {\tanh (e+f x) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{2 f} \end {gather*}

Int[Sech[e + f*x]^3*Sqrt[a + b*Sinh[e + f*x]^2],x]

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

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

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(175\) vs. \(2(86)=172\).
time = 1.13, size = 175, normalized size = 2.03 \begin {gather*} \frac {\sinh (e+f x) \left (\sqrt {2} a \tanh ^{-1}\left (\frac {\sqrt {-\frac {(a-b) \sinh ^2(e+f x)}{a}}}{\sqrt {1+\frac {b \sinh ^2(e+f x)}{a}}}\right ) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}}+(2 a-b+b \cosh (2 (e+f x))) \text {sech}^2(e+f x) \sqrt {-\frac {(a-b) \sinh ^2(e+f x)}{a}}\right )}{4 f \sqrt {-\frac {(a-b) \sinh ^2(e+f x)}{a}} \sqrt {a+b \sinh ^2(e+f x)}} \end {gather*}

Integrate[Sech[e + f*x]^3*Sqrt[a + b*Sinh[e + f*x]^2],x]

(Sinh[e + f*x]*(Sqrt[2]*a*ArcTanh[Sqrt[-(((a - b)*Sinh[e + f*x]^2)/a)]/Sqrt[1 + (b*Sinh[e + f*x]^2)/a]]*Sqrt[(

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

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

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.15, size = 35, normalized size = 0.41


method	result	size

default	\(\frac {\mathit {`\,int/indef0`\,}\left (\frac {\sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}{\cosh \left (f x +e \right )^{4}}, \sinh \left (f x +e \right )\right )}{f}\)	\(35\)

int(sech(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(1/2),x,method=_RETURNVERBOSE)

`int/indef0`(1/cosh(f*x+e)^4*(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)^3*(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)^3, x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 605 vs. \(2 (74) = 148\).
time = 0.44, size = 1327, normalized size = 15.43 \begin {gather*} \text {Too large to display} \end {gather*}

integrate(sech(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="fricas")

[-1/4*((a*cosh(f*x + e)^4 + 4*a*cosh(f*x + e)*sinh(f*x + e)^3 + a*sinh(f*x + e)^4 + 2*a*cosh(f*x + e)^2 + 2*(3

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

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

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

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

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

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

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

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

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

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

(a - b)*f + 4*((a - b)*f*cosh(f*x + e)^3 + (a - b)*f*cosh(f*x + e))*sinh(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}^{3}{\left (e + f x \right )}\, dx \end {gather*}

integrate(sech(f*x+e)**3*(a+b*sinh(f*x+e)**2)**(1/2),x)

Integral(sqrt(a + b*sinh(e + f*x)**2)*sech(e + f*x)**3, 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)^3*(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 )}^3} \,d x \end {gather*}

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

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

________________________________________________________________________________________

3.4.55 \(\int \text {sech}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx\) [355]