\(\int \text {sech}^4(e+f x) (a+b \sinh ^2(e+f x))^{3/2} \, dx\) [336]

Optimal result

Integrand size = 25, antiderivative size = 193 \[ \int \text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\frac {2 (a+b) E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {b \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(a-b) \text {sech}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 f} \] Output:

2/3*(a+b)*EllipticE(sinh(f*x+e)/(1+sinh(f*x+e)^2)^(1/2),(1-b/a)^(1/2))*sec 
h(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a) 
^(1/2)-1/3*b*InverseJacobiAM(arctan(sinh(f*x+e)),(1-b/a)^(1/2))*sech(f*x+e 
)*(a+b*sinh(f*x+e)^2)^(1/2)/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)+ 
1/3*(a-b)*sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)/f
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.60 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.02 \[ \int \text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\frac {4 i a (a+b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )-2 i a (2 a+b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} \operatorname {EllipticF}\left (i (e+f x),\frac {b}{a}\right )+\frac {\left (8 a^2-3 a b+b^2+\left (4 a^2+6 a b-2 b^2\right ) \cosh (2 (e+f x))+b (a+b) \cosh (4 (e+f x))\right ) \text {sech}^2(e+f x) \tanh (e+f x)}{\sqrt {2}}}{6 f \sqrt {2 a-b+b \cosh (2 (e+f x))}} \] Input:

Integrate[Sech[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^(3/2),x]
 

Output:

((4*I)*a*(a + b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + 
f*x), b/a] - (2*I)*a*(2*a + b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*Ell 
ipticF[I*(e + f*x), b/a] + ((8*a^2 - 3*a*b + b^2 + (4*a^2 + 6*a*b - 2*b^2) 
*Cosh[2*(e + f*x)] + b*(a + b)*Cosh[4*(e + f*x)])*Sech[e + f*x]^2*Tanh[e + 
 f*x])/Sqrt[2])/(6*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.23, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3042, 3671, 315, 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.

\(\displaystyle \int \text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a-b \sin (i e+i f x)^2\right )^{3/2}}{\cos (i e+i f x)^4}dx\)

\(\Big \downarrow \) 3671

\(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \int \frac {\left (b \sinh ^2(e+f x)+a\right )^{3/2}}{\left (\sinh ^2(e+f x)+1\right )^{5/2}}d\sinh (e+f x)}{f}\)

\(\Big \downarrow \) 315

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

\(\Big \downarrow \) 400

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

\(\Big \downarrow \) 313

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

\(\Big \downarrow \) 320

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

Input:

Int[Sech[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^(3/2),x]
 

Output:

(Sqrt[Cosh[e + f*x]^2]*Sech[e + f*x]*(((a - b)*Sinh[e + f*x]*Sqrt[a + b*Si 
nh[e + f*x]^2])/(3*(1 + Sinh[e + f*x]^2)^(3/2)) + ((2*(a + b)*EllipticE[Ar 
cTan[Sinh[e + f*x]], 1 - b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(Sqrt[1 + Sinh[ 
e + f*x]^2]*Sqrt[(a + b*Sinh[e + f*x]^2)/(a*(1 + Sinh[e + f*x]^2))]) - (b* 
EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(Sq 
rt[1 + Sinh[e + f*x]^2]*Sqrt[(a + b*Sinh[e + f*x]^2)/(a*(1 + Sinh[e + f*x] 
^2))]))/3))/f
 

Defintions of rubi rules used

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 315

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), 
x] - Simp[1/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S 
imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) 
*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 
1] && GtQ[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 400

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]
 

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 = 5.04 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.68

method	result	size
default	\(\frac {\left (2 \sqrt {-\frac {b}{a}}\, a b +2 \sqrt {-\frac {b}{a}}\, b^{2}\right ) \cosh \left (f x +e \right )^{4} \sinh \left (f x +e \right )+\left (2 \sqrt {-\frac {b}{a}}\, a^{2}+\sqrt {-\frac {b}{a}}\, a b -3 \sqrt {-\frac {b}{a}}\, b^{2}\right ) \cosh \left (f x +e \right )^{2} \sinh \left (f x +e \right )+\sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, b \left (a \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )+2 b \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-2 \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a -2 b \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )\right ) \cosh \left (f x +e \right )^{2}+\left (\sqrt {-\frac {b}{a}}\, a^{2}-2 \sqrt {-\frac {b}{a}}\, a b +\sqrt {-\frac {b}{a}}\, b^{2}\right ) \sinh \left (f x +e \right )}{3 \cosh \left (f x +e \right )^{3} \sqrt {-\frac {b}{a}}\, \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\)	\(324\)

Input:

int(sech(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/3*((2*(-b/a)^(1/2)*a*b+2*(-b/a)^(1/2)*b^2)*cosh(f*x+e)^4*sinh(f*x+e)+(2* 
(-b/a)^(1/2)*a^2+(-b/a)^(1/2)*a*b-3*(-b/a)^(1/2)*b^2)*cosh(f*x+e)^2*sinh(f 
*x+e)+(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*b*(a*Ellipti 
cF(sinh(f*x+e)*(-b/a)^(1/2),(1/b*a)^(1/2))+2*b*EllipticF(sinh(f*x+e)*(-b/a 
)^(1/2),(1/b*a)^(1/2))-2*EllipticE(sinh(f*x+e)*(-b/a)^(1/2),(1/b*a)^(1/2)) 
*a-2*b*EllipticE(sinh(f*x+e)*(-b/a)^(1/2),(1/b*a)^(1/2)))*cosh(f*x+e)^2+(( 
-b/a)^(1/2)*a^2-2*(-b/a)^(1/2)*a*b+(-b/a)^(1/2)*b^2)*sinh(f*x+e))/cosh(f*x 
+e)^3/(-b/a)^(1/2)/(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. 2071 vs. \(2 (193) = 386\).

Time = 0.12 (sec) , antiderivative size = 2071, normalized size of antiderivative = 10.73 \[ \int \text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Too large to display} \] Input:

integrate(sech(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="fricas")
 

Output:

-2/3*(((2*a^2 + a*b - b^2)*cosh(f*x + e)^6 + 6*(2*a^2 + a*b - b^2)*cosh(f* 
x + e)*sinh(f*x + e)^5 + (2*a^2 + a*b - b^2)*sinh(f*x + e)^6 + 3*(2*a^2 + 
a*b - b^2)*cosh(f*x + e)^4 + 3*(5*(2*a^2 + a*b - b^2)*cosh(f*x + e)^2 + 2* 
a^2 + a*b - b^2)*sinh(f*x + e)^4 + 4*(5*(2*a^2 + a*b - b^2)*cosh(f*x + e)^ 
3 + 3*(2*a^2 + a*b - b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 3*(2*a^2 + a*b 
- b^2)*cosh(f*x + e)^2 + 3*(5*(2*a^2 + a*b - b^2)*cosh(f*x + e)^4 + 6*(2*a 
^2 + a*b - b^2)*cosh(f*x + e)^2 + 2*a^2 + a*b - b^2)*sinh(f*x + e)^2 + 2*a 
^2 + a*b - b^2 + 6*((2*a^2 + a*b - b^2)*cosh(f*x + e)^5 + 2*(2*a^2 + a*b - 
 b^2)*cosh(f*x + e)^3 + (2*a^2 + a*b - b^2)*cosh(f*x + e))*sinh(f*x + e) - 
 2*((a*b + b^2)*cosh(f*x + e)^6 + 6*(a*b + b^2)*cosh(f*x + e)*sinh(f*x + e 
)^5 + (a*b + b^2)*sinh(f*x + e)^6 + 3*(a*b + b^2)*cosh(f*x + e)^4 + 3*(5*( 
a*b + b^2)*cosh(f*x + e)^2 + a*b + b^2)*sinh(f*x + e)^4 + 4*(5*(a*b + b^2) 
*cosh(f*x + e)^3 + 3*(a*b + b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 3*(a*b + 
 b^2)*cosh(f*x + e)^2 + 3*(5*(a*b + b^2)*cosh(f*x + e)^4 + 6*(a*b + b^2)*c 
osh(f*x + e)^2 + a*b + b^2)*sinh(f*x + e)^2 + a*b + b^2 + 6*((a*b + b^2)*c 
osh(f*x + e)^5 + 2*(a*b + b^2)*cosh(f*x + e)^3 + (a*b + 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^2 - 8*a*b + b^2 + 4*(2*a* 
b - b^2)*sqrt((a^2 - a*b)/b^2))/b^2) - ((2*a^2 - a*b)*cosh(f*x + e)^6 +...
 

Sympy [F(-1)]

Timed out. \[ \int \text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \] Input:

integrate(sech(f*x+e)**4*(a+b*sinh(f*x+e)**2)**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \operatorname {sech}\left (f x + e\right )^{4} \,d x } \] Input:

integrate(sech(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="maxima")
 

Output:

integrate((b*sinh(f*x + e)^2 + a)^(3/2)*sech(f*x + e)^4, x)
 

Giac [F]

\[ \int \text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \operatorname {sech}\left (f x + e\right )^{4} \,d x } \] Input:

integrate(sech(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")
 

Output:

integrate((b*sinh(f*x + e)^2 + a)^(3/2)*sech(f*x + e)^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int \frac {{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}}{{\mathrm {cosh}\left (e+f\,x\right )}^4} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\left (\int \sqrt {\sinh \left (f x +e \right )^{2} b +a}\, \mathrm {sech}\left (f x +e \right )^{4} \sinh \left (f x +e \right )^{2}d x \right ) b +\left (\int \sqrt {\sinh \left (f x +e \right )^{2} b +a}\, \mathrm {sech}\left (f x +e \right )^{4}d x \right ) a \] Input:

int(sech(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(3/2),x)
 

Output:

int(sqrt(sinh(e + f*x)**2*b + a)*sech(e + f*x)**4*sinh(e + f*x)**2,x)*b + 
int(sqrt(sinh(e + f*x)**2*b + a)*sech(e + f*x)**4,x)*a