\(\int \text {sech}^2(e+f x) (a+b \sinh ^2(e+f x))^{3/2} \, dx\) [335]

Optimal result

Integrand size = 25, antiderivative size = 174 \[ \int \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\frac {(a-2 b) E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {b \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {b \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{f} \] Output:

(a-2*b)*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)/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 
*sinh(f*x+e)^2)^(1/2)/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)+b*(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 = 0.93 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.92 \[ \int \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\frac {2 i a (a-2 b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )+(a-b) \left (-2 i a \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} \operatorname {EllipticF}\left (i (e+f x),\frac {b}{a}\right )+\sqrt {2} (2 a-b+b \cosh (2 (e+f x))) \tanh (e+f x)\right )}{2 f \sqrt {2 a-b+b \cosh (2 (e+f x))}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.60, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 3671, 315, 27, 406, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3671

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

\(\Big \downarrow \) 315

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

Input:

Int[Sech[e + f*x]^2*(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])/Sqrt[1 + Sinh[e + f*x]^2] + b*((EllipticF[ArcTan[Sinh[e + 
f*x]], 1 - b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(Sqrt[1 + Sinh[e + f*x]^2]*Sq 
rt[(a + b*Sinh[e + f*x]^2)/(a*(1 + Sinh[e + f*x]^2))]) - (a - 2*b)*((Sinh[ 
e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(b*Sqrt[1 + Sinh[e + f*x]^2]) - (Ell 
ipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(b*Sqr 
t[1 + Sinh[e + f*x]^2]*Sqrt[(a + b*Sinh[e + f*x]^2)/(a*(1 + Sinh[e + f*x]^ 
2))])))))/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 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 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 406

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[e   Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim 
p[f   Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, 
f, p, q}, x]
 

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 = 3.85 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.92

method	result	size
default	\(\frac {\sqrt {-\frac {b}{a}}\, a b \sinh \left (f x +e \right )^{3}-\sqrt {-\frac {b}{a}}\, b^{2} \sinh \left (f x +e \right )^{3}+2 a \sqrt {\frac {a +b \sinh \left (f x +e \right )^{2}}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b -2 \sqrt {\frac {a +b \sinh \left (f x +e \right )^{2}}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}-\sqrt {\frac {a +b \sinh \left (f x +e \right )^{2}}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b +2 \sqrt {\frac {a +b \sinh \left (f x +e \right )^{2}}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}+\sqrt {-\frac {b}{a}}\, a^{2} \sinh \left (f x +e \right )-\sqrt {-\frac {b}{a}}\, a b \sinh \left (f x +e \right )}{\sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\)	\(334\)

Input:

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

Output:

((-b/a)^(1/2)*a*b*sinh(f*x+e)^3-(-b/a)^(1/2)*b^2*sinh(f*x+e)^3+2*a*((a+b*s 
inh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-b/a)^ 
(1/2),(1/b*a)^(1/2))*b-2*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/ 
2)*EllipticF(sinh(f*x+e)*(-b/a)^(1/2),(1/b*a)^(1/2))*b^2-((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))*a*b+2*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*Ellip 
ticE(sinh(f*x+e)*(-b/a)^(1/2),(1/b*a)^(1/2))*b^2+(-b/a)^(1/2)*a^2*sinh(f*x 
+e)-(-b/a)^(1/2)*a*b*sinh(f*x+e))/(-b/a)^(1/2)/cosh(f*x+e)/(a+b*sinh(f*x+e 
)^2)^(1/2)/f
 

Fricas [F]

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

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

Output:

integral((b*sech(f*x + e)^2*sinh(f*x + e)^2 + a*sech(f*x + e)^2)*sqrt(b*si 
nh(f*x + e)^2 + a), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate(sech(f*x+e)^2*(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)^2, x)
 

Giac [F]

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

integrate(sech(f*x+e)^2*(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)^2, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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