\(\int \text {sech}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx\) [324]

Optimal result

Integrand size = 25, antiderivative size = 70 \[ \int \text {sech}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\frac {E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}} \] Output:

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)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.55 (sec) , antiderivative size = 148, normalized size of antiderivative = 2.11 \[ \int \text {sech}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\frac {2 i a \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )-2 i a \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} \operatorname {EllipticF}\left (i (e+f x),\frac {b}{a}\right )+\sqrt {2} (2 a-b+b \cosh (2 (e+f x))) \tanh (e+f x)}{2 f \sqrt {2 a-b+b \cosh (2 (e+f x))}} \] Input:

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

Output:

((2*I)*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/ 
a] - (2*I)*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[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 = 0.34 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.43, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {3042, 3671, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3671

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

\(\Big \downarrow \) 313

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

Input:

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

Output:

(Sqrt[Cosh[e + f*x]^2]*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + 
f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(f*Sqrt[1 + Sinh[e + f*x]^2]*Sqrt[(a + b 
*Sinh[e + f*x]^2)/(a*(1 + Sinh[e + f*x]^2))])
 

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 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(176\) vs. \(2(80)=160\).

Time = 2.54 (sec) , antiderivative size = 177, normalized size of antiderivative = 2.53

method	result	size
default	\(\frac {\sqrt {-\frac {b}{a}}\, b \sinh \left (f x +e \right )^{3}+b \sqrt {\frac {a +b \sinh \left (f x +e \right )^{2}}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-b \sqrt {\frac {a +b \sinh \left (f x +e \right )^{2}}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )+\sqrt {-\frac {b}{a}}\, a \sinh \left (f x +e \right )}{\sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\)	\(177\)

Input:

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

Output:

((-b/a)^(1/2)*b*sinh(f*x+e)^3+b*((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*((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))+(-b/a)^(1/2)*a*sinh(f*x+e))/(-b/a)^(1/2)/cosh(f*x+e)/(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. 535 vs. \(2 (80) = 160\).

Time = 0.11 (sec) , antiderivative size = 535, normalized size of antiderivative = 7.64 \[ \int \text {sech}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=-\frac {4 \, {\left (b \cosh \left (f x + e\right )^{2} + 2 \, b \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + b \sinh \left (f x + e\right )^{2} + b\right )} \sqrt {b} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}} \sqrt {\frac {a^{2} - a b}{b^{2}}} F(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}} {\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} - 8 \, a b + b^{2} + 4 \, {\left (2 \, a b - b^{2}\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}}{b^{2}}) + {\left ({\left (2 \, a - b\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (2 \, a - b\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + {\left (2 \, a - b\right )} \sinh \left (f x + e\right )^{2} - 2 \, {\left (b \cosh \left (f x + e\right )^{2} + 2 \, b \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + b \sinh \left (f x + e\right )^{2} + b\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}} + 2 \, a - b\right )} \sqrt {b} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}} E(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}} {\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} - 8 \, a b + b^{2} + 4 \, {\left (2 \, a b - b^{2}\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}}{b^{2}}) - \sqrt {2} {\left (b \cosh \left (f x + e\right ) + b \sinh \left (f x + e\right )\right )} \sqrt {\frac {b \cosh \left (f x + e\right )^{2} + b \sinh \left (f x + e\right )^{2} + 2 \, a - b}{\cosh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2}}}}{b f \cosh \left (f x + e\right )^{2} + 2 \, b f \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + b f \sinh \left (f x + e\right )^{2} + b f} \] Input:

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

Output:

-(4*(b*cosh(f*x + e)^2 + 2*b*cosh(f*x + e)*sinh(f*x + e) + b*sinh(f*x + e) 
^2 + b)*sqrt(b)*sqrt((2*b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*sqrt((a^2 - 
a*b)/b^2)*elliptic_f(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)*s 
qrt((a^2 - a*b)/b^2))/b^2) + ((2*a - b)*cosh(f*x + e)^2 + 2*(2*a - b)*cosh 
(f*x + e)*sinh(f*x + e) + (2*a - b)*sinh(f*x + e)^2 - 2*(b*cosh(f*x + e)^2 
 + 2*b*cosh(f*x + e)*sinh(f*x + e) + b*sinh(f*x + e)^2 + b)*sqrt((a^2 - a* 
b)/b^2) + 2*a - b)*sqrt(b)*sqrt((2*b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*e 
lliptic_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) - sqrt(2)*(b*cosh(f*x + e) + b*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)))/(b*f*cosh(f*x + e)^2 + 2*b*f*cosh(f 
*x + e)*sinh(f*x + e) + b*f*sinh(f*x + e)^2 + b*f)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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