\(\int \text {sech}^4(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx\) [325]

Optimal result

Integrand size = 25, antiderivative size = 206 \[ \int \text {sech}^4(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\frac {(2 a-b) E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 (a-b) f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {b \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 (a-b) f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {\text {sech}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 f} \] Output:

1/3*(2*a-b)*EllipticE(sinh(f*x+e)/(1+sinh(f*x+e)^2)^(1/2),(1-b/a)^(1/2))*s 
ech(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/(a-b)/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))*se 
ch(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/(a-b)/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e 
)^2)/a)^(1/2)+1/3*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 = 2.14 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.99 \[ \int \text {sech}^4(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\frac {8 i a (2 a-b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )-16 i a (a-b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} \operatorname {EllipticF}\left (i (e+f x),\frac {b}{a}\right )+\sqrt {2} \left (\left (8 a^2-4 b^2\right ) \cosh (2 (e+f x))+(2 a-b) (8 a-5 b+b \cosh (4 (e+f x)))\right ) \text {sech}^2(e+f x) \tanh (e+f x)}{24 (a-b) f \sqrt {2 a-b+b \cosh (2 (e+f x))}} \] Input:

Integrate[Sech[e + f*x]^4*Sqrt[a + b*Sinh[e + f*x]^2],x]
 

Output:

((8*I)*a*(2*a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e 
+ f*x), b/a] - (16*I)*a*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*El 
lipticF[I*(e + f*x), b/a] + Sqrt[2]*((8*a^2 - 4*b^2)*Cosh[2*(e + f*x)] + ( 
2*a - b)*(8*a - 5*b + b*Cosh[4*(e + f*x)]))*Sech[e + f*x]^2*Tanh[e + f*x]) 
/(24*(a - b)*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.21, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3042, 3671, 314, 25, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3671

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

\(\Big \downarrow \) 314

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 400

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

\(\Big \downarrow \) 313

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

\(\Big \downarrow \) 320

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

Input:

Int[Sech[e + f*x]^4*Sqrt[a + b*Sinh[e + f*x]^2],x]
 

Output:

(Sqrt[Cosh[e + f*x]^2]*Sech[e + f*x]*((Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f 
*x]^2])/(3*(1 + Sinh[e + f*x]^2)^(3/2)) + (((2*a - b)*EllipticE[ArcTan[Sin 
h[e + f*x]], 1 - b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/((a - b)*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])/((a 
 - b)*Sqrt[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 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 314

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] + Simp[1/(2*a* 
(p + 1))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(2*p + 3) + d 
*(2*(p + q + 1) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && LtQ[p, -1] && LtQ[0, 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 = 6.08 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.54

method	result	size
default	\(\frac {\left (2 \sqrt {-\frac {b}{a}}\, a b -\sqrt {-\frac {b}{a}}\, b^{2}\right ) \cosh \left (f x +e \right )^{4} \sinh \left (f x +e \right )+\left (2 \sqrt {-\frac {b}{a}}\, a^{2}-2 \sqrt {-\frac {b}{a}}\, a b \right ) \cosh \left (f x +e \right )^{2} \sinh \left (f x +e \right )+\sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, b \left (a \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-b \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-2 \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a +b \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )\right ) \cosh \left (f x +e \right )^{2}+\left (\sqrt {-\frac {b}{a}}\, a^{2}-2 \sqrt {-\frac {b}{a}}\, a b +\sqrt {-\frac {b}{a}}\, b^{2}\right ) \sinh \left (f x +e \right )}{3 \cosh \left (f x +e \right )^{3} \left (a -b \right ) \sqrt {-\frac {b}{a}}\, \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\)	\(318\)
risch	\(\text {Expression too large to display}\)	\(243643\)

Input:

int(sech(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/3*((2*(-b/a)^(1/2)*a*b-(-b/a)^(1/2)*b^2)*cosh(f*x+e)^4*sinh(f*x+e)+(2*(- 
b/a)^(1/2)*a^2-2*(-b/a)^(1/2)*a*b)*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*EllipticF(sinh(f*x+e)*(-b/ 
a)^(1/2),(1/b*a)^(1/2))-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+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/(a-b)/(-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. 2257 vs. \(2 (206) = 412\).

Time = 0.12 (sec) , antiderivative size = 2257, normalized size of antiderivative = 10.96 \[ \int \text {sech}^4(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\text {Too large to display} \] Input:

integrate(sech(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="fricas")
 

Output:

-1/3*(((4*a^2 - 4*a*b + b^2)*cosh(f*x + e)^6 + 6*(4*a^2 - 4*a*b + b^2)*cos 
h(f*x + e)*sinh(f*x + e)^5 + (4*a^2 - 4*a*b + b^2)*sinh(f*x + e)^6 + 3*(4* 
a^2 - 4*a*b + b^2)*cosh(f*x + e)^4 + 3*(5*(4*a^2 - 4*a*b + b^2)*cosh(f*x + 
 e)^2 + 4*a^2 - 4*a*b + b^2)*sinh(f*x + e)^4 + 4*(5*(4*a^2 - 4*a*b + b^2)* 
cosh(f*x + e)^3 + 3*(4*a^2 - 4*a*b + b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 
 3*(4*a^2 - 4*a*b + b^2)*cosh(f*x + e)^2 + 3*(5*(4*a^2 - 4*a*b + b^2)*cosh 
(f*x + e)^4 + 6*(4*a^2 - 4*a*b + b^2)*cosh(f*x + e)^2 + 4*a^2 - 4*a*b + b^ 
2)*sinh(f*x + e)^2 + 4*a^2 - 4*a*b + b^2 + 6*((4*a^2 - 4*a*b + b^2)*cosh(f 
*x + e)^5 + 2*(4*a^2 - 4*a*b + b^2)*cosh(f*x + e)^3 + (4*a^2 - 4*a*b + b^2 
)*cosh(f*x + e))*sinh(f*x + e) - 2*((2*a*b - b^2)*cosh(f*x + e)^6 + 6*(2*a 
*b - b^2)*cosh(f*x + e)*sinh(f*x + e)^5 + (2*a*b - b^2)*sinh(f*x + e)^6 + 
3*(2*a*b - b^2)*cosh(f*x + e)^4 + 3*(5*(2*a*b - b^2)*cosh(f*x + e)^2 + 2*a 
*b - b^2)*sinh(f*x + e)^4 + 4*(5*(2*a*b - b^2)*cosh(f*x + e)^3 + 3*(2*a*b 
- b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 3*(2*a*b - b^2)*cosh(f*x + e)^2 + 
3*(5*(2*a*b - b^2)*cosh(f*x + e)^4 + 6*(2*a*b - b^2)*cosh(f*x + e)^2 + 2*a 
*b - b^2)*sinh(f*x + e)^2 + 2*a*b - b^2 + 6*((2*a*b - b^2)*cosh(f*x + e)^5 
 + 2*(2*a*b - b^2)*cosh(f*x + e)^3 + (2*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...
 

Sympy [F]

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

integrate(sech(f*x+e)**4*(a+b*sinh(f*x+e)**2)**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate(sech(f*x+e)^4*(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)^4, x)
 

Giac [F]

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

integrate(sech(f*x+e)^4*(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)^4, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int(sech(f*x+e)^4*(a+b*sinh(f*x+e)^2)^(1/2),x)
 

Output:

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