\(\int \cosh ^4(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx\) [321]

Optimal result

Integrand size = 25, antiderivative size = 300 \[ \int \cosh ^4(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\frac {(a+3 b) \cosh (e+f x) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{15 b f}+\frac {\cosh ^3(e+f x) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{5 f}+\frac {\left (2 a^2-7 a b-3 b^2\right ) E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{15 b^2 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {(a-9 b) \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{15 b f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {\left (2 a^2-7 a b-3 b^2\right ) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{15 b^2 f} \] Output:

1/15*(a+3*b)*cosh(f*x+e)*sinh(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/b/f+1/5*cos 
h(f*x+e)^3*sinh(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/f+1/15*(2*a^2-7*a*b-3*b^2 
)*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)/b^2/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/ 
2)-1/15*(a-9*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)/b/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^( 
1/2)-1/15*(2*a^2-7*a*b-3*b^2)*(a+b*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)/b^2/f
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.21 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.70 \[ \int \cosh ^4(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\frac {16 i a \left (2 a^2-7 a b-3 b^2\right ) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )-32 i a \left (a^2-4 a b+3 b^2\right ) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} \operatorname {EllipticF}\left (i (e+f x),\frac {b}{a}\right )+\sqrt {2} b \left (8 a^2+32 a b-15 b^2+4 b (4 a+3 b) \cosh (2 (e+f x))+3 b^2 \cosh (4 (e+f x))\right ) \sinh (2 (e+f x))}{240 b^2 f \sqrt {2 a-b+b \cosh (2 (e+f x))}} \] Input:

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

Output:

((16*I)*a*(2*a^2 - 7*a*b - 3*b^2)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]* 
EllipticE[I*(e + f*x), b/a] - (32*I)*a*(a^2 - 4*a*b + 3*b^2)*Sqrt[(2*a - b 
 + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + Sqrt[2]*b*(8*a^2 
+ 32*a*b - 15*b^2 + 4*b*(4*a + 3*b)*Cosh[2*(e + f*x)] + 3*b^2*Cosh[4*(e + 
f*x)])*Sinh[2*(e + f*x)])/(240*b^2*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.16, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3042, 3671, 318, 25, 403, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3671

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

\(\Big \downarrow \) 318

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 403

\(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1} \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{5 b}-\frac {\frac {1}{3} \int \frac {\left (2 a^2-7 b a-3 b^2\right ) \sinh ^2(e+f x)+a (a-9 b)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+\frac {2}{3} (a-3 b) \sqrt {\sinh ^2(e+f x)+1} \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{5 b}\right )}{f}\)

\(\Big \downarrow \) 406

\(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1} \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{5 b}-\frac {\frac {1}{3} \left (\left (2 a^2-7 a b-3 b^2\right ) \int \frac {\sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+a (a-9 b) \int \frac {1}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)\right )+\frac {2}{3} (a-3 b) \sqrt {\sinh ^2(e+f x)+1} \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{5 b}\right )}{f}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1} \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{5 b}-\frac {\frac {1}{3} \left (\left (2 a^2-7 a b-3 b^2\right ) \int \frac {\sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+\frac {(a-9 b) \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}\right )+\frac {2}{3} (a-3 b) \sqrt {\sinh ^2(e+f x)+1} \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{5 b}\right )}{f}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1} \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{5 b}-\frac {\frac {1}{3} \left (\left (2 a^2-7 a b-3 b^2\right ) \left (\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{b \sqrt {\sinh ^2(e+f x)+1}}-\frac {\int \frac {\sqrt {b \sinh ^2(e+f x)+a}}{\left (\sinh ^2(e+f x)+1\right )^{3/2}}d\sinh (e+f x)}{b}\right )+\frac {(a-9 b) \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}\right )+\frac {2}{3} (a-3 b) \sqrt {\sinh ^2(e+f x)+1} \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{5 b}\right )}{f}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1} \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{5 b}-\frac {\frac {1}{3} \left (\left (2 a^2-7 a b-3 b^2\right ) \left (\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{b \sqrt {\sinh ^2(e+f x)+1}}-\frac {\sqrt {a+b \sinh ^2(e+f x)} E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{b \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}\right )+\frac {(a-9 b) \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}\right )+\frac {2}{3} (a-3 b) \sqrt {\sinh ^2(e+f x)+1} \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{5 b}\right )}{f}\)

Input:

Int[Cosh[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[1 + Sinh[e + f*x 
]^2]*(a + b*Sinh[e + f*x]^2)^(3/2))/(5*b) - ((2*(a - 3*b)*Sinh[e + f*x]*Sq 
rt[1 + Sinh[e + f*x]^2]*Sqrt[a + b*Sinh[e + f*x]^2])/3 + (((a - 9*b)*Ellip 
ticF[ArcTan[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))] 
) + (2*a^2 - 7*a*b - 3*b^2)*((Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/( 
b*Sqrt[1 + Sinh[e + f*x]^2]) - (EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]* 
Sqrt[a + b*Sinh[e + f*x]^2])/(b*Sqrt[1 + Sinh[e + f*x]^2]*Sqrt[(a + b*Sinh 
[e + f*x]^2)/(a*(1 + Sinh[e + f*x]^2))])))/3)/(5*b)))/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 318

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S 
imp[1/(b*(2*(p + q) + 1))   Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b 
*c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 
 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G 
tQ[q, 1] && NeQ[2*(p + q) + 1, 0] &&  !IGtQ[p, 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 403

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

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 = 6.22 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.74

method	result	size
default	\(\frac {3 \sqrt {-\frac {b}{a}}\, b^{2} \cosh \left (f x +e \right )^{6} \sinh \left (f x +e \right )+4 \sqrt {-\frac {b}{a}}\, a b \cosh \left (f x +e \right )^{4} \sinh \left (f x +e \right )+\left (\sqrt {-\frac {b}{a}}\, a^{2}+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 )+a^{2} \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}}\, \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )+2 a \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}}\, \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b -3 \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}}\, \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}-2 \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}}\, \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{2}+7 \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}}\, \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b +3 \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}}\, \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}}{15 b \sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\)	\(521\)

Input:

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

Output:

1/15*(3*(-b/a)^(1/2)*b^2*cosh(f*x+e)^6*sinh(f*x+e)+4*(-b/a)^(1/2)*a*b*cosh 
(f*x+e)^4*sinh(f*x+e)+((-b/a)^(1/2)*a^2+2*(-b/a)^(1/2)*a*b-3*(-b/a)^(1/2)* 
b^2)*cosh(f*x+e)^2*sinh(f*x+e)+a^2*(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh 
(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-b/a)^(1/2),(1/b*a)^(1/2))+2*a*(b/ 
a*cosh(f*x+e)^2+(a-b)/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-3*(b/a*cosh(f*x+e)^2+(a-b)/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^2-2*(b/ 
a*cosh(f*x+e)^2+(a-b)/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^2+7*(b/a*cosh(f*x+e)^2+(a-b)/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+3*( 
b/a*cosh(f*x+e)^2+(a-b)/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^2)/b/(-b/a)^(1/2)/cosh(f*x+e)/(a+b*sinh(f 
*x+e)^2)^(1/2)/f
                                                                                    
                                                                                    
 

Fricas [F]

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

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

Output:

integral(sqrt(b*sinh(f*x + e)^2 + a)*cosh(f*x + e)^4, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Giac [F]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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