\(\int \cosh ^2(e+f x) (a+b \sinh ^2(e+f x))^{3/2} \, dx\) [333]

Optimal result

Integrand size = 25, antiderivative size = 294 \[ \int \cosh ^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\frac {(3 a+b) \cosh (e+f x) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{15 f}+\frac {\cosh (e+f x) \sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{5 f}-\frac {\left (3 a^2+7 a b-2 b^2\right ) E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{15 b f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(9 a-b) \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{15 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {\left (3 a^2+7 a b-2 b^2\right ) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{15 b f} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.15 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.72 \[ \int \cosh ^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\frac {-16 i a \left (3 a^2+7 a b-2 b^2\right ) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )+16 i a \left (3 a^2-2 a b-b^2\right ) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} \operatorname {EllipticF}\left (i (e+f x),\frac {b}{a}\right )+\sqrt {2} b \left (48 a^2-28 a b+5 b^2+4 (9 a-2 b) b \cosh (2 (e+f x))+3 b^2 \cosh (4 (e+f x))\right ) \sinh (2 (e+f x))}{240 b f \sqrt {2 a-b+b \cosh (2 (e+f x))}} \] Input:

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

Output:

((-16*I)*a*(3*a^2 + 7*a*b - 2*b^2)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a] 
*EllipticE[I*(e + f*x), b/a] + (16*I)*a*(3*a^2 - 2*a*b - b^2)*Sqrt[(2*a - 
b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a] + Sqrt[2]*b*(48*a^ 
2 - 28*a*b + 5*b^2 + 4*(9*a - 2*b)*b*Cosh[2*(e + f*x)] + 3*b^2*Cosh[4*(e + 
 f*x)])*Sinh[2*(e + f*x)])/(240*b*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.18, 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, 403, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3671

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

\(\Big \downarrow \) 318

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

Input:

Int[Cosh[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]*((b*Sinh[e + f*x]*(1 + Sinh[e + f*x]^ 
2)^(3/2)*Sqrt[a + b*Sinh[e + f*x]^2])/5 + ((2*(3*a - b)*Sinh[e + f*x]*Sqrt 
[1 + Sinh[e + f*x]^2]*Sqrt[a + b*Sinh[e + f*x]^2])/3 + (((9*a - b)*Ellipti 
cF[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))]) 
+ (3*a^2 + 7*a*b - 2*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]*Sq 
rt[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))/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 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 = 5.87 (sec) , antiderivative size = 535, normalized size of antiderivative = 1.82

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 )+\left (9 \sqrt {-\frac {b}{a}}\, a b -5 \sqrt {-\frac {b}{a}}\, b^{2}\right ) \cosh \left (f x +e \right )^{4} \sinh \left (f x +e \right )+\left (6 \sqrt {-\frac {b}{a}}\, a^{2}-8 \sqrt {-\frac {b}{a}}\, a b +2 \sqrt {-\frac {b}{a}}\, b^{2}\right ) \cosh \left (f x +e \right )^{2} \sinh \left (f x +e \right )+6 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 )-8 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 +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 ) b^{2}+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 ) 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 -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 ) b^{2}}{15 \sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\)	\(535\)

Input:

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

Output:

1/15*(3*(-b/a)^(1/2)*b^2*cosh(f*x+e)^6*sinh(f*x+e)+(9*(-b/a)^(1/2)*a*b-5*( 
-b/a)^(1/2)*b^2)*cosh(f*x+e)^4*sinh(f*x+e)+(6*(-b/a)^(1/2)*a^2-8*(-b/a)^(1 
/2)*a*b+2*(-b/a)^(1/2)*b^2)*cosh(f*x+e)^2*sinh(f*x+e)+6*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))-8*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+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))*b^2+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))*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-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))*b^2)/(-b/a)^(1/2)/ 
cosh(f*x+e)/(a+b*sinh(f*x+e)^2)^(1/2)/f
                                                                                    
                                                                                    
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Giac [F]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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