\(\int (a+b \cosh (x))^{3/2} (A+B \cosh (x)) \, dx\) [108]

Optimal result

Integrand size = 17, antiderivative size = 181 \[ \int (a+b \cosh (x))^{3/2} (A+B \cosh (x)) \, dx=-\frac {2 i \left (20 a A b+3 a^2 B+9 b^2 B\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{15 b \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2 i \left (a^2-b^2\right ) (5 A b+3 a B) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{15 b \sqrt {a+b \cosh (x)}}+\frac {2}{15} (5 A b+3 a B) \sqrt {a+b \cosh (x)} \sinh (x)+\frac {2}{5} B (a+b \cosh (x))^{3/2} \sinh (x) \] Output:

-2/15*I*(20*A*a*b+3*B*a^2+9*B*b^2)*(a+b*cosh(x))^(1/2)*EllipticE(I*sinh(1/ 
2*x),2^(1/2)*(b/(a+b))^(1/2))/b/((a+b*cosh(x))/(a+b))^(1/2)+2/15*I*(a^2-b^ 
2)*(5*A*b+3*B*a)*((a+b*cosh(x))/(a+b))^(1/2)*InverseJacobiAM(1/2*I*x,2^(1/ 
2)*(b/(a+b))^(1/2))/b/(a+b*cosh(x))^(1/2)+2/15*(5*A*b+3*B*a)*(a+b*cosh(x)) 
^(1/2)*sinh(x)+2/5*B*(a+b*cosh(x))^(3/2)*sinh(x)
 

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.69 \[ \int (a+b \cosh (x))^{3/2} (A+B \cosh (x)) \, dx=\frac {2}{15} \sqrt {a+b \cosh (x)} \left (-\frac {i \left (\left (20 a A b+3 a^2 B+9 b^2 B\right ) E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )-(a-b) (5 A b+3 a B) \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}+(5 A b+6 a B+3 b B \cosh (x)) \sinh (x)\right ) \] Input:

Integrate[(a + b*Cosh[x])^(3/2)*(A + B*Cosh[x]),x]
 

Output:

(2*Sqrt[a + b*Cosh[x]]*(((-I)*((20*a*A*b + 3*a^2*B + 9*b^2*B)*EllipticE[(I 
/2)*x, (2*b)/(a + b)] - (a - b)*(5*A*b + 3*a*B)*EllipticF[(I/2)*x, (2*b)/( 
a + b)]))/(b*Sqrt[(a + b*Cosh[x])/(a + b)]) + (5*A*b + 6*a*B + 3*b*B*Cosh[ 
x])*Sinh[x]))/15
 

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.882, Rules used = {3042, 3232, 27, 3042, 3232, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}

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.

\(\displaystyle \int (a+b \cosh (x))^{3/2} (A+B \cosh (x)) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (a+b \sin \left (\frac {\pi }{2}+i x\right )\right )^{3/2} \left (A+B \sin \left (\frac {\pi }{2}+i x\right )\right )dx\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {2}{5} \int \frac {1}{2} \sqrt {a+b \cosh (x)} (5 a A+3 b B+(5 A b+3 a B) \cosh (x))dx+\frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} \int \sqrt {a+b \cosh (x)} (5 a A+3 b B+(5 A b+3 a B) \cosh (x))dx+\frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2}+\frac {1}{5} \int \sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )} \left (5 a A+3 b B+(5 A b+3 a B) \sin \left (i x+\frac {\pi }{2}\right )\right )dx\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {1}{5} \left (\frac {2}{3} \int \frac {15 A a^2+12 b B a+5 A b^2+\left (3 B a^2+20 A b a+9 b^2 B\right ) \cosh (x)}{2 \sqrt {a+b \cosh (x)}}dx+\frac {2}{3} \sinh (x) (3 a B+5 A b) \sqrt {a+b \cosh (x)}\right )+\frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} \left (\frac {1}{3} \int \frac {15 A a^2+12 b B a+5 A b^2+\left (3 B a^2+20 A b a+9 b^2 B\right ) \cosh (x)}{\sqrt {a+b \cosh (x)}}dx+\frac {2}{3} \sinh (x) (3 a B+5 A b) \sqrt {a+b \cosh (x)}\right )+\frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \sinh (x) (3 a B+5 A b) \sqrt {a+b \cosh (x)}+\frac {1}{3} \int \frac {15 A a^2+12 b B a+5 A b^2+\left (3 B a^2+20 A b a+9 b^2 B\right ) \sin \left (i x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx\right )\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (3 a^2 B+20 a A b+9 b^2 B\right ) \int \sqrt {a+b \cosh (x)}dx}{b}-\frac {\left (a^2-b^2\right ) (3 a B+5 A b) \int \frac {1}{\sqrt {a+b \cosh (x)}}dx}{b}\right )+\frac {2}{3} \sinh (x) (3 a B+5 A b) \sqrt {a+b \cosh (x)}\right )+\frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \sinh (x) (3 a B+5 A b) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (\frac {\left (3 a^2 B+20 a A b+9 b^2 B\right ) \int \sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}dx}{b}-\frac {\left (a^2-b^2\right ) (3 a B+5 A b) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx}{b}\right )\right )\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \sinh (x) (3 a B+5 A b) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (\frac {\left (3 a^2 B+20 a A b+9 b^2 B\right ) \sqrt {a+b \cosh (x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}}dx}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {\left (a^2-b^2\right ) (3 a B+5 A b) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx}{b}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \sinh (x) (3 a B+5 A b) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (\frac {\left (3 a^2 B+20 a A b+9 b^2 B\right ) \sqrt {a+b \cosh (x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (i x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {\left (a^2-b^2\right ) (3 a B+5 A b) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx}{b}\right )\right )\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \sinh (x) (3 a B+5 A b) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (-\frac {\left (a^2-b^2\right ) (3 a B+5 A b) \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 i \left (3 a^2 B+20 a A b+9 b^2 B\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}\right )\right )\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \sinh (x) (3 a B+5 A b) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (-\frac {\left (a^2-b^2\right ) (3 a B+5 A b) \sqrt {\frac {a+b \cosh (x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}}}dx}{b \sqrt {a+b \cosh (x)}}-\frac {2 i \left (3 a^2 B+20 a A b+9 b^2 B\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \sinh (x) (3 a B+5 A b) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (-\frac {\left (a^2-b^2\right ) (3 a B+5 A b) \sqrt {\frac {a+b \cosh (x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (i x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cosh (x)}}-\frac {2 i \left (3 a^2 B+20 a A b+9 b^2 B\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}\right )\right )\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2}+\frac {1}{5} \left (\frac {2}{3} \sinh (x) (3 a B+5 A b) \sqrt {a+b \cosh (x)}+\frac {1}{3} \left (\frac {2 i \left (a^2-b^2\right ) (3 a B+5 A b) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{b \sqrt {a+b \cosh (x)}}-\frac {2 i \left (3 a^2 B+20 a A b+9 b^2 B\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}\right )\right )\)

Input:

Int[(a + b*Cosh[x])^(3/2)*(A + B*Cosh[x]),x]
 

Output:

(2*B*(a + b*Cosh[x])^(3/2)*Sinh[x])/5 + ((((-2*I)*(20*a*A*b + 3*a^2*B + 9* 
b^2*B)*Sqrt[a + b*Cosh[x]]*EllipticE[(I/2)*x, (2*b)/(a + b)])/(b*Sqrt[(a + 
 b*Cosh[x])/(a + b)]) + ((2*I)*(a^2 - b^2)*(5*A*b + 3*a*B)*Sqrt[(a + b*Cos 
h[x])/(a + b)]*EllipticF[(I/2)*x, (2*b)/(a + b)])/(b*Sqrt[a + b*Cosh[x]])) 
/3 + (2*(5*A*b + 3*a*B)*Sqrt[a + b*Cosh[x]]*Sinh[x])/3)/5
 

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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3132

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a 
 + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, 
b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3134

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + 
b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)]   Int[Sqrt[a/(a + b) + ( 
b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 
, 0] &&  !GtQ[a + b, 0]
 

rule 3140

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S 
qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ 
{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3142

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a 
 + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]]   Int[1/Sqrt[a/(a + b) 
 + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - 
 b^2, 0] &&  !GtQ[a + b, 0]
 

rule 3231

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

rule 3232

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(972\) vs. \(2(166)=332\).

Time = 18.36 (sec) , antiderivative size = 973, normalized size of antiderivative = 5.38

method	result	size
default	\(\text {Expression too large to display}\)	\(973\)
parts	\(\text {Expression too large to display}\)	\(1066\)

Input:

int((a+b*cosh(x))^(3/2)*(A+B*cosh(x)),x,method=_RETURNVERBOSE)
 

Output:

2/15*(24*B*cosh(1/2*x)*(-2*b/(a-b))^(1/2)*sinh(1/2*x)^6*b^2+(20*A*(-2*b/(a 
-b))^(1/2)*b^2+36*B*(-2*b/(a-b))^(1/2)*a*b+24*B*(-2*b/(a-b))^(1/2)*b^2)*si 
nh(1/2*x)^4*cosh(1/2*x)+(10*A*(-2*b/(a-b))^(1/2)*a*b+10*A*(-2*b/(a-b))^(1/ 
2)*b^2+12*B*(-2*b/(a-b))^(1/2)*a^2+18*B*(-2*b/(a-b))^(1/2)*a*b+6*B*(-2*b/( 
a-b))^(1/2)*b^2)*sinh(1/2*x)^2*cosh(1/2*x)+15*A*a^2*(2*b/(a-b)*sinh(1/2*x) 
^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticF(cosh(1/2*x)*(-2*b/( 
a-b))^(1/2),1/2*(-2/b*(a-b))^(1/2))+20*A*a*b*(2*b/(a-b)*sinh(1/2*x)^2+(a+b 
)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^( 
1/2),1/2*(-2/b*(a-b))^(1/2))+5*A*b^2*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b)) 
^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2 
*(-2/b*(a-b))^(1/2))-40*A*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-si 
nh(1/2*x)^2)^(1/2)*EllipticE(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2/b*(a-b 
))^(1/2))*a*b+3*B*a^2*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1 
/2*x)^2)^(1/2)*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2/b*(a-b))^( 
1/2))+12*B*a*b*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2 
)^(1/2)*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2/b*(a-b))^(1/2))+9 
*B*b^2*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)* 
EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2/b*(a-b))^(1/2))-6*B*(2*b/ 
(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticE(co 
sh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2/b*(a-b))^(1/2))*a^2-18*B*(2*b/(a-b...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 623 vs. \(2 (164) = 328\).

Time = 0.10 (sec) , antiderivative size = 623, normalized size of antiderivative = 3.44 \[ \int (a+b \cosh (x))^{3/2} (A+B \cosh (x)) \, dx =\text {Too large to display} \] Input:

integrate((a+b*cosh(x))^(3/2)*(A+B*cosh(x)),x, algorithm="fricas")
 

Output:

-1/90*(8*sqrt(1/2)*((6*B*a^3 - 5*A*a^2*b - 18*B*a*b^2 - 15*A*b^3)*cosh(x)^ 
2 + 2*(6*B*a^3 - 5*A*a^2*b - 18*B*a*b^2 - 15*A*b^3)*cosh(x)*sinh(x) + (6*B 
*a^3 - 5*A*a^2*b - 18*B*a*b^2 - 15*A*b^3)*sinh(x)^2)*sqrt(b)*weierstrassPI 
nverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cosh 
(x) + 3*b*sinh(x) + 2*a)/b) + 24*sqrt(1/2)*((3*B*a^2*b + 20*A*a*b^2 + 9*B* 
b^3)*cosh(x)^2 + 2*(3*B*a^2*b + 20*A*a*b^2 + 9*B*b^3)*cosh(x)*sinh(x) + (3 
*B*a^2*b + 20*A*a*b^2 + 9*B*b^3)*sinh(x)^2)*sqrt(b)*weierstrassZeta(4/3*(4 
*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPInverse(4/3*(4 
*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cosh(x) + 3*b*sin 
h(x) + 2*a)/b)) - 3*(3*B*b^3*cosh(x)^4 + 3*B*b^3*sinh(x)^4 - 3*B*b^3 + 2*( 
6*B*a*b^2 + 5*A*b^3)*cosh(x)^3 + 2*(6*B*b^3*cosh(x) + 6*B*a*b^2 + 5*A*b^3) 
*sinh(x)^3 - 4*(3*B*a^2*b + 20*A*a*b^2 + 9*B*b^3)*cosh(x)^2 + 2*(9*B*b^3*c 
osh(x)^2 - 6*B*a^2*b - 40*A*a*b^2 - 18*B*b^3 + 3*(6*B*a*b^2 + 5*A*b^3)*cos 
h(x))*sinh(x)^2 - 2*(6*B*a*b^2 + 5*A*b^3)*cosh(x) + 2*(6*B*b^3*cosh(x)^3 - 
 6*B*a*b^2 - 5*A*b^3 + 3*(6*B*a*b^2 + 5*A*b^3)*cosh(x)^2 - 4*(3*B*a^2*b + 
20*A*a*b^2 + 9*B*b^3)*cosh(x))*sinh(x))*sqrt(b*cosh(x) + a))/(b^2*cosh(x)^ 
2 + 2*b^2*cosh(x)*sinh(x) + b^2*sinh(x)^2)
 

Sympy [F]

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

integrate((a+b*cosh(x))**(3/2)*(A+B*cosh(x)),x)
 

Output:

Integral((A + B*cosh(x))*(a + b*cosh(x))**(3/2), x)
 

Maxima [F]

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

integrate((a+b*cosh(x))^(3/2)*(A+B*cosh(x)),x, algorithm="maxima")
 

Output:

integrate((B*cosh(x) + A)*(b*cosh(x) + a)^(3/2), x)
 

Giac [F]

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

integrate((a+b*cosh(x))^(3/2)*(A+B*cosh(x)),x, algorithm="giac")
 

Output:

integrate((B*cosh(x) + A)*(b*cosh(x) + a)^(3/2), x)
 

Mupad [F(-1)]

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

int((A + B*cosh(x))*(a + b*cosh(x))^(3/2),x)
 

Output:

int((A + B*cosh(x))*(a + b*cosh(x))^(3/2), x)
 

Reduce [F]

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

int((a+b*cosh(x))^(3/2)*(A+B*cosh(x)),x)
 

Output:

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