\(\int (a+b \cosh (x))^{5/2} (A+B \cosh (x)) \, dx\) [107]

Optimal result

Integrand size = 17, antiderivative size = 233 \[ \int (a+b \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=-\frac {2 i \left (161 a^2 A b+63 A b^3+15 a^3 B+145 a b^2 B\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{105 b \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2 i \left (a^2-b^2\right ) \left (56 a A b+15 a^2 B+25 b^2 B\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{105 b \sqrt {a+b \cosh (x)}}+\frac {2}{105} \left (56 a A b+15 a^2 B+25 b^2 B\right ) \sqrt {a+b \cosh (x)} \sinh (x)+\frac {2}{35} (7 A b+5 a B) (a+b \cosh (x))^{3/2} \sinh (x)+\frac {2}{7} B (a+b \cosh (x))^{5/2} \sinh (x) \]

[Out]

2/35*(7*A*b+5*B*a)*(a+b*cosh(x))^(3/2)*sinh(x)+2/7*B*(a+b*cosh(x))^(5/2)*sinh(x)+2/105*(56*A*a*b+15*B*a^2+25*B
*b^2)*sinh(x)*(a+b*cosh(x))^(1/2)-2/105*I*(161*A*a^2*b+63*A*b^3+15*B*a^3+145*B*a*b^2)*(cosh(1/2*x)^2)^(1/2)/co
sh(1/2*x)*EllipticE(I*sinh(1/2*x),2^(1/2)*(b/(a+b))^(1/2))*(a+b*cosh(x))^(1/2)/b/((a+b*cosh(x))/(a+b))^(1/2)+2
/105*I*(a^2-b^2)*(56*A*a*b+15*B*a^2+25*B*b^2)*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticF(I*sinh(1/2*x),2^(1/2
)*(b/(a+b))^(1/2))*((a+b*cosh(x))/(a+b))^(1/2)/b/(a+b*cosh(x))^(1/2)

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2832, 2831, 2742, 2740, 2734, 2732} \[ \int (a+b \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\frac {2}{105} \sinh (x) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sqrt {a+b \cosh (x)}+\frac {2 i \left (a^2-b^2\right ) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{105 b \sqrt {a+b \cosh (x)}}-\frac {2 i \left (15 a^3 B+161 a^2 A b+145 a b^2 B+63 A b^3\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{105 b \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2}{35} \sinh (x) (5 a B+7 A b) (a+b \cosh (x))^{3/2}+\frac {2}{7} B \sinh (x) (a+b \cosh (x))^{5/2} \]

[In]

Int[(a + b*Cosh[x])^(5/2)*(A + B*Cosh[x]),x]

[Out]

(((-2*I)/105)*(161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*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)/105)*(a^2 - b^2)*(56*a*A*b + 15*a^2*B + 25*b^2*B)*Sqrt[(a
+ b*Cosh[x])/(a + b)]*EllipticF[(I/2)*x, (2*b)/(a + b)])/(b*Sqrt[a + b*Cosh[x]]) + (2*(56*a*A*b + 15*a^2*B +
25*b^2*B)*Sqrt[a + b*Cosh[x]]*Sinh[x])/105 + (2*(7*A*b + 5*a*B)*(a + b*Cosh[x])^(3/2)*Sinh[x])/35 + (2*B*(a +
b*Cosh[x])^(5/2)*Sinh[x])/7

Rule 2732

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 2734

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[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 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/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 2742

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[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 2831

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[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 2832

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] + Dist[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Sim
p[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]

Rubi steps \begin{align*} \text {integral}& = \frac {2}{7} B (a+b \cosh (x))^{5/2} \sinh (x)+\frac {2}{7} \int (a+b \cosh (x))^{3/2} \left (\frac {1}{2} (7 a A+5 b B)+\frac {1}{2} (7 A b+5 a B) \cosh (x)\right ) \, dx \\ & = \frac {2}{35} (7 A b+5 a B) (a+b \cosh (x))^{3/2} \sinh (x)+\frac {2}{7} B (a+b \cosh (x))^{5/2} \sinh (x)+\frac {4}{35} \int \sqrt {a+b \cosh (x)} \left (\frac {1}{4} \left (35 a^2 A+21 A b^2+40 a b B\right )+\frac {1}{4} \left (56 a A b+15 a^2 B+25 b^2 B\right ) \cosh (x)\right ) \, dx \\ & = \frac {2}{105} \left (56 a A b+15 a^2 B+25 b^2 B\right ) \sqrt {a+b \cosh (x)} \sinh (x)+\frac {2}{35} (7 A b+5 a B) (a+b \cosh (x))^{3/2} \sinh (x)+\frac {2}{7} B (a+b \cosh (x))^{5/2} \sinh (x)+\frac {8}{105} \int \frac {\frac {1}{8} \left (105 a^3 A+119 a A b^2+135 a^2 b B+25 b^3 B\right )+\frac {1}{8} \left (161 a^2 A b+63 A b^3+15 a^3 B+145 a b^2 B\right ) \cosh (x)}{\sqrt {a+b \cosh (x)}} \, dx \\ & = \frac {2}{105} \left (56 a A b+15 a^2 B+25 b^2 B\right ) \sqrt {a+b \cosh (x)} \sinh (x)+\frac {2}{35} (7 A b+5 a B) (a+b \cosh (x))^{3/2} \sinh (x)+\frac {2}{7} B (a+b \cosh (x))^{5/2} \sinh (x)-\frac {\left (\left (a^2-b^2\right ) \left (56 a A b+15 a^2 B+25 b^2 B\right )\right ) \int \frac {1}{\sqrt {a+b \cosh (x)}} \, dx}{105 b}+\frac {\left (161 a^2 A b+63 A b^3+15 a^3 B+145 a b^2 B\right ) \int \sqrt {a+b \cosh (x)} \, dx}{105 b} \\ & = \frac {2}{105} \left (56 a A b+15 a^2 B+25 b^2 B\right ) \sqrt {a+b \cosh (x)} \sinh (x)+\frac {2}{35} (7 A b+5 a B) (a+b \cosh (x))^{3/2} \sinh (x)+\frac {2}{7} B (a+b \cosh (x))^{5/2} \sinh (x)+\frac {\left (\left (161 a^2 A b+63 A b^3+15 a^3 B+145 a b^2 B\right ) \sqrt {a+b \cosh (x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}} \, dx}{105 b \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {\left (\left (a^2-b^2\right ) \left (56 a A b+15 a^2 B+25 b^2 B\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}}} \, dx}{105 b \sqrt {a+b \cosh (x)}} \\ & = -\frac {2 i \left (161 a^2 A b+63 A b^3+15 a^3 B+145 a b^2 B\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{105 b \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2 i \left (a^2-b^2\right ) \left (56 a A b+15 a^2 B+25 b^2 B\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{105 b \sqrt {a+b \cosh (x)}}+\frac {2}{105} \left (56 a A b+15 a^2 B+25 b^2 B\right ) \sqrt {a+b \cosh (x)} \sinh (x)+\frac {2}{35} (7 A b+5 a B) (a+b \cosh (x))^{3/2} \sinh (x)+\frac {2}{7} B (a+b \cosh (x))^{5/2} \sinh (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.87 \[ \int (a+b \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\frac {-\frac {2 i \sqrt {\frac {a+b \cosh (x)}{a+b}} \left (b \left (105 a^3 A+119 a A b^2+135 a^2 b B+25 b^3 B\right ) \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )+\left (161 a^2 A b+63 A b^3+15 a^3 B+145 a b^2 B\right ) \left ((a+b) E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )\right )\right )}{b}+(a+b \cosh (x)) \left (154 a A b+90 a^2 B+65 b^2 B+6 b (7 A b+15 a B) \cosh (x)+15 b^2 B \cosh (2 x)\right ) \sinh (x)}{105 \sqrt {a+b \cosh (x)}} \]

[In]

Integrate[(a + b*Cosh[x])^(5/2)*(A + B*Cosh[x]),x]

[Out]

(((-2*I)*Sqrt[(a + b*Cosh[x])/(a + b)]*(b*(105*a^3*A + 119*a*A*b^2 + 135*a^2*b*B + 25*b^3*B)*EllipticF[(I/2)*x
, (2*b)/(a + b)] + (161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*((a + b)*EllipticE[(I/2)*x, (2*b)/(a + b)
] - a*EllipticF[(I/2)*x, (2*b)/(a + b)])))/b + (a + b*Cosh[x])*(154*a*A*b + 90*a^2*B + 65*b^2*B + 6*b*(7*A*b +
15*a*B)*Cosh[x] + 15*b^2*B*Cosh[2*x])*Sinh[x])/(105*Sqrt[a + b*Cosh[x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1364\) vs. \(2(245)=490\).

Time = 6.79 (sec) , antiderivative size = 1365, normalized size of antiderivative = 5.86

method	result	size
default	\(\text {Expression too large to display}\)	\(1365\)
parts	\(\text {Expression too large to display}\)	\(1454\)

[In]

int((a+b*cosh(x))^(5/2)*(A+B*cosh(x)),x,method=_RETURNVERBOSE)

[Out]

2/105*(240*B*cosh(1/2*x)*(-2*b/(a-b))^(1/2)*sinh(1/2*x)^8*b^3+(168*A*(-2*b/(a-b))^(1/2)*b^3+480*B*(-2*b/(a-b))
^(1/2)*a*b^2+360*B*(-2*b/(a-b))^(1/2)*b^3)*sinh(1/2*x)^6*cosh(1/2*x)+(392*A*(-2*b/(a-b))^(1/2)*a*b^2+168*A*(-2
*b/(a-b))^(1/2)*b^3+360*B*(-2*b/(a-b))^(1/2)*a^2*b+480*B*(-2*b/(a-b))^(1/2)*a*b^2+280*B*(-2*b/(a-b))^(1/2)*b^3
)*sinh(1/2*x)^4*cosh(1/2*x)+(154*A*(-2*b/(a-b))^(1/2)*a^2*b+196*A*(-2*b/(a-b))^(1/2)*a*b^2+42*A*(-2*b/(a-b))^(
1/2)*b^3+90*B*(-2*b/(a-b))^(1/2)*a^3+180*B*(-2*b/(a-b))^(1/2)*a^2*b+170*B*(-2*b/(a-b))^(1/2)*a*b^2+80*B*(-2*b/
(a-b))^(1/2)*b^3)*sinh(1/2*x)^2*cosh(1/2*x)+105*A*a^3*(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*(a-b)/b)^(1/2))+161*A*a^2*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*(a-b)/b)^(1/2))+
119*A*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*(a-b)/b)^(1/2))+63*A*b^3*(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*(a-b)/b)^(1/2))-322*A*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(
1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticE(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))*a^2*b-126*A*(2*b/
(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticE(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2
*(a-b)/b)^(1/2))*b^3+15*a^3*B*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticF(cos
h(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))+135*B*a^2*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*(a-b)/b)^(1/2))+145*B*a*b^2*(2*b/(a-b)*s
inh(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*(a-b)/
b)^(1/2))+25*B*b^3*(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*(a-b)/b)^(1/2))-30*B*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/
2)*EllipticE(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))*a^3-290*B*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(
a-b))^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticE(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))*a*b^2)*((2
*cosh(1/2*x)^2*b+a-b)*sinh(1/2*x)^2)^(1/2)/(-2*b/(a-b))^(1/2)/(2*sinh(1/2*x)^4*b+(a+b)*sinh(1/2*x)^2)^(1/2)/si
nh(1/2*x)/(2*sinh(1/2*x)^2*b+a+b)^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.10 (sec) , antiderivative size = 1141, normalized size of antiderivative = 4.90 \[ \int (a+b \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\text {Too large to display} \]

[In]

integrate((a+b*cosh(x))^(5/2)*(A+B*cosh(x)),x, algorithm="fricas")

[Out]

-1/1260*(8*(sqrt(2)*(30*B*a^4 + 7*A*a^3*b - 115*B*a^2*b^2 - 231*A*a*b^3 - 75*B*b^4)*cosh(x)^3 + 3*sqrt(2)*(30*
B*a^4 + 7*A*a^3*b - 115*B*a^2*b^2 - 231*A*a*b^3 - 75*B*b^4)*cosh(x)^2*sinh(x) + 3*sqrt(2)*(30*B*a^4 + 7*A*a^3*
b - 115*B*a^2*b^2 - 231*A*a*b^3 - 75*B*b^4)*cosh(x)*sinh(x)^2 + sqrt(2)*(30*B*a^4 + 7*A*a^3*b - 115*B*a^2*b^2
- 231*A*a*b^3 - 75*B*b^4)*sinh(x)^3)*sqrt(b)*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*sinh(x) + 2*a)/b) + 24*(sqrt(2)*(15*B*a^3*b + 161*A*a^2*b^2 + 145*B*a*b^3 + 63
*A*b^4)*cosh(x)^3 + 3*sqrt(2)*(15*B*a^3*b + 161*A*a^2*b^2 + 145*B*a*b^3 + 63*A*b^4)*cosh(x)^2*sinh(x) + 3*sqrt
(2)*(15*B*a^3*b + 161*A*a^2*b^2 + 145*B*a*b^3 + 63*A*b^4)*cosh(x)*sinh(x)^2 + sqrt(2)*(15*B*a^3*b + 161*A*a^2*
b^2 + 145*B*a*b^3 + 63*A*b^4)*sinh(x)^3)*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*sinh
(x) + 2*a)/b)) - 3*(15*B*b^4*cosh(x)^6 + 15*B*b^4*sinh(x)^6 + 6*(15*B*a*b^3 + 7*A*b^4)*cosh(x)^5 + 6*(15*B*b^4
*cosh(x) + 15*B*a*b^3 + 7*A*b^4)*sinh(x)^5 - 15*B*b^4 + (180*B*a^2*b^2 + 308*A*a*b^3 + 115*B*b^4)*cosh(x)^4 +
(225*B*b^4*cosh(x)^2 + 180*B*a^2*b^2 + 308*A*a*b^3 + 115*B*b^4 + 30*(15*B*a*b^3 + 7*A*b^4)*cosh(x))*sinh(x)^4
- 8*(15*B*a^3*b + 161*A*a^2*b^2 + 145*B*a*b^3 + 63*A*b^4)*cosh(x)^3 + 4*(75*B*b^4*cosh(x)^3 - 30*B*a^3*b - 322
*A*a^2*b^2 - 290*B*a*b^3 - 126*A*b^4 + 15*(15*B*a*b^3 + 7*A*b^4)*cosh(x)^2 + (180*B*a^2*b^2 + 308*A*a*b^3 + 11
5*B*b^4)*cosh(x))*sinh(x)^3 - (180*B*a^2*b^2 + 308*A*a*b^3 + 115*B*b^4)*cosh(x)^2 + (225*B*b^4*cosh(x)^4 - 180
*B*a^2*b^2 - 308*A*a*b^3 - 115*B*b^4 + 60*(15*B*a*b^3 + 7*A*b^4)*cosh(x)^3 + 6*(180*B*a^2*b^2 + 308*A*a*b^3 +
115*B*b^4)*cosh(x)^2 - 24*(15*B*a^3*b + 161*A*a^2*b^2 + 145*B*a*b^3 + 63*A*b^4)*cosh(x))*sinh(x)^2 - 6*(15*B*a
*b^3 + 7*A*b^4)*cosh(x) + 2*(45*B*b^4*cosh(x)^5 - 45*B*a*b^3 - 21*A*b^4 + 15*(15*B*a*b^3 + 7*A*b^4)*cosh(x)^4
+ 2*(180*B*a^2*b^2 + 308*A*a*b^3 + 115*B*b^4)*cosh(x)^3 - 12*(15*B*a^3*b + 161*A*a^2*b^2 + 145*B*a*b^3 + 63*A*
b^4)*cosh(x)^2 - (180*B*a^2*b^2 + 308*A*a*b^3 + 115*B*b^4)*cosh(x))*sinh(x))*sqrt(b*cosh(x) + a))/(b^2*cosh(x)
^3 + 3*b^2*cosh(x)^2*sinh(x) + 3*b^2*cosh(x)*sinh(x)^2 + b^2*sinh(x)^3)

Sympy [F(-1)]

Timed out. \[ \int (a+b \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\text {Timed out} \]

[In]

integrate((a+b*cosh(x))**(5/2)*(A+B*cosh(x)),x)

[Out]

Timed out

Maxima [F]

\[ \int (a+b \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\int { {\left (B \cosh \left (x\right ) + A\right )} {\left (b \cosh \left (x\right ) + a\right )}^{\frac {5}{2}} \,d x } \]

[In]

integrate((a+b*cosh(x))^(5/2)*(A+B*cosh(x)),x, algorithm="maxima")

[Out]

integrate((B*cosh(x) + A)*(b*cosh(x) + a)^(5/2), x)

Giac [F]

\[ \int (a+b \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\int { {\left (B \cosh \left (x\right ) + A\right )} {\left (b \cosh \left (x\right ) + a\right )}^{\frac {5}{2}} \,d x } \]

[In]

integrate((a+b*cosh(x))^(5/2)*(A+B*cosh(x)),x, algorithm="giac")

[Out]

integrate((B*cosh(x) + A)*(b*cosh(x) + a)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int (a+b \cosh (x))^{5/2} (A+B \cosh (x)) \, dx=\int \left (A+B\,\mathrm {cosh}\left (x\right )\right )\,{\left (a+b\,\mathrm {cosh}\left (x\right )\right )}^{5/2} \,d x \]

[In]

int((A + B*cosh(x))*(a + b*cosh(x))^(5/2),x)

[Out]

int((A + B*cosh(x))*(a + b*cosh(x))^(5/2), x)