3.8.66 \(\int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{5/2}} \, dx\) [766]

3.8.66.1 Optimal result

Integrand size = 14, antiderivative size = 322 \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{5/2}} \, dx=-\frac {2 (c \cosh (x)+b \sinh (x))}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{3/2}}-\frac {8 (a c \cosh (x)+a b \sinh (x))}{3 \left (a^2-b^2+c^2\right )^2 \sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {8 i a E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}{3 \left (a^2-b^2+c^2\right )^2 \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}+\frac {2 i \operatorname {EllipticF}\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right ),\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}{3 \left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}} \]

output

-2/3*(c*cosh(x)+b*sinh(x))/(a^2-b^2+c^2)/(a+b*cosh(x)+c*sinh(x))^(3/2)-8/3 
*(a*c*cosh(x)+a*b*sinh(x))/(a^2-b^2+c^2)^2/(a+b*cosh(x)+c*sinh(x))^(1/2)-8 
/3*I*a*(cos(1/2*I*x-1/2*arctan(b,-I*c))^2)^(1/2)/cos(1/2*I*x-1/2*arctan(b, 
-I*c))*EllipticE(sin(1/2*I*x-1/2*arctan(b,-I*c)),2^(1/2)*((b^2-c^2)^(1/2)/ 
(a+(b^2-c^2)^(1/2)))^(1/2))*(a+b*cosh(x)+c*sinh(x))^(1/2)/(a^2-b^2+c^2)^2/ 
((a+b*cosh(x)+c*sinh(x))/(a+(b^2-c^2)^(1/2)))^(1/2)+2/3*I*(cos(1/2*I*x-1/2 
*arctan(b,-I*c))^2)^(1/2)/cos(1/2*I*x-1/2*arctan(b,-I*c))*EllipticF(sin(1/ 
2*I*x-1/2*arctan(b,-I*c)),2^(1/2)*((b^2-c^2)^(1/2)/(a+(b^2-c^2)^(1/2)))^(1 
/2))*((a+b*cosh(x)+c*sinh(x))/(a+(b^2-c^2)^(1/2)))^(1/2)/(a^2-b^2+c^2)/(a+ 
b*cosh(x)+c*sinh(x))^(1/2)
 

3.8.66.2 Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 6.23 (sec) , antiderivative size = 2492, normalized size of antiderivative = 7.74 \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{5/2}} \, dx=\text {Result too large to show} \]

input

Integrate[(a + b*Cosh[x] + c*Sinh[x])^(-5/2),x]
 

output

Sqrt[a + b*Cosh[x] + c*Sinh[x]]*((8*a*(b^2 - c^2))/(3*b*c*(a^2 - b^2 + c^2 
)^2) - (2*(a*c - b^2*Sinh[x] + c^2*Sinh[x]))/(3*b*(-a^2 + b^2 - c^2)*(a + 
b*Cosh[x] + c*Sinh[x])^2) - (2*(-3*a^2*c - b^2*c + c^3 + 4*a*b^2*Sinh[x] - 
 4*a*c^2*Sinh[x]))/(3*b*(-a^2 + b^2 - c^2)^2*(a + b*Cosh[x] + c*Sinh[x]))) 
 + (2*a^2*AppellF1[1/2, 1/2, 1/2, 3/2, ((-I)*(a + Sqrt[1 - b^2/c^2]*c*Sinh 
[x + ArcTanh[b/c]]))/(Sqrt[1 - b^2/c^2]*(1 - (I*a)/(Sqrt[1 - b^2/c^2]*c))* 
c), ((-I)*(a + Sqrt[1 - b^2/c^2]*c*Sinh[x + ArcTanh[b/c]]))/(Sqrt[1 - b^2/ 
c^2]*(-1 - (I*a)/(Sqrt[1 - b^2/c^2]*c))*c)]*Sech[x + ArcTanh[b/c]]*Sqrt[-1 
 + I*Sinh[x + ArcTanh[b/c]]]*Sqrt[(c*Sqrt[(-b^2 + c^2)/c^2] - I*c*Sqrt[(-b 
^2 + c^2)/c^2]*Sinh[x + ArcTanh[b/c]])/(I*a + c*Sqrt[(-b^2 + c^2)/c^2])]*S 
qrt[(c*Sqrt[(-b^2 + c^2)/c^2] + I*c*Sqrt[(-b^2 + c^2)/c^2]*Sinh[x + ArcTan 
h[b/c]])/((-I)*a + c*Sqrt[(-b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(-b^2 + c^2) 
/c^2]*Sinh[x + ArcTanh[b/c]]])/(Sqrt[1 - b^2/c^2]*c*(a^2 - b^2 + c^2)^2*Sq 
rt[I*(I + Sinh[x + ArcTanh[b/c]])]) + (2*b^2*AppellF1[1/2, 1/2, 1/2, 3/2, 
((-I)*(a + Sqrt[1 - b^2/c^2]*c*Sinh[x + ArcTanh[b/c]]))/(Sqrt[1 - b^2/c^2] 
*(1 - (I*a)/(Sqrt[1 - b^2/c^2]*c))*c), ((-I)*(a + Sqrt[1 - b^2/c^2]*c*Sinh 
[x + ArcTanh[b/c]]))/(Sqrt[1 - b^2/c^2]*(-1 - (I*a)/(Sqrt[1 - b^2/c^2]*c)) 
*c)]*Sech[x + ArcTanh[b/c]]*Sqrt[-1 + I*Sinh[x + ArcTanh[b/c]]]*Sqrt[(c*Sq 
rt[(-b^2 + c^2)/c^2] - I*c*Sqrt[(-b^2 + c^2)/c^2]*Sinh[x + ArcTanh[b/c]])/ 
(I*a + c*Sqrt[(-b^2 + c^2)/c^2])]*Sqrt[(c*Sqrt[(-b^2 + c^2)/c^2] + I*c*...
 

3.8.66.3 Rubi [A] (verified)

Time = 1.38 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.071, Rules used = {3042, 3608, 27, 3042, 3635, 27, 3042, 3628, 3042, 3598, 3042, 3132, 3606, 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.

\(\displaystyle \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(a+b \cos (i x)-i c \sin (i x))^{5/2}}dx\)

\(\Big \downarrow \) 3608

\(\displaystyle -\frac {2 \int -\frac {3 a-b \cosh (x)-c \sinh (x)}{2 (a+b \cosh (x)+c \sinh (x))^{3/2}}dx}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (b \sinh (x)+c \cosh (x))}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {3 a-b \cosh (x)-c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^{3/2}}dx}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (b \sinh (x)+c \cosh (x))}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 (b \sinh (x)+c \cosh (x))}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{3/2}}+\frac {\int \frac {3 a-b \cos (i x)+i c \sin (i x)}{(a+b \cos (i x)-i c \sin (i x))^{3/2}}dx}{3 \left (a^2-b^2+c^2\right )}\)

\(\Big \downarrow \) 3635

\(\displaystyle \frac {-\frac {2 \int -\frac {3 a^2+4 b \cosh (x) a+4 c \sinh (x) a+b^2-c^2}{2 \sqrt {a+b \cosh (x)+c \sinh (x)}}dx}{a^2-b^2+c^2}-\frac {8 (a b \sinh (x)+a c \cosh (x))}{\left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (b \sinh (x)+c \cosh (x))}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {3 a^2+4 b \cosh (x) a+4 c \sinh (x) a+b^2-c^2}{\sqrt {a+b \cosh (x)+c \sinh (x)}}dx}{a^2-b^2+c^2}-\frac {8 (a b \sinh (x)+a c \cosh (x))}{\left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (b \sinh (x)+c \cosh (x))}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 (b \sinh (x)+c \cosh (x))}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{3/2}}+\frac {-\frac {8 (a b \sinh (x)+a c \cosh (x))}{\left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}+\frac {\int \frac {3 a^2+4 b \cos (i x) a-4 i c \sin (i x) a+b^2-c^2}{\sqrt {a+b \cos (i x)-i c \sin (i x)}}dx}{a^2-b^2+c^2}}{3 \left (a^2-b^2+c^2\right )}\)

\(\Big \downarrow \) 3628

\(\displaystyle \frac {\frac {4 a \int \sqrt {a+b \cosh (x)+c \sinh (x)}dx-\left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {a+b \cosh (x)+c \sinh (x)}}dx}{a^2-b^2+c^2}-\frac {8 (a b \sinh (x)+a c \cosh (x))}{\left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (b \sinh (x)+c \cosh (x))}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 (b \sinh (x)+c \cosh (x))}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{3/2}}+\frac {-\frac {8 (a b \sinh (x)+a c \cosh (x))}{\left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}+\frac {4 a \int \sqrt {a+b \cos (i x)-i c \sin (i x)}dx-\left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {a+b \cos (i x)-i c \sin (i x)}}dx}{a^2-b^2+c^2}}{3 \left (a^2-b^2+c^2\right )}\)

\(\Big \downarrow \) 3598

\(\displaystyle -\frac {2 (b \sinh (x)+c \cosh (x))}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{3/2}}+\frac {-\frac {8 (a b \sinh (x)+a c \cosh (x))}{\left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}+\frac {\frac {4 a \sqrt {a+b \cosh (x)+c \sinh (x)} \int \sqrt {\frac {a}{a+\sqrt {b^2-c^2}}+\frac {\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}{a+\sqrt {b^2-c^2}}}dx}{\sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}-\left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {a+b \cos (i x)-i c \sin (i x)}}dx}{a^2-b^2+c^2}}{3 \left (a^2-b^2+c^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 (b \sinh (x)+c \cosh (x))}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{3/2}}+\frac {-\frac {8 (a b \sinh (x)+a c \cosh (x))}{\left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}+\frac {\frac {4 a \sqrt {a+b \cosh (x)+c \sinh (x)} \int \sqrt {\frac {a}{a+\sqrt {b^2-c^2}}+\frac {\sqrt {b^2-c^2} \sin \left (i x-\tan ^{-1}(b,-i c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2-c^2}}}dx}{\sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}-\left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {a+b \cos (i x)-i c \sin (i x)}}dx}{a^2-b^2+c^2}}{3 \left (a^2-b^2+c^2\right )}\)

\(\Big \downarrow \) 3132

\(\displaystyle -\frac {2 (b \sinh (x)+c \cosh (x))}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{3/2}}+\frac {-\frac {8 (a b \sinh (x)+a c \cosh (x))}{\left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}+\frac {-\left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {a+b \cos (i x)-i c \sin (i x)}}dx-\frac {8 i a \sqrt {a+b \cosh (x)+c \sinh (x)} E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{\sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}}{a^2-b^2+c^2}}{3 \left (a^2-b^2+c^2\right )}\)

\(\Big \downarrow \) 3606

\(\displaystyle -\frac {2 (b \sinh (x)+c \cosh (x))}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{3/2}}+\frac {-\frac {8 (a b \sinh (x)+a c \cosh (x))}{\left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}+\frac {-\frac {\left (a^2-b^2+c^2\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2-c^2}}+\frac {\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}{a+\sqrt {b^2-c^2}}}}dx}{\sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {8 i a \sqrt {a+b \cosh (x)+c \sinh (x)} E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{\sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}}{a^2-b^2+c^2}}{3 \left (a^2-b^2+c^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 (b \sinh (x)+c \cosh (x))}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{3/2}}+\frac {-\frac {8 (a b \sinh (x)+a c \cosh (x))}{\left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}+\frac {-\frac {\left (a^2-b^2+c^2\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2-c^2}}+\frac {\sqrt {b^2-c^2} \sin \left (i x-\tan ^{-1}(b,-i c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2-c^2}}}}dx}{\sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {8 i a \sqrt {a+b \cosh (x)+c \sinh (x)} E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{\sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}}{a^2-b^2+c^2}}{3 \left (a^2-b^2+c^2\right )}\)

\(\Big \downarrow \) 3140

\(\displaystyle -\frac {2 (b \sinh (x)+c \cosh (x))}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^{3/2}}+\frac {-\frac {8 (a b \sinh (x)+a c \cosh (x))}{\left (a^2-b^2+c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}+\frac {\frac {2 i \left (a^2-b^2+c^2\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}} \operatorname {EllipticF}\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right ),\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{\sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {8 i a \sqrt {a+b \cosh (x)+c \sinh (x)} E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{\sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}}{a^2-b^2+c^2}}{3 \left (a^2-b^2+c^2\right )}\)

input

Int[(a + b*Cosh[x] + c*Sinh[x])^(-5/2),x]
 

output

(-2*(c*Cosh[x] + b*Sinh[x]))/(3*(a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[ 
x])^(3/2)) + ((-8*(a*c*Cosh[x] + a*b*Sinh[x]))/((a^2 - b^2 + c^2)*Sqrt[a + 
 b*Cosh[x] + c*Sinh[x]]) + (((-8*I)*a*EllipticE[(I*x - ArcTan[b, (-I)*c])/ 
2, (2*Sqrt[b^2 - c^2])/(a + Sqrt[b^2 - c^2])]*Sqrt[a + b*Cosh[x] + c*Sinh[ 
x]])/Sqrt[(a + b*Cosh[x] + c*Sinh[x])/(a + Sqrt[b^2 - c^2])] + ((2*I)*(a^2 
 - b^2 + c^2)*EllipticF[(I*x - ArcTan[b, (-I)*c])/2, (2*Sqrt[b^2 - c^2])/( 
a + Sqrt[b^2 - c^2])]*Sqrt[(a + b*Cosh[x] + c*Sinh[x])/(a + Sqrt[b^2 - c^2 
])])/Sqrt[a + b*Cosh[x] + c*Sinh[x]])/(a^2 - b^2 + c^2))/(3*(a^2 - b^2 + c 
^2))
 

3.8.66.3.1 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 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 3598

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

rule 3606

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

rule 3608

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ 
(n_), x_Symbol] :> Simp[((-c)*Cos[d + e*x] + b*Sin[d + e*x])*((a + b*Cos[d 
+ e*x] + c*Sin[d + e*x])^(n + 1)/(e*(n + 1)*(a^2 - b^2 - c^2))), x] + Simp[ 
1/((n + 1)*(a^2 - b^2 - c^2))   Int[(a*(n + 1) - b*(n + 2)*Cos[d + e*x] - c 
*(n + 2)*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1), x], x 
] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && LtQ[n, -1] && 
NeQ[n, -3/2]
 

rule 3628

Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)]) 
/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]] 
, x_Symbol] :> Simp[B/b   Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x] 
, x] + Simp[(A*b - a*B)/b   Int[1/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]] 
, x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && EqQ[B*c - b*C, 0] && NeQ[ 
A*b - a*B, 0]
 

rule 3635

Int[((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]) 
^(n_)*((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_) 
]), x_Symbol] :> Simp[(-(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B - b*A) 
*Sin[d + e*x]))*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/(e*(n + 1)*( 
a^2 - b^2 - c^2))), x] + Simp[1/((n + 1)*(a^2 - b^2 - c^2))   Int[(a + b*Co 
s[d + e*x] + c*Sin[d + e*x])^(n + 1)*Simp[(n + 1)*(a*A - b*B - c*C) + (n + 
2)*(a*B - b*A)*Cos[d + e*x] + (n + 2)*(a*C - c*A)*Sin[d + e*x], x], x], x] 
/; FreeQ[{a, b, c, d, e, A, B, C}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 
 0] && NeQ[n, -2]
 

3.8.66.4 Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2159\) vs. \(2(356)=712\).

Time = 2.51 (sec) , antiderivative size = 2160, normalized size of antiderivative = 6.71

method	result	size
default	\(\text {Expression too large to display}\)	\(2160\)

input

int(1/(a+b*cosh(x)+c*sinh(x))^(5/2),x,method=_RETURNVERBOSE)
 

output

2/((-b^2*sinh(x)+sinh(x)*c^2+a*(b^2-c^2)^(1/2))/(b^2-c^2)^(1/2))^(1/2)*a*( 
b^2-c^2)^(1/2)*(-1/2*cosh(x)/(a^2+b^2-c^2)/(sinh(x)^2*(b^2-c^2)-a^2)+1/2/( 
a^2+b^2-c^2)/((a^2+b^2-c^2)*(b^2-c^2))^(1/2)*arctanh((b^2-c^2)*cosh(x)/((a 
^2+b^2-c^2)*(b^2-c^2))^(1/2)))+((-b^2*sinh(x)+sinh(x)*c^2+a*(b^2-c^2)^(1/2 
))/(b^2-c^2)^(1/2)*sinh(x)^2)^(1/2)*(1/(b^2-c^2)*a^2/(2*a^2+2*b^2-2*c^2)*( 
1/(sinh(x)*(b^2-c^2)^(1/2)-a)/a^2*(b^2-c^2)/(cosh(x)+((a^2+b^2-c^2)*(b-c)* 
(b+c))^(1/2)/(b^2-c^2))*((-b^2*sinh(x)+sinh(x)*c^2+a*(b^2-c^2)^(1/2))/(b^2 
-c^2)^(1/2)*sinh(x)^2)^(1/2)-(b^2*sinh(x)-sinh(x)*c^2-a*(b^2-c^2)^(1/2))/( 
b^2-c^2)^(1/2)*((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)/(sinh(x)*(b^2-c^2)^(1/2)- 
a)/a^2/((-sinh(x)*(b^2-c^2)^(1/2)+a)*a^2/(b^2-c^2))^(1/2)*ln((-2*(sinh(x)* 
(b^2-c^2)^(1/2)-a)*a^2/(b^2-c^2)+2*(b^2*sinh(x)-sinh(x)*c^2-a*(b^2-c^2)^(1 
/2))/(b^2-c^2)^(3/2)*((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)*(cosh(x)+((a^2+b^2- 
c^2)*(b-c)*(b+c))^(1/2)/(b^2-c^2))+2*((-sinh(x)*(b^2-c^2)^(1/2)+a)*a^2/(b^ 
2-c^2))^(1/2)*((-b^2*sinh(x)+sinh(x)*c^2+a*(b^2-c^2)^(1/2))/(b^2-c^2)^(1/2 
)*sinh(x)^2)^(1/2))/(cosh(x)+((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)/(b^2-c^2))) 
)+(b^2-c^2)*a^2/(2*a^2+2*b^2-2*c^2)/(-b^2+c^2)^2*(1/(sinh(x)*(b^2-c^2)^(1/ 
2)-a)/a^2*(b^2-c^2)/(cosh(x)+((a^2+b^2-c^2)*(b-c)*(b+c))^(1/2)/(-b^2+c^2)) 
*((-b^2*sinh(x)+sinh(x)*c^2+a*(b^2-c^2)^(1/2))/(b^2-c^2)^(1/2)*sinh(x)^2)^ 
(1/2)+(b^2*sinh(x)-sinh(x)*c^2-a*(b^2-c^2)^(1/2))/(b^2-c^2)^(1/2)*((a^2+b^ 
2-c^2)*(b-c)*(b+c))^(1/2)/(sinh(x)*(b^2-c^2)^(1/2)-a)/a^2/((-sinh(x)*(b...
 

3.8.66.5 Fricas [C] (verification not implemented)

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

Time = 0.17 (sec) , antiderivative size = 3730, normalized size of antiderivative = 11.58 \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{5/2}} \, dx=\text {Too large to display} \]

input

integrate(1/(a+b*cosh(x)+c*sinh(x))^(5/2),x, algorithm="fricas")
 

output

2/9*((sqrt(2)*(a^2*b^2 + 3*b^4 + a^2*c^2 - 6*b*c^3 - 3*c^4 + 2*(a^2*b + 3* 
b^3)*c)*cosh(x)^4 + sqrt(2)*(a^2*b^2 + 3*b^4 + a^2*c^2 - 6*b*c^3 - 3*c^4 + 
 2*(a^2*b + 3*b^3)*c)*sinh(x)^4 + 4*sqrt(2)*(a^3*b + 3*a*b^3 - 3*a*b*c^2 - 
 3*a*c^3 + (a^3 + 3*a*b^2)*c)*cosh(x)^3 + 4*(sqrt(2)*(a^2*b^2 + 3*b^4 + a^ 
2*c^2 - 6*b*c^3 - 3*c^4 + 2*(a^2*b + 3*b^3)*c)*cosh(x) + sqrt(2)*(a^3*b + 
3*a*b^3 - 3*a*b*c^2 - 3*a*c^3 + (a^3 + 3*a*b^2)*c))*sinh(x)^3 + 2*sqrt(2)* 
(2*a^4 + 7*a^2*b^2 + 3*b^4 + 3*c^4 - (7*a^2 + 6*b^2)*c^2)*cosh(x)^2 + 2*(3 
*sqrt(2)*(a^2*b^2 + 3*b^4 + a^2*c^2 - 6*b*c^3 - 3*c^4 + 2*(a^2*b + 3*b^3)* 
c)*cosh(x)^2 + 6*sqrt(2)*(a^3*b + 3*a*b^3 - 3*a*b*c^2 - 3*a*c^3 + (a^3 + 3 
*a*b^2)*c)*cosh(x) + sqrt(2)*(2*a^4 + 7*a^2*b^2 + 3*b^4 + 3*c^4 - (7*a^2 + 
 6*b^2)*c^2))*sinh(x)^2 + 4*sqrt(2)*(a^3*b + 3*a*b^3 - 3*a*b*c^2 + 3*a*c^3 
 - (a^3 + 3*a*b^2)*c)*cosh(x) + 4*(sqrt(2)*(a^2*b^2 + 3*b^4 + a^2*c^2 - 6* 
b*c^3 - 3*c^4 + 2*(a^2*b + 3*b^3)*c)*cosh(x)^3 + 3*sqrt(2)*(a^3*b + 3*a*b^ 
3 - 3*a*b*c^2 - 3*a*c^3 + (a^3 + 3*a*b^2)*c)*cosh(x)^2 + sqrt(2)*(2*a^4 + 
7*a^2*b^2 + 3*b^4 + 3*c^4 - (7*a^2 + 6*b^2)*c^2)*cosh(x) + sqrt(2)*(a^3*b 
+ 3*a*b^3 - 3*a*b*c^2 + 3*a*c^3 - (a^3 + 3*a*b^2)*c))*sinh(x) + sqrt(2)*(a 
^2*b^2 + 3*b^4 + a^2*c^2 + 6*b*c^3 - 3*c^4 - 2*(a^2*b + 3*b^3)*c))*sqrt(b 
+ c)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2 + 3*c^2)/(b^2 + 2*b*c + c^2), 
-8/27*(8*a^3 - 9*a*b^2 + 9*a*c^2)/(b^3 + 3*b^2*c + 3*b*c^2 + c^3), 1/3*(3* 
(b + c)*cosh(x) + 3*(b + c)*sinh(x) + 2*a)/(b + c)) - 12*(sqrt(2)*(a*b^...
 

3.8.66.6 Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{5/2}} \, dx=\text {Timed out} \]

input

integrate(1/(a+b*cosh(x)+c*sinh(x))**(5/2),x)
 

output

Timed out
 

3.8.66.7 Maxima [F]

\[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \cosh \left (x\right ) + c \sinh \left (x\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

input

integrate(1/(a+b*cosh(x)+c*sinh(x))^(5/2),x, algorithm="maxima")
 

output

integrate((b*cosh(x) + c*sinh(x) + a)^(-5/2), x)
 

3.8.66.8 Giac [F]

\[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \cosh \left (x\right ) + c \sinh \left (x\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

input

integrate(1/(a+b*cosh(x)+c*sinh(x))^(5/2),x, algorithm="giac")
 

output

integrate((b*cosh(x) + c*sinh(x) + a)^(-5/2), x)
 

3.8.66.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{5/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {cosh}\left (x\right )+c\,\mathrm {sinh}\left (x\right )\right )}^{5/2}} \,d x \]

input

int(1/(a + b*cosh(x) + c*sinh(x))^(5/2),x)
 

output

int(1/(a + b*cosh(x) + c*sinh(x))^(5/2), x)