3.8.61 \(\int (a+b \cosh (x)+c \sinh (x))^{5/2} \, dx\) [761]

3.8.61.1 Optimal result

Integrand size = 14, antiderivative size = 294 \[ \int (a+b \cosh (x)+c \sinh (x))^{5/2} \, dx=\frac {16}{15} (a c \cosh (x)+a b \sinh (x)) \sqrt {a+b \cosh (x)+c \sinh (x)}+\frac {2}{5} (c \cosh (x)+b \sinh (x)) (a+b \cosh (x)+c \sinh (x))^{3/2}-\frac {2 i \left (23 a^2+9 b^2-9 c^2\right ) E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}{15 \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}+\frac {16 i a \left (a^2-b^2+c^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right ),\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}{15 \sqrt {a+b \cosh (x)+c \sinh (x)}} \]

output

2/5*(c*cosh(x)+b*sinh(x))*(a+b*cosh(x)+c*sinh(x))^(3/2)+16/15*(a*c*cosh(x) 
+a*b*sinh(x))*(a+b*cosh(x)+c*sinh(x))^(1/2)-2/15*I*(23*a^2+9*b^2-9*c^2)*(c 
os(1/2*I*x-1/2*arctan(b,-I*c))^2)^(1/2)/cos(1/2*I*x-1/2*arctan(b,-I*c))*El 
lipticE(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+b*cosh(x)+c*sinh(x))/ 
(a+(b^2-c^2)^(1/2)))^(1/2)+16/15*I*a*(a^2-b^2+c^2)*(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+b*cosh(x)+c*sinh(x))^ 
(1/2)
 

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

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

Time = 6.28 (sec) , antiderivative size = 3775, normalized size of antiderivative = 12.84 \[ \int (a+b \cosh (x)+c \sinh (x))^{5/2} \, dx=\text {Result too large to show} \]

input

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

output

Sqrt[a + b*Cosh[x] + c*Sinh[x]]*((2*b*(23*a^2 + 9*b^2 - 9*c^2))/(15*c) + ( 
22*a*c*Cosh[x])/15 + (2*b*c*Cosh[2*x])/5 + (22*a*b*Sinh[x])/15 + ((b^2 + c 
^2)*Sinh[2*x])/5) + (2*a^3*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 + ArcT 
anh[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])]*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])]*Sqrt[a + c* 
Sqrt[(-b^2 + c^2)/c^2]*Sinh[x + ArcTanh[b/c]]])/(Sqrt[1 - b^2/c^2]*c*Sqrt[ 
I*(I + Sinh[x + ArcTanh[b/c]])]) + (34*a*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*Sqr 
t[(-b^2 + c^2)/c^2]*Sinh[x + ArcTanh[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]]])/(15*S...
 

3.8.61.3 Rubi [A] (verified)

Time = 1.49 (sec) , antiderivative size = 306, 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, 3599, 27, 3042, 3625, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3599

\(\displaystyle \frac {2}{5} \int \frac {1}{2} \sqrt {a+b \cosh (x)+c \sinh (x)} \left (5 a^2+8 b \cosh (x) a+8 c \sinh (x) a+3 b^2-3 c^2\right )dx+\frac {2}{5} (b \sinh (x)+c \cosh (x)) (a+b \cosh (x)+c \sinh (x))^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} \int \sqrt {a+b \cosh (x)+c \sinh (x)} \left (5 a^2+8 b \cosh (x) a+8 c \sinh (x) a+3 b^2-3 c^2\right )dx+\frac {2}{5} (b \sinh (x)+c \cosh (x)) (a+b \cosh (x)+c \sinh (x))^{3/2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2}{5} (b \sinh (x)+c \cosh (x)) (a+b \cosh (x)+c \sinh (x))^{3/2}+\frac {1}{5} \int \sqrt {a+b \cos (i x)-i c \sin (i x)} \left (5 a^2+8 b \cos (i x) a-8 i c \sin (i x) a+3 b^2-3 c^2\right )dx\)

\(\Big \downarrow \) 3625

\(\displaystyle \frac {1}{5} \left (\frac {2 \int \frac {\left (15 a^2+17 b^2-17 c^2\right ) a^2+b \left (23 a^2+9 b^2-9 c^2\right ) \cosh (x) a+c \left (23 a^2+9 b^2-9 c^2\right ) \sinh (x) a}{2 \sqrt {a+b \cosh (x)+c \sinh (x)}}dx}{3 a}+\frac {16}{3} \sqrt {a+b \cosh (x)+c \sinh (x)} (a b \sinh (x)+a c \cosh (x))\right )+\frac {2}{5} (b \sinh (x)+c \cosh (x)) (a+b \cosh (x)+c \sinh (x))^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} \left (\frac {\int \frac {\left (15 a^2+17 \left (b^2-c^2\right )\right ) a^2+b \left (23 a^2+9 b^2-9 c^2\right ) \cosh (x) a+c \left (23 a^2+9 b^2-9 c^2\right ) \sinh (x) a}{\sqrt {a+b \cosh (x)+c \sinh (x)}}dx}{3 a}+\frac {16}{3} \sqrt {a+b \cosh (x)+c \sinh (x)} (a b \sinh (x)+a c \cosh (x))\right )+\frac {2}{5} (b \sinh (x)+c \cosh (x)) (a+b \cosh (x)+c \sinh (x))^{3/2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2}{5} (b \sinh (x)+c \cosh (x)) (a+b \cosh (x)+c \sinh (x))^{3/2}+\frac {1}{5} \left (\frac {16}{3} \sqrt {a+b \cosh (x)+c \sinh (x)} (a b \sinh (x)+a c \cosh (x))+\frac {\int \frac {\left (15 a^2+17 \left (b^2-c^2\right )\right ) a^2+b \left (23 a^2+9 b^2-9 c^2\right ) \cos (i x) a-i c \left (23 a^2+9 b^2-9 c^2\right ) \sin (i x) a}{\sqrt {a+b \cos (i x)-i c \sin (i x)}}dx}{3 a}\right )\)

\(\Big \downarrow \) 3628

\(\displaystyle \frac {1}{5} \left (\frac {a \left (23 a^2+9 b^2-9 c^2\right ) \int \sqrt {a+b \cosh (x)+c \sinh (x)}dx-8 a^2 \left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {a+b \cosh (x)+c \sinh (x)}}dx}{3 a}+\frac {16}{3} \sqrt {a+b \cosh (x)+c \sinh (x)} (a b \sinh (x)+a c \cosh (x))\right )+\frac {2}{5} (b \sinh (x)+c \cosh (x)) (a+b \cosh (x)+c \sinh (x))^{3/2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2}{5} (b \sinh (x)+c \cosh (x)) (a+b \cosh (x)+c \sinh (x))^{3/2}+\frac {1}{5} \left (\frac {16}{3} \sqrt {a+b \cosh (x)+c \sinh (x)} (a b \sinh (x)+a c \cosh (x))+\frac {a \left (23 a^2+9 b^2-9 c^2\right ) \int \sqrt {a+b \cos (i x)-i c \sin (i x)}dx-8 a^2 \left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {a+b \cos (i x)-i c \sin (i x)}}dx}{3 a}\right )\)

\(\Big \downarrow \) 3598

\(\displaystyle \frac {2}{5} (b \sinh (x)+c \cosh (x)) (a+b \cosh (x)+c \sinh (x))^{3/2}+\frac {1}{5} \left (\frac {16}{3} \sqrt {a+b \cosh (x)+c \sinh (x)} (a b \sinh (x)+a c \cosh (x))+\frac {\frac {a \left (23 a^2+9 b^2-9 c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)} \int \sqrt {\frac {a}{a+\sqrt {b^2-c^2}}+\frac {\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}{a+\sqrt {b^2-c^2}}}dx}{\sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}-8 a^2 \left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {a+b \cos (i x)-i c \sin (i x)}}dx}{3 a}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2}{5} (b \sinh (x)+c \cosh (x)) (a+b \cosh (x)+c \sinh (x))^{3/2}+\frac {1}{5} \left (\frac {16}{3} \sqrt {a+b \cosh (x)+c \sinh (x)} (a b \sinh (x)+a c \cosh (x))+\frac {\frac {a \left (23 a^2+9 b^2-9 c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)} \int \sqrt {\frac {a}{a+\sqrt {b^2-c^2}}+\frac {\sqrt {b^2-c^2} \sin \left (i x-\tan ^{-1}(b,-i c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2-c^2}}}dx}{\sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}-8 a^2 \left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {a+b \cos (i x)-i c \sin (i x)}}dx}{3 a}\right )\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {2}{5} (b \sinh (x)+c \cosh (x)) (a+b \cosh (x)+c \sinh (x))^{3/2}+\frac {1}{5} \left (\frac {16}{3} \sqrt {a+b \cosh (x)+c \sinh (x)} (a b \sinh (x)+a c \cosh (x))+\frac {-8 a^2 \left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {a+b \cos (i x)-i c \sin (i x)}}dx-\frac {2 i a \left (23 a^2+9 b^2-9 c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)} E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{\sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}}{3 a}\right )\)

\(\Big \downarrow \) 3606

\(\displaystyle \frac {2}{5} (b \sinh (x)+c \cosh (x)) (a+b \cosh (x)+c \sinh (x))^{3/2}+\frac {1}{5} \left (\frac {16}{3} \sqrt {a+b \cosh (x)+c \sinh (x)} (a b \sinh (x)+a c \cosh (x))+\frac {-\frac {8 a^2 \left (a^2-b^2+c^2\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2-c^2}}+\frac {\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}{a+\sqrt {b^2-c^2}}}}dx}{\sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {2 i a \left (23 a^2+9 b^2-9 c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)} E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{\sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}}{3 a}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2}{5} (b \sinh (x)+c \cosh (x)) (a+b \cosh (x)+c \sinh (x))^{3/2}+\frac {1}{5} \left (\frac {16}{3} \sqrt {a+b \cosh (x)+c \sinh (x)} (a b \sinh (x)+a c \cosh (x))+\frac {-\frac {8 a^2 \left (a^2-b^2+c^2\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2-c^2}}+\frac {\sqrt {b^2-c^2} \sin \left (i x-\tan ^{-1}(b,-i c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2-c^2}}}}dx}{\sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {2 i a \left (23 a^2+9 b^2-9 c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)} E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{\sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}}{3 a}\right )\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {2}{5} (b \sinh (x)+c \cosh (x)) (a+b \cosh (x)+c \sinh (x))^{3/2}+\frac {1}{5} \left (\frac {16}{3} \sqrt {a+b \cosh (x)+c \sinh (x)} (a b \sinh (x)+a c \cosh (x))+\frac {\frac {16 i a^2 \left (a^2-b^2+c^2\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}} \operatorname {EllipticF}\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right ),\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{\sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {2 i a \left (23 a^2+9 b^2-9 c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)} E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{\sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}}{3 a}\right )\)

input

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

output

(2*(c*Cosh[x] + b*Sinh[x])*(a + b*Cosh[x] + c*Sinh[x])^(3/2))/5 + ((16*(a* 
c*Cosh[x] + a*b*Sinh[x])*Sqrt[a + b*Cosh[x] + c*Sinh[x]])/3 + (((-2*I)*a*( 
23*a^2 + 9*b^2 - 9*c^2)*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])] + ((16*I)*a^2*(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]])/(3*a))/5
 

3.8.61.3.1 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 3599

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)), x] + Simp[1/n   Int[Simp[n*a^2 + ( 
n - 1)*(b^2 + c^2) + a*b*(2*n - 1)*Cos[d + e*x] + a*c*(2*n - 1)*Sin[d + e*x 
], x]*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 2), x], x] /; FreeQ[{a, b, 
 c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && GtQ[n, 1]
 

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 3625

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

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]
 

3.8.61.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(898\) vs. \(2(328)=656\).

Time = 1.74 (sec) , antiderivative size = 899, normalized size of antiderivative = 3.06

method	result	size
default	\(-\frac {\sqrt {\left (b -c \right ) \left (b +c \right )}\, \left (\frac {\cosh \left (x \right )^{3} b^{2}}{3}-\frac {\cosh \left (x \right )^{3} c^{2}}{3}+3 \cosh \left (x \right ) a^{2}-\cosh \left (x \right ) b^{2}+\cosh \left (x \right ) c^{2}\right )}{\sqrt {\frac {-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}+\frac {\sqrt {\frac {\left (-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \sinh \left (x \right )^{2}}{\sqrt {b^{2}-c^{2}}}}\, \left (\frac {a^{3} \ln \left (\frac {\left (-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \cosh \left (x \right )}{\sqrt {b^{2}-c^{2}}\, \sqrt {\frac {-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}+\sqrt {\frac {\left (-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \sinh \left (x \right )^{2}}{\sqrt {b^{2}-c^{2}}}}\right )}{\sqrt {\frac {-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}+\frac {a \,c^{2} \ln \left (\frac {\left (-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \cosh \left (x \right )}{\sqrt {b^{2}-c^{2}}\, \sqrt {\frac {-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}+\sqrt {\frac {\left (-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \sinh \left (x \right )^{2}}{\sqrt {b^{2}-c^{2}}}}\right )}{2 \sqrt {\frac {-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}-\frac {a \,b^{2} \ln \left (\frac {\left (-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \cosh \left (x \right )}{\sqrt {b^{2}-c^{2}}\, \sqrt {\frac {-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}+\sqrt {\frac {\left (-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \sinh \left (x \right )^{2}}{\sqrt {b^{2}-c^{2}}}}\right )}{2 \sqrt {\frac {-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}+\frac {a \cosh \left (x \right ) \sqrt {b^{2}-c^{2}}\, \sqrt {\frac {\left (-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \sinh \left (x \right )^{2}}{\sqrt {b^{2}-c^{2}}}}\, b^{2}}{-2 b^{2} \sinh \left (x \right )+2 \sinh \left (x \right ) c^{2}+2 a \sqrt {b^{2}-c^{2}}}-\frac {a \cosh \left (x \right ) \sqrt {b^{2}-c^{2}}\, \sqrt {\frac {\left (-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \sinh \left (x \right )^{2}}{\sqrt {b^{2}-c^{2}}}}\, c^{2}}{2 \left (-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right )}\right )}{\sinh \left (x \right ) \sqrt {\frac {-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}\)	\(899\)

input

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

output

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

3.8.61.5 Fricas [C] (verification not implemented)

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

Time = 0.10 (sec) , antiderivative size = 928, normalized size of antiderivative = 3.16 \[ \int (a+b \cosh (x)+c \sinh (x))^{5/2} \, dx=\text {Too large to display} \]

input

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

output

-1/90*(4*(sqrt(2)*(a^3 - 33*a*b^2 + 33*a*c^2)*cosh(x)^2 + 2*sqrt(2)*(a^3 - 
 33*a*b^2 + 33*a*c^2)*cosh(x)*sinh(x) + sqrt(2)*(a^3 - 33*a*b^2 + 33*a*c^2 
)*sinh(x)^2)*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)*(23*a^2*b + 9*b^3 - 9*b*c^2 - 9*c^3 + (23*a^2 + 9*b^2)*c)*cosh 
(x)^2 + 2*sqrt(2)*(23*a^2*b + 9*b^3 - 9*b*c^2 - 9*c^3 + (23*a^2 + 9*b^2)*c 
)*cosh(x)*sinh(x) + sqrt(2)*(23*a^2*b + 9*b^3 - 9*b*c^2 - 9*c^3 + (23*a^2 
+ 9*b^2)*c)*sinh(x)^2)*sqrt(b + c)*weierstrassZeta(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), 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))) - 3*(3* 
(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^4 + 3*(b^3 + 3*b^2*c + 3*b*c^2 + c 
^3)*sinh(x)^4 + 22*(a*b^2 + 2*a*b*c + a*c^2)*cosh(x)^3 + 2*(11*a*b^2 + 22* 
a*b*c + 11*a*c^2 + 6*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x))*sinh(x)^3 - 
3*b^3 + 3*b^2*c + 3*b*c^2 - 3*c^3 - 4*(23*a^2*b + 9*b^3 - 9*b*c^2 - 9*c^3 
+ (23*a^2 + 9*b^2)*c)*cosh(x)^2 - 2*(46*a^2*b + 18*b^3 - 18*b*c^2 - 18*c^3 
 - 9*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^2 + 2*(23*a^2 + 9*b^2)*c - 33 
*(a*b^2 + 2*a*b*c + a*c^2)*cosh(x))*sinh(x)^2 - 22*(a*b^2 - a*c^2)*cosh...
 

3.8.61.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.8.61.7 Maxima [F]

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

input

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

output

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

3.8.61.8 Giac [F]

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

input

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

output

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

3.8.61.9 Mupad [F(-1)]

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

input

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

output

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