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

Optimal result

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

[Out]

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)*(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+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*ar
ctan(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*sin
h(x))^(1/2)

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3199, 3225, 3228, 3198, 2732, 3206, 2740} \[ \int (a+b \cosh (x)+c \sinh (x))^{5/2} \, dx=\frac {16 i a \left (a^2-b^2+c^2\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}} \operatorname {EllipticF}\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right ),\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{15 \sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {2 i \left (23 a^2+9 b^2-9 c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)} E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{15 \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}+\frac {2}{5} (b \sinh (x)+c \cosh (x)) (a+b \cosh (x)+c \sinh (x))^{3/2}+\frac {16}{15} (a b \sinh (x)+a c \cosh (x)) \sqrt {a+b \cosh (x)+c \sinh (x)} \]

[In]

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

[Out]

(16*(a*c*Cosh[x] + a*b*Sinh[x])*Sqrt[a + b*Cosh[x] + c*Sinh[x]])/15 + (2*(c*Cosh[x] + b*Sinh[x])*(a + b*Cosh[x
] + c*Sinh[x])^(3/2))/5 - (((2*I)/15)*(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)/15)*a*(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]]

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 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 3198

Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*C
os[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
+ 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 3199

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] + Dist[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*S
in[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 3206

Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a +
b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[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] /; Free
Q[{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 3225

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] + Dist[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 3228

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] :> Dist[B/b, Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]
, x], x] + Dist[(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]

Rubi steps \begin{align*} \text {integral}& = \frac {2}{5} (c \cosh (x)+b \sinh (x)) (a+b \cosh (x)+c \sinh (x))^{3/2}+\frac {2}{5} \int \sqrt {a+b \cosh (x)+c \sinh (x)} \left (\frac {1}{2} \left (5 a^2+3 b^2-3 c^2\right )+4 a b \cosh (x)+4 a c \sinh (x)\right ) \, 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 {4 \int \frac {\frac {1}{4} a^2 \left (15 a^2+17 b^2-17 c^2\right )+\frac {1}{4} a b \left (23 a^2+9 b^2-9 c^2\right ) \cosh (x)+\frac {1}{4} a c \left (23 a^2+9 b^2-9 c^2\right ) \sinh (x)}{\sqrt {a+b \cosh (x)+c \sinh (x)}} \, dx}{15 a} \\ & = \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 {1}{15} \left (23 a^2+9 b^2-9 c^2\right ) \int \sqrt {a+b \cosh (x)+c \sinh (x)} \, dx-\frac {1}{15} \left (8 a \left (a^2-b^2+c^2\right )\right ) \int \frac {1}{\sqrt {a+b \cosh (x)+c \sinh (x)}} \, 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 {\left (\left (23 a^2+9 b^2-9 c^2\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}\right ) \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}{15 \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}-\frac {\left (8 a \left (a^2-b^2+c^2\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}\right ) \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}{15 \sqrt {a+b \cosh (x)+c \sinh (x)}} \\ & = \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)}} \\ \end{align*}

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 \[ \int (a+b \cosh (x)+c \sinh (x))^{5/2} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

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 + Sq
rt[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])]*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*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]]])/(15*Sqrt[1 - b^2/c^2]*c*Sqrt[I*(I + Sinh[x + ArcTanh[b/c]])]) - (34*
a*c*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*Sqr
t[(-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])]*Sqr
t[(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]]])/(15*Sqrt[1 - b^2/c^2]*Sqrt[I*(I + Sinh[x +
ArcTanh[b/c]])]) - (23*a^2*b^2*((c*AppellF1[-1/2, -1/2, -1/2, 1/2, (a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c
/b]])/(b*Sqrt[1 - c^2/b^2]*(1 + a/(b*Sqrt[1 - c^2/b^2]))), (a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]])/(b
*Sqrt[1 - c^2/b^2]*(-1 + a/(b*Sqrt[1 - c^2/b^2])))]*Sinh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]*Sqrt[(b*Sqrt[
(b^2 - c^2)/b^2] - b*Sqrt[(b^2 - c^2)/b^2]*Cosh[x + ArcTanh[c/b]])/(a + b*Sqrt[(b^2 - c^2)/b^2])]*Sqrt[a + b*S
qrt[(b^2 - c^2)/b^2]*Cosh[x + ArcTanh[c/b]]]*Sqrt[(b*Sqrt[(b^2 - c^2)/b^2] + b*Sqrt[(b^2 - c^2)/b^2]*Cosh[x +
ArcTanh[c/b]])/(-a + b*Sqrt[(b^2 - c^2)/b^2])]) - ((-2*b*(a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]]))/(b^
2 - c^2) + (c*Sinh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]))/Sqrt[a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b
]]]))/(15*c) - (3*b^4*((c*AppellF1[-1/2, -1/2, -1/2, 1/2, (a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]])/(b*
Sqrt[1 - c^2/b^2]*(1 + a/(b*Sqrt[1 - c^2/b^2]))), (a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]])/(b*Sqrt[1 -
c^2/b^2]*(-1 + a/(b*Sqrt[1 - c^2/b^2])))]*Sinh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]*Sqrt[(b*Sqrt[(b^2 - c^
2)/b^2] - b*Sqrt[(b^2 - c^2)/b^2]*Cosh[x + ArcTanh[c/b]])/(a + b*Sqrt[(b^2 - c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2
- c^2)/b^2]*Cosh[x + ArcTanh[c/b]]]*Sqrt[(b*Sqrt[(b^2 - c^2)/b^2] + b*Sqrt[(b^2 - c^2)/b^2]*Cosh[x + ArcTanh[c
/b]])/(-a + b*Sqrt[(b^2 - c^2)/b^2])]) - ((-2*b*(a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]]))/(b^2 - c^2)
+ (c*Sinh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]))/Sqrt[a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]]]))/(5*
c) + (23*a^2*c*((c*AppellF1[-1/2, -1/2, -1/2, 1/2, (a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]])/(b*Sqrt[1
- c^2/b^2]*(1 + a/(b*Sqrt[1 - c^2/b^2]))), (a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^
2]*(-1 + a/(b*Sqrt[1 - c^2/b^2])))]*Sinh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]*Sqrt[(b*Sqrt[(b^2 - c^2)/b^2]
- b*Sqrt[(b^2 - c^2)/b^2]*Cosh[x + ArcTanh[c/b]])/(a + b*Sqrt[(b^2 - c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 - c^2)/
b^2]*Cosh[x + ArcTanh[c/b]]]*Sqrt[(b*Sqrt[(b^2 - c^2)/b^2] + b*Sqrt[(b^2 - c^2)/b^2]*Cosh[x + ArcTanh[c/b]])/(
-a + b*Sqrt[(b^2 - c^2)/b^2])]) - ((-2*b*(a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]]))/(b^2 - c^2) + (c*Si
nh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]))/Sqrt[a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]]]))/15 + (6*b^
2*c*((c*AppellF1[-1/2, -1/2, -1/2, 1/2, (a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]*
(1 + a/(b*Sqrt[1 - c^2/b^2]))), (a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]*(-1 + a/
(b*Sqrt[1 - c^2/b^2])))]*Sinh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]*Sqrt[(b*Sqrt[(b^2 - c^2)/b^2] - b*Sqrt[(
b^2 - c^2)/b^2]*Cosh[x + ArcTanh[c/b]])/(a + b*Sqrt[(b^2 - c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 - c^2)/b^2]*Cosh[x
+ ArcTanh[c/b]]]*Sqrt[(b*Sqrt[(b^2 - c^2)/b^2] + b*Sqrt[(b^2 - c^2)/b^2]*Cosh[x + ArcTanh[c/b]])/(-a + b*Sqrt
[(b^2 - c^2)/b^2])]) - ((-2*b*(a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]]))/(b^2 - c^2) + (c*Sinh[x + ArcT
anh[c/b]])/(b*Sqrt[1 - c^2/b^2]))/Sqrt[a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]]]))/5 - (3*c^3*((c*Appell
F1[-1/2, -1/2, -1/2, 1/2, (a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]*(1 + a/(b*Sqrt
[1 - c^2/b^2]))), (a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]*(-1 + a/(b*Sqrt[1 - c^
2/b^2])))]*Sinh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]*Sqrt[(b*Sqrt[(b^2 - c^2)/b^2] - b*Sqrt[(b^2 - c^2)/b^2
]*Cosh[x + ArcTanh[c/b]])/(a + b*Sqrt[(b^2 - c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 - c^2)/b^2]*Cosh[x + ArcTanh[c/b
]]]*Sqrt[(b*Sqrt[(b^2 - c^2)/b^2] + b*Sqrt[(b^2 - c^2)/b^2]*Cosh[x + ArcTanh[c/b]])/(-a + b*Sqrt[(b^2 - c^2)/b
^2])]) - ((-2*b*(a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]]))/(b^2 - c^2) + (c*Sinh[x + ArcTanh[c/b]])/(b*
Sqrt[1 - c^2/b^2]))/Sqrt[a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]]]))/5

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}}}}}\)	\(899\)

[In]

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

[Out]

-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*sinh(x)+sinh(x)*c^2+a*(b^2-c^2)^(1/2))/(b^2-c^2)^(1/2))^(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 = 928, normalized size of antiderivative = 3.16 \[ \int (a+b \cosh (x)+c \sinh (x))^{5/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-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), weierstrass
PInverse(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(x) + 2*(6*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^3 - 11*a*b^2
+ 11*a*c^2 + 33*(a*b^2 + 2*a*b*c + a*c^2)*cosh(x)^2 - 4*(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(b*cosh(x) + c*sinh(x) + a))/((b + c)*cosh(x)^2 + 2*(b + c)*cosh(x)*sinh(x) + (b +
c)*sinh(x)^2)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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