\(\int (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x))^{5/2} \, dx\) [768]

Optimal result

Integrand size = 26, antiderivative size = 140 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2} \, dx=\frac {64 \left (b^2-c^2\right ) (c \cosh (x)+b \sinh (x))}{15 \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}}+\frac {16}{15} \sqrt {b^2-c^2} (c \cosh (x)+b \sinh (x)) \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}+\frac {2}{5} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2} \]

[Out]

2/5*(c*cosh(x)+b*sinh(x))*(b*cosh(x)+c*sinh(x)+(b^2-c^2)^(1/2))^(3/2)+64/15*(b^2-c^2)*(c*cosh(x)+b*sinh(x))/(b
*cosh(x)+c*sinh(x)+(b^2-c^2)^(1/2))^(1/2)+16/15*(c*cosh(x)+b*sinh(x))*(b^2-c^2)^(1/2)*(b*cosh(x)+c*sinh(x)+(b^
2-c^2)^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3192, 3191} \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2} \, dx=\frac {2}{5} (b \sinh (x)+c \cosh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}+\frac {16}{15} \sqrt {b^2-c^2} (b \sinh (x)+c \cosh (x)) \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}+\frac {64 \left (b^2-c^2\right ) (b \sinh (x)+c \cosh (x))}{15 \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}} \]

[In]

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

[Out]

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

Rule 3191

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

Rule 3192

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

Rubi steps \begin{align*} \text {integral}& = \frac {2}{5} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}+\frac {1}{5} \left (8 \sqrt {b^2-c^2}\right ) \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2} \, dx \\ & = \frac {16}{15} \sqrt {b^2-c^2} (c \cosh (x)+b \sinh (x)) \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}+\frac {2}{5} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}+\frac {1}{15} \left (32 \left (b^2-c^2\right )\right ) \int \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} \, dx \\ & = \frac {64 \left (b^2-c^2\right ) (c \cosh (x)+b \sinh (x))}{15 \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}}+\frac {16}{15} \sqrt {b^2-c^2} (c \cosh (x)+b \sinh (x)) \sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}+\frac {2}{5} (c \cosh (x)+b \sinh (x)) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 56.31 (sec) , antiderivative size = 4500, normalized size of antiderivative = 32.14 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

Sqrt[b^2 - c^2]*((4*b*Sqrt[b^2 - c^2])/(3*c) + (4*c*Cosh[x])/3 + (4*b*Sinh[x])/3)*Sqrt[Sqrt[b^2 - c^2] + b*Cos
h[x] + c*Sinh[x]] + Sqrt[Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]]*((44*b*(b^2 - c^2))/(15*c) + (2*c*Sqrt[b^2 -
c^2]*Cosh[x])/15 + (2*b*c*Cosh[2*x])/5 + (2*b*Sqrt[b^2 - c^2]*Sinh[x])/15 + ((b^2 + c^2)*Sinh[2*x])/5) + (256
*b*(-b + c)*(b + c)^2*Sqrt[b^2 - c^2]*(EllipticF[ArcSin[Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((
-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1] - 2*EllipticPi[-1, ArcSin[Sqrt[-(((-b - c + Sqrt[b^2 - c^2]
)*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1])*Sqrt[Sqrt[(b - c)*(b + c)] + b*Cosh[x
] + c*Sinh[x]]*(-1 + Tanh[x/2])*Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2]
)*(-1 + Tanh[x/2])))]*(-c + (-b + Sqrt[b^2 - c^2])*Tanh[x/2]))/(15*(b + c - Sqrt[b^2 - c^2])^2*(b + c + Sqrt[b
^2 - c^2])*(1 + Cosh[x])*Sqrt[(Sqrt[(b - c)*(b + c)] + b*Cosh[x] + c*Sinh[x])/(1 + Cosh[x])^2]*Sqrt[(-1 + Tanh
[x/2]^2)*(-2*c*Tanh[x/2] + Sqrt[b^2 - c^2]*(-1 + Tanh[x/2]^2) - b*(1 + Tanh[x/2]^2))]) + (128*(b - c)^2*(b + c
)^2*Sqrt[Sqrt[(b - c)*(b + c)] + b*Cosh[x] + c*Sinh[x]]*(2*b^3*c^2 + 3*b^2*c^3 - c^5 - 2*b^2*c^2*Sqrt[b^2 - c^
2] - 3*b*c^3*Sqrt[b^2 - c^2] - c^4*Sqrt[b^2 - c^2] + 8*b^4*c*Tanh[x/2] + 12*b^3*c^2*Tanh[x/2] - 2*b^2*c^3*Tanh
[x/2] - 8*b*c^4*Tanh[x/2] - 2*c^5*Tanh[x/2] - 8*b^3*c*Sqrt[b^2 - c^2]*Tanh[x/2] - 12*b^2*c^2*Sqrt[b^2 - c^2]*T
anh[x/2] - 2*b*c^3*Sqrt[b^2 - c^2]*Tanh[x/2] + 2*c^4*Sqrt[b^2 - c^2]*Tanh[x/2] + 8*b^5*Tanh[x/2]^2 + 12*b^4*c*
Tanh[x/2]^2 - 4*b^3*c^2*Tanh[x/2]^2 - 11*b^2*c^3*Tanh[x/2]^2 - 2*b*c^4*Tanh[x/2]^2 + c^5*Tanh[x/2]^2 - 8*b^4*S
qrt[b^2 - c^2]*Tanh[x/2]^2 - 12*b^3*c*Sqrt[b^2 - c^2]*Tanh[x/2]^2 + 5*b*c^3*Sqrt[b^2 - c^2]*Tanh[x/2]^2 + c^4*
Sqrt[b^2 - c^2]*Tanh[x/2]^2 - 8*b^4*c*EllipticPi[-1, ArcSin[Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2])
)/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1]*Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-
b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] + 8*b^2*c^3*EllipticPi[-1, ArcSin[Sqrt[-(((-b - c + Sqrt[b^2 - c^
2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1]*Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(
1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] + 8*b^3*c*Sqrt[b^2 - c^2]*EllipticPi[-1, ArcSi
n[Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1]*Sqr
t[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] - 4*b*c^3*Sqr
t[b^2 - c^2]*EllipticPi[-1, ArcSin[Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c
^2])*(-1 + Tanh[x/2])))]], 1]*Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*
(-1 + Tanh[x/2])))] - 16*b^5*EllipticPi[-1, ArcSin[Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b +
c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1]*Tanh[x/2]*Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((
-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] + 8*b^4*c*EllipticPi[-1, ArcSin[Sqrt[-(((-b - c + Sqrt[b^2 - c^2
])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1]*Tanh[x/2]*Sqrt[-(((-b - c + Sqrt[b^2
- c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] + 20*b^3*c^2*EllipticPi[-1, ArcSin[Sq
rt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1]*Tanh[x/
2]*Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] - 8*b^2
*c^3*EllipticPi[-1, ArcSin[Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1
+ Tanh[x/2])))]], 1]*Tanh[x/2]*Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2]
)*(-1 + Tanh[x/2])))] - 4*b*c^4*EllipticPi[-1, ArcSin[Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b
+ c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1]*Tanh[x/2]*Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))
/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] + 16*b^4*Sqrt[b^2 - c^2]*EllipticPi[-1, ArcSin[Sqrt[-(((-b -
c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1]*Tanh[x/2]*Sqrt[-(((
-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] - 8*b^3*c*Sqrt[b^2
- c^2]*EllipticPi[-1, ArcSin[Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(
-1 + Tanh[x/2])))]], 1]*Tanh[x/2]*Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^
2])*(-1 + Tanh[x/2])))] - 12*b^2*c^2*Sqrt[b^2 - c^2]*EllipticPi[-1, ArcSin[Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*
(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1]*Tanh[x/2]*Sqrt[-(((-b - c + Sqrt[b^2 - c
^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] + 4*b*c^3*Sqrt[b^2 - c^2]*EllipticPi[-1,
ArcSin[Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]],
1]*Tanh[x/2]*Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2]))
)] + 16*b^5*EllipticPi[-1, ArcSin[Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^
2])*(-1 + Tanh[x/2])))]], 1]*Tanh[x/2]^2*Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b
^2 - c^2])*(-1 + Tanh[x/2])))] - 20*b^3*c^2*EllipticPi[-1, ArcSin[Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh
[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1]*Tanh[x/2]^2*Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1
+ Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] + 4*b*c^4*EllipticPi[-1, ArcSin[Sqrt[-(((-b - c
+ Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1]*Tanh[x/2]^2*Sqrt[-((
(-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] - 16*b^4*Sqrt[b^2
- c^2]*EllipticPi[-1, ArcSin[Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(
-1 + Tanh[x/2])))]], 1]*Tanh[x/2]^2*Sqrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 -
c^2])*(-1 + Tanh[x/2])))] + 12*b^2*c^2*Sqrt[b^2 - c^2]*EllipticPi[-1, ArcSin[Sqrt[-(((-b - c + Sqrt[b^2 - c^2]
)*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1]*Tanh[x/2]^2*Sqrt[-(((-b - c + Sqrt[b^2
- c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] + 2*c*EllipticE[ArcSin[Sqrt[-(((-b -
c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1]*(-1 + Tanh[x/2])*S
qrt[-(((-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]*(4*b^4*Tanh
[x/2] + c^3*(Sqrt[b^2 - c^2] + c*Tanh[x/2]) - b^2*c*(2*Sqrt[b^2 - c^2] + 5*c*Tanh[x/2]) + b^3*(2*c - 4*Sqrt[b^
2 - c^2]*Tanh[x/2]) + b*c^2*(-2*c + 3*Sqrt[b^2 - c^2]*Tanh[x/2])) + 2*b*EllipticF[ArcSin[Sqrt[-(((-b - c + Sqr
t[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1]*(-1 + Tanh[x/2])*Sqrt[-(((
-b - c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((-b + c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]*(-4*b^4*Tanh[x/2] -
c^3*(Sqrt[b^2 - c^2] + c*Tanh[x/2]) + b^2*c*(2*Sqrt[b^2 - c^2] + 5*c*Tanh[x/2]) + b*c^2*(2*c - 3*Sqrt[b^2 - c
^2]*Tanh[x/2]) + b^3*(-2*c + 4*Sqrt[b^2 - c^2]*Tanh[x/2]))))/(15*c*Sqrt[b^2 - c^2]*(b + c - Sqrt[b^2 - c^2])^2
*(-b + Sqrt[b^2 - c^2])*(-b + c + Sqrt[b^2 - c^2])*(1 + Cosh[x])*Sqrt[(Sqrt[(b - c)*(b + c)] + b*Cosh[x] + c*S
inh[x])/(1 + Cosh[x])^2]*Sqrt[(-1 + Tanh[x/2]^2)*(-2*c*Tanh[x/2] + Sqrt[b^2 - c^2]*(-1 + Tanh[x/2]^2) - b*(1 +
Tanh[x/2]^2))])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(288\) vs. \(2(120)=240\).

Time = 0.60 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.06

method	result	size
default	\(-\frac {\left (b -c \right )^{2} \left (b +c \right )^{2} \left (\frac {\cosh \left (x \right )^{3}}{3}+2 \cosh \left (x \right )\right )}{\sqrt {-\frac {\sinh \left (x \right ) b^{2}-\sinh \left (x \right ) c^{2}-b^{2}+c^{2}}{\sqrt {b^{2}-c^{2}}}}\, \sqrt {\left (b -c \right ) \left (b +c \right )}}-\frac {\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )-1\right ) \sinh \left (x \right )^{2}}\, \sqrt {b^{2}-c^{2}}\, \left (-b^{2}+c^{2}\right ) \left (-\frac {\cosh \left (x \right ) \sqrt {b^{2}-c^{2}}\, \sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )-1\right ) \sinh \left (x \right )^{2}}}{2 \left (\sinh \left (x \right ) b^{2}-\sinh \left (x \right ) c^{2}-b^{2}+c^{2}\right )}+\frac {\arctan \left (\frac {\sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )-1\right )}\, \cosh \left (x \right )}{\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )-1\right ) \sinh \left (x \right )^{2}}}\right )}{2 \sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )-1\right )}}\right )}{\sinh \left (x \right ) \sqrt {-\frac {\sinh \left (x \right ) b^{2}-\sinh \left (x \right ) c^{2}-b^{2}+c^{2}}{\sqrt {b^{2}-c^{2}}}}}\)	\(289\)

[In]

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

[Out]

-1/(-(sinh(x)*b^2-sinh(x)*c^2-b^2+c^2)/(b^2-c^2)^(1/2))^(1/2)*(b-c)^2*(b+c)^2/((b-c)*(b+c))^(1/2)*(1/3*cosh(x)
^3+2*cosh(x))-(-(b^2-c^2)^(1/2)*(sinh(x)-1)*sinh(x)^2)^(1/2)*(b^2-c^2)^(1/2)*(-b^2+c^2)*(-1/2*cosh(x)/(sinh(x)
*b^2-sinh(x)*c^2-b^2+c^2)*(b^2-c^2)^(1/2)*(-(b^2-c^2)^(1/2)*(sinh(x)-1)*sinh(x)^2)^(1/2)+1/2/((b^2-c^2)^(1/2)*
(sinh(x)-1))^(1/2)*arctan(((b^2-c^2)^(1/2)*(sinh(x)-1))^(1/2)*cosh(x)/(-(b^2-c^2)^(1/2)*(sinh(x)-1)*sinh(x)^2)
^(1/2)))/sinh(x)/(-(sinh(x)*b^2-sinh(x)*c^2-b^2+c^2)/(b^2-c^2)^(1/2))^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 784 vs. \(2 (120) = 240\).

Time = 0.32 (sec) , antiderivative size = 784, normalized size of antiderivative = 5.60 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/30*sqrt(1/2)*(3*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^6 + 18*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)*sinh(
x)^5 + 3*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*sinh(x)^6 + 125*(b^3 + b^2*c - b*c^2 - c^3)*cosh(x)^4 + 5*(25*b^3 + 2
5*b^2*c - 25*b*c^2 - 25*c^3 + 9*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^2)*sinh(x)^4 + 20*(3*(b^3 + 3*b^2*c +
3*b*c^2 + c^3)*cosh(x)^3 + 25*(b^3 + b^2*c - b*c^2 - c^3)*cosh(x))*sinh(x)^3 + 3*b^3 - 9*b^2*c + 9*b*c^2 - 3*c
^3 + 125*(b^3 - b^2*c - b*c^2 + c^3)*cosh(x)^2 + 5*(9*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^4 + 25*b^3 - 25*
b^2*c - 25*b*c^2 + 25*c^3 + 150*(b^3 + b^2*c - b*c^2 - c^3)*cosh(x)^2)*sinh(x)^2 + 2*(9*(b^3 + 3*b^2*c + 3*b*c
^2 + c^3)*cosh(x)^5 + 250*(b^3 + b^2*c - b*c^2 - c^3)*cosh(x)^3 + 125*(b^3 - b^2*c - b*c^2 + c^3)*cosh(x))*sin
h(x) + 2*(11*(b^2 + 2*b*c + c^2)*cosh(x)^5 + 55*(b^2 + 2*b*c + c^2)*cosh(x)*sinh(x)^4 + 11*(b^2 + 2*b*c + c^2)
*sinh(x)^5 - 150*(b^2 - c^2)*cosh(x)^3 + 10*(11*(b^2 + 2*b*c + c^2)*cosh(x)^2 - 15*b^2 + 15*c^2)*sinh(x)^3 + 1
0*(11*(b^2 + 2*b*c + c^2)*cosh(x)^3 - 45*(b^2 - c^2)*cosh(x))*sinh(x)^2 + 11*(b^2 - 2*b*c + c^2)*cosh(x) + (55
*(b^2 + 2*b*c + c^2)*cosh(x)^4 - 450*(b^2 - c^2)*cosh(x)^2 + 11*b^2 - 22*b*c + 11*c^2)*sinh(x))*sqrt(b^2 - c^2
))*sqrt(((b + c)*cosh(x)^2 + 2*(b + c)*cosh(x)*sinh(x) + (b + c)*sinh(x)^2 + 2*sqrt(b^2 - c^2)*(cosh(x) + sinh
(x)) + b - c)/(cosh(x) + sinh(x)))/((b + c)*cosh(x)^4 + 4*(b + c)*cosh(x)*sinh(x)^3 + (b + c)*sinh(x)^4 - (b -
c)*cosh(x)^2 + (6*(b + c)*cosh(x)^2 - b + c)*sinh(x)^2 + 2*(2*(b + c)*cosh(x)^3 - (b - c)*cosh(x))*sinh(x))

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1783 vs. \(2 (120) = 240\).

Time = 2.15 (sec) , antiderivative size = 1783, normalized size of antiderivative = 12.74 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/20*sqrt(2)*(sqrt(b + c)*sqrt(b - c)*b^2 + 2*sqrt(b + c)*sqrt(b - c)*b*c + sqrt(b + c)*sqrt(b - c)*c^2)*(2*sq
rt(b + c)*sqrt(b - c)*e^(-x) + (b - c)*e^(-2*x) + b + c)^(5/2)*e^(5/2*x)/(sqrt(b + c)*sqrt(b - c)*b^2 + 2*sqrt
(b + c)*sqrt(b - c)*b*c + sqrt(b + c)*sqrt(b - c)*c^2 + 5*(b^3 + b^2*c - b*c^2 - c^3)*e^(-x) + 10*(sqrt(b + c)
*sqrt(b - c)*b^2 - sqrt(b + c)*sqrt(b - c)*c^2)*e^(-2*x) + 10*(b^3 - b^2*c - b*c^2 + c^3)*e^(-3*x) + 5*(sqrt(b
+ c)*sqrt(b - c)*b^2 - 2*sqrt(b + c)*sqrt(b - c)*b*c + sqrt(b + c)*sqrt(b - c)*c^2)*e^(-4*x) + (b^3 - 3*b^2*c
+ 3*b*c^2 - c^3)*e^(-5*x)) + 5/12*sqrt(2)*(b^3 + b^2*c - b*c^2 - c^3)*(2*sqrt(b + c)*sqrt(b - c)*e^(-x) + (b
- c)*e^(-2*x) + b + c)^(5/2)*e^(3/2*x)/(sqrt(b + c)*sqrt(b - c)*b^2 + 2*sqrt(b + c)*sqrt(b - c)*b*c + sqrt(b +
c)*sqrt(b - c)*c^2 + 5*(b^3 + b^2*c - b*c^2 - c^3)*e^(-x) + 10*(sqrt(b + c)*sqrt(b - c)*b^2 - sqrt(b + c)*sqr
t(b - c)*c^2)*e^(-2*x) + 10*(b^3 - b^2*c - b*c^2 + c^3)*e^(-3*x) + 5*(sqrt(b + c)*sqrt(b - c)*b^2 - 2*sqrt(b +
c)*sqrt(b - c)*b*c + sqrt(b + c)*sqrt(b - c)*c^2)*e^(-4*x) + (b^3 - 3*b^2*c + 3*b*c^2 - c^3)*e^(-5*x)) + 5/2*
sqrt(2)*(sqrt(b + c)*sqrt(b - c)*b^2 - sqrt(b + c)*sqrt(b - c)*c^2)*(2*sqrt(b + c)*sqrt(b - c)*e^(-x) + (b - c
)*e^(-2*x) + b + c)^(5/2)*e^(1/2*x)/(sqrt(b + c)*sqrt(b - c)*b^2 + 2*sqrt(b + c)*sqrt(b - c)*b*c + sqrt(b + c)
*sqrt(b - c)*c^2 + 5*(b^3 + b^2*c - b*c^2 - c^3)*e^(-x) + 10*(sqrt(b + c)*sqrt(b - c)*b^2 - sqrt(b + c)*sqrt(b
- c)*c^2)*e^(-2*x) + 10*(b^3 - b^2*c - b*c^2 + c^3)*e^(-3*x) + 5*(sqrt(b + c)*sqrt(b - c)*b^2 - 2*sqrt(b + c)
*sqrt(b - c)*b*c + sqrt(b + c)*sqrt(b - c)*c^2)*e^(-4*x) + (b^3 - 3*b^2*c + 3*b*c^2 - c^3)*e^(-5*x)) - 5/2*sqr
t(2)*(b^3 - b^2*c - b*c^2 + c^3)*(2*sqrt(b + c)*sqrt(b - c)*e^(-x) + (b - c)*e^(-2*x) + b + c)^(5/2)*e^(-1/2*x
)/(sqrt(b + c)*sqrt(b - c)*b^2 + 2*sqrt(b + c)*sqrt(b - c)*b*c + sqrt(b + c)*sqrt(b - c)*c^2 + 5*(b^3 + b^2*c
- b*c^2 - c^3)*e^(-x) + 10*(sqrt(b + c)*sqrt(b - c)*b^2 - sqrt(b + c)*sqrt(b - c)*c^2)*e^(-2*x) + 10*(b^3 - b^
2*c - b*c^2 + c^3)*e^(-3*x) + 5*(sqrt(b + c)*sqrt(b - c)*b^2 - 2*sqrt(b + c)*sqrt(b - c)*b*c + sqrt(b + c)*sqr
t(b - c)*c^2)*e^(-4*x) + (b^3 - 3*b^2*c + 3*b*c^2 - c^3)*e^(-5*x)) - 5/12*sqrt(2)*(sqrt(b + c)*sqrt(b - c)*b^2
- 2*sqrt(b + c)*sqrt(b - c)*b*c + sqrt(b + c)*sqrt(b - c)*c^2)*(2*sqrt(b + c)*sqrt(b - c)*e^(-x) + (b - c)*e^
(-2*x) + b + c)^(5/2)*e^(-3/2*x)/(sqrt(b + c)*sqrt(b - c)*b^2 + 2*sqrt(b + c)*sqrt(b - c)*b*c + sqrt(b + c)*sq
rt(b - c)*c^2 + 5*(b^3 + b^2*c - b*c^2 - c^3)*e^(-x) + 10*(sqrt(b + c)*sqrt(b - c)*b^2 - sqrt(b + c)*sqrt(b -
c)*c^2)*e^(-2*x) + 10*(b^3 - b^2*c - b*c^2 + c^3)*e^(-3*x) + 5*(sqrt(b + c)*sqrt(b - c)*b^2 - 2*sqrt(b + c)*sq
rt(b - c)*b*c + sqrt(b + c)*sqrt(b - c)*c^2)*e^(-4*x) + (b^3 - 3*b^2*c + 3*b*c^2 - c^3)*e^(-5*x)) - 1/20*sqrt(
2)*(b^3 - 3*b^2*c + 3*b*c^2 - c^3)*(2*sqrt(b + c)*sqrt(b - c)*e^(-x) + (b - c)*e^(-2*x) + b + c)^(5/2)*e^(-5/2
*x)/(sqrt(b + c)*sqrt(b - c)*b^2 + 2*sqrt(b + c)*sqrt(b - c)*b*c + sqrt(b + c)*sqrt(b - c)*c^2 + 5*(b^3 + b^2*
c - b*c^2 - c^3)*e^(-x) + 10*(sqrt(b + c)*sqrt(b - c)*b^2 - sqrt(b + c)*sqrt(b - c)*c^2)*e^(-2*x) + 10*(b^3 -
b^2*c - b*c^2 + c^3)*e^(-3*x) + 5*(sqrt(b + c)*sqrt(b - c)*b^2 - 2*sqrt(b + c)*sqrt(b - c)*b*c + sqrt(b + c)*s
qrt(b - c)*c^2)*e^(-4*x) + (b^3 - 3*b^2*c + 3*b*c^2 - c^3)*e^(-5*x))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (120) = 240\).

Time = 0.30 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.25 \[ \int \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2} \, dx=-\frac {\sqrt {2} {\left (150 \, {\left (b^{2} - c^{2}\right )}^{\frac {3}{2}} e^{\left (\frac {1}{2} \, x\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) + 3 \, {\left (\sqrt {b^{2} - c^{2}} b^{2} + 2 \, \sqrt {b^{2} - c^{2}} b c + \sqrt {b^{2} - c^{2}} c^{2}\right )} e^{\left (\frac {5}{2} \, x\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) + 25 \, {\left (b^{3} + b^{2} c - b c^{2} - c^{3}\right )} e^{\left (\frac {3}{2} \, x\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) - {\left (25 \, {\left (b^{2} - 2 \, b c + c^{2}\right )} \sqrt {b^{2} - c^{2}} e^{x} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) + 150 \, {\left (b^{3} - b^{2} c - b c^{2} + c^{3}\right )} e^{\left (2 \, x\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) + 3 \, {\left (b^{3} - 3 \, b^{2} c + 3 \, b c^{2} - c^{3}\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right )\right )} e^{\left (-\frac {5}{2} \, x\right )}\right )}}{60 \, \sqrt {b - c}} \]

[In]

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

[Out]

-1/60*sqrt(2)*(150*(b^2 - c^2)^(3/2)*e^(1/2*x)*sgn(-sqrt(b^2 - c^2)*e^x - b + c) + 3*(sqrt(b^2 - c^2)*b^2 + 2*
sqrt(b^2 - c^2)*b*c + sqrt(b^2 - c^2)*c^2)*e^(5/2*x)*sgn(-sqrt(b^2 - c^2)*e^x - b + c) + 25*(b^3 + b^2*c - b*c
^2 - c^3)*e^(3/2*x)*sgn(-sqrt(b^2 - c^2)*e^x - b + c) - (25*(b^2 - 2*b*c + c^2)*sqrt(b^2 - c^2)*e^x*sgn(-sqrt(
b^2 - c^2)*e^x - b + c) + 150*(b^3 - b^2*c - b*c^2 + c^3)*e^(2*x)*sgn(-sqrt(b^2 - c^2)*e^x - b + c) + 3*(b^3 -
3*b^2*c + 3*b*c^2 - c^3)*sgn(-sqrt(b^2 - c^2)*e^x - b + c))*e^(-5/2*x))/sqrt(b - c)

Mupad [F(-1)]

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

[In]

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

[Out]

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