### 3.761 $$\int (a+b \cosh (x)+c \sinh (x))^{5/2} \, dx$$

Optimal. Leaf size=294 $\frac{16 i a \left (a^2-b^2+c^2\right ) \sqrt{\frac{a+b \cosh (x)+c \sinh (x)}{a+\sqrt{b^2-c^2}}} \text{EllipticF}\left (\frac{1}{2} \left (i x-\tan ^{-1}(b,-i c)\right ),\frac{2 \sqrt{b^2-c^2}}{a+\sqrt{b^2-c^2}}\right )}{15 \sqrt{a+b \cosh (x)+c \sinh (x)}}-\frac{2 i \left (23 a^2+9 b^2-9 c^2\right ) \sqrt{a+b \cosh (x)+c \sinh (x)} E\left (\frac{1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac{2 \sqrt{b^2-c^2}}{a+\sqrt{b^2-c^2}}\right )}{15 \sqrt{\frac{a+b \cosh (x)+c \sinh (x)}{a+\sqrt{b^2-c^2}}}}+\frac{2}{5} (b \sinh (x)+c \cosh (x)) (a+b \cosh (x)+c \sinh (x))^{3/2}+\frac{16}{15} (a b \sinh (x)+a c \cosh (x)) \sqrt{a+b \cosh (x)+c \sinh (x)}$

[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]]

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Rubi [A]  time = 0.482871, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 14, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.5, Rules used = {3120, 3146, 3149, 3119, 2653, 3127, 2661} $\frac{16 i a \left (a^2-b^2+c^2\right ) \sqrt{\frac{a+b \cosh (x)+c \sinh (x)}{a+\sqrt{b^2-c^2}}} F\left (\frac{1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac{2 \sqrt{b^2-c^2}}{a+\sqrt{b^2-c^2}}\right )}{15 \sqrt{a+b \cosh (x)+c \sinh (x)}}-\frac{2 i \left (23 a^2+9 b^2-9 c^2\right ) \sqrt{a+b \cosh (x)+c \sinh (x)} E\left (\frac{1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac{2 \sqrt{b^2-c^2}}{a+\sqrt{b^2-c^2}}\right )}{15 \sqrt{\frac{a+b \cosh (x)+c \sinh (x)}{a+\sqrt{b^2-c^2}}}}+\frac{2}{5} (b \sinh (x)+c \cosh (x)) (a+b \cosh (x)+c \sinh (x))^{3/2}+\frac{16}{15} (a b \sinh (x)+a c \cosh (x)) \sqrt{a+b \cosh (x)+c \sinh (x)}$

Antiderivative was successfully veriﬁed.

[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 3120

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*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 3146

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 3149

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]

Rule 3119

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]*Cos[d + e*x - ArcTan[b, c]])/(a + Sqrt[b^2 + c^2])], 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 2653

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 3127

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]*Cos[d + e*x - ArcTan[b, c]])/(a + Sqrt[b^2 + c^2])], 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 2661

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rubi steps

\begin{align*} \int (a+b \cosh (x)+c \sinh (x))^{5/2} \, dx &=\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 ) F\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]  time = 6.33537, size = 3775, normalized size = 12.84 $\text{Result too large to show}$

Warning: Unable to verify antiderivative.

[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

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Maple [B]  time = 0.879, size = 1036, normalized size = 3.5 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6/(-b^2*sinh(x)^2+c^2*sinh(x)^2+2*sinh(x)*a*(b^2-c^2)^(1/2)-a^2)/sinh(x)*(-6*ln((cosh(x)*sinh(x)*(-b^2+c^2)+
cosh(x)*(b^2-c^2)^(1/2)*a+((-b^2+c^2)/(b^2-c^2)^(1/2)*sinh(x)^3+a*sinh(x)^2)^(1/2)*(b^2-c^2)^(1/2)*((-b^2+c^2)
/(b^2-c^2)^(1/2)*sinh(x)+a)^(1/2))/(b^2-c^2)^(1/2)/((-b^2+c^2)/(b^2-c^2)^(1/2)*sinh(x)+a)^(1/2))*((-b^2+c^2)/(
b^2-c^2)^(1/2)*sinh(x)^3+a*sinh(x)^2)^(1/2)*a^4+3*ln((cosh(x)*sinh(x)*(-b^2+c^2)+cosh(x)*(b^2-c^2)^(1/2)*a+((-
b^2+c^2)/(b^2-c^2)^(1/2)*sinh(x)^3+a*sinh(x)^2)^(1/2)*(b^2-c^2)^(1/2)*((-b^2+c^2)/(b^2-c^2)^(1/2)*sinh(x)+a)^(
1/2))/(b^2-c^2)^(1/2)/((-b^2+c^2)/(b^2-c^2)^(1/2)*sinh(x)+a)^(1/2))*((-b^2+c^2)/(b^2-c^2)^(1/2)*sinh(x)^3+a*si
nh(x)^2)^(1/2)*a^2*b^2-3*ln((cosh(x)*sinh(x)*(-b^2+c^2)+cosh(x)*(b^2-c^2)^(1/2)*a+((-b^2+c^2)/(b^2-c^2)^(1/2)*
sinh(x)^3+a*sinh(x)^2)^(1/2)*(b^2-c^2)^(1/2)*((-b^2+c^2)/(b^2-c^2)^(1/2)*sinh(x)+a)^(1/2))/(b^2-c^2)^(1/2)/((-
b^2+c^2)/(b^2-c^2)^(1/2)*sinh(x)+a)^(1/2))*((-b^2+c^2)/(b^2-c^2)^(1/2)*sinh(x)^3+a*sinh(x)^2)^(1/2)*a^2*c^2+5*
sinh(x)^3*cosh(x)*(b^2-c^2)^(3/2)*((-b^2+c^2)/(b^2-c^2)^(1/2)*sinh(x)+a)^(1/2)*a-2*(b^4-2*b^2*c^2+c^4)*cosh(x)
*sinh(x)^4*((-b^2+c^2)/(b^2-c^2)^(1/2)*sinh(x)+a)^(1/2)+3*(2*a^2-b^2+c^2)*sinh(x)*ln((cosh(x)*sinh(x)*(-b^2+c^
2)+cosh(x)*(b^2-c^2)^(1/2)*a+((-b^2+c^2)/(b^2-c^2)^(1/2)*sinh(x)^3+a*sinh(x)^2)^(1/2)*(b^2-c^2)^(1/2)*((-b^2+c
^2)/(b^2-c^2)^(1/2)*sinh(x)+a)^(1/2))/(b^2-c^2)^(1/2)/((-b^2+c^2)/(b^2-c^2)^(1/2)*sinh(x)+a)^(1/2))*(b^2-c^2)^
(1/2)*((-b^2+c^2)/(b^2-c^2)^(1/2)*sinh(x)^3+a*sinh(x)^2)^(1/2)*a+2*sinh(x)*cosh(x)*(9*a^2-2*b^2+2*c^2)*(b^2-c^
2)^(1/2)*((-b^2+c^2)/(b^2-c^2)^(1/2)*sinh(x)+a)^(1/2)*a-(21*a^2*b^2-21*a^2*c^2-4*b^4+8*b^2*c^2-4*c^4)*cosh(x)*
sinh(x)^2*((-b^2+c^2)/(b^2-c^2)^(1/2)*sinh(x)+a)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \cosh \left (x\right ) + c \sinh \left (x\right ) + a\right )}^{\frac{5}{2}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{2} \cosh \left (x\right )^{2} + c^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + a^{2} + 2 \,{\left (b c \cosh \left (x\right ) + a c\right )} \sinh \left (x\right )\right )} \sqrt{b \cosh \left (x\right ) + c \sinh \left (x\right ) + a}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*cosh(x)^2 + c^2*sinh(x)^2 + 2*a*b*cosh(x) + a^2 + 2*(b*c*cosh(x) + a*c)*sinh(x))*sqrt(b*cosh(x)
+ c*sinh(x) + a), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \cosh \left (x\right ) + c \sinh \left (x\right ) + a\right )}^{\frac{5}{2}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)