3.189 \(\int (d-c^2 d x^2)^{5/2} (a+b \cosh ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=486 \[ \frac{b d^2 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{18 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{48 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{48 b c \sqrt{c x-1} \sqrt{c x+1}}+\frac{5}{16} d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{5 b c d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{6} x \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{5}{24} d x \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{108} b^2 d^2 x (1-c x)^2 (c x+1)^2 \sqrt{d-c^2 d x^2}+\frac{245 b^2 d^2 x \sqrt{d-c^2 d x^2}}{1152}+\frac{65 b^2 d^2 x (1-c x) (c x+1) \sqrt{d-c^2 d x^2}}{1728}+\frac{115 b^2 d^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{1152 c \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

(245*b^2*d^2*x*Sqrt[d - c^2*d*x^2])/1152 + (65*b^2*d^2*x*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2])/1728 + (b^2*
d^2*x*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2])/108 + (115*b^2*d^2*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(1152*
c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5*b*c*d^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(16*Sqrt[-1 + c*x]*
Sqrt[1 + c*x]) + (5*b*d^2*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(48*c*Sqrt[-1 + c*x]*Sqrt[
1 + c*x]) + (b*d^2*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(18*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x
]) + (5*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/16 + (5*d*x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x
])^2)/24 + (x*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2)/6 - (5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]
)^3)/(48*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

________________________________________________________________________________________

Rubi [A]  time = 0.867642, antiderivative size = 517, normalized size of antiderivative = 1.06, number of steps used = 17, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {5713, 5685, 5683, 5676, 5662, 90, 52, 5716, 38} \[ \frac{b d^2 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{18 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{48 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{48 b c \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{6} d^2 x (1-c x)^2 (c x+1)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{5}{16} d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{5}{24} d^2 x (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{5 b c d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{108} b^2 d^2 x (1-c x)^2 (c x+1)^2 \sqrt{d-c^2 d x^2}+\frac{245 b^2 d^2 x \sqrt{d-c^2 d x^2}}{1152}+\frac{65 b^2 d^2 x (1-c x) (c x+1) \sqrt{d-c^2 d x^2}}{1728}+\frac{115 b^2 d^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{1152 c \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(245*b^2*d^2*x*Sqrt[d - c^2*d*x^2])/1152 + (65*b^2*d^2*x*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2])/1728 + (b^2*
d^2*x*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2])/108 + (115*b^2*d^2*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(1152*
c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5*b*c*d^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(16*Sqrt[-1 + c*x]*
Sqrt[1 + c*x]) + (5*b*d^2*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(48*c*Sqrt[-1 + c*x]*Sqrt[
1 + c*x]) + (b*d^2*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(18*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x
]) + (5*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/16 + (5*d^2*x*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2
]*(a + b*ArcCosh[c*x])^2)/24 + (d^2*x*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/6 -
(5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^3)/(48*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 5713

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist[((-d)^IntPart[p]*(
d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(1 + c*x)^p*(-1 + c*x)^p*(a + b*Ar
cCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[c^2*d + e, 0] &&  !IntegerQ[p]

Rule 5685

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbo
l] :> Simp[(x*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n)/(2*p + 1), x] + (Dist[(2*d1*d2*p)/(2*p + 1),
Int[(d1 + e1*x)^(p - 1)*(d2 + e2*x)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[(b*c*n*(-(d1*d2))^(p - 1/2)
*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/((2*p + 1)*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[x*(-1 + c^2*x^2)^(p - 1/2)*(a
+ b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)]
&& GtQ[n, 0] && GtQ[p, 0] && IntegerQ[p - 1/2]

Rule 5683

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)], x_Symbol] :
> Simp[(x*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/2, x] + (-Dist[(Sqrt[d1 + e1*x]*Sqrt[d2 + e2
*x])/(2*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Dis
t[(b*c*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(2*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[x*(a + b*ArcCosh[c*x])^(n - 1)
, x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[n, 0]

Rule 5676

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n},
x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0] && NeQ[n, -1]

Rule 5662

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
osh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 52

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

Rule 5716

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)
^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e*(p + 1)), x] - Dist[(b*n*(-d)^p)/(2*c*(p + 1)), Int[(1 + c*x)^(p + 1/2)*
(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0]
&& GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p]

Rule 38

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(x*(a + b*x)^m*(c + d*x)^m)/(2*m + 1)
, x] + Dist[(2*a*c*m)/(2*m + 1), Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] &&
EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0]

Rubi steps

\begin{align*} \int \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{6} d^2 x (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{\left (5 d^2 \sqrt{d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{6 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \int x \left (-1+c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b d^2 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{18 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5}{24} d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{6} d^2 x (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{\left (5 d^2 \sqrt{d-c^2 d x^2}\right ) \int \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{8 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int (-1+c x)^{5/2} (1+c x)^{5/2} \, dx}{18 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (5 b c d^2 \sqrt{d-c^2 d x^2}\right ) \int x \left (-1+c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{12 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{108} b^2 d^2 x (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2}+\frac{5 b d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{48 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d^2 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{18 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5}{16} d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{5}{24} d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{6} d^2 x (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{\left (5 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{16 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx}{108 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx}{48 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 b c d^2 \sqrt{d-c^2 d x^2}\right ) \int x \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{8 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{65 b^2 d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}{1728}+\frac{1}{108} b^2 d^2 x (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2}-\frac{5 b c d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5 b d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{48 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d^2 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{18 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5}{16} d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{5}{24} d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{6} d^2 x (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{48 b c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (5 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \sqrt{-1+c x} \sqrt{1+c x} \, dx}{144 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (5 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \sqrt{-1+c x} \sqrt{1+c x} \, dx}{64 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (5 b^2 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{16 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{245 b^2 d^2 x \sqrt{d-c^2 d x^2}}{1152}+\frac{65 b^2 d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}{1728}+\frac{1}{108} b^2 d^2 x (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2}-\frac{5 b c d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5 b d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{48 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d^2 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{18 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5}{16} d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{5}{24} d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{6} d^2 x (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{48 b c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{288 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{128 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (5 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{32 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{245 b^2 d^2 x \sqrt{d-c^2 d x^2}}{1152}+\frac{65 b^2 d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}{1728}+\frac{1}{108} b^2 d^2 x (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2}+\frac{115 b^2 d^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{1152 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{5 b c d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5 b d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{48 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d^2 \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{18 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5}{16} d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{5}{24} d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{6} d^2 x (1-c x)^2 (1+c x)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{48 b c \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [A]  time = 3.3673, size = 740, normalized size = 1.52 \[ \frac{d^2 \left (2304 a^2 c^6 x^6 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2}+2304 a^2 c^5 x^5 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2}-7488 a^2 c^4 x^4 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2}-7488 a^2 c^3 x^3 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2}+9504 a^2 c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2}+9504 a^2 c x \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2}-4320 a^2 c \sqrt{d} x \sqrt{\frac{c x-1}{c x+1}} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-4320 a^2 \sqrt{d} \sqrt{\frac{c x-1}{c x+1}} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-3240 a b \sqrt{d-c^2 d x^2} \cosh \left (2 \cosh ^{-1}(c x)\right )+324 a b \sqrt{d-c^2 d x^2} \cosh \left (4 \cosh ^{-1}(c x)\right )-24 a b \sqrt{d-c^2 d x^2} \cosh \left (6 \cosh ^{-1}(c x)\right )-12 b \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) \left (-540 a \sinh \left (2 \cosh ^{-1}(c x)\right )+108 a \sinh \left (4 \cosh ^{-1}(c x)\right )-12 a \sinh \left (6 \cosh ^{-1}(c x)\right )+270 b \cosh \left (2 \cosh ^{-1}(c x)\right )-27 b \cosh \left (4 \cosh ^{-1}(c x)\right )+2 b \cosh \left (6 \cosh ^{-1}(c x)\right )\right )+72 b \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)^2 \left (-60 a+45 b \sinh \left (2 \cosh ^{-1}(c x)\right )-9 b \sinh \left (4 \cosh ^{-1}(c x)\right )+b \sinh \left (6 \cosh ^{-1}(c x)\right )\right )-1440 b^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)^3+1620 b^2 \sqrt{d-c^2 d x^2} \sinh \left (2 \cosh ^{-1}(c x)\right )-81 b^2 \sqrt{d-c^2 d x^2} \sinh \left (4 \cosh ^{-1}(c x)\right )+4 b^2 \sqrt{d-c^2 d x^2} \sinh \left (6 \cosh ^{-1}(c x)\right )\right )}{13824 c \sqrt{\frac{c x-1}{c x+1}} (c x+1)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(d^2*(9504*a^2*c*x*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] + 9504*a^2*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)
]*Sqrt[d - c^2*d*x^2] - 7488*a^2*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] - 7488*a^2*c^4*x^4*Sqr
t[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] + 2304*a^2*c^5*x^5*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2]
+ 2304*a^2*c^6*x^6*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] - 1440*b^2*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]^
3 - 4320*a^2*Sqrt[d]*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 4
320*a^2*c*Sqrt[d]*x*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 32
40*a*b*Sqrt[d - c^2*d*x^2]*Cosh[2*ArcCosh[c*x]] + 324*a*b*Sqrt[d - c^2*d*x^2]*Cosh[4*ArcCosh[c*x]] - 24*a*b*Sq
rt[d - c^2*d*x^2]*Cosh[6*ArcCosh[c*x]] + 1620*b^2*Sqrt[d - c^2*d*x^2]*Sinh[2*ArcCosh[c*x]] - 81*b^2*Sqrt[d - c
^2*d*x^2]*Sinh[4*ArcCosh[c*x]] + 4*b^2*Sqrt[d - c^2*d*x^2]*Sinh[6*ArcCosh[c*x]] - 12*b*Sqrt[d - c^2*d*x^2]*Arc
Cosh[c*x]*(270*b*Cosh[2*ArcCosh[c*x]] - 27*b*Cosh[4*ArcCosh[c*x]] + 2*b*Cosh[6*ArcCosh[c*x]] - 540*a*Sinh[2*Ar
cCosh[c*x]] + 108*a*Sinh[4*ArcCosh[c*x]] - 12*a*Sinh[6*ArcCosh[c*x]]) + 72*b*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]^
2*(-60*a + 45*b*Sinh[2*ArcCosh[c*x]] - 9*b*Sinh[4*ArcCosh[c*x]] + b*Sinh[6*ArcCosh[c*x]])))/(13824*c*Sqrt[(-1
+ c*x)/(1 + c*x)]*(1 + c*x))

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Maple [B]  time = 0.339, size = 1053, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x)

[Out]

1/6*x*(-c^2*d*x^2+d)^(5/2)*a^2+5/24*a^2*d*x*(-c^2*d*x^2+d)^(3/2)+5/16*a^2*d^2*x*(-c^2*d*x^2+d)^(1/2)+5/16*a^2*
d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-299/1152*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)
/(c*x-1)*x+59/48*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)/(c*x-1)*c^2*arccosh(c*x)^2*x^3+1/6*b^2*(-d*(c^2*x^2-1)
)^(1/2)*d^2/(c*x+1)/(c*x-1)*c^6*arccosh(c*x)^2*x^7-17/24*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)/(c*x-1)*c^4*ar
ccosh(c*x)^2*x^5-5/16*a*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c*arccosh(c*x)^2*d^2-11/8*a*b*(-d
*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)/(c*x-1)*arccosh(c*x)*x-1/18*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)/(c*x-
1)^(1/2)*c^5*arccosh(c*x)*x^6+13/48*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^3*arccosh(c*x
)*x^4-11/16*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c*arccosh(c*x)*x^2-1/18*a*b*(-d*(c^2*x^
2-1))^(1/2)*d^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^5*x^6+13/48*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)/(c*x-1)
^(1/2)*c^3*x^4-11/16*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c*x^2+59/24*a*b*(-d*(c^2*x^2-1
))^(1/2)*d^2/(c*x+1)/(c*x-1)*c^2*arccosh(c*x)*x^3+1/3*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)/(c*x-1)*c^6*arcco
sh(c*x)*x^7-17/12*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)/(c*x-1)*c^4*arccosh(c*x)*x^5+299/1152*a*b*(-d*(c^2*x^
2-1))^(1/2)*d^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)/c+299/1152*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)/(c*x-1)^(1
/2)/c*arccosh(c*x)+1/108*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)/(c*x-1)*c^6*x^7-113/1728*b^2*(-d*(c^2*x^2-1))^
(1/2)*d^2/(c*x+1)/(c*x-1)*c^4*x^5+1091/3456*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)/(c*x-1)*c^2*x^3-5/48*b^2*(-
d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c*arccosh(c*x)^3*d^2-11/16*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*
x+1)/(c*x-1)*arccosh(c*x)^2*x

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{2} c^{4} d^{2} x^{4} - 2 \, a^{2} c^{2} d^{2} x^{2} + a^{2} d^{2} +{\left (b^{2} c^{4} d^{2} x^{4} - 2 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \operatorname{arcosh}\left (c x\right )^{2} + 2 \,{\left (a b c^{4} d^{2} x^{4} - 2 \, a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \operatorname{arcosh}\left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x, algorithm="fricas")

[Out]

integral((a^2*c^4*d^2*x^4 - 2*a^2*c^2*d^2*x^2 + a^2*d^2 + (b^2*c^4*d^2*x^4 - 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arcc
osh(c*x)^2 + 2*(a*b*c^4*d^2*x^4 - 2*a*b*c^2*d^2*x^2 + a*b*d^2)*arccosh(c*x))*sqrt(-c^2*d*x^2 + d), 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((-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x))**2,x)

[Out]

Timed out

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Giac [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((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x, algorithm="giac")

[Out]

Timed out