3.179 \(\int x^2 (d-c^2 d x^2)^{3/2} (a+b \cosh ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=441 \[ \frac{b c^3 d x^6 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{18 \sqrt{c x-1} \sqrt{c x+1}}-\frac{7 b c d x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{48 \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{8} d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{b d x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 c^2}-\frac{d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{48 b c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{1}{108} b^2 c^2 d x^5 \sqrt{d-c^2 d x^2}+\frac{43 b^2 d x^3 \sqrt{d-c^2 d x^2}}{1728}+\frac{7 b^2 d x \sqrt{d-c^2 d x^2}}{1152 c^2}+\frac{7 b^2 d \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{1152 c^3 \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

(7*b^2*d*x*Sqrt[d - c^2*d*x^2])/(1152*c^2) + (43*b^2*d*x^3*Sqrt[d - c^2*d*x^2])/1728 - (b^2*c^2*d*x^5*Sqrt[d -
c^2*d*x^2])/108 + (7*b^2*d*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(1152*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d*x
^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(16*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (7*b*c*d*x^4*Sqrt[d - c^2*d
*x^2]*(a + b*ArcCosh[c*x]))/(48*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*x^6*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos
h[c*x]))/(18*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(16*c^2) + (d*x^
3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/8 + (x^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/6 - (d*Sq
rt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^3)/(48*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

________________________________________________________________________________________

Rubi [A]  time = 1.4774, antiderivative size = 453, normalized size of antiderivative = 1.03, number of steps used = 20, number of rules used = 13, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.448, Rules used = {5798, 5745, 5743, 5759, 5676, 5662, 90, 52, 100, 12, 14, 5731, 460} \[ \frac{b c^3 d x^6 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{18 \sqrt{c x-1} \sqrt{c x+1}}-\frac{7 b c d x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{48 \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{8} d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{6} d x^3 (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{b d x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 c^2}-\frac{d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{48 b c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{1}{108} b^2 c^2 d x^5 \sqrt{d-c^2 d x^2}+\frac{43 b^2 d x^3 \sqrt{d-c^2 d x^2}}{1728}+\frac{7 b^2 d x \sqrt{d-c^2 d x^2}}{1152 c^2}+\frac{7 b^2 d \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{1152 c^3 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2,x]

[Out]

(7*b^2*d*x*Sqrt[d - c^2*d*x^2])/(1152*c^2) + (43*b^2*d*x^3*Sqrt[d - c^2*d*x^2])/1728 - (b^2*c^2*d*x^5*Sqrt[d -
c^2*d*x^2])/108 + (7*b^2*d*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(1152*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d*x
^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(16*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (7*b*c*d*x^4*Sqrt[d - c^2*d
*x^2]*(a + b*ArcCosh[c*x]))/(48*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*x^6*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos
h[c*x]))/(18*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(16*c^2) + (d*x^
3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/8 + (d*x^3*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos
h[c*x])^2)/6 - (d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^3)/(48*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 5798

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((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[(f*x)^m*(1 + c*
x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[c^2*d + e, 0]
&&  !IntegerQ[p]

Rule 5745

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_
))^(p_), x_Symbol] :> Simp[((f*x)^(m + 1)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n)/(f*(m + 2*p + 1)
), x] + (Dist[(2*d1*d2*p)/(m + 2*p + 1), Int[(f*x)^m*(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])/(f*(m + 2*p + 1)*Sqrt[1 + c*
x]*Sqrt[-1 + c*x]), Int[(f*x)^(m + 1)*(-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, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[p, 0] &&  !L
tQ[m, -1] && IntegerQ[p - 1/2] && (RationalQ[m] || EqQ[n, 1])

Rule 5743

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*
(x_)], x_Symbol] :> Simp[((f*x)^(m + 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(f*(m + 2)), x
] + (-Dist[(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/((m + 2)*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[((f*x)^m*(a + b*ArcCo
sh[c*x])^n)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Dist[(b*c*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(f*(m + 2)*S
qrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e
1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] &&  !LtQ[m, -1] && (RationalQ[m] |
| EqQ[n, 1])

Rule 5759

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

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 100

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 5731

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[1
+ c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 460

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.)*((c_) + (d_.)
*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1))/(b1*b2*e*
(m + n*(p + 1) + 1)), x] - Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(b1*b2*(m + n*(p + 1) + 1)), I
nt[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] &&
EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]

Rubi steps

\begin{align*} \int x^2 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int x^2 (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{6} d x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \int x^3 \left (-1+c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{12 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d x^6 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{18 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{1}{8} d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{6} d x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \int x^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4 \left (-3+2 c^2 x^2\right )}{12 \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{7 b c d x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{48 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d x^6 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{18 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 c^2}+\frac{1}{8} d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{6} d x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{\left (d \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 c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b d \sqrt{d-c^2 d x^2}\right ) \int x \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{8 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4 \left (-3+2 c^2 x^2\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{36 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{16 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{64} b^2 d x^3 \sqrt{d-c^2 d x^2}-\frac{1}{108} b^2 c^2 d x^5 \sqrt{d-c^2 d x^2}+\frac{b d x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{7 b c d x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{48 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d x^6 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{18 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 c^2}+\frac{1}{8} d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{6} d x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{48 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{3 x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{64 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b^2 d \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{\left (b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{27 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b^2 d x \sqrt{d-c^2 d x^2}}{32 c^2}+\frac{43 b^2 d x^3 \sqrt{d-c^2 d x^2}}{1728}-\frac{1}{108} b^2 c^2 d x^5 \sqrt{d-c^2 d x^2}+\frac{b d x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{7 b c d x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{48 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d x^6 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{18 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 c^2}+\frac{1}{8} d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{6} d x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{48 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{3 x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{108 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 b^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{64 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{32 c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b^2 d x \sqrt{d-c^2 d x^2}}{128 c^2}+\frac{43 b^2 d x^3 \sqrt{d-c^2 d x^2}}{1728}-\frac{1}{108} b^2 c^2 d x^5 \sqrt{d-c^2 d x^2}-\frac{b^2 d \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{32 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{7 b c d x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{48 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d x^6 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{18 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 c^2}+\frac{1}{8} d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{6} d x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{48 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{36 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 b^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{128 c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{7 b^2 d x \sqrt{d-c^2 d x^2}}{1152 c^2}+\frac{43 b^2 d x^3 \sqrt{d-c^2 d x^2}}{1728}-\frac{1}{108} b^2 c^2 d x^5 \sqrt{d-c^2 d x^2}-\frac{b^2 d \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{128 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{7 b c d x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{48 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d x^6 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{18 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 c^2}+\frac{1}{8} d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{6} d x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{48 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{72 c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{7 b^2 d x \sqrt{d-c^2 d x^2}}{1152 c^2}+\frac{43 b^2 d x^3 \sqrt{d-c^2 d x^2}}{1728}-\frac{1}{108} b^2 c^2 d x^5 \sqrt{d-c^2 d x^2}+\frac{7 b^2 d \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{1152 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{7 b c d x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{48 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d x^6 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{18 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 c^2}+\frac{1}{8} d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{6} d x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{48 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [A]  time = 4.28346, size = 485, normalized size = 1.1 \[ \frac{-864 a^2 d^{3/2} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-288 a^2 c d x \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (8 c^4 x^4-14 c^2 x^2+3\right ) \sqrt{d-c^2 d x^2}-216 a b d \sqrt{d-c^2 d x^2} \left (8 \cosh ^{-1}(c x)^2+\cosh \left (4 \cosh ^{-1}(c x)\right )-4 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )\right )-12 a b d \sqrt{d-c^2 d x^2} \left (-72 \cosh ^{-1}(c x)^2+18 \cosh \left (2 \cosh ^{-1}(c x)\right )-9 \cosh \left (4 \cosh ^{-1}(c x)\right )-2 \cosh \left (6 \cosh ^{-1}(c x)\right )+12 \cosh ^{-1}(c x) \left (-3 \sinh \left (2 \cosh ^{-1}(c x)\right )+3 \sinh \left (4 \cosh ^{-1}(c x)\right )+\sinh \left (6 \cosh ^{-1}(c x)\right )\right )\right )-18 b^2 d \sqrt{d-c^2 d x^2} \left (32 \cosh ^{-1}(c x)^3+12 \cosh \left (4 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)-3 \left (8 \cosh ^{-1}(c x)^2+1\right ) \sinh \left (4 \cosh ^{-1}(c x)\right )\right )+b^2 d \sqrt{d-c^2 d x^2} \left (288 \cosh ^{-1}(c x)^3+12 \left (-18 \cosh \left (2 \cosh ^{-1}(c x)\right )+9 \cosh \left (4 \cosh ^{-1}(c x)\right )+2 \cosh \left (6 \cosh ^{-1}(c x)\right )\right ) \cosh ^{-1}(c x)-72 \cosh ^{-1}(c x)^2 \left (-3 \sinh \left (2 \cosh ^{-1}(c x)\right )+3 \sinh \left (4 \cosh ^{-1}(c x)\right )+\sinh \left (6 \cosh ^{-1}(c x)\right )\right )+108 \sinh \left (2 \cosh ^{-1}(c x)\right )-27 \sinh \left (4 \cosh ^{-1}(c x)\right )-4 \sinh \left (6 \cosh ^{-1}(c x)\right )\right )}{13824 c^3 \sqrt{\frac{c x-1}{c x+1}} (c x+1)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2,x]

[Out]

(-288*a^2*c*d*x*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(3 - 14*c^2*x^2 + 8*c^4*x^4) - 864*a^
2*d^(3/2)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 21
6*a*b*d*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]) -
18*b^2*d*Sqrt[d - c^2*d*x^2]*(32*ArcCosh[c*x]^3 + 12*ArcCosh[c*x]*Cosh[4*ArcCosh[c*x]] - 3*(1 + 8*ArcCosh[c*x]
^2)*Sinh[4*ArcCosh[c*x]]) - 12*a*b*d*Sqrt[d - c^2*d*x^2]*(-72*ArcCosh[c*x]^2 + 18*Cosh[2*ArcCosh[c*x]] - 9*Cos
h[4*ArcCosh[c*x]] - 2*Cosh[6*ArcCosh[c*x]] + 12*ArcCosh[c*x]*(-3*Sinh[2*ArcCosh[c*x]] + 3*Sinh[4*ArcCosh[c*x]]
+ Sinh[6*ArcCosh[c*x]])) + b^2*d*Sqrt[d - c^2*d*x^2]*(288*ArcCosh[c*x]^3 + 12*ArcCosh[c*x]*(-18*Cosh[2*ArcCos
h[c*x]] + 9*Cosh[4*ArcCosh[c*x]] + 2*Cosh[6*ArcCosh[c*x]]) + 108*Sinh[2*ArcCosh[c*x]] - 27*Sinh[4*ArcCosh[c*x]
] - 4*Sinh[6*ArcCosh[c*x]] - 72*ArcCosh[c*x]^2*(-3*Sinh[2*ArcCosh[c*x]] + 3*Sinh[4*ArcCosh[c*x]] + Sinh[6*ArcC
osh[c*x]])))/(13824*c^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))

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Maple [B]  time = 0.444, size = 1021, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2,x)

[Out]

1/16*a^2/c^2*d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+1/18*a*b*(-d*(c^2*x^2-1))^(1/2)*d/
(c*x+1)^(1/2)*c^3/(c*x-1)^(1/2)*x^6-7/48*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)^(1/2)*c/(c*x-1)^(1/2)*x^4+11/24*
b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)*c^2/(c*x-1)*arccosh(c*x)^2*x^5-1/6*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)*c
^4/(c*x-1)*arccosh(c*x)^2*x^7-1/16*a*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c^3*arccosh(c*x)^2*d
-17/24*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)/(c*x-1)*arccosh(c*x)*x^3+1/16*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)
/c^2/(c*x-1)*arccosh(c*x)^2*x+1/16*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)^(1/2)/c/(c*x-1)^(1/2)*x^2+1/18*b^2*(-d
*(c^2*x^2-1))^(1/2)*d/(c*x+1)^(1/2)*c^3/(c*x-1)^(1/2)*arccosh(c*x)*x^6-7/48*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+
1)^(1/2)*c/(c*x-1)^(1/2)*arccosh(c*x)*x^4+1/16*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)^(1/2)/c/(c*x-1)^(1/2)*arcc
osh(c*x)*x^2+59/1728*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)*c^2/(c*x-1)*x^5-7/1152*b^2*(-d*(c^2*x^2-1))^(1/2)*d/
(c*x+1)/c^2/(c*x-1)*x-17/48*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)/(c*x-1)*arccosh(c*x)^2*x^3+1/24*a^2/c^2*x*(-c
^2*d*x^2+d)^(3/2)+1/16*a^2/c^2*d*x*(-c^2*d*x^2+d)^(1/2)+1/8*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)/c^2/(c*x-1)*a
rccosh(c*x)*x-1/3*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)*c^4/(c*x-1)*arccosh(c*x)*x^7+11/12*a*b*(-d*(c^2*x^2-1))
^(1/2)*d/(c*x+1)*c^2/(c*x-1)*arccosh(c*x)*x^5+7/1152*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)^(1/2)/c^3/(c*x-1)^(1
/2)*arccosh(c*x)+7/1152*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)^(1/2)/c^3/(c*x-1)^(1/2)-1/48*b^2*(-d*(c^2*x^2-1))
^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c^3*arccosh(c*x)^3*d-1/108*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)*c^4/(c*x-1)
*x^7-1/6*a^2*x*(-c^2*d*x^2+d)^(5/2)/c^2/d-65/3456*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)/(c*x-1)*x^3

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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(x^2*(-c^2*d*x^2+d)^(3/2)*(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^{2} d x^{4} - a^{2} d x^{2} +{\left (b^{2} c^{2} d x^{4} - b^{2} d x^{2}\right )} \operatorname{arcosh}\left (c x\right )^{2} + 2 \,{\left (a b c^{2} d x^{4} - a b d x^{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(x^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2,x, algorithm="fricas")

[Out]

integral(-(a^2*c^2*d*x^4 - a^2*d*x^2 + (b^2*c^2*d*x^4 - b^2*d*x^2)*arccosh(c*x)^2 + 2*(a*b*c^2*d*x^4 - a*b*d*x
^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(x**2*(-c**2*d*x**2+d)**(3/2)*(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(x^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2,x, algorithm="giac")

[Out]

Timed out