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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.48, 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 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 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 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 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 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]

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 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 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 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 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 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 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]

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*}

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Mathematica [A]  time = 4.37, size = 485, normalized size = 1.10 \[ \frac {-864 a^2 d^{3/2} \sqrt {\frac {c x-1}{c x+1}} (c x+1) \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )-288 a^2 c d x \sqrt {\frac {c x-1}{c x+1}} (c x+1) \left (8 c^4 x^4-14 c^2 x^2+3\right ) \sqrt {d-c^2 d x^2}-216 a b d \sqrt {d-c^2 d x^2} \left (8 \cosh ^{-1}(c x)^2+\cosh \left (4 \cosh ^{-1}(c x)\right )-4 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )\right )-12 a b d \sqrt {d-c^2 d x^2} \left (-72 \cosh ^{-1}(c x)^2+18 \cosh \left (2 \cosh ^{-1}(c x)\right )-9 \cosh \left (4 \cosh ^{-1}(c x)\right )-2 \cosh \left (6 \cosh ^{-1}(c x)\right )+12 \cosh ^{-1}(c x) \left (-3 \sinh \left (2 \cosh ^{-1}(c x)\right )+3 \sinh \left (4 \cosh ^{-1}(c x)\right )+\sinh \left (6 \cosh ^{-1}(c x)\right )\right )\right )-18 b^2 d \sqrt {d-c^2 d x^2} \left (32 \cosh ^{-1}(c x)^3+12 \cosh \left (4 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)-3 \left (8 \cosh ^{-1}(c x)^2+1\right ) \sinh \left (4 \cosh ^{-1}(c x)\right )\right )+b^2 d \sqrt {d-c^2 d x^2} \left (288 \cosh ^{-1}(c x)^3+12 \left (-18 \cosh \left (2 \cosh ^{-1}(c x)\right )+9 \cosh \left (4 \cosh ^{-1}(c x)\right )+2 \cosh \left (6 \cosh ^{-1}(c x)\right )\right ) \cosh ^{-1}(c x)-72 \cosh ^{-1}(c x)^2 \left (-3 \sinh \left (2 \cosh ^{-1}(c x)\right )+3 \sinh \left (4 \cosh ^{-1}(c x)\right )+\sinh \left (6 \cosh ^{-1}(c x)\right )\right )+108 \sinh \left (2 \cosh ^{-1}(c x)\right )-27 \sinh \left (4 \cosh ^{-1}(c x)\right )-4 \sinh \left (6 \cosh ^{-1}(c x)\right )\right )}{13824 c^3 \sqrt {\frac {c x-1}{c x+1}} (c x+1)} \]

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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fricas [F]  time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{2} c^{2} d x^{4} - a^{2} d x^{2} + {\left (b^{2} c^{2} d x^{4} - b^{2} d x^{2}\right )} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, {\left (a b c^{2} d x^{4} - a b d x^{2}\right )} \operatorname {arcosh}\left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{2}\,{d x} \]

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]

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

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maple [B]  time = 0.96, size = 1021, normalized size = 2.32 \[ \text {result too large to display} \]

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]

1/8*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)/c^2/(c*x-1)*arccosh(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+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+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^2-1/6*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+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+1/16*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)/c^2/(c*x-1)*arccosh(c*x)^2*x-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*a*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c^3*arcco
sh(c*x)^2*d+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)*arccosh(c*x)*x^2-1/6*a^2*x*(-c^2*d*x^2+d)^(5/2)/c^2/d+1/16*a^2/c^2*d*x*(-c^2*d*x^2+d)^(1/2
)+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))+1/24*a^2/c^2*x*(-c^2*d*x^2+d)^(3
/2)+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)+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)-1/48*b^2*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c^3*arc
cosh(c*x)^3*d-17/48*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)/(c*x-1)*arccosh(c*x)^2*x^3-1/108*b^2*(-d*(c^2*x^2-1))
^(1/2)*d/(c*x+1)*c^4/(c*x-1)*x^7+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-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]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{48} \, a^{2} {\left (\frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x}{c^{2}} - \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x}{c^{2} d} + \frac {3 \, \sqrt {-c^{2} d x^{2} + d} d x}{c^{2}} + \frac {3 \, d^{\frac {3}{2}} \arcsin \left (c x\right )}{c^{3}}\right )} + \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} b^{2} x^{2} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2} + 2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a b x^{2} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )\,{d x} \]

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]

1/48*a^2*(2*(-c^2*d*x^2 + d)^(3/2)*x/c^2 - 8*(-c^2*d*x^2 + d)^(5/2)*x/(c^2*d) + 3*sqrt(-c^2*d*x^2 + d)*d*x/c^2
+ 3*d^(3/2)*arcsin(c*x)/c^3) + integrate((-c^2*d*x^2 + d)^(3/2)*b^2*x^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)
)^2 + 2*(-c^2*d*x^2 + d)^(3/2)*a*b*x^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}\, dx \]

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]

Integral(x**2*(-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*acosh(c*x))**2, x)

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