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

Optimal. Leaf size=495 \[ \frac{4 a b d x \sqrt{d-c^2 d x^2}}{35 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b c^3 d x^7 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{49 \sqrt{c x-1} \sqrt{c x+1}}-\frac{16 b c d x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{175 \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{3}{35} d x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{2 b d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{d x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^2}-\frac{2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^4}-\frac{2}{343} b^2 c^2 d x^6 \sqrt{d-c^2 d x^2}+\frac{484 b^2 d x^4 \sqrt{d-c^2 d x^2}}{42875}+\frac{3358 b^2 d x^2 \sqrt{d-c^2 d x^2}}{385875 c^2}-\frac{37384 b^2 d \sqrt{d-c^2 d x^2}}{385875 c^4}+\frac{4 b^2 d x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{35 c^3 \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

(-37384*b^2*d*Sqrt[d - c^2*d*x^2])/(385875*c^4) + (3358*b^2*d*x^2*Sqrt[d - c^2*d*x^2])/(385875*c^2) + (484*b^2
*d*x^4*Sqrt[d - c^2*d*x^2])/42875 - (2*b^2*c^2*d*x^6*Sqrt[d - c^2*d*x^2])/343 + (4*a*b*d*x*Sqrt[d - c^2*d*x^2]
)/(35*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*b^2*d*x*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(35*c^3*Sqrt[-1 + c*x]*
Sqrt[1 + c*x]) + (2*b*d*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(105*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (
16*b*c*d*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(175*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c^3*d*x^7*Sqr
t[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(49*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*d*Sqrt[d - c^2*d*x^2]*(a + b*Arc
Cosh[c*x])^2)/(35*c^4) - (d*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(35*c^2) + (3*d*x^4*Sqrt[d - c^2*d
*x^2]*(a + b*ArcCosh[c*x])^2)/35 + (x^4*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/7

________________________________________________________________________________________

Rubi [A]  time = 1.67374, antiderivative size = 507, normalized size of antiderivative = 1.02, number of steps used = 26, number of rules used = 13, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.448, Rules used = {5798, 5745, 5743, 5759, 5718, 5654, 74, 5662, 100, 12, 14, 5731, 460} \[ \frac{4 a b d x \sqrt{d-c^2 d x^2}}{35 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b c^3 d x^7 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{49 \sqrt{c x-1} \sqrt{c x+1}}-\frac{16 b c d x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{175 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3}{35} d x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{7} d x^4 (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{2 b d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{d x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^2}-\frac{2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^4}-\frac{2}{343} b^2 c^2 d x^6 \sqrt{d-c^2 d x^2}+\frac{484 b^2 d x^4 \sqrt{d-c^2 d x^2}}{42875}+\frac{3358 b^2 d x^2 \sqrt{d-c^2 d x^2}}{385875 c^2}-\frac{37384 b^2 d \sqrt{d-c^2 d x^2}}{385875 c^4}+\frac{4 b^2 d x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{35 c^3 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-37384*b^2*d*Sqrt[d - c^2*d*x^2])/(385875*c^4) + (3358*b^2*d*x^2*Sqrt[d - c^2*d*x^2])/(385875*c^2) + (484*b^2
*d*x^4*Sqrt[d - c^2*d*x^2])/42875 - (2*b^2*c^2*d*x^6*Sqrt[d - c^2*d*x^2])/343 + (4*a*b*d*x*Sqrt[d - c^2*d*x^2]
)/(35*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*b^2*d*x*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(35*c^3*Sqrt[-1 + c*x]*
Sqrt[1 + c*x]) + (2*b*d*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(105*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (
16*b*c*d*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(175*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c^3*d*x^7*Sqr
t[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(49*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*d*Sqrt[d - c^2*d*x^2]*(a + b*Arc
Cosh[c*x])^2)/(35*c^4) - (d*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(35*c^2) + (3*d*x^4*Sqrt[d - c^2*d
*x^2]*(a + b*ArcCosh[c*x])^2)/35 + (d*x^4*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/7

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 5718

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

Rule 5654

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcCosh[c*x])^(n - 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 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 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^3 \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^3 (-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}{7} d x^4 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{\left (3 d \sqrt{d-c^2 d x^2}\right ) \int x^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{7 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 b c d \sqrt{d-c^2 d x^2}\right ) \int x^4 \left (-1+c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{7 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{2 b c d x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c^3 d x^7 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{49 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3}{35} d x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{7} d x^4 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{\left (3 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{35 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (6 b c d \sqrt{d-c^2 d x^2}\right ) \int x^4 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{35 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x^5 \left (-7+5 c^2 x^2\right )}{35 \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{7 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{16 b c d x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{175 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c^3 d x^7 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{49 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^2}+\frac{3}{35} d x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{7} d x^4 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{\left (2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{35 c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 b d \sqrt{d-c^2 d x^2}\right ) \int x^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{35 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x^5 \left (-7+5 c^2 x^2\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{245 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (6 b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x^5}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{175 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{6}{875} b^2 d x^4 \sqrt{d-c^2 d x^2}-\frac{2}{343} b^2 c^2 d x^6 \sqrt{d-c^2 d x^2}+\frac{2 b d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{16 b c d x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{175 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c^3 d x^7 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{49 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^4}-\frac{d x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^2}+\frac{3}{35} d x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{7} d x^4 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{\left (6 b^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{4 x^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{875 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{105 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (4 b d \sqrt{d-c^2 d x^2}\right ) \int \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{35 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (38 b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x^5}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{1715 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{2 b^2 d x^2 \sqrt{d-c^2 d x^2}}{315 c^2}+\frac{484 b^2 d x^4 \sqrt{d-c^2 d x^2}}{42875}-\frac{2}{343} b^2 c^2 d x^6 \sqrt{d-c^2 d x^2}+\frac{4 a b d x \sqrt{d-c^2 d x^2}}{35 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{16 b c d x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{175 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c^3 d x^7 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{49 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^4}-\frac{d x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^2}+\frac{3}{35} d x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{7} d x^4 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{\left (38 b^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{4 x^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{8575 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (24 b^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{875 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (4 b^2 d \sqrt{d-c^2 d x^2}\right ) \int \cosh ^{-1}(c x) \, dx}{35 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{2 x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{315 c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{22 b^2 d x^2 \sqrt{d-c^2 d x^2}}{7875 c^2}+\frac{484 b^2 d x^4 \sqrt{d-c^2 d x^2}}{42875}-\frac{2}{343} b^2 c^2 d x^6 \sqrt{d-c^2 d x^2}+\frac{4 a b d x \sqrt{d-c^2 d x^2}}{35 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{4 b^2 d x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{35 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{16 b c d x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{175 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c^3 d x^7 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{49 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^4}-\frac{d x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^2}+\frac{3}{35} d x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{7} d x^4 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{\left (152 b^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{8575 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (8 b^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{2 x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{875 c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (4 b^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{315 c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (4 b^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{35 c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{8 b^2 d \sqrt{d-c^2 d x^2}}{63 c^4}+\frac{3358 b^2 d x^2 \sqrt{d-c^2 d x^2}}{385875 c^2}+\frac{484 b^2 d x^4 \sqrt{d-c^2 d x^2}}{42875}-\frac{2}{343} b^2 c^2 d x^6 \sqrt{d-c^2 d x^2}+\frac{4 a b d x \sqrt{d-c^2 d x^2}}{35 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{4 b^2 d x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{35 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{16 b c d x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{175 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c^3 d x^7 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{49 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^4}-\frac{d x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^2}+\frac{3}{35} d x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{7} d x^4 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{\left (152 b^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{2 x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{25725 c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (16 b^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{875 c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{856 b^2 d \sqrt{d-c^2 d x^2}}{7875 c^4}+\frac{3358 b^2 d x^2 \sqrt{d-c^2 d x^2}}{385875 c^2}+\frac{484 b^2 d x^4 \sqrt{d-c^2 d x^2}}{42875}-\frac{2}{343} b^2 c^2 d x^6 \sqrt{d-c^2 d x^2}+\frac{4 a b d x \sqrt{d-c^2 d x^2}}{35 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{4 b^2 d x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{35 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{16 b c d x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{175 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c^3 d x^7 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{49 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^4}-\frac{d x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^2}+\frac{3}{35} d x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{7} d x^4 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{\left (304 b^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{25725 c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{37384 b^2 d \sqrt{d-c^2 d x^2}}{385875 c^4}+\frac{3358 b^2 d x^2 \sqrt{d-c^2 d x^2}}{385875 c^2}+\frac{484 b^2 d x^4 \sqrt{d-c^2 d x^2}}{42875}-\frac{2}{343} b^2 c^2 d x^6 \sqrt{d-c^2 d x^2}+\frac{4 a b d x \sqrt{d-c^2 d x^2}}{35 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{4 b^2 d x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{35 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{16 b c d x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{175 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c^3 d x^7 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{49 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^4}-\frac{d x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{35 c^2}+\frac{3}{35} d x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{7} d x^4 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2\\ \end{align*}

Mathematica [A]  time = 0.587769, size = 262, normalized size = 0.53 \[ -\frac{d \sqrt{d-c^2 d x^2} \left (11025 a^2 \left (5 c^2 x^2+2\right ) \left (c^2 x^2-1\right )^3-210 a b c x \sqrt{c x-1} \sqrt{c x+1} \left (75 c^6 x^6-168 c^4 x^4+35 c^2 x^2+210\right )-210 b \cosh ^{-1}(c x) \left (b c x \sqrt{c x-1} \sqrt{c x+1} \left (75 c^6 x^6-168 c^4 x^4+35 c^2 x^2+210\right )-105 a \left (c^2 x^2-1\right )^3 \left (5 c^2 x^2+2\right )\right )+2 b^2 \left (1125 c^8 x^8-3303 c^6 x^6+499 c^4 x^4+20371 c^2 x^2-18692\right )+11025 b^2 \left (5 c^2 x^2+2\right ) \left (c^2 x^2-1\right )^3 \cosh ^{-1}(c x)^2\right )}{385875 c^4 \left (c^2 x^2-1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*Sqrt[d - c^2*d*x^2]*(11025*a^2*(-1 + c^2*x^2)^3*(2 + 5*c^2*x^2) - 210*a*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]
*(210 + 35*c^2*x^2 - 168*c^4*x^4 + 75*c^6*x^6) + 2*b^2*(-18692 + 20371*c^2*x^2 + 499*c^4*x^4 - 3303*c^6*x^6 +
1125*c^8*x^8) - 210*b*(-105*a*(-1 + c^2*x^2)^3*(2 + 5*c^2*x^2) + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(210 + 35*
c^2*x^2 - 168*c^4*x^4 + 75*c^6*x^6))*ArcCosh[c*x] + 11025*b^2*(-1 + c^2*x^2)^3*(2 + 5*c^2*x^2)*ArcCosh[c*x]^2)
)/(385875*c^4*(-1 + c^2*x^2))

________________________________________________________________________________________

Maple [B]  time = 0.536, size = 1952, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*(-1/7*x^2*(-c^2*d*x^2+d)^(5/2)/c^2/d-2/35/d/c^4*(-c^2*d*x^2+d)^(5/2))+b^2*(-1/43904*(-d*(c^2*x^2-1))^(1/2)
*(64*c^8*x^8-144*c^6*x^6+64*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^7*c^7+104*c^4*x^4-112*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^
5*c^5-25*c^2*x^2+56*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-7*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+1)*(49*arccosh(c*x)^
2-14*arccosh(c*x)+2)*d/(c*x+1)/c^4/(c*x-1)+1/16000*(-d*(c^2*x^2-1))^(1/2)*(16*c^6*x^6-28*c^4*x^4+16*(c*x+1)^(1
/2)*(c*x-1)^(1/2)*x^5*c^5+13*c^2*x^2-20*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+5*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-
1)*(25*arccosh(c*x)^2-10*arccosh(c*x)+2)*d/(c*x+1)/c^4/(c*x-1)+1/1152*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*
x^2+4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+1)*(9*arccosh(c*x)^2-6*arccosh(c*x
)+2)*d/(c*x+1)/c^4/(c*x-1)-3/128*(-d*(c^2*x^2-1))^(1/2)*((c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*(arccosh(c
*x)^2-2*arccosh(c*x)+2)*d/(c*x+1)/c^4/(c*x-1)-3/128*(-d*(c^2*x^2-1))^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c
^2*x^2-1)*(arccosh(c*x)^2+2*arccosh(c*x)+2)*d/(c*x+1)/c^4/(c*x-1)+1/1152*(-d*(c^2*x^2-1))^(1/2)*(-4*(c*x+1)^(1
/2)*(c*x-1)^(1/2)*x^3*c^3+4*c^4*x^4+3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-5*c^2*x^2+1)*(9*arccosh(c*x)^2+6*arccosh
(c*x)+2)*d/(c*x+1)/c^4/(c*x-1)+1/16000*(-d*(c^2*x^2-1))^(1/2)*(-16*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5+16*c^6*
x^6+20*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-28*c^4*x^4-5*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+13*c^2*x^2-1)*(25*arcc
osh(c*x)^2+10*arccosh(c*x)+2)*d/(c*x+1)/c^4/(c*x-1)-1/43904*(-d*(c^2*x^2-1))^(1/2)*(-64*(c*x+1)^(1/2)*(c*x-1)^
(1/2)*x^7*c^7+64*c^8*x^8+112*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5-144*c^6*x^6-56*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^
3*c^3+104*c^4*x^4+7*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-25*c^2*x^2+1)*(49*arccosh(c*x)^2+14*arccosh(c*x)+2)*d/(c*x
+1)/c^4/(c*x-1))+2*a*b*(-1/6272*(-d*(c^2*x^2-1))^(1/2)*(64*c^8*x^8-144*c^6*x^6+64*(c*x+1)^(1/2)*(c*x-1)^(1/2)*
x^7*c^7+104*c^4*x^4-112*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5-25*c^2*x^2+56*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-
7*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+1)*(-1+7*arccosh(c*x))*d/(c*x+1)/c^4/(c*x-1)+1/3200*(-d*(c^2*x^2-1))^(1/2)*(
16*c^6*x^6-28*c^4*x^4+16*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5+13*c^2*x^2-20*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3
+5*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-1)*(-1+5*arccosh(c*x))*d/(c*x+1)/c^4/(c*x-1)+1/384*(-d*(c^2*x^2-1))^(1/2)*(
4*c^4*x^4-5*c^2*x^2+4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+1)*(-1+3*arccosh(c
*x))*d/(c*x+1)/c^4/(c*x-1)-3/128*(-d*(c^2*x^2-1))^(1/2)*((c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*(-1+arccos
h(c*x))*d/(c*x+1)/c^4/(c*x-1)-3/128*(-d*(c^2*x^2-1))^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*(1+arc
cosh(c*x))*d/(c*x+1)/c^4/(c*x-1)+1/384*(-d*(c^2*x^2-1))^(1/2)*(-4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+4*c^4*x^
4+3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-5*c^2*x^2+1)*(1+3*arccosh(c*x))*d/(c*x+1)/c^4/(c*x-1)+1/3200*(-d*(c^2*x^2-
1))^(1/2)*(-16*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5+16*c^6*x^6+20*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-28*c^4*x^
4-5*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+13*c^2*x^2-1)*(1+5*arccosh(c*x))*d/(c*x+1)/c^4/(c*x-1)-1/6272*(-d*(c^2*x^2
-1))^(1/2)*(-64*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^7*c^7+64*c^8*x^8+112*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5-144*c^6
*x^6-56*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+104*c^4*x^4+7*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-25*c^2*x^2+1)*(1+7*a
rccosh(c*x))*d/(c*x+1)/c^4/(c*x-1))

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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^3*(-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 [A]  time = 2.37675, size = 996, normalized size = 2.01 \begin{align*} -\frac{11025 \,{\left (5 \, b^{2} c^{8} d x^{8} - 13 \, b^{2} c^{6} d x^{6} + 9 \, b^{2} c^{4} d x^{4} + b^{2} c^{2} d x^{2} - 2 \, b^{2} d\right )} \sqrt{-c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )^{2} - 210 \,{\left (75 \, a b c^{7} d x^{7} - 168 \, a b c^{5} d x^{5} + 35 \, a b c^{3} d x^{3} + 210 \, a b c d x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} - 210 \,{\left ({\left (75 \, b^{2} c^{7} d x^{7} - 168 \, b^{2} c^{5} d x^{5} + 35 \, b^{2} c^{3} d x^{3} + 210 \, b^{2} c d x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} - 105 \,{\left (5 \, a b c^{8} d x^{8} - 13 \, a b c^{6} d x^{6} + 9 \, a b c^{4} d x^{4} + a b c^{2} d x^{2} - 2 \, a b d\right )} \sqrt{-c^{2} d x^{2} + d}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) +{\left (1125 \,{\left (49 \, a^{2} + 2 \, b^{2}\right )} c^{8} d x^{8} - 9 \,{\left (15925 \, a^{2} + 734 \, b^{2}\right )} c^{6} d x^{6} +{\left (99225 \, a^{2} + 998 \, b^{2}\right )} c^{4} d x^{4} +{\left (11025 \, a^{2} + 40742 \, b^{2}\right )} c^{2} d x^{2} - 2 \,{\left (11025 \, a^{2} + 18692 \, b^{2}\right )} d\right )} \sqrt{-c^{2} d x^{2} + d}}{385875 \,{\left (c^{6} x^{2} - c^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/385875*(11025*(5*b^2*c^8*d*x^8 - 13*b^2*c^6*d*x^6 + 9*b^2*c^4*d*x^4 + b^2*c^2*d*x^2 - 2*b^2*d)*sqrt(-c^2*d*
x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1))^2 - 210*(75*a*b*c^7*d*x^7 - 168*a*b*c^5*d*x^5 + 35*a*b*c^3*d*x^3 + 210*a
*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 210*((75*b^2*c^7*d*x^7 - 168*b^2*c^5*d*x^5 + 35*b^2*c^3*d*x
^3 + 210*b^2*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 105*(5*a*b*c^8*d*x^8 - 13*a*b*c^6*d*x^6 + 9*a*b*c
^4*d*x^4 + a*b*c^2*d*x^2 - 2*a*b*d)*sqrt(-c^2*d*x^2 + d))*log(c*x + sqrt(c^2*x^2 - 1)) + (1125*(49*a^2 + 2*b^2
)*c^8*d*x^8 - 9*(15925*a^2 + 734*b^2)*c^6*d*x^6 + (99225*a^2 + 998*b^2)*c^4*d*x^4 + (11025*a^2 + 40742*b^2)*c^
2*d*x^2 - 2*(11025*a^2 + 18692*b^2)*d)*sqrt(-c^2*d*x^2 + d))/(c^6*x^2 - c^4)

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError