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

Optimal. Leaf size=371 \[ \frac{4 a b x \sqrt{d-c^2 d x^2}}{15 c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b c x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{5} x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{2 b x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{45 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^2}-\frac{2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4}+\frac{2}{125} b^2 x^4 \sqrt{d-c^2 d x^2}+\frac{22 b^2 x^2 \sqrt{d-c^2 d x^2}}{3375 c^2}-\frac{856 b^2 \sqrt{d-c^2 d x^2}}{3375 c^4}+\frac{4 b^2 x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{15 c^3 \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

(-856*b^2*Sqrt[d - c^2*d*x^2])/(3375*c^4) + (22*b^2*x^2*Sqrt[d - c^2*d*x^2])/(3375*c^2) + (2*b^2*x^4*Sqrt[d -
c^2*d*x^2])/125 + (4*a*b*x*Sqrt[d - c^2*d*x^2])/(15*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*b^2*x*Sqrt[d - c^2*
d*x^2]*ArcCosh[c*x])/(15*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])
)/(45*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(25*Sqrt[-1 + c*x
]*Sqrt[1 + c*x]) - (2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(15*c^4) - (x^2*Sqrt[d - c^2*d*x^2]*(a + b*A
rcCosh[c*x])^2)/(15*c^2) + (x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/5

________________________________________________________________________________________

Rubi [A]  time = 1.05567, antiderivative size = 371, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 9, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31, Rules used = {5798, 5743, 5759, 5718, 5654, 74, 5662, 100, 12} \[ \frac{4 a b x \sqrt{d-c^2 d x^2}}{15 c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b c x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{5} x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{2 b x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{45 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^2}-\frac{2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4}+\frac{2}{125} b^2 x^4 \sqrt{d-c^2 d x^2}+\frac{22 b^2 x^2 \sqrt{d-c^2 d x^2}}{3375 c^2}-\frac{856 b^2 \sqrt{d-c^2 d x^2}}{3375 c^4}+\frac{4 b^2 x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{15 c^3 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-856*b^2*Sqrt[d - c^2*d*x^2])/(3375*c^4) + (22*b^2*x^2*Sqrt[d - c^2*d*x^2])/(3375*c^2) + (2*b^2*x^4*Sqrt[d -
c^2*d*x^2])/125 + (4*a*b*x*Sqrt[d - c^2*d*x^2])/(15*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*b^2*x*Sqrt[d - c^2*
d*x^2]*ArcCosh[c*x])/(15*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])
)/(45*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(25*Sqrt[-1 + c*x
]*Sqrt[1 + c*x]) - (2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(15*c^4) - (x^2*Sqrt[d - c^2*d*x^2]*(a + b*A
rcCosh[c*x])^2)/(15*c^2) + (x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/5

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 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]]

Rubi steps

\begin{align*} \int x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=\frac{\sqrt{d-c^2 d x^2} \int x^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{5} x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{\sqrt{d-c^2 d x^2} \int \frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{5 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b c \sqrt{d-c^2 d x^2}\right ) \int x^4 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{5 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{2 b c x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^2}+\frac{1}{5} x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{\left (2 \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}{15 c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 b \sqrt{d-c^2 d x^2}\right ) \int x^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{15 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 b^2 c^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^5}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{25 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{2}{125} b^2 x^4 \sqrt{d-c^2 d x^2}+\frac{2 b x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{45 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4}-\frac{x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^2}+\frac{1}{5} x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{\left (2 b^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{4 x^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{125 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{45 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (4 b \sqrt{d-c^2 d x^2}\right ) \int \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{15 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{2 b^2 x^2 \sqrt{d-c^2 d x^2}}{135 c^2}+\frac{2}{125} b^2 x^4 \sqrt{d-c^2 d x^2}+\frac{4 a b x \sqrt{d-c^2 d x^2}}{15 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{45 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4}-\frac{x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^2}+\frac{1}{5} x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{\left (8 b^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{125 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (4 b^2 \sqrt{d-c^2 d x^2}\right ) \int \cosh ^{-1}(c x) \, dx}{15 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{2 x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{135 c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{22 b^2 x^2 \sqrt{d-c^2 d x^2}}{3375 c^2}+\frac{2}{125} b^2 x^4 \sqrt{d-c^2 d x^2}+\frac{4 a b x \sqrt{d-c^2 d x^2}}{15 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{4 b^2 x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{15 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{45 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4}-\frac{x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^2}+\frac{1}{5} x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{\left (8 b^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{2 x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{375 c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (4 b^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{135 c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (4 b^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{15 c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{8 b^2 \sqrt{d-c^2 d x^2}}{27 c^4}+\frac{22 b^2 x^2 \sqrt{d-c^2 d x^2}}{3375 c^2}+\frac{2}{125} b^2 x^4 \sqrt{d-c^2 d x^2}+\frac{4 a b x \sqrt{d-c^2 d x^2}}{15 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{4 b^2 x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{15 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{45 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4}-\frac{x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^2}+\frac{1}{5} x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{\left (16 b^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{375 c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{856 b^2 \sqrt{d-c^2 d x^2}}{3375 c^4}+\frac{22 b^2 x^2 \sqrt{d-c^2 d x^2}}{3375 c^2}+\frac{2}{125} b^2 x^4 \sqrt{d-c^2 d x^2}+\frac{4 a b x \sqrt{d-c^2 d x^2}}{15 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{4 b^2 x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{15 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{45 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c x^5 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4}-\frac{x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^2}+\frac{1}{5} x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2\\ \end{align*}

Mathematica [A]  time = 0.427265, size = 237, normalized size = 0.64 \[ \frac{\sqrt{d-c^2 d x^2} \left (225 a^2 \left (c^2 x^2-1\right )^2 \left (3 c^2 x^2+2\right )-30 a b c x \sqrt{c x-1} \sqrt{c x+1} \left (9 c^4 x^4-5 c^2 x^2-30\right )+30 b \cosh ^{-1}(c x) \left (15 a \left (3 c^2 x^2+2\right ) \left (c^2 x^2-1\right )^2+b c x \sqrt{c x-1} \sqrt{c x+1} \left (-9 c^4 x^4+5 c^2 x^2+30\right )\right )+2 b^2 \left (27 c^6 x^6-16 c^4 x^4-439 c^2 x^2+428\right )+225 b^2 \left (c^2 x^2-1\right )^2 \left (3 c^2 x^2+2\right ) \cosh ^{-1}(c x)^2\right )}{3375 c^4 \left (c^2 x^2-1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d - c^2*d*x^2]*(225*a^2*(-1 + c^2*x^2)^2*(2 + 3*c^2*x^2) - 30*a*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-30
- 5*c^2*x^2 + 9*c^4*x^4) + 2*b^2*(428 - 439*c^2*x^2 - 16*c^4*x^4 + 27*c^6*x^6) + 30*b*(15*a*(-1 + c^2*x^2)^2*(
2 + 3*c^2*x^2) + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(30 + 5*c^2*x^2 - 9*c^4*x^4))*ArcCosh[c*x] + 225*b^2*(-1 +
c^2*x^2)^2*(2 + 3*c^2*x^2)*ArcCosh[c*x]^2))/(3375*c^4*(-1 + c^2*x^2))

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Maple [B]  time = 0.544, size = 1284, normalized size = 3.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2),x)

[Out]

a^2*(-1/5*x^2*(-c^2*d*x^2+d)^(3/2)/c^2/d-2/15/d/c^4*(-c^2*d*x^2+d)^(3/2))+b^2*(1/4000*(-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)/(c*x+1)/c^4/(c*x-1)+1/864*(-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)/(c*x+1)/c^4/(c*x-1)-1/16*(-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)/(c*x+1)/c^4/(c*x-1)-1/16*(-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)/(c*x+1)/c^4/(c*x-1)+1/864*(-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*a
rccosh(c*x)^2+6*arccosh(c*x)+2)/(c*x+1)/c^4/(c*x-1)+1/4000*(-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*arccosh(c*x)^2+10*arccosh(c*x)+2)/(c*x+1)/c^4/(c*x-1))+2*a*b*(1/800*(-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))/(c*x+1)/c^4/(c*x-1)+1/288*(-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))
/(c*x+1)/c^4/(c*x-1)-1/16*(-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+arccosh(c*x))
/(c*x+1)/c^4/(c*x-1)-1/16*(-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+arccosh(c*x))
/(c*x+1)/c^4/(c*x-1)+1/288*(-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))/(c*x+1)/c^4/(c*x-1)+1/800*(-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))/(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*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.27616, size = 753, normalized size = 2.03 \begin{align*} \frac{225 \,{\left (3 \, b^{2} c^{6} x^{6} - 4 \, b^{2} c^{4} x^{4} - b^{2} c^{2} x^{2} + 2 \, b^{2}\right )} \sqrt{-c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )^{2} - 30 \,{\left (9 \, a b c^{5} x^{5} - 5 \, a b c^{3} x^{3} - 30 \, a b c x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} - 30 \,{\left ({\left (9 \, b^{2} c^{5} x^{5} - 5 \, b^{2} c^{3} x^{3} - 30 \, b^{2} c x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} - 15 \,{\left (3 \, a b c^{6} x^{6} - 4 \, a b c^{4} x^{4} - a b c^{2} x^{2} + 2 \, a b\right )} \sqrt{-c^{2} d x^{2} + d}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) +{\left (27 \,{\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{6} x^{6} - 4 \,{\left (225 \, a^{2} + 8 \, b^{2}\right )} c^{4} x^{4} -{\left (225 \, a^{2} + 878 \, b^{2}\right )} c^{2} x^{2} + 450 \, a^{2} + 856 \, b^{2}\right )} \sqrt{-c^{2} d x^{2} + d}}{3375 \,{\left (c^{6} x^{2} - c^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

1/3375*(225*(3*b^2*c^6*x^6 - 4*b^2*c^4*x^4 - b^2*c^2*x^2 + 2*b^2)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2
- 1))^2 - 30*(9*a*b*c^5*x^5 - 5*a*b*c^3*x^3 - 30*a*b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 30*((9*b^2*
c^5*x^5 - 5*b^2*c^3*x^3 - 30*b^2*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 15*(3*a*b*c^6*x^6 - 4*a*b*c^4*x
^4 - a*b*c^2*x^2 + 2*a*b)*sqrt(-c^2*d*x^2 + d))*log(c*x + sqrt(c^2*x^2 - 1)) + (27*(25*a^2 + 2*b^2)*c^6*x^6 -
4*(225*a^2 + 8*b^2)*c^4*x^4 - (225*a^2 + 878*b^2)*c^2*x^2 + 450*a^2 + 856*b^2)*sqrt(-c^2*d*x^2 + d))/(c^6*x^2
- c^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \sqrt{- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname{acosh}{\left (c x \right )}\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError