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

Optimal. Leaf size=421 \[ -\frac{16 a b x \sqrt{c x-1} \sqrt{c x+1}}{15 c^5 \sqrt{d-c^2 d x^2}}-\frac{2 b x^5 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{25 c \sqrt{d-c^2 d x^2}}-\frac{x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2 d}-\frac{8 b x^3 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{45 c^3 \sqrt{d-c^2 d x^2}}-\frac{4 x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4 d}-\frac{8 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^6 d}-\frac{2 b^2 x^4 (1-c x) (c x+1)}{125 c^2 \sqrt{d-c^2 d x^2}}-\frac{272 b^2 x^2 (1-c x) (c x+1)}{3375 c^4 \sqrt{d-c^2 d x^2}}-\frac{4144 b^2 (1-c x) (c x+1)}{3375 c^6 \sqrt{d-c^2 d x^2}}-\frac{16 b^2 x \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{15 c^5 \sqrt{d-c^2 d x^2}} \]

[Out]

(-16*a*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(15*c^5*Sqrt[d - c^2*d*x^2]) - (4144*b^2*(1 - c*x)*(1 + c*x))/(3375*c
^6*Sqrt[d - c^2*d*x^2]) - (272*b^2*x^2*(1 - c*x)*(1 + c*x))/(3375*c^4*Sqrt[d - c^2*d*x^2]) - (2*b^2*x^4*(1 - c
*x)*(1 + c*x))/(125*c^2*Sqrt[d - c^2*d*x^2]) - (16*b^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(15*c^5*Sq
rt[d - c^2*d*x^2]) - (8*b*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(45*c^3*Sqrt[d - c^2*d*x^2])
- (2*b*x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(25*c*Sqrt[d - c^2*d*x^2]) - (8*Sqrt[d - c^2*d*x
^2]*(a + b*ArcCosh[c*x])^2)/(15*c^6*d) - (4*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(15*c^4*d) - (x^4*
Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(5*c^2*d)

________________________________________________________________________________________

Rubi [A]  time = 1.13135, antiderivative size = 445, normalized size of antiderivative = 1.06, number of steps used = 17, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {5798, 5759, 5718, 5654, 74, 5662, 100, 12} \[ -\frac{16 a b x \sqrt{c x-1} \sqrt{c x+1}}{15 c^5 \sqrt{d-c^2 d x^2}}-\frac{2 b x^5 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{25 c \sqrt{d-c^2 d x^2}}-\frac{x^4 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2 \sqrt{d-c^2 d x^2}}-\frac{8 b x^3 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{45 c^3 \sqrt{d-c^2 d x^2}}-\frac{4 x^2 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4 \sqrt{d-c^2 d x^2}}-\frac{8 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^6 \sqrt{d-c^2 d x^2}}-\frac{2 b^2 x^4 (1-c x) (c x+1)}{125 c^2 \sqrt{d-c^2 d x^2}}-\frac{272 b^2 x^2 (1-c x) (c x+1)}{3375 c^4 \sqrt{d-c^2 d x^2}}-\frac{4144 b^2 (1-c x) (c x+1)}{3375 c^6 \sqrt{d-c^2 d x^2}}-\frac{16 b^2 x \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{15 c^5 \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-16*a*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(15*c^5*Sqrt[d - c^2*d*x^2]) - (4144*b^2*(1 - c*x)*(1 + c*x))/(3375*c
^6*Sqrt[d - c^2*d*x^2]) - (272*b^2*x^2*(1 - c*x)*(1 + c*x))/(3375*c^4*Sqrt[d - c^2*d*x^2]) - (2*b^2*x^4*(1 - c
*x)*(1 + c*x))/(125*c^2*Sqrt[d - c^2*d*x^2]) - (16*b^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(15*c^5*Sq
rt[d - c^2*d*x^2]) - (8*b*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(45*c^3*Sqrt[d - c^2*d*x^2])
- (2*b*x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(25*c*Sqrt[d - c^2*d*x^2]) - (8*(1 - c*x)*(1 + c
*x)*(a + b*ArcCosh[c*x])^2)/(15*c^6*Sqrt[d - c^2*d*x^2]) - (4*x^2*(1 - c*x)*(1 + c*x)*(a + b*ArcCosh[c*x])^2)/
(15*c^4*Sqrt[d - c^2*d*x^2]) - (x^4*(1 - c*x)*(1 + c*x)*(a + b*ArcCosh[c*x])^2)/(5*c^2*Sqrt[d - c^2*d*x^2])

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 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 \frac{x^5 \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{d-c^2 d x^2}} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^5 \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{d-c^2 d x^2}}\\ &=-\frac{x^4 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2 \sqrt{d-c^2 d x^2}}+\frac{\left (4 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{5 c^2 \sqrt{d-c^2 d x^2}}-\frac{\left (2 b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int x^4 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{5 c \sqrt{d-c^2 d x^2}}\\ &=-\frac{2 b x^5 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c \sqrt{d-c^2 d x^2}}-\frac{4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4 \sqrt{d-c^2 d x^2}}-\frac{x^4 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2 \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^5}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{25 \sqrt{d-c^2 d x^2}}+\frac{\left (8 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{15 c^4 \sqrt{d-c^2 d x^2}}-\frac{\left (8 b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int x^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{15 c^3 \sqrt{d-c^2 d x^2}}\\ &=-\frac{2 b^2 x^4 (1-c x) (1+c x)}{125 c^2 \sqrt{d-c^2 d x^2}}-\frac{8 b x^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{45 c^3 \sqrt{d-c^2 d x^2}}-\frac{2 b x^5 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c \sqrt{d-c^2 d x^2}}-\frac{8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^6 \sqrt{d-c^2 d x^2}}-\frac{4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4 \sqrt{d-c^2 d x^2}}-\frac{x^4 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2 \sqrt{d-c^2 d x^2}}-\frac{\left (16 b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{15 c^5 \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{4 x^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{125 c^2 \sqrt{d-c^2 d x^2}}+\frac{\left (8 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{45 c^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{16 a b x \sqrt{-1+c x} \sqrt{1+c x}}{15 c^5 \sqrt{d-c^2 d x^2}}-\frac{8 b^2 x^2 (1-c x) (1+c x)}{135 c^4 \sqrt{d-c^2 d x^2}}-\frac{2 b^2 x^4 (1-c x) (1+c x)}{125 c^2 \sqrt{d-c^2 d x^2}}-\frac{8 b x^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{45 c^3 \sqrt{d-c^2 d x^2}}-\frac{2 b x^5 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c \sqrt{d-c^2 d x^2}}-\frac{8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^6 \sqrt{d-c^2 d x^2}}-\frac{4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4 \sqrt{d-c^2 d x^2}}-\frac{x^4 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2 \sqrt{d-c^2 d x^2}}-\frac{\left (16 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \cosh ^{-1}(c x) \, dx}{15 c^5 \sqrt{d-c^2 d x^2}}+\frac{\left (8 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{2 x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{135 c^4 \sqrt{d-c^2 d x^2}}+\frac{\left (8 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{125 c^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{16 a b x \sqrt{-1+c x} \sqrt{1+c x}}{15 c^5 \sqrt{d-c^2 d x^2}}-\frac{272 b^2 x^2 (1-c x) (1+c x)}{3375 c^4 \sqrt{d-c^2 d x^2}}-\frac{2 b^2 x^4 (1-c x) (1+c x)}{125 c^2 \sqrt{d-c^2 d x^2}}-\frac{16 b^2 x \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{15 c^5 \sqrt{d-c^2 d x^2}}-\frac{8 b x^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{45 c^3 \sqrt{d-c^2 d x^2}}-\frac{2 b x^5 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c \sqrt{d-c^2 d x^2}}-\frac{8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^6 \sqrt{d-c^2 d x^2}}-\frac{4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4 \sqrt{d-c^2 d x^2}}-\frac{x^4 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2 \sqrt{d-c^2 d x^2}}+\frac{\left (8 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{2 x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{375 c^4 \sqrt{d-c^2 d x^2}}+\frac{\left (16 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{135 c^4 \sqrt{d-c^2 d x^2}}+\frac{\left (16 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{15 c^4 \sqrt{d-c^2 d x^2}}\\ &=-\frac{16 a b x \sqrt{-1+c x} \sqrt{1+c x}}{15 c^5 \sqrt{d-c^2 d x^2}}-\frac{32 b^2 (1-c x) (1+c x)}{27 c^6 \sqrt{d-c^2 d x^2}}-\frac{272 b^2 x^2 (1-c x) (1+c x)}{3375 c^4 \sqrt{d-c^2 d x^2}}-\frac{2 b^2 x^4 (1-c x) (1+c x)}{125 c^2 \sqrt{d-c^2 d x^2}}-\frac{16 b^2 x \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{15 c^5 \sqrt{d-c^2 d x^2}}-\frac{8 b x^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{45 c^3 \sqrt{d-c^2 d x^2}}-\frac{2 b x^5 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c \sqrt{d-c^2 d x^2}}-\frac{8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^6 \sqrt{d-c^2 d x^2}}-\frac{4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4 \sqrt{d-c^2 d x^2}}-\frac{x^4 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2 \sqrt{d-c^2 d x^2}}+\frac{\left (16 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{375 c^4 \sqrt{d-c^2 d x^2}}\\ &=-\frac{16 a b x \sqrt{-1+c x} \sqrt{1+c x}}{15 c^5 \sqrt{d-c^2 d x^2}}-\frac{4144 b^2 (1-c x) (1+c x)}{3375 c^6 \sqrt{d-c^2 d x^2}}-\frac{272 b^2 x^2 (1-c x) (1+c x)}{3375 c^4 \sqrt{d-c^2 d x^2}}-\frac{2 b^2 x^4 (1-c x) (1+c x)}{125 c^2 \sqrt{d-c^2 d x^2}}-\frac{16 b^2 x \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{15 c^5 \sqrt{d-c^2 d x^2}}-\frac{8 b x^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{45 c^3 \sqrt{d-c^2 d x^2}}-\frac{2 b x^5 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c \sqrt{d-c^2 d x^2}}-\frac{8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^6 \sqrt{d-c^2 d x^2}}-\frac{4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{15 c^4 \sqrt{d-c^2 d x^2}}-\frac{x^4 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{5 c^2 \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 0.53885, size = 255, normalized size = 0.61 \[ \frac{\sqrt{d-c^2 d x^2} \left (-225 a^2 \left (3 c^6 x^6+c^4 x^4+4 c^2 x^2-8\right )+30 a b c x \sqrt{c x-1} \sqrt{c x+1} \left (9 c^4 x^4+20 c^2 x^2+120\right )+30 b \cosh ^{-1}(c x) \left (b c x \sqrt{c x-1} \sqrt{c x+1} \left (9 c^4 x^4+20 c^2 x^2+120\right )-15 a \left (3 c^6 x^6+c^4 x^4+4 c^2 x^2-8\right )\right )-2 b^2 \left (27 c^6 x^6+109 c^4 x^4+1936 c^2 x^2-2072\right )-225 b^2 \left (3 c^6 x^6+c^4 x^4+4 c^2 x^2-8\right ) \cosh ^{-1}(c x)^2\right )}{3375 c^6 d (c x-1) (c x+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d - c^2*d*x^2]*(30*a*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(120 + 20*c^2*x^2 + 9*c^4*x^4) - 225*a^2*(-8 + 4
*c^2*x^2 + c^4*x^4 + 3*c^6*x^6) - 2*b^2*(-2072 + 1936*c^2*x^2 + 109*c^4*x^4 + 27*c^6*x^6) + 30*b*(b*c*x*Sqrt[-
1 + c*x]*Sqrt[1 + c*x]*(120 + 20*c^2*x^2 + 9*c^4*x^4) - 15*a*(-8 + 4*c^2*x^2 + c^4*x^4 + 3*c^6*x^6))*ArcCosh[c
*x] - 225*b^2*(-8 + 4*c^2*x^2 + c^4*x^4 + 3*c^6*x^6)*ArcCosh[c*x]^2))/(3375*c^6*d*(-1 + c*x)*(1 + c*x))

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Maple [B]  time = 0.477, size = 1314, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(1/2),x)

[Out]

a^2*(-1/5*x^4/c^2/d*(-c^2*d*x^2+d)^(1/2)+4/5/c^2*(-1/3*x^2/c^2/d*(-c^2*d*x^2+d)^(1/2)-2/3/d/c^4*(-c^2*d*x^2+d)
^(1/2)))+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*arcc
osh(c*x)+2)/c^6/d/(c^2*x^2-1)-5/864*(-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^6/d/(c^2*x^2-1)-5/16*(-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^6/d/(c^2*x^2-
1)-5/16*(-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^6/d/(c^2*x^2-1)-5/864*(-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*arccosh(c*x)^2+6*arccosh(c*x)+2)/c^6/d/(c^2*x^2-1)-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^6/d/(c^2*x^2-1))+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^6/d/(c^2*x^2-1)-5
/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^6/d/(c^2*x^2-1)-5/16*(-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^6/d/(c^2*x^2-1)-5/16*(-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^6/d/(c^2*x^2-1)-5/288*(-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^6/d/(c^2*x^2-1)-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^6/d/(c^2*x^2-1))

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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^5*(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 = 2.17923, size = 771, normalized size = 1.83 \begin{align*} -\frac{225 \,{\left (3 \, b^{2} c^{6} x^{6} + b^{2} c^{4} x^{4} + 4 \, b^{2} c^{2} x^{2} - 8 \, 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} + 20 \, a b c^{3} x^{3} + 120 \, 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} + 20 \, b^{2} c^{3} x^{3} + 120 \, 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} + a b c^{4} x^{4} + 4 \, a b c^{2} x^{2} - 8 \, 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} +{\left (225 \, a^{2} + 218 \, b^{2}\right )} c^{4} x^{4} + 4 \,{\left (225 \, a^{2} + 968 \, b^{2}\right )} c^{2} x^{2} - 1800 \, a^{2} - 4144 \, b^{2}\right )} \sqrt{-c^{2} d x^{2} + d}}{3375 \,{\left (c^{8} d x^{2} - c^{6} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(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 + b^2*c^4*x^4 + 4*b^2*c^2*x^2 - 8*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 + 20*a*b*c^3*x^3 + 120*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 + 20*b^2*c^3*x^3 + 120*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 + a*b*c^
4*x^4 + 4*a*b*c^2*x^2 - 8*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 + (225*a^2 + 218*b^2)*c^4*x^4 + 4*(225*a^2 + 968*b^2)*c^2*x^2 - 1800*a^2 - 4144*b^2)*sqrt(-c^2*d*x^2 + d))/
(c^8*d*x^2 - c^6*d)

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{2} x^{5}}{\sqrt{-c^{2} d x^{2} + d}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

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