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

Optimal. Leaf size=319 \[ -\frac{b c x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{4} x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{b x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{8 c^2}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{24 b c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{32} b^2 x^3 \sqrt{d-c^2 d x^2}-\frac{b^2 x \sqrt{d-c^2 d x^2}}{64 c^2}-\frac{b^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{64 c^3 \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

-(b^2*x*Sqrt[d - c^2*d*x^2])/(64*c^2) + (b^2*x^3*Sqrt[d - c^2*d*x^2])/32 - (b^2*Sqrt[d - c^2*d*x^2]*ArcCosh[c*
x])/(64*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(8*c*Sqrt[-1 + c*
x]*Sqrt[1 + c*x]) - (b*c*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(8*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (x*S
qrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(8*c^2) + (x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/4 - (Sqr
t[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^3)/(24*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

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Rubi [A]  time = 0.93039, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31, Rules used = {5798, 5743, 5759, 5676, 5662, 90, 52, 100, 12} \[ -\frac{b c x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{4} x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{b x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{8 c^2}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{24 b c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{32} b^2 x^3 \sqrt{d-c^2 d x^2}-\frac{b^2 x \sqrt{d-c^2 d x^2}}{64 c^2}-\frac{b^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{64 c^3 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^2*x*Sqrt[d - c^2*d*x^2])/(64*c^2) + (b^2*x^3*Sqrt[d - c^2*d*x^2])/32 - (b^2*Sqrt[d - c^2*d*x^2]*ArcCosh[c*
x])/(64*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(8*c*Sqrt[-1 + c*
x]*Sqrt[1 + c*x]) - (b*c*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(8*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (x*S
qrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(8*c^2) + (x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/4 - (Sqr
t[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^3)/(24*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 5798

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

Rule 5743

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

Rule 5759

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

Rule 5676

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

Rule 5662

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
osh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rule 100

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

Rule 12

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

Rubi steps

\begin{align*} \int x^2 \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^2 \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}{4} x^3 \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^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c \sqrt{d-c^2 d x^2}\right ) \int x^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{8 c^2}+\frac{1}{4} x^3 \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{\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{8 c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b \sqrt{d-c^2 d x^2}\right ) \int x \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{4 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b^2 c^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{8 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{32} b^2 x^3 \sqrt{d-c^2 d x^2}+\frac{b x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{8 c^2}+\frac{1}{4} x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{24 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{3 x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{32 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{8 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b^2 x \sqrt{d-c^2 d x^2}}{16 c^2}+\frac{1}{32} b^2 x^3 \sqrt{d-c^2 d x^2}+\frac{b x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{8 c^2}+\frac{1}{4} x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{24 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 b^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{32 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{16 c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b^2 x \sqrt{d-c^2 d x^2}}{64 c^2}+\frac{1}{32} b^2 x^3 \sqrt{d-c^2 d x^2}-\frac{b^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{16 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{8 c^2}+\frac{1}{4} x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{24 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 b^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{64 c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b^2 x \sqrt{d-c^2 d x^2}}{64 c^2}+\frac{1}{32} b^2 x^3 \sqrt{d-c^2 d x^2}-\frac{b^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{64 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{8 c^2}+\frac{1}{4} x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{24 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [A]  time = 1.99127, size = 241, normalized size = 0.76 \[ -\frac{-96 a^2 c x \left (2 c^2 x^2-1\right ) \sqrt{d-c^2 d x^2}+96 a^2 \sqrt{d} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )+\frac{12 a b \sqrt{d-c^2 d x^2} \left (8 \cosh ^{-1}(c x)^2+\cosh \left (4 \cosh ^{-1}(c x)\right )-4 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )\right )}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\frac{b^2 \sqrt{d-c^2 d x^2} \left (32 \cosh ^{-1}(c x)^3+12 \cosh \left (4 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)-3 \left (8 \cosh ^{-1}(c x)^2+1\right ) \sinh \left (4 \cosh ^{-1}(c x)\right )\right )}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}}{768 c^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(-96*a^2*c*x*(-1 + 2*c^2*x^2)*Sqrt[d - c^2*d*x^2] + 96*a^2*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*
(-1 + c^2*x^2))] + (12*a*b*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]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (b^2*Sqrt[d - c^2*d*x^2]*(32*ArcCosh[c*x]^3 + 12*Ar
cCosh[c*x]*Cosh[4*ArcCosh[c*x]] - 3*(1 + 8*ArcCosh[c*x]^2)*Sinh[4*ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*
(1 + c*x)))/(768*c^3)

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Maple [B]  time = 0.376, size = 767, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*a^2*x*(-c^2*d*x^2+d)^(3/2)/c^2/d+1/8*a^2/c^2*x*(-c^2*d*x^2+d)^(1/2)+1/8*a^2/c^2*d/(c^2*d)^(1/2)*arctan((c
^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+1/8*b^2*(-d*(c^2*x^2-1))^(1/2)/(c*x+1)^(1/2)/c/(c*x-1)^(1/2)*arccosh(c*x)*
x^2-1/8*b^2*(-d*(c^2*x^2-1))^(1/2)/(c*x+1)^(1/2)*c/(c*x-1)^(1/2)*arccosh(c*x)*x^4-1/24*b^2*(-d*(c^2*x^2-1))^(1
/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c^3*arccosh(c*x)^3-1/64*b^2*(-d*(c^2*x^2-1))^(1/2)/(c*x+1)^(1/2)/c^3/(c*x-1)^(
1/2)*arccosh(c*x)+1/32*b^2*(-d*(c^2*x^2-1))^(1/2)/(c*x+1)*c^2/(c*x-1)*x^5-3/64*b^2*(-d*(c^2*x^2-1))^(1/2)/(c*x
+1)/(c*x-1)*x^3+1/64*b^2*(-d*(c^2*x^2-1))^(1/2)/(c*x+1)/c^2/(c*x-1)*x+1/4*b^2*(-d*(c^2*x^2-1))^(1/2)/(c*x+1)*c
^2/(c*x-1)*arccosh(c*x)^2*x^5-3/8*b^2*(-d*(c^2*x^2-1))^(1/2)/(c*x+1)/(c*x-1)*arccosh(c*x)^2*x^3+1/8*b^2*(-d*(c
^2*x^2-1))^(1/2)/(c*x+1)/c^2/(c*x-1)*arccosh(c*x)^2*x-1/8*a*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/
2)/c^3*arccosh(c*x)^2-1/8*a*b*(-d*(c^2*x^2-1))^(1/2)/(c*x+1)^(1/2)*c/(c*x-1)^(1/2)*x^4+1/8*a*b*(-d*(c^2*x^2-1)
)^(1/2)/(c*x+1)^(1/2)/c/(c*x-1)^(1/2)*x^2+1/2*a*b*(-d*(c^2*x^2-1))^(1/2)/(c*x+1)*c^2/(c*x-1)*arccosh(c*x)*x^5-
3/4*a*b*(-d*(c^2*x^2-1))^(1/2)/(c*x+1)/(c*x-1)*arccosh(c*x)*x^3+1/4*a*b*(-d*(c^2*x^2-1))^(1/2)/(c*x+1)/c^2/(c*
x-1)*arccosh(c*x)*x-1/64*a*b*(-d*(c^2*x^2-1))^(1/2)/(c*x+1)^(1/2)/c^3/(c*x-1)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*x^2*arccosh(c*x)^2 + 2*a*b*x^2*arccosh(c*x) + a^2*x^2)*sqrt(-c^2*d*x^2 + d), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \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**2*(a+b*acosh(c*x))**2*(-c**2*d*x**2+d)**(1/2),x)

[Out]

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

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out