\(\int (d-c^2 d x^2)^{3/2} (a+b \text {arccosh}(c x))^2 \, dx\) [181]

Optimal result

Integrand size = 26, antiderivative size = 336 \[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {15}{64} b^2 d x \sqrt {d-c^2 d x^2}+\frac {1}{32} b^2 d x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}+\frac {9 b^2 d \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{64 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c d x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3}{8} d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{8 b c \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

1/4*x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2+15/64*b^2*d*x*(-c^2*d*x^2+d)^(1/2)+1/32*b^2*d*x*(-c*x+1)*(c*x+
1)*(-c^2*d*x^2+d)^(1/2)+3/8*d*x*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)+9/64*b^2*d*arccosh(c*x)*(-c^2*d*x^2+
d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-3/8*b*c*d*x^2*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*
x+1)^(1/2)+1/8*b*d*(-c^2*x^2+1)^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)-1/8*d*
(a+b*arccosh(c*x))^3*(-c^2*d*x^2+d)^(1/2)/b/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {5897, 5895, 5893, 5883, 92, 54, 5912, 5914, 38} \[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{8 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {3}{8} d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {b d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 b c d x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {c x-1} \sqrt {c x+1}}+\frac {9 b^2 d \text {arccosh}(c x) \sqrt {d-c^2 d x^2}}{64 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {15}{64} b^2 d x \sqrt {d-c^2 d x^2}+\frac {1}{32} b^2 d x (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \]

[In]

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

[Out]

(15*b^2*d*x*Sqrt[d - c^2*d*x^2])/64 + (b^2*d*x*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2])/32 + (9*b^2*d*Sqrt[d -
c^2*d*x^2]*ArcCosh[c*x])/(64*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*b*c*d*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos
h[c*x]))/(8*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(8*
c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/8 + (x*(d - c^2*d*x^2)^(3
/2)*(a + b*ArcCosh[c*x])^2)/4 - (d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^3)/(8*b*c*Sqrt[-1 + c*x]*Sqrt[1 +
c*x])

Rule 38

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[x*(a + b*x)^m*((c + d*x)^m/(2*m + 1))
, x] + Dist[2*a*c*(m/(2*m + 1)), Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] &&
EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0]

Rule 54

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 92

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 5883

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)/(Sqrt
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5893

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

Rule 5895

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[x*Sqrt[d + e*x^2]*(
(a + b*ArcCosh[c*x])^n/2), x] + (-Dist[(1/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])], Int[(a + b*
ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Dist[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sq
rt[-1 + c*x])], Int[x*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
&& GtQ[n, 0]

Rule 5897

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

Rule 5912

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

Rule 5914

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{4} (3 d) \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int x (-1+c x) (1+c x) (a+b \text {arccosh}(c x)) \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {3}{8} d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {\left (3 d \sqrt {d-c^2 d x^2}\right ) \int \frac {(a+b \text {arccosh}(c x))^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 \left (-1+c^2 x^2\right ) (a+b \text {arccosh}(c x)) \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 b c d \sqrt {d-c^2 d x^2}\right ) \int x (a+b \text {arccosh}(c x)) \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {3 b c d x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3}{8} d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{8 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b^2 d \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b^2 c^2 d \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 {3}{16} b^2 d x \sqrt {d-c^2 d x^2}+\frac {1}{32} b^2 d x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}-\frac {3 b c d x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3}{8} d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{8 b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx}{32 \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}{16 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {15}{64} b^2 d x \sqrt {d-c^2 d x^2}+\frac {1}{32} b^2 d x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}+\frac {3 b^2 d \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{16 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c d x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3}{8} d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{8 b c \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}{64 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {15}{64} b^2 d x \sqrt {d-c^2 d x^2}+\frac {1}{32} b^2 d x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}+\frac {9 b^2 d \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{64 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c d x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3}{8} d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{8 b c \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 3.35 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.11 \[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {-96 a^2 c d x \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (-5+2 c^2 x^2\right ) \sqrt {d-c^2 d x^2}-288 a^2 d^{3/2} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-192 a b d \sqrt {d-c^2 d x^2} (\cosh (2 \text {arccosh}(c x))+2 \text {arccosh}(c x) (\text {arccosh}(c x)-\sinh (2 \text {arccosh}(c x))))-32 b^2 d \sqrt {d-c^2 d x^2} \left (4 \text {arccosh}(c x)^3+6 \text {arccosh}(c x) \cosh (2 \text {arccosh}(c x))-3 \left (1+2 \text {arccosh}(c x)^2\right ) \sinh (2 \text {arccosh}(c x))\right )+12 a b d \sqrt {d-c^2 d x^2} \left (8 \text {arccosh}(c x)^2+\cosh (4 \text {arccosh}(c x))-4 \text {arccosh}(c x) \sinh (4 \text {arccosh}(c x))\right )+b^2 d \sqrt {d-c^2 d x^2} \left (32 \text {arccosh}(c x)^3+12 \text {arccosh}(c x) \cosh (4 \text {arccosh}(c x))-3 \left (1+8 \text {arccosh}(c x)^2\right ) \sinh (4 \text {arccosh}(c x))\right )}{768 c \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \]

[In]

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

[Out]

(-96*a^2*c*d*x*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(-5 + 2*c^2*x^2)*Sqrt[d - c^2*d*x^2] - 288*a^2*d^(3/2)*Sqr
t[(-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))] - 192*a*b*d*Sqrt[
d - c^2*d*x^2]*(Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCosh[c*x] - Sinh[2*ArcCosh[c*x]])) - 32*b^2*d*Sqrt[d
- c^2*d*x^2]*(4*ArcCosh[c*x]^3 + 6*ArcCosh[c*x]*Cosh[2*ArcCosh[c*x]] - 3*(1 + 2*ArcCosh[c*x]^2)*Sinh[2*ArcCos
h[c*x]]) + 12*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*ArcCo
sh[c*x]]) + 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*Arc
Cosh[c*x]^2)*Sinh[4*ArcCosh[c*x]]))/(768*c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1060\) vs. \(2(288)=576\).

Time = 1.05 (sec) , antiderivative size = 1061, normalized size of antiderivative = 3.16

method	result	size
default	\(\text {Expression too large to display}\)	\(1061\)
parts	\(\text {Expression too large to display}\)	\(1061\)

[In]

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

[Out]

1/4*x*(-c^2*d*x^2+d)^(3/2)*a^2+3/8*a^2*d*x*(-c^2*d*x^2+d)^(1/2)+3/8*a^2*d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)
*x/(-c^2*d*x^2+d)^(1/2))+b^2*(-1/8*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c*arccosh(c*x)^3*d-1/512
*(-d*(c^2*x^2-1))^(1/2)*(8*c^5*x^5-12*c^3*x^3+8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+4*c*x-8*(c*x-1)^(1/2)*(c*x
+1)^(1/2)*c^2*x^2+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(8*arccosh(c*x)^2-4*arccosh(c*x)+1)*d/(c*x-1)/(c*x+1)/c+1/16*(-
d*(c^2*x^2-1))^(1/2)*(2*c^3*x^3-2*c*x+2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c*x+1)^(1/2))*(2*ar
ccosh(c*x)^2-2*arccosh(c*x)+1)*d/(c*x-1)/(c*x+1)/c+1/16*(-d*(c^2*x^2-1))^(1/2)*(-2*(c*x-1)^(1/2)*(c*x+1)^(1/2)
*c^2*x^2+2*c^3*x^3+(c*x-1)^(1/2)*(c*x+1)^(1/2)-2*c*x)*(2*arccosh(c*x)^2+2*arccosh(c*x)+1)*d/(c*x-1)/(c*x+1)/c-
1/512*(-d*(c^2*x^2-1))^(1/2)*(-8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+8*c^5*x^5+8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c
^2*x^2-12*c^3*x^3-(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*c*x)*(8*arccosh(c*x)^2+4*arccosh(c*x)+1)*d/(c*x-1)/(c*x+1)/c)+
2*a*b*(-3/16*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c*arccosh(c*x)^2*d-1/256*(-d*(c^2*x^2-1))^(1/2
)*(8*c^5*x^5-12*c^3*x^3+8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+4*c*x-8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+(c*x
-1)^(1/2)*(c*x+1)^(1/2))*(-1+4*arccosh(c*x))*d/(c*x-1)/(c*x+1)/c+1/16*(-d*(c^2*x^2-1))^(1/2)*(2*c^3*x^3-2*c*x+
2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c*x+1)^(1/2))*(-1+2*arccosh(c*x))*d/(c*x-1)/(c*x+1)/c+1/1
6*(-d*(c^2*x^2-1))^(1/2)*(-2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+2*c^3*x^3+(c*x-1)^(1/2)*(c*x+1)^(1/2)-2*c*x)*
(1+2*arccosh(c*x))*d/(c*x-1)/(c*x+1)/c-1/256*(-d*(c^2*x^2-1))^(1/2)*(-8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*x^4+8*
c^5*x^5+8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-12*c^3*x^3-(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*c*x)*(1+4*arccosh(c*x))
*d/(c*x-1)/(c*x+1)/c)

Fricas [F]

\[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \,d x } \]

[In]

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

Sympy [F]

\[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\int \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]

[In]

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

[Out]

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