\(\int x^2 (d-c^2 d x^2)^3 (a+b \text {arccosh}(c x)) \, dx\) [21]

Optimal result

Integrand size = 25, antiderivative size = 227 \[ \int x^2 \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=-\frac {16 b d^3 \sqrt {-1+c x} \sqrt {1+c x}}{315 c^3}+\frac {8 b d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}{945 c^3}-\frac {2 b d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{525 c^3}+\frac {b d^3 (-1+c x)^{7/2} (1+c x)^{7/2}}{441 c^3}+\frac {b d^3 (-1+c x)^{9/2} (1+c x)^{9/2}}{81 c^3}+\frac {1}{3} d^3 x^3 (a+b \text {arccosh}(c x))-\frac {3}{5} c^2 d^3 x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} c^4 d^3 x^7 (a+b \text {arccosh}(c x))-\frac {1}{9} c^6 d^3 x^9 (a+b \text {arccosh}(c x)) \] Output:

-16/315*b*d^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3+8/945*b*d^3*(c*x-1)^(3/2)*(c 
*x+1)^(3/2)/c^3-2/525*b*d^3*(c*x-1)^(5/2)*(c*x+1)^(5/2)/c^3+1/441*b*d^3*(c 
*x-1)^(7/2)*(c*x+1)^(7/2)/c^3+1/81*b*d^3*(c*x-1)^(9/2)*(c*x+1)^(9/2)/c^3+1 
/3*d^3*x^3*(a+b*arccosh(c*x))-3/5*c^2*d^3*x^5*(a+b*arccosh(c*x))+3/7*c^4*d 
^3*x^7*(a+b*arccosh(c*x))-1/9*c^6*d^3*x^9*(a+b*arccosh(c*x))
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.61 \[ \int x^2 \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=-\frac {d^3 \left (315 a c^3 x^3 \left (-105+189 c^2 x^2-135 c^4 x^4+35 c^6 x^6\right )+b \sqrt {-1+c x} \sqrt {1+c x} \left (5258+2629 c^2 x^2-6297 c^4 x^4+4675 c^6 x^6-1225 c^8 x^8\right )+315 b c^3 x^3 \left (-105+189 c^2 x^2-135 c^4 x^4+35 c^6 x^6\right ) \text {arccosh}(c x)\right )}{99225 c^3} \] Input:

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

Output:

-1/99225*(d^3*(315*a*c^3*x^3*(-105 + 189*c^2*x^2 - 135*c^4*x^4 + 35*c^6*x^ 
6) + b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(5258 + 2629*c^2*x^2 - 6297*c^4*x^4 + 
4675*c^6*x^6 - 1225*c^8*x^8) + 315*b*c^3*x^3*(-105 + 189*c^2*x^2 - 135*c^4 
*x^4 + 35*c^6*x^6)*ArcCosh[c*x]))/c^3
 

Rubi [A] (verified)

Time = 1.19 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6336, 27, 2113, 2331, 2123, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^2 \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx\)

\(\Big \downarrow \) 6336

\(\displaystyle -b c \int \frac {d^3 x^3 \left (-35 c^6 x^6+135 c^4 x^4-189 c^2 x^2+105\right )}{315 \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{9} c^6 d^3 x^9 (a+b \text {arccosh}(c x))+\frac {3}{7} c^4 d^3 x^7 (a+b \text {arccosh}(c x))-\frac {3}{5} c^2 d^3 x^5 (a+b \text {arccosh}(c x))+\frac {1}{3} d^3 x^3 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{315} b c d^3 \int \frac {x^3 \left (-35 c^6 x^6+135 c^4 x^4-189 c^2 x^2+105\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{9} c^6 d^3 x^9 (a+b \text {arccosh}(c x))+\frac {3}{7} c^4 d^3 x^7 (a+b \text {arccosh}(c x))-\frac {3}{5} c^2 d^3 x^5 (a+b \text {arccosh}(c x))+\frac {1}{3} d^3 x^3 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 2113

\(\displaystyle -\frac {b c d^3 \sqrt {c^2 x^2-1} \int \frac {x^3 \left (-35 c^6 x^6+135 c^4 x^4-189 c^2 x^2+105\right )}{\sqrt {c^2 x^2-1}}dx}{315 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{9} c^6 d^3 x^9 (a+b \text {arccosh}(c x))+\frac {3}{7} c^4 d^3 x^7 (a+b \text {arccosh}(c x))-\frac {3}{5} c^2 d^3 x^5 (a+b \text {arccosh}(c x))+\frac {1}{3} d^3 x^3 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 2331

\(\displaystyle -\frac {b c d^3 \sqrt {c^2 x^2-1} \int \frac {x^2 \left (-35 c^6 x^6+135 c^4 x^4-189 c^2 x^2+105\right )}{\sqrt {c^2 x^2-1}}dx^2}{630 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{9} c^6 d^3 x^9 (a+b \text {arccosh}(c x))+\frac {3}{7} c^4 d^3 x^7 (a+b \text {arccosh}(c x))-\frac {3}{5} c^2 d^3 x^5 (a+b \text {arccosh}(c x))+\frac {1}{3} d^3 x^3 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 2123

\(\displaystyle -\frac {b c d^3 \sqrt {c^2 x^2-1} \int \left (-\frac {35 \left (c^2 x^2-1\right )^{7/2}}{c^2}-\frac {5 \left (c^2 x^2-1\right )^{5/2}}{c^2}+\frac {6 \left (c^2 x^2-1\right )^{3/2}}{c^2}-\frac {8 \sqrt {c^2 x^2-1}}{c^2}+\frac {16}{c^2 \sqrt {c^2 x^2-1}}\right )dx^2}{630 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{9} c^6 d^3 x^9 (a+b \text {arccosh}(c x))+\frac {3}{7} c^4 d^3 x^7 (a+b \text {arccosh}(c x))-\frac {3}{5} c^2 d^3 x^5 (a+b \text {arccosh}(c x))+\frac {1}{3} d^3 x^3 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {1}{9} c^6 d^3 x^9 (a+b \text {arccosh}(c x))+\frac {3}{7} c^4 d^3 x^7 (a+b \text {arccosh}(c x))-\frac {3}{5} c^2 d^3 x^5 (a+b \text {arccosh}(c x))+\frac {1}{3} d^3 x^3 (a+b \text {arccosh}(c x))-\frac {b c d^3 \sqrt {c^2 x^2-1} \left (-\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^4}-\frac {10 \left (c^2 x^2-1\right )^{7/2}}{7 c^4}+\frac {12 \left (c^2 x^2-1\right )^{5/2}}{5 c^4}-\frac {16 \left (c^2 x^2-1\right )^{3/2}}{3 c^4}+\frac {32 \sqrt {c^2 x^2-1}}{c^4}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

-1/630*(b*c*d^3*Sqrt[-1 + c^2*x^2]*((32*Sqrt[-1 + c^2*x^2])/c^4 - (16*(-1 
+ c^2*x^2)^(3/2))/(3*c^4) + (12*(-1 + c^2*x^2)^(5/2))/(5*c^4) - (10*(-1 + 
c^2*x^2)^(7/2))/(7*c^4) - (70*(-1 + c^2*x^2)^(9/2))/(9*c^4)))/(Sqrt[-1 + c 
*x]*Sqrt[1 + c*x]) + (d^3*x^3*(a + b*ArcCosh[c*x]))/3 - (3*c^2*d^3*x^5*(a 
+ b*ArcCosh[c*x]))/5 + (3*c^4*d^3*x^7*(a + b*ArcCosh[c*x]))/7 - (c^6*d^3*x 
^9*(a + b*ArcCosh[c*x]))/9
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2113

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

rule 2123

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] 
:> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c 
, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
 

rule 2331

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2   S 
ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; 
 FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
 

rule 6336

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]}, Simp 
[(a + b*ArcCosh[c*x])   u, x] - Simp[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] && E 
qQ[c^2*d + e, 0] && IGtQ[p, 0]
 

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.64

method	result	size
parts	\(-d^{3} a \left (\frac {1}{9} c^{6} x^{9}-\frac {3}{7} c^{4} x^{7}+\frac {3}{5} c^{2} x^{5}-\frac {1}{3} x^{3}\right )-\frac {d^{3} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{9} x^{9}}{9}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 c^{8} x^{8}-4675 c^{6} x^{6}+6297 c^{4} x^{4}-2629 c^{2} x^{2}-5258\right )}{99225}\right )}{c^{3}}\)	\(146\)
derivativedivides	\(\frac {-d^{3} a \left (\frac {1}{9} c^{9} x^{9}-\frac {3}{7} c^{7} x^{7}+\frac {3}{5} c^{5} x^{5}-\frac {1}{3} c^{3} x^{3}\right )-d^{3} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{9} x^{9}}{9}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 c^{8} x^{8}-4675 c^{6} x^{6}+6297 c^{4} x^{4}-2629 c^{2} x^{2}-5258\right )}{99225}\right )}{c^{3}}\)	\(150\)
default	\(\frac {-d^{3} a \left (\frac {1}{9} c^{9} x^{9}-\frac {3}{7} c^{7} x^{7}+\frac {3}{5} c^{5} x^{5}-\frac {1}{3} c^{3} x^{3}\right )-d^{3} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{9} x^{9}}{9}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 c^{8} x^{8}-4675 c^{6} x^{6}+6297 c^{4} x^{4}-2629 c^{2} x^{2}-5258\right )}{99225}\right )}{c^{3}}\)	\(150\)
orering	\(\frac {\left (20825 c^{10} x^{10}-82375 c^{8} x^{8}+119261 c^{6} x^{6}-66701 c^{4} x^{4}-36806 c^{2} x^{2}+10516\right ) \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{99225 c^{4} \left (c x -1\right )^{2} \left (c x +1\right )^{2} x \left (c^{2} x^{2}-1\right )}-\frac {\left (1225 c^{8} x^{8}-4675 c^{6} x^{6}+6297 c^{4} x^{4}-2629 c^{2} x^{2}-5258\right ) \left (2 x \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )-6 x^{3} \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) c^{2} d +\frac {x^{2} \left (-c^{2} d \,x^{2}+d \right )^{3} b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{99225 c^{4} \left (c x -1\right )^{2} \left (c x +1\right )^{2} x^{2}}\)	\(242\)

Input:

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

Output:

-d^3*a*(1/9*c^6*x^9-3/7*c^4*x^7+3/5*c^2*x^5-1/3*x^3)-d^3*b/c^3*(1/9*arccos 
h(c*x)*c^9*x^9-3/7*arccosh(c*x)*c^7*x^7+3/5*arccosh(c*x)*c^5*x^5-1/3*c^3*x 
^3*arccosh(c*x)-1/99225*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(1225*c^8*x^8-4675*c^6 
*x^6+6297*c^4*x^4-2629*c^2*x^2-5258))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.83 \[ \int x^2 \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=-\frac {11025 \, a c^{9} d^{3} x^{9} - 42525 \, a c^{7} d^{3} x^{7} + 59535 \, a c^{5} d^{3} x^{5} - 33075 \, a c^{3} d^{3} x^{3} + 315 \, {\left (35 \, b c^{9} d^{3} x^{9} - 135 \, b c^{7} d^{3} x^{7} + 189 \, b c^{5} d^{3} x^{5} - 105 \, b c^{3} d^{3} x^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (1225 \, b c^{8} d^{3} x^{8} - 4675 \, b c^{6} d^{3} x^{6} + 6297 \, b c^{4} d^{3} x^{4} - 2629 \, b c^{2} d^{3} x^{2} - 5258 \, b d^{3}\right )} \sqrt {c^{2} x^{2} - 1}}{99225 \, c^{3}} \] Input:

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

Output:

-1/99225*(11025*a*c^9*d^3*x^9 - 42525*a*c^7*d^3*x^7 + 59535*a*c^5*d^3*x^5 
- 33075*a*c^3*d^3*x^3 + 315*(35*b*c^9*d^3*x^9 - 135*b*c^7*d^3*x^7 + 189*b* 
c^5*d^3*x^5 - 105*b*c^3*d^3*x^3)*log(c*x + sqrt(c^2*x^2 - 1)) - (1225*b*c^ 
8*d^3*x^8 - 4675*b*c^6*d^3*x^6 + 6297*b*c^4*d^3*x^4 - 2629*b*c^2*d^3*x^2 - 
 5258*b*d^3)*sqrt(c^2*x^2 - 1))/c^3
 

Sympy [F]

\[ \int x^2 \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=- d^{3} \left (\int \left (- a x^{2}\right )\, dx + \int 3 a c^{2} x^{4}\, dx + \int \left (- 3 a c^{4} x^{6}\right )\, dx + \int a c^{6} x^{8}\, dx + \int \left (- b x^{2} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int 3 b c^{2} x^{4} \operatorname {acosh}{\left (c x \right )}\, dx + \int \left (- 3 b c^{4} x^{6} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int b c^{6} x^{8} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \] Input:

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

Output:

-d**3*(Integral(-a*x**2, x) + Integral(3*a*c**2*x**4, x) + Integral(-3*a*c 
**4*x**6, x) + Integral(a*c**6*x**8, x) + Integral(-b*x**2*acosh(c*x), x) 
+ Integral(3*b*c**2*x**4*acosh(c*x), x) + Integral(-3*b*c**4*x**6*acosh(c* 
x), x) + Integral(b*c**6*x**8*acosh(c*x), x))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 388 vs. \(2 (189) = 378\).

Time = 0.04 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.71 \[ \int x^2 \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=-\frac {1}{9} \, a c^{6} d^{3} x^{9} + \frac {3}{7} \, a c^{4} d^{3} x^{7} - \frac {1}{2835} \, {\left (315 \, x^{9} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {35 \, \sqrt {c^{2} x^{2} - 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} - 1}}{c^{10}}\right )} c\right )} b c^{6} d^{3} - \frac {3}{5} \, a c^{2} d^{3} x^{5} + \frac {3}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b c^{4} d^{3} - \frac {1}{25} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b c^{2} d^{3} + \frac {1}{3} \, a d^{3} x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b d^{3} \] Input:

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

Output:

-1/9*a*c^6*d^3*x^9 + 3/7*a*c^4*d^3*x^7 - 1/2835*(315*x^9*arccosh(c*x) - (3 
5*sqrt(c^2*x^2 - 1)*x^8/c^2 + 40*sqrt(c^2*x^2 - 1)*x^6/c^4 + 48*sqrt(c^2*x 
^2 - 1)*x^4/c^6 + 64*sqrt(c^2*x^2 - 1)*x^2/c^8 + 128*sqrt(c^2*x^2 - 1)/c^1 
0)*c)*b*c^6*d^3 - 3/5*a*c^2*d^3*x^5 + 3/245*(35*x^7*arccosh(c*x) - (5*sqrt 
(c^2*x^2 - 1)*x^6/c^2 + 6*sqrt(c^2*x^2 - 1)*x^4/c^4 + 8*sqrt(c^2*x^2 - 1)* 
x^2/c^6 + 16*sqrt(c^2*x^2 - 1)/c^8)*c)*b*c^4*d^3 - 1/25*(15*x^5*arccosh(c* 
x) - (3*sqrt(c^2*x^2 - 1)*x^4/c^2 + 4*sqrt(c^2*x^2 - 1)*x^2/c^4 + 8*sqrt(c 
^2*x^2 - 1)/c^6)*c)*b*c^2*d^3 + 1/3*a*d^3*x^3 + 1/9*(3*x^3*arccosh(c*x) - 
c*(sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1)/c^4))*b*d^3
 

Giac [F(-2)]

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

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.82 \[ \int x^2 \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {d^{3} \left (-11025 \mathit {acosh} \left (c x \right ) b \,c^{9} x^{9}+42525 \mathit {acosh} \left (c x \right ) b \,c^{7} x^{7}-59535 \mathit {acosh} \left (c x \right ) b \,c^{5} x^{5}+33075 \mathit {acosh} \left (c x \right ) b \,c^{3} x^{3}+1225 \sqrt {c^{2} x^{2}-1}\, b \,c^{8} x^{8}-4675 \sqrt {c^{2} x^{2}-1}\, b \,c^{6} x^{6}+6297 \sqrt {c^{2} x^{2}-1}\, b \,c^{4} x^{4}-2629 \sqrt {c^{2} x^{2}-1}\, b \,c^{2} x^{2}-5258 \sqrt {c^{2} x^{2}-1}\, b -11025 a \,c^{9} x^{9}+42525 a \,c^{7} x^{7}-59535 a \,c^{5} x^{5}+33075 a \,c^{3} x^{3}\right )}{99225 c^{3}} \] Input:

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

Output:

(d**3*( - 11025*acosh(c*x)*b*c**9*x**9 + 42525*acosh(c*x)*b*c**7*x**7 - 59 
535*acosh(c*x)*b*c**5*x**5 + 33075*acosh(c*x)*b*c**3*x**3 + 1225*sqrt(c**2 
*x**2 - 1)*b*c**8*x**8 - 4675*sqrt(c**2*x**2 - 1)*b*c**6*x**6 + 6297*sqrt( 
c**2*x**2 - 1)*b*c**4*x**4 - 2629*sqrt(c**2*x**2 - 1)*b*c**2*x**2 - 5258*s 
qrt(c**2*x**2 - 1)*b - 11025*a*c**9*x**9 + 42525*a*c**7*x**7 - 59535*a*c** 
5*x**5 + 33075*a*c**3*x**3))/(99225*c**3)