\(\int x^4 (d-c^2 d x^2)^2 (a+b \text {arccosh}(c x)) \, dx\) [10]

Optimal result

Integrand size = 25, antiderivative size = 206 \[ \int x^4 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=-\frac {8 b d^2 \sqrt {-1+c x} \sqrt {1+c x}}{315 c^5}+\frac {4 b d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{945 c^5}-\frac {b d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{525 c^5}-\frac {10 b d^2 (-1+c x)^{7/2} (1+c x)^{7/2}}{441 c^5}-\frac {b d^2 (-1+c x)^{9/2} (1+c x)^{9/2}}{81 c^5}+\frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 d^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{9} c^4 d^2 x^9 (a+b \text {arccosh}(c x)) \] Output:

-8/315*b*d^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^5+4/945*b*d^2*(c*x-1)^(3/2)*(c* 
x+1)^(3/2)/c^5-1/525*b*d^2*(c*x-1)^(5/2)*(c*x+1)^(5/2)/c^5-10/441*b*d^2*(c 
*x-1)^(7/2)*(c*x+1)^(7/2)/c^5-1/81*b*d^2*(c*x-1)^(9/2)*(c*x+1)^(9/2)/c^5+1 
/5*d^2*x^5*(a+b*arccosh(c*x))-2/7*c^2*d^2*x^7*(a+b*arccosh(c*x))+1/9*c^4*d 
^2*x^9*(a+b*arccosh(c*x))
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.60 \[ \int x^4 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {d^2 \left (315 a c^5 x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right )-b \sqrt {-1+c x} \sqrt {1+c x} \left (2104+1052 c^2 x^2+789 c^4 x^4-2650 c^6 x^6+1225 c^8 x^8\right )+315 b c^5 x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right ) \text {arccosh}(c x)\right )}{99225 c^5} \] Input:

Integrate[x^4*(d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x]),x]
 

Output:

(d^2*(315*a*c^5*x^5*(63 - 90*c^2*x^2 + 35*c^4*x^4) - b*Sqrt[-1 + c*x]*Sqrt 
[1 + c*x]*(2104 + 1052*c^2*x^2 + 789*c^4*x^4 - 2650*c^6*x^6 + 1225*c^8*x^8 
) + 315*b*c^5*x^5*(63 - 90*c^2*x^2 + 35*c^4*x^4)*ArcCosh[c*x]))/(99225*c^5 
)
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 200, 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, 1905, 1578, 1195, 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.

\(\displaystyle \int x^4 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx\)

\(\Big \downarrow \) 6336

\(\displaystyle -b c \int \frac {d^2 x^5 \left (35 c^4 x^4-90 c^2 x^2+63\right )}{315 \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{9} c^4 d^2 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 d^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{315} b c d^2 \int \frac {x^5 \left (35 c^4 x^4-90 c^2 x^2+63\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{9} c^4 d^2 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 d^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 1905

\(\displaystyle -\frac {b c d^2 \sqrt {c^2 x^2-1} \int \frac {x^5 \left (35 c^4 x^4-90 c^2 x^2+63\right )}{\sqrt {c^2 x^2-1}}dx}{315 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{9} c^4 d^2 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 d^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 1578

\(\displaystyle -\frac {b c d^2 \sqrt {c^2 x^2-1} \int \frac {x^4 \left (35 c^4 x^4-90 c^2 x^2+63\right )}{\sqrt {c^2 x^2-1}}dx^2}{630 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{9} c^4 d^2 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 d^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 1195

\(\displaystyle -\frac {b c d^2 \sqrt {c^2 x^2-1} \int \left (\frac {35 \left (c^2 x^2-1\right )^{7/2}}{c^4}+\frac {50 \left (c^2 x^2-1\right )^{5/2}}{c^4}+\frac {3 \left (c^2 x^2-1\right )^{3/2}}{c^4}-\frac {4 \sqrt {c^2 x^2-1}}{c^4}+\frac {8}{c^4 \sqrt {c^2 x^2-1}}\right )dx^2}{630 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{9} c^4 d^2 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 d^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{9} c^4 d^2 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 d^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} d^2 x^5 (a+b \text {arccosh}(c x))-\frac {b c d^2 \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

Int[x^4*(d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x]),x]
 

Output:

-1/630*(b*c*d^2*Sqrt[-1 + c^2*x^2]*((16*Sqrt[-1 + c^2*x^2])/c^6 - (8*(-1 + 
 c^2*x^2)^(3/2))/(3*c^6) + (6*(-1 + c^2*x^2)^(5/2))/(5*c^6) + (100*(-1 + c 
^2*x^2)^(7/2))/(7*c^6) + (70*(-1 + c^2*x^2)^(9/2))/(9*c^6)))/(Sqrt[-1 + c* 
x]*Sqrt[1 + c*x]) + (d^2*x^5*(a + b*ArcCosh[c*x]))/5 - (2*c^2*d^2*x^7*(a + 
 b*ArcCosh[c*x]))/7 + (c^4*d^2*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 1195

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && IGtQ[p, 0]
 

rule 1578

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ 
)^4)^(p_.), x_Symbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a 
+ b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int 
egerQ[(m - 1)/2]
 

rule 1905

Int[((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_)^(non2_.))^(q_.)*((d2_) + (e2_.) 
*(x_)^(non2_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_.), x 
_Symbol] :> Simp[(d1 + e1*x^(n/2))^FracPart[q]*((d2 + e2*x^(n/2))^FracPart[ 
q]/(d1*d2 + e1*e2*x^n)^FracPart[q])   Int[(f*x)^m*(d1*d2 + e1*e2*x^n)^q*(a 
+ b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, n, p, 
q}, x] && EqQ[n2, 2*n] && EqQ[non2, n/2] && EqQ[d2*e1 + d1*e2, 0]
 

rule 2009

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

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 = 124, normalized size of antiderivative = 0.60

method	result	size
parts	\(d^{2} a \left (\frac {1}{9} c^{4} x^{9}-\frac {2}{7} c^{2} x^{7}+\frac {1}{5} x^{5}\right )+\frac {d^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{9} x^{9}}{9}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 c^{8} x^{8}-2650 c^{6} x^{6}+789 c^{4} x^{4}+1052 c^{2} x^{2}+2104\right )}{99225}\right )}{c^{5}}\)	\(124\)
derivativedivides	\(\frac {d^{2} a \left (\frac {1}{9} c^{9} x^{9}-\frac {2}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+d^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{9} x^{9}}{9}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 c^{8} x^{8}-2650 c^{6} x^{6}+789 c^{4} x^{4}+1052 c^{2} x^{2}+2104\right )}{99225}\right )}{c^{5}}\)	\(128\)
default	\(\frac {d^{2} a \left (\frac {1}{9} c^{9} x^{9}-\frac {2}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+d^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{9} x^{9}}{9}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (1225 c^{8} x^{8}-2650 c^{6} x^{6}+789 c^{4} x^{4}+1052 c^{2} x^{2}+2104\right )}{99225}\right )}{c^{5}}\)	\(128\)
orering	\(\frac {\left (20825 c^{10} x^{10}-54450 c^{8} x^{8}+36757 c^{6} x^{6}+5260 c^{4} x^{4}+12624 c^{2} x^{2}-8416\right ) \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{99225 c^{6} \left (c x -1\right ) \left (c x +1\right ) x \left (c^{2} x^{2}-1\right )}-\frac {\left (1225 c^{8} x^{8}-2650 c^{6} x^{6}+789 c^{4} x^{4}+1052 c^{2} x^{2}+2104\right ) \left (4 x^{3} \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )-4 x^{5} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) c^{2} d +\frac {x^{4} \left (-c^{2} d \,x^{2}+d \right )^{2} b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{99225 c^{6} \left (c x -1\right ) \left (c x +1\right ) x^{4}}\)	\(242\)

Input:

int(x^4*(-c^2*d*x^2+d)^2*(a+b*arccosh(c*x)),x,method=_RETURNVERBOSE)
 

Output:

d^2*a*(1/9*c^4*x^9-2/7*c^2*x^7+1/5*x^5)+d^2*b/c^5*(1/9*arccosh(c*x)*c^9*x^ 
9-2/7*arccosh(c*x)*c^7*x^7+1/5*arccosh(c*x)*c^5*x^5-1/99225*(c*x-1)^(1/2)* 
(c*x+1)^(1/2)*(1225*c^8*x^8-2650*c^6*x^6+789*c^4*x^4+1052*c^2*x^2+2104))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.80 \[ \int x^4 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {11025 \, a c^{9} d^{2} x^{9} - 28350 \, a c^{7} d^{2} x^{7} + 19845 \, a c^{5} d^{2} x^{5} + 315 \, {\left (35 \, b c^{9} d^{2} x^{9} - 90 \, b c^{7} d^{2} x^{7} + 63 \, b c^{5} d^{2} x^{5}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (1225 \, b c^{8} d^{2} x^{8} - 2650 \, b c^{6} d^{2} x^{6} + 789 \, b c^{4} d^{2} x^{4} + 1052 \, b c^{2} d^{2} x^{2} + 2104 \, b d^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{99225 \, c^{5}} \] Input:

integrate(x^4*(-c^2*d*x^2+d)^2*(a+b*arccosh(c*x)),x, algorithm="fricas")
 

Output:

1/99225*(11025*a*c^9*d^2*x^9 - 28350*a*c^7*d^2*x^7 + 19845*a*c^5*d^2*x^5 + 
 315*(35*b*c^9*d^2*x^9 - 90*b*c^7*d^2*x^7 + 63*b*c^5*d^2*x^5)*log(c*x + sq 
rt(c^2*x^2 - 1)) - (1225*b*c^8*d^2*x^8 - 2650*b*c^6*d^2*x^6 + 789*b*c^4*d^ 
2*x^4 + 1052*b*c^2*d^2*x^2 + 2104*b*d^2)*sqrt(c^2*x^2 - 1))/c^5
 

Sympy [F]

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

integrate(x**4*(-c**2*d*x**2+d)**2*(a+b*acosh(c*x)),x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.55 \[ \int x^4 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{9} \, a c^{4} d^{2} x^{9} - \frac {2}{7} \, a c^{2} d^{2} x^{7} + \frac {1}{2835} \, {\left (315 \, x^{9} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {35 \, \sqrt {c^{2} x^{2} - 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} - 1}}{c^{10}}\right )} c\right )} b c^{4} d^{2} + \frac {1}{5} \, a d^{2} x^{5} - \frac {2}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b c^{2} d^{2} + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b d^{2} \] Input:

integrate(x^4*(-c^2*d*x^2+d)^2*(a+b*arccosh(c*x)),x, algorithm="maxima")
 

Output:

1/9*a*c^4*d^2*x^9 - 2/7*a*c^2*d^2*x^7 + 1/2835*(315*x^9*arccosh(c*x) - (35 
*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^10 
)*c)*b*c^4*d^2 + 1/5*a*d^2*x^5 - 2/245*(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^2*d^2 + 1/75*(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*d^2
 

Giac [F(-2)]

Exception generated. \[ \int x^4 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^4*(-c^2*d*x^2+d)^2*(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. \[ \int x^4 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\int x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^2 \,d x \] Input:

int(x^4*(a + b*acosh(c*x))*(d - c^2*d*x^2)^2,x)
 

Output:

int(x^4*(a + b*acosh(c*x))*(d - c^2*d*x^2)^2, x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.80 \[ \int x^4 \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {d^{2} \left (11025 \mathit {acosh} \left (c x \right ) b \,c^{9} x^{9}-28350 \mathit {acosh} \left (c x \right ) b \,c^{7} x^{7}+19845 \mathit {acosh} \left (c x \right ) b \,c^{5} x^{5}-1225 \sqrt {c^{2} x^{2}-1}\, b \,c^{8} x^{8}+2650 \sqrt {c^{2} x^{2}-1}\, b \,c^{6} x^{6}-789 \sqrt {c^{2} x^{2}-1}\, b \,c^{4} x^{4}-1052 \sqrt {c^{2} x^{2}-1}\, b \,c^{2} x^{2}-2104 \sqrt {c^{2} x^{2}-1}\, b +11025 a \,c^{9} x^{9}-28350 a \,c^{7} x^{7}+19845 a \,c^{5} x^{5}\right )}{99225 c^{5}} \] Input:

int(x^4*(-c^2*d*x^2+d)^2*(a+b*acosh(c*x)),x)
 

Output:

(d**2*(11025*acosh(c*x)*b*c**9*x**9 - 28350*acosh(c*x)*b*c**7*x**7 + 19845 
*acosh(c*x)*b*c**5*x**5 - 1225*sqrt(c**2*x**2 - 1)*b*c**8*x**8 + 2650*sqrt 
(c**2*x**2 - 1)*b*c**6*x**6 - 789*sqrt(c**2*x**2 - 1)*b*c**4*x**4 - 1052*s 
qrt(c**2*x**2 - 1)*b*c**2*x**2 - 2104*sqrt(c**2*x**2 - 1)*b + 11025*a*c**9 
*x**9 - 28350*a*c**7*x**7 + 19845*a*c**5*x**5))/(99225*c**5)