\(\int \frac {(d-c^2 d x^2)^{3/2} (a+b \text {arccosh}(c x))}{x^8} \, dx\) [74]

Optimal result

Integrand size = 27, antiderivative size = 247 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=-\frac {b c d \sqrt {d-c^2 d x^2}}{42 x^6 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 d \sqrt {d-c^2 d x^2}}{35 x^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{70 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{7 d x^7}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{35 d x^5}+\frac {2 b c^7 d \sqrt {d-c^2 d x^2} \log (x)}{35 \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-1/42*b*c*d*(-c^2*d*x^2+d)^(1/2)/x^6/(c*x-1)^(1/2)/(c*x+1)^(1/2)+2/35*b*c^ 
3*d*(-c^2*d*x^2+d)^(1/2)/x^4/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/70*b*c^5*d*(-c^ 
2*d*x^2+d)^(1/2)/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/7*(-c^2*d*x^2+d)^(5/2)* 
(a+b*arccosh(c*x))/d/x^7-2/35*c^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/ 
d/x^5+2/35*b*c^7*d*(-c^2*d*x^2+d)^(1/2)*ln(x)/(c*x-1)^(1/2)/(c*x+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.55 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=-\frac {d \sqrt {d-c^2 d x^2} \left (30 (-1+c x)^{5/2} (1+c x)^{5/2} (a+b \text {arccosh}(c x))+12 c^2 x^2 (-1+c x)^{5/2} (1+c x)^{5/2} (a+b \text {arccosh}(c x))+b c x \left (5-12 c^2 x^2+3 c^4 x^4-12 c^6 x^6 \log (x)\right )\right )}{210 x^7 \sqrt {-1+c x} \sqrt {1+c x}} \] Input:

Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^8,x]
 

Output:

-1/210*(d*Sqrt[d - c^2*d*x^2]*(30*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b* 
ArcCosh[c*x]) + 12*c^2*x^2*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh 
[c*x]) + b*c*x*(5 - 12*c^2*x^2 + 3*c^4*x^4 - 12*c^6*x^6*Log[x])))/(x^7*Sqr 
t[-1 + c*x]*Sqrt[1 + c*x])
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.58, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6337, 27, 354, 85, 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.

\(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^8} \, dx\)

\(\Big \downarrow \) 6337

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 354

\(\displaystyle \frac {b c d \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2 \left (2 c^2 x^2+5\right )}{x^8}dx^2}{70 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{7 d x^7}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{35 d x^5}\)

\(\Big \downarrow \) 85

\(\displaystyle \frac {b c d \sqrt {d-c^2 d x^2} \int \left (\frac {2 c^6}{x^2}+\frac {c^4}{x^4}-\frac {8 c^2}{x^6}+\frac {5}{x^8}\right )dx^2}{70 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{7 d x^7}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{35 d x^5}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{7 d x^7}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{35 d x^5}+\frac {b c d \sqrt {d-c^2 d x^2} \left (2 c^6 \log \left (x^2\right )-\frac {c^4}{x^2}+\frac {4 c^2}{x^4}-\frac {5}{3 x^6}\right )}{70 \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

Int[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^8,x]
 

Output:

-1/7*((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(d*x^7) - (2*c^2*(d - c^ 
2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(35*d*x^5) + (b*c*d*Sqrt[d - c^2*d*x^ 
2]*(-5/(3*x^6) + (4*c^2)/x^4 - c^4/x^2 + 2*c^6*Log[x^2]))/(70*Sqrt[-1 + c* 
x]*Sqrt[1 + c*x])
 

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 85

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : 
> Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, 
 d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && NeQ[b*e + a* 
f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n 
 + p + 2, 0] && RationalQ[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 
1])
 

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

rule 2009

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

rule 6337

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_ 
), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcCo 
sh[c*x])   u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c 
*x])]   Int[SimplifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b 
, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] 
 && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3144\) vs. \(2(207)=414\).

Time = 0.48 (sec) , antiderivative size = 3145, normalized size of antiderivative = 12.73

method	result	size
default	\(\text {Expression too large to display}\)	\(3145\)
parts	\(\text {Expression too large to display}\)	\(3145\)

Input:

int((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^8,x,method=_RETURNVERBOSE)
 

Output:

-116/105*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+15 
4*c^4*x^4-105*c^2*x^2+25)*x^5*c^12-5/21*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^1 
0*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x*c^8-2/35*b*(-d* 
(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c 
^2*x^2+25)*x^11*c^18+20/21*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8 
*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^3*c^10+1/5*b*(-d*(c^2*x^2-1) 
)^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)* 
x^9*c^16+26/105*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6 
*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^7*c^14-2*b*(-d*(c^2*x^2-1))^(1/2)*d/(35 
*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^11/(c*x-1)/ 
(c*x+1)*arccosh(c*x)*c^18+3*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^ 
8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x^9/(c*x-1)/(c*x+1)*arccosh(c 
*x)*c^16+12*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6 
+154*c^4*x^4-105*c^2*x^2+25)*x^7/(c*x-1)/(c*x+1)*arccosh(c*x)*c^14-164/5*b 
*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4- 
105*c^2*x^2+25)*x^5/(c*x-1)/(c*x+1)*arccosh(c*x)*c^12+52/5*b*(-d*(c^2*x^2- 
1))^(1/2)*d/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25 
)*x^3/(c*x-1)/(c*x+1)*arccosh(c*x)*c^10+1966/35*b*(-d*(c^2*x^2-1))^(1/2)*d 
/(35*c^10*x^10-35*c^8*x^8-70*c^6*x^6+154*c^4*x^4-105*c^2*x^2+25)*x/(c*x-1) 
/(c*x+1)*arccosh(c*x)*c^8-3272/35*b*(-d*(c^2*x^2-1))^(1/2)*d/(35*c^10*x...
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 649, normalized size of antiderivative = 2.63 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=\left [-\frac {6 \, {\left (2 \, b c^{8} d x^{8} - b c^{6} d x^{6} - 9 \, b c^{4} d x^{4} + 13 \, b c^{2} d x^{2} - 5 \, b d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 6 \, {\left (b c^{9} d x^{9} - b c^{7} d x^{7}\right )} \sqrt {-d} \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} - \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{4} - 1\right )} \sqrt {-d} - d}{c^{2} x^{4} - x^{2}}\right ) + {\left (3 \, b c^{5} d x^{5} - {\left (3 \, b c^{5} - 12 \, b c^{3} + 5 \, b c\right )} d x^{7} - 12 \, b c^{3} d x^{3} + 5 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 6 \, {\left (2 \, a c^{8} d x^{8} - a c^{6} d x^{6} - 9 \, a c^{4} d x^{4} + 13 \, a c^{2} d x^{2} - 5 \, a d\right )} \sqrt {-c^{2} d x^{2} + d}}{210 \, {\left (c^{2} x^{9} - x^{7}\right )}}, \frac {12 \, {\left (b c^{9} d x^{9} - b c^{7} d x^{7}\right )} \sqrt {d} \arctan \left (\frac {\sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{2} - 1\right )} \sqrt {d}}{c^{2} d x^{4} + {\left (c^{2} - 1\right )} d x^{2} - d}\right ) - 6 \, {\left (2 \, b c^{8} d x^{8} - b c^{6} d x^{6} - 9 \, b c^{4} d x^{4} + 13 \, b c^{2} d x^{2} - 5 \, b d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (3 \, b c^{5} d x^{5} - {\left (3 \, b c^{5} - 12 \, b c^{3} + 5 \, b c\right )} d x^{7} - 12 \, b c^{3} d x^{3} + 5 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 6 \, {\left (2 \, a c^{8} d x^{8} - a c^{6} d x^{6} - 9 \, a c^{4} d x^{4} + 13 \, a c^{2} d x^{2} - 5 \, a d\right )} \sqrt {-c^{2} d x^{2} + d}}{210 \, {\left (c^{2} x^{9} - x^{7}\right )}}\right ] \] Input:

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^8,x, algorithm="fricas 
")
 

Output:

[-1/210*(6*(2*b*c^8*d*x^8 - b*c^6*d*x^6 - 9*b*c^4*d*x^4 + 13*b*c^2*d*x^2 - 
 5*b*d)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - 6*(b*c^9*d*x^9 
 - b*c^7*d*x^7)*sqrt(-d)*log((c^2*d*x^6 + c^2*d*x^2 - d*x^4 - sqrt(-c^2*d* 
x^2 + d)*sqrt(c^2*x^2 - 1)*(x^4 - 1)*sqrt(-d) - d)/(c^2*x^4 - x^2)) + (3*b 
*c^5*d*x^5 - (3*b*c^5 - 12*b*c^3 + 5*b*c)*d*x^7 - 12*b*c^3*d*x^3 + 5*b*c*d 
*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 6*(2*a*c^8*d*x^8 - a*c^6*d*x^ 
6 - 9*a*c^4*d*x^4 + 13*a*c^2*d*x^2 - 5*a*d)*sqrt(-c^2*d*x^2 + d))/(c^2*x^9 
 - x^7), 1/210*(12*(b*c^9*d*x^9 - b*c^7*d*x^7)*sqrt(d)*arctan(sqrt(-c^2*d* 
x^2 + d)*sqrt(c^2*x^2 - 1)*(x^2 - 1)*sqrt(d)/(c^2*d*x^4 + (c^2 - 1)*d*x^2 
- d)) - 6*(2*b*c^8*d*x^8 - b*c^6*d*x^6 - 9*b*c^4*d*x^4 + 13*b*c^2*d*x^2 - 
5*b*d)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - (3*b*c^5*d*x^5 
- (3*b*c^5 - 12*b*c^3 + 5*b*c)*d*x^7 - 12*b*c^3*d*x^3 + 5*b*c*d*x)*sqrt(-c 
^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 6*(2*a*c^8*d*x^8 - a*c^6*d*x^6 - 9*a*c^4 
*d*x^4 + 13*a*c^2*d*x^2 - 5*a*d)*sqrt(-c^2*d*x^2 + d))/(c^2*x^9 - x^7)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=\text {Timed out} \] Input:

integrate((-c**2*d*x**2+d)**(3/2)*(a+b*acosh(c*x))/x**8,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.66 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=\frac {1}{210} \, {\left (12 \, c^{6} \sqrt {-d} d \log \left (x\right ) - \frac {3 \, c^{4} \sqrt {-d} d x^{4} - 12 \, c^{2} \sqrt {-d} d x^{2} + 5 \, \sqrt {-d} d}{x^{6}}\right )} b c - \frac {1}{35} \, b {\left (\frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{5}} + \frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{7}}\right )} \operatorname {arcosh}\left (c x\right ) - \frac {1}{35} \, a {\left (\frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{5}} + \frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{7}}\right )} \] Input:

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^8,x, algorithm="maxima 
")
 

Output:

1/210*(12*c^6*sqrt(-d)*d*log(x) - (3*c^4*sqrt(-d)*d*x^4 - 12*c^2*sqrt(-d)* 
d*x^2 + 5*sqrt(-d)*d)/x^6)*b*c - 1/35*b*(2*(-c^2*d*x^2 + d)^(5/2)*c^2/(d*x 
^5) + 5*(-c^2*d*x^2 + d)^(5/2)/(d*x^7))*arccosh(c*x) - 1/35*a*(2*(-c^2*d*x 
^2 + d)^(5/2)*c^2/(d*x^5) + 5*(-c^2*d*x^2 + d)^(5/2)/(d*x^7))
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^8,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^8} \,d x \] Input:

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(3/2))/x^8,x)
 

Output:

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(3/2))/x^8, x)
 

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=\frac {\sqrt {d}\, d \left (-2 \sqrt {-c^{2} x^{2}+1}\, a \,c^{6} x^{6}-\sqrt {-c^{2} x^{2}+1}\, a \,c^{4} x^{4}+8 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} x^{2}-5 \sqrt {-c^{2} x^{2}+1}\, a +35 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )}{x^{8}}d x \right ) b \,x^{7}-35 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )}{x^{6}}d x \right ) b \,c^{2} x^{7}\right )}{35 x^{7}} \] Input:

int((-c^2*d*x^2+d)^(3/2)*(a+b*acosh(c*x))/x^8,x)
 

Output:

(sqrt(d)*d*( - 2*sqrt( - c**2*x**2 + 1)*a*c**6*x**6 - sqrt( - c**2*x**2 + 
1)*a*c**4*x**4 + 8*sqrt( - c**2*x**2 + 1)*a*c**2*x**2 - 5*sqrt( - c**2*x** 
2 + 1)*a + 35*int((sqrt( - c**2*x**2 + 1)*acosh(c*x))/x**8,x)*b*x**7 - 35* 
int((sqrt( - c**2*x**2 + 1)*acosh(c*x))/x**6,x)*b*c**2*x**7))/(35*x**7)