\(\int \frac {(d-c^2 d x^2)^{5/2} (a+b \text {arccosh}(c x))}{x^8} \, dx\) [90]

Optimal result

Integrand size = 27, antiderivative size = 219 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{42 x^6 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c^3 d^2 \sqrt {d-c^2 d x^2}}{28 x^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c^5 d^2 \sqrt {d-c^2 d x^2}}{14 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 d x^7}-\frac {b c^7 d^2 \sqrt {d-c^2 d x^2} \log (x)}{7 \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-1/42*b*c*d^2*(-c^2*d*x^2+d)^(1/2)/x^6/(c*x-1)^(1/2)/(c*x+1)^(1/2)+3/28*b* 
c^3*d^2*(-c^2*d*x^2+d)^(1/2)/x^4/(c*x-1)^(1/2)/(c*x+1)^(1/2)-3/14*b*c^5*d^ 
2*(-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)^ 
(7/2)*(a+b*arccosh(c*x))/d/x^7-1/7*b*c^7*d^2*(-c^2*d*x^2+d)^(1/2)*ln(x)/(c 
*x-1)^(1/2)/(c*x+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.48 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (12 (-1+c x)^{7/2} (1+c x)^{7/2} (a+b \text {arccosh}(c x))-b c x \left (2-9 c^2 x^2+18 c^4 x^4+12 c^6 x^6 \log (x)\right )\right )}{84 x^7 \sqrt {-1+c x} \sqrt {1+c x}} \] Input:

Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^8,x]
 

Output:

(d^2*Sqrt[d - c^2*d*x^2]*(12*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcCo 
sh[c*x]) - b*c*x*(2 - 9*c^2*x^2 + 18*c^4*x^4 + 12*c^6*x^6*Log[x])))/(84*x^ 
7*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.51, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6332, 25, 82, 243, 49, 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.

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

\(\Big \downarrow \) 6332

\(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \int -\frac {(1-c x)^3 (c x+1)^3}{x^7}dx}{7 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 d x^7}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {(1-c x)^3 (c x+1)^3}{x^7}dx}{7 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 d x^7}\)

\(\Big \downarrow \) 82

\(\displaystyle \frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^3}{x^7}dx}{7 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 d x^7}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^3}{x^8}dx^2}{14 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 d x^7}\)

\(\Big \downarrow \) 49

\(\displaystyle \frac {b c d^2 \sqrt {d-c^2 d x^2} \int \left (-\frac {c^6}{x^2}+\frac {3 c^4}{x^4}-\frac {3 c^2}{x^6}+\frac {1}{x^8}\right )dx^2}{14 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 d x^7}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^6 \left (-\log \left (x^2\right )\right )-\frac {3 c^4}{x^2}+\frac {3 c^2}{2 x^4}-\frac {1}{3 x^6}\right )}{14 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 d x^7}\)

Input:

Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^8,x]
 

Output:

-1/7*((d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(d*x^7) + (b*c*d^2*Sqrt[ 
d - c^2*d*x^2]*(-1/3*1/x^6 + (3*c^2)/(2*x^4) - (3*c^4)/x^2 - c^6*Log[x^2]) 
)/(14*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 49

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

rule 82

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

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 2009

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

rule 6332

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3774\) vs. \(2(183)=366\).

Time = 0.48 (sec) , antiderivative size = 3775, normalized size of antiderivative = 17.24

method	result	size
default	\(\text {Expression too large to display}\)	\(3775\)
parts	\(\text {Expression too large to display}\)	\(3775\)

Input:

int((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^8,x,method=_RETURNVERBOSE)
 

Output:

55/12*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35 
*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*c^7+2/7*b*(-d 
*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*d^2*c^7-1/7*b 
*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*ln(1+(c*x+(c*x-1)^(1/2 
)*(c*x+1)^(1/2))^2)*d^2*c^7-1/7*a/d/x^7*(-c^2*d*x^2+d)^(7/2)-3/2*b*(-d*(c^ 
2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4 
*x^4-7*c^2*x^2+1)*x^10/(c*x-1)^(1/2)/(c*x+1)^(1/2)*c^17+73/42*b*(-d*(c^2*x 
^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^ 
4-7*c^2*x^2+1)*x^9/(c*x-1)/(c*x+1)*c^16-67/42*b*(-d*(c^2*x^2-1))^(1/2)*d^2 
/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x 
^7/(c*x-1)/(c*x+1)*c^14+11/14*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21 
*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^5/(c*x-1)/(c*x+ 
1)*c^12-17/84*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^ 
8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^3/(c*x-1)/(c*x+1)*c^10+1/42*b*( 
-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+ 
21*c^4*x^4-7*c^2*x^2+1)*x/(c*x-1)/(c*x+1)*c^8+47/4*b*(-d*(c^2*x^2-1))^(1/2 
)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2 
+1)*x^4/(c*x-1)^(1/2)/(c*x+1)^(1/2)*c^11-119/12*b*(-d*(c^2*x^2-1))^(1/2)*d 
^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1) 
*x^6/(c*x-1)^(1/2)/(c*x+1)^(1/2)*c^13+23/84*b*(-d*(c^2*x^2-1))^(1/2)*d^...
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 704, normalized size of antiderivative = 3.21 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=\left [\frac {12 \, {\left (b c^{8} d^{2} x^{8} - 4 \, b c^{6} d^{2} x^{6} + 6 \, b c^{4} d^{2} x^{4} - 4 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 6 \, {\left (b c^{9} d^{2} x^{9} - b c^{7} d^{2} x^{7}\right )} \sqrt {-d} \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} + \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{4} - 1\right )} \sqrt {-d} - d}{c^{2} x^{4} - x^{2}}\right ) - {\left (18 \, b c^{5} d^{2} x^{5} - {\left (18 \, b c^{5} - 9 \, b c^{3} + 2 \, b c\right )} d^{2} x^{7} - 9 \, b c^{3} d^{2} x^{3} + 2 \, b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 12 \, {\left (a c^{8} d^{2} x^{8} - 4 \, a c^{6} d^{2} x^{6} + 6 \, a c^{4} d^{2} x^{4} - 4 \, a c^{2} d^{2} x^{2} + a d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{84 \, {\left (c^{2} x^{9} - x^{7}\right )}}, -\frac {12 \, {\left (b c^{9} d^{2} x^{9} - b c^{7} d^{2} x^{7}\right )} \sqrt {d} \arctan \left (\frac {\sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{2} - 1\right )} \sqrt {d}}{c^{2} d x^{4} + {\left (c^{2} - 1\right )} d x^{2} - d}\right ) - 12 \, {\left (b c^{8} d^{2} x^{8} - 4 \, b c^{6} d^{2} x^{6} + 6 \, b c^{4} d^{2} x^{4} - 4 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (18 \, b c^{5} d^{2} x^{5} - {\left (18 \, b c^{5} - 9 \, b c^{3} + 2 \, b c\right )} d^{2} x^{7} - 9 \, b c^{3} d^{2} x^{3} + 2 \, b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 12 \, {\left (a c^{8} d^{2} x^{8} - 4 \, a c^{6} d^{2} x^{6} + 6 \, a c^{4} d^{2} x^{4} - 4 \, a c^{2} d^{2} x^{2} + a d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{84 \, {\left (c^{2} x^{9} - x^{7}\right )}}\right ] \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^8,x, algorithm="fricas 
")
 

Output:

[1/84*(12*(b*c^8*d^2*x^8 - 4*b*c^6*d^2*x^6 + 6*b*c^4*d^2*x^4 - 4*b*c^2*d^2 
*x^2 + b*d^2)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) + 6*(b*c^9 
*d^2*x^9 - b*c^7*d^2*x^7)*sqrt(-d)*log((c^2*d*x^6 + c^2*d*x^2 - d*x^4 + sq 
rt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*(x^4 - 1)*sqrt(-d) - d)/(c^2*x^4 - x^ 
2)) - (18*b*c^5*d^2*x^5 - (18*b*c^5 - 9*b*c^3 + 2*b*c)*d^2*x^7 - 9*b*c^3*d 
^2*x^3 + 2*b*c*d^2*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 12*(a*c^8*d 
^2*x^8 - 4*a*c^6*d^2*x^6 + 6*a*c^4*d^2*x^4 - 4*a*c^2*d^2*x^2 + a*d^2)*sqrt 
(-c^2*d*x^2 + d))/(c^2*x^9 - x^7), -1/84*(12*(b*c^9*d^2*x^9 - b*c^7*d^2*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)) - 12*(b*c^8*d^2*x^8 - 4*b*c^6*d^2*x^6 
+ 6*b*c^4*d^2*x^4 - 4*b*c^2*d^2*x^2 + b*d^2)*sqrt(-c^2*d*x^2 + d)*log(c*x 
+ sqrt(c^2*x^2 - 1)) + (18*b*c^5*d^2*x^5 - (18*b*c^5 - 9*b*c^3 + 2*b*c)*d^ 
2*x^7 - 9*b*c^3*d^2*x^3 + 2*b*c*d^2*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 
 1) - 12*(a*c^8*d^2*x^8 - 4*a*c^6*d^2*x^6 + 6*a*c^4*d^2*x^4 - 4*a*c^2*d^2* 
x^2 + a*d^2)*sqrt(-c^2*d*x^2 + d))/(c^2*x^9 - x^7)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=\text {Timed out} \] Input:

integrate((-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x))/x**8,x)
 

Output:

Timed out
 

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.14 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.02 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=\frac {{\left (6 \, c^{8} d^{4} \sqrt {-\frac {1}{c^{4} d}} \log \left (x^{2} - \frac {1}{c^{2}}\right ) + 6 i \, \left (-1\right )^{-2 \, c^{2} d x^{2} + 2 \, d} c^{6} d^{\frac {7}{2}} \log \left (-2 \, c^{2} d + \frac {2 \, d}{x^{2}}\right ) + \frac {11 \, \sqrt {-c^{4} d x^{4} + 2 \, c^{2} d x^{2} - d} c^{4} d^{3}}{x^{2}} - \frac {7 \, \sqrt {-c^{4} d x^{4} + 2 \, c^{2} d x^{2} - d} c^{2} d^{3}}{x^{4}} + \frac {2 \, \sqrt {-c^{4} d x^{4} + 2 \, c^{2} d x^{2} - d} d^{3}}{x^{6}}\right )} b c}{84 \, d} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} b \operatorname {arcosh}\left (c x\right )}{7 \, d x^{7}} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} a}{7 \, d x^{7}} \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^8,x, algorithm="maxima 
")
                                                                                    
                                                                                    
 

Output:

1/84*(6*c^8*d^4*sqrt(-1/(c^4*d))*log(x^2 - 1/c^2) + 6*I*(-1)^(-2*c^2*d*x^2 
 + 2*d)*c^6*d^(7/2)*log(-2*c^2*d + 2*d/x^2) + 11*sqrt(-c^4*d*x^4 + 2*c^2*d 
*x^2 - d)*c^4*d^3/x^2 - 7*sqrt(-c^4*d*x^4 + 2*c^2*d*x^2 - d)*c^2*d^3/x^4 + 
 2*sqrt(-c^4*d*x^4 + 2*c^2*d*x^2 - d)*d^3/x^6)*b*c/d - 1/7*(-c^2*d*x^2 + d 
)^(7/2)*b*arccosh(c*x)/(d*x^7) - 1/7*(-c^2*d*x^2 + d)^(7/2)*a/(d*x^7)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(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. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^8} \,d x \] Input:

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2))/x^8,x)
 

Output:

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2))/x^8, x)
 

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^8} \, dx=\frac {\sqrt {d}\, d^{2} \left (\sqrt {-c^{2} x^{2}+1}\, a \,c^{6} x^{6}-3 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} x^{4}+3 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}\, a +7 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )}{x^{8}}d x \right ) b \,x^{7}-14 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )}{x^{6}}d x \right ) b \,c^{2} x^{7}+7 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )}{x^{4}}d x \right ) b \,c^{4} x^{7}\right )}{7 x^{7}} \] Input:

int((-c^2*d*x^2+d)^(5/2)*(a+b*acosh(c*x))/x^8,x)
 

Output:

(sqrt(d)*d**2*(sqrt( - c**2*x**2 + 1)*a*c**6*x**6 - 3*sqrt( - c**2*x**2 + 
1)*a*c**4*x**4 + 3*sqrt( - c**2*x**2 + 1)*a*c**2*x**2 - sqrt( - c**2*x**2 
+ 1)*a + 7*int((sqrt( - c**2*x**2 + 1)*acosh(c*x))/x**8,x)*b*x**7 - 14*int 
((sqrt( - c**2*x**2 + 1)*acosh(c*x))/x**6,x)*b*c**2*x**7 + 7*int((sqrt( - 
c**2*x**2 + 1)*acosh(c*x))/x**4,x)*b*c**4*x**7))/(7*x**7)