\(\int \frac {a+b \text {arccosh}(c x)}{(\pi -c^2 \pi x^2)^{7/2}} \, dx\) [44]

Optimal result

Integrand size = 24, antiderivative size = 225 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (\pi -c^2 \pi x^2\right )^{7/2}} \, dx=\frac {b \sqrt {-1+c x}}{20 c \pi ^{7/2} \sqrt {1-c x} \left (1-c^2 x^2\right )^2}+\frac {2 b \sqrt {-1+c x}}{15 c \pi ^{7/2} \sqrt {1-c x} \left (1-c^2 x^2\right )}+\frac {x (a+b \text {arccosh}(c x))}{5 \pi \left (\pi -c^2 \pi x^2\right )^{5/2}}+\frac {4 x (a+b \text {arccosh}(c x))}{15 \pi ^2 \left (\pi -c^2 \pi x^2\right )^{3/2}}+\frac {8 x (a+b \text {arccosh}(c x))}{15 \pi ^3 \sqrt {\pi -c^2 \pi x^2}}-\frac {4 b \sqrt {-1+c x} \log \left (1-c^2 x^2\right )}{15 c \pi ^{7/2} \sqrt {1-c x}} \] Output:

1/20*b*(c*x-1)^(1/2)/c/Pi^(7/2)/(-c*x+1)^(1/2)/(-c^2*x^2+1)^2+2/15*b*(c*x- 
1)^(1/2)/c/Pi^(7/2)/(-c*x+1)^(1/2)/(-c^2*x^2+1)+1/5*x*(a+b*arccosh(c*x))/P 
i/(-Pi*c^2*x^2+Pi)^(5/2)+4/15*x*(a+b*arccosh(c*x))/Pi^2/(-Pi*c^2*x^2+Pi)^( 
3/2)+8/15*x*(a+b*arccosh(c*x))/Pi^3/(-Pi*c^2*x^2+Pi)^(1/2)-4/15*b*(c*x-1)^ 
(1/2)*ln(-c^2*x^2+1)/c/Pi^(7/2)/(-c*x+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.75 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (\pi -c^2 \pi x^2\right )^{7/2}} \, dx=\frac {60 a c x-80 a c^3 x^3+32 a c^5 x^5+11 b \sqrt {-1+c x} \sqrt {1+c x}-8 b c^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}+4 b c x \left (15-20 c^2 x^2+8 c^4 x^4\right ) \text {arccosh}(c x)-16 b \sqrt {-1+c x} \sqrt {1+c x} \left (-1+c^2 x^2\right )^2 \log \left (1-c^2 x^2\right )}{60 c \pi ^{7/2} \left (1-c^2 x^2\right )^{5/2}} \] Input:

Integrate[(a + b*ArcCosh[c*x])/(Pi - c^2*Pi*x^2)^(7/2),x]
 

Output:

(60*a*c*x - 80*a*c^3*x^3 + 32*a*c^5*x^5 + 11*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x 
] - 8*b*c^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 4*b*c*x*(15 - 20*c^2*x^2 + 
8*c^4*x^4)*ArcCosh[c*x] - 16*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-1 + c^2*x^2) 
^2*Log[1 - c^2*x^2])/(60*c*Pi^(7/2)*(1 - c^2*x^2)^(5/2))
 

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.22, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6316, 25, 82, 241, 6316, 82, 241, 6314, 240}

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.

\(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{\left (\pi -\pi c^2 x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 6316

\(\displaystyle \frac {4 \int \frac {a+b \text {arccosh}(c x)}{\left (\pi -c^2 \pi x^2\right )^{5/2}}dx}{5 \pi }-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \int -\frac {x}{(1-c x)^3 (c x+1)^3}dx}{5 \pi ^3 \sqrt {\pi -\pi c^2 x^2}}+\frac {x (a+b \text {arccosh}(c x))}{5 \pi \left (\pi -\pi c^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {4 \int \frac {a+b \text {arccosh}(c x)}{\left (\pi -c^2 \pi x^2\right )^{5/2}}dx}{5 \pi }+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x}{(1-c x)^3 (c x+1)^3}dx}{5 \pi ^3 \sqrt {\pi -\pi c^2 x^2}}+\frac {x (a+b \text {arccosh}(c x))}{5 \pi \left (\pi -\pi c^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 82

\(\displaystyle \frac {4 \int \frac {a+b \text {arccosh}(c x)}{\left (\pi -c^2 \pi x^2\right )^{5/2}}dx}{5 \pi }+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x}{\left (1-c^2 x^2\right )^3}dx}{5 \pi ^3 \sqrt {\pi -\pi c^2 x^2}}+\frac {x (a+b \text {arccosh}(c x))}{5 \pi \left (\pi -\pi c^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 241

\(\displaystyle \frac {4 \int \frac {a+b \text {arccosh}(c x)}{\left (\pi -c^2 \pi x^2\right )^{5/2}}dx}{5 \pi }+\frac {x (a+b \text {arccosh}(c x))}{5 \pi \left (\pi -\pi c^2 x^2\right )^{5/2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{20 \pi ^3 c \left (1-c^2 x^2\right )^2 \sqrt {\pi -\pi c^2 x^2}}\)

\(\Big \downarrow \) 6316

\(\displaystyle \frac {4 \left (\frac {2 \int \frac {a+b \text {arccosh}(c x)}{\left (\pi -c^2 \pi x^2\right )^{3/2}}dx}{3 \pi }+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x}{(1-c x)^2 (c x+1)^2}dx}{3 \pi ^2 \sqrt {\pi -\pi c^2 x^2}}+\frac {x (a+b \text {arccosh}(c x))}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}\right )}{5 \pi }+\frac {x (a+b \text {arccosh}(c x))}{5 \pi \left (\pi -\pi c^2 x^2\right )^{5/2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{20 \pi ^3 c \left (1-c^2 x^2\right )^2 \sqrt {\pi -\pi c^2 x^2}}\)

\(\Big \downarrow \) 82

\(\displaystyle \frac {4 \left (\frac {2 \int \frac {a+b \text {arccosh}(c x)}{\left (\pi -c^2 \pi x^2\right )^{3/2}}dx}{3 \pi }+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x}{\left (1-c^2 x^2\right )^2}dx}{3 \pi ^2 \sqrt {\pi -\pi c^2 x^2}}+\frac {x (a+b \text {arccosh}(c x))}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}\right )}{5 \pi }+\frac {x (a+b \text {arccosh}(c x))}{5 \pi \left (\pi -\pi c^2 x^2\right )^{5/2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{20 \pi ^3 c \left (1-c^2 x^2\right )^2 \sqrt {\pi -\pi c^2 x^2}}\)

\(\Big \downarrow \) 241

\(\displaystyle \frac {4 \left (\frac {2 \int \frac {a+b \text {arccosh}(c x)}{\left (\pi -c^2 \pi x^2\right )^{3/2}}dx}{3 \pi }+\frac {x (a+b \text {arccosh}(c x))}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{6 \pi ^2 c \left (1-c^2 x^2\right ) \sqrt {\pi -\pi c^2 x^2}}\right )}{5 \pi }+\frac {x (a+b \text {arccosh}(c x))}{5 \pi \left (\pi -\pi c^2 x^2\right )^{5/2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{20 \pi ^3 c \left (1-c^2 x^2\right )^2 \sqrt {\pi -\pi c^2 x^2}}\)

\(\Big \downarrow \) 6314

\(\displaystyle \frac {4 \left (\frac {2 \left (\frac {b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x}{1-c^2 x^2}dx}{\pi \sqrt {\pi -\pi c^2 x^2}}+\frac {x (a+b \text {arccosh}(c x))}{\pi \sqrt {\pi -\pi c^2 x^2}}\right )}{3 \pi }+\frac {x (a+b \text {arccosh}(c x))}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{6 \pi ^2 c \left (1-c^2 x^2\right ) \sqrt {\pi -\pi c^2 x^2}}\right )}{5 \pi }+\frac {x (a+b \text {arccosh}(c x))}{5 \pi \left (\pi -\pi c^2 x^2\right )^{5/2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{20 \pi ^3 c \left (1-c^2 x^2\right )^2 \sqrt {\pi -\pi c^2 x^2}}\)

\(\Big \downarrow \) 240

\(\displaystyle \frac {x (a+b \text {arccosh}(c x))}{5 \pi \left (\pi -\pi c^2 x^2\right )^{5/2}}+\frac {4 \left (\frac {x (a+b \text {arccosh}(c x))}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))}{\pi \sqrt {\pi -\pi c^2 x^2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \log \left (1-c^2 x^2\right )}{2 \pi c \sqrt {\pi -\pi c^2 x^2}}\right )}{3 \pi }+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{6 \pi ^2 c \left (1-c^2 x^2\right ) \sqrt {\pi -\pi c^2 x^2}}\right )}{5 \pi }+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{20 \pi ^3 c \left (1-c^2 x^2\right )^2 \sqrt {\pi -\pi c^2 x^2}}\)

Input:

Int[(a + b*ArcCosh[c*x])/(Pi - c^2*Pi*x^2)^(7/2),x]
 

Output:

(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(20*c*Pi^3*(1 - c^2*x^2)^2*Sqrt[Pi - c^2* 
Pi*x^2]) + (x*(a + b*ArcCosh[c*x]))/(5*Pi*(Pi - c^2*Pi*x^2)^(5/2)) + (4*(( 
b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*c*Pi^2*(1 - c^2*x^2)*Sqrt[Pi - c^2*Pi*x 
^2]) + (x*(a + b*ArcCosh[c*x]))/(3*Pi*(Pi - c^2*Pi*x^2)^(3/2)) + (2*((x*(a 
 + b*ArcCosh[c*x]))/(Pi*Sqrt[Pi - c^2*Pi*x^2]) - (b*Sqrt[-1 + c*x]*Sqrt[1 
+ c*x]*Log[1 - c^2*x^2])/(2*c*Pi*Sqrt[Pi - c^2*Pi*x^2])))/(3*Pi)))/(5*Pi)
 

Defintions of rubi rules used

rule 25

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

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 240

Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x 
^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
 

rule 241

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ 
(2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
 

rule 6314

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), 
x_Symbol] :> Simp[x*((a + b*ArcCosh[c*x])^n/(d*Sqrt[d + e*x^2])), x] + Simp 
[b*c*(n/d)*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])]   Int[x*((a 
 + b*ArcCosh[c*x])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, 
 x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
 

rule 6316

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x 
_Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*d*(p + 
 1))), x] + (Simp[(2*p + 3)/(2*d*(p + 1))   Int[(d + e*x^2)^(p + 1)*(a + b* 
ArcCosh[c*x])^n, x], x] - Simp[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] && LtQ[p, -1] && NeQ[p, -3/2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2224\) vs. \(2(189)=378\).

Time = 0.32 (sec) , antiderivative size = 2225, normalized size of antiderivative = 9.89

method	result	size
default	\(\text {Expression too large to display}\)	\(2225\)
parts	\(\text {Expression too large to display}\)	\(2225\)

Input:

int((a+b*arccosh(c*x))/(-Pi*c^2*x^2+Pi)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

-64*b*(-c^2*x^2+1)^(1/2)/Pi^(7/2)/(40*c^10*x^10-215*c^8*x^8+469*c^6*x^6-51 
7*c^4*x^4+287*c^2*x^2-64)*arccosh(c*x)*x+128/15*b*(-c^2*x^2+1)^(1/2)/Pi^(7 
/2)/(40*c^10*x^10-215*c^8*x^8+469*c^6*x^6-517*c^4*x^4+287*c^2*x^2-64)*c^12 
*x^13-176/3*b*(-c^2*x^2+1)^(1/2)/Pi^(7/2)/(40*c^10*x^10-215*c^8*x^8+469*c^ 
6*x^6-517*c^4*x^4+287*c^2*x^2-64)*c^10*x^11+2552/15*b*(-c^2*x^2+1)^(1/2)/P 
i^(7/2)/(40*c^10*x^10-215*c^8*x^8+469*c^6*x^6-517*c^4*x^4+287*c^2*x^2-64)* 
c^8*x^9-3986/15*b*(-c^2*x^2+1)^(1/2)/Pi^(7/2)/(40*c^10*x^10-215*c^8*x^8+46 
9*c^6*x^6-517*c^4*x^4+287*c^2*x^2-64)*c^6*x^7+3526/15*b*(-c^2*x^2+1)^(1/2) 
/Pi^(7/2)/(40*c^10*x^10-215*c^8*x^8+469*c^6*x^6-517*c^4*x^4+287*c^2*x^2-64 
)*c^4*x^5-334/3*b*(-c^2*x^2+1)^(1/2)/Pi^(7/2)/(40*c^10*x^10-215*c^8*x^8+46 
9*c^6*x^6-517*c^4*x^4+287*c^2*x^2-64)*c^2*x^3+a*(1/5/Pi*x/(-Pi*c^2*x^2+Pi) 
^(5/2)+4/5/Pi*(1/3/Pi*x/(-Pi*c^2*x^2+Pi)^(3/2)+2/3/Pi^2*x/(-Pi*c^2*x^2+Pi) 
^(1/2)))+64/3*b*(-c^2*x^2+1)^(1/2)/Pi^(7/2)/(40*c^10*x^10-215*c^8*x^8+469* 
c^6*x^6-517*c^4*x^4+287*c^2*x^2-64)*c^7*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccos 
h(c*x)*x^8-280/3*b*(-c^2*x^2+1)^(1/2)/Pi^(7/2)/(40*c^10*x^10-215*c^8*x^8+4 
69*c^6*x^6-517*c^4*x^4+287*c^2*x^2-64)*c^5*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arc 
cosh(c*x)*x^6+784/5*b*(-c^2*x^2+1)^(1/2)/Pi^(7/2)/(40*c^10*x^10-215*c^8*x^ 
8+469*c^6*x^6-517*c^4*x^4+287*c^2*x^2-64)*c^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)* 
arccosh(c*x)*x^4-1784/15*b*(-c^2*x^2+1)^(1/2)/Pi^(7/2)/(40*c^10*x^10-215*c 
^8*x^8+469*c^6*x^6-517*c^4*x^4+287*c^2*x^2-64)*c*(c*x-1)^(1/2)*(c*x+1)^...
 

Fricas [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{\left (\pi -c^2 \pi x^2\right )^{7/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (\pi - \pi c^{2} x^{2}\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((a+b*arccosh(c*x))/(-pi*c^2*x^2+pi)^(7/2),x, algorithm="fricas")
 

Output:

integral(sqrt(pi - pi*c^2*x^2)*(b*arccosh(c*x) + a)/(pi^4*c^8*x^8 - 4*pi^4 
*c^6*x^6 + 6*pi^4*c^4*x^4 - 4*pi^4*c^2*x^2 + pi^4), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{\left (\pi -c^2 \pi x^2\right )^{7/2}} \, dx=\text {Timed out} \] Input:

integrate((a+b*acosh(c*x))/(-pi*c**2*x**2+pi)**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{\left (\pi -c^2 \pi x^2\right )^{7/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (\pi - \pi c^{2} x^{2}\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((a+b*arccosh(c*x))/(-pi*c^2*x^2+pi)^(7/2),x, algorithm="maxima")
 

Output:

1/15*a*(3*x/(pi*(pi - pi*c^2*x^2)^(5/2)) + 4*x/(pi^2*(pi - pi*c^2*x^2)^(3/ 
2)) + 8*x/(pi^3*sqrt(pi - pi*c^2*x^2))) + b*integrate(log(c*x + sqrt(c*x + 
 1)*sqrt(c*x - 1))/(pi - pi*c^2*x^2)^(7/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {a+b \text {arccosh}(c x)}{\left (\pi -c^2 \pi x^2\right )^{7/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a+b*arccosh(c*x))/(-pi*c^2*x^2+pi)^(7/2),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. \[ \int \frac {a+b \text {arccosh}(c x)}{\left (\pi -c^2 \pi x^2\right )^{7/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{{\left (\Pi -\Pi \,c^2\,x^2\right )}^{7/2}} \,d x \] Input:

int((a + b*acosh(c*x))/(Pi - Pi*c^2*x^2)^(7/2),x)
 

Output:

int((a + b*acosh(c*x))/(Pi - Pi*c^2*x^2)^(7/2), x)
 

Reduce [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{\left (\pi -c^2 \pi x^2\right )^{7/2}} \, dx=\frac {-15 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{6} x^{6}-3 \sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}+3 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{4} x^{4}+30 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{6} x^{6}-3 \sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}+3 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{2} x^{2}-15 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{6} x^{6}-3 \sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}+3 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}}d x \right ) b +8 a \,c^{4} x^{5}-20 a \,c^{2} x^{3}+15 a x}{15 \sqrt {\pi }\, \sqrt {-c^{2} x^{2}+1}\, \pi ^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )} \] Input:

int((a+b*acosh(c*x))/(-Pi*c^2*x^2+Pi)^(7/2),x)
 

Output:

( - 15*sqrt( - c**2*x**2 + 1)*int(acosh(c*x)/(sqrt( - c**2*x**2 + 1)*c**6* 
x**6 - 3*sqrt( - c**2*x**2 + 1)*c**4*x**4 + 3*sqrt( - c**2*x**2 + 1)*c**2* 
x**2 - sqrt( - c**2*x**2 + 1)),x)*b*c**4*x**4 + 30*sqrt( - c**2*x**2 + 1)* 
int(acosh(c*x)/(sqrt( - c**2*x**2 + 1)*c**6*x**6 - 3*sqrt( - c**2*x**2 + 1 
)*c**4*x**4 + 3*sqrt( - c**2*x**2 + 1)*c**2*x**2 - sqrt( - c**2*x**2 + 1)) 
,x)*b*c**2*x**2 - 15*sqrt( - c**2*x**2 + 1)*int(acosh(c*x)/(sqrt( - c**2*x 
**2 + 1)*c**6*x**6 - 3*sqrt( - c**2*x**2 + 1)*c**4*x**4 + 3*sqrt( - c**2*x 
**2 + 1)*c**2*x**2 - sqrt( - c**2*x**2 + 1)),x)*b + 8*a*c**4*x**5 - 20*a*c 
**2*x**3 + 15*a*x)/(15*sqrt(pi)*sqrt( - c**2*x**2 + 1)*pi**3*(c**4*x**4 - 
2*c**2*x**2 + 1))