\(\int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{(d-c^2 d x^2)^{3/2}} \, dx\) [59]

Optimal result

Integrand size = 31, antiderivative size = 390 \[ \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {b g^3 x \sqrt {-1+c x} \sqrt {1+c x}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^3 (1-c x) (a+b \text {arccosh}(c x))}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 (1+c x) (a+b \text {arccosh}(c x))}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 (1-c x) (1+c x) (a+b \text {arccosh}(c x))}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}-\frac {b (c f+g)^3 \sqrt {-1+c x} \sqrt {1-c^2 x^2} \log (1-c x)}{2 c^4 d \sqrt {1-c x} \sqrt {d-c^2 d x^2}}-\frac {b (c f-g)^3 \sqrt {-1+c x} \sqrt {1-c^2 x^2} \log (1+c x)}{2 c^4 d \sqrt {1-c x} \sqrt {d-c^2 d x^2}} \] Output:

b*g^3*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3/d/(-c^2*d*x^2+d)^(1/2)-1/2*(c*f-g) 
^3*(-c*x+1)*(a+b*arccosh(c*x))/c^4/d/(-c^2*d*x^2+d)^(1/2)+1/2*(c*f+g)^3*(c 
*x+1)*(a+b*arccosh(c*x))/c^4/d/(-c^2*d*x^2+d)^(1/2)+g^3*(-c*x+1)*(c*x+1)*( 
a+b*arccosh(c*x))/c^4/d/(-c^2*d*x^2+d)^(1/2)-3/2*f*g^2*(c*x-1)^(1/2)*(c*x+ 
1)^(1/2)*(a+b*arccosh(c*x))^2/b/c^3/d/(-c^2*d*x^2+d)^(1/2)-1/2*b*(c*f+g)^3 
*(c*x-1)^(1/2)*(-c^2*x^2+1)^(1/2)*ln(-c*x+1)/c^4/d/(-c*x+1)^(1/2)/(-c^2*d* 
x^2+d)^(1/2)-1/2*b*(c*f-g)^3*(c*x-1)^(1/2)*(-c^2*x^2+1)^(1/2)*ln(c*x+1)/c^ 
4/d/(-c*x+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 1.97 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.28 \[ \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\sqrt {-d \left (-1+c^2 x^2\right )} \left (\frac {a g^3}{c^4 d^2}-\frac {a \left (3 c^2 f^2 g+g^3+c^4 f^3 x+3 c^2 f g^2 x\right )}{c^4 d^2 \left (-1+c^2 x^2\right )}\right )+\frac {3 a f g^2 \arctan \left (\frac {c x \sqrt {-d \left (-1+c^2 x^2\right )}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )}{c^3 d^{3/2}}-\frac {b f^3 \left (-c x \text {arccosh}(c x)+\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)\right )\right )}{c d \sqrt {-d (-1+c x) (1+c x)}}-\frac {3 b f g^2 \left (-2 c x \text {arccosh}(c x)+\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\text {arccosh}(c x)^2+2 \log \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)\right )\right )\right )}{2 c^3 d \sqrt {-d (-1+c x) (1+c x)}}+\frac {3 b f^2 g \left (\text {arccosh}(c x)+\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-\log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )\right )\right )}{c^2 d \sqrt {-d (-1+c x) (1+c x)}}-\frac {b g^3 \left (-3 \text {arccosh}(c x)+\text {arccosh}(c x) \cosh (2 \text {arccosh}(c x))-2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-\log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )\right )-\sinh (2 \text {arccosh}(c x))\right )}{2 c^4 d \sqrt {-d (-1+c x) (1+c x)}} \] Input:

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

Output:

Sqrt[-(d*(-1 + c^2*x^2))]*((a*g^3)/(c^4*d^2) - (a*(3*c^2*f^2*g + g^3 + c^4 
*f^3*x + 3*c^2*f*g^2*x))/(c^4*d^2*(-1 + c^2*x^2))) + (3*a*f*g^2*ArcTan[(c* 
x*Sqrt[-(d*(-1 + c^2*x^2))])/(Sqrt[d]*(-1 + c^2*x^2))])/(c^3*d^(3/2)) - (b 
*f^3*(-(c*x*ArcCosh[c*x]) + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Sqrt[ 
(-1 + c*x)/(1 + c*x)]*(1 + c*x)]))/(c*d*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]) - 
 (3*b*f*g^2*(-2*c*x*ArcCosh[c*x] + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(A 
rcCosh[c*x]^2 + 2*Log[Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)])))/(2*c^3*d*Sq 
rt[-(d*(-1 + c*x)*(1 + c*x))]) + (3*b*f^2*g*(ArcCosh[c*x] + Sqrt[(-1 + c*x 
)/(1 + c*x)]*(1 + c*x)*(Log[Cosh[ArcCosh[c*x]/2]] - Log[Sinh[ArcCosh[c*x]/ 
2]])))/(c^2*d*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]) - (b*g^3*(-3*ArcCosh[c*x] + 
 ArcCosh[c*x]*Cosh[2*ArcCosh[c*x]] - 2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x 
)*(Log[Cosh[ArcCosh[c*x]/2]] - Log[Sinh[ArcCosh[c*x]/2]]) - Sinh[2*ArcCosh 
[c*x]]))/(2*c^4*d*Sqrt[-(d*(-1 + c*x)*(1 + c*x))])
 

Rubi [A] (warning: unable to verify)

Time = 1.82 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.22, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {6387, 6396, 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.

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

\(\Big \downarrow \) 6387

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

\(\Big \downarrow \) 6396

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((b*g^3*x)/c^3) - ((c*f - g)^3*Sqrt[-1 + 
 c*x]*(a + b*ArcCosh[c*x]))/(2*c^4*Sqrt[1 + c*x]) - ((c*f + g)^3*Sqrt[1 + 
c*x]*(a + b*ArcCosh[c*x]))/(2*c^4*Sqrt[-1 + c*x]) + (g^3*Sqrt[-1 + c*x]*Sq 
rt[1 + c*x]*(a + b*ArcCosh[c*x]))/c^4 + (3*f*g^2*(a + b*ArcCosh[c*x])^2)/( 
2*b*c^3) - (b*(c*f + g)^3*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[ 
Sqrt[-((1 - c*x)/(1 + c*x))]])/(c^4*Sqrt[-1 + c*x]*Sqrt[-((1 - c*x)/(1 + c 
*x))]*(1 + c*x)^(3/2)) + (b*(c*f - g)^3*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - 
 c^2*x^2]*Log[2/(1 + c*x)])/(2*c^4*Sqrt[-1 + c*x]*Sqrt[-((1 - c*x)/(1 + c* 
x))]*(1 + c*x)^(3/2)) + (b*(c*f + g)^3*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - 
c^2*x^2]*Log[2/(1 + c*x)])/(2*c^4*Sqrt[-1 + c*x]*Sqrt[-((1 - c*x)/(1 + c*x 
))]*(1 + c*x)^(3/2))))/(d*Sqrt[d - c^2*d*x^2]))
 

Defintions of rubi rules used

rule 2009

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

rule 6387

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d 
_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-d)^IntPart[p]*((d + e*x^2)^Fra 
cPart[p]/((-1 + c*x)^FracPart[p]*(1 + c*x)^FracPart[p]))   Int[(f + g*x)^m* 
(-1 + c*x)^p*(1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, 
d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && IntegerQ[m]
 

rule 6396

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_)*(( 
d2_) + (e2_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(m_.), x_Symbol] :> Int[Expand 
Integrand[(a + b*ArcCosh[c*x])^n/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), (f + g* 
x)^m*(d1 + e1*x)^(p + 1/2)*(d2 + e2*x)^(p + 1/2), x], x] /; FreeQ[{a, b, c, 
 d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && Int 
egerQ[m] && ILtQ[p + 1/2, 0] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1231\) vs. \(2(346)=692\).

Time = 0.75 (sec) , antiderivative size = 1232, normalized size of antiderivative = 3.16

method	result	size
default	\(\text {Expression too large to display}\)	\(1232\)
parts	\(\text {Expression too large to display}\)	\(1232\)

Input:

int((g*x+f)^3*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2),x,method=_RETURNVERB 
OSE)
 

Output:

a*(f^3*x/d/(-c^2*d*x^2+d)^(1/2)+g^3*(-x^2/c^2/d/(-c^2*d*x^2+d)^(1/2)+2/d/c 
^4/(-c^2*d*x^2+d)^(1/2))+3*f*g^2*(x/c^2/d/(-c^2*d*x^2+d)^(1/2)-1/c^2/d/(c^ 
2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2)))+3*f^2*g/c^2/d/(-c 
^2*d*x^2+d)^(1/2))+3/2*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2 
)/d^2/c^3/(c^2*x^2-1)*arccosh(c*x)^2*f*g^2-b*(-d*(c^2*x^2-1))^(1/2)*g^3/c^ 
3/d^2/(c^2*x^2-1)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x-b*(-d*(c^2*x^2-1))^(1/2)*( 
c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/c/(c^2*x^2-1)*f^3*arccosh(c*x)-3*b*(-d*(c^2 
*x^2-1))^(1/2)*arccosh(c*x)/d^2/(c^2*x^2-1)*x^2*f^2*g+b*(-d*(c^2*x^2-1))^( 
1/2)*arccosh(c*x)/c^4/d^2/(c^2*x^2-1)*(c*x-1)*(c*x+1)*g^3-3*b*(-d*(c^2*x^2 
-1))^(1/2)*arccosh(c*x)/c^2/d^2/(c^2*x^2-1)*x*f*g^2+b*(-d*(c^2*x^2-1))^(1/ 
2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c/d^2/(c^2*x^2-1)*ln((c*x-1)^(1/2)*(c*x+1)^ 
(1/2)+c*x-1)*f^3+b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^4/ 
d^2/(c^2*x^2-1)*ln((c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x-1)*g^3+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)) 
/c/d^2/(c^2*x^2-1)*f^3-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))/c^4/d^2/(c^2*x^2-1)*g^3-b*(-d*(c^2 
*x^2-1))^(1/2)*arccosh(c*x)/d^2/(c^2*x^2-1)*x*f^3-3*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))/c^2/d 
^2/(c^2*x^2-1)*f^2*g+3*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))/c^3/d^2/(c^2*x^2-1)*f*g^2-3*b*(...
 

Fricas [F]

\[ \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

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

Output:

integral((a*g^3*x^3 + 3*a*f*g^2*x^2 + 3*a*f^2*g*x + a*f^3 + (b*g^3*x^3 + 3 
*b*f*g^2*x^2 + 3*b*f^2*g*x + b*f^3)*arccosh(c*x))*sqrt(-c^2*d*x^2 + d)/(c^ 
4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x)
 

Sympy [F]

\[ \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (f + g x\right )^{3}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:

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

Output:

Integral((a + b*acosh(c*x))*(f + g*x)**3/(-d*(c*x - 1)*(c*x + 1))**(3/2), 
x)
 

Maxima [F]

\[ \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

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

Output:

-1/2*b*c*f^3*sqrt(-1/(c^4*d))*log(x^2 - 1/c^2)/d - a*g^3*(x^2/(sqrt(-c^2*d 
*x^2 + d)*c^2*d) - 2/(sqrt(-c^2*d*x^2 + d)*c^4*d)) + 3*a*f*g^2*(x/(sqrt(-c 
^2*d*x^2 + d)*c^2*d) - arcsin(c*x)/(c^3*d^(3/2))) + b*f^3*x*arccosh(c*x)/( 
sqrt(-c^2*d*x^2 + d)*d) + a*f^3*x/(sqrt(-c^2*d*x^2 + d)*d) + 3*a*f^2*g/(sq 
rt(-c^2*d*x^2 + d)*c^2*d) + integrate(b*g^3*x^3*log(c*x + sqrt(c*x + 1)*sq 
rt(c*x - 1))/(-c^2*d*x^2 + d)^(3/2) + 3*b*f*g^2*x^2*log(c*x + sqrt(c*x + 1 
)*sqrt(c*x - 1))/(-c^2*d*x^2 + d)^(3/2) + 3*b*f^2*g*x*log(c*x + sqrt(c*x + 
 1)*sqrt(c*x - 1))/(-c^2*d*x^2 + d)^(3/2), x)
 

Giac [F]

\[ \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

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

Output:

integrate((g*x + f)^3*(b*arccosh(c*x) + a)/(-c^2*d*x^2 + d)^(3/2), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 3*sqrt( - c**2*x**2 + 1)*asin(c*x)*a*c*f*g**2 - sqrt( - c**2*x**2 + 1) 
*int(acosh(c*x)/(sqrt( - c**2*x**2 + 1)*c**2*x**2 - sqrt( - c**2*x**2 + 1) 
),x)*b*c**4*f**3 - sqrt( - c**2*x**2 + 1)*int((acosh(c*x)*x**3)/(sqrt( - c 
**2*x**2 + 1)*c**2*x**2 - sqrt( - c**2*x**2 + 1)),x)*b*c**4*g**3 - 3*sqrt( 
 - c**2*x**2 + 1)*int((acosh(c*x)*x**2)/(sqrt( - c**2*x**2 + 1)*c**2*x**2 
- sqrt( - c**2*x**2 + 1)),x)*b*c**4*f*g**2 - 3*sqrt( - c**2*x**2 + 1)*int( 
(acosh(c*x)*x)/(sqrt( - c**2*x**2 + 1)*c**2*x**2 - sqrt( - c**2*x**2 + 1)) 
,x)*b*c**4*f**2*g + a*c**4*f**3*x + 3*a*c**2*f**2*g + 3*a*c**2*f*g**2*x - 
a*c**2*g**3*x**2 + 2*a*g**3)/(sqrt(d)*sqrt( - c**2*x**2 + 1)*c**4*d)