\(\int \frac {a+b \text {arccosh}(c x)}{(f+g x) (d-c^2 d x^2)^{3/2}} \, dx\) [62]

Optimal result

Integrand size = 31, antiderivative size = 617 \[ \int \frac {a+b \text {arccosh}(c x)}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {(1-c x) (a+b \text {arccosh}(c x))}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {(1+c x) (a+b \text {arccosh}(c x))}{2 d (c f+g) \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {g^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {-1+c x} \sqrt {1-c^2 x^2} \log (1-c x)}{2 d (c f+g) \sqrt {1-c x} \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {-1+c x} \sqrt {1-c^2 x^2} \log (1+c x)}{2 d (c f-g) \sqrt {1-c x} \sqrt {d-c^2 d x^2}}-\frac {b g^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {b g^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 8.69 (sec) , antiderivative size = 1173, normalized size of antiderivative = 1.90 \[ \int \frac {a+b \text {arccosh}(c x)}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

((-(a*g) + a*c^2*f*x)*Sqrt[-(d*(-1 + c^2*x^2))])/(d^2*(-(c^2*f^2) + g^2)*( 
-1 + c^2*x^2)) + (a*g^2*Log[f + g*x])/(d^(3/2)*(-(c*f) + g)*(c*f + g)*Sqrt 
[-(c^2*f^2) + g^2]) - (a*g^2*Log[d*g + c^2*d*f*x + Sqrt[d]*Sqrt[-(c^2*f^2) 
 + g^2]*Sqrt[-(d*(-1 + c^2*x^2))]])/(d^(3/2)*(-(c*f) + g)*(c*f + g)*Sqrt[- 
(c^2*f^2) + g^2]) - (b*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(-((ArcCosh[c* 
x]*Coth[ArcCosh[c*x]/2])/(c*f + g)) + (2*Log[Cosh[ArcCosh[c*x]/2]])/(c*f - 
 g) + (2*Log[Sinh[ArcCosh[c*x]/2]])/(c*f + g) + (2*g^2*(2*ArcCosh[c*x]*Arc 
Tan[((c*f + g)*Coth[ArcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] - (2*I)*ArcCo 
s[-((c*f)/g)]*ArcTan[((-(c*f) + g)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + 
 g^2]] + (ArcCos[-((c*f)/g)] + 2*(ArcTan[((c*f + g)*Coth[ArcCosh[c*x]/2])/ 
Sqrt[-(c^2*f^2) + g^2]] + ArcTan[((-(c*f) + g)*Tanh[ArcCosh[c*x]/2])/Sqrt[ 
-(c^2*f^2) + g^2]]))*Log[Sqrt[-(c^2*f^2) + g^2]/(Sqrt[2]*E^(ArcCosh[c*x]/2 
)*Sqrt[g]*Sqrt[c*f + c*g*x])] + (ArcCos[-((c*f)/g)] - 2*(ArcTan[((c*f + g) 
*Coth[ArcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + ArcTan[((-(c*f) + g)*Tanh 
[ArcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]]))*Log[(E^(ArcCosh[c*x]/2)*Sqrt[- 
(c^2*f^2) + g^2])/(Sqrt[2]*Sqrt[g]*Sqrt[c*f + c*g*x])] - (ArcCos[-((c*f)/g 
)] + 2*ArcTan[((-(c*f) + g)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]]) 
*Log[((c*f + g)*(c*f - g + I*Sqrt[-(c^2*f^2) + g^2])*(-1 + Tanh[ArcCosh[c* 
x]/2]))/(g*(c*f + g + I*Sqrt[-(c^2*f^2) + g^2]*Tanh[ArcCosh[c*x]/2]))] - ( 
ArcCos[-((c*f)/g)] - 2*ArcTan[((-(c*f) + g)*Tanh[ArcCosh[c*x]/2])/Sqrt[...
 

Rubi [A] (warning: unable to verify)

Time = 2.38 (sec) , antiderivative size = 636, normalized size of antiderivative = 1.03, 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.

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

\(\Big \downarrow \) 6387

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

\(\Big \downarrow \) 6396

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-1/2*(Sqrt[-1 + c*x]*(a + b*ArcCosh[c*x]) 
)/((c*f - g)*Sqrt[1 + c*x]) - (Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(2*(c*f 
 + g)*Sqrt[-1 + c*x]) + (g^2*(a + b*ArcCosh[c*x])*Log[1 + (E^ArcCosh[c*x]* 
g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(c^2*f^2 - g^2)^(3/2) - (g^2*(a + b*ArcCo 
sh[c*x])*Log[1 + (E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(c^2*f^2 
 - g^2)^(3/2) - (b*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[Sqrt[-( 
(1 - c*x)/(1 + c*x))]])/((c*f + g)*Sqrt[-1 + c*x]*Sqrt[-((1 - c*x)/(1 + c* 
x))]*(1 + c*x)^(3/2)) + (b*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log 
[2/(1 + c*x)])/(2*(c*f - g)*Sqrt[-1 + c*x]*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 
 + c*x)^(3/2)) + (b*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[2/(1 + 
 c*x)])/(2*(c*f + g)*Sqrt[-1 + c*x]*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x) 
^(3/2)) + (b*g^2*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2 
]))])/(c^2*f^2 - g^2)^(3/2) - (b*g^2*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f 
+ Sqrt[c^2*f^2 - g^2]))])/(c^2*f^2 - g^2)^(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. \(1925\) vs. \(2(601)=1202\).

Time = 1.04 (sec) , antiderivative size = 1926, normalized size of antiderivative = 3.12

method	result	size
default	\(\text {Expression too large to display}\)	\(1926\)
parts	\(\text {Expression too large to display}\)	\(1926\)

Input:

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

Output:

-a*g/d/(c^2*f^2-g^2)/(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2) 
/g^2)^(1/2)+a*f/(c^2*f^2-g^2)/d/(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c 
^2*f^2-g^2)/g^2)^(1/2)*x*c^2+a*g/d/(c^2*f^2-g^2)/(-d*(c^2*f^2-g^2)/g^2)^(1 
/2)*ln((-2*d*(c^2*f^2-g^2)/g^2+2*c^2*d*f/g*(x+f/g)+2*(-d*(c^2*f^2-g^2)/g^2 
)^(1/2)*(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2))/ 
(x+f/g))-b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/(c^2*x^2-1)/d^2/(c^2*f^2-g^ 
2)*(c*x-1)*(c*x+1)*g+b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/(c^2*x^2-1)/d^2 
/(c^2*f^2-g^2)*x^2*c^2*g+b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/(c^2*x^2-1) 
/d^2/(c^2*f^2-g^2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*f-b*(-d*(c^2*x^2-1))^(1/2 
)*arccosh(c*x)/(c^2*x^2-1)/d^2/(c^2*f^2-g^2)*x*c^2*f+b*(c^2*f^2-g^2)^(1/2) 
*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^6*f^4*x^2-2*c^4*f^2 
*g^2*x^2-c^4*f^4+c^2*g^4*x^2+2*c^2*f^2*g^2-g^4)/d^2*ln((-(c*x+(c*x-1)^(1/2 
)*(c*x+1)^(1/2))*g-f*c+(c^2*f^2-g^2)^(1/2))/(-f*c+(c^2*f^2-g^2)^(1/2)))*ar 
ccosh(c*x)*g^2-b*(c^2*f^2-g^2)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)* 
(c*x+1)^(1/2)/(c^6*f^4*x^2-2*c^4*f^2*g^2*x^2-c^4*f^4+c^2*g^4*x^2+2*c^2*f^2 
*g^2-g^4)/d^2*ln(((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g+f*c+(c^2*f^2-g^2)^(1 
/2))/(f*c+(c^2*f^2-g^2)^(1/2)))*arccosh(c*x)*g^2+b*(c^2*f^2-g^2)*(-d*(c^2* 
x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^6*f^4*x^2-2*c^4*f^2*g^2*x^2-c 
^4*f^4+c^2*g^4*x^2+2*c^2*f^2*g^2-g^4)/d^2*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^( 
1/2))*c*f-2*b*(c^2*f^2-g^2)*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1...
 

Fricas [F]

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

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

Output:

integral(sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)/(c^4*d^2*g*x^5 + c^4*d^ 
2*f*x^4 - 2*c^2*d^2*g*x^3 - 2*c^2*d^2*f*x^2 + d^2*g*x + d^2*f), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 2*sqrt(c**2*f**2 - g**2)*sqrt( - c**2*x**2 + 1)*atan((tan(asin(c*x)/2) 
*c*f + g)/sqrt(c**2*f**2 - g**2))*a*g**2 - sqrt( - c**2*x**2 + 1)*int(acos 
h(c*x)/(sqrt( - c**2*x**2 + 1)*c**2*f*x**2 + sqrt( - c**2*x**2 + 1)*c**2*g 
*x**3 - sqrt( - c**2*x**2 + 1)*f - sqrt( - c**2*x**2 + 1)*g*x),x)*b*c**4*f 
**4 + 2*sqrt( - c**2*x**2 + 1)*int(acosh(c*x)/(sqrt( - c**2*x**2 + 1)*c**2 
*f*x**2 + sqrt( - c**2*x**2 + 1)*c**2*g*x**3 - sqrt( - c**2*x**2 + 1)*f - 
sqrt( - c**2*x**2 + 1)*g*x),x)*b*c**2*f**2*g**2 - sqrt( - c**2*x**2 + 1)*i 
nt(acosh(c*x)/(sqrt( - c**2*x**2 + 1)*c**2*f*x**2 + sqrt( - c**2*x**2 + 1) 
*c**2*g*x**3 - sqrt( - c**2*x**2 + 1)*f - sqrt( - c**2*x**2 + 1)*g*x),x)*b 
*g**4 + sqrt( - c**2*x**2 + 1)*a*c**2*f**2*g - sqrt( - c**2*x**2 + 1)*a*g* 
*3 + a*c**4*f**3*x - a*c**2*f**2*g - a*c**2*f*g**2*x + a*g**3)/(sqrt(d)*sq 
rt( - c**2*x**2 + 1)*d*(c**4*f**4 - 2*c**2*f**2*g**2 + g**4))