\(\int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{(f+g x)^2} \, dx\) [45]

Optimal result

Integrand size = 31, antiderivative size = 908 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{(f+g x)^2} \, dx=-\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}+\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \sqrt {-\frac {1-c x}{1+c x}} \sqrt {1+c x} \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{g \sqrt {-1+c x} (f+g x)}+\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c (f+g x)^2}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}-\frac {2 a c^2 f \sqrt {d-c^2 d x^2} \text {arctanh}\left (\frac {\sqrt {c f+g} \sqrt {1+c x}}{\sqrt {c f-g} \sqrt {-1+c x}}\right )}{\sqrt {c f-g} g^2 \sqrt {c f+g} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^2 f \sqrt {d-c^2 d x^2} \text {arccosh}(c x) \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^2 f \sqrt {d-c^2 d x^2} \text {arccosh}(c x) \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c \sqrt {d-c^2 d x^2} \log (f+g x)}{g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^2 f \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^2 f \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 4.84 (sec) , antiderivative size = 1139, normalized size of antiderivative = 1.25 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{(f+g x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

((-2*a*g*Sqrt[d - c^2*d*x^2])/(f + g*x) + 2*a*c*Sqrt[d]*ArcTan[(c*x*Sqrt[d 
 - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + (2*a*c^2*Sqrt[d]*f*Log[f + g*x] 
)/Sqrt[-(c^2*f^2) + g^2] - (2*a*c^2*Sqrt[d]*f*Log[d*(g + c^2*f*x) + Sqrt[d 
]*Sqrt[-(c^2*f^2) + g^2]*Sqrt[d - c^2*d*x^2]])/Sqrt[-(c^2*f^2) + g^2] + b* 
c*Sqrt[d - c^2*d*x^2]*((-2*g*ArcCosh[c*x])/(c*f + c*g*x) + ArcCosh[c*x]^2/ 
(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (2*Log[1 + (g*x)/f])/(Sqrt[(-1 + 
c*x)/(1 + c*x)]*(1 + c*x)) + (2*c*f*(2*ArcCosh[c*x]*ArcTan[((c*f + g)*Coth 
[ArcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] - (2*I)*ArcCos[-((c*f)/g)]*ArcTa 
n[((-(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 
 + 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])/S 
qrt[-(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 + 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[-(c^2*f^2) + g^2]])...
 

Rubi [A] (verified)

Time = 4.12 (sec) , antiderivative size = 625, normalized size of antiderivative = 0.69, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {6387, 6391, 27, 6384, 27, 6408, 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.

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

\(\Big \downarrow \) 6387

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

\(\Big \downarrow \) 6391

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6384

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6408

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(Sqrt[d - c^2*d*x^2]*(-1/2*((1 - c^2*x^2)*(a + b*ArcCosh[c*x])^2)/(b*c*(f 
+ g*x)^2) - (((g + c^2*f*x)^2*(a + b*ArcCosh[c*x])^2)/(2*(c^2*f^2 - g^2)*( 
f + g*x)^2) - (b*c*(-((a*(c*f - g)*(c*f + g)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) 
/(g*(f + g*x))) + (a*c^3*f^2*ArcCosh[c*x])/g^2 - (b*(c*f - g)*(c*f + g)*Sq 
rt[-((1 - c*x)/(1 + c*x))]*(1 + c*x)*ArcCosh[c*x])/(g*(f + g*x)) + (b*c^3* 
f^2*ArcCosh[c*x]^2)/(2*g^2) - (2*a*c^2*f*Sqrt[c*f - g]*Sqrt[c*f + g]*ArcTa 
nh[(Sqrt[c*f + g]*Sqrt[1 + c*x])/(Sqrt[c*f - g]*Sqrt[-1 + c*x])])/g^2 - (b 
*c^2*f*Sqrt[c^2*f^2 - g^2]*ArcCosh[c*x]*Log[1 + (E^ArcCosh[c*x]*g)/(c*f - 
Sqrt[c^2*f^2 - g^2])])/g^2 + (b*c^2*f*Sqrt[c^2*f^2 - g^2]*ArcCosh[c*x]*Log 
[1 + (E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/g^2 - b*c*(1 - (c^2* 
f^2)/g^2)*Log[f + g*x] - (b*c^2*f*Sqrt[c^2*f^2 - g^2]*PolyLog[2, -((E^ArcC 
osh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/g^2 + (b*c^2*f*Sqrt[c^2*f^2 - g 
^2]*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/g^2))/( 
c^2*f^2 - g^2))/(b*c)))/(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 2009

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

rule 6384

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_))^(m_)*((f_. 
) + (g_.)*(x_))^(p_.), x_Symbol] :> With[{u = IntHide[(f + g*x)^p*(d + e*x) 
^m, x]}, Simp[(a + b*ArcCosh[c*x])^n   u, x] - Simp[b*c*n   Int[SimplifyInt 
egrand[u*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])), x], 
 x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[n, 0] && IGtQ[p, 0] && 
ILtQ[m, 0] && LtQ[m + p + 1, 0]
 

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 6391

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

rule 6408

Int[(ArcCosh[(c_.)*(x_)]*(b_.) + (a_))^(n_.)*(RFx_)*((d1_) + (e1_.)*(x_))^( 
p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(d1 + e1*x 
)^p*(d2 + e2*x)^p, RFx*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d1 
, e1, d2, e2}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[e1 - c* 
d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p - 1/2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1822\) vs. \(2(856)=1712\).

Time = 0.60 (sec) , antiderivative size = 1823, normalized size of antiderivative = 2.01

method	result	size
default	\(\text {Expression too large to display}\)	\(1823\)
parts	\(\text {Expression too large to display}\)	\(1823\)

Input:

int((-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/(g*x+f)^2,x,method=_RETURNVERB 
OSE)
 

Output:

a/g^2*(1/d/(c^2*f^2-g^2)*g^2/(x+f/g)*(-(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)^(3/2)-c^2*f*g/(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)+c^2*d*f/g/(c^2*d)^(1/2)*arctan(( 
c^2*d)^(1/2)*x/(-(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))+d*(c^2*f^2-g^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)))+2*c^2/(c^2*f 
^2-g^2)*g^2*(-1/4*(-2*(x+f/g)*c^2*d+2*c^2*d*f/g)/c^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)-1/8*(4*c^2*d^2*(c^2*f^2-g^2) 
/g^2-4*c^4*d^2*f^2/g^2)/c^2/d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-(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))))+1/2*b*(-d*(c^ 
2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)^2*c/g^2+b*(-d*(c^ 
2*x^2-1))^(1/2)*arccosh(c*x)/g^2/(g*x+f)*x*c^2*f-b*(-d*(c^2*x^2-1))^(1/2)* 
arccosh(c*x)/(c*x-1)/(c*x+1)/g^2/(g*x+f)*x^3*c^4*f+b*(-d*(c^2*x^2-1))^(1/2 
)*arccosh(c*x)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g/(g*x+f)*x*c-b*(-d*(c^2*x^2-1) 
)^(1/2)*arccosh(c*x)/(c*x-1)/(c*x+1)/g/(g*x+f)*x^2*c^2+b*(-d*(c^2*x^2-1))^ 
(1/2)*arccosh(c*x)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^2/(g*x+f)*c*f+b*(-d*(c^2* 
x^2-1))^(1/2)*arccosh(c*x)/(c*x-1)/(c*x+1)/g^2/(g*x+f)*x*c^2*f+b*(-d*(c^2* 
x^2-1))^(1/2)*arccosh(c*x)/(c*x-1)/(c*x+1)/g/(g*x+f)-b*(-d*(c^2*x^2-1))^(1 
/2)*c^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^2/(c^2*f^2-g^2)^(1/2)*arccosh(c*x...
 

Fricas [F]

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

integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/(g*x+f)^2,x, algorithm=" 
fricas")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{(f+g x)^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/(g*x+f)^2,x, algorithm=" 
maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(g-c*f>0)', see `assume?` for mor 
e details)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{(f+g x)^2} \, dx=\text {Exception raised: TypeError} \] Input:

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{(f+g x)^2} \, dx=\frac {\sqrt {d}\, \left (-\mathit {asin} \left (c x \right ) a \,c^{3} f^{3}-\mathit {asin} \left (c x \right ) a \,c^{3} f^{2} g x +\mathit {asin} \left (c x \right ) a c f \,g^{2}+\mathit {asin} \left (c x \right ) a c \,g^{3} x +2 \sqrt {c^{2} f^{2}-g^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right ) c f +g}{\sqrt {c^{2} f^{2}-g^{2}}}\right ) a \,c^{2} f^{2}+2 \sqrt {c^{2} f^{2}-g^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right ) c f +g}{\sqrt {c^{2} f^{2}-g^{2}}}\right ) a \,c^{2} f g x -\sqrt {-c^{2} x^{2}+1}\, a \,c^{2} f^{2} g +\sqrt {-c^{2} x^{2}+1}\, a \,g^{3}+\left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )}{g^{2} x^{2}+2 f g x +f^{2}}d x \right ) b \,c^{2} f^{3} g^{2}+\left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )}{g^{2} x^{2}+2 f g x +f^{2}}d x \right ) b \,c^{2} f^{2} g^{3} x -\left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )}{g^{2} x^{2}+2 f g x +f^{2}}d x \right ) b f \,g^{4}-\left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )}{g^{2} x^{2}+2 f g x +f^{2}}d x \right ) b \,g^{5} x \right )}{g^{2} \left (c^{2} f^{2} g x +c^{2} f^{3}-g^{3} x -f \,g^{2}\right )} \] Input:

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

Output:

(sqrt(d)*( - asin(c*x)*a*c**3*f**3 - asin(c*x)*a*c**3*f**2*g*x + asin(c*x) 
*a*c*f*g**2 + asin(c*x)*a*c*g**3*x + 2*sqrt(c**2*f**2 - g**2)*atan((tan(as 
in(c*x)/2)*c*f + g)/sqrt(c**2*f**2 - g**2))*a*c**2*f**2 + 2*sqrt(c**2*f**2 
 - g**2)*atan((tan(asin(c*x)/2)*c*f + g)/sqrt(c**2*f**2 - g**2))*a*c**2*f* 
g*x - sqrt( - c**2*x**2 + 1)*a*c**2*f**2*g + sqrt( - c**2*x**2 + 1)*a*g**3 
 + int((sqrt( - c**2*x**2 + 1)*acosh(c*x))/(f**2 + 2*f*g*x + g**2*x**2),x) 
*b*c**2*f**3*g**2 + int((sqrt( - c**2*x**2 + 1)*acosh(c*x))/(f**2 + 2*f*g* 
x + g**2*x**2),x)*b*c**2*f**2*g**3*x - int((sqrt( - c**2*x**2 + 1)*acosh(c 
*x))/(f**2 + 2*f*g*x + g**2*x**2),x)*b*f*g**4 - int((sqrt( - c**2*x**2 + 1 
)*acosh(c*x))/(f**2 + 2*f*g*x + g**2*x**2),x)*b*g**5*x))/(g**2*(c**2*f**3 
+ c**2*f**2*g*x - f*g**2 - g**3*x))