\(\int \frac {a+b \text {arccosh}(c x)}{(f+g x) \sqrt {d-c^2 d x^2}} \, dx\) [57]

Optimal result

Integrand size = 31, antiderivative size = 365 \[ \int \frac {a+b \text {arccosh}(c x)}{(f+g x) \sqrt {d-c^2 d x^2}} \, dx=\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}} \] Output:

(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)))/(c^2*f^2-g^2)^(1/2)/(-c^2*d*x^2+d 
)^(1/2)-(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)))/(c^2*f^2-g^2)^(1/2)/(-c^2 
*d*x^2+d)^(1/2)+b*(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)))/(c^2*f^2-g^2)^(1/2)/(-c^2*d* 
x^2+d)^(1/2)-b*(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)))/(c^2*f^2-g^2)^(1/2)/(-c^2*d*x^2 
+d)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.50 (sec) , antiderivative size = 932, normalized size of antiderivative = 2.55 \[ \int \frac {a+b \text {arccosh}(c x)}{(f+g x) \sqrt {d-c^2 d x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

((a*Log[f + g*x])/Sqrt[d] - (a*Log[d*(g + c^2*f*x) + Sqrt[d]*Sqrt[-(c^2*f^ 
2) + g^2]*Sqrt[d - c^2*d*x^2]])/Sqrt[d] - (b*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 
 + c*x)*(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)]*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 + 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 + 
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)*Tan 
h[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]))] + I*(PolyLog[2, ((c*f - I*Sqrt[- 
(c^2*f^2) + g^2])*(c*f + g - I*Sqrt[-(c^2*f^2) + g^2]*Tanh[ArcCosh[c*x]/2] 
))/(g*(c*f + g + I*Sqrt[-(c^2*f^2) + g^2]*Tanh[ArcCosh[c*x]/2]))] - Pol...
 

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.70, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {6387, 6395, 3042, 3801, 2694, 27, 2620, 2715, 2838}

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.

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

\(\Big \downarrow \) 6387

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

\(\Big \downarrow \) 6395

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3801

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

\(\Big \downarrow \) 2694

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

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 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2694

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) 
*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q)   Int 
[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q)   Int[(f + g*x) 
^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ 
v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3801

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (Comple 
x[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Simp[2   Int[((c + d*x)^m*(E^((-I)*e 
+ f*fz*x)/(b + (2*a*E^((-I)*e + f*fz*x))/E^(I*Pi*(k - 1/2)) - (b*E^(2*((-I) 
*e + f*fz*x)))/E^(2*I*k*Pi))))/E^(I*Pi*(k - 1/2)), x], x] /; FreeQ[{a, b, c 
, d, e, f, fz}, x] && IntegerQ[2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 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 6395

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

Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 591, normalized size of antiderivative = 1.62

method	result	size
default	\(-\frac {a \ln \left (\frac {-\frac {2 d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}+\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}\, \sqrt {-\left (x +\frac {f}{g}\right )^{2} c^{2} d +\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}{x +\frac {f}{g}}\right )}{g \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \left (\ln \left (\frac {-\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g -f c +\sqrt {c^{2} f^{2}-g^{2}}}{-f c +\sqrt {c^{2} f^{2}-g^{2}}}\right )-\ln \left (\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g +f c +\sqrt {c^{2} f^{2}-g^{2}}}{f c +\sqrt {c^{2} f^{2}-g^{2}}}\right )\right )}{d \left (c^{4} f^{2} x^{2}-c^{2} g^{2} x^{2}-c^{2} f^{2}+g^{2}\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (\operatorname {dilog}\left (\frac {-\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g -f c +\sqrt {c^{2} f^{2}-g^{2}}}{-f c +\sqrt {c^{2} f^{2}-g^{2}}}\right )-\operatorname {dilog}\left (\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g +f c +\sqrt {c^{2} f^{2}-g^{2}}}{f c +\sqrt {c^{2} f^{2}-g^{2}}}\right )\right )}{d \left (c^{4} f^{2} x^{2}-c^{2} g^{2} x^{2}-c^{2} f^{2}+g^{2}\right )}\right )\)	\(591\)
parts	\(-\frac {a \ln \left (\frac {-\frac {2 d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}+\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}\, \sqrt {-\left (x +\frac {f}{g}\right )^{2} c^{2} d +\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}{x +\frac {f}{g}}\right )}{g \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \left (\ln \left (\frac {-\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g -f c +\sqrt {c^{2} f^{2}-g^{2}}}{-f c +\sqrt {c^{2} f^{2}-g^{2}}}\right )-\ln \left (\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g +f c +\sqrt {c^{2} f^{2}-g^{2}}}{f c +\sqrt {c^{2} f^{2}-g^{2}}}\right )\right )}{d \left (c^{4} f^{2} x^{2}-c^{2} g^{2} x^{2}-c^{2} f^{2}+g^{2}\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (\operatorname {dilog}\left (\frac {-\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g -f c +\sqrt {c^{2} f^{2}-g^{2}}}{-f c +\sqrt {c^{2} f^{2}-g^{2}}}\right )-\operatorname {dilog}\left (\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g +f c +\sqrt {c^{2} f^{2}-g^{2}}}{f c +\sqrt {c^{2} f^{2}-g^{2}}}\right )\right )}{d \left (c^{4} f^{2} x^{2}-c^{2} g^{2} x^{2}-c^{2} f^{2}+g^{2}\right )}\right )\)	\(591\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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