\(\int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f+g x} \, dx\) [44]

Optimal result

Integrand size = 31, antiderivative size = 743 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\frac {a \sqrt {d-c^2 d x^2}}{g}-\frac {b c x \sqrt {d-c^2 d x^2}}{g \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{g}-\frac {c x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b g \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c (f+g x)}+\frac {2 a \sqrt {c f-g} \sqrt {c f+g} \sqrt {d-c^2 d x^2} \text {arctanh}\left (\frac {\sqrt {c f+g} \sqrt {1+c x}}{\sqrt {c f-g} \sqrt {-1+c x}}\right )}{g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \text {arccosh}(c x) \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \text {arccosh}(c x) \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

a*(-c^2*d*x^2+d)^(1/2)/g-b*c*x*(-c^2*d*x^2+d)^(1/2)/g/(c*x-1)^(1/2)/(c*x+1 
)^(1/2)+b*(-c^2*d*x^2+d)^(1/2)*arccosh(c*x)/g-1/2*c*x*(-c^2*d*x^2+d)^(1/2) 
*(a+b*arccosh(c*x))^2/b/g/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/2*(1-c^2*f^2/g^2)* 
(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2/b/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)/ 
(g*x+f)+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*a*(c*f-g)^(1/2)*(c*f+g)^(1/2)*(-c^2*d*x^2+d)^(1/2)*a 
rctanh((c*f+g)^(1/2)*(c*x+1)^(1/2)/(c*f-g)^(1/2)/(c*x-1)^(1/2))/g^2/(c*x-1 
)^(1/2)/(c*x+1)^(1/2)+b*(c^2*f^2-g^2)^(1/2)*(-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*x-1)^(1/2)/(c*x+1)^(1/2)-b*(c^2*f^2-g^2)^(1/2)*(-c^2*d*x^2+d)^(1/2)*a 
rccosh(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*x-1)^(1/2)/(c*x+1)^(1/2)+b*(c^2*f^2-g^2)^(1/2)*(-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*x-1)^(1/2)/(c*x+1)^(1/2)-b*(c^2*f^2-g^2)^(1/2)*(-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*x-1)^(1/2)/(c*x+1)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 2.81 (sec) , antiderivative size = 1121, normalized size of antiderivative = 1.51 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{f+g x} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*a*g*Sqrt[d - c^2*d*x^2] - 2*a*c*Sqrt[d]*f*ArcTan[(c*x*Sqrt[d - c^2*d*x^ 
2])/(Sqrt[d]*(-1 + c^2*x^2))] + 2*a*Sqrt[d]*Sqrt[-(c^2*f^2) + g^2]*Log[f + 
 g*x] - 2*a*Sqrt[d]*Sqrt[-(c^2*f^2) + g^2]*Log[d*(g + c^2*f*x) + Sqrt[d]*S 
qrt[-(c^2*f^2) + g^2]*Sqrt[d - c^2*d*x^2]] + b*Sqrt[d - c^2*d*x^2]*((2*c*g 
*x*Sqrt[(-1 + c*x)/(1 + c*x)])/(1 - c*x) + 2*g*ArcCosh[c*x] + (c*f*Sqrt[(- 
1 + c*x)/(1 + c*x)]*ArcCosh[c*x]^2)/(1 - c*x) + (2*(-(c*f) + g)*(c*f + g)* 
(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)*Cot 
h[ArcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + ArcTan[((-(c*f) + g)*Tanh[Arc 
Cosh[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^(Arc 
Cosh[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[Arc 
Cosh[c*x]/2]))] - (ArcCos[-((c*f)/g)] - 2*ArcTan[((-(c*f) + g)*Tanh[ArcCos 
h[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[((c*f + g)*(-(c*f) + g + I*Sqrt...
 

Rubi [A] (verified)

Time = 4.44 (sec) , antiderivative size = 538, normalized size of antiderivative = 0.72, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {6387, 6391, 6385, 25, 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.

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

\(\Big \downarrow \) 6387

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

\(\Big \downarrow \) 6391

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

\(\Big \downarrow \) 6385

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 6408

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

\(\Big \downarrow \) 2009

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

Input:

Int[(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f + g*x),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)) - ((c^2*x*(a + b*ArcCosh[c*x])^2)/g - ((1 - (c^2*f^2)/g^2)*(a + b* 
ArcCosh[c*x])^2)/(f + g*x) + 2*b*c*((b*c*x)/g + (a*(1 - c^2*x^2))/(g*Sqrt[ 
-1 + c*x]*Sqrt[1 + c*x]) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/g 
 - (a*Sqrt[c^2*f^2 - g^2]*Sqrt[-1 + c^2*x^2]*ArcTanh[(g + c^2*f*x)/(Sqrt[c 
^2*f^2 - g^2]*Sqrt[-1 + c^2*x^2])])/(g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ( 
b*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*Sqrt[c^2*f^2 - g^2]*ArcCosh[c*x]*Log[1 + (E^Arc 
Cosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/g^2 - (b*Sqrt[c^2*f^2 - g^2]*Po 
lyLog[2, -((E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/g^2 + (b*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))/(2*b*c)))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])
 

Defintions of rubi rules used

rule 25

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

rule 2009

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

rule 6385

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.) + (g_.)*(x_) + (h_.)*( 
x_)^2)^(p_.))/((d_) + (e_.)*(x_))^2, x_Symbol] :> With[{u = IntHide[(f + g* 
x + h*x^2)^p/(d + e*x)^2, x]}, Simp[(a + b*ArcCosh[c*x])^n   u, x] - Simp[b 
*c*n   Int[SimplifyIntegrand[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, h}, x] && IGt 
Q[n, 0] && IGtQ[p, 0] && EqQ[e*g - 2*d*h, 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 [A] (verified)

Time = 0.56 (sec) , antiderivative size = 925, normalized size of antiderivative = 1.24

method	result	size
default	\(\frac {a \left (\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}}}+\frac {c^{2} d f \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\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}}}}\right )}{g \sqrt {c^{2} d}}+\frac {d \left (c^{2} f^{2}-g^{2}\right ) \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^{2} \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}\right )}{g}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2} f c}{2 \sqrt {c x -1}\, \sqrt {c x +1}\, g^{2}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x^{2} c^{2}}{\left (c x -1\right ) \left (c x +1\right ) g}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x c}{\sqrt {c x -1}\, \sqrt {c x +1}\, g}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )}{\left (c x -1\right ) \left (c x +1\right ) g}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \operatorname {arccosh}\left (c x \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 )}{\sqrt {c x -1}\, \sqrt {c x +1}\, g^{2}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \operatorname {arccosh}\left (c x \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 )}{\sqrt {c x -1}\, \sqrt {c x +1}\, g^{2}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \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 )}{\sqrt {c x -1}\, \sqrt {c x +1}\, g^{2}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \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 )}{\sqrt {c x -1}\, \sqrt {c x +1}\, g^{2}}\)	\(925\)
parts	\(\frac {a \left (\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}}}+\frac {c^{2} d f \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\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}}}}\right )}{g \sqrt {c^{2} d}}+\frac {d \left (c^{2} f^{2}-g^{2}\right ) \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^{2} \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}\right )}{g}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2} f c}{2 \sqrt {c x -1}\, \sqrt {c x +1}\, g^{2}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x^{2} c^{2}}{\left (c x -1\right ) \left (c x +1\right ) g}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x c}{\sqrt {c x -1}\, \sqrt {c x +1}\, g}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )}{\left (c x -1\right ) \left (c x +1\right ) g}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \operatorname {arccosh}\left (c x \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 )}{\sqrt {c x -1}\, \sqrt {c x +1}\, g^{2}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \operatorname {arccosh}\left (c x \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 )}{\sqrt {c x -1}\, \sqrt {c x +1}\, g^{2}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \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 )}{\sqrt {c x -1}\, \sqrt {c x +1}\, g^{2}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \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 )}{\sqrt {c x -1}\, \sqrt {c x +1}\, g^{2}}\)	\(925\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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