\(\int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{f+g x} \, dx\) [4]

Optimal result

Integrand size = 31, antiderivative size = 725 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{f+g x} \, dx=\frac {a \sqrt {d-c^2 d x^2}}{g}+\frac {b c x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}+\frac {b \sqrt {d-c^2 d x^2} \arccos (c x)}{g}-\frac {c x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b g \sqrt {1-c^2 x^2}}+\frac {\left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x) \sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x)}-\frac {a \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \arctan \left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}-\frac {i b \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \arccos (c x) \log \left (1+\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {i b \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \arccos (c x) \log \left (1+\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}-\frac {b \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {b \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}} \] Output:

a*(-c^2*d*x^2+d)^(1/2)/g+b*c*x*(-c^2*d*x^2+d)^(1/2)/g/(-c^2*x^2+1)^(1/2)+b 
*(-c^2*d*x^2+d)^(1/2)*arccos(c*x)/g-1/2*c*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arcc 
os(c*x))^2/b/g/(-c^2*x^2+1)^(1/2)+1/2*(1-c^2*f^2/g^2)*(-c^2*d*x^2+d)^(1/2) 
*(a+b*arccos(c*x))^2/b/c/(g*x+f)/(-c^2*x^2+1)^(1/2)-1/2*(-c^2*x^2+1)^(1/2) 
*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2/b/c/(g*x+f)-a*(c^2*f^2-g^2)^(1/2 
)*(-c^2*d*x^2+d)^(1/2)*arctan((c^2*f*x+g)/(c^2*f^2-g^2)^(1/2)/(-c^2*x^2+1) 
^(1/2))/g^2/(-c^2*x^2+1)^(1/2)-I*b*(c^2*f^2-g^2)^(1/2)*(-c^2*d*x^2+d)^(1/2 
)*arccos(c*x)*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2))) 
/g^2/(-c^2*x^2+1)^(1/2)+I*b*(c^2*f^2-g^2)^(1/2)*(-c^2*d*x^2+d)^(1/2)*arcco 
s(c*x)*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))/g^2/(- 
c^2*x^2+1)^(1/2)-b*(c^2*f^2-g^2)^(1/2)*(-c^2*d*x^2+d)^(1/2)*polylog(2,-(c* 
x+I*(-c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))/g^2/(-c^2*x^2+1)^(1/2 
)+b*(c^2*f^2-g^2)^(1/2)*(-c^2*d*x^2+d)^(1/2)*polylog(2,-(c*x+I*(-c^2*x^2+1 
)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))/g^2/(-c^2*x^2+1)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 2.49 (sec) , antiderivative size = 1095, normalized size of antiderivative = 1.51 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{f+g x} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/2*(-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]*L 
og[f + g*x] + 2*a*Sqrt[d]*Sqrt[-(c^2*f^2) + g^2]*Log[d*(g + c^2*f*x) + Sqr 
t[d]*Sqrt[-(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^2*x^2] - 2*g*ArcCos[c*x] + (c*f*ArcCos[c*x]^2)/Sqrt[ 
1 - c^2*x^2] + (2*(-(c*f) + g)*(c*f + g)*(2*ArcCos[c*x]*ArcTanh[((c*f + g) 
*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] - 2*ArcCos[-((c*f)/g)]*ArcTan 
h[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + (ArcCos[-((c 
*f)/g)] - (2*I)*ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g 
^2]] + (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g 
^2]])*Log[Sqrt[-(c^2*f^2) + g^2]/(Sqrt[2]*E^((I/2)*ArcCos[c*x])*Sqrt[g]*Sq 
rt[c*(f + g*x)])] + (ArcCos[-((c*f)/g)] + (2*I)*(ArcTanh[((c*f + g)*Cot[Ar 
cCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] - ArcTanh[((-(c*f) + g)*Tan[ArcCos[c 
*x]/2])/Sqrt[-(c^2*f^2) + g^2]]))*Log[(E^((I/2)*ArcCos[c*x])*Sqrt[-(c^2*f^ 
2) + g^2])/(Sqrt[2]*Sqrt[g]*Sqrt[c*(f + g*x)])] - (ArcCos[-((c*f)/g)] - (2 
*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log 
[((c*f + g)*((-I)*c*f + I*g + Sqrt[-(c^2*f^2) + g^2])*(-I + Tan[ArcCos[c*x 
]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))] - (ArcCo 
s[-((c*f)/g)] + (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2 
*f^2) + g^2]])*Log[((c*f + g)*(I*c*f - I*g + Sqrt[-(c^2*f^2) + g^2])*(I...
 

Rubi [A] (verified)

Time = 2.32 (sec) , antiderivative size = 506, normalized size of antiderivative = 0.70, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {5277, 5265, 25, 5257, 25, 5299, 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.

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

\(\Big \downarrow \) 5277

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

\(\Big \downarrow \) 5265

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 5257

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 5299

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(Sqrt[d - c^2*d*x^2]*(-1/2*((1 - c^2*x^2)*(a + b*ArcCos[c*x])^2)/(b*c*(f + 
 g*x)) - ((c^2*x*(a + b*ArcCos[c*x])^2)/g - ((1 - (c^2*f^2)/g^2)*(a + b*Ar 
cCos[c*x])^2)/(f + g*x) - 2*b*c*((b*c*x)/g + (a*Sqrt[1 - c^2*x^2])/g + (b* 
Sqrt[1 - c^2*x^2]*ArcCos[c*x])/g - (a*Sqrt[c^2*f^2 - g^2]*ArcTan[(g + c^2* 
f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])])/g^2 - (I*b*Sqrt[c^2*f^2 - g 
^2]*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] 
)/g^2 + (I*b*Sqrt[c^2*f^2 - g^2]*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[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^(I*ArcCos[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^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/g 
^2))/(2*b*c)))/Sqrt[1 - c^2*x^2]
 

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 5257

Int[(((a_.) + ArcCos[(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*ArcCos[c*x])^n   u, x] + Simp[b*c 
*n   Int[SimplifyIntegrand[u*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2] 
), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[n, 0] && IGtQ[ 
p, 0] && EqQ[e*g - 2*d*h, 0]
 

rule 5265

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_)*Sqrt[ 
(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(-(f + g*x)^m)*(d + e*x^2)*((a + b* 
ArcCos[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1))), x] + Simp[1/(b*c*Sqrt[d]*(n + 
1))   Int[(d*g*m + 2*e*f*x + e*g*(m + 2)*x^2)*(f + g*x)^(m - 1)*(a + b*ArcC 
os[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + 
e, 0] && ILtQ[m, 0] && GtQ[d, 0] && IGtQ[n, 0]
 

rule 5277

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ 
) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[Simp[(d + e*x^2)^p/(1 - c^2*x^2)^ 
p]   Int[(f + g*x)^m*(1 - c^2*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] /; FreeQ 
[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && Intege 
rQ[p - 1/2] &&  !GtQ[d, 0]
 

rule 5299

Int[(ArcCos[(c_.)*(x_)]*(b_.) + (a_))^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p 
_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p, RFx*(a + b*ArcCos[c*x]) 
^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && RationalFunctionQ[RFx, x] && IGt 
Q[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]
 

Maple [A] (verified)

Time = 0.81 (sec) , antiderivative size = 816, normalized size of antiderivative = 1.13

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}+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} f c}{2 \left (c^{2} x^{2}-1\right ) g^{2}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )+i\right )}{2 \left (c^{2} x^{2}-1\right ) g}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )-i\right )}{2 \left (c^{2} x^{2}-1\right ) g}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arccos \left (c x \right ) \ln \left (\frac {-\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}-g^{2}}}{-c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )-i \arccos \left (c x \right ) \ln \left (\frac {\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}-g^{2}}}{c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )-\operatorname {dilog}\left (\frac {\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}-g^{2}}}{c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )+\operatorname {dilog}\left (\frac {-\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}-g^{2}}}{-c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )\right )}{\left (c^{2} x^{2}-1\right ) g^{2}}\right )\)	\(816\)
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}+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} f c}{2 \left (c^{2} x^{2}-1\right ) g^{2}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )+i\right )}{2 \left (c^{2} x^{2}-1\right ) g}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arccos \left (c x \right )-i\right )}{2 \left (c^{2} x^{2}-1\right ) g}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arccos \left (c x \right ) \ln \left (\frac {-\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}-g^{2}}}{-c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )-i \arccos \left (c x \right ) \ln \left (\frac {\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}-g^{2}}}{c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )-\operatorname {dilog}\left (\frac {\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}-g^{2}}}{c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )+\operatorname {dilog}\left (\frac {-\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}-g^{2}}}{-c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )\right )}{\left (c^{2} x^{2}-1\right ) g^{2}}\right )\)	\(816\)

Input:

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

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)))+b*(1/2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2- 
1)*arccos(c*x)^2*f*c/g^2+1/2*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)* 
c*x+c^2*x^2-1)*(arccos(c*x)+I)/(c^2*x^2-1)/g+1/2*(-d*(c^2*x^2-1))^(1/2)*(- 
I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(arccos(c*x)-I)/(c^2*x^2-1)/g+(-d*(c^2 
*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)*(-c^2*x^2+1)^(1/2)*(I*arccos(c*x)*ln((- 
(c*x+I*(-c^2*x^2+1)^(1/2))*g-c*f+(c^2*f^2-g^2)^(1/2))/(-c*f+(c^2*f^2-g^2)^ 
(1/2)))-I*arccos(c*x)*ln(((c*x+I*(-c^2*x^2+1)^(1/2))*g+c*f+(c^2*f^2-g^2)^( 
1/2))/(c*f+(c^2*f^2-g^2)^(1/2)))-dilog(((c*x+I*(-c^2*x^2+1)^(1/2))*g+c*f+( 
c^2*f^2-g^2)^(1/2))/(c*f+(c^2*f^2-g^2)^(1/2)))+dilog((-(c*x+I*(-c^2*x^2+1) 
^(1/2))*g-c*f+(c^2*f^2-g^2)^(1/2))/(-c*f+(c^2*f^2-g^2)^(1/2))))/(c^2*x^2-1 
)/g^2)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

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

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

Output:

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

Reduce [F]

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

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