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

Optimal. Leaf size=725 \[ \frac {\sqrt {d-c^2 d x^2} \left (1-\frac {c^2 f^2}{g^2}\right ) \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \sqrt {1-c^2 x^2} (f+g x)}-\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac {c x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {a \sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^2} \tan ^{-1}\left (\frac {c^2 f x+g}{\sqrt {1-c^2 x^2} \sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {a \sqrt {d-c^2 d x^2}}{g}-\frac {b \sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^2} \text {Li}_2\left (-\frac {e^{i \cos ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {b \sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^2} \text {Li}_2\left (-\frac {e^{i \cos ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}-\frac {i b \sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^2} \cos ^{-1}(c x) \log \left (1+\frac {g e^{i \cos ^{-1}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {i b \sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^2} \cos ^{-1}(c x) \log \left (1+\frac {g e^{i \cos ^{-1}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {b c x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}+\frac {b \sqrt {d-c^2 d x^2} \cos ^{-1}(c x)}{g} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.86, antiderivative size = 725, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 19, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.613, Rules used = {4778, 4766, 683, 4758, 6742, 725, 204, 1654, 12, 4800, 4798, 4678, 8, 4774, 3321, 2264, 2190, 2279, 2391} \[ -\frac {b \sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^2} \text {PolyLog}\left (2,-\frac {g e^{i \cos ^{-1}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {b \sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^2} \text {PolyLog}\left (2,-\frac {g e^{i \cos ^{-1}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \left (1-\frac {c^2 f^2}{g^2}\right ) \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \sqrt {1-c^2 x^2} (f+g x)}-\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac {c x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g \sqrt {1-c^2 x^2}}-\frac {a \sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^2} \tan ^{-1}\left (\frac {c^2 f x+g}{\sqrt {1-c^2 x^2} \sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {a \sqrt {d-c^2 d x^2}}{g}-\frac {i b \sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^2} \cos ^{-1}(c x) \log \left (1+\frac {g e^{i \cos ^{-1}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {i b \sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^2} \cos ^{-1}(c x) \log \left (1+\frac {g e^{i \cos ^{-1}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {b c x \sqrt {d-c^2 d x^2}}{g \sqrt {1-c^2 x^2}}+\frac {b \sqrt {d-c^2 d x^2} \cos ^{-1}(c x)}{g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 1654

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + c*x^2)^(p + 1))/(c*e^(q - 1)*(m + q + 2*p + 1)), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 0]))

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(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 2264

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]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(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 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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 2391

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 3321

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

Rule 4678

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^
(p + 1)*(a + b*ArcCos[c*x])^n)/(2*e*(p + 1)), x] - Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[p]), Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x] /; FreeQ[{a, b,
c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rule 4758

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]}, Dist[(a + b*ArcCos[c*x])^n, u, x] + Dist
[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 4766

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] + Dist[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*ArcCos[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 4774

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

Rule 4778

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Dist[(d^IntPart[p]*(d + e*x^2)^FracPart[p])/(1 - c^2*x^2)^FracPart[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] && IntegerQ
[p - 1/2] &&  !GtQ[d, 0]

Rule 4798

Int[ArcCos[(c_.)*(x_)]^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = ExpandIntegrand[(d + e*
x^2)^p*ArcCos[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{c, d, e}, x] && RationalFunctionQ[RFx, x] && I
GtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]

Rule 4800

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

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {\sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{f+g x} \, dx &=\frac {\sqrt {d-c^2 d x^2} \int \frac {\sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )}{f+g x} \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c (f+g x)}+\frac {\sqrt {d-c^2 d x^2} \int \frac {\left (-g-2 c^2 f x-c^2 g x^2\right ) \left (a+b \cos ^{-1}(c x)\right )^2}{(f+g x)^2} \, dx}{2 b c \sqrt {1-c^2 x^2}}\\ &=-\frac {c x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^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} \left (a+b \cos ^{-1}(c x)\right )^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} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c (f+g x)}+\frac {\sqrt {d-c^2 d x^2} \int \frac {\left (\frac {1}{f+g x}-\frac {c^2 \left (g x+\frac {f^2}{f+g x}\right )}{g^2}\right ) \left (a+b \cos ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {c x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^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} \left (a+b \cos ^{-1}(c x)\right )^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} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c (f+g x)}+\frac {\sqrt {d-c^2 d x^2} \int \left (-\frac {a \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right )}{g^2 (f+g x) \sqrt {1-c^2 x^2}}-\frac {b \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \cos ^{-1}(c x)}{g^2 (f+g x) \sqrt {1-c^2 x^2}}\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {c x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^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} \left (a+b \cos ^{-1}(c x)\right )^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} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac {\left (a \sqrt {d-c^2 d x^2}\right ) \int \frac {c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^2 \sqrt {1-c^2 x^2}}-\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \cos ^{-1}(c x)}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^2 \sqrt {1-c^2 x^2}}\\ &=\frac {a \sqrt {d-c^2 d x^2}}{g}-\frac {c x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^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} \left (a+b \cos ^{-1}(c x)\right )^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} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac {\left (a \sqrt {d-c^2 d x^2}\right ) \int \frac {c^2 g^2 \left (c^2 f^2-g^2\right )}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{c^2 g^4 \sqrt {1-c^2 x^2}}-\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int \left (\frac {c^2 g x \cos ^{-1}(c x)}{\sqrt {1-c^2 x^2}}+\frac {\left (c^2 f^2-g^2\right ) \cos ^{-1}(c x)}{(f+g x) \sqrt {1-c^2 x^2}}\right ) \, dx}{g^2 \sqrt {1-c^2 x^2}}\\ &=\frac {a \sqrt {d-c^2 d x^2}}{g}-\frac {c x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^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} \left (a+b \cos ^{-1}(c x)\right )^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} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac {\left (b c^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x \cos ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{g \sqrt {1-c^2 x^2}}-\frac {\left (a (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^2 \sqrt {1-c^2 x^2}}-\frac {\left (b (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \int \frac {\cos ^{-1}(c x)}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{g^2 \sqrt {1-c^2 x^2}}\\ &=\frac {a \sqrt {d-c^2 d x^2}}{g}+\frac {b \sqrt {d-c^2 d x^2} \cos ^{-1}(c x)}{g}-\frac {c x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^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} \left (a+b \cos ^{-1}(c x)\right )^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} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c (f+g x)}+\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int 1 \, dx}{g \sqrt {1-c^2 x^2}}+\frac {\left (a (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-c^2 f^2+g^2-x^2} \, dx,x,\frac {g+c^2 f x}{\sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {\left (b (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{c f+g \cos (x)} \, dx,x,\cos ^{-1}(c x)\right )}{g^2 \sqrt {1-c^2 x^2}}\\ &=\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} \cos ^{-1}(c x)}{g}-\frac {c x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^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} \left (a+b \cos ^{-1}(c x)\right )^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} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac {a (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {\left (2 b (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x}{2 c e^{i x} f+g+e^{2 i x} g} \, dx,x,\cos ^{-1}(c x)\right )}{g^2 \sqrt {1-c^2 x^2}}\\ &=\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} \cos ^{-1}(c x)}{g}-\frac {c x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^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} \left (a+b \cos ^{-1}(c x)\right )^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} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac {a (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {\left (2 b (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x}{2 c f+2 e^{i x} g-2 \sqrt {c^2 f^2-g^2}} \, dx,x,\cos ^{-1}(c x)\right )}{g \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {\left (2 b (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x}{2 c f+2 e^{i x} g+2 \sqrt {c^2 f^2-g^2}} \, dx,x,\cos ^{-1}(c x)\right )}{g \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\\ &=\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} \cos ^{-1}(c x)}{g}-\frac {c x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^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} \left (a+b \cos ^{-1}(c x)\right )^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} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac {a (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {i b (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \cos ^{-1}(c x) \log \left (1+\frac {e^{i \cos ^{-1}(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^2 x^2}}+\frac {i b (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \cos ^{-1}(c x) \log \left (1+\frac {e^{i \cos ^{-1}(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^2 x^2}}+\frac {\left (i b (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+\frac {2 e^{i x} g}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {\left (i b (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+\frac {2 e^{i x} g}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\\ &=\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} \cos ^{-1}(c x)}{g}-\frac {c x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^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} \left (a+b \cos ^{-1}(c x)\right )^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} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac {a (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {i b (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \cos ^{-1}(c x) \log \left (1+\frac {e^{i \cos ^{-1}(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^2 x^2}}+\frac {i b (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \cos ^{-1}(c x) \log \left (1+\frac {e^{i \cos ^{-1}(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^2 x^2}}+\frac {\left (b (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 g x}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \cos ^{-1}(c x)}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {\left (b (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 g x}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \cos ^{-1}(c x)}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\\ &=\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} \cos ^{-1}(c x)}{g}-\frac {c x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^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} \left (a+b \cos ^{-1}(c x)\right )^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} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac {a (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {i b (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \cos ^{-1}(c x) \log \left (1+\frac {e^{i \cos ^{-1}(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^2 x^2}}+\frac {i b (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \cos ^{-1}(c x) \log \left (1+\frac {e^{i \cos ^{-1}(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^2 x^2}}-\frac {b (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \text {Li}_2\left (-\frac {e^{i \cos ^{-1}(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^2 x^2}}+\frac {b (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \text {Li}_2\left (-\frac {e^{i \cos ^{-1}(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^2 x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 3.69, size = 1095, normalized size = 1.51 \[ -\frac {-2 a \sqrt {d-c^2 d x^2} g+2 a c \sqrt {d} f \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )-2 a \sqrt {d} \sqrt {g^2-c^2 f^2} \log (f+g x)+2 a \sqrt {d} \sqrt {g^2-c^2 f^2} \log \left (d \left (f x c^2+g\right )+\sqrt {d} \sqrt {g^2-c^2 f^2} \sqrt {d-c^2 d x^2}\right )+b \sqrt {d-c^2 d x^2} \left (\frac {c f \cos ^{-1}(c x)^2}{\sqrt {1-c^2 x^2}}-2 g \cos ^{-1}(c x)+\frac {2 (g-c f) (c f+g) \left (2 \cos ^{-1}(c x) \tanh ^{-1}\left (\frac {(c f+g) \cot \left (\frac {1}{2} \cos ^{-1}(c x)\right )}{\sqrt {g^2-c^2 f^2}}\right )-2 \cos ^{-1}\left (-\frac {c f}{g}\right ) \tanh ^{-1}\left (\frac {(g-c f) \tan \left (\frac {1}{2} \cos ^{-1}(c x)\right )}{\sqrt {g^2-c^2 f^2}}\right )+\left (\cos ^{-1}\left (-\frac {c f}{g}\right )-2 i \tanh ^{-1}\left (\frac {(c f+g) \cot \left (\frac {1}{2} \cos ^{-1}(c x)\right )}{\sqrt {g^2-c^2 f^2}}\right )+2 i \tanh ^{-1}\left (\frac {(g-c f) \tan \left (\frac {1}{2} \cos ^{-1}(c x)\right )}{\sqrt {g^2-c^2 f^2}}\right )\right ) \log \left (\frac {e^{-\frac {1}{2} i \cos ^{-1}(c x)} \sqrt {g^2-c^2 f^2}}{\sqrt {2} \sqrt {g} \sqrt {c (f+g x)}}\right )+\left (\cos ^{-1}\left (-\frac {c f}{g}\right )+2 i \left (\tanh ^{-1}\left (\frac {(c f+g) \cot \left (\frac {1}{2} \cos ^{-1}(c x)\right )}{\sqrt {g^2-c^2 f^2}}\right )-\tanh ^{-1}\left (\frac {(g-c f) \tan \left (\frac {1}{2} \cos ^{-1}(c x)\right )}{\sqrt {g^2-c^2 f^2}}\right )\right )\right ) \log \left (\frac {e^{\frac {1}{2} i \cos ^{-1}(c x)} \sqrt {g^2-c^2 f^2}}{\sqrt {2} \sqrt {g} \sqrt {c (f+g x)}}\right )-\left (\cos ^{-1}\left (-\frac {c f}{g}\right )-2 i \tanh ^{-1}\left (\frac {(g-c f) \tan \left (\frac {1}{2} \cos ^{-1}(c x)\right )}{\sqrt {g^2-c^2 f^2}}\right )\right ) \log \left (\frac {(c f+g) \left (-i c f+i g+\sqrt {g^2-c^2 f^2}\right ) \left (\tan \left (\frac {1}{2} \cos ^{-1}(c x)\right )-i\right )}{g \left (c f+g+\sqrt {g^2-c^2 f^2} \tan \left (\frac {1}{2} \cos ^{-1}(c x)\right )\right )}\right )-\left (\cos ^{-1}\left (-\frac {c f}{g}\right )+2 i \tanh ^{-1}\left (\frac {(g-c f) \tan \left (\frac {1}{2} \cos ^{-1}(c x)\right )}{\sqrt {g^2-c^2 f^2}}\right )\right ) \log \left (\frac {(c f+g) \left (i c f-i g+\sqrt {g^2-c^2 f^2}\right ) \left (\tan \left (\frac {1}{2} \cos ^{-1}(c x)\right )+i\right )}{g \left (c f+g+\sqrt {g^2-c^2 f^2} \tan \left (\frac {1}{2} \cos ^{-1}(c x)\right )\right )}\right )+i \left (\text {Li}_2\left (\frac {\left (c f-i \sqrt {g^2-c^2 f^2}\right ) \left (c f+g-\sqrt {g^2-c^2 f^2} \tan \left (\frac {1}{2} \cos ^{-1}(c x)\right )\right )}{g \left (c f+g+\sqrt {g^2-c^2 f^2} \tan \left (\frac {1}{2} \cos ^{-1}(c x)\right )\right )}\right )-\text {Li}_2\left (\frac {\left (c f+i \sqrt {g^2-c^2 f^2}\right ) \left (c f+g-\sqrt {g^2-c^2 f^2} \tan \left (\frac {1}{2} \cos ^{-1}(c x)\right )\right )}{g \left (c f+g+\sqrt {g^2-c^2 f^2} \tan \left (\frac {1}{2} \cos ^{-1}(c x)\right )\right )}\right )\right )\right )}{\sqrt {g^2-c^2 f^2} \sqrt {1-c^2 x^2}}-\frac {2 c g x}{\sqrt {1-c^2 x^2}}\right )}{2 g^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-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]*Log[f + g*x] + 2*a*Sqrt[d]*Sqrt[-(c^2*f^2) + g^2]*Log[d*(g + c^2*f*x) + S
qrt[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)]*ArcTanh[((-(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]*Sqrt[c*(f + g*x)])] + (ArcCos[-((c*f)/g)] + (2*I)*(ArcTanh[((c*f + g
)*Cot[ArcCos[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]
))] - (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]*Ta
n[ArcCos[c*x]/2]))] + I*(PolyLog[2, ((c*f - I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Tan[Ar
cCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))] - PolyLog[2, ((c*f + I*Sqrt[-(c^2*f^
2) + g^2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[Arc
Cos[c*x]/2]))])))/(Sqrt[-(c^2*f^2) + g^2]*Sqrt[1 - c^2*x^2])))/g^2

________________________________________________________________________________________

fricas [F]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \arccos \left (c x\right ) + a\right )}}{g x + f}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

________________________________________________________________________________________

maple [A]  time = 0.96, size = 1209, normalized size = 1.67 \[ \frac {a \sqrt {-c^{2} d \left (x +\frac {f}{g}\right )^{2}+\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}}}}{g}+\frac {a \,c^{2} d f \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \left (x +\frac {f}{g}\right )^{2}+\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^{2} \sqrt {c^{2} d}}+\frac {a d \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 {-c^{2} d \left (x +\frac {f}{g}\right )^{2}+\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 ) c^{2} f^{2}}{g^{3} \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}-\frac {a d \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 {-c^{2} d \left (x +\frac {f}{g}\right )^{2}+\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}}}}+\frac {b \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 {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, x c}{\left (c^{2} x^{2}-1\right ) g}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) x^{2} c^{2}}{\left (c^{2} x^{2}-1\right ) g}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right )}{\left (c^{2} x^{2}-1\right ) g}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \sqrt {-c^{2} x^{2}+1}\, \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 ) \arccos \left (c x \right )}{\left (c^{2} x^{2}-1\right ) g^{2}}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \sqrt {-c^{2} x^{2}+1}\, \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 ) \arccos \left (c x \right )}{\left (c^{2} x^{2}-1\right ) g^{2}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \sqrt {-c^{2} x^{2}+1}\, \dilog \left (-\frac {\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) g}{-c f +\sqrt {c^{2} f^{2}-g^{2}}}-\frac {c f}{-c f +\sqrt {c^{2} f^{2}-g^{2}}}+\frac {\sqrt {c^{2} f^{2}-g^{2}}}{-c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )}{\left (c^{2} x^{2}-1\right ) g^{2}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \sqrt {-c^{2} x^{2}+1}\, \dilog \left (\frac {\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) g}{c f +\sqrt {c^{2} f^{2}-g^{2}}}+\frac {c f}{c f +\sqrt {c^{2} f^{2}-g^{2}}}+\frac {\sqrt {c^{2} f^{2}-g^{2}}}{c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )}{\left (c^{2} x^{2}-1\right ) g^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a/g*(-c^2*d*(x+f/g)^2+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2)+a/g^2*c^2*d*f/(c^2*d)^(1/2)*arctan((c^2*d
)^(1/2)*x/(-c^2*d*(x+f/g)^2+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2))+a/g^3*d/(-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)*(-c^2*d*(x+f/g)^2+2*c^2*d*f/g
*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2))/(x+f/g))*c^2*f^2-a/g*d/(-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)*(-c^2*d*(x+f/g)^2+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^2*x^2+1)^(1/2)/(c^2*x^2-1)*arccos(c*x)^2*f*c/g^2-b*(-d*
(c^2*x^2-1))^(1/2)/(c^2*x^2-1)/g*(-c^2*x^2+1)^(1/2)*x*c+b*(-d*(c^2*x^2-1))^(1/2)/(c^2*x^2-1)/g*arccos(c*x)*x^2
*c^2-b*(-d*(c^2*x^2-1))^(1/2)/(c^2*x^2-1)/g*arccos(c*x)+I*b*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)*(-c^2*x
^2+1)^(1/2)/(c^2*x^2-1)/g^2*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)))*arccos(c*x)-I*b*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)/g^2*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)))*arccos(c*x)+b*(-d*(c^2*x^2-1))^(1/2)
*(c^2*f^2-g^2)^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)/g^2*dilog(-1/(-c*f+(c^2*f^2-g^2)^(1/2))*(c*x+I*(-c^2*x^2+1
)^(1/2))*g-1/(-c*f+(c^2*f^2-g^2)^(1/2))*c*f+1/(-c*f+(c^2*f^2-g^2)^(1/2))*(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^2*x^2+1)^(1/2)/(c^2*x^2-1)/g^2*dilog((c*x+I*(-c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f
^2-g^2)^(1/2))+1/(c*f+(c^2*f^2-g^2)^(1/2))*c*f+1/(c*f+(c^2*f^2-g^2)^(1/2))*(c^2*f^2-g^2)^(1/2))

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2}}{f+g\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acos}{\left (c x \right )}\right )}{f + g x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________