3.1.5 \(\int \frac {\sqrt {d-c^2 d x^2} (a+b \text {ArcCos}(c x))}{(f+g x)^2} \, dx\) [5]

Optimal. Leaf size=851 \[ -\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}-\frac {b \sqrt {d-c^2 d x^2} \text {ArcCos}(c x)}{g (f+g x)}+\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \text {ArcCos}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcCos}(c x))^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \text {ArcCos}(c x))^2}{2 b c (f+g x)^2}-\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}+\frac {a c^2 f \sqrt {d-c^2 d x^2} \text {ArcTan}\left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {i b c^2 f \sqrt {d-c^2 d x^2} \text {ArcCos}(c x) \log \left (1+\frac {e^{i \text {ArcCos}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {i b c^2 f \sqrt {d-c^2 d x^2} \text {ArcCos}(c x) \log \left (1+\frac {e^{i \text {ArcCos}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {b c \sqrt {d-c^2 d x^2} \log (f+g x)}{g^2 \sqrt {1-c^2 x^2}}+\frac {b c^2 f \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,-\frac {e^{i \text {ArcCos}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {b c^2 f \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,-\frac {e^{i \text {ArcCos}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 1.92, antiderivative size = 851, normalized size of antiderivative = 1.00, number of steps used = 35, number of rules used = 22, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.710, Rules used = {4862, 4850, 37, 4840, 12, 1665, 858, 222, 739, 210, 4884, 4882, 4738, 4858, 3405, 3402, 2296, 2221, 2317, 2438, 2747, 31} \begin {gather*} \frac {b f^2 \sqrt {d-c^2 d x^2} \text {ArcCos}(c x)^2 c^3}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {a f^2 \sqrt {d-c^2 d x^2} \text {ArcSin}(c x) c^3}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}+\frac {a f \sqrt {d-c^2 d x^2} \text {ArcTan}\left (\frac {f x c^2+g}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right ) c^2}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {i b f \sqrt {d-c^2 d x^2} \text {ArcCos}(c x) \log \left (\frac {e^{i \text {ArcCos}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}+1\right ) c^2}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {i b f \sqrt {d-c^2 d x^2} \text {ArcCos}(c x) \log \left (\frac {e^{i \text {ArcCos}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}+1\right ) c^2}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {b f \sqrt {d-c^2 d x^2} \text {Li}_2\left (-\frac {e^{i \text {ArcCos}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right ) c^2}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {b f \sqrt {d-c^2 d x^2} \text {Li}_2\left (-\frac {e^{i \text {ArcCos}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right ) c^2}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {b \sqrt {d-c^2 d x^2} \log (f+g x) c}{g^2 \sqrt {1-c^2 x^2}}-\frac {b \sqrt {d-c^2 d x^2} \text {ArcCos}(c x)}{g (f+g x)}-\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}-\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \text {ArcCos}(c x))^2}{2 b (f+g x)^2 c}-\frac {\left (f x c^2+g\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcCos}(c x))^2}{2 b \left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt {1-c^2 x^2} c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*Sqrt[d - c^2*d*x^2])/(g*(f + g*x))) - (b*Sqrt[d - c^2*d*x^2]*ArcCos[c*x])/(g*(f + g*x)) + (b*c^3*f^2*Sqrt
[d - c^2*d*x^2]*ArcCos[c*x]^2)/(2*g^2*(c^2*f^2 - g^2)*Sqrt[1 - c^2*x^2]) - ((g + c^2*f*x)^2*Sqrt[d - c^2*d*x^2
]*(a + b*ArcCos[c*x])^2)/(2*b*c*(c^2*f^2 - g^2)*(f + g*x)^2*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)^2) - (a*c^3*f^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(g^2*(c^2*f
^2 - g^2)*Sqrt[1 - c^2*x^2]) + (a*c^2*f*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[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2]) + (I*b*c^2*f*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[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2]) - (I*b*c^2*f*S
qrt[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[c^2*f^2 -
g^2]*Sqrt[1 - c^2*x^2]) - (b*c*Sqrt[d - c^2*d*x^2]*Log[f + g*x])/(g^2*Sqrt[1 - c^2*x^2]) + (b*c^2*f*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[c^2*f^2 - g^2]*Sqrt[1 -
c^2*x^2]) - (b*c^2*f*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[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])

Rule 12

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 210

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

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 739

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 858

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 1665

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, d
+ e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1
)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*ExpandToSum[(m
+ 1)*(c*d^2 + a*e^2)*Q + c*d*R*(m + 1) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e, p}, x] && Po
lyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

Rule 2221

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]
- 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 2296

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 2317

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 2438

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 2747

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 3402

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 3405

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

Rule 4738

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

Rule 4840

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(p_.), x_Symbol] :>
With[{u = IntHide[(f + g*x)^p*(d + e*x)^m, x]}, Dist[(a + b*ArcCos[c*x])^n, u, x] + Dist[b*c*n, Int[SimplifyIn
tegrand[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}, x] && I
GtQ[n, 0] && IGtQ[p, 0] && ILtQ[m, 0] && LtQ[m + p + 1, 0]

Rule 4850

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] /; Fre
eQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && ILtQ[m, 0] && GtQ[d, 0] && IGtQ[n, 0]

Rule 4858

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 4862

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Dist[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] /; F
reeQ[{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 4882

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 4884

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]

Rubi steps

\begin {align*} \int \frac {\sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{(f+g x)^2} \, 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)^2} \, 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)^2}+\frac {\sqrt {d-c^2 d x^2} \int \frac {\left (-2 g-2 c^2 f x\right ) \left (a+b \cos ^{-1}(c x)\right )^2}{(f+g x)^3} \, dx}{2 b c \sqrt {1-c^2 x^2}}\\ &=-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \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)^2}-\frac {\sqrt {d-c^2 d x^2} \int \frac {\left (g+c^2 f x\right )^2 \left (a+b \cos ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt {1-c^2 x^2}} \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \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)^2}-\frac {\sqrt {d-c^2 d x^2} \int \frac {\left (g+c^2 f x\right )^2 \left (a+b \cos ^{-1}(c x)\right )}{(f+g x)^2 \sqrt {1-c^2 x^2}} \, dx}{\left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}\\ &=-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \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)^2}-\frac {\sqrt {d-c^2 d x^2} \int \left (\frac {a \left (g+c^2 f x\right )^2}{(f+g x)^2 \sqrt {1-c^2 x^2}}+\frac {b \left (g+c^2 f x\right )^2 \cos ^{-1}(c x)}{(f+g x)^2 \sqrt {1-c^2 x^2}}\right ) \, dx}{\left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}\\ &=-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \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)^2}-\frac {\left (a \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (g+c^2 f x\right )^2}{(f+g x)^2 \sqrt {1-c^2 x^2}} \, dx}{\left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (g+c^2 f x\right )^2 \cos ^{-1}(c x)}{(f+g x)^2 \sqrt {1-c^2 x^2}} \, dx}{\left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}\\ &=-\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \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)^2}-\frac {\left (a \sqrt {d-c^2 d x^2}\right ) \int \frac {c^2 f \left (c^2 f^2-g^2\right )+c^4 f^2 \left (\frac {c^2 f^2}{g}-g\right ) x}{(f+g x) \sqrt {1-c^2 x^2}} \, dx}{\left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2}}-\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int \left (\frac {c^4 f^2 \cos ^{-1}(c x)}{g^2 \sqrt {1-c^2 x^2}}+\frac {\left (-c^2 f^2+g^2\right )^2 \cos ^{-1}(c x)}{g^2 (f+g x)^2 \sqrt {1-c^2 x^2}}+\frac {2 c^2 f \left (-c^2 f^2+g^2\right ) \cos ^{-1}(c x)}{g^2 (f+g x) \sqrt {1-c^2 x^2}}\right ) \, dx}{\left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}\\ &=-\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \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)^2}+\frac {\left (a c^2 f \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 (a c^4 f^2 \left (1-\frac {c^2 f^2}{g^2}\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{\left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2}}-\frac {\left (b c^4 f^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\cos ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {\left (b \left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {\cos ^{-1}(c x)}{(f+g x)^2 \sqrt {1-c^2 x^2}} \, dx}{g^2 \sqrt {1-c^2 x^2}}-\frac {\left (2 b c^2 f \left (-c^2 f^2+g^2\right ) \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 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}\\ &=-\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}+\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \cos ^{-1}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \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)^2}-\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {\left (a c^2 f \sqrt {d-c^2 d x^2}\right ) \text {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 \left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {x}{(c f+g \cos (x))^2} \, dx,x,\cos ^{-1}(c x)\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {\left (2 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {x}{c f+g \cos (x)} \, dx,x,\cos ^{-1}(c x)\right )}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}\\ &=-\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}-\frac {b \sqrt {d-c^2 d x^2} \cos ^{-1}(c x)}{g (f+g x)}+\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \cos ^{-1}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \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)^2}-\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}+\frac {a c^2 f \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 (b c^2 f \sqrt {d-c^2 d x^2}\right ) \text {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 {\left (b c \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\sin (x)}{c f+g \cos (x)} \, dx,x,\cos ^{-1}(c x)\right )}{g \sqrt {1-c^2 x^2}}+\frac {\left (4 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt {d-c^2 d x^2}\right ) \text {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 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}\\ &=-\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}-\frac {b \sqrt {d-c^2 d x^2} \cos ^{-1}(c x)}{g (f+g x)}+\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \cos ^{-1}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \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)^2}-\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}+\frac {a c^2 f \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 (b c \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {1}{c f+x} \, dx,x,c g x\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {\left (2 b c^2 f \sqrt {d-c^2 d x^2}\right ) \text {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 {\left (4 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt {d-c^2 d x^2}\right ) \text {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 \left (c^2 f^2-g^2\right )^{3/2} \sqrt {1-c^2 x^2}}-\frac {\left (4 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt {d-c^2 d x^2}\right ) \text {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 \left (c^2 f^2-g^2\right )^{3/2} \sqrt {1-c^2 x^2}}\\ &=-\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}-\frac {b \sqrt {d-c^2 d x^2} \cos ^{-1}(c x)}{g (f+g x)}+\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \cos ^{-1}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \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)^2}-\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}+\frac {a c^2 f \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 {2 i b c^2 f \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 {2 i b c^2 f \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 \sqrt {d-c^2 d x^2} \log (f+g x)}{g^2 \sqrt {1-c^2 x^2}}+\frac {\left (2 b c^2 f \sqrt {d-c^2 d x^2}\right ) \text {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^2 f \sqrt {d-c^2 d x^2}\right ) \text {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 i b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt {d-c^2 d x^2}\right ) \text {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 \left (c^2 f^2-g^2\right )^{3/2} \sqrt {1-c^2 x^2}}-\frac {\left (2 i b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt {d-c^2 d x^2}\right ) \text {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 \left (c^2 f^2-g^2\right )^{3/2} \sqrt {1-c^2 x^2}}\\ &=-\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}-\frac {b \sqrt {d-c^2 d x^2} \cos ^{-1}(c x)}{g (f+g x)}+\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \cos ^{-1}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \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)^2}-\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}+\frac {a c^2 f \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^2 f \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^2 f \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 \sqrt {d-c^2 d x^2} \log (f+g x)}{g^2 \sqrt {1-c^2 x^2}}+\frac {\left (i b c^2 f \sqrt {d-c^2 d x^2}\right ) \text {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^2 f \sqrt {d-c^2 d x^2}\right ) \text {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 (2 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt {d-c^2 d x^2}\right ) \text {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 \left (c^2 f^2-g^2\right )^{3/2} \sqrt {1-c^2 x^2}}-\frac {\left (2 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt {d-c^2 d x^2}\right ) \text {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 \left (c^2 f^2-g^2\right )^{3/2} \sqrt {1-c^2 x^2}}\\ &=-\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}-\frac {b \sqrt {d-c^2 d x^2} \cos ^{-1}(c x)}{g (f+g x)}+\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \cos ^{-1}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \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)^2}-\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}+\frac {a c^2 f \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^2 f \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^2 f \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 \sqrt {d-c^2 d x^2} \log (f+g x)}{g^2 \sqrt {1-c^2 x^2}}+\frac {2 b c^2 f \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 {2 b c^2 f \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 {\left (b c^2 f \sqrt {d-c^2 d x^2}\right ) \text {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^2 f \sqrt {d-c^2 d x^2}\right ) \text {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 (f+g x)}-\frac {b \sqrt {d-c^2 d x^2} \cos ^{-1}(c x)}{g (f+g x)}+\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \cos ^{-1}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \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)^2}-\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}+\frac {a c^2 f \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^2 f \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^2 f \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 \sqrt {d-c^2 d x^2} \log (f+g x)}{g^2 \sqrt {1-c^2 x^2}}+\frac {b c^2 f \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^2 f \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*}

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Mathematica [A]
time = 6.94, size = 1116, normalized size = 1.31 \begin {gather*} -\frac {\frac {2 a g \sqrt {d-c^2 d x^2}}{f+g x}-2 a c \sqrt {d} \text {ArcTan}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-\frac {2 a c^2 \sqrt {d} f \log (f+g x)}{\sqrt {-c^2 f^2+g^2}}+\frac {2 a c^2 \sqrt {d} f \log \left (d \left (g+c^2 f x\right )+\sqrt {d} \sqrt {-c^2 f^2+g^2} \sqrt {d-c^2 d x^2}\right )}{\sqrt {-c^2 f^2+g^2}}+b c \sqrt {d-c^2 d x^2} \left (\frac {2 g \text {ArcCos}(c x)}{c f+c g x}-\frac {\text {ArcCos}(c x)^2}{\sqrt {1-c^2 x^2}}+\frac {2 \log \left (1+\frac {g x}{f}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 c f \left (2 \text {ArcCos}(c x) \tanh ^{-1}\left (\frac {(c f+g) \cot \left (\frac {1}{2} \text {ArcCos}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )-2 \text {ArcCos}\left (-\frac {c f}{g}\right ) \tanh ^{-1}\left (\frac {(-c f+g) \tan \left (\frac {1}{2} \text {ArcCos}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )+\left (\text {ArcCos}\left (-\frac {c f}{g}\right )-2 i \tanh ^{-1}\left (\frac {(c f+g) \cot \left (\frac {1}{2} \text {ArcCos}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )+2 i \tanh ^{-1}\left (\frac {(-c f+g) \tan \left (\frac {1}{2} \text {ArcCos}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )\right ) \log \left (\frac {e^{-\frac {1}{2} i \text {ArcCos}(c x)} \sqrt {-c^2 f^2+g^2}}{\sqrt {2} \sqrt {g} \sqrt {c (f+g x)}}\right )+\left (\text {ArcCos}\left (-\frac {c f}{g}\right )+2 i \left (\tanh ^{-1}\left (\frac {(c f+g) \cot \left (\frac {1}{2} \text {ArcCos}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )-\tanh ^{-1}\left (\frac {(-c f+g) \tan \left (\frac {1}{2} \text {ArcCos}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )\right )\right ) \log \left (\frac {e^{\frac {1}{2} i \text {ArcCos}(c x)} \sqrt {-c^2 f^2+g^2}}{\sqrt {2} \sqrt {g} \sqrt {c (f+g x)}}\right )-\left (\text {ArcCos}\left (-\frac {c f}{g}\right )-2 i \tanh ^{-1}\left (\frac {(-c f+g) \tan \left (\frac {1}{2} \text {ArcCos}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )\right ) \log \left (\frac {(c f+g) \left (-i c f+i g+\sqrt {-c^2 f^2+g^2}\right ) \left (-i+\tan \left (\frac {1}{2} \text {ArcCos}(c x)\right )\right )}{g \left (c f+g+\sqrt {-c^2 f^2+g^2} \tan \left (\frac {1}{2} \text {ArcCos}(c x)\right )\right )}\right )-\left (\text {ArcCos}\left (-\frac {c f}{g}\right )+2 i \tanh ^{-1}\left (\frac {(-c f+g) \tan \left (\frac {1}{2} \text {ArcCos}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )\right ) \log \left (\frac {(c f+g) \left (i c f-i g+\sqrt {-c^2 f^2+g^2}\right ) \left (i+\tan \left (\frac {1}{2} \text {ArcCos}(c x)\right )\right )}{g \left (c f+g+\sqrt {-c^2 f^2+g^2} \tan \left (\frac {1}{2} \text {ArcCos}(c x)\right )\right )}\right )+i \left (\text {PolyLog}\left (2,\frac {\left (c f-i \sqrt {-c^2 f^2+g^2}\right ) \left (c f+g-\sqrt {-c^2 f^2+g^2} \tan \left (\frac {1}{2} \text {ArcCos}(c x)\right )\right )}{g \left (c f+g+\sqrt {-c^2 f^2+g^2} \tan \left (\frac {1}{2} \text {ArcCos}(c x)\right )\right )}\right )-\text {PolyLog}\left (2,\frac {\left (c f+i \sqrt {-c^2 f^2+g^2}\right ) \left (c f+g-\sqrt {-c^2 f^2+g^2} \tan \left (\frac {1}{2} \text {ArcCos}(c x)\right )\right )}{g \left (c f+g+\sqrt {-c^2 f^2+g^2} \tan \left (\frac {1}{2} \text {ArcCos}(c x)\right )\right )}\right )\right )\right )}{\sqrt {-c^2 f^2+g^2} \sqrt {1-c^2 x^2}}\right )}{2 g^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/2*((2*a*g*Sqrt[d - c^2*d*x^2])/(f + g*x) - 2*a*c*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^
2*x^2))] - (2*a*c^2*Sqrt[d]*f*Log[f + g*x])/Sqrt[-(c^2*f^2) + g^2] + (2*a*c^2*Sqrt[d]*f*Log[d*(g + c^2*f*x) +
Sqrt[d]*Sqrt[-(c^2*f^2) + g^2]*Sqrt[d - c^2*d*x^2]])/Sqrt[-(c^2*f^2) + g^2] + b*c*Sqrt[d - c^2*d*x^2]*((2*g*Ar
cCos[c*x])/(c*f + c*g*x) - ArcCos[c*x]^2/Sqrt[1 - c^2*x^2] + (2*Log[1 + (g*x)/f])/Sqrt[1 - c^2*x^2] + (2*c*f*(
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[A
rcCos[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]])*L
og[((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]*Tan[ArcCos[c*x]/2]))] + I*(PolyLog[2, ((c*f - I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - S
qrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))] - PolyLo
g[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 + Sq
rt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))])))/(Sqrt[-(c^2*f^2) + g^2]*Sqrt[1 - c^2*x^2])))/g^2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.30, size = 1573, normalized size = 1.85

method	result	size
default	\(\text {Expression too large to display}\)	\(1573\)

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)^2,x,method=_RETURNVERBOSE)

[Out]

a/d/(c^2*f^2-g^2)/(x+f/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)^(3/2)-a/g*c^2*f/(c^2*f^2-
g^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)-a/g^2*c^4*f^2/(c^2*f^2-g^2)*d/(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*c^4*f^3/(c^
2*f^2-g^2)*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))+a/g*c^2*f/(c^2*f^2-g^2)*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))+a*c^2/(c^2*f^2-g^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+a*c^2/(c^2*f^2-g^2)*d/(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))+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*c/g^2-(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*arccos(c*x)*(
c^2*f*x+g+I*(-c^2*x^2+1)^(1/2)*c*f)/(c^2*x^2-1)/g^2/(g*x+f)-c*(I*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)*(c^2*f^2-g^2)^(1/2)*c*f-I*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)*(c^2*f^2-g^2)^(1/2)*c*f+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))*(c^2*f^2-g^2)^(1/2)*c*f-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))*(c^2*f^2-g^2)^(1/2)*c*f-ln((c*x+I*(
-c^2*x^2+1)^(1/2))^2*g+2*c*f*(c*x+I*(-c^2*x^2+1)^(1/2))+g)*c^2*f^2+ln((c*x+I*(-c^2*x^2+1)^(1/2))^2*g+2*c*f*(c*
x+I*(-c^2*x^2+1)^(1/2))+g)*g^2-2*Im(arccos(c*x))*c^2*f^2+2*Im(arccos(c*x))*g^2+2*ln(exp(I*Re(arccos(c*x))))*c^
2*f^2-2*ln(exp(I*Re(arccos(c*x))))*g^2)*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*f^2-g^2)/(c^2*x^2-1)/g^
2)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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)^2,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)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acos}{\left (c x \right )}\right )}{\left (f + g x\right )^{2}}\, dx \end {gather*}

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)**2,x)

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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

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)^2,x)

[Out]

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

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