3.9 \(\int \frac{(d-c^2 d x^2)^{3/2} (a+b \cos ^{-1}(c x))}{f+g x} \, dx\)

Optimal. Leaf size=1064 \[ -\frac{b d x^3 \sqrt{d-c^2 d x^2} c^3}{9 g \sqrt{1-c^2 x^2}}+\frac{b d f x^2 \sqrt{d-c^2 d x^2} c^3}{4 g^2 \sqrt{1-c^2 x^2}}+\frac{d f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) c^2}{2 g^2}+\frac{d (c f-g) (c f+g) x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2 c}{2 b g^3 \sqrt{1-c^2 x^2}}-\frac{d f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2 c}{4 b g^2 \sqrt{1-c^2 x^2}}-\frac{b d (c f-g) (c f+g) x \sqrt{d-c^2 d x^2} c}{g^3 \sqrt{1-c^2 x^2}}+\frac{b d x \sqrt{d-c^2 d x^2} c}{3 g \sqrt{1-c^2 x^2}}-\frac{b d (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)}{g^3}+\frac{d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}+\frac{a d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{f x c^2+g}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right )}{g^4 \sqrt{1-c^2 x^2}}+\frac{i b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left (\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}+1\right )}{g^4 \sqrt{1-c^2 x^2}}-\frac{i b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left (\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}+1\right )}{g^4 \sqrt{1-c^2 x^2}}+\frac{b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^4 \sqrt{1-c^2 x^2}}-\frac{b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^4 \sqrt{1-c^2 x^2}}-\frac{a d (c f-g) (c f+g) \sqrt{d-c^2 d x^2}}{g^3}+\frac{d (c f-g) (c f+g) \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^2 (f+g x) c}+\frac{d \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^4 (f+g x) \sqrt{1-c^2 x^2} c} \]

[Out]

-((a*d*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2])/g^3) + (b*c*d*x*Sqrt[d - c^2*d*x^2])/(3*g*Sqrt[1 - c^2*x^2]) -
(b*c*d*(c*f - g)*(c*f + g)*x*Sqrt[d - c^2*d*x^2])/(g^3*Sqrt[1 - c^2*x^2]) + (b*c^3*d*f*x^2*Sqrt[d - c^2*d*x^2
])/(4*g^2*Sqrt[1 - c^2*x^2]) - (b*c^3*d*x^3*Sqrt[d - c^2*d*x^2])/(9*g*Sqrt[1 - c^2*x^2]) - (b*d*(c*f - g)*(c*f
+ g)*Sqrt[d - c^2*d*x^2]*ArcCos[c*x])/g^3 + (c^2*d*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(2*g^2) + (d*
(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(3*g) - (c*d*f*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^
2)/(4*b*g^2*Sqrt[1 - c^2*x^2]) + (c*d*(c*f - g)*(c*f + g)*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(2*b*g^
3*Sqrt[1 - c^2*x^2]) + (d*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(2*b*c*g^4*(f + g*x)*Sq
rt[1 - c^2*x^2]) + (d*(c*f - g)*(c*f + g)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(2*b*c*
g^2*(f + g*x)) + (a*d*(c^2*f^2 - g^2)^(3/2)*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^4*Sqrt[1 - c^2*x^2]) + (I*b*d*(c^2*f^2 - g^2)^(3/2)*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^4*Sqrt[1 - c^2*x^2]) - (I*b*d*(c^2*f^2 - g^2)^(3/2)*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^4*Sqrt[1 - c^2*x
^2]) + (b*d*(c^2*f^2 - g^2)^(3/2)*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^4*Sqrt[1 - c^2*x^2]) - (b*d*(c^2*f^2 - g^2)^(3/2)*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^4*Sqrt[1 - c^2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 2.28, antiderivative size = 1064, normalized size of antiderivative = 1., number of steps used = 29, number of rules used = 23, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.742, Rules used = {4778, 4768, 4648, 4642, 30, 4678, 4766, 683, 4758, 6742, 725, 204, 1654, 12, 4800, 4798, 8, 4774, 3321, 2264, 2190, 2279, 2391} \[ -\frac{b d x^3 \sqrt{d-c^2 d x^2} c^3}{9 g \sqrt{1-c^2 x^2}}+\frac{b d f x^2 \sqrt{d-c^2 d x^2} c^3}{4 g^2 \sqrt{1-c^2 x^2}}+\frac{d f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) c^2}{2 g^2}+\frac{d (c f-g) (c f+g) x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2 c}{2 b g^3 \sqrt{1-c^2 x^2}}-\frac{d f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2 c}{4 b g^2 \sqrt{1-c^2 x^2}}-\frac{b d (c f-g) (c f+g) x \sqrt{d-c^2 d x^2} c}{g^3 \sqrt{1-c^2 x^2}}+\frac{b d x \sqrt{d-c^2 d x^2} c}{3 g \sqrt{1-c^2 x^2}}-\frac{b d (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)}{g^3}+\frac{d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}+\frac{a d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{f x c^2+g}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right )}{g^4 \sqrt{1-c^2 x^2}}+\frac{i b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left (\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}+1\right )}{g^4 \sqrt{1-c^2 x^2}}-\frac{i b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left (\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}+1\right )}{g^4 \sqrt{1-c^2 x^2}}+\frac{b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^4 \sqrt{1-c^2 x^2}}-\frac{b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^4 \sqrt{1-c^2 x^2}}-\frac{a d (c f-g) (c f+g) \sqrt{d-c^2 d x^2}}{g^3}+\frac{d (c f-g) (c f+g) \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^2 (f+g x) c}+\frac{d \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b g^4 (f+g x) \sqrt{1-c^2 x^2} c} \]

Antiderivative was successfully verified.

[In]

Int[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x]))/(f + g*x),x]

[Out]

-((a*d*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2])/g^3) + (b*c*d*x*Sqrt[d - c^2*d*x^2])/(3*g*Sqrt[1 - c^2*x^2]) -
(b*c*d*(c*f - g)*(c*f + g)*x*Sqrt[d - c^2*d*x^2])/(g^3*Sqrt[1 - c^2*x^2]) + (b*c^3*d*f*x^2*Sqrt[d - c^2*d*x^2
])/(4*g^2*Sqrt[1 - c^2*x^2]) - (b*c^3*d*x^3*Sqrt[d - c^2*d*x^2])/(9*g*Sqrt[1 - c^2*x^2]) - (b*d*(c*f - g)*(c*f
+ g)*Sqrt[d - c^2*d*x^2]*ArcCos[c*x])/g^3 + (c^2*d*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(2*g^2) + (d*
(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(3*g) - (c*d*f*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^
2)/(4*b*g^2*Sqrt[1 - c^2*x^2]) + (c*d*(c*f - g)*(c*f + g)*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(2*b*g^
3*Sqrt[1 - c^2*x^2]) + (d*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(2*b*c*g^4*(f + g*x)*Sq
rt[1 - c^2*x^2]) + (d*(c*f - g)*(c*f + g)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(2*b*c*
g^2*(f + g*x)) + (a*d*(c^2*f^2 - g^2)^(3/2)*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^4*Sqrt[1 - c^2*x^2]) + (I*b*d*(c^2*f^2 - g^2)^(3/2)*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^4*Sqrt[1 - c^2*x^2]) - (I*b*d*(c^2*f^2 - g^2)^(3/2)*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^4*Sqrt[1 - c^2*x
^2]) + (b*d*(c^2*f^2 - g^2)^(3/2)*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^4*Sqrt[1 - c^2*x^2]) - (b*d*(c^2*f^2 - g^2)^(3/2)*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^4*Sqrt[1 - c^2*x^2])

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 4768

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

Rule 4648

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

Rule 4642

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 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 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 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 6742

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

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 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 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 12

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

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 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 8

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

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 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 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 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 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]

Rubi steps

\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right )}{f+g x} \, dx &=\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (1-c^2 x^2\right )^{3/2} \left (a+b \cos ^{-1}(c x)\right )}{f+g x} \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int \left (\frac{c^2 f \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )}{g^2}-\frac{c^2 x \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )}{g}+\frac{\left (-c^2 f^2+g^2\right ) \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )}{g^2 (f+g x)}\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (d \left (1-\frac{c^2 f^2}{g^2}\right ) \sqrt{d-c^2 d x^2}\right ) \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{\left (c^2 d f \sqrt{d-c^2 d x^2}\right ) \int \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{g^2 \sqrt{1-c^2 x^2}}-\frac{\left (c^2 d \sqrt{d-c^2 d x^2}\right ) \int x \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{g \sqrt{1-c^2 x^2}}\\ &=\frac{c^2 d f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^2}+\frac{d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac{d \left (1-\frac{c^2 f^2}{g^2}\right ) \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 (d \left (1-\frac{c^2 f^2}{g^2}\right ) \sqrt{d-c^2 d x^2}\right ) \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{\left (c^2 d f \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cos ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{2 g^2 \sqrt{1-c^2 x^2}}+\frac{\left (b c^3 d f \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{2 g^2 \sqrt{1-c^2 x^2}}+\frac{\left (b c d \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right ) \, dx}{3 g \sqrt{1-c^2 x^2}}\\ &=\frac{b c d x \sqrt{d-c^2 d x^2}}{3 g \sqrt{1-c^2 x^2}}+\frac{b c^3 d f x^2 \sqrt{d-c^2 d x^2}}{4 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^3 d x^3 \sqrt{d-c^2 d x^2}}{9 g \sqrt{1-c^2 x^2}}+\frac{c^2 d f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^2}+\frac{d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac{c d f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{1-c^2 x^2}}-\frac{c d \left (1-\frac{c^2 f^2}{g^2}\right ) 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{d \left (1-\frac{c^2 f^2}{g^2}\right )^2 \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{d \left (1-\frac{c^2 f^2}{g^2}\right ) \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 (d \left (1-\frac{c^2 f^2}{g^2}\right ) \sqrt{d-c^2 d x^2}\right ) \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{b c d x \sqrt{d-c^2 d x^2}}{3 g \sqrt{1-c^2 x^2}}+\frac{b c^3 d f x^2 \sqrt{d-c^2 d x^2}}{4 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^3 d x^3 \sqrt{d-c^2 d x^2}}{9 g \sqrt{1-c^2 x^2}}+\frac{c^2 d f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^2}+\frac{d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac{c d f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{1-c^2 x^2}}-\frac{c d \left (1-\frac{c^2 f^2}{g^2}\right ) 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{d \left (1-\frac{c^2 f^2}{g^2}\right )^2 \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{d \left (1-\frac{c^2 f^2}{g^2}\right ) \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 (d \left (1-\frac{c^2 f^2}{g^2}\right ) \sqrt{d-c^2 d x^2}\right ) \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{b c d x \sqrt{d-c^2 d x^2}}{3 g \sqrt{1-c^2 x^2}}+\frac{b c^3 d f x^2 \sqrt{d-c^2 d x^2}}{4 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^3 d x^3 \sqrt{d-c^2 d x^2}}{9 g \sqrt{1-c^2 x^2}}+\frac{c^2 d f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^2}+\frac{d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac{c d f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{1-c^2 x^2}}-\frac{c d \left (1-\frac{c^2 f^2}{g^2}\right ) 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{d \left (1-\frac{c^2 f^2}{g^2}\right )^2 \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{d \left (1-\frac{c^2 f^2}{g^2}\right ) \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 d \left (1-\frac{c^2 f^2}{g^2}\right ) \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 d \left (1-\frac{c^2 f^2}{g^2}\right ) \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 d (c f-g) (c f+g) \sqrt{d-c^2 d x^2}}{g^3}+\frac{b c d x \sqrt{d-c^2 d x^2}}{3 g \sqrt{1-c^2 x^2}}+\frac{b c^3 d f x^2 \sqrt{d-c^2 d x^2}}{4 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^3 d x^3 \sqrt{d-c^2 d x^2}}{9 g \sqrt{1-c^2 x^2}}+\frac{c^2 d f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^2}+\frac{d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac{c d f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{1-c^2 x^2}}-\frac{c d \left (1-\frac{c^2 f^2}{g^2}\right ) 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{d \left (1-\frac{c^2 f^2}{g^2}\right )^2 \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{d \left (1-\frac{c^2 f^2}{g^2}\right ) \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 d \left (1-\frac{c^2 f^2}{g^2}\right ) \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 d \left (1-\frac{c^2 f^2}{g^2}\right ) \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 d (c f-g) (c f+g) \sqrt{d-c^2 d x^2}}{g^3}+\frac{b c d x \sqrt{d-c^2 d x^2}}{3 g \sqrt{1-c^2 x^2}}+\frac{b c^3 d f x^2 \sqrt{d-c^2 d x^2}}{4 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^3 d x^3 \sqrt{d-c^2 d x^2}}{9 g \sqrt{1-c^2 x^2}}+\frac{c^2 d f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^2}+\frac{d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac{c d f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{1-c^2 x^2}}-\frac{c d \left (1-\frac{c^2 f^2}{g^2}\right ) 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{d \left (1-\frac{c^2 f^2}{g^2}\right )^2 \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{d \left (1-\frac{c^2 f^2}{g^2}\right ) \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 d \left (1-\frac{c^2 f^2}{g^2}\right ) \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 d \left (1-\frac{c^2 f^2}{g^2}\right ) (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 d \left (1-\frac{c^2 f^2}{g^2}\right ) (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 d (c f-g) (c f+g) \sqrt{d-c^2 d x^2}}{g^3}+\frac{b c d x \sqrt{d-c^2 d x^2}}{3 g \sqrt{1-c^2 x^2}}+\frac{b c^3 d f x^2 \sqrt{d-c^2 d x^2}}{4 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^3 d x^3 \sqrt{d-c^2 d x^2}}{9 g \sqrt{1-c^2 x^2}}-\frac{b d (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)}{g^3}+\frac{c^2 d f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^2}+\frac{d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac{c d f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{1-c^2 x^2}}-\frac{c d \left (1-\frac{c^2 f^2}{g^2}\right ) 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{d \left (1-\frac{c^2 f^2}{g^2}\right )^2 \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{d \left (1-\frac{c^2 f^2}{g^2}\right ) \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 d \left (1-\frac{c^2 f^2}{g^2}\right ) \sqrt{d-c^2 d x^2}\right ) \int 1 \, dx}{g \sqrt{1-c^2 x^2}}+\frac{\left (a d \left (1-\frac{c^2 f^2}{g^2}\right ) (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 d \left (1-\frac{c^2 f^2}{g^2}\right ) (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 d (c f-g) (c f+g) \sqrt{d-c^2 d x^2}}{g^3}+\frac{b c d x \sqrt{d-c^2 d x^2}}{3 g \sqrt{1-c^2 x^2}}+\frac{b c d \left (1-\frac{c^2 f^2}{g^2}\right ) x \sqrt{d-c^2 d x^2}}{g \sqrt{1-c^2 x^2}}+\frac{b c^3 d f x^2 \sqrt{d-c^2 d x^2}}{4 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^3 d x^3 \sqrt{d-c^2 d x^2}}{9 g \sqrt{1-c^2 x^2}}-\frac{b d (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)}{g^3}+\frac{c^2 d f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^2}+\frac{d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac{c d f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{1-c^2 x^2}}-\frac{c d \left (1-\frac{c^2 f^2}{g^2}\right ) 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{d \left (1-\frac{c^2 f^2}{g^2}\right )^2 \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{d \left (1-\frac{c^2 f^2}{g^2}\right ) \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 d (c f-g)^2 (c f+g)^2 \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^4 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{\left (2 b d \left (1-\frac{c^2 f^2}{g^2}\right ) (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 d (c f-g) (c f+g) \sqrt{d-c^2 d x^2}}{g^3}+\frac{b c d x \sqrt{d-c^2 d x^2}}{3 g \sqrt{1-c^2 x^2}}+\frac{b c d \left (1-\frac{c^2 f^2}{g^2}\right ) x \sqrt{d-c^2 d x^2}}{g \sqrt{1-c^2 x^2}}+\frac{b c^3 d f x^2 \sqrt{d-c^2 d x^2}}{4 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^3 d x^3 \sqrt{d-c^2 d x^2}}{9 g \sqrt{1-c^2 x^2}}-\frac{b d (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)}{g^3}+\frac{c^2 d f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^2}+\frac{d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac{c d f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{1-c^2 x^2}}-\frac{c d \left (1-\frac{c^2 f^2}{g^2}\right ) 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{d \left (1-\frac{c^2 f^2}{g^2}\right )^2 \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{d \left (1-\frac{c^2 f^2}{g^2}\right ) \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 d (c f-g)^2 (c f+g)^2 \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^4 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{\left (2 b d \left (1-\frac{c^2 f^2}{g^2}\right ) (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 d \left (1-\frac{c^2 f^2}{g^2}\right ) (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 d (c f-g) (c f+g) \sqrt{d-c^2 d x^2}}{g^3}+\frac{b c d x \sqrt{d-c^2 d x^2}}{3 g \sqrt{1-c^2 x^2}}+\frac{b c d \left (1-\frac{c^2 f^2}{g^2}\right ) x \sqrt{d-c^2 d x^2}}{g \sqrt{1-c^2 x^2}}+\frac{b c^3 d f x^2 \sqrt{d-c^2 d x^2}}{4 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^3 d x^3 \sqrt{d-c^2 d x^2}}{9 g \sqrt{1-c^2 x^2}}-\frac{b d (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)}{g^3}+\frac{c^2 d f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^2}+\frac{d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac{c d f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{1-c^2 x^2}}-\frac{c d \left (1-\frac{c^2 f^2}{g^2}\right ) 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{d \left (1-\frac{c^2 f^2}{g^2}\right )^2 \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{d \left (1-\frac{c^2 f^2}{g^2}\right ) \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 d (c f-g)^2 (c f+g)^2 \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^4 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{i b d (c f-g)^2 (c f+g)^2 \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^4 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{i b d (c f-g)^2 (c f+g)^2 \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^4 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{\left (i b d \left (1-\frac{c^2 f^2}{g^2}\right ) (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 d \left (1-\frac{c^2 f^2}{g^2}\right ) (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 d (c f-g) (c f+g) \sqrt{d-c^2 d x^2}}{g^3}+\frac{b c d x \sqrt{d-c^2 d x^2}}{3 g \sqrt{1-c^2 x^2}}+\frac{b c d \left (1-\frac{c^2 f^2}{g^2}\right ) x \sqrt{d-c^2 d x^2}}{g \sqrt{1-c^2 x^2}}+\frac{b c^3 d f x^2 \sqrt{d-c^2 d x^2}}{4 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^3 d x^3 \sqrt{d-c^2 d x^2}}{9 g \sqrt{1-c^2 x^2}}-\frac{b d (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)}{g^3}+\frac{c^2 d f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^2}+\frac{d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac{c d f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{1-c^2 x^2}}-\frac{c d \left (1-\frac{c^2 f^2}{g^2}\right ) 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{d \left (1-\frac{c^2 f^2}{g^2}\right )^2 \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{d \left (1-\frac{c^2 f^2}{g^2}\right ) \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 d (c f-g)^2 (c f+g)^2 \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^4 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{i b d (c f-g)^2 (c f+g)^2 \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^4 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{i b d (c f-g)^2 (c f+g)^2 \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^4 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{\left (b d \left (1-\frac{c^2 f^2}{g^2}\right ) (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 d \left (1-\frac{c^2 f^2}{g^2}\right ) (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 d (c f-g) (c f+g) \sqrt{d-c^2 d x^2}}{g^3}+\frac{b c d x \sqrt{d-c^2 d x^2}}{3 g \sqrt{1-c^2 x^2}}+\frac{b c d \left (1-\frac{c^2 f^2}{g^2}\right ) x \sqrt{d-c^2 d x^2}}{g \sqrt{1-c^2 x^2}}+\frac{b c^3 d f x^2 \sqrt{d-c^2 d x^2}}{4 g^2 \sqrt{1-c^2 x^2}}-\frac{b c^3 d x^3 \sqrt{d-c^2 d x^2}}{9 g \sqrt{1-c^2 x^2}}-\frac{b d (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)}{g^3}+\frac{c^2 d f x \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{2 g^2}+\frac{d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 g}-\frac{c d f \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{1-c^2 x^2}}-\frac{c d \left (1-\frac{c^2 f^2}{g^2}\right ) 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{d \left (1-\frac{c^2 f^2}{g^2}\right )^2 \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{d \left (1-\frac{c^2 f^2}{g^2}\right ) \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 d (c f-g)^2 (c f+g)^2 \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^4 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{i b d (c f-g)^2 (c f+g)^2 \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^4 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{i b d (c f-g)^2 (c f+g)^2 \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^4 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{b d (c f-g)^2 (c f+g)^2 \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^4 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{b d (c f-g)^2 (c f+g)^2 \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^4 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\\ \end{align*}

Mathematica [B]  time = 12.5042, size = 3034, normalized size = 2.85 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x]))/(f + g*x),x]

[Out]

Sqrt[-(d*(-1 + c^2*x^2))]*((a*d*(-3*c^2*f^2 + 4*g^2))/(3*g^3) + (a*c^2*d*f*x)/(2*g^2) - (a*c^2*d*x^2)/(3*g)) +
(a*c*d^(3/2)*f*(2*c^2*f^2 - 3*g^2)*ArcTan[(c*x*Sqrt[-(d*(-1 + c^2*x^2))])/(Sqrt[d]*(-1 + c^2*x^2))])/(2*g^4)
+ (a*d^(3/2)*(-(c^2*f^2) + g^2)^(3/2)*Log[f + g*x])/g^4 - (a*d^(3/2)*(-(c^2*f^2) + g^2)^(3/2)*Log[d*g + c^2*d*
f*x + Sqrt[d]*Sqrt[-(c^2*f^2) + g^2]*Sqrt[-(d*(-1 + c^2*x^2))]])/g^4 - (b*d*Sqrt[d*(1 - c^2*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*Arc
Cos[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]])*L
og[Sqrt[-(c^2*f^2) + g^2]/(Sqrt[2]*E^((I/2)*ArcCos[c*x])*Sqrt[g]*Sqrt[c*f + c*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 + c*
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 + Sq
rt[-(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 - Sqrt[-
(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]))] - 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[ArcCos[c*x]/2]))])))/(Sqrt[-(c^2*f^2) + g^2]*Sqrt[1 - c^2*x^2])))/(2*g^2) + (b*d*Sqrt[d*(1
- c^2*x^2)]*((9*(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)*ArcTan
h[((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])/Sqr
t[-(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 + c*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 + c*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[ArcC
os[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 - Sqrt[-(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]))] - 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[ArcCos[c*x]/2]))])))/Sqrt[-(c^2*f^2) + g^2] + (18*c*g*(-4*c^2*f^2 +
g^2)*x + 18*g*(-4*c^2*f^2 + g^2)*Sqrt[1 - c^2*x^2]*ArcCos[c*x] + 18*c*f*(2*c^2*f^2 - g^2)*ArcCos[c*x]^2 + 9*c
*f*g^2*Cos[2*ArcCos[c*x]] - 2*g^3*Cos[3*ArcCos[c*x]] - (9*(8*c^4*f^4 - 8*c^2*f^2*g^2 + g^4)*(2*ArcCos[c*x]*Arc
Tanh[((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[A
rcCos[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])/Sq
rt[-(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 + c*g*x])] + (ArcCos[-((c*f)/g)] + (2*I)*(ArcTan
h[((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 + c*g*x])] - (Ar
cCos[-((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[Arc
Cos[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 - Sqrt[-(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]))] - PolyLog[2, ((c*f + I*Sq
rt[-(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[ArcCos[c*x]/2]))])))/Sqrt[-(c^2*f^2) + g^2] + 18*c*f*g^2*ArcCos[c*x]*Sin[2*ArcCos[c*x]] - 6*g^3*ArcCos
[c*x]*Sin[3*ArcCos[c*x]])/g^4))/(72*Sqrt[1 - c^2*x^2])

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Maple [B]  time = 0.398, size = 2132, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))/(g*x+f),x)

[Out]

a/g*d*(-d*c^2*(x+f/g)^2+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2)+1/8*b*(-d*(c^2*x^2-1))^(1/2)*f*d*c/(c^2
*x^2-1)/g^2*(-c^2*x^2+1)^(1/2)+1/9*b*(-d*(c^2*x^2-1))^(1/2)*d/(c^2*x^2-1)/g*(-c^2*x^2+1)^(1/2)*x^3*c^3-4/3*b*(
-d*(c^2*x^2-1))^(1/2)*d/(c^2*x^2-1)/g*(-c^2*x^2+1)^(1/2)*x*c+b*(-d*(c^2*x^2-1))^(1/2)*d/(c^2*x^2-1)/g^3*arccos
(c*x)*c^2*f^2-b*(c^2*f^2-g^2)^(3/2)*d*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)/g^4*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*(c^2*f^2-g^2)^(3/2)*d*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)/g^4*dilo
g((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))-1/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(c^2*x^2-1)/g*arccos(c*x)*x^4*c^4+5/3*b*(-d*(c^2*x^
2-1))^(1/2)*d/(c^2*x^2-1)/g*arccos(c*x)*x^2*c^2+1/3*a/g*(-d*c^2*(x+f/g)^2+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/
g^2)^(3/2)+2*a/g^3*d^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)*(-d*c^2*(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+1/2*a/g
^2*c^2*d*f*(-d*c^2*(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+3/2*a/g^2*c^2*d^2*f/(c^2*d)^(1/2
)*arctan((c^2*d)^(1/2)*x/(-d*c^2*(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^4*d^2*c^4*f^3/(
c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-d*c^2*(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^5*d^
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)*(
-d*c^2*(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^4*f^4-a/g^3*d*(-d*c^2*(x+f/g)^2+2*
c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2)*c^2*f^2-4/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(c^2*x^2-1)/g*arccos(c*x)+
I*b*(c^2*f^2-g^2)^(3/2)*d*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)/g^4*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*b*(c^2*f^2-g^2)^(3/2)*d*(-d*(c^2*x^2-
1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)/g^4*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)))-1/4*b*(-d*(c^2*x^2-1))^(1/2)*f*d*c^3/(c^2*x^2-1)/g^2*(-c^2*x^2+1)^(1/2)*x^2+
b*(-d*(c^2*x^2-1))^(1/2)*d/(c^2*x^2-1)/g^3*(-c^2*x^2+1)^(1/2)*x*c^3*f^2-1/2*b*(-d*(c^2*x^2-1))^(1/2)*f*d*c^2/(
c^2*x^2-1)/g^2*arccos(c*x)*x-b*(-d*(c^2*x^2-1))^(1/2)*d/(c^2*x^2-1)/g^3*arccos(c*x)*x^2*c^4*f^2-1/2*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^3*d*c^3/g^4+3/4*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*d*c/g^2+1/2*b*(-d*(c^2*x^2-1))^(1/2)*f*d*c^4/(c^2*x^2-1)/g^2*arccos(c*
x)*x^3-a/g*d^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)*(-d*c^2*(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))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))/(g*x+f),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (a c^{2} d x^{2} - a d +{\left (b c^{2} d x^{2} - b d\right )} \arccos \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{g x + f}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))/(g*x+f),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{3}{2}} \left (a + b \operatorname{acos}{\left (c x \right )}\right )}{f + g x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*d*x**2+d)**(3/2)*(a+b*acos(c*x))/(g*x+f),x)

[Out]

Integral((-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*acos(c*x))/(f + g*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\left (b \arccos \left (c x\right ) + a\right )}}{g x + f}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))/(g*x+f),x, algorithm="giac")

[Out]

integrate((-c^2*d*x^2 + d)^(3/2)*(b*arccos(c*x) + a)/(g*x + f), x)