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

Optimal. Leaf size=496 \[ \frac{b c^2 f \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-\frac{g e^{i \cos ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac{b c^2 f \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-\frac{g e^{i \cos ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac{g \left (1-c^2 x^2\right ) \left (a+b \cos ^{-1}(c x)\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right ) (f+g x)}+\frac{i c^2 f \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \log \left (1+\frac{g e^{i \cos ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac{i c^2 f \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \log \left (1+\frac{g e^{i \cos ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac{b c \sqrt{1-c^2 x^2} \log (f+g x)}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )} \]

[Out]

(g*(1 - c^2*x^2)*(a + b*ArcCos[c*x]))/((c^2*f^2 - g^2)*(f + g*x)*Sqrt[d - c^2*d*x^2]) + (I*c^2*f*Sqrt[1 - c^2*
x^2]*(a + b*ArcCos[c*x])*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/((c^2*f^2 - g^2)^(3/2)*Sq
rt[d - c^2*d*x^2]) - (I*c^2*f*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[
c^2*f^2 - g^2])])/((c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[1 - c^2*x^2]*Log[f + g*x])/((c^2*f^2
- g^2)*Sqrt[d - c^2*d*x^2]) + (b*c^2*f*Sqrt[1 - c^2*x^2]*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f
^2 - g^2]))])/((c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) - (b*c^2*f*Sqrt[1 - c^2*x^2]*PolyLog[2, -((E^(I*ArcC
os[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/((c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.721545, antiderivative size = 496, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.323, Rules used = {4778, 4774, 3324, 3321, 2264, 2190, 2279, 2391, 2668, 31} \[ \frac{b c^2 f \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-\frac{g e^{i \cos ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac{b c^2 f \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-\frac{g e^{i \cos ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac{g \left (1-c^2 x^2\right ) \left (a+b \cos ^{-1}(c x)\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right ) (f+g x)}+\frac{i c^2 f \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \log \left (1+\frac{g e^{i \cos ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac{i c^2 f \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \log \left (1+\frac{g e^{i \cos ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac{b c \sqrt{1-c^2 x^2} \log (f+g x)}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g*(1 - c^2*x^2)*(a + b*ArcCos[c*x]))/((c^2*f^2 - g^2)*(f + g*x)*Sqrt[d - c^2*d*x^2]) + (I*c^2*f*Sqrt[1 - c^2*
x^2]*(a + b*ArcCos[c*x])*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/((c^2*f^2 - g^2)^(3/2)*Sq
rt[d - c^2*d*x^2]) - (I*c^2*f*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[
c^2*f^2 - g^2])])/((c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[1 - c^2*x^2]*Log[f + g*x])/((c^2*f^2
- g^2)*Sqrt[d - c^2*d*x^2]) + (b*c^2*f*Sqrt[1 - c^2*x^2]*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f
^2 - g^2]))])/((c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) - (b*c^2*f*Sqrt[1 - c^2*x^2]*PolyLog[2, -((E^(I*ArcC
os[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/((c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*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 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 3324

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 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]

Rule 2668

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 31

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

Rubi steps

\begin{align*} \int \frac{a+b \cos ^{-1}(c x)}{(f+g x)^2 \sqrt{d-c^2 d x^2}} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{a+b \cos ^{-1}(c x)}{(f+g x)^2 \sqrt{1-c^2 x^2}} \, dx}{\sqrt{d-c^2 d x^2}}\\ &=-\frac{\left (c \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{a+b x}{(c f+g \cos (x))^2} \, dx,x,\cos ^{-1}(c x)\right )}{\sqrt{d-c^2 d x^2}}\\ &=\frac{g \left (1-c^2 x^2\right ) \left (a+b \cos ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt{d-c^2 d x^2}}-\frac{\left (c^2 f \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{a+b x}{c f+g \cos (x)} \, dx,x,\cos ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) \sqrt{d-c^2 d x^2}}-\frac{\left (b c g \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (x)}{c f+g \cos (x)} \, dx,x,\cos ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) \sqrt{d-c^2 d x^2}}\\ &=\frac{g \left (1-c^2 x^2\right ) \left (a+b \cos ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt{d-c^2 d x^2}}+\frac{\left (b c \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{c f+x} \, dx,x,c g x\right )}{\left (c^2 f^2-g^2\right ) \sqrt{d-c^2 d x^2}}-\frac{\left (2 c^2 f \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{2 c e^{i x} f+g+e^{2 i x} g} \, dx,x,\cos ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) \sqrt{d-c^2 d x^2}}\\ &=\frac{g \left (1-c^2 x^2\right ) \left (a+b \cos ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{1-c^2 x^2} \log (f+g x)}{\left (c^2 f^2-g^2\right ) \sqrt{d-c^2 d x^2}}-\frac{\left (2 c^2 f g \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{2 c f+2 e^{i x} g-2 \sqrt{c^2 f^2-g^2}} \, dx,x,\cos ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{\left (2 c^2 f g \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{2 c f+2 e^{i x} g+2 \sqrt{c^2 f^2-g^2}} \, dx,x,\cos ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}\\ &=\frac{g \left (1-c^2 x^2\right ) \left (a+b \cos ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt{d-c^2 d x^2}}+\frac{i c^2 f \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \log \left (1+\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}-\frac{i c^2 f \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \log \left (1+\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{1-c^2 x^2} \log (f+g x)}{\left (c^2 f^2-g^2\right ) \sqrt{d-c^2 d x^2}}-\frac{\left (i b c^2 f \sqrt{1-c^2 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 )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{\left (i b c^2 f \sqrt{1-c^2 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 )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}\\ &=\frac{g \left (1-c^2 x^2\right ) \left (a+b \cos ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt{d-c^2 d x^2}}+\frac{i c^2 f \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \log \left (1+\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}-\frac{i c^2 f \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \log \left (1+\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{1-c^2 x^2} \log (f+g x)}{\left (c^2 f^2-g^2\right ) \sqrt{d-c^2 d x^2}}-\frac{\left (b c^2 f \sqrt{1-c^2 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 )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{\left (b c^2 f \sqrt{1-c^2 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 )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}\\ &=\frac{g \left (1-c^2 x^2\right ) \left (a+b \cos ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt{d-c^2 d x^2}}+\frac{i c^2 f \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \log \left (1+\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}-\frac{i c^2 f \sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \log \left (1+\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{1-c^2 x^2} \log (f+g x)}{\left (c^2 f^2-g^2\right ) \sqrt{d-c^2 d x^2}}+\frac{b c^2 f \sqrt{1-c^2 x^2} \text{Li}_2\left (-\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}-\frac{b c^2 f \sqrt{1-c^2 x^2} \text{Li}_2\left (-\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [B]  time = 4.98671, size = 1108, normalized size = 2.23 \[ -\frac{a f \log (f+g x) c^2}{\sqrt{d} \left (g^2-c^2 f^2\right )^{3/2}}-\frac{a f \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 ) c^2}{\sqrt{d} (c f-g) (c f+g) \sqrt{g^2-c^2 f^2}}-\frac{b \sqrt{1-c^2 x^2} \left (-\frac{g \sqrt{1-c^2 x^2} \cos ^{-1}(c x)}{(c f-g) (c f+g) (c f+c g x)}-\frac{\log \left (\frac{g x}{f}+1\right )}{c^2 f^2-g^2}-\frac{c f \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{PolyLog}\left (2,\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{PolyLog}\left (2,\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 )}{\left (g^2-c^2 f^2\right )^{3/2}}\right ) c}{\sqrt{d-c^2 d x^2}}-\frac{a g \sqrt{d-c^2 d x^2}}{d \left (g^2-c^2 f^2\right ) (f+g x)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((a*g*Sqrt[d - c^2*d*x^2])/(d*(-(c^2*f^2) + g^2)*(f + g*x))) - (a*c^2*f*Log[f + g*x])/(Sqrt[d]*(-(c^2*f^2) +
g^2)^(3/2)) - (a*c^2*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[d]*(c*
f - g)*(c*f + g)*Sqrt[-(c^2*f^2) + g^2]) - (b*c*Sqrt[1 - c^2*x^2]*(-((g*Sqrt[1 - c^2*x^2]*ArcCos[c*x])/((c*f -
g)*(c*f + g)*(c*f + c*g*x))) - Log[1 + (g*x)/f]/(c^2*f^2 - g^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[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[Arc
Cos[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]))] - (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[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]))])))/(-(c^2*f^2) + g^2)^(3/2)))/Sqrt[d - c^2*d*x^2]

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Maple [B]  time = 0.305, size = 1622, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a/d/(c^2*f^2-g^2)/(x+f/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)^(1/2)-a/g*c^2*f/(c^2*f^2-
g^2)/(-d*(c^2*f^2-g^2)/g^2)^(1/2)*ln((-2*d*(c^2*f^2-g^2)/g^2+2*c^2*d*f/g*(x+f/g)+2*(-d*(c^2*f^2-g^2)/g^2)^(1/2
)*(-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))+b*(-d*(c^2*x^2-1))^(1/2)*arccos(c
*x)/d/(c^2*x^2-1)/(c^2*f^2-g^2)/(g*x+f)*(-c^2*x^2+1)*x*c^2*f+b*(-d*(c^2*x^2-1))^(1/2)*arccos(c*x)/d/(c^2*x^2-1
)/(c^2*f^2-g^2)/(g*x+f)*x^3*c^4*f+I*b*c^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d/(c^2*x^2-1)/(c^2*f^2-g^2
)^(3/2)*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)*f+b*(
-d*(c^2*x^2-1))^(1/2)*arccos(c*x)/d/(c^2*x^2-1)/(c^2*f^2-g^2)/(g*x+f)*x^2*c^2*g-I*b*(-d*(c^2*x^2-1))^(1/2)*arc
cos(c*x)/d/(c^2*x^2-1)/(c^2*f^2-g^2)/(g*x+f)*(-c^2*x^2+1)^(1/2)*x*c*g-b*(-d*(c^2*x^2-1))^(1/2)*arccos(c*x)/d/(
c^2*x^2-1)/(c^2*f^2-g^2)/(g*x+f)*x*c^2*f-b*(-d*(c^2*x^2-1))^(1/2)*arccos(c*x)/d/(c^2*x^2-1)/(c^2*f^2-g^2)/(g*x
+f)*g-I*b*(-d*(c^2*x^2-1))^(1/2)*arccos(c*x)/d/(c^2*x^2-1)/(c^2*f^2-g^2)/(g*x+f)*(-c^2*x^2+1)^(1/2)*c*f-I*b*c^
2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d/(c^2*x^2-1)/(c^2*f^2-g^2)^(3/2)*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)*f-b*c^3*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)
^(1/2)/d/(c^2*x^2-1)/(c^2*f^2-g^2)^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)*f^2
+2*b*c^3*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d/(c^2*x^2-1)/(c^2*f^2-g^2)^2*ln(c*x+I*(-c^2*x^2+1)^(1/2))*
f^2-b*c^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d/(c^2*x^2-1)/(c^2*f^2-g^2)^(3/2)*dilog((-(c*x+I*(-c^2*x^2
+1)^(1/2))*g-c*f+(c^2*f^2-g^2)^(1/2))/(-c*f+(c^2*f^2-g^2)^(1/2)))*f+b*c^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^
(1/2)/d/(c^2*x^2-1)/(c^2*f^2-g^2)^(3/2)*dilog(((c*x+I*(-c^2*x^2+1)^(1/2))*g+c*f+(c^2*f^2-g^2)^(1/2))/(c*f+(c^2
*f^2-g^2)^(1/2)))*f+b*c*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d/(c^2*x^2-1)/(c^2*f^2-g^2)^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*b*c*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d/
(c^2*x^2-1)/(c^2*f^2-g^2)^2*ln(c*x+I*(-c^2*x^2+1)^(1/2))*g^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c^{2} d x^{2} + d}{\left (b \arccos \left (c x\right ) + a\right )}}{c^{2} d g^{2} x^{4} + 2 \, c^{2} d f g x^{3} - 2 \, d f g x - d f^{2} +{\left (c^{2} d f^{2} - d g^{2}\right )} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-c^2*d*x^2 + d)*(b*arccos(c*x) + a)/(c^2*d*g^2*x^4 + 2*c^2*d*f*g*x^3 - 2*d*f*g*x - d*f^2 + (c^2
*d*f^2 - d*g^2)*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError