3.28 \(\int \frac{\cos ^{-1}(a x)}{(c+d x^2)^2} \, dx\)

Optimal. Leaf size=727 \[ -\frac{i \text{PolyLog}\left (2,-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{i \text{PolyLog}\left (2,\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{i \text{PolyLog}\left (2,-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{i \text{PolyLog}\left (2,\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{d}-a^2 \sqrt{-c} x}{\sqrt{1-a^2 x^2} \sqrt{a^2 c+d}}\right )}{4 c \sqrt{d} \sqrt{a^2 c+d}}-\frac{a \tanh ^{-1}\left (\frac{a^2 \sqrt{-c} x+\sqrt{d}}{\sqrt{1-a^2 x^2} \sqrt{a^2 c+d}}\right )}{4 c \sqrt{d} \sqrt{a^2 c+d}}-\frac{\cos ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\cos ^{-1}(a x) \log \left (1+\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{\cos ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\cos ^{-1}(a x) \log \left (1+\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{\cos ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}-\sqrt{d} x\right )}+\frac{\cos ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}+\sqrt{d} x\right )} \]

[Out]

-ArcCos[a*x]/(4*c*Sqrt[d]*(Sqrt[-c] - Sqrt[d]*x)) + ArcCos[a*x]/(4*c*Sqrt[d]*(Sqrt[-c] + Sqrt[d]*x)) - (a*ArcT
anh[(Sqrt[d] - a^2*Sqrt[-c]*x)/(Sqrt[a^2*c + d]*Sqrt[1 - a^2*x^2])])/(4*c*Sqrt[d]*Sqrt[a^2*c + d]) - (a*ArcTan
h[(Sqrt[d] + a^2*Sqrt[-c]*x)/(Sqrt[a^2*c + d]*Sqrt[1 - a^2*x^2])])/(4*c*Sqrt[d]*Sqrt[a^2*c + d]) - (ArcCos[a*x
]*Log[1 - (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/(4*(-c)^(3/2)*Sqrt[d]) + (ArcCos[a*x]
*Log[1 + (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/(4*(-c)^(3/2)*Sqrt[d]) - (ArcCos[a*x]*
Log[1 - (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/(4*(-c)^(3/2)*Sqrt[d]) + (ArcCos[a*x]*L
og[1 + (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/(4*(-c)^(3/2)*Sqrt[d]) - ((I/4)*PolyLog[
2, -((Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d]))])/((-c)^(3/2)*Sqrt[d]) + ((I/4)*PolyLog[2,
(Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/((-c)^(3/2)*Sqrt[d]) - ((I/4)*PolyLog[2, -((Sqr
t[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d]))])/((-c)^(3/2)*Sqrt[d]) + ((I/4)*PolyLog[2, (Sqrt[d]*
E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/((-c)^(3/2)*Sqrt[d])

________________________________________________________________________________________

Rubi [A]  time = 1.06906, antiderivative size = 727, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {4668, 4744, 725, 206, 4742, 4522, 2190, 2279, 2391} \[ -\frac{i \text{PolyLog}\left (2,-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{i \text{PolyLog}\left (2,\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{i \text{PolyLog}\left (2,-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{i \text{PolyLog}\left (2,\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{d}-a^2 \sqrt{-c} x}{\sqrt{1-a^2 x^2} \sqrt{a^2 c+d}}\right )}{4 c \sqrt{d} \sqrt{a^2 c+d}}-\frac{a \tanh ^{-1}\left (\frac{a^2 \sqrt{-c} x+\sqrt{d}}{\sqrt{1-a^2 x^2} \sqrt{a^2 c+d}}\right )}{4 c \sqrt{d} \sqrt{a^2 c+d}}-\frac{\cos ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\cos ^{-1}(a x) \log \left (1+\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{\cos ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\cos ^{-1}(a x) \log \left (1+\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{\cos ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}-\sqrt{d} x\right )}+\frac{\cos ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}+\sqrt{d} x\right )} \]

Antiderivative was successfully verified.

[In]

Int[ArcCos[a*x]/(c + d*x^2)^2,x]

[Out]

-ArcCos[a*x]/(4*c*Sqrt[d]*(Sqrt[-c] - Sqrt[d]*x)) + ArcCos[a*x]/(4*c*Sqrt[d]*(Sqrt[-c] + Sqrt[d]*x)) - (a*ArcT
anh[(Sqrt[d] - a^2*Sqrt[-c]*x)/(Sqrt[a^2*c + d]*Sqrt[1 - a^2*x^2])])/(4*c*Sqrt[d]*Sqrt[a^2*c + d]) - (a*ArcTan
h[(Sqrt[d] + a^2*Sqrt[-c]*x)/(Sqrt[a^2*c + d]*Sqrt[1 - a^2*x^2])])/(4*c*Sqrt[d]*Sqrt[a^2*c + d]) - (ArcCos[a*x
]*Log[1 - (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/(4*(-c)^(3/2)*Sqrt[d]) + (ArcCos[a*x]
*Log[1 + (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/(4*(-c)^(3/2)*Sqrt[d]) - (ArcCos[a*x]*
Log[1 - (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/(4*(-c)^(3/2)*Sqrt[d]) + (ArcCos[a*x]*L
og[1 + (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/(4*(-c)^(3/2)*Sqrt[d]) - ((I/4)*PolyLog[
2, -((Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d]))])/((-c)^(3/2)*Sqrt[d]) + ((I/4)*PolyLog[2,
(Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/((-c)^(3/2)*Sqrt[d]) - ((I/4)*PolyLog[2, -((Sqr
t[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d]))])/((-c)^(3/2)*Sqrt[d]) + ((I/4)*PolyLog[2, (Sqrt[d]*
E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/((-c)^(3/2)*Sqrt[d])

Rule 4668

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcCos[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p]
&& (GtQ[p, 0] || IGtQ[n, 0])

Rule 4744

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

Rule 725

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

Rule 206

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

Rule 4742

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Subst[Int[((a + b*x)^n*Sin[x])
/(c*d + e*Cos[x]), x], x, ArcCos[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]

Rule 4522

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

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{\cos ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx &=\int \left (-\frac{d \cos ^{-1}(a x)}{4 c \left (\sqrt{-c} \sqrt{d}-d x\right )^2}-\frac{d \cos ^{-1}(a x)}{4 c \left (\sqrt{-c} \sqrt{d}+d x\right )^2}-\frac{d \cos ^{-1}(a x)}{2 c \left (-c d-d^2 x^2\right )}\right ) \, dx\\ &=-\frac{d \int \frac{\cos ^{-1}(a x)}{\left (\sqrt{-c} \sqrt{d}-d x\right )^2} \, dx}{4 c}-\frac{d \int \frac{\cos ^{-1}(a x)}{\left (\sqrt{-c} \sqrt{d}+d x\right )^2} \, dx}{4 c}-\frac{d \int \frac{\cos ^{-1}(a x)}{-c d-d^2 x^2} \, dx}{2 c}\\ &=-\frac{\cos ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}-\sqrt{d} x\right )}+\frac{\cos ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}+\sqrt{d} x\right )}-\frac{a \int \frac{1}{\left (\sqrt{-c} \sqrt{d}-d x\right ) \sqrt{1-a^2 x^2}} \, dx}{4 c}+\frac{a \int \frac{1}{\left (\sqrt{-c} \sqrt{d}+d x\right ) \sqrt{1-a^2 x^2}} \, dx}{4 c}-\frac{d \int \left (-\frac{\sqrt{-c} \cos ^{-1}(a x)}{2 c d \left (\sqrt{-c}-\sqrt{d} x\right )}-\frac{\sqrt{-c} \cos ^{-1}(a x)}{2 c d \left (\sqrt{-c}+\sqrt{d} x\right )}\right ) \, dx}{2 c}\\ &=-\frac{\cos ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}-\sqrt{d} x\right )}+\frac{\cos ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}+\sqrt{d} x\right )}+\frac{\int \frac{\cos ^{-1}(a x)}{\sqrt{-c}-\sqrt{d} x} \, dx}{4 (-c)^{3/2}}+\frac{\int \frac{\cos ^{-1}(a x)}{\sqrt{-c}+\sqrt{d} x} \, dx}{4 (-c)^{3/2}}+\frac{a \operatorname{Subst}\left (\int \frac{1}{a^2 c d+d^2-x^2} \, dx,x,\frac{-d+a^2 \sqrt{-c} \sqrt{d} x}{\sqrt{1-a^2 x^2}}\right )}{4 c}-\frac{a \operatorname{Subst}\left (\int \frac{1}{a^2 c d+d^2-x^2} \, dx,x,\frac{d+a^2 \sqrt{-c} \sqrt{d} x}{\sqrt{1-a^2 x^2}}\right )}{4 c}\\ &=-\frac{\cos ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}-\sqrt{d} x\right )}+\frac{\cos ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}+\sqrt{d} x\right )}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{d}-a^2 \sqrt{-c} x}{\sqrt{a^2 c+d} \sqrt{1-a^2 x^2}}\right )}{4 c \sqrt{d} \sqrt{a^2 c+d}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{d}+a^2 \sqrt{-c} x}{\sqrt{a^2 c+d} \sqrt{1-a^2 x^2}}\right )}{4 c \sqrt{d} \sqrt{a^2 c+d}}-\frac{\operatorname{Subst}\left (\int \frac{x \sin (x)}{a \sqrt{-c}-\sqrt{d} \cos (x)} \, dx,x,\cos ^{-1}(a x)\right )}{4 (-c)^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{x \sin (x)}{a \sqrt{-c}+\sqrt{d} \cos (x)} \, dx,x,\cos ^{-1}(a x)\right )}{4 (-c)^{3/2}}\\ &=-\frac{\cos ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}-\sqrt{d} x\right )}+\frac{\cos ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}+\sqrt{d} x\right )}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{d}-a^2 \sqrt{-c} x}{\sqrt{a^2 c+d} \sqrt{1-a^2 x^2}}\right )}{4 c \sqrt{d} \sqrt{a^2 c+d}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{d}+a^2 \sqrt{-c} x}{\sqrt{a^2 c+d} \sqrt{1-a^2 x^2}}\right )}{4 c \sqrt{d} \sqrt{a^2 c+d}}-\frac{\operatorname{Subst}\left (\int \frac{e^{i x} x}{i a \sqrt{-c}-\sqrt{a^2 c+d}-i \sqrt{d} e^{i x}} \, dx,x,\cos ^{-1}(a x)\right )}{4 (-c)^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{e^{i x} x}{i a \sqrt{-c}+\sqrt{a^2 c+d}-i \sqrt{d} e^{i x}} \, dx,x,\cos ^{-1}(a x)\right )}{4 (-c)^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{e^{i x} x}{i a \sqrt{-c}-\sqrt{a^2 c+d}+i \sqrt{d} e^{i x}} \, dx,x,\cos ^{-1}(a x)\right )}{4 (-c)^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{e^{i x} x}{i a \sqrt{-c}+\sqrt{a^2 c+d}+i \sqrt{d} e^{i x}} \, dx,x,\cos ^{-1}(a x)\right )}{4 (-c)^{3/2}}\\ &=-\frac{\cos ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}-\sqrt{d} x\right )}+\frac{\cos ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}+\sqrt{d} x\right )}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{d}-a^2 \sqrt{-c} x}{\sqrt{a^2 c+d} \sqrt{1-a^2 x^2}}\right )}{4 c \sqrt{d} \sqrt{a^2 c+d}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{d}+a^2 \sqrt{-c} x}{\sqrt{a^2 c+d} \sqrt{1-a^2 x^2}}\right )}{4 c \sqrt{d} \sqrt{a^2 c+d}}-\frac{\cos ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\cos ^{-1}(a x) \log \left (1+\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{\cos ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\cos ^{-1}(a x) \log \left (1+\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\operatorname{Subst}\left (\int \log \left (1-\frac{i \sqrt{d} e^{i x}}{i a \sqrt{-c}-\sqrt{a^2 c+d}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{\operatorname{Subst}\left (\int \log \left (1+\frac{i \sqrt{d} e^{i x}}{i a \sqrt{-c}-\sqrt{a^2 c+d}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\operatorname{Subst}\left (\int \log \left (1-\frac{i \sqrt{d} e^{i x}}{i a \sqrt{-c}+\sqrt{a^2 c+d}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{\operatorname{Subst}\left (\int \log \left (1+\frac{i \sqrt{d} e^{i x}}{i a \sqrt{-c}+\sqrt{a^2 c+d}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{4 (-c)^{3/2} \sqrt{d}}\\ &=-\frac{\cos ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}-\sqrt{d} x\right )}+\frac{\cos ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}+\sqrt{d} x\right )}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{d}-a^2 \sqrt{-c} x}{\sqrt{a^2 c+d} \sqrt{1-a^2 x^2}}\right )}{4 c \sqrt{d} \sqrt{a^2 c+d}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{d}+a^2 \sqrt{-c} x}{\sqrt{a^2 c+d} \sqrt{1-a^2 x^2}}\right )}{4 c \sqrt{d} \sqrt{a^2 c+d}}-\frac{\cos ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\cos ^{-1}(a x) \log \left (1+\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{\cos ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\cos ^{-1}(a x) \log \left (1+\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{i \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i \sqrt{d} x}{i a \sqrt{-c}-\sqrt{a^2 c+d}}\right )}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{i \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{i \sqrt{d} x}{i a \sqrt{-c}-\sqrt{a^2 c+d}}\right )}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{i \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i \sqrt{d} x}{i a \sqrt{-c}+\sqrt{a^2 c+d}}\right )}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{i \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{i \sqrt{d} x}{i a \sqrt{-c}+\sqrt{a^2 c+d}}\right )}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )}{4 (-c)^{3/2} \sqrt{d}}\\ &=-\frac{\cos ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}-\sqrt{d} x\right )}+\frac{\cos ^{-1}(a x)}{4 c \sqrt{d} \left (\sqrt{-c}+\sqrt{d} x\right )}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{d}-a^2 \sqrt{-c} x}{\sqrt{a^2 c+d} \sqrt{1-a^2 x^2}}\right )}{4 c \sqrt{d} \sqrt{a^2 c+d}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{d}+a^2 \sqrt{-c} x}{\sqrt{a^2 c+d} \sqrt{1-a^2 x^2}}\right )}{4 c \sqrt{d} \sqrt{a^2 c+d}}-\frac{\cos ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\cos ^{-1}(a x) \log \left (1+\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{\cos ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{\cos ^{-1}(a x) \log \left (1+\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{i \text{Li}_2\left (-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{i \text{Li}_2\left (\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}-\frac{i \text{Li}_2\left (-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}+\frac{i \text{Li}_2\left (\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt{d}}\\ \end{align*}

Mathematica [A]  time = 2.29308, size = 1065, normalized size = 1.46 \[ \frac{4 \sin ^{-1}\left (\frac{\sqrt{1-\frac{i a \sqrt{c}}{\sqrt{d}}}}{\sqrt{2}}\right ) \tan ^{-1}\left (\frac{\left (a \sqrt{c}-i \sqrt{d}\right ) \tan \left (\frac{1}{2} \cos ^{-1}(a x)\right )}{\sqrt{c a^2+d}}\right )-4 \sin ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c} a}{\sqrt{d}}+1}}{\sqrt{2}}\right ) \tan ^{-1}\left (\frac{\left (\sqrt{c} a+i \sqrt{d}\right ) \tan \left (\frac{1}{2} \cos ^{-1}(a x)\right )}{\sqrt{c a^2+d}}\right )+i \cos ^{-1}(a x) \log \left (1-\frac{i \left (\sqrt{c a^2+d}-a \sqrt{c}\right ) e^{i \cos ^{-1}(a x)}}{\sqrt{d}}\right )+2 i \sin ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c} a}{\sqrt{d}}+1}}{\sqrt{2}}\right ) \log \left (1-\frac{i \left (\sqrt{c a^2+d}-a \sqrt{c}\right ) e^{i \cos ^{-1}(a x)}}{\sqrt{d}}\right )-i \cos ^{-1}(a x) \log \left (\frac{i e^{i \cos ^{-1}(a x)} \left (\sqrt{c a^2+d}-a \sqrt{c}\right )}{\sqrt{d}}+1\right )-2 i \sin ^{-1}\left (\frac{\sqrt{1-\frac{i a \sqrt{c}}{\sqrt{d}}}}{\sqrt{2}}\right ) \log \left (\frac{i e^{i \cos ^{-1}(a x)} \left (\sqrt{c a^2+d}-a \sqrt{c}\right )}{\sqrt{d}}+1\right )-i \cos ^{-1}(a x) \log \left (1-\frac{i \left (\sqrt{c} a+\sqrt{c a^2+d}\right ) e^{i \cos ^{-1}(a x)}}{\sqrt{d}}\right )+2 i \sin ^{-1}\left (\frac{\sqrt{1-\frac{i a \sqrt{c}}{\sqrt{d}}}}{\sqrt{2}}\right ) \log \left (1-\frac{i \left (\sqrt{c} a+\sqrt{c a^2+d}\right ) e^{i \cos ^{-1}(a x)}}{\sqrt{d}}\right )+i \cos ^{-1}(a x) \log \left (\frac{i e^{i \cos ^{-1}(a x)} \left (\sqrt{c} a+\sqrt{c a^2+d}\right )}{\sqrt{d}}+1\right )-2 i \sin ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c} a}{\sqrt{d}}+1}}{\sqrt{2}}\right ) \log \left (\frac{i e^{i \cos ^{-1}(a x)} \left (\sqrt{c} a+\sqrt{c a^2+d}\right )}{\sqrt{d}}+1\right )+\sqrt{c} \left (\frac{\cos ^{-1}(a x)}{\sqrt{d} x-i \sqrt{c}}-\frac{a \log \left (\frac{2 d \left (-i \sqrt{c} x a^2+\sqrt{d}+\sqrt{c a^2+d} \sqrt{1-a^2 x^2}\right )}{a \sqrt{c a^2+d} \left (\sqrt{d} x-i \sqrt{c}\right )}\right )}{\sqrt{c a^2+d}}\right )+\sqrt{c} \left (\frac{\cos ^{-1}(a x)}{\sqrt{d} x+i \sqrt{c}}-\frac{a \log \left (-\frac{2 d \left (i \sqrt{c} x a^2+\sqrt{d}+\sqrt{c a^2+d} \sqrt{1-a^2 x^2}\right )}{a \sqrt{c a^2+d} \left (\sqrt{d} x+i \sqrt{c}\right )}\right )}{\sqrt{c a^2+d}}\right )-\text{PolyLog}\left (2,-\frac{i \left (\sqrt{c a^2+d}-a \sqrt{c}\right ) e^{i \cos ^{-1}(a x)}}{\sqrt{d}}\right )+\text{PolyLog}\left (2,\frac{i \left (\sqrt{c a^2+d}-a \sqrt{c}\right ) e^{i \cos ^{-1}(a x)}}{\sqrt{d}}\right )+\text{PolyLog}\left (2,-\frac{i \left (\sqrt{c} a+\sqrt{c a^2+d}\right ) e^{i \cos ^{-1}(a x)}}{\sqrt{d}}\right )-\text{PolyLog}\left (2,\frac{i \left (\sqrt{c} a+\sqrt{c a^2+d}\right ) e^{i \cos ^{-1}(a x)}}{\sqrt{d}}\right )}{4 c^{3/2} \sqrt{d}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCos[a*x]/(c + d*x^2)^2,x]

[Out]

(4*ArcSin[Sqrt[1 - (I*a*Sqrt[c])/Sqrt[d]]/Sqrt[2]]*ArcTan[((a*Sqrt[c] - I*Sqrt[d])*Tan[ArcCos[a*x]/2])/Sqrt[a^
2*c + d]] - 4*ArcSin[Sqrt[1 + (I*a*Sqrt[c])/Sqrt[d]]/Sqrt[2]]*ArcTan[((a*Sqrt[c] + I*Sqrt[d])*Tan[ArcCos[a*x]/
2])/Sqrt[a^2*c + d]] + I*ArcCos[a*x]*Log[1 - (I*(-(a*Sqrt[c]) + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] +
(2*I)*ArcSin[Sqrt[1 + (I*a*Sqrt[c])/Sqrt[d]]/Sqrt[2]]*Log[1 - (I*(-(a*Sqrt[c]) + Sqrt[a^2*c + d])*E^(I*ArcCos
[a*x]))/Sqrt[d]] - I*ArcCos[a*x]*Log[1 + (I*(-(a*Sqrt[c]) + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] - (2*
I)*ArcSin[Sqrt[1 - (I*a*Sqrt[c])/Sqrt[d]]/Sqrt[2]]*Log[1 + (I*(-(a*Sqrt[c]) + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x
]))/Sqrt[d]] - I*ArcCos[a*x]*Log[1 - (I*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] + (2*I)*ArcS
in[Sqrt[1 - (I*a*Sqrt[c])/Sqrt[d]]/Sqrt[2]]*Log[1 - (I*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d
]] + I*ArcCos[a*x]*Log[1 + (I*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] - (2*I)*ArcSin[Sqrt[1
+ (I*a*Sqrt[c])/Sqrt[d]]/Sqrt[2]]*Log[1 + (I*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] + Sqrt[
c]*(ArcCos[a*x]/((-I)*Sqrt[c] + Sqrt[d]*x) - (a*Log[(2*d*(Sqrt[d] - I*a^2*Sqrt[c]*x + Sqrt[a^2*c + d]*Sqrt[1 -
a^2*x^2]))/(a*Sqrt[a^2*c + d]*((-I)*Sqrt[c] + Sqrt[d]*x))])/Sqrt[a^2*c + d]) + Sqrt[c]*(ArcCos[a*x]/(I*Sqrt[c
] + Sqrt[d]*x) - (a*Log[(-2*d*(Sqrt[d] + I*a^2*Sqrt[c]*x + Sqrt[a^2*c + d]*Sqrt[1 - a^2*x^2]))/(a*Sqrt[a^2*c +
d]*(I*Sqrt[c] + Sqrt[d]*x))])/Sqrt[a^2*c + d]) - PolyLog[2, ((-I)*(-(a*Sqrt[c]) + Sqrt[a^2*c + d])*E^(I*ArcCo
s[a*x]))/Sqrt[d]] + PolyLog[2, (I*(-(a*Sqrt[c]) + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] + PolyLog[2, ((
-I)*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] - PolyLog[2, (I*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^
(I*ArcCos[a*x]))/Sqrt[d]])/(4*c^(3/2)*Sqrt[d])

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Maple [C]  time = 0.913, size = 1654, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccos(a*x)/(d*x^2+c)^2,x)

[Out]

1/2*a^2*arccos(a*x)*x/c/(a^2*d*x^2+a^2*c)+1/2*I*a*((2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2)*arctan(d*(I*
(-a^2*x^2+1)^(1/2)+a*x)/((2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2))/c/d^2+1/2*I*a*((2*a^2*c+2*(a^2*c*(a^2
*c+d))^(1/2)+d)*d)^(1/2)*arctan(d*(I*(-a^2*x^2+1)^(1/2)+a*x)/((2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2))/
c/(a^2*c+d)/d^2*(a^2*c*(a^2*c+d))^(1/2)-I*a^3*(-(2*a^2*c-2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2)*arctanh(d*(I*(-
a^2*x^2+1)^(1/2)+a*x)/((-2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)-d)*d)^(1/2))/(a^2*c+d)/d^3*(a^2*c*(a^2*c+d))^(1/2)+
I*a^3*(-(2*a^2*c-2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2)*arctanh(d*(I*(-a^2*x^2+1)^(1/2)+a*x)/((-2*a^2*c+2*(a^2*
c*(a^2*c+d))^(1/2)-d)*d)^(1/2))/d^3+I*a^3*((2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2)*arctan(d*(I*(-a^2*x^
2+1)^(1/2)+a*x)/((2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2))/d^3-I*a^3*((2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)
+d)*d)^(1/2)*arctan(d*(I*(-a^2*x^2+1)^(1/2)+a*x)/((2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2))/(a^2*c+d)/d^
2-I*a*((2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2)*arctan(d*(I*(-a^2*x^2+1)^(1/2)+a*x)/((2*a^2*c+2*(a^2*c*(
a^2*c+d))^(1/2)+d)*d)^(1/2))/c/d^3*(a^2*c*(a^2*c+d))^(1/2)-I*a^5*((2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/
2)*arctan(d*(I*(-a^2*x^2+1)^(1/2)+a*x)/((2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2))*c/(a^2*c+d)/d^3+I*a*(-
(2*a^2*c-2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2)*arctanh(d*(I*(-a^2*x^2+1)^(1/2)+a*x)/((-2*a^2*c+2*(a^2*c*(a^2*c
+d))^(1/2)-d)*d)^(1/2))/c/d^3*(a^2*c*(a^2*c+d))^(1/2)+1/2*I*a*(-(2*a^2*c-2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2)
*arctanh(d*(I*(-a^2*x^2+1)^(1/2)+a*x)/((-2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)-d)*d)^(1/2))/c/d^2-I*a^5*(-(2*a^2*c
-2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2)*arctanh(d*(I*(-a^2*x^2+1)^(1/2)+a*x)/((-2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/
2)-d)*d)^(1/2))*c/(a^2*c+d)/d^3+I*a^3*((2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2)*arctan(d*(I*(-a^2*x^2+1)
^(1/2)+a*x)/((2*a^2*c+2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2))/(a^2*c+d)/d^3*(a^2*c*(a^2*c+d))^(1/2)-I*a^3*(-(2*
a^2*c-2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2)*arctanh(d*(I*(-a^2*x^2+1)^(1/2)+a*x)/((-2*a^2*c+2*(a^2*c*(a^2*c+d)
)^(1/2)-d)*d)^(1/2))/(a^2*c+d)/d^2+1/4*I*a/c*sum(1/_R1/(_R1^2*d+2*a^2*c+d)*(I*arccos(a*x)*ln((_R1-a*x-I*(-a^2*
x^2+1)^(1/2))/_R1)+dilog((_R1-a*x-I*(-a^2*x^2+1)^(1/2))/_R1)),_R1=RootOf(d*_Z^4+(4*a^2*c+2*d)*_Z^2+d))-1/2*I*a
*(-(2*a^2*c-2*(a^2*c*(a^2*c+d))^(1/2)+d)*d)^(1/2)*arctanh(d*(I*(-a^2*x^2+1)^(1/2)+a*x)/((-2*a^2*c+2*(a^2*c*(a^
2*c+d))^(1/2)-d)*d)^(1/2))/c/(a^2*c+d)/d^2*(a^2*c*(a^2*c+d))^(1/2)-1/4*I*a/c*sum(_R1/(_R1^2*d+2*a^2*c+d)*(I*ar
ccos(a*x)*ln((_R1-a*x-I*(-a^2*x^2+1)^(1/2))/_R1)+dilog((_R1-a*x-I*(-a^2*x^2+1)^(1/2))/_R1)),_R1=RootOf(d*_Z^4+
(4*a^2*c+2*d)*_Z^2+d))

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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(arccos(a*x)/(d*x^2+c)^2,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{\arccos \left (a x\right )}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)/(d*x^2+c)^2,x, algorithm="fricas")

[Out]

integral(arccos(a*x)/(d^2*x^4 + 2*c*d*x^2 + c^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acos(a*x)/(d*x**2+c)**2,x)

[Out]

Integral(acos(a*x)/(c + d*x**2)**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)/(d*x^2+c)^2,x, algorithm="giac")

[Out]

integrate(arccos(a*x)/(d*x^2 + c)^2, x)