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

Optimal. Leaf size=521 \[ \frac{i \text{PolyLog}\left (2,-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right )}{2 \sqrt{-c} \sqrt{d}}-\frac{i \text{PolyLog}\left (2,\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right )}{2 \sqrt{-c} \sqrt{d}}+\frac{i \text{PolyLog}\left (2,-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right )}{2 \sqrt{-c} \sqrt{d}}-\frac{i \text{PolyLog}\left (2,\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right )}{2 \sqrt{-c} \sqrt{d}}+\frac{\cos ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right )}{2 \sqrt{-c} \sqrt{d}}-\frac{\cos ^{-1}(a x) \log \left (1+\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right )}{2 \sqrt{-c} \sqrt{d}}+\frac{\cos ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right )}{2 \sqrt{-c} \sqrt{d}}-\frac{\cos ^{-1}(a x) \log \left (1+\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right )}{2 \sqrt{-c} \sqrt{d}} \]

[Out]

(ArcCos[a*x]*Log[1 - (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/(2*Sqrt[-c]*Sqrt[d]) - (Ar
cCos[a*x]*Log[1 + (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/(2*Sqrt[-c]*Sqrt[d]) + (ArcCo
s[a*x]*Log[1 - (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/(2*Sqrt[-c]*Sqrt[d]) - (ArcCos[a
*x]*Log[1 + (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/(2*Sqrt[-c]*Sqrt[d]) + ((I/2)*PolyL
og[2, -((Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d]))])/(Sqrt[-c]*Sqrt[d]) - ((I/2)*PolyLog[2,
(Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/(Sqrt[-c]*Sqrt[d]) + ((I/2)*PolyLog[2, -((Sqrt
[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d]))])/(Sqrt[-c]*Sqrt[d]) - ((I/2)*PolyLog[2, (Sqrt[d]*E^(
I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/(Sqrt[-c]*Sqrt[d])

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Rubi [A]  time = 0.809598, antiderivative size = 521, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4668, 4742, 4522, 2190, 2279, 2391} \[ \frac{i \text{PolyLog}\left (2,-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right )}{2 \sqrt{-c} \sqrt{d}}-\frac{i \text{PolyLog}\left (2,\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right )}{2 \sqrt{-c} \sqrt{d}}+\frac{i \text{PolyLog}\left (2,-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right )}{2 \sqrt{-c} \sqrt{d}}-\frac{i \text{PolyLog}\left (2,\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right )}{2 \sqrt{-c} \sqrt{d}}+\frac{\cos ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right )}{2 \sqrt{-c} \sqrt{d}}-\frac{\cos ^{-1}(a x) \log \left (1+\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right )}{2 \sqrt{-c} \sqrt{d}}+\frac{\cos ^{-1}(a x) \log \left (1-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right )}{2 \sqrt{-c} \sqrt{d}}-\frac{\cos ^{-1}(a x) \log \left (1+\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right )}{2 \sqrt{-c} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]

Int[ArcCos[a*x]/(c + d*x^2),x]

[Out]

(ArcCos[a*x]*Log[1 - (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/(2*Sqrt[-c]*Sqrt[d]) - (Ar
cCos[a*x]*Log[1 + (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/(2*Sqrt[-c]*Sqrt[d]) + (ArcCo
s[a*x]*Log[1 - (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/(2*Sqrt[-c]*Sqrt[d]) - (ArcCos[a
*x]*Log[1 + (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/(2*Sqrt[-c]*Sqrt[d]) + ((I/2)*PolyL
og[2, -((Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d]))])/(Sqrt[-c]*Sqrt[d]) - ((I/2)*PolyLog[2,
(Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/(Sqrt[-c]*Sqrt[d]) + ((I/2)*PolyLog[2, -((Sqrt
[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d]))])/(Sqrt[-c]*Sqrt[d]) - ((I/2)*PolyLog[2, (Sqrt[d]*E^(
I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/(Sqrt[-c]*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 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)}{c+d x^2} \, dx &=\int \left (\frac{\sqrt{-c} \cos ^{-1}(a x)}{2 c \left (\sqrt{-c}-\sqrt{d} x\right )}+\frac{\sqrt{-c} \cos ^{-1}(a x)}{2 c \left (\sqrt{-c}+\sqrt{d} x\right )}\right ) \, dx\\ &=-\frac{\int \frac{\cos ^{-1}(a x)}{\sqrt{-c}-\sqrt{d} x} \, dx}{2 \sqrt{-c}}-\frac{\int \frac{\cos ^{-1}(a x)}{\sqrt{-c}+\sqrt{d} x} \, dx}{2 \sqrt{-c}}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x \sin (x)}{a \sqrt{-c}-\sqrt{d} \cos (x)} \, dx,x,\cos ^{-1}(a x)\right )}{2 \sqrt{-c}}+\frac{\operatorname{Subst}\left (\int \frac{x \sin (x)}{a \sqrt{-c}+\sqrt{d} \cos (x)} \, dx,x,\cos ^{-1}(a x)\right )}{2 \sqrt{-c}}\\ &=\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 )}{2 \sqrt{-c}}+\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 )}{2 \sqrt{-c}}+\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 )}{2 \sqrt{-c}}+\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 )}{2 \sqrt{-c}}\\ &=\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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \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 )}{2 \sqrt{-c} \sqrt{d}}\\ \end{align*}

Mathematica [A]  time = 1.09452, size = 811, normalized size = 1.56 \[ \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 )-\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 )}{2 \sqrt{c} \sqrt{d}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCos[a*x]/(c + d*x^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]] - PolyL
og[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]))/Sq
rt[d]] - PolyLog[2, (I*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]])/(2*Sqrt[c]*Sqrt[d])

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Maple [C]  time = 0.535, size = 216, normalized size = 0.4 \begin{align*} -{\frac{i}{2}}a\sum _{{\it \_R1}={\it RootOf} \left ( d{{\it \_Z}}^{4}+ \left ( 4\,{a}^{2}c+2\,d \right ){{\it \_Z}}^{2}+d \right ) }{\frac{{\it \_R1}}{{{\it \_R1}}^{2}d+2\,{a}^{2}c+d} \left ( i\arccos \left ( ax \right ) \ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-ax-i\sqrt{-{a}^{2}{x}^{2}+1} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-ax-i\sqrt{-{a}^{2}{x}^{2}+1} \right ) } \right ) \right ) }+{\frac{i}{2}}a\sum _{{\it \_R1}={\it RootOf} \left ( d{{\it \_Z}}^{4}+ \left ( 4\,{a}^{2}c+2\,d \right ){{\it \_Z}}^{2}+d \right ) }{\frac{1}{{\it \_R1}\, \left ({{\it \_R1}}^{2}d+2\,{a}^{2}c+d \right ) } \left ( i\arccos \left ( ax \right ) \ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-ax-i\sqrt{-{a}^{2}{x}^{2}+1} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-ax-i\sqrt{-{a}^{2}{x}^{2}+1} \right ) } \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*I*a*sum(_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*sum(1/_R1/(_R1^2*d+2*a^2*c+d)*(I*arcc
os(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),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 x^{2} + c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccos(a*x)/(d*x^2 + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acos(a*x)/(c + d*x**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccos(a*x)/(d*x^2 + c), x)