Integrand size = 14, antiderivative size = 727 \[ \int \frac {\arccos (a x)}{\left (c+d x^2\right )^2} \, dx=-\frac {\arccos (a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\arccos (a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )}-\frac {a \text {arctanh}\left (\frac {\sqrt {d}-a^2 \sqrt {-c} x}{\sqrt {a^2 c+d} \sqrt {1-a^2 x^2}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {a \text {arctanh}\left (\frac {\sqrt {d}+a^2 \sqrt {-c} x}{\sqrt {a^2 c+d} \sqrt {1-a^2 x^2}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {\arccos (a x) \log \left (1-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\arccos (a x) \log \left (1+\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\arccos (a x) \log \left (1-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\arccos (a x) \log \left (1+\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}} \]
-1/4*arccos(a*x)*ln(1-(a*x+I*(-a^2*x^2+1)^(1/2))*d^(1/2)/(a*(-c)^(1/2)-I*( a^2*c+d)^(1/2)))/(-c)^(3/2)/d^(1/2)+1/4*arccos(a*x)*ln(1+(a*x+I*(-a^2*x^2+ 1)^(1/2))*d^(1/2)/(a*(-c)^(1/2)-I*(a^2*c+d)^(1/2)))/(-c)^(3/2)/d^(1/2)-1/4 *arccos(a*x)*ln(1-(a*x+I*(-a^2*x^2+1)^(1/2))*d^(1/2)/(a*(-c)^(1/2)+I*(a^2* c+d)^(1/2)))/(-c)^(3/2)/d^(1/2)+1/4*arccos(a*x)*ln(1+(a*x+I*(-a^2*x^2+1)^( 1/2))*d^(1/2)/(a*(-c)^(1/2)+I*(a^2*c+d)^(1/2)))/(-c)^(3/2)/d^(1/2)-1/4*I*p olylog(2,-(a*x+I*(-a^2*x^2+1)^(1/2))*d^(1/2)/(a*(-c)^(1/2)-I*(a^2*c+d)^(1/ 2)))/(-c)^(3/2)/d^(1/2)+1/4*I*polylog(2,(a*x+I*(-a^2*x^2+1)^(1/2))*d^(1/2) /(a*(-c)^(1/2)-I*(a^2*c+d)^(1/2)))/(-c)^(3/2)/d^(1/2)-1/4*I*polylog(2,-(a* x+I*(-a^2*x^2+1)^(1/2))*d^(1/2)/(a*(-c)^(1/2)+I*(a^2*c+d)^(1/2)))/(-c)^(3/ 2)/d^(1/2)+1/4*I*polylog(2,(a*x+I*(-a^2*x^2+1)^(1/2))*d^(1/2)/(a*(-c)^(1/2 )+I*(a^2*c+d)^(1/2)))/(-c)^(3/2)/d^(1/2)-1/4*arccos(a*x)/c/d^(1/2)/((-c)^( 1/2)-x*d^(1/2))+1/4*arccos(a*x)/c/d^(1/2)/((-c)^(1/2)+x*d^(1/2))-1/4*a*arc tanh((-a^2*x*(-c)^(1/2)+d^(1/2))/(a^2*c+d)^(1/2)/(-a^2*x^2+1)^(1/2))/c/d^( 1/2)/(a^2*c+d)^(1/2)-1/4*a*arctanh((a^2*x*(-c)^(1/2)+d^(1/2))/(a^2*c+d)^(1 /2)/(-a^2*x^2+1)^(1/2))/c/d^(1/2)/(a^2*c+d)^(1/2)
Time = 2.83 (sec) , antiderivative size = 1065, normalized size of antiderivative = 1.46 \[ \int \frac {\arccos (a x)}{\left (c+d x^2\right )^2} \, dx=\frac {4 \arcsin \left (\frac {\sqrt {1-\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \arctan \left (\frac {\left (a \sqrt {c}-i \sqrt {d}\right ) \tan \left (\frac {1}{2} \arccos (a x)\right )}{\sqrt {a^2 c+d}}\right )-4 \arcsin \left (\frac {\sqrt {1+\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \arctan \left (\frac {\left (a \sqrt {c}+i \sqrt {d}\right ) \tan \left (\frac {1}{2} \arccos (a x)\right )}{\sqrt {a^2 c+d}}\right )+i \arccos (a x) \log \left (1-\frac {i \left (-a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )+2 i \arcsin \left (\frac {\sqrt {1+\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1-\frac {i \left (-a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )-i \arccos (a x) \log \left (1+\frac {i \left (-a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )-2 i \arcsin \left (\frac {\sqrt {1-\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {i \left (-a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )-i \arccos (a x) \log \left (1-\frac {i \left (a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )+2 i \arcsin \left (\frac {\sqrt {1-\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1-\frac {i \left (a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )+i \arccos (a x) \log \left (1+\frac {i \left (a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )-2 i \arcsin \left (\frac {\sqrt {1+\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {i \left (a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )+\sqrt {c} \left (\frac {\arccos (a x)}{-i \sqrt {c}+\sqrt {d} x}-\frac {a \log \left (\frac {2 d \left (\sqrt {d}-i a^2 \sqrt {c} x+\sqrt {a^2 c+d} \sqrt {1-a^2 x^2}\right )}{a \sqrt {a^2 c+d} \left (-i \sqrt {c}+\sqrt {d} x\right )}\right )}{\sqrt {a^2 c+d}}\right )+\sqrt {c} \left (\frac {\arccos (a x)}{i \sqrt {c}+\sqrt {d} x}-\frac {a \log \left (-\frac {2 d \left (\sqrt {d}+i a^2 \sqrt {c} x+\sqrt {a^2 c+d} \sqrt {1-a^2 x^2}\right )}{a \sqrt {a^2 c+d} \left (i \sqrt {c}+\sqrt {d} x\right )}\right )}{\sqrt {a^2 c+d}}\right )-\operatorname {PolyLog}\left (2,-\frac {i \left (-a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )+\operatorname {PolyLog}\left (2,\frac {i \left (-a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )+\operatorname {PolyLog}\left (2,-\frac {i \left (a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )-\operatorname {PolyLog}\left (2,\frac {i \left (a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \arccos (a x)}}{\sqrt {d}}\right )}{4 c^{3/2} \sqrt {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]] - 4*ArcSin[Sqrt[1 + (I*a*Sqr t[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])/Sqr t[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]))/Sq rt[d]] - I*ArcCos[a*x]*Log[1 - (I*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^(I*ArcCo s[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*A rcCos[a*x]*Log[1 + (I*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqr t[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] + S qrt[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])...
Time = 1.41 (sec) , antiderivative size = 727, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5173, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\arccos (a x)}{\left (c+d x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 5173 |
| \(\displaystyle \int \left (-\frac {d \arccos (a x)}{2 c \left (-c d-d^2 x^2\right )}-\frac {d \arccos (a x)}{4 c \left (\sqrt {-c} \sqrt {d}-d x\right )^2}-\frac {d \arccos (a x)}{4 c \left (\sqrt {-c} \sqrt {d}+d x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {i \operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {c a^2+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {c a^2+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{i \arccos (a x)}}{\sqrt {-c} a+i \sqrt {c a^2+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{i \arccos (a x)}}{\sqrt {-c} a+i \sqrt {c a^2+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\arccos (a x) \log \left (1-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\arccos (a x) \log \left (1+\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\arccos (a x) \log \left (1-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\arccos (a x) \log \left (1+\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {a \text {arctanh}\left (\frac {\sqrt {d}-a^2 \sqrt {-c} x}{\sqrt {1-a^2 x^2} \sqrt {a^2 c+d}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {a \text {arctanh}\left (\frac {a^2 \sqrt {-c} x+\sqrt {d}}{\sqrt {1-a^2 x^2} \sqrt {a^2 c+d}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {\arccos (a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\arccos (a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )}\) |
-1/4*ArcCos[a*x]/(c*Sqrt[d]*(Sqrt[-c] - Sqrt[d]*x)) + ArcCos[a*x]/(4*c*Sqr t[d]*(Sqrt[-c] + Sqrt[d]*x)) - (a*ArcTanh[(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*Sq rt[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]*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]) - (ArcCos[a*x]*Log[1 - (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*S qrt[-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]) - ((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, -((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*Arc Cos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/((-c)^(3/2)*Sqrt[d])
3.1.28.3.1 Defintions of rubi rules used
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] && (G tQ[p, 0] || IGtQ[n, 0])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 5.75 (sec) , antiderivative size = 796, normalized size of antiderivative = 1.09
\[\frac {\frac {\arccos \left (a x \right ) a^{3} x}{2 c \left (a^{2} d \,x^{2}+c \,a^{2}\right )}-\frac {i \sqrt {\left (2 c \,a^{2}+2 \sqrt {c \,a^{2} \left (c \,a^{2}+d \right )}+d \right ) d}\, \left (2 a^{4} c^{2}-2 \sqrt {c \,a^{2} \left (c \,a^{2}+d \right )}\, a^{2} c +2 a^{2} c d -d \sqrt {c \,a^{2} \left (c \,a^{2}+d \right )}\right ) a^{2} \arctan \left (\frac {d \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )}{\sqrt {\left (2 c \,a^{2}+2 \sqrt {c \,a^{2} \left (c \,a^{2}+d \right )}+d \right ) d}}\right )}{2 c \left (c \,a^{2}+d \right ) d^{3}}+\frac {i \sqrt {\left (2 c \,a^{2}+2 \sqrt {c \,a^{2} \left (c \,a^{2}+d \right )}+d \right ) d}\, \left (2 c \,a^{2}-2 \sqrt {c \,a^{2} \left (c \,a^{2}+d \right )}+d \right ) \arctan \left (\frac {d \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )}{\sqrt {\left (2 c \,a^{2}+2 \sqrt {c \,a^{2} \left (c \,a^{2}+d \right )}+d \right ) d}}\right ) a^{2}}{2 c \,d^{3}}-\frac {i \sqrt {-\left (2 c \,a^{2}-2 \sqrt {c \,a^{2} \left (c \,a^{2}+d \right )}+d \right ) d}\, \left (2 a^{4} c^{2}+2 \sqrt {c \,a^{2} \left (c \,a^{2}+d \right )}\, a^{2} c +2 a^{2} c d +d \sqrt {c \,a^{2} \left (c \,a^{2}+d \right )}\right ) a^{2} \operatorname {arctanh}\left (\frac {d \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )}{\sqrt {\left (-2 c \,a^{2}+2 \sqrt {c \,a^{2} \left (c \,a^{2}+d \right )}-d \right ) d}}\right )}{2 c \left (c \,a^{2}+d \right ) d^{3}}+\frac {i \sqrt {-\left (2 c \,a^{2}-2 \sqrt {c \,a^{2} \left (c \,a^{2}+d \right )}+d \right ) d}\, \left (2 c \,a^{2}+2 \sqrt {c \,a^{2} \left (c \,a^{2}+d \right )}+d \right ) \operatorname {arctanh}\left (\frac {d \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )}{\sqrt {\left (-2 c \,a^{2}+2 \sqrt {c \,a^{2} \left (c \,a^{2}+d \right )}-d \right ) d}}\right ) a^{2}}{2 c \,d^{3}}-\frac {i a^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+\left (4 c \,a^{2}+2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {\textit {\_R1} \left (i \arccos \left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -i \sqrt {-a^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -a x -i \sqrt {-a^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} d +2 c \,a^{2}+d}\right )}{4 c}+\frac {i a^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+\left (4 c \,a^{2}+2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {i \arccos \left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -i \sqrt {-a^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -a x -i \sqrt {-a^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} d +2 c \,a^{2}+d \right )}\right )}{4 c}}{a}\]
1/a*(1/2*arccos(a*x)*a^3*x/c/(a^2*d*x^2+a^2*c)-1/2*I*((2*c*a^2+2*(c*a^2*(a ^2*c+d))^(1/2)+d)*d)^(1/2)*(2*a^4*c^2-2*(c*a^2*(a^2*c+d))^(1/2)*a^2*c+2*a^ 2*c*d-d*(c*a^2*(a^2*c+d))^(1/2))*a^2*arctan(d*(I*(-a^2*x^2+1)^(1/2)+a*x)/( (2*c*a^2+2*(c*a^2*(a^2*c+d))^(1/2)+d)*d)^(1/2))/c/(a^2*c+d)/d^3+1/2*I*((2* c*a^2+2*(c*a^2*(a^2*c+d))^(1/2)+d)*d)^(1/2)*(2*c*a^2-2*(c*a^2*(a^2*c+d))^( 1/2)+d)*arctan(d*(I*(-a^2*x^2+1)^(1/2)+a*x)/((2*c*a^2+2*(c*a^2*(a^2*c+d))^ (1/2)+d)*d)^(1/2))*a^2/c/d^3-1/2*I*(-(2*c*a^2-2*(c*a^2*(a^2*c+d))^(1/2)+d) *d)^(1/2)*(2*a^4*c^2+2*(c*a^2*(a^2*c+d))^(1/2)*a^2*c+2*a^2*c*d+d*(c*a^2*(a ^2*c+d))^(1/2))*a^2*arctanh(d*(I*(-a^2*x^2+1)^(1/2)+a*x)/((-2*c*a^2+2*(c*a ^2*(a^2*c+d))^(1/2)-d)*d)^(1/2))/c/(a^2*c+d)/d^3+1/2*I*(-(2*c*a^2-2*(c*a^2 *(a^2*c+d))^(1/2)+d)*d)^(1/2)*(2*c*a^2+2*(c*a^2*(a^2*c+d))^(1/2)+d)*arctan h(d*(I*(-a^2*x^2+1)^(1/2)+a*x)/((-2*c*a^2+2*(c*a^2*(a^2*c+d))^(1/2)-d)*d)^ (1/2))*a^2/c/d^3-1/4*I/c*a^2*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))+1/4*I/c*a^2*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)))
\[ \int \frac {\arccos (a x)}{\left (c+d x^2\right )^2} \, dx=\int { \frac {\arccos \left (a x\right )}{{\left (d x^{2} + c\right )}^{2}} \,d x } \]
\[ \int \frac {\arccos (a x)}{\left (c+d x^2\right )^2} \, dx=\int \frac {\operatorname {acos}{\left (a x \right )}}{\left (c + d x^{2}\right )^{2}}\, dx \]
\[ \int \frac {\arccos (a x)}{\left (c+d x^2\right )^2} \, dx=\int { \frac {\arccos \left (a x\right )}{{\left (d x^{2} + c\right )}^{2}} \,d x } \]
\[ \int \frac {\arccos (a x)}{\left (c+d x^2\right )^2} \, dx=\int { \frac {\arccos \left (a x\right )}{{\left (d x^{2} + c\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\arccos (a x)}{\left (c+d x^2\right )^2} \, dx=\int \frac {\mathrm {acos}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^2} \,d x \]