3.1.27 \(\int \frac {\arccos (a x)}{c+d x^2} \, dx\) [27]

3.1.27.1 Optimal result

Integrand size = 14, antiderivative size = 521 \[ \int \frac {\arccos (a x)}{c+d x^2} \, dx=\frac {\arccos (a x) \log \left (1-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\arccos (a x) \log \left (1+\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\arccos (a x) \log \left (1-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\arccos (a x) \log \left (1+\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}} \]

output

1/2*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)^(1/2)/d^(1/2)-1/2*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)^(1/2)/d^(1/2)+1/2* 
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)^(1/2)/d^(1/2)-1/2*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)^(1/2)/d^(1/2)+1/2*I*po 
lylog(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)^(1/2)/d^(1/2)-1/2*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)^(1/2)/d^(1/2)+1/2*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)^(1/2 
)/d^(1/2)-1/2*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)^(1/2)/d^(1/2)
 

3.1.27.2 Mathematica [A] (verified)

Time = 1.81 (sec) , antiderivative size = 811, normalized size of antiderivative = 1.56 \[ \int \frac {\arccos (a x)}{c+d x^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 )-\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 )}{2 \sqrt {c} \sqrt {d}} \]

input

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

output

(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]] - 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]] + PolyLo 
g[2, ((-I)*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] - Pol 
yLog[2, (I*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]])/(2*S 
qrt[c]*Sqrt[d])
 

3.1.27.3 Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 521, 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.

\(\displaystyle \int \frac {\arccos (a x)}{c+d x^2} \, dx\)

\(\Big \downarrow \) 5173

\(\displaystyle \int \left (\frac {\sqrt {-c} \arccos (a x)}{2 c \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\sqrt {-c} \arccos (a x)}{2 c \left (\sqrt {-c}+\sqrt {d} x\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {i \operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {c a^2+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {c a^2+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {i \operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{i \arccos (a x)}}{\sqrt {-c} a+i \sqrt {c a^2+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{i \arccos (a x)}}{\sqrt {-c} a+i \sqrt {c a^2+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\arccos (a x) \log \left (1-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\arccos (a x) \log \left (1+\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\arccos (a x) \log \left (1-\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\arccos (a x) \log \left (1+\frac {\sqrt {d} e^{i \arccos (a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}\)

input

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

output

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

3.1.27.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5173

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])
 

3.1.27.4 Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 4.62 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.41

\[-\frac {i a \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+\left (4 c \,a^{2}+2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {\textit {\_R1} \left (i \arccos \left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -i \sqrt {-a^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -a x -i \sqrt {-a^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} d +2 c \,a^{2}+d}\right )}{2}+\frac {i a \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+\left (4 c \,a^{2}+2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {i \arccos \left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -i \sqrt {-a^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -a x -i \sqrt {-a^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} d +2 c \,a^{2}+d \right )}\right )}{2}\]

input

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

output

-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*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))
 

3.1.27.5 Fricas [F]

\[ \int \frac {\arccos (a x)}{c+d x^2} \, dx=\int { \frac {\arccos \left (a x\right )}{d x^{2} + c} \,d x } \]

input

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

output

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

3.1.27.6 Sympy [F]

\[ \int \frac {\arccos (a x)}{c+d x^2} \, dx=\int \frac {\operatorname {acos}{\left (a x \right )}}{c + d x^{2}}\, dx \]

input

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

output

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

3.1.27.7 Maxima [F]

\[ \int \frac {\arccos (a x)}{c+d x^2} \, dx=\int { \frac {\arccos \left (a x\right )}{d x^{2} + c} \,d x } \]

input

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

output

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

3.1.27.8 Giac [F]

\[ \int \frac {\arccos (a x)}{c+d x^2} \, dx=\int { \frac {\arccos \left (a x\right )}{d x^{2} + c} \,d x } \]

input

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

output

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

3.1.27.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\arccos (a x)}{c+d x^2} \, dx=\int \frac {\mathrm {acos}\left (a\,x\right )}{d\,x^2+c} \,d x \]

input

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

output

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