\(\int \frac {a+b \arccos (c x)}{d+e x^2} \, dx\) [630]

Optimal result

Integrand size = 18, antiderivative size = 541 \[ \int \frac {a+b \arccos (c x)}{d+e x^2} \, dx=\frac {(a+b \arccos (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \arccos (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {(a+b \arccos (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \arccos (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}} \] Output:

1/2*(a+b*arccos(c*x))*ln(1-e^(1/2)*(c*x+I*(-c^2*x^2+1)^(1/2))/(I*c*(-d)^(1 
/2)-(c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1/2)-1/2*(a+b*arccos(c*x))*ln(1+e^(1/2 
)*(c*x+I*(-c^2*x^2+1)^(1/2))/(I*c*(-d)^(1/2)-(c^2*d+e)^(1/2)))/(-d)^(1/2)/ 
e^(1/2)+1/2*(a+b*arccos(c*x))*ln(1-e^(1/2)*(c*x+I*(-c^2*x^2+1)^(1/2))/(I*c 
*(-d)^(1/2)+(c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1/2)-1/2*(a+b*arccos(c*x))*ln( 
1+e^(1/2)*(c*x+I*(-c^2*x^2+1)^(1/2))/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/(-d 
)^(1/2)/e^(1/2)+1/2*I*b*polylog(2,-e^(1/2)*(c*x+I*(-c^2*x^2+1)^(1/2))/(I*c 
*(-d)^(1/2)-(c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1/2)-1/2*I*b*polylog(2,e^(1/2) 
*(c*x+I*(-c^2*x^2+1)^(1/2))/(I*c*(-d)^(1/2)-(c^2*d+e)^(1/2)))/(-d)^(1/2)/e 
^(1/2)+1/2*I*b*polylog(2,-e^(1/2)*(c*x+I*(-c^2*x^2+1)^(1/2))/(I*c*(-d)^(1/ 
2)+(c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1/2)-1/2*I*b*polylog(2,e^(1/2)*(c*x+I*( 
-c^2*x^2+1)^(1/2))/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1/2)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 843, normalized size of antiderivative = 1.56 \[ \int \frac {a+b \arccos (c x)}{d+e x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + 4*b*ArcSin[Sqrt[1 - (I*c*Sqrt[d])/Sqrt[ 
e]]/Sqrt[2]]*ArcTan[((c*Sqrt[d] - I*Sqrt[e])*Tan[ArcCos[c*x]/2])/Sqrt[c^2* 
d + e]] - 4*b*ArcSin[Sqrt[1 + (I*c*Sqrt[d])/Sqrt[e]]/Sqrt[2]]*ArcTan[((c*S 
qrt[d] + I*Sqrt[e])*Tan[ArcCos[c*x]/2])/Sqrt[c^2*d + e]] + I*b*ArcCos[c*x] 
*Log[1 - (I*(-(c*Sqrt[d]) + Sqrt[c^2*d + e])*E^(I*ArcCos[c*x]))/Sqrt[e]] + 
 (2*I)*b*ArcSin[Sqrt[1 + (I*c*Sqrt[d])/Sqrt[e]]/Sqrt[2]]*Log[1 - (I*(-(c*S 
qrt[d]) + Sqrt[c^2*d + e])*E^(I*ArcCos[c*x]))/Sqrt[e]] - I*b*ArcCos[c*x]*L 
og[1 + (I*(-(c*Sqrt[d]) + Sqrt[c^2*d + e])*E^(I*ArcCos[c*x]))/Sqrt[e]] - ( 
2*I)*b*ArcSin[Sqrt[1 - (I*c*Sqrt[d])/Sqrt[e]]/Sqrt[2]]*Log[1 + (I*(-(c*Sqr 
t[d]) + Sqrt[c^2*d + e])*E^(I*ArcCos[c*x]))/Sqrt[e]] - I*b*ArcCos[c*x]*Log 
[1 - (I*(c*Sqrt[d] + Sqrt[c^2*d + e])*E^(I*ArcCos[c*x]))/Sqrt[e]] + (2*I)* 
b*ArcSin[Sqrt[1 - (I*c*Sqrt[d])/Sqrt[e]]/Sqrt[2]]*Log[1 - (I*(c*Sqrt[d] + 
Sqrt[c^2*d + e])*E^(I*ArcCos[c*x]))/Sqrt[e]] + I*b*ArcCos[c*x]*Log[1 + (I* 
(c*Sqrt[d] + Sqrt[c^2*d + e])*E^(I*ArcCos[c*x]))/Sqrt[e]] - (2*I)*b*ArcSin 
[Sqrt[1 + (I*c*Sqrt[d])/Sqrt[e]]/Sqrt[2]]*Log[1 + (I*(c*Sqrt[d] + Sqrt[c^2 
*d + e])*E^(I*ArcCos[c*x]))/Sqrt[e]] - b*PolyLog[2, ((-I)*(-(c*Sqrt[d]) + 
Sqrt[c^2*d + e])*E^(I*ArcCos[c*x]))/Sqrt[e]] + b*PolyLog[2, (I*(-(c*Sqrt[d 
]) + Sqrt[c^2*d + e])*E^(I*ArcCos[c*x]))/Sqrt[e]] + b*PolyLog[2, ((-I)*(c* 
Sqrt[d] + Sqrt[c^2*d + e])*E^(I*ArcCos[c*x]))/Sqrt[e]] - b*PolyLog[2, (I*( 
c*Sqrt[d] + Sqrt[c^2*d + e])*E^(I*ArcCos[c*x]))/Sqrt[e]])/(2*Sqrt[d]*Sq...
 

Rubi [A] (verified)

Time = 1.24 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.

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

\(\Big \downarrow \) 5173

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((a + b*ArcCos[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[-d] - I*S 
qrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcCos[c*x])*Log[1 + (Sqr 
t[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[-d] - I*Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqr 
t[e]) + ((a + b*ArcCos[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[- 
d] + I*Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcCos[c*x])*Log[ 
1 + (Sqrt[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[-d] + I*Sqrt[c^2*d + e])])/(2*Sqrt 
[-d]*Sqrt[e]) + ((I/2)*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[ 
-d] - I*Sqrt[c^2*d + e]))])/(Sqrt[-d]*Sqrt[e]) - ((I/2)*b*PolyLog[2, (Sqrt 
[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[-d] - I*Sqrt[c^2*d + e])])/(Sqrt[-d]*Sqrt[e 
]) + ((I/2)*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[-d] + I*Sqr 
t[c^2*d + e]))])/(Sqrt[-d]*Sqrt[e]) - ((I/2)*b*PolyLog[2, (Sqrt[e]*E^(I*Ar 
cCos[c*x]))/(c*Sqrt[-d] + I*Sqrt[c^2*d + e])])/(Sqrt[-d]*Sqrt[e])
 

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])
 

Maple [C] (verified)

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

Time = 1.37 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.43

method	result	size
parts	\(\frac {a \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}}-\frac {i b c \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (i \arccos \left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{2}+\frac {i b c \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {i \arccos \left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right )}{2}\)	\(234\)
derivativedivides	\(\frac {\frac {a c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}}+b \,c^{2} \left (-\frac {i \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (i \arccos \left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{2}+\frac {i \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {i \arccos \left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right )}{2}\right )}{c}\)	\(241\)
default	\(\frac {\frac {a c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}}+b \,c^{2} \left (-\frac {i \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (i \arccos \left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{2}+\frac {i \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {i \arccos \left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -i \sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right )}{2}\right )}{c}\)	\(241\)

Input:

int((a+b*arccos(c*x))/(e*x^2+d),x,method=_RETURNVERBOSE)
 

Output:

a/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))-1/2*I*b*c*sum(_R1/(_R1^2*e+2*c^2*d+e 
)*(I*arccos(c*x)*ln((_R1-c*x-I*(-c^2*x^2+1)^(1/2))/_R1)+dilog((_R1-c*x-I*( 
-c^2*x^2+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(4*c^2*d+2*e)*_Z^2+e))+1/2*I*b* 
c*sum(1/_R1/(_R1^2*e+2*c^2*d+e)*(I*arccos(c*x)*ln((_R1-c*x-I*(-c^2*x^2+1)^ 
(1/2))/_R1)+dilog((_R1-c*x-I*(-c^2*x^2+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+( 
4*c^2*d+2*e)*_Z^2+e))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \arccos (c x)}{d+e x^2} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F(-2)]

Exception generated. \[ \int \frac {a+b \arccos (c x)}{d+e x^2} \, dx=\text {Exception raised: RuntimeError} \] Input:

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \arccos (c x)}{d+e x^2} \, dx=\frac {\sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a +\left (\int \frac {\mathit {acos} \left (c x \right )}{e \,x^{2}+d}d x \right ) b d e}{d e} \] Input:

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

Output:

(sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a + int(acos(c*x)/(d + e*x* 
*2),x)*b*d*e)/(d*e)