\(\int \frac {(a+b \arccos (c x))^2}{d+e x^2} \, dx\) [90]

Optimal result

Integrand size = 20, antiderivative size = 821 \[ \int \frac {(a+b \arccos (c x))^2}{d+e x^2} \, dx=\frac {(a+b \arccos (c x))^2 \log \left (1-\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}-i \sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \arccos (c x))^2 \log \left (1+\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}-i \sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {(a+b \arccos (c x))^2 \log \left (1-\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}+i \sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \arccos (c x))^2 \log \left (1+\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}+i \sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}-i \sqrt {c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}-i \sqrt {c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}+i \sqrt {c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}+i \sqrt {c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}-i \sqrt {c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}-i \sqrt {c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}+i \sqrt {c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}+i \sqrt {c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}} \] Output:

1/2*(a+b*arccos(c*x))^2*ln(1-e^(1/2)*(c*x+I*(-c^2*x^2+1)^(1/2))/(c*(-d)^(1 
/2)-I*(c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1/2)-1/2*(a+b*arccos(c*x))^2*ln(1+e^ 
(1/2)*(c*x+I*(-c^2*x^2+1)^(1/2))/(c*(-d)^(1/2)-I*(c^2*d+e)^(1/2)))/(-d)^(1 
/2)/e^(1/2)+1/2*(a+b*arccos(c*x))^2*ln(1-e^(1/2)*(c*x+I*(-c^2*x^2+1)^(1/2) 
)/(c*(-d)^(1/2)+I*(c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1/2)-1/2*(a+b*arccos(c*x 
))^2*ln(1+e^(1/2)*(c*x+I*(-c^2*x^2+1)^(1/2))/(c*(-d)^(1/2)+I*(c^2*d+e)^(1/ 
2)))/(-d)^(1/2)/e^(1/2)+I*b*(a+b*arccos(c*x))*polylog(2,-e^(1/2)*(c*x+I*(- 
c^2*x^2+1)^(1/2))/(c*(-d)^(1/2)-I*(c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1/2)-I*b 
*(a+b*arccos(c*x))*polylog(2,e^(1/2)*(c*x+I*(-c^2*x^2+1)^(1/2))/(c*(-d)^(1 
/2)-I*(c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1/2)+I*b*(a+b*arccos(c*x))*polylog(2 
,-e^(1/2)*(c*x+I*(-c^2*x^2+1)^(1/2))/(c*(-d)^(1/2)+I*(c^2*d+e)^(1/2)))/(-d 
)^(1/2)/e^(1/2)-I*b*(a+b*arccos(c*x))*polylog(2,e^(1/2)*(c*x+I*(-c^2*x^2+1 
)^(1/2))/(c*(-d)^(1/2)+I*(c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1/2)-b^2*polylog( 
3,-e^(1/2)*(c*x+I*(-c^2*x^2+1)^(1/2))/(c*(-d)^(1/2)-I*(c^2*d+e)^(1/2)))/(- 
d)^(1/2)/e^(1/2)+b^2*polylog(3,e^(1/2)*(c*x+I*(-c^2*x^2+1)^(1/2))/(c*(-d)^ 
(1/2)-I*(c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1/2)-b^2*polylog(3,-e^(1/2)*(c*x+I 
*(-c^2*x^2+1)^(1/2))/(c*(-d)^(1/2)+I*(c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1/2)+ 
b^2*polylog(3,e^(1/2)*(c*x+I*(-c^2*x^2+1)^(1/2))/(c*(-d)^(1/2)+I*(c^2*d+e) 
^(1/2)))/(-d)^(1/2)/e^(1/2)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 1.85 (sec) , antiderivative size = 821, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.

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

\(\Big \downarrow \) 5173

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\operatorname {PolyLog}\left (3,-\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}-i \sqrt {d c^2+e}}\right ) b^2}{\sqrt {-d} \sqrt {e}}+\frac {\operatorname {PolyLog}\left (3,\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}-i \sqrt {d c^2+e}}\right ) b^2}{\sqrt {-d} \sqrt {e}}-\frac {\operatorname {PolyLog}\left (3,-\frac {\sqrt {e} e^{i \arccos (c x)}}{\sqrt {-d} c+i \sqrt {d c^2+e}}\right ) b^2}{\sqrt {-d} \sqrt {e}}+\frac {\operatorname {PolyLog}\left (3,\frac {\sqrt {e} e^{i \arccos (c x)}}{\sqrt {-d} c+i \sqrt {d c^2+e}}\right ) b^2}{\sqrt {-d} \sqrt {e}}+\frac {i (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}-i \sqrt {d c^2+e}}\right ) b}{\sqrt {-d} \sqrt {e}}-\frac {i (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}-i \sqrt {d c^2+e}}\right ) b}{\sqrt {-d} \sqrt {e}}+\frac {i (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arccos (c x)}}{\sqrt {-d} c+i \sqrt {d c^2+e}}\right ) b}{\sqrt {-d} \sqrt {e}}-\frac {i (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arccos (c x)}}{\sqrt {-d} c+i \sqrt {d c^2+e}}\right ) b}{\sqrt {-d} \sqrt {e}}+\frac {(a+b \arccos (c x))^2 \log \left (1-\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}-i \sqrt {d c^2+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \arccos (c x))^2 \log \left (\frac {e^{i \arccos (c x)} \sqrt {e}}{c \sqrt {-d}-i \sqrt {d c^2+e}}+1\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {(a+b \arccos (c x))^2 \log \left (1-\frac {\sqrt {e} e^{i \arccos (c x)}}{\sqrt {-d} c+i \sqrt {d c^2+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \arccos (c x))^2 \log \left (\frac {e^{i \arccos (c x)} \sqrt {e}}{\sqrt {-d} c+i \sqrt {d c^2+e}}+1\right )}{2 \sqrt {-d} \sqrt {e}}\)

Input:

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

Output:

((a + b*ArcCos[c*x])^2*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])^2*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])^2*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] 
)^2*Log[1 + (Sqrt[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[-d] + I*Sqrt[c^2*d + e])]) 
/(2*Sqrt[-d]*Sqrt[e]) + (I*b*(a + b*ArcCos[c*x])*PolyLog[2, -((Sqrt[e]*E^( 
I*ArcCos[c*x]))/(c*Sqrt[-d] - I*Sqrt[c^2*d + e]))])/(Sqrt[-d]*Sqrt[e]) - ( 
I*b*(a + b*ArcCos[c*x])*PolyLog[2, (Sqrt[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[-d] 
 - I*Sqrt[c^2*d + e])])/(Sqrt[-d]*Sqrt[e]) + (I*b*(a + b*ArcCos[c*x])*Poly 
Log[2, -((Sqrt[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[-d] + I*Sqrt[c^2*d + e]))])/( 
Sqrt[-d]*Sqrt[e]) - (I*b*(a + b*ArcCos[c*x])*PolyLog[2, (Sqrt[e]*E^(I*ArcC 
os[c*x]))/(c*Sqrt[-d] + I*Sqrt[c^2*d + e])])/(Sqrt[-d]*Sqrt[e]) - (b^2*Pol 
yLog[3, -((Sqrt[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[-d] - I*Sqrt[c^2*d + e]))])/ 
(Sqrt[-d]*Sqrt[e]) + (b^2*PolyLog[3, (Sqrt[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[- 
d] - I*Sqrt[c^2*d + e])])/(Sqrt[-d]*Sqrt[e]) - (b^2*PolyLog[3, -((Sqrt[e]* 
E^(I*ArcCos[c*x]))/(c*Sqrt[-d] + I*Sqrt[c^2*d + e]))])/(Sqrt[-d]*Sqrt[e]) 
+ (b^2*PolyLog[3, (Sqrt[e]*E^(I*ArcCos[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 [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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