\(\int \frac {a+b \arccos (c x)}{(d+e x^2)^3} \, dx\) [650]

Optimal result

Integrand size = 18, antiderivative size = 1092 \[ \int \frac {a+b \arccos (c x)}{\left (d+e x^2\right )^3} \, dx =\text {Too large to display} \] Output:

1/16*b*c*(-c^2*x^2+1)^(1/2)/(-d)^(3/2)/(c^2*d+e)/((-d)^(1/2)-e^(1/2)*x)+1/ 
16*b*c*(-c^2*x^2+1)^(1/2)/(-d)^(3/2)/(c^2*d+e)/((-d)^(1/2)+e^(1/2)*x)-1/16 
*(a+b*arccos(c*x))/(-d)^(3/2)/e^(1/2)/((-d)^(1/2)-e^(1/2)*x)^2-3/16*(a+b*a 
rccos(c*x))/d^2/e^(1/2)/((-d)^(1/2)-e^(1/2)*x)+1/16*(a+b*arccos(c*x))/(-d) 
^(3/2)/e^(1/2)/((-d)^(1/2)+e^(1/2)*x)^2+3/16*(a+b*arccos(c*x))/d^2/e^(1/2) 
/((-d)^(1/2)+e^(1/2)*x)+1/16*b*c^3*arctanh((e^(1/2)-c^2*(-d)^(1/2)*x)/(c^2 
*d+e)^(1/2)/(-c^2*x^2+1)^(1/2))/d/e^(1/2)/(c^2*d+e)^(3/2)+3/16*b*c*arctanh 
((e^(1/2)-c^2*(-d)^(1/2)*x)/(c^2*d+e)^(1/2)/(-c^2*x^2+1)^(1/2))/d^2/e^(1/2 
)/(c^2*d+e)^(1/2)+1/16*b*c^3*arctanh((e^(1/2)+c^2*(-d)^(1/2)*x)/(c^2*d+e)^ 
(1/2)/(-c^2*x^2+1)^(1/2))/d/e^(1/2)/(c^2*d+e)^(3/2)+3/16*b*c*arctanh((e^(1 
/2)+c^2*(-d)^(1/2)*x)/(c^2*d+e)^(1/2)/(-c^2*x^2+1)^(1/2))/d^2/e^(1/2)/(c^2 
*d+e)^(1/2)+3/16*(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)^(5/2)/e^(1/2)-3/16*(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)^(5/2)/e^(1/2)+3/16*(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)^(5/2)/e^(1/2)-3/16*(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)^(5/2)/e^(1/2)+3/16*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)^(5/2)/e^(1/2)+3/16*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...
 

Mathematica [A] (warning: unable to verify)

Time = 5.65 (sec) , antiderivative size = 1547, normalized size of antiderivative = 1.42 \[ \int \frac {a+b \arccos (c x)}{\left (d+e x^2\right )^3} \, dx =\text {Too large to display} \] Input:

Integrate[(a + b*ArcCos[c*x])/(d + e*x^2)^3,x]
 

Output:

((8*a*d^(3/2)*x)/(d + e*x^2)^2 + (12*a*Sqrt[d]*x)/(d + e*x^2) + (12*a*ArcT 
an[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] + (6*b*Sqrt[d]*(ArcCos[c*x]/((-I)*Sqrt[d] 
 + Sqrt[e]*x) - (c*Log[(2*e*(Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*S 
qrt[1 - c^2*x^2]))/(c*Sqrt[c^2*d + e]*((-I)*Sqrt[d] + Sqrt[e]*x))])/Sqrt[c 
^2*d + e]))/Sqrt[e] + (2*I)*b*d*((c*Sqrt[1 - c^2*x^2])/((c^2*d + e)*((-I)* 
Sqrt[d] + Sqrt[e]*x)) - ArcCos[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) 
 + (I*c^3*Sqrt[d]*Log[(-4*e*Sqrt[c^2*d + e]*(Sqrt[e] - I*c^2*Sqrt[d]*x + S 
qrt[c^2*d + e]*Sqrt[1 - c^2*x^2]))/(c^3*(d + I*Sqrt[d]*Sqrt[e]*x))])/(Sqrt 
[e]*(c^2*d + e)^(3/2))) - (6*b*Sqrt[d]*(-(ArcCos[c*x]/(I*Sqrt[d] + Sqrt[e] 
*x)) + (c*Log[(-2*e*(Sqrt[e] + I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[1 - 
c^2*x^2]))/(c*Sqrt[c^2*d + e]*(I*Sqrt[d] + Sqrt[e]*x))])/Sqrt[c^2*d + e])) 
/Sqrt[e] - (2*I)*b*d*((c*Sqrt[1 - c^2*x^2])/((c^2*d + e)*(I*Sqrt[d] + Sqrt 
[e]*x)) - ArcCos[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) - (I*c^3*Sqrt[d] 
*Log[(-4*e*Sqrt[c^2*d + e]*(Sqrt[e] + I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sq 
rt[1 - c^2*x^2]))/(c^3*(d - I*Sqrt[d]*Sqrt[e]*x))])/(Sqrt[e]*(c^2*d + e)^( 
3/2))) + (3*b*(ArcCos[c*x]^2 - 8*ArcSin[Sqrt[1 + (I*c*Sqrt[d])/Sqrt[e]]/Sq 
rt[2]]*ArcTan[((c*Sqrt[d] + I*Sqrt[e])*Tan[ArcCos[c*x]/2])/Sqrt[c^2*d + e] 
] + (2*I)*ArcCos[c*x]*Log[1 - (I*(-(c*Sqrt[d]) + Sqrt[c^2*d + e])*E^(I*Arc 
Cos[c*x]))/Sqrt[e]] + (4*I)*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...
 

Rubi [A] (verified)

Time = 1.96 (sec) , antiderivative size = 1092, 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.

\(\displaystyle \int \frac {a+b \arccos (c x)}{\left (d+e x^2\right )^3} \, dx\)

\(\Big \downarrow \) 5173

\(\displaystyle \int \left (-\frac {3 e (a+b \arccos (c x))}{8 d^2 \left (-d e-e^2 x^2\right )}-\frac {3 e (a+b \arccos (c x))}{16 d^2 \left (\sqrt {-d} \sqrt {e}-e x\right )^2}-\frac {3 e (a+b \arccos (c x))}{16 d^2 \left (\sqrt {-d} \sqrt {e}+e x\right )^2}-\frac {e^{3/2} (a+b \arccos (c x))}{8 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}-e x\right )^3}-\frac {e^{3/2} (a+b \arccos (c x))}{8 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}+e x\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right ) c^3}{16 d \sqrt {e} \left (d c^2+e\right )^{3/2}}-\frac {b \text {arctanh}\left (\frac {\sqrt {-d} x c^2+\sqrt {e}}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right ) c^3}{16 d \sqrt {e} \left (d c^2+e\right )^{3/2}}-\frac {3 b \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right ) c}{16 d^2 \sqrt {e} \sqrt {d c^2+e}}-\frac {3 b \text {arctanh}\left (\frac {\sqrt {-d} x c^2+\sqrt {e}}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right ) c}{16 d^2 \sqrt {e} \sqrt {d c^2+e}}-\frac {b \sqrt {1-c^2 x^2} c}{16 (-d)^{3/2} \left (d c^2+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {b \sqrt {1-c^2 x^2} c}{16 (-d)^{3/2} \left (d c^2+e\right ) \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {3 (a+b \arccos (c x))}{16 d^2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {3 (a+b \arccos (c x))}{16 d^2 \sqrt {e} \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {a+b \arccos (c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )^2}+\frac {a+b \arccos (c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {e} x+\sqrt {-d}\right )^2}+\frac {3 (a+b \arccos (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}-i \sqrt {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 (a+b \arccos (c x)) \log \left (\frac {e^{i \arccos (c x)} \sqrt {e}}{c \sqrt {-d}-i \sqrt {d c^2+e}}+1\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 (a+b \arccos (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arccos (c x)}}{\sqrt {-d} c+i \sqrt {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 (a+b \arccos (c x)) \log \left (\frac {e^{i \arccos (c x)} \sqrt {e}}{\sqrt {-d} c+i \sqrt {d c^2+e}}+1\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}-i \sqrt {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}-i \sqrt {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arccos (c x)}}{\sqrt {-d} c+i \sqrt {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arccos (c x)}}{\sqrt {-d} c+i \sqrt {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}\)

Input:

Int[(a + b*ArcCos[c*x])/(d + e*x^2)^3,x]
 

Output:

-1/16*(b*c*Sqrt[1 - c^2*x^2])/((-d)^(3/2)*(c^2*d + e)*(Sqrt[-d] - Sqrt[e]* 
x)) - (b*c*Sqrt[1 - c^2*x^2])/(16*(-d)^(3/2)*(c^2*d + e)*(Sqrt[-d] + Sqrt[ 
e]*x)) - (a + b*ArcCos[c*x])/(16*(-d)^(3/2)*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x) 
^2) - (3*(a + b*ArcCos[c*x]))/(16*d^2*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x)) + (a 
 + b*ArcCos[c*x])/(16*(-d)^(3/2)*Sqrt[e]*(Sqrt[-d] + Sqrt[e]*x)^2) + (3*(a 
 + b*ArcCos[c*x]))/(16*d^2*Sqrt[e]*(Sqrt[-d] + Sqrt[e]*x)) - (b*c^3*ArcTan 
h[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d*S 
qrt[e]*(c^2*d + e)^(3/2)) - (3*b*c*ArcTanh[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqr 
t[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d^2*Sqrt[e]*Sqrt[c^2*d + e]) - (b*c^ 
3*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])]) 
/(16*d*Sqrt[e]*(c^2*d + e)^(3/2)) - (3*b*c*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d] 
*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d^2*Sqrt[e]*Sqrt[c^2*d + e]) 
 + (3*(a + b*ArcCos[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[-d] 
- I*Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*(a + b*ArcCos[c*x])*Lo 
g[1 + (Sqrt[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[-d] - I*Sqrt[c^2*d + e])])/(16*( 
-d)^(5/2)*Sqrt[e]) + (3*(a + b*ArcCos[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcCos[c 
*x]))/(c*Sqrt[-d] + I*Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*(a + 
 b*ArcCos[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[-d] + I*Sqrt[c 
^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) + (((3*I)/16)*b*PolyLog[2, -((Sqrt[e] 
*E^(I*ArcCos[c*x]))/(c*Sqrt[-d] - I*Sqrt[c^2*d + e]))])/((-d)^(5/2)*Sqr...
 

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] (warning: unable to verify)

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

Time = 5.82 (sec) , antiderivative size = 1786, normalized size of antiderivative = 1.64

method	result	size
parts	\(\text {Expression too large to display}\)	\(1786\)
derivativedivides	\(\text {Expression too large to display}\)	\(1811\)
default	\(\text {Expression too large to display}\)	\(1811\)

Input:

int((a+b*arccos(c*x))/(e*x^2+d)^3,x,method=_RETURNVERBOSE)
 

Output:

1/4*a*x/d/(e*x^2+d)^2+3/8*a/d^2*x/(e*x^2+d)+3/8*a/d^2/(d*e)^(1/2)*arctan(e 
*x/(d*e)^(1/2))+b/c*(1/8*c^2*(5*arccos(c*x)*d^2*c^5*x+3*arccos(c*x)*c^5*d* 
e*x^3-d^2*c^4*(-c^2*x^2+1)^(1/2)-c^4*d*e*x^2*(-c^2*x^2+1)^(1/2)+5*arccos(c 
*x)*c^3*d*e*x+3*arccos(c*x)*e^2*c^3*x^3)/d^2/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2 
+3/8*I*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*(2*c^2*d+2*(c^2*d* 
(c^2*d+e))^(1/2)+e)*arctanh(e*(c*x+I*(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(c^2 
*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))*c^2/d^2/(c^2*d+e)/e^2+1/2*I*((2*c^2*d+2*( 
c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*a 
rctan(e*(c*x+I*(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)* 
e)^(1/2))*c^4/e^3/d/(c^2*d+e)-3/16*I/(c^2*d+e)*c^2/d^2*e*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((_R 
1-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)) 
-3/16*I/(c^2*d+e)/d*c^4*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))+3/16*I/(c^2*d+e)*c^2/d^2*e*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))-1/2*I*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*(2*c 
^4*d^2+2*d*c^2*(c^2*d*(c^2*d+e))^(1/2)+2*c^2*d*e+(c^2*d*(c^2*d+e))^(1/2)*e 
)*c^4*arctanh(e*(c*x+I*(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(c^2*d*(c^2*d+e...
 

Fricas [F]

\[ \int \frac {a+b \arccos (c x)}{\left (d+e x^2\right )^3} \, dx=\int { \frac {b \arccos \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:

integrate((a+b*arccos(c*x))/(e*x^2+d)^3,x, algorithm="fricas")
 

Output:

integral((b*arccos(c*x) + a)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), 
x)
 

Sympy [F]

\[ \int \frac {a+b \arccos (c x)}{\left (d+e x^2\right )^3} \, dx=\int \frac {a + b \operatorname {acos}{\left (c x \right )}}{\left (d + e x^{2}\right )^{3}}\, dx \] Input:

integrate((a+b*acos(c*x))/(e*x**2+d)**3,x)
 

Output:

Integral((a + b*acos(c*x))/(d + e*x**2)**3, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \arccos (c x)}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a+b*arccos(c*x))/(e*x^2+d)^3,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. \[ \int \frac {a+b \arccos (c x)}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((a+b*arccos(c*x))/(e*x^2+d)^3,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. \[ \int \frac {a+b \arccos (c x)}{\left (d+e x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^3} \,d x \] Input:

int((a + b*acos(c*x))/(d + e*x^2)^3,x)
 

Output:

int((a + b*acos(c*x))/(d + e*x^2)^3, x)
 

Reduce [F]

\[ \int \frac {a+b \arccos (c x)}{\left (d+e x^2\right )^3} \, dx=\frac {3 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a \,d^{2}+6 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a d e \,x^{2}+3 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a \,e^{2} x^{4}+8 \left (\int \frac {\mathit {acos} \left (c x \right )}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{5} e +16 \left (\int \frac {\mathit {acos} \left (c x \right )}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{4} e^{2} x^{2}+8 \left (\int \frac {\mathit {acos} \left (c x \right )}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{3} e^{3} x^{4}+5 a \,d^{2} e x +3 a d \,e^{2} x^{3}}{8 d^{3} e \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:

int((a+b*acos(c*x))/(e*x^2+d)^3,x)
 

Output:

(3*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*d**2 + 6*sqrt(e)*sqrt(d 
)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*d*e*x**2 + 3*sqrt(e)*sqrt(d)*atan((e*x)/ 
(sqrt(e)*sqrt(d)))*a*e**2*x**4 + 8*int(acos(c*x)/(d**3 + 3*d**2*e*x**2 + 3 
*d*e**2*x**4 + e**3*x**6),x)*b*d**5*e + 16*int(acos(c*x)/(d**3 + 3*d**2*e* 
x**2 + 3*d*e**2*x**4 + e**3*x**6),x)*b*d**4*e**2*x**2 + 8*int(acos(c*x)/(d 
**3 + 3*d**2*e*x**2 + 3*d*e**2*x**4 + e**3*x**6),x)*b*d**3*e**3*x**4 + 5*a 
*d**2*e*x + 3*a*d*e**2*x**3)/(8*d**3*e*(d**2 + 2*d*e*x**2 + e**2*x**4))