\(\int \frac {a+b \arccos (c x)}{x^3 (d+e x^2)^3} \, dx\) [647]

Optimal result

Integrand size = 21, antiderivative size = 783 \[ \int \frac {a+b \arccos (c x)}{x^3 \left (d+e x^2\right )^3} \, dx=-\frac {b c \sqrt {1-c^2 x^2}}{2 d^3 x}+\frac {b c e^2 x \sqrt {1-c^2 x^2}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \arccos (c x)}{2 d^3 x^2}-\frac {e (a+b \arccos (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \arccos (c x))}{d^3 \left (d+e x^2\right )}+\frac {b c e \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{d^{7/2} \sqrt {c^2 d+e}}+\frac {b c e \left (2 c^2 d+e\right ) \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{7/2} \left (c^2 d+e\right )^{3/2}}+\frac {3 e (a+b \arccos (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^4}+\frac {3 e (a+b \arccos (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^4}+\frac {3 e (a+b \arccos (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^4}+\frac {3 e (a+b \arccos (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^4}-\frac {3 e (a+b \arccos (c x)) \log \left (1-e^{2 i \arccos (c x)}\right )}{d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arccos (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^4}+\frac {3 i b e \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{2 d^4} \] Output:

-1/2*b*c*(-c^2*x^2+1)^(1/2)/d^3/x+1/8*b*c*e^2*x*(-c^2*x^2+1)^(1/2)/d^3/(c^ 
2*d+e)/(e*x^2+d)-1/2*(a+b*arccos(c*x))/d^3/x^2-1/4*e*(a+b*arccos(c*x))/d^2 
/(e*x^2+d)^2-e*(a+b*arccos(c*x))/d^3/(e*x^2+d)+b*c*e*arctan((c^2*d+e)^(1/2 
)*x/d^(1/2)/(-c^2*x^2+1)^(1/2))/d^(7/2)/(c^2*d+e)^(1/2)+1/8*b*c*e*(2*c^2*d 
+e)*arctan((c^2*d+e)^(1/2)*x/d^(1/2)/(-c^2*x^2+1)^(1/2))/d^(7/2)/(c^2*d+e) 
^(3/2)+3/2*e*(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^4+3/2*e*(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^4+3/2*e*(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^4+3/2*e*(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^4-3*e*(a+b*arccos(c*x))*ln(1-(c 
*x+I*(-c^2*x^2+1)^(1/2))^2)/d^4-3/2*I*b*e*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^4-3/2*I*b*e*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^4-3/ 
2*I*b*e*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^4+3/2*I*b*e*polylog(2,(c*x+I*(-c^2*x^2+1)^(1/2))^2)/d^4-3/2 
*I*b*e*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^4
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1660\) vs. \(2(783)=1566\).

Time = 6.06 (sec) , antiderivative size = 1660, normalized size of antiderivative = 2.12 \[ \int \frac {a+b \arccos (c x)}{x^3 \left (d+e x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/2*a/(d^3*x^2) - (a*e)/(4*d^2*(d + e*x^2)^2) - (a*e)/(d^3*(d + e*x^2)) - 
 (3*a*e*Log[x])/d^4 + (3*a*e*Log[d + e*x^2])/(2*d^4) + b*((c*x*Sqrt[1 - c^ 
2*x^2] - ArcCos[c*x])/(2*d^3*x^2) + (((9*I)/16)*e*(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]))/d^(7/2) - (e^(3/2)*((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]*Sqrt[1 - c^2*x^2]))/(c^3*(d + I*Sqrt[d]*Sqrt[e]*x))])/(Sqr 
t[e]*(c^2*d + e)^(3/2))))/(16*d^3) + (((9*I)/16)*e*(-(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]))/d^(7/2) - (e^(3/2)*((c*Sqrt[1 - c^2*x^2])/((c^2*d + e)*(I*Sqr 
t[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]*Sqrt[1 - c^2*x^2]))/(c^3*(d - I*Sqrt[d]*Sqrt[e]*x))])/(Sqrt[e]*(c 
^2*d + e)^(3/2))))/(16*d^3) - (((3*I)/4)*e*(ArcCos[c*x]^2 - 8*ArcSin[Sqrt[ 
1 + (I*c*Sqrt[d])/Sqrt[e]]/Sqrt[2]]*ArcTan[((c*Sqrt[d] + I*Sqrt[e])*Tan[Ar 
cCos[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*ArcCos[c*x]))/Sqrt[e]] + (4*I)*ArcSin[Sqrt[1 ...
 

Rubi [A] (verified)

Time = 1.74 (sec) , antiderivative size = 784, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5233, 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.

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

\(\Big \downarrow \) 5233

\(\displaystyle \int \left (\frac {3 e^2 x (a+b \arccos (c x))}{d^4 \left (d+e x^2\right )}-\frac {3 e (a+b \arccos (c x))}{d^4 x}+\frac {2 e^2 x (a+b \arccos (c x))}{d^3 \left (d+e x^2\right )^2}+\frac {a+b \arccos (c x)}{d^3 x^3}+\frac {e^2 x (a+b \arccos (c x))}{d^2 \left (d+e x^2\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 e (a+b \arccos (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}-i \sqrt {c^2 d+e}}\right )}{2 d^4}+\frac {3 e (a+b \arccos (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}-i \sqrt {c^2 d+e}}\right )}{2 d^4}+\frac {3 e (a+b \arccos (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}+i \sqrt {c^2 d+e}}\right )}{2 d^4}+\frac {3 e (a+b \arccos (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}+i \sqrt {c^2 d+e}}\right )}{2 d^4}-\frac {3 e \log \left (1+e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))}{d^4}-\frac {e (a+b \arccos (c x))}{d^3 \left (d+e x^2\right )}-\frac {a+b \arccos (c x)}{2 d^3 x^2}-\frac {e (a+b \arccos (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {3 i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}-i \sqrt {d c^2+e}}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}-i \sqrt {d c^2+e}}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arccos (c x)}}{\sqrt {-d} c+i \sqrt {d c^2+e}}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arccos (c x)}}{\sqrt {-d} c+i \sqrt {d c^2+e}}\right )}{2 d^4}+\frac {3 i b e \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{2 d^4}-\frac {b c e \left (2 c^2 d+e\right ) \arctan \left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{7/2} \left (c^2 d+e\right )^{3/2}}-\frac {b c e \arctan \left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{d^{7/2} \sqrt {c^2 d+e}}-\frac {b c e^2 x \sqrt {1-c^2 x^2}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}+\frac {b c \sqrt {1-c^2 x^2}}{2 d^3 x}\)

Input:

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

Output:

(b*c*Sqrt[1 - c^2*x^2])/(2*d^3*x) - (b*c*e^2*x*Sqrt[1 - c^2*x^2])/(8*d^3*( 
c^2*d + e)*(d + e*x^2)) - (a + b*ArcCos[c*x])/(2*d^3*x^2) - (e*(a + b*ArcC 
os[c*x]))/(4*d^2*(d + e*x^2)^2) - (e*(a + b*ArcCos[c*x]))/(d^3*(d + e*x^2) 
) - (b*c*e*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(d^(7/ 
2)*Sqrt[c^2*d + e]) - (b*c*e*(2*c^2*d + e)*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqr 
t[d]*Sqrt[1 - c^2*x^2])])/(8*d^(7/2)*(c^2*d + e)^(3/2)) + (3*e*(a + b*ArcC 
os[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[-d] - I*Sqrt[c^2*d + 
e])])/(2*d^4) + (3*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*d^4) + (3*e*(a + b*ArcCos[c*x])*L 
og[1 - (Sqrt[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[-d] + I*Sqrt[c^2*d + e])])/(2*d 
^4) + (3*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*d^4) - (3*e*(a + b*ArcCos[c*x])*Log[1 + E^( 
(2*I)*ArcCos[c*x])])/d^4 - (((3*I)/2)*b*e*PolyLog[2, -((Sqrt[e]*E^(I*ArcCo 
s[c*x]))/(c*Sqrt[-d] - I*Sqrt[c^2*d + e]))])/d^4 - (((3*I)/2)*b*e*PolyLog[ 
2, (Sqrt[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[-d] - I*Sqrt[c^2*d + e])])/d^4 - (( 
(3*I)/2)*b*e*PolyLog[2, -((Sqrt[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[-d] + I*Sqrt 
[c^2*d + e]))])/d^4 - (((3*I)/2)*b*e*PolyLog[2, (Sqrt[e]*E^(I*ArcCos[c*x]) 
)/(c*Sqrt[-d] + I*Sqrt[c^2*d + e])])/d^4 + (((3*I)/2)*b*e*PolyLog[2, -E^(( 
2*I)*ArcCos[c*x])])/d^4
 

Defintions of rubi rules used

rule 2009

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

rule 5233

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcCos[c*x])^n, ( 
f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[c^2*d + 
 e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]
 

Maple [C] (warning: unable to verify)

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

Time = 1.70 (sec) , antiderivative size = 1459, normalized size of antiderivative = 1.86

method	result	size
parts	\(\text {Expression too large to display}\)	\(1459\)
derivativedivides	\(\text {Expression too large to display}\)	\(1519\)
default	\(\text {Expression too large to display}\)	\(1519\)

Input:

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

Output:

-1/2*a/d^3/x^2-3*a/d^4*e*ln(x)-1/4*a*e/d^2/(e*x^2+d)^2+3/2*a*e/d^4*ln(e*x^ 
2+d)-a*e/d^3/(e*x^2+d)+b*c^2*(-1/8*(-4*I*c^8*d^3*x^2-4*I*c^8*d*e^2*x^6-8*I 
*c^8*d^2*e*x^4-4*(-c^2*x^2+1)^(1/2)*c^7*d^3*x-8*(-c^2*x^2+1)^(1/2)*c^7*d^2 
*e*x^3-4*(-c^2*x^2+1)^(1/2)*c^7*d*e^2*x^5-6*I*c^6*e^2*x^4*d-3*I*e^3*c^6*x^ 
6-3*I*c^6*e*x^2*d^2+4*c^6*arccos(c*x)*d^3+18*c^6*arccos(c*x)*e*x^2*d^2+12* 
arccos(c*x)*c^6*d*e^2*x^4-4*c^5*(-c^2*x^2+1)^(1/2)*e*x*d^2-7*c^5*(-c^2*x^2 
+1)^(1/2)*e^2*x^3*d-3*(-c^2*x^2+1)^(1/2)*e^3*c^5*x^5+4*c^4*e*arccos(c*x)*d 
^2+18*c^4*arccos(c*x)*e^2*x^2*d+12*arccos(c*x)*e^3*c^4*x^4)/c^2/x^2/(c^2*d 
+e)/(c^2*e*x^2+c^2*d)^2/d^3-3/(c^2*d+e)/d^3*e*arccos(c*x)*ln(1+I*(c*x+I*(- 
c^2*x^2+1)^(1/2)))-3/(c^2*d+e)/d^3*e*arccos(c*x)*ln(1-I*(c*x+I*(-c^2*x^2+1 
)^(1/2)))-3/(c^2*d+e)/d^4*e^2/c^2*arccos(c*x)*ln(1+I*(c*x+I*(-c^2*x^2+1)^( 
1/2)))-3/(c^2*d+e)/d^4*e^2/c^2*arccos(c*x)*ln(1-I*(c*x+I*(-c^2*x^2+1)^(1/2 
)))+3*I/(c^2*d+e)/d^3*e*dilog(1+I*(c*x+I*(-c^2*x^2+1)^(1/2)))+3*I/(c^2*d+e 
)/d^3*e*dilog(1-I*(c*x+I*(-c^2*x^2+1)^(1/2)))-3/4*I/(c^2*d+e)/d^3*e*sum((_ 
R1^2*e+4*c^2*d+e)/(_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))-3/4*I/(c^2*d+e)/d^3*e^2*sum((_R1^2+1)/(_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) 
)+5/4*I*(c^2*d*(c^2*d+e))^(1/2)/(c^2*d+e)^2/d^3*arctanh(1/4*(4*c^2*d+2*...
 

Fricas [F]

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

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

Output:

integral((b*arccos(c*x) + a)/(e^3*x^9 + 3*d*e^2*x^7 + 3*d^2*e*x^5 + d^3*x^ 
3), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/4*a*((6*e^2*x^4 + 9*d*e*x^2 + 2*d^2)/(d^3*e^2*x^6 + 2*d^4*e*x^4 + d^5*x 
^2) - 6*e*log(e*x^2 + d)/d^4 + 12*e*log(x)/d^4) + b*integrate(arctan2(sqrt 
(c*x + 1)*sqrt(-c*x + 1), c*x)/(e^3*x^9 + 3*d*e^2*x^7 + 3*d^2*e*x^5 + d^3* 
x^3), x)
 

Giac [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \arccos (c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\frac {4 \left (\int \frac {\mathit {acos} \left (c x \right )}{e^{3} x^{9}+3 d \,e^{2} x^{7}+3 d^{2} e \,x^{5}+d^{3} x^{3}}d x \right ) b \,d^{6} x^{2}+8 \left (\int \frac {\mathit {acos} \left (c x \right )}{e^{3} x^{9}+3 d \,e^{2} x^{7}+3 d^{2} e \,x^{5}+d^{3} x^{3}}d x \right ) b \,d^{5} e \,x^{4}+4 \left (\int \frac {\mathit {acos} \left (c x \right )}{e^{3} x^{9}+3 d \,e^{2} x^{7}+3 d^{2} e \,x^{5}+d^{3} x^{3}}d x \right ) b \,d^{4} e^{2} x^{6}+6 \,\mathrm {log}\left (e \,x^{2}+d \right ) a \,d^{2} e \,x^{2}+12 \,\mathrm {log}\left (e \,x^{2}+d \right ) a d \,e^{2} x^{4}+6 \,\mathrm {log}\left (e \,x^{2}+d \right ) a \,e^{3} x^{6}-12 \,\mathrm {log}\left (x \right ) a \,d^{2} e \,x^{2}-24 \,\mathrm {log}\left (x \right ) a d \,e^{2} x^{4}-12 \,\mathrm {log}\left (x \right ) a \,e^{3} x^{6}-2 a \,d^{3}-6 a \,d^{2} e \,x^{2}+3 a \,e^{3} x^{6}}{4 d^{4} x^{2} \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:

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

Output:

(4*int(acos(c*x)/(d**3*x**3 + 3*d**2*e*x**5 + 3*d*e**2*x**7 + e**3*x**9),x 
)*b*d**6*x**2 + 8*int(acos(c*x)/(d**3*x**3 + 3*d**2*e*x**5 + 3*d*e**2*x**7 
 + e**3*x**9),x)*b*d**5*e*x**4 + 4*int(acos(c*x)/(d**3*x**3 + 3*d**2*e*x** 
5 + 3*d*e**2*x**7 + e**3*x**9),x)*b*d**4*e**2*x**6 + 6*log(d + e*x**2)*a*d 
**2*e*x**2 + 12*log(d + e*x**2)*a*d*e**2*x**4 + 6*log(d + e*x**2)*a*e**3*x 
**6 - 12*log(x)*a*d**2*e*x**2 - 24*log(x)*a*d*e**2*x**4 - 12*log(x)*a*e**3 
*x**6 - 2*a*d**3 - 6*a*d**2*e*x**2 + 3*a*e**3*x**6)/(4*d**4*x**2*(d**2 + 2 
*d*e*x**2 + e**2*x**4))