\(\int \frac {(d+e x^2)^3 (a+b \arccos (c x))}{x^3} \, dx\) [623]

Optimal result

Integrand size = 21, antiderivative size = 262 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arccos (c x))}{x^3} \, dx=-\frac {b c d^3 \sqrt {1-c^2 x^2}}{2 x}+\frac {3 b e^2 \left (8 c^2 d+e\right ) x \sqrt {1-c^2 x^2}}{32 c^3}+\frac {b e^3 x^3 \sqrt {1-c^2 x^2}}{16 c}-\frac {3 b e^2 \left (8 c^2 d+e\right ) \arccos (c x)}{32 c^4}-\frac {3}{2} i b d^2 e \arccos (c x)^2-\frac {d^3 (a+b \arccos (c x))}{2 x^2}+\frac {3}{2} d e^2 x^2 (a+b \arccos (c x))+\frac {1}{4} e^3 x^4 (a+b \arccos (c x))+3 b d^2 e \arccos (c x) \log \left (1-e^{2 i \arccos (c x)}\right )-3 b d^2 e \arccos (c x) \log (x)+3 d^2 e (a+b \arccos (c x)) \log (x)-\frac {3}{2} i b d^2 e \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right ) \] Output:

-1/2*b*c*d^3*(-c^2*x^2+1)^(1/2)/x+3/32*b*e^2*(8*c^2*d+e)*x*(-c^2*x^2+1)^(1 
/2)/c^3+1/16*b*e^3*x^3*(-c^2*x^2+1)^(1/2)/c-3/32*b*e^2*(8*c^2*d+e)*arccos( 
c*x)/c^4-3/2*I*b*d^2*e*arccos(c*x)^2-1/2*d^3*(a+b*arccos(c*x))/x^2+3/2*d*e 
^2*x^2*(a+b*arccos(c*x))+1/4*e^3*x^4*(a+b*arccos(c*x))+3*b*d^2*e*arccos(c* 
x)*ln(1-(c*x+I*(-c^2*x^2+1)^(1/2))^2)-3*b*d^2*e*arccos(c*x)*ln(x)+3*d^2*e* 
(a+b*arccos(c*x))*ln(x)-3/2*I*b*d^2*e*polylog(2,(c*x+I*(-c^2*x^2+1)^(1/2)) 
^2)
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.02 \[ \int \frac {\left (d+e x^2\right )^3 (a+b \arccos (c x))}{x^3} \, dx=\frac {1}{4} \left (-\frac {2 a d^3}{x^2}+6 a d e^2 x^2+a e^3 x^4+\frac {2 b d^3 \left (c x \sqrt {1-c^2 x^2}-\arccos (c x)\right )}{x^2}+6 b d e^2 x^2 \arccos (c x)+b e^3 x^4 \arccos (c x)-\frac {b e^3 \left (c x \sqrt {1-c^2 x^2} \left (3+2 c^2 x^2\right )-6 \arctan \left (\frac {c x}{-1+\sqrt {1-c^2 x^2}}\right )\right )}{8 c^4}-\frac {3 b d e^2 \left (c x \sqrt {1-c^2 x^2}-2 \arctan \left (\frac {c x}{-1+\sqrt {1-c^2 x^2}}\right )\right )}{c^2}+12 a d^2 e \log (x)-6 i b d^2 e \left (\arccos (c x) \left (\arccos (c x)+2 i \log \left (1+e^{2 i \arccos (c x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )\right )\right ) \] Input:

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

Output:

((-2*a*d^3)/x^2 + 6*a*d*e^2*x^2 + a*e^3*x^4 + (2*b*d^3*(c*x*Sqrt[1 - c^2*x 
^2] - ArcCos[c*x]))/x^2 + 6*b*d*e^2*x^2*ArcCos[c*x] + b*e^3*x^4*ArcCos[c*x 
] - (b*e^3*(c*x*Sqrt[1 - c^2*x^2]*(3 + 2*c^2*x^2) - 6*ArcTan[(c*x)/(-1 + S 
qrt[1 - c^2*x^2])]))/(8*c^4) - (3*b*d*e^2*(c*x*Sqrt[1 - c^2*x^2] - 2*ArcTa 
n[(c*x)/(-1 + Sqrt[1 - c^2*x^2])]))/c^2 + 12*a*d^2*e*Log[x] - (6*I)*b*d^2* 
e*(ArcCos[c*x]*(ArcCos[c*x] + (2*I)*Log[1 + E^((2*I)*ArcCos[c*x])]) + Poly 
Log[2, -E^((2*I)*ArcCos[c*x])]))/4
 

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5231, 27, 7293, 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.

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

\(\Big \downarrow \) 5231

\(\displaystyle b c \int -\frac {-e^3 x^6-6 d e^2 x^4-12 d^2 e \log (x) x^2+2 d^3}{4 x^2 \sqrt {1-c^2 x^2}}dx-\frac {d^3 (a+b \arccos (c x))}{2 x^2}+3 d^2 e \log (x) (a+b \arccos (c x))+\frac {3}{2} d e^2 x^2 (a+b \arccos (c x))+\frac {1}{4} e^3 x^4 (a+b \arccos (c x))\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{4} b c \int \frac {-e^3 x^6-6 d e^2 x^4-12 d^2 e \log (x) x^2+2 d^3}{x^2 \sqrt {1-c^2 x^2}}dx-\frac {d^3 (a+b \arccos (c x))}{2 x^2}+3 d^2 e \log (x) (a+b \arccos (c x))+\frac {3}{2} d e^2 x^2 (a+b \arccos (c x))+\frac {1}{4} e^3 x^4 (a+b \arccos (c x))\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{4} b c \int \left (\frac {-e^3 x^6-6 d e^2 x^4+2 d^3}{x^2 \sqrt {1-c^2 x^2}}-\frac {12 d^2 e \log (x)}{\sqrt {1-c^2 x^2}}\right )dx-\frac {d^3 (a+b \arccos (c x))}{2 x^2}+3 d^2 e \log (x) (a+b \arccos (c x))+\frac {3}{2} d e^2 x^2 (a+b \arccos (c x))+\frac {1}{4} e^3 x^4 (a+b \arccos (c x))\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {d^3 (a+b \arccos (c x))}{2 x^2}+3 d^2 e \log (x) (a+b \arccos (c x))+\frac {3}{2} d e^2 x^2 (a+b \arccos (c x))+\frac {1}{4} e^3 x^4 (a+b \arccos (c x))-\frac {1}{4} b c \left (-\frac {3 e^2 \arcsin (c x) \left (8 c^2 d+e\right )}{8 c^5}-\frac {6 i d^2 e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{c}-\frac {6 i d^2 e \arcsin (c x)^2}{c}+\frac {12 d^2 e \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{c}-\frac {12 d^2 e \log (x) \arcsin (c x)}{c}-\frac {2 d^3 \sqrt {1-c^2 x^2}}{x}+\frac {e^3 x^3 \sqrt {1-c^2 x^2}}{4 c^2}+\frac {3 e^2 x \sqrt {1-c^2 x^2} \left (8 c^2 d+e\right )}{8 c^4}\right )\)

Input:

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

Output:

-1/2*(d^3*(a + b*ArcCos[c*x]))/x^2 + (3*d*e^2*x^2*(a + b*ArcCos[c*x]))/2 + 
 (e^3*x^4*(a + b*ArcCos[c*x]))/4 + 3*d^2*e*(a + b*ArcCos[c*x])*Log[x] - (b 
*c*((-2*d^3*Sqrt[1 - c^2*x^2])/x + (3*e^2*(8*c^2*d + e)*x*Sqrt[1 - c^2*x^2 
])/(8*c^4) + (e^3*x^3*Sqrt[1 - c^2*x^2])/(4*c^2) - (3*e^2*(8*c^2*d + e)*Ar 
cSin[c*x])/(8*c^5) - ((6*I)*d^2*e*ArcSin[c*x]^2)/c + (12*d^2*e*ArcSin[c*x] 
*Log[1 - E^((2*I)*ArcSin[c*x])])/c - (12*d^2*e*ArcSin[c*x]*Log[x])/c - ((6 
*I)*d^2*e*PolyLog[2, E^((2*I)*ArcSin[c*x])])/c))/4
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

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

rule 5231

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_ 
)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Simp 
[(a + b*ArcCos[c*x])   u, x] + Simp[b*c   Int[SimplifyIntegrand[u/Sqrt[1 - 
c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 
0] && IntegerQ[p] && (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [A] (verified)

Time = 2.32 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.16

method	result	size
parts	\(a \left (\frac {x^{4} e^{3}}{4}+\frac {3 d \,e^{2} x^{2}}{2}+3 d^{2} e \ln \left (x \right )-\frac {d^{3}}{2 x^{2}}\right )-\frac {3 i b \,d^{2} e \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\frac {3 b \,e^{2} \arccos \left (c x \right ) x^{2} d}{2}-\frac {b \,e^{3} \sqrt {-c^{2} x^{2}+1}\, x}{8 c^{3}}+\frac {b \arccos \left (c x \right ) e^{3} \cos \left (4 \arccos \left (c x \right )\right )}{32 c^{4}}+\frac {b \,e^{3} \arccos \left (c x \right ) x^{2}}{4 c^{2}}-\frac {3 b \,e^{2} d \arccos \left (c x \right )}{4 c^{2}}+\frac {i b \,c^{2} d^{3}}{2}+\frac {b c \,d^{3} \sqrt {-c^{2} x^{2}+1}}{2 x}-\frac {b \,d^{3} \arccos \left (c x \right )}{2 x^{2}}+3 b \,d^{2} e \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {3 i b \,d^{2} e \arccos \left (c x \right )^{2}}{2}-\frac {3 b \,e^{2} \sqrt {-c^{2} x^{2}+1}\, x d}{4 c}-\frac {b \,e^{3} \sin \left (4 \arccos \left (c x \right )\right )}{128 c^{4}}-\frac {b \,e^{3} \arccos \left (c x \right )}{8 c^{4}}\)	\(303\)
derivativedivides	\(c^{2} \left (\frac {a \left (\frac {3 c^{4} d \,e^{2} x^{2}}{2}+\frac {c^{4} e^{3} x^{4}}{4}+3 c^{4} d^{2} e \ln \left (c x \right )-\frac {c^{4} d^{3}}{2 x^{2}}\right )}{c^{6}}+\frac {b \,e^{3} \arccos \left (c x \right ) \cos \left (4 \arccos \left (c x \right )\right )}{32 c^{6}}-\frac {3 b d \,e^{2} x \sqrt {-c^{2} x^{2}+1}}{4 c^{3}}+\frac {b \arccos \left (c x \right ) e^{3} x^{2}}{4 c^{4}}+\frac {i d^{3} b}{2}+\frac {b \,d^{3} \sqrt {-c^{2} x^{2}+1}}{2 c x}-\frac {b \,e^{3} x \sqrt {-c^{2} x^{2}+1}}{8 c^{5}}-\frac {3 b d \,e^{2} \arccos \left (c x \right )}{4 c^{4}}+\frac {3 b \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) d^{2} e \arccos \left (c x \right )}{c^{2}}-\frac {3 i b \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) d^{2} e}{2 c^{2}}+\frac {3 b \arccos \left (c x \right ) d \,e^{2} x^{2}}{2 c^{2}}-\frac {b \arccos \left (c x \right ) d^{3}}{2 c^{2} x^{2}}-\frac {b \,e^{3} \sin \left (4 \arccos \left (c x \right )\right )}{128 c^{6}}-\frac {3 i b \,d^{2} e \arccos \left (c x \right )^{2}}{2 c^{2}}-\frac {b \,e^{3} \arccos \left (c x \right )}{8 c^{6}}\right )\)	\(338\)
default	\(c^{2} \left (\frac {a \left (\frac {3 c^{4} d \,e^{2} x^{2}}{2}+\frac {c^{4} e^{3} x^{4}}{4}+3 c^{4} d^{2} e \ln \left (c x \right )-\frac {c^{4} d^{3}}{2 x^{2}}\right )}{c^{6}}+\frac {b \,e^{3} \arccos \left (c x \right ) \cos \left (4 \arccos \left (c x \right )\right )}{32 c^{6}}-\frac {3 b d \,e^{2} x \sqrt {-c^{2} x^{2}+1}}{4 c^{3}}+\frac {b \arccos \left (c x \right ) e^{3} x^{2}}{4 c^{4}}+\frac {i d^{3} b}{2}+\frac {b \,d^{3} \sqrt {-c^{2} x^{2}+1}}{2 c x}-\frac {b \,e^{3} x \sqrt {-c^{2} x^{2}+1}}{8 c^{5}}-\frac {3 b d \,e^{2} \arccos \left (c x \right )}{4 c^{4}}+\frac {3 b \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) d^{2} e \arccos \left (c x \right )}{c^{2}}-\frac {3 i b \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) d^{2} e}{2 c^{2}}+\frac {3 b \arccos \left (c x \right ) d \,e^{2} x^{2}}{2 c^{2}}-\frac {b \arccos \left (c x \right ) d^{3}}{2 c^{2} x^{2}}-\frac {b \,e^{3} \sin \left (4 \arccos \left (c x \right )\right )}{128 c^{6}}-\frac {3 i b \,d^{2} e \arccos \left (c x \right )^{2}}{2 c^{2}}-\frac {b \,e^{3} \arccos \left (c x \right )}{8 c^{6}}\right )\)	\(338\)

Input:

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

Output:

a*(1/4*x^4*e^3+3/2*d*e^2*x^2+3*d^2*e*ln(x)-1/2*d^3/x^2)-3/2*I*b*d^2*e*poly 
log(2,-(c*x+I*(-c^2*x^2+1)^(1/2))^2)+3/2*b*e^2*arccos(c*x)*x^2*d-1/8*b/c^3 
*e^3*(-c^2*x^2+1)^(1/2)*x+1/32*b/c^4*arccos(c*x)*e^3*cos(4*arccos(c*x))+1/ 
4*b/c^2*e^3*arccos(c*x)*x^2-3/4*b/c^2*e^2*d*arccos(c*x)+1/2*I*b*c^2*d^3+1/ 
2*b*c*d^3*(-c^2*x^2+1)^(1/2)/x-1/2*b*d^3/x^2*arccos(c*x)+3*b*d^2*e*arccos( 
c*x)*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))^2)-3/2*I*b*d^2*e*arccos(c*x)^2-3/4*b/ 
c*e^2*(-c^2*x^2+1)^(1/2)*x*d-1/128*b/c^4*e^3*sin(4*arccos(c*x))-1/8*b/c^4* 
e^3*arccos(c*x)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/4*a*e^3*x^4 + 3/2*a*d*e^2*x^2 + 1/2*b*d^3*(sqrt(-c^2*x^2 + 1)*c/x - arcc 
os(c*x)/x^2) + 3*a*d^2*e*log(x) - 1/2*a*d^3/x^2 + integrate((b*e^3*x^4 + 3 
*b*d*e^2*x^2 + 3*b*d^2*e)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)/x, x)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d+e x^2\right )^3 (a+b \arccos (c x))}{x^3} \, dx=\frac {-16 \mathit {acos} \left (c x \right ) b \,c^{4} d^{3}+48 \mathit {acos} \left (c x \right ) b \,c^{4} d \,e^{2} x^{4}+8 \mathit {acos} \left (c x \right ) b \,c^{4} e^{3} x^{6}+24 \mathit {asin} \left (c x \right ) b \,c^{2} d \,e^{2} x^{2}+3 \mathit {asin} \left (c x \right ) b \,e^{3} x^{2}+16 \sqrt {-c^{2} x^{2}+1}\, b \,c^{5} d^{3} x -24 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} d \,e^{2} x^{3}-2 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} e^{3} x^{5}-3 \sqrt {-c^{2} x^{2}+1}\, b c \,e^{3} x^{3}+96 \left (\int \frac {\mathit {acos} \left (c x \right )}{x}d x \right ) b \,c^{4} d^{2} e \,x^{2}+96 \,\mathrm {log}\left (x \right ) a \,c^{4} d^{2} e \,x^{2}-16 a \,c^{4} d^{3}+48 a \,c^{4} d \,e^{2} x^{4}+8 a \,c^{4} e^{3} x^{6}}{32 c^{4} x^{2}} \] Input:

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

Output:

( - 16*acos(c*x)*b*c**4*d**3 + 48*acos(c*x)*b*c**4*d*e**2*x**4 + 8*acos(c* 
x)*b*c**4*e**3*x**6 + 24*asin(c*x)*b*c**2*d*e**2*x**2 + 3*asin(c*x)*b*e**3 
*x**2 + 16*sqrt( - c**2*x**2 + 1)*b*c**5*d**3*x - 24*sqrt( - c**2*x**2 + 1 
)*b*c**3*d*e**2*x**3 - 2*sqrt( - c**2*x**2 + 1)*b*c**3*e**3*x**5 - 3*sqrt( 
 - c**2*x**2 + 1)*b*c*e**3*x**3 + 96*int(acos(c*x)/x,x)*b*c**4*d**2*e*x**2 
 + 96*log(x)*a*c**4*d**2*e*x**2 - 16*a*c**4*d**3 + 48*a*c**4*d*e**2*x**4 + 
 8*a*c**4*e**3*x**6)/(32*c**4*x**2)