\(\int \frac {(d+e x^2)^2 (a+b \arccos (c x))}{x} \, dx\) [612]

Optimal result

Integrand size = 21, antiderivative size = 229 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \arccos (c x))}{x} \, dx=\frac {b d e x \sqrt {1-c^2 x^2}}{2 c}+\frac {3 b e^2 x \sqrt {1-c^2 x^2}}{32 c^3}+\frac {b e^2 x^3 \sqrt {1-c^2 x^2}}{16 c}-\frac {b d e \arccos (c x)}{2 c^2}-\frac {3 b e^2 \arccos (c x)}{32 c^4}-\frac {1}{2} i b d^2 \arccos (c x)^2+d e x^2 (a+b \arccos (c x))+\frac {1}{4} e^2 x^4 (a+b \arccos (c x))+b d^2 \arccos (c x) \log \left (1-e^{2 i \arccos (c x)}\right )-b d^2 \arccos (c x) \log (x)+d^2 (a+b \arccos (c x)) \log (x)-\frac {1}{2} i b d^2 \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.95 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \arccos (c x))}{x} \, dx=a d e x^2+\frac {1}{4} a e^2 x^4+b d e x^2 \arccos (c x)+\frac {1}{4} b e^2 x^4 \arccos (c x)-\frac {b e^2 \left (c x \sqrt {1-c^2 x^2} \left (3+2 c^2 x^2\right )-6 \arctan \left (\frac {c x}{-1+\sqrt {1-c^2 x^2}}\right )\right )}{32 c^4}+\frac {b d e \left (-\frac {1}{2} c x \sqrt {1-c^2 x^2}+\arctan \left (\frac {c x}{-1+\sqrt {1-c^2 x^2}}\right )\right )}{c^2}+a d^2 \log (x)-\frac {1}{2} i b d^2 \left (\arccos (c x) \left (\arccos (c x)+2 i \log \left (1+e^{2 i \arccos (c x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )\right ) \] Input:

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

Output:

a*d*e*x^2 + (a*e^2*x^4)/4 + b*d*e*x^2*ArcCos[c*x] + (b*e^2*x^4*ArcCos[c*x] 
)/4 - (b*e^2*(c*x*Sqrt[1 - c^2*x^2]*(3 + 2*c^2*x^2) - 6*ArcTan[(c*x)/(-1 + 
 Sqrt[1 - c^2*x^2])]))/(32*c^4) + (b*d*e*(-1/2*(c*x*Sqrt[1 - c^2*x^2]) + A 
rcTan[(c*x)/(-1 + Sqrt[1 - c^2*x^2])]))/c^2 + a*d^2*Log[x] - (I/2)*b*d^2*( 
ArcCos[c*x]*(ArcCos[c*x] + (2*I)*Log[1 + E^((2*I)*ArcCos[c*x])]) + PolyLog 
[2, -E^((2*I)*ArcCos[c*x])])
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.01, 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.

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

\(\Big \downarrow \) 5231

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

d*e*x^2*(a + b*ArcCos[c*x]) + (e^2*x^4*(a + b*ArcCos[c*x]))/4 + d^2*(a + b 
*ArcCos[c*x])*Log[x] + (b*c*((-2*d*e*x*Sqrt[1 - c^2*x^2])/c^2 - (3*e^2*x*S 
qrt[1 - c^2*x^2])/(8*c^4) - (e^2*x^3*Sqrt[1 - c^2*x^2])/(4*c^2) + (2*d*e*A 
rcSin[c*x])/c^3 + (3*e^2*ArcSin[c*x])/(8*c^5) + ((2*I)*d^2*ArcSin[c*x]^2)/ 
c - (4*d^2*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])])/c + (4*d^2*ArcSin[c 
*x]*Log[x])/c + ((2*I)*d^2*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 = 1.12 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.02

method	result	size
parts	\(a \left (\frac {e^{2} x^{4}}{4}+d e \,x^{2}+d^{2} \ln \left (x \right )\right )+b \arccos \left (c x \right ) d e \,x^{2}-\frac {b \,e^{2} x \sqrt {-c^{2} x^{2}+1}}{8 c^{3}}+\frac {b \,e^{2} \arccos \left (c x \right ) \cos \left (4 \arccos \left (c x \right )\right )}{32 c^{4}}+\frac {b \arccos \left (c x \right ) e^{2} x^{2}}{4 c^{2}}-\frac {b d e \arccos \left (c x \right )}{2 c^{2}}-\frac {i b \,d^{2} \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\frac {b \,e^{2} \sin \left (4 \arccos \left (c x \right )\right )}{128 c^{4}}+b \,d^{2} \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i b \,d^{2} \arccos \left (c x \right )^{2}}{2}-\frac {b d e x \sqrt {-c^{2} x^{2}+1}}{2 c}-\frac {b \,e^{2} \arccos \left (c x \right )}{8 c^{4}}\)	\(234\)
derivativedivides	\(a d e \,x^{2}+\frac {a \,e^{2} x^{4}}{4}+a \,d^{2} \ln \left (c x \right )+b \,d^{2} \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i b \,d^{2} \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\frac {b \,e^{2} x \sqrt {-c^{2} x^{2}+1}}{8 c^{3}}-\frac {b d e \arccos \left (c x \right )}{2 c^{2}}-\frac {b \,e^{2} \sin \left (4 \arccos \left (c x \right )\right )}{128 c^{4}}+b \arccos \left (c x \right ) d e \,x^{2}-\frac {b d e x \sqrt {-c^{2} x^{2}+1}}{2 c}-\frac {i b \,d^{2} \arccos \left (c x \right )^{2}}{2}-\frac {b \,e^{2} \arccos \left (c x \right )}{8 c^{4}}+\frac {b \,e^{2} \arccos \left (c x \right ) \cos \left (4 \arccos \left (c x \right )\right )}{32 c^{4}}+\frac {b \arccos \left (c x \right ) e^{2} x^{2}}{4 c^{2}}\)	\(236\)
default	\(a d e \,x^{2}+\frac {a \,e^{2} x^{4}}{4}+a \,d^{2} \ln \left (c x \right )+b \,d^{2} \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i b \,d^{2} \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\frac {b \,e^{2} x \sqrt {-c^{2} x^{2}+1}}{8 c^{3}}-\frac {b d e \arccos \left (c x \right )}{2 c^{2}}-\frac {b \,e^{2} \sin \left (4 \arccos \left (c x \right )\right )}{128 c^{4}}+b \arccos \left (c x \right ) d e \,x^{2}-\frac {b d e x \sqrt {-c^{2} x^{2}+1}}{2 c}-\frac {i b \,d^{2} \arccos \left (c x \right )^{2}}{2}-\frac {b \,e^{2} \arccos \left (c x \right )}{8 c^{4}}+\frac {b \,e^{2} \arccos \left (c x \right ) \cos \left (4 \arccos \left (c x \right )\right )}{32 c^{4}}+\frac {b \arccos \left (c x \right ) e^{2} x^{2}}{4 c^{2}}\)	\(236\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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