\(\int \frac {(d+e x^2) (a+b \arccos (c x))}{x} \, dx\) [20]

Optimal result

Integrand size = 19, antiderivative size = 132 \[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x} \, dx=-\frac {b e x \sqrt {1-c^2 x^2}}{4 c}+\frac {1}{2} e x^2 (a+b \arccos (c x))+\frac {b e \arcsin (c x)}{4 c^2}+\frac {1}{2} i b d \arcsin (c x)^2-b d \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+d (a+b \arccos (c x)) \log (x)+b d \arcsin (c x) \log (x)+\frac {1}{2} i b d \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) \] Output:

-1/4*b*e*x*(-c^2*x^2+1)^(1/2)/c+1/2*e*x^2*(a+b*arccos(c*x))+1/4*b*e*arcsin 
(c*x)/c^2+1/2*I*b*d*arcsin(c*x)^2-b*d*arcsin(c*x)*ln(1-(I*c*x+(-c^2*x^2+1) 
^(1/2))^2)+d*(a+b*arccos(c*x))*ln(x)+b*d*arcsin(c*x)*ln(x)+1/2*I*b*d*polyl 
og(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.92 \[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x} \, dx=\frac {1}{2} a e x^2-\frac {b e x \sqrt {1-c^2 x^2}}{4 c}+\frac {1}{2} b e x^2 \arccos (c x)-\frac {1}{2} i b d \arccos (c x)^2+\frac {b e \arcsin (c x)}{4 c^2}+b d \arccos (c x) \log \left (1+e^{2 i \arccos (c x)}\right )+a d \log (x)-\frac {1}{2} i b d \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right ) \] Input:

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

Output:

(a*e*x^2)/2 - (b*e*x*Sqrt[1 - c^2*x^2])/(4*c) + (b*e*x^2*ArcCos[c*x])/2 - 
(I/2)*b*d*ArcCos[c*x]^2 + (b*e*ArcSin[c*x])/(4*c^2) + b*d*ArcCos[c*x]*Log[ 
1 + E^((2*I)*ArcCos[c*x])] + a*d*Log[x] - (I/2)*b*d*PolyLog[2, -E^((2*I)*A 
rcCos[c*x])]
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, 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.

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

\(\Big \downarrow \) 5231

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(e*x^2*(a + b*ArcCos[c*x]))/2 + d*(a + b*ArcCos[c*x])*Log[x] + (b*c*(-1/2* 
(e*x*Sqrt[1 - c^2*x^2])/c^2 + (e*ArcSin[c*x])/(2*c^3) + (I*d*ArcSin[c*x]^2 
)/c - (2*d*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])])/c + (2*d*ArcSin[c*x 
]*Log[x])/c + (I*d*PolyLog[2, E^((2*I)*ArcSin[c*x])])/c))/2
 

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 = 0.40 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.97

method	result	size
parts	\(\frac {a e \,x^{2}}{2}+a d \ln \left (x \right )-\frac {i b d \arccos \left (c x \right )^{2}}{2}-\frac {b e x \sqrt {-c^{2} x^{2}+1}}{4 c}+\frac {\arccos \left (c x \right ) b e \,x^{2}}{2}-\frac {b e \arccos \left (c x \right )}{4 c^{2}}+d b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i d b \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\)	\(128\)
derivativedivides	\(\frac {a e \,x^{2}}{2}+a d \ln \left (c x \right )-\frac {i b d \arccos \left (c x \right )^{2}}{2}-\frac {b e x \sqrt {-c^{2} x^{2}+1}}{4 c}+\frac {\arccos \left (c x \right ) b e \,x^{2}}{2}-\frac {b e \arccos \left (c x \right )}{4 c^{2}}+d b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i d b \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\)	\(130\)
default	\(\frac {a e \,x^{2}}{2}+a d \ln \left (c x \right )-\frac {i b d \arccos \left (c x \right )^{2}}{2}-\frac {b e x \sqrt {-c^{2} x^{2}+1}}{4 c}+\frac {\arccos \left (c x \right ) b e \,x^{2}}{2}-\frac {b e \arccos \left (c x \right )}{4 c^{2}}+d b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i d b \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\)	\(130\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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