\(\int \frac {a+b \arccos (c x)}{x^2 (d-c^2 d x^2)^{3/2}} \, dx\) [128]

Optimal result

Integrand size = 27, antiderivative size = 150 \[ \int \frac {a+b \arccos (c x)}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {a+b \arccos (c x)}{d x \sqrt {d-c^2 d x^2}}+\frac {2 c^2 x (a+b \arccos (c x))}{d \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \log (x)}{d^2 \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \log \left (1-c^2 x^2\right )}{2 d^2 \sqrt {1-c^2 x^2}} \] Output:

-(a+b*arccos(c*x))/d/x/(-c^2*d*x^2+d)^(1/2)+2*c^2*x*(a+b*arccos(c*x))/d/(- 
c^2*d*x^2+d)^(1/2)+b*c*(-c^2*d*x^2+d)^(1/2)*ln(x)/d^2/(-c^2*x^2+1)^(1/2)+1 
/2*b*c*(-c^2*d*x^2+d)^(1/2)*ln(-c^2*x^2+1)/d^2/(-c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.14 \[ \int \frac {a+b \arccos (c x)}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {\sqrt {d-c^2 d x^2} \left (2 a \sqrt {1-c^2 x^2}-4 a c^2 x^2 \sqrt {1-c^2 x^2}-2 b \sqrt {1-c^2 x^2} \left (-1+2 c^2 x^2\right ) \arccos (c x)+b c x \left (-1+c^2 x^2\right ) \log \left (1-\frac {1}{c^2 x^2}\right )+2 b c x \log \left (1-c^2 x^2\right )-2 b c^3 x^3 \log \left (1-c^2 x^2\right )\right )}{2 d^2 x \left (1-c^2 x^2\right )^{3/2}} \] Input:

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

Output:

-1/2*(Sqrt[d - c^2*d*x^2]*(2*a*Sqrt[1 - c^2*x^2] - 4*a*c^2*x^2*Sqrt[1 - c^ 
2*x^2] - 2*b*Sqrt[1 - c^2*x^2]*(-1 + 2*c^2*x^2)*ArcCos[c*x] + b*c*x*(-1 + 
c^2*x^2)*Log[1 - 1/(c^2*x^2)] + 2*b*c*x*Log[1 - c^2*x^2] - 2*b*c^3*x^3*Log 
[1 - c^2*x^2]))/(d^2*x*(1 - c^2*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.79, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5195, 25, 27, 354, 86, 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.

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

\(\Big \downarrow \) 5195

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((a + b*ArcCos[c*x])/(d*x*Sqrt[d - c^2*d*x^2])) + (2*c^2*x*(a + b*ArcCos[ 
c*x]))/(d*Sqrt[d - c^2*d*x^2]) - (b*c*Sqrt[d - c^2*d*x^2]*(Log[x^2] + Log[ 
1 - c^2*x^2]))/(2*d^2*Sqrt[1 - c^2*x^2])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 
] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p 
+ 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
 

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

rule 2009

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

rule 5195

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_) 
, x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcCos 
[c*x])   u, x] + Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[Sim 
plifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] 
&& EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 
1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.70 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.53

method	result	size
default	\(a \left (-\frac {1}{d x \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2 c^{2} x}{d \sqrt {-c^{2} d \,x^{2}+d}}\right )+b \left (-\frac {4 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x c +2 c^{2} x^{2}-1\right ) \arccos \left (c x \right )}{d^{2} x \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{4}-1\right ) c}{d^{2} \left (c^{2} x^{2}-1\right )}\right )\)	\(229\)
parts	\(a \left (-\frac {1}{d x \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2 c^{2} x}{d \sqrt {-c^{2} d \,x^{2}+d}}\right )+b \left (-\frac {4 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x c +2 c^{2} x^{2}-1\right ) \arccos \left (c x \right )}{d^{2} x \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{4}-1\right ) c}{d^{2} \left (c^{2} x^{2}-1\right )}\right )\)	\(229\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int \frac {a+b \arccos (c x)}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a + b \operatorname {acos}{\left (c x \right )}}{x^{2} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:

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

Output:

Integral((a + b*acos(c*x))/(x**2*(-d*(c*x - 1)*(c*x + 1))**(3/2)), x)
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.86 \[ \int \frac {a+b \arccos (c x)}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {1}{2} \, b c {\left (\frac {\log \left (c x + 1\right )}{d^{\frac {3}{2}}} + \frac {\log \left (c x - 1\right )}{d^{\frac {3}{2}}} + \frac {2 \, \log \left (x\right )}{d^{\frac {3}{2}}}\right )} + {\left (\frac {2 \, c^{2} x}{\sqrt {-c^{2} d x^{2} + d} d} - \frac {1}{\sqrt {-c^{2} d x^{2} + d} d x}\right )} b \arccos \left (c x\right ) + {\left (\frac {2 \, c^{2} x}{\sqrt {-c^{2} d x^{2} + d} d} - \frac {1}{\sqrt {-c^{2} d x^{2} + d} d x}\right )} a \] Input:

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

Output:

-1/2*b*c*(log(c*x + 1)/d^(3/2) + log(c*x - 1)/d^(3/2) + 2*log(x)/d^(3/2)) 
+ (2*c^2*x/(sqrt(-c^2*d*x^2 + d)*d) - 1/(sqrt(-c^2*d*x^2 + d)*d*x))*b*arcc 
os(c*x) + (2*c^2*x/(sqrt(-c^2*d*x^2 + d)*d) - 1/(sqrt(-c^2*d*x^2 + d)*d*x) 
)*a
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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