\(\int \frac {a+b \arccos (c x)}{x^3 \sqrt {d-c^2 d x^2}} \, dx\) [119]

Optimal result

Integrand size = 27, antiderivative size = 229 \[ \int \frac {a+b \arccos (c x)}{x^3 \sqrt {d-c^2 d x^2}} \, dx=-\frac {b c \sqrt {1-c^2 x^2}}{2 x \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{2 d x^2}-\frac {c^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {i b c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{2 \sqrt {d-c^2 d x^2}}-\frac {i b c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 \sqrt {d-c^2 d x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.59 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.07 \[ \int \frac {a+b \arccos (c x)}{x^3 \sqrt {d-c^2 d x^2}} \, dx=-\frac {\frac {2 a \sqrt {d-c^2 d x^2}}{x^2}-2 a c^2 \sqrt {d} \log (x)+2 a c^2 \sqrt {d} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {2 b d \sqrt {1-c^2 x^2} \left (-c x+\sqrt {1-c^2 x^2} \arccos (c x)+c^2 x^2 \arccos (c x) \log \left (1-i e^{i \arccos (c x)}\right )-c^2 x^2 \arccos (c x) \log \left (1+i e^{i \arccos (c x)}\right )+i c^2 x^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )-i c^2 x^2 \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )\right )}{x^2 \sqrt {d-c^2 d x^2}}}{4 d} \] Input:

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

Output:

-1/4*((2*a*Sqrt[d - c^2*d*x^2])/x^2 - 2*a*c^2*Sqrt[d]*Log[x] + 2*a*c^2*Sqr 
t[d]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (2*b*d*Sqrt[1 - c^2*x^2]*(-(c* 
x) + Sqrt[1 - c^2*x^2]*ArcCos[c*x] + c^2*x^2*ArcCos[c*x]*Log[1 - I*E^(I*Ar 
cCos[c*x])] - c^2*x^2*ArcCos[c*x]*Log[1 + I*E^(I*ArcCos[c*x])] + I*c^2*x^2 
*PolyLog[2, (-I)*E^(I*ArcCos[c*x])] - I*c^2*x^2*PolyLog[2, I*E^(I*ArcCos[c 
*x])]))/(x^2*Sqrt[d - c^2*d*x^2]))/d
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.76, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {5205, 15, 5219, 3042, 4669, 2715, 2838}

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.

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

\(\Big \downarrow \) 5205

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 5219

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

(b*c*Sqrt[1 - c^2*x^2])/(2*x*Sqrt[d - c^2*d*x^2]) - (Sqrt[d - c^2*d*x^2]*( 
a + b*ArcCos[c*x]))/(2*d*x^2) - (c^2*Sqrt[1 - c^2*x^2]*((-2*I)*(a + b*ArcC 
os[c*x])*ArcTan[E^(I*ArcCos[c*x])] + I*b*PolyLog[2, (-I)*E^(I*ArcCos[c*x]) 
] - I*b*PolyLog[2, I*E^(I*ArcCos[c*x])]))/(2*Sqrt[d - c^2*d*x^2])
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4669

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol 
] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si 
mp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], 
 x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x 
))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
 

rule 5205

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b 
*ArcCos[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1)) 
)   Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] + Simp[b* 
c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Int[(f*x)^(m + 1)*( 
1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, 
 c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]
 

rule 5219

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)* 
(x_)^2], x_Symbol] :> Simp[(-(c^(m + 1))^(-1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[ 
d + e*x^2]]   Subst[Int[(a + b*x)^n*Cos[x]^m, x], x, ArcCos[c*x]], x] /; Fr 
eeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]
 

Maple [A] (verified)

Time = 0.72 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.28

method	result	size
default	\(-\frac {a \sqrt {-c^{2} d \,x^{2}+d}}{2 d \,x^{2}}-\frac {a \,c^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{2 \sqrt {d}}+b \left (-\frac {\left (c^{2} x^{2} \arccos \left (c x \right )+c x \sqrt {-c^{2} x^{2}+1}-\arccos \left (c x \right )\right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{2 x^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-\arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-i \operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+i \operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )\right ) c^{2}}{2 d \left (c^{2} x^{2}-1\right )}\right )\)	\(292\)
parts	\(-\frac {a \sqrt {-c^{2} d \,x^{2}+d}}{2 d \,x^{2}}-\frac {a \,c^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{2 \sqrt {d}}+b \left (-\frac {\left (c^{2} x^{2} \arccos \left (c x \right )+c x \sqrt {-c^{2} x^{2}+1}-\arccos \left (c x \right )\right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{2 x^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-\arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )-i \operatorname {dilog}\left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )+i \operatorname {dilog}\left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )\right ) c^{2}}{2 d \left (c^{2} x^{2}-1\right )}\right )\)	\(292\)

Input:

int((a+b*arccos(c*x))/x^3/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

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

integrate((a+b*arccos(c*x))/x^3/(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas" 
)
 

Output:

integral(-sqrt(-c^2*d*x^2 + d)*(b*arccos(c*x) + a)/(c^2*d*x^5 - d*x^3), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((a+b*arccos(c*x))/x^3/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima" 
)
                                                                                    
                                                                                    
 

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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