\(\int \frac {a+b \arccos (c x)}{(d-c^2 d x^2)^2} \, dx\) [5]

Optimal result

Integrand size = 22, antiderivative size = 145 \[ \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {b}{2 c d^2 \sqrt {1-c^2 x^2}}+\frac {x (a+b \arccos (c x))}{2 d^2 \left (1-c^2 x^2\right )}+\frac {(2 a+b \pi -b (\pi -2 \arccos (c x))) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{2 c d^2}-\frac {i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{2 c d^2}+\frac {i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 c d^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.52 \[ \int \frac {a+b \arccos (c x)}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\frac {b \sqrt {1-c^2 x^2}}{c-c^2 x}+\frac {b \sqrt {1-c^2 x^2}}{c+c^2 x}-\frac {2 a x}{-1+c^2 x^2}+\frac {b \arccos (c x)}{c-c^2 x}-\frac {b \arccos (c x)}{c+c^2 x}-\frac {2 b \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right )}{c}+\frac {2 b \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )}{c}-\frac {a \log (1-c x)}{c}+\frac {a \log (1+c x)}{c}-\frac {2 i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{c}+\frac {2 i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{c}}{4 d^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.84, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {5163, 27, 241, 5165, 3042, 4671, 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.

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

\(\Big \downarrow \) 5163

\(\displaystyle \frac {\int \frac {a+b \arccos (c x)}{d \left (1-c^2 x^2\right )}dx}{2 d}+\frac {b c \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}}dx}{2 d^2}+\frac {x (a+b \arccos (c x))}{2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {a+b \arccos (c x)}{1-c^2 x^2}dx}{2 d^2}+\frac {b c \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}}dx}{2 d^2}+\frac {x (a+b \arccos (c x))}{2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 5165

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

b/(2*c*d^2*Sqrt[1 - c^2*x^2]) + (x*(a + b*ArcCos[c*x]))/(2*d^2*(1 - c^2*x^ 
2)) - (-2*(a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])] + I*b*PolyLog[2, 
-E^(I*ArcCos[c*x])] - I*b*PolyLog[2, E^(I*ArcCos[c*x])])/(2*c*d^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 241

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ 
(2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -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 4671

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

rule 5163

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_ 
Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(2*d*(p + 1 
))), x] + (Simp[(2*p + 3)/(2*d*(p + 1))   Int[(d + e*x^2)^(p + 1)*(a + b*Ar 
cCos[c*x])^n, x], x] - Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2 
*x^2)^p]   Int[x*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x 
]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, 
 -1] && NeQ[p, -3/2]
 

rule 5165

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbo 
l] :> Simp[-(c*d)^(-1)   Subst[Int[(a + b*x)^n*Csc[x], x], x, ArcCos[c*x]], 
 x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
 

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.30

method	result	size
derivativedivides	\(\frac {\frac {a \left (-\frac {1}{4 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b \left (-\frac {c x \arccos \left (c x \right )+\sqrt {-c^{2} x^{2}+1}}{2 \left (c^{2} x^{2}-1\right )}-\frac {\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2}+\frac {i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2}+\frac {\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2}-\frac {i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2}\right )}{d^{2}}}{c}\)	\(189\)
default	\(\frac {\frac {a \left (-\frac {1}{4 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b \left (-\frac {c x \arccos \left (c x \right )+\sqrt {-c^{2} x^{2}+1}}{2 \left (c^{2} x^{2}-1\right )}-\frac {\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2}+\frac {i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2}+\frac {\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2}-\frac {i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2}\right )}{d^{2}}}{c}\)	\(189\)
parts	\(\frac {a \left (-\frac {1}{4 c \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{4 c}-\frac {1}{4 c \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{4 c}\right )}{d^{2}}+\frac {b \left (-\frac {c x \arccos \left (c x \right )+\sqrt {-c^{2} x^{2}+1}}{2 \left (c^{2} x^{2}-1\right )}-\frac {\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2}+\frac {i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2}+\frac {\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2}-\frac {i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2}\right )}{d^{2} c}\)	\(200\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

(4*int(acos(c*x)/(c**4*x**4 - 2*c**2*x**2 + 1),x)*b*c**3*x**2 - 4*int(acos 
(c*x)/(c**4*x**4 - 2*c**2*x**2 + 1),x)*b*c - log(c**2*x - c)*a*c**2*x**2 + 
 log(c**2*x - c)*a + log(c**2*x + c)*a*c**2*x**2 - log(c**2*x + c)*a - 2*a 
*c*x)/(4*c*d**2*(c**2*x**2 - 1))