\(\int \frac {x (a+b \arccos (c x))}{d-c^2 d x^2} \, dx\) [33]

Optimal result

Integrand size = 23, antiderivative size = 82 \[ \int \frac {x (a+b \arccos (c x))}{d-c^2 d x^2} \, dx=\frac {i (a+b \arccos (c x))^2}{2 b c^2 d}-\frac {(a+b \arccos (c x)) \log \left (1+e^{2 i \arccos (c x)}\right )}{c^2 d}+\frac {i b \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{2 c^2 d} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.40 \[ \int \frac {x (a+b \arccos (c x))}{d-c^2 d x^2} \, dx=\frac {i \left (b \arccos (c x)^2+2 i b \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right )+2 i b \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )+i a \log \left (1-c^2 x^2\right )+2 b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )+2 b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )\right )}{2 c^2 d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.96, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {5181, 3042, 25, 4200, 25, 2620, 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.

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

\(\Big \downarrow \) 5181

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4200

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

-((((-1/2*I)*(a + b*ArcCos[c*x])^2)/b - (2*I)*((I/2)*(a + b*ArcCos[c*x])*L 
og[1 - E^((2*I)*ArcCos[c*x])] + (b*PolyLog[2, E^((2*I)*ArcCos[c*x])])/4))/ 
(c^2*d))
 

Defintions of rubi rules used

rule 25

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

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

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 4200

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol 
] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I   Int[(c + d*x)^ 
m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))), x] 
, x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]
 

rule 5181

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), 
x_Symbol] :> Simp[1/e   Subst[Int[(a + b*x)^n*Cot[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 = 0.33 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.73

method	result	size
parts	\(-\frac {a \ln \left (c^{2} x^{2}-1\right )}{2 d \,c^{2}}-\frac {b \left (-\frac {i \arccos \left (c x \right )^{2}}{2}+\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d \,c^{2}}\)	\(142\)
derivativedivides	\(\frac {-\frac {a \left (\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \left (-\frac {i \arccos \left (c x \right )^{2}}{2}+\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d}}{c^{2}}\)	\(147\)
default	\(\frac {-\frac {a \left (\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \left (-\frac {i \arccos \left (c x \right )^{2}}{2}+\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d}}{c^{2}}\)	\(147\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/2*(2*c^2*d*integrate(1/2*(e^(1/2*log(c*x + 1) + 1/2*log(-c*x + 1))*log(c 
*x + 1) + e^(1/2*log(c*x + 1) + 1/2*log(-c*x + 1))*log(-c*x + 1))/(c^5*d*x 
^4 - c^3*d*x^2 + (c^3*d*x^2 - c*d)*e^(log(c*x + 1) + log(-c*x + 1))), x) - 
 (log(c*x + 1) + log(-c*x + 1))*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) 
)*b/(c^2*d) - 1/2*a*log(c^2*d*x^2 - d)/(c^2*d)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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