\(\int \frac {x (a+b \arccos (c x))^2}{d-c^2 d x^2} \, dx\) [188]

Optimal result

Integrand size = 25, antiderivative size = 117 \[ \int \frac {x (a+b \arccos (c x))^2}{d-c^2 d x^2} \, dx=\frac {i (a+b \arccos (c x))^3}{3 b c^2 d}-\frac {(a+b \arccos (c x))^2 \log \left (1+e^{2 i \arccos (c x)}\right )}{c^2 d}+\frac {i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{c^2 d}-\frac {b^2 \operatorname {PolyLog}\left (3,-e^{2 i \arccos (c x)}\right )}{2 c^2 d} \] Output:

1/3*I*(a+b*arccos(c*x))^3/b/c^2/d-(a+b*arccos(c*x))^2*ln(1+(c*x+I*(-c^2*x^ 
2+1)^(1/2))^2)/c^2/d+I*b*(a+b*arccos(c*x))*polylog(2,-(c*x+I*(-c^2*x^2+1)^ 
(1/2))^2)/c^2/d-1/2*b^2*polylog(3,-(c*x+I*(-c^2*x^2+1)^(1/2))^2)/c^2/d
 

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.79 \[ \int \frac {x (a+b \arccos (c x))^2}{d-c^2 d x^2} \, dx=\frac {i \left (b^2 \pi ^3+24 a b \arccos (c x)^2-8 b^2 \arccos (c x)^3+48 i a b \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right )+48 i a b \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )+24 i b^2 \arccos (c x)^2 \log \left (1-e^{-2 i \arccos (c x)}\right )+12 i a^2 \log \left (1-c^2 x^2\right )+48 a b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )+48 a b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )-24 b^2 \arccos (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arccos (c x)}\right )+12 i b^2 \operatorname {PolyLog}\left (3,e^{-2 i \arccos (c x)}\right )\right )}{24 c^2 d} \] Input:

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

Output:

((I/24)*(b^2*Pi^3 + 24*a*b*ArcCos[c*x]^2 - 8*b^2*ArcCos[c*x]^3 + (48*I)*a* 
b*ArcCos[c*x]*Log[1 - E^(I*ArcCos[c*x])] + (48*I)*a*b*ArcCos[c*x]*Log[1 + 
E^(I*ArcCos[c*x])] + (24*I)*b^2*ArcCos[c*x]^2*Log[1 - E^((-2*I)*ArcCos[c*x 
])] + (12*I)*a^2*Log[1 - c^2*x^2] + 48*a*b*PolyLog[2, -E^(I*ArcCos[c*x])] 
+ 48*a*b*PolyLog[2, E^(I*ArcCos[c*x])] - 24*b^2*ArcCos[c*x]*PolyLog[2, E^( 
(-2*I)*ArcCos[c*x])] + (12*I)*b^2*PolyLog[3, E^((-2*I)*ArcCos[c*x])]))/(c^ 
2*d)
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.97, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {5181, 3042, 25, 4200, 25, 2620, 3011, 2720, 7143}

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.

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

\(\Big \downarrow \) 5181

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4200

\(\displaystyle -\frac {2 i \int -\frac {e^{2 i \arccos (c x)} (a+b \arccos (c x))^2}{1-e^{2 i \arccos (c x)}}d\arccos (c x)-\frac {i (a+b \arccos (c x))^3}{3 b}}{c^2 d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-2 i \int \frac {e^{2 i \arccos (c x)} (a+b \arccos (c x))^2}{1-e^{2 i \arccos (c x)}}d\arccos (c x)-\frac {i (a+b \arccos (c x))^3}{3 b}}{c^2 d}\)

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {1}{4} b \int e^{-2 i \arccos (c x)} \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )de^{2 i \arccos (c x)}\right )\right )-\frac {i (a+b \arccos (c x))^3}{3 b}}{c^2 d}\)

\(\Big \downarrow \) 7143

\(\displaystyle -\frac {-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (3,e^{2 i \arccos (c x)}\right )\right )\right )-\frac {i (a+b \arccos (c x))^3}{3 b}}{c^2 d}\)

Input:

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

Output:

-((((-1/3*I)*(a + b*ArcCos[c*x])^3)/b - (2*I)*((I/2)*(a + b*ArcCos[c*x])^2 
*Log[1 - E^((2*I)*ArcCos[c*x])] - I*b*((I/2)*(a + b*ArcCos[c*x])*PolyLog[2 
, E^((2*I)*ArcCos[c*x])] - (b*PolyLog[3, 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 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

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]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (144 ) = 288\).

Time = 0.30 (sec) , antiderivative size = 327, normalized size of antiderivative = 2.79

method	result	size
parts	\(-\frac {a^{2} \ln \left (c^{2} x^{2}-1\right )}{2 d \,c^{2}}-\frac {b^{2} \left (-\frac {i \arccos \left (c x \right )^{3}}{3}+\arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+\arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d \,c^{2}}-\frac {2 a b \left (-\frac {i \arccos \left (c x \right )^{2}}{2}+\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d \,c^{2}}\)	\(327\)
derivativedivides	\(\frac {-\frac {a^{2} \left (\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b^{2} \left (-\frac {i \arccos \left (c x \right )^{3}}{3}+\arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+\arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {2 a b \left (-\frac {i \arccos \left (c x \right )^{2}}{2}+\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d}}{c^{2}}\)	\(329\)
default	\(\frac {-\frac {a^{2} \left (\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b^{2} \left (-\frac {i \arccos \left (c x \right )^{3}}{3}+\arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+\arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {2 a b \left (-\frac {i \arccos \left (c x \right )^{2}}{2}+\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d}}{c^{2}}\)	\(329\)

Input:

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

Output:

-1/2*a^2/d/c^2*ln(c^2*x^2-1)-b^2/d/c^2*(-1/3*I*arccos(c*x)^3+arccos(c*x)^2 
*ln(1+c*x+I*(-c^2*x^2+1)^(1/2))-2*I*arccos(c*x)*polylog(2,-c*x-I*(-c^2*x^2 
+1)^(1/2))+2*polylog(3,-c*x-I*(-c^2*x^2+1)^(1/2))+arccos(c*x)^2*ln(1-c*x-I 
*(-c^2*x^2+1)^(1/2))-2*I*arccos(c*x)*polylog(2,c*x+I*(-c^2*x^2+1)^(1/2))+2 
*polylog(3,c*x+I*(-c^2*x^2+1)^(1/2)))-2*a*b/d/c^2*(-1/2*I*arccos(c*x)^2+ar 
ccos(c*x)*ln(1+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))-I*polylog(2,c*x+I*(-c^2*x 
^2+1)^(1/2)))
                                                                                    
                                                                                    
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/2*a^2*log(c^2*d*x^2 - d)/(c^2*d) + 1/2*(2*c^2*d*integrate(-(2*a*b*c*x*a 
rctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) - (b^2*log(c*x + 1) + b^2*log(-c 
*x + 1))*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1) 
, c*x))/(c^3*d*x^2 - c*d), x) - (b^2*log(c*x + 1) + b^2*log(-c*x + 1))*arc 
tan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2)/(c^2*d)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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