\(\int \frac {(a+b \arccos (c x))^2}{d-c^2 d x^2} \, dx\) [10]

Optimal result

Integrand size = 24, antiderivative size = 174 \[ \int \frac {(a+b \arccos (c x))^2}{d-c^2 d x^2} \, dx=\frac {(2 a+b \pi -b (\pi -2 \arccos (c x)))^2 \text {arctanh}\left (e^{i \arccos (c x)}\right )}{2 c d}-\frac {i b (2 a+b \pi -b (\pi -2 \arccos (c x))) \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{c d}+\frac {i b (2 a+b \pi -b (\pi -2 \arccos (c x))) \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{c d}+\frac {2 b^2 \operatorname {PolyLog}\left (3,-e^{i \arccos (c x)}\right )}{c d}-\frac {2 b^2 \operatorname {PolyLog}\left (3,e^{i \arccos (c x)}\right )}{c d} \] Output:

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

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.24 \[ \int \frac {(a+b \arccos (c x))^2}{d-c^2 d x^2} \, dx=\frac {-4 a b \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right )-2 b^2 \arccos (c x)^2 \log \left (1-e^{i \arccos (c x)}\right )+4 a b \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )+2 b^2 \arccos (c x)^2 \log \left (1+e^{i \arccos (c x)}\right )-a^2 \log (1-c x)+a^2 \log (1+c x)-4 i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )+4 i b (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )+4 b^2 \operatorname {PolyLog}\left (3,-e^{i \arccos (c x)}\right )-4 b^2 \operatorname {PolyLog}\left (3,e^{i \arccos (c x)}\right )}{2 c d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.70, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5165, 3042, 4671, 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.

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

\(\Big \downarrow \) 5165

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

-((-2*(a + b*ArcCos[c*x])^2*ArcTanh[E^(I*ArcCos[c*x])] + 2*b*(I*(a + b*Arc 
Cos[c*x])*PolyLog[2, -E^(I*ArcCos[c*x])] - b*PolyLog[3, -E^(I*ArcCos[c*x]) 
]) - 2*b*(I*(a + b*ArcCos[c*x])*PolyLog[2, E^(I*ArcCos[c*x])] - b*PolyLog[ 
3, E^(I*ArcCos[c*x])]))/(c*d))
 

Defintions of rubi rules used

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 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 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]
 

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 [A] (verified)

Time = 0.31 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.80

method	result	size
derivativedivides	\(\frac {\frac {a^{2} \operatorname {arctanh}\left (c x \right )}{d}-\frac {b^{2} \left (\arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )-\arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {2 a b \left (-\operatorname {arctanh}\left (c x \right ) \arccos \left (c x \right )-i \operatorname {arctanh}\left (c x \right ) \left (\ln \left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\ln \left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )\right )+i \operatorname {dilog}\left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )\right )}{d}}{c}\)	\(314\)
default	\(\frac {\frac {a^{2} \operatorname {arctanh}\left (c x \right )}{d}-\frac {b^{2} \left (\arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )-\arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {2 a b \left (-\operatorname {arctanh}\left (c x \right ) \arccos \left (c x \right )-i \operatorname {arctanh}\left (c x \right ) \left (\ln \left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\ln \left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )\right )+i \operatorname {dilog}\left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )\right )}{d}}{c}\)	\(314\)
parts	\(-\frac {a^{2} \ln \left (c x -1\right )}{2 d c}+\frac {a^{2} \ln \left (c x +1\right )}{2 d c}-\frac {b^{2} \left (\arccos \left (c x \right )^{2} \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, c x +i \sqrt {-c^{2} x^{2}+1}\right )-\arccos \left (c x \right )^{2} \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 i \arccos \left (c x \right ) \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -c x -i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d c}-\frac {2 a b \left (-\operatorname {arctanh}\left (c x \right ) \arccos \left (c x \right )-i \operatorname {arctanh}\left (c x \right ) \left (\ln \left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-\ln \left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )\right )+i \operatorname {dilog}\left (1+\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1-\frac {i \left (c x +1\right )}{\sqrt {-c^{2} x^{2}+1}}\right )\right )}{d c}\)	\(339\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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