\(\int \frac {(a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx\) [184]

Optimal result

Integrand size = 24, antiderivative size = 156 \[ \int \frac {(a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx=-\frac {2 i (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{c d}+\frac {2 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c d}-\frac {2 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c d}-\frac {2 b^2 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{c d}+\frac {2 b^2 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{c d} \] Output:

-2*I*(a+b*arcsin(c*x))^2*arctan(I*c*x+(-c^2*x^2+1)^(1/2))/c/d+2*I*b*(a+b*a 
rcsin(c*x))*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c/d-2*I*b*(a+b*arcsin 
(c*x))*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c/d-2*b^2*polylog(3,-I*(I*c 
*x+(-c^2*x^2+1)^(1/2)))/c/d+2*b^2*polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/ 
c/d
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.94 \[ \int \frac {(a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx=-\frac {2 i a b \pi \arcsin (c x)+4 i b^2 \arcsin (c x)^2 \arctan \left (e^{i \arcsin (c x)}\right )-2 a b \pi \log \left (1-i e^{i \arcsin (c x)}\right )-4 a b \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )-2 a b \pi \log \left (1+i e^{i \arcsin (c x)}\right )+4 a b \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )+a^2 \log (1-c x)-a^2 \log (1+c x)+2 a b \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+2 a b \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-4 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+4 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )+4 b^2 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )-4 b^2 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{2 c d} \] Input:

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

Output:

-1/2*((2*I)*a*b*Pi*ArcSin[c*x] + (4*I)*b^2*ArcSin[c*x]^2*ArcTan[E^(I*ArcSi 
n[c*x])] - 2*a*b*Pi*Log[1 - I*E^(I*ArcSin[c*x])] - 4*a*b*ArcSin[c*x]*Log[1 
 - I*E^(I*ArcSin[c*x])] - 2*a*b*Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 4*a*b*Ar 
cSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + a^2*Log[1 - c*x] - a^2*Log[1 + c* 
x] + 2*a*b*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 2*a*b*Pi*Log[Sin[(Pi + 2 
*ArcSin[c*x])/4]] - (4*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSi 
n[c*x])] + (4*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])] + 4 
*b^2*PolyLog[3, (-I)*E^(I*ArcSin[c*x])] - 4*b^2*PolyLog[3, I*E^(I*ArcSin[c 
*x])])/(c*d)
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.87, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5164, 3042, 4669, 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.

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

\(\Big \downarrow \) 5164

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

((-2*I)*(a + b*ArcSin[c*x])^2*ArcTan[E^(I*ArcSin[c*x])] + 2*b*(I*(a + b*Ar 
cSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - b*PolyLog[3, (-I)*E^(I*Arc 
Sin[c*x])]) - 2*b*(I*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])] - 
 b*PolyLog[3, I*E^(I*ArcSin[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 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 5164

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbo 
l] :> Simp[1/(c*d)   Subst[Int[(a + b*x)^n*Sec[x], x], x, ArcSin[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 = 0.00 (sec) , antiderivative size = 375, normalized size of antiderivative = 2.40

method	result	size
derivativedivides	\(\frac {\frac {b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {2 i b^{2} \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}+\frac {2 b^{2} \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}+\frac {2 i b^{2} \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {2 b^{2} \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}+\frac {2 a b \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {2 i a b \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {2 a b \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}+\frac {2 i a b \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {2 i a^{2} \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}}{c}\)	\(375\)
default	\(\frac {\frac {b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {2 i b^{2} \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}+\frac {2 b^{2} \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}+\frac {2 i b^{2} \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {2 b^{2} \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}+\frac {2 a b \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {2 i a b \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {2 a b \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}+\frac {2 i a b \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d}-\frac {2 i a^{2} \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}}{c}\)	\(375\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 4*int(asin(c*x)/(c**2*x**2 - 1),x)*a*b*c - 2*int(asin(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)