3.2.89 \(\int \frac {(a+b \arcsin (c x))^2}{x^2 (d-c^2 d x^2)} \, dx\) [189]

3.2.89.1 Optimal result

Integrand size = 27, antiderivative size = 238 \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )} \, dx=-\frac {(a+b \arcsin (c x))^2}{d x}-\frac {2 i c (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{d}-\frac {4 b c (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d}+\frac {2 i b^2 c \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d}+\frac {2 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d}-\frac {2 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d}-\frac {2 i b^2 c \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d}-\frac {2 b^2 c \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{d}+\frac {2 b^2 c \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{d} \]

output

-(a+b*arcsin(c*x))^2/d/x-2*I*c*(a+b*arcsin(c*x))^2*arctan(I*c*x+(-c^2*x^2+ 
1)^(1/2))/d-4*b*c*(a+b*arcsin(c*x))*arctanh(I*c*x+(-c^2*x^2+1)^(1/2))/d+2* 
I*b^2*c*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))/d+2*I*b*c*(a+b*arcsin(c*x))*p 
olylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/d-2*I*b*c*(a+b*arcsin(c*x))*polylo 
g(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/d-2*I*b^2*c*polylog(2,I*c*x+(-c^2*x^2+1) 
^(1/2))/d-2*b^2*c*polylog(3,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/d+2*b^2*c*polyl 
og(3,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/d
 

3.2.89.2 Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(525\) vs. \(2(238)=476\).

Time = 1.17 (sec) , antiderivative size = 525, normalized size of antiderivative = 2.21 \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )} \, dx=-\frac {2 a^2+4 a b \arcsin (c x)+2 i a b c \pi x \arcsin (c x)+2 b^2 \arcsin (c x)^2+4 a b c x \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-4 b^2 c x \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )-2 a b c \pi x \log \left (1-i e^{i \arcsin (c x)}\right )-4 a b c x \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )-2 b^2 c x \arcsin (c x)^2 \log \left (1-i e^{i \arcsin (c x)}\right )-2 a b c \pi x \log \left (1+i e^{i \arcsin (c x)}\right )+4 a b c x \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )+2 b^2 c x \arcsin (c x)^2 \log \left (1+i e^{i \arcsin (c x)}\right )+4 b^2 c x \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )+a^2 c x \log (1-c x)-a^2 c x \log (1+c x)+2 a b c \pi x \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+2 a b c \pi x \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-4 i b^2 c x \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-4 i b c x (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+4 i a b c x \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )+4 i b^2 c x \arcsin (c x) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )+4 i b^2 c x \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )+4 b^2 c x \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )-4 b^2 c x \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{2 d x} \]

input

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

output

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

3.2.89.3 Rubi [A] (verified)

Time = 1.48 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.91, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {5204, 27, 5164, 3042, 4669, 3011, 2720, 5218, 3042, 4671, 2715, 2838, 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.

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

\(\Big \downarrow \) 5204

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5164

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

\(\displaystyle \frac {c \left (2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{d}+\frac {2 b c \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx}{d}-\frac {(a+b \arcsin (c x))^2}{d x}\)

\(\Big \downarrow \) 5218

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 7143

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

input

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

output

-((a + b*ArcSin[c*x])^2/(d*x)) + (2*b*c*(-2*(a + b*ArcSin[c*x])*ArcTanh[E^ 
(I*ArcSin[c*x])] + I*b*PolyLog[2, -E^(I*ArcSin[c*x])] - I*b*PolyLog[2, E^( 
I*ArcSin[c*x])]))/d + (c*((-2*I)*(a + b*ArcSin[c*x])^2*ArcTan[E^(I*ArcSin[ 
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])]) - 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])])))/d
 

3.2.89.3.1 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 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 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 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 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 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 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 5204

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b 
*ArcSin[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1)) 
)   Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] - Simp[b* 
c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Int[(f*x)^(m + 1)*( 
1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, 
 c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]
 

rule 5218

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)* 
(x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e* 
x^2]]   Subst[Int[(a + b*x)^n*Sin[x]^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a 
, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]
 

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]
 

3.2.89.4 Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.99

method	result	size
parts	\(-\frac {a^{2} \left (\frac {c \ln \left (c x -1\right )}{2}-\frac {c \ln \left (c x +1\right )}{2}+\frac {1}{x}\right )}{d}-\frac {b^{2} c \left (\frac {\arcsin \left (c x \right )^{2}}{c x}-2 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-2 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{d}-\frac {2 a b c \left (\frac {\arcsin \left (c x \right )}{c x}+\arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{d}\)	\(474\)
derivativedivides	\(c \left (-\frac {a^{2} \left (\frac {1}{c x}+\frac {\ln \left (c x -1\right )}{2}-\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b^{2} \left (\frac {\arcsin \left (c x \right )^{2}}{c x}-2 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-2 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{d}-\frac {2 a b \left (\frac {\arcsin \left (c x \right )}{c x}+\arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{d}\right )\)	\(476\)
default	\(c \left (-\frac {a^{2} \left (\frac {1}{c x}+\frac {\ln \left (c x -1\right )}{2}-\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b^{2} \left (\frac {\arcsin \left (c x \right )^{2}}{c x}-2 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-2 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{d}-\frac {2 a b \left (\frac {\arcsin \left (c x \right )}{c x}+\arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{d}\right )\)	\(476\)

input

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

output

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

3.2.89.5 Fricas [F]

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

input

integrate((a+b*arcsin(c*x))^2/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^4 - d*x^2 
), x)
 

3.2.89.6 Sympy [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )} \, dx=- \frac {\int \frac {a^{2}}{c^{2} x^{4} - x^{2}}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{2} x^{4} - x^{2}}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{c^{2} x^{4} - x^{2}}\, dx}{d} \]

input

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

output

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

3.2.89.7 Maxima [F]

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

input

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

output

1/2*a^2*(c*log(c*x + 1)/d - c*log(c*x - 1)/d - 2/(d*x)) + 1/2*(b^2*c*x*arc 
tan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2*log(c*x + 1) - b^2*c*x*arctan2(c 
*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2*log(-c*x + 1) - 2*b^2*arctan2(c*x, sqr 
t(c*x + 1)*sqrt(-c*x + 1))^2 + 2*d*x*integrate(-(2*a*b*arctan2(c*x, sqrt(c 
*x + 1)*sqrt(-c*x + 1)) - (b^2*c^2*x^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c* 
x + 1))*log(c*x + 1) - b^2*c^2*x^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 
1))*log(-c*x + 1) - 2*b^2*c*x*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))* 
sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^2*d*x^4 - d*x^2), x))/(d*x)
 

3.2.89.8 Giac [F]

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

input

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

output

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

3.2.89.9 Mupad [F(-1)]

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

input

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

output

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