3.2.93 \(\int \frac {x^3 (a+b \arcsin (c x))^2}{(d-c^2 d x^2)^2} \, dx\) [193]

3.2.93.1 Optimal result

Integrand size = 27, antiderivative size = 227 \[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {b x (a+b \arcsin (c x))}{c^3 d^2 \sqrt {1-c^2 x^2}}+\frac {(a+b \arcsin (c x))^2}{2 c^4 d^2}+\frac {x^2 (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {i (a+b \arcsin (c x))^3}{3 b c^4 d^2}+\frac {(a+b \arcsin (c x))^2 \log \left (1+e^{2 i \arcsin (c x)}\right )}{c^4 d^2}-\frac {b^2 \log \left (1-c^2 x^2\right )}{2 c^4 d^2}-\frac {i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{c^4 d^2}+\frac {b^2 \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )}{2 c^4 d^2} \]

output

1/2*(a+b*arcsin(c*x))^2/c^4/d^2+1/2*x^2*(a+b*arcsin(c*x))^2/c^2/d^2/(-c^2* 
x^2+1)-1/3*I*(a+b*arcsin(c*x))^3/b/c^4/d^2+(a+b*arcsin(c*x))^2*ln(1+(I*c*x 
+(-c^2*x^2+1)^(1/2))^2)/c^4/d^2-1/2*b^2*ln(-c^2*x^2+1)/c^4/d^2-I*b*(a+b*ar 
csin(c*x))*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)/c^4/d^2+1/2*b^2*polylo 
g(3,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)/c^4/d^2-b*x*(a+b*arcsin(c*x))/c^3/d^2/( 
-c^2*x^2+1)^(1/2)
 

3.2.93.2 Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(502\) vs. \(2(227)=454\).

Time = 1.21 (sec) , antiderivative size = 502, normalized size of antiderivative = 2.21 \[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\frac {3 a b \sqrt {1-c^2 x^2}}{-1+c x}+\frac {3 a b \sqrt {1-c^2 x^2}}{1+c x}-\frac {3 a^2}{-1+c^2 x^2}+12 i a b \pi \arcsin (c x)-\frac {3 a b \arcsin (c x)}{-1+c x}+\frac {3 a b \arcsin (c x)}{1+c x}-\frac {6 b^2 c x \arcsin (c x)}{\sqrt {1-c^2 x^2}}-6 i a b \arcsin (c x)^2+\frac {3 b^2 \arcsin (c x)^2}{1-c^2 x^2}-2 i b^2 \arcsin (c x)^3+24 a b \pi \log \left (1+e^{-i \arcsin (c x)}\right )+6 a b \pi \log \left (1-i e^{i \arcsin (c x)}\right )+12 a b \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )-6 a b \pi \log \left (1+i e^{i \arcsin (c x)}\right )+12 a b \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )+6 b^2 \arcsin (c x)^2 \log \left (1+e^{2 i \arcsin (c x)}\right )+3 a^2 \log \left (1-c^2 x^2\right )-3 b^2 \log \left (1-c^2 x^2\right )-24 a b \pi \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )+6 a b \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-6 a b \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-12 i a b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-12 i a b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )-6 i b^2 \arcsin (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )+3 b^2 \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )}{6 c^4 d^2} \]

input

Integrate[(x^3*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^2,x]
 

output

((3*a*b*Sqrt[1 - c^2*x^2])/(-1 + c*x) + (3*a*b*Sqrt[1 - c^2*x^2])/(1 + c*x 
) - (3*a^2)/(-1 + c^2*x^2) + (12*I)*a*b*Pi*ArcSin[c*x] - (3*a*b*ArcSin[c*x 
])/(-1 + c*x) + (3*a*b*ArcSin[c*x])/(1 + c*x) - (6*b^2*c*x*ArcSin[c*x])/Sq 
rt[1 - c^2*x^2] - (6*I)*a*b*ArcSin[c*x]^2 + (3*b^2*ArcSin[c*x]^2)/(1 - c^2 
*x^2) - (2*I)*b^2*ArcSin[c*x]^3 + 24*a*b*Pi*Log[1 + E^((-I)*ArcSin[c*x])] 
+ 6*a*b*Pi*Log[1 - I*E^(I*ArcSin[c*x])] + 12*a*b*ArcSin[c*x]*Log[1 - I*E^( 
I*ArcSin[c*x])] - 6*a*b*Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 12*a*b*ArcSin[c* 
x]*Log[1 + I*E^(I*ArcSin[c*x])] + 6*b^2*ArcSin[c*x]^2*Log[1 + E^((2*I)*Arc 
Sin[c*x])] + 3*a^2*Log[1 - c^2*x^2] - 3*b^2*Log[1 - c^2*x^2] - 24*a*b*Pi*L 
og[Cos[ArcSin[c*x]/2]] + 6*a*b*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] - 6*a* 
b*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (12*I)*a*b*PolyLog[2, (-I)*E^(I*Ar 
cSin[c*x])] - (12*I)*a*b*PolyLog[2, I*E^(I*ArcSin[c*x])] - (6*I)*b^2*ArcSi 
n[c*x]*PolyLog[2, -E^((2*I)*ArcSin[c*x])] + 3*b^2*PolyLog[3, -E^((2*I)*Arc 
Sin[c*x])])/(6*c^4*d^2)
 

3.2.93.3 Rubi [A] (verified)

Time = 1.51 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5206, 27, 5180, 3042, 4202, 2620, 3011, 2720, 5206, 240, 5152, 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.

\(\displaystyle \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx\)

\(\Big \downarrow \) 5206

\(\displaystyle -\frac {b \int \frac {x^2 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c d^2}-\frac {\int \frac {x (a+b \arcsin (c x))^2}{d \left (1-c^2 x^2\right )}dx}{c^2 d}+\frac {x^2 (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b \int \frac {x^2 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c d^2}-\frac {\int \frac {x (a+b \arcsin (c x))^2}{1-c^2 x^2}dx}{c^2 d^2}+\frac {x^2 (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 5180

\(\displaystyle -\frac {b \int \frac {x^2 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c d^2}-\frac {\int \frac {c x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{c^4 d^2}+\frac {x^2 (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int (a+b \arcsin (c x))^2 \tan (\arcsin (c x))d\arcsin (c x)}{c^4 d^2}-\frac {b \int \frac {x^2 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c d^2}+\frac {x^2 (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 4202

\(\displaystyle -\frac {\frac {i (a+b \arcsin (c x))^3}{3 b}-2 i \int \frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))^2}{1+e^{2 i \arcsin (c x)}}d\arcsin (c x)}{c^4 d^2}-\frac {b \int \frac {x^2 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c d^2}+\frac {x^2 (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 2620

\(\displaystyle -\frac {\frac {i (a+b \arcsin (c x))^3}{3 b}-2 i \left (i b \int (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )d\arcsin (c x)-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{c^4 d^2}-\frac {b \int \frac {x^2 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx}{c d^2}+\frac {x^2 (a+b \arcsin (c x))^2}{2 c^2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 5206

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

\(\Big \downarrow \) 240

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

\(\Big \downarrow \) 5152

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

\(\Big \downarrow \) 7143

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

input

Int[(x^3*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^2,x]
 

output

(x^2*(a + b*ArcSin[c*x])^2)/(2*c^2*d^2*(1 - c^2*x^2)) - (b*((x*(a + b*ArcS 
in[c*x]))/(c^2*Sqrt[1 - c^2*x^2]) - (a + b*ArcSin[c*x])^2/(2*b*c^3) + (b*L 
og[1 - c^2*x^2])/(2*c^3)))/(c*d^2) - (((I/3)*(a + b*ArcSin[c*x])^3)/b - (2 
*I)*((-1/2*I)*(a + b*ArcSin[c*x])^2*Log[1 + E^((2*I)*ArcSin[c*x])] + I*b*( 
(I/2)*(a + b*ArcSin[c*x])*PolyLog[2, -E^((2*I)*ArcSin[c*x])] - (b*PolyLog[ 
3, -E^((2*I)*ArcSin[c*x])])/4)))/(c^4*d^2)
 

3.2.93.3.1 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 240

Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x 
^2, x]]/(2*b), x] /; FreeQ[{a, b}, 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 4202

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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*( 
e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt 
Q[m, 0]
 

rule 5152

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

rule 5180

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

rule 5206

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

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]
 

3.2.93.4 Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.60

method	result	size
derivativedivides	\(\frac {\frac {a^{2} \left (-\frac {1}{4 \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2}+\frac {1}{4 c x +4}+\frac {\ln \left (c x +1\right )}{2}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {i \arcsin \left (c x \right )^{3}}{3}-\frac {\left (2 i c^{2} x^{2}-2 c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )-2 i\right ) \arcsin \left (c x \right )}{2 \left (c^{2} x^{2}-1\right )}+\arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {2 a b \left (-\frac {i \arcsin \left (c x \right )^{2}}{2}-\frac {i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )-i}{2 \left (c^{2} x^{2}-1\right )}+\arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d^{2}}}{c^{4}}\)	\(364\)
default	\(\frac {\frac {a^{2} \left (-\frac {1}{4 \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2}+\frac {1}{4 c x +4}+\frac {\ln \left (c x +1\right )}{2}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {i \arcsin \left (c x \right )^{3}}{3}-\frac {\left (2 i c^{2} x^{2}-2 c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )-2 i\right ) \arcsin \left (c x \right )}{2 \left (c^{2} x^{2}-1\right )}+\arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {2 a b \left (-\frac {i \arcsin \left (c x \right )^{2}}{2}-\frac {i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )-i}{2 \left (c^{2} x^{2}-1\right )}+\arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d^{2}}}{c^{4}}\)	\(364\)
parts	\(\frac {a^{2} \left (-\frac {1}{4 c^{4} \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2 c^{4}}+\frac {1}{4 c^{4} \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{2 c^{4}}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {i \arcsin \left (c x \right )^{3}}{3}-\frac {\left (2 i c^{2} x^{2}-2 c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )-2 i\right ) \arcsin \left (c x \right )}{2 \left (c^{2} x^{2}-1\right )}+\arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{4}}+\frac {2 a b \left (-\frac {i \arcsin \left (c x \right )^{2}}{2}-\frac {i c^{2} x^{2}-c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )-i}{2 \left (c^{2} x^{2}-1\right )}+\arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d^{2} c^{4}}\)	\(378\)

input

int(x^3*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^2,x,method=_RETURNVERBOSE)
 

output

1/c^4*(a^2/d^2*(-1/4/(c*x-1)+1/2*ln(c*x-1)+1/4/(c*x+1)+1/2*ln(c*x+1))+b^2/ 
d^2*(-1/3*I*arcsin(c*x)^3-1/2*(2*I*c^2*x^2-2*c*x*(-c^2*x^2+1)^(1/2)+arcsin 
(c*x)-2*I)*arcsin(c*x)/(c^2*x^2-1)+arcsin(c*x)^2*ln(1+(I*c*x+(-c^2*x^2+1)^ 
(1/2))^2)-I*arcsin(c*x)*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)+1/2*polyl 
og(3,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)-ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)+2*l 
n(I*c*x+(-c^2*x^2+1)^(1/2)))+2*a*b/d^2*(-1/2*I*arcsin(c*x)^2-1/2*(I*c^2*x^ 
2-c*x*(-c^2*x^2+1)^(1/2)+arcsin(c*x)-I)/(c^2*x^2-1)+arcsin(c*x)*ln(1+(I*c* 
x+(-c^2*x^2+1)^(1/2))^2)-1/2*I*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)))
 

3.2.93.5 Fricas [F]

\[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{3}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]

input

integrate(x^3*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^2,x, algorithm="fricas")
 

output

integral((b^2*x^3*arcsin(c*x)^2 + 2*a*b*x^3*arcsin(c*x) + a^2*x^3)/(c^4*d^ 
2*x^4 - 2*c^2*d^2*x^2 + d^2), x)
 

3.2.93.6 Sympy [F]

\[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a^{2} x^{3}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b^{2} x^{3} \operatorname {asin}^{2}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {2 a b x^{3} \operatorname {asin}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]

input

integrate(x**3*(a+b*asin(c*x))**2/(-c**2*d*x**2+d)**2,x)
 

output

(Integral(a**2*x**3/(c**4*x**4 - 2*c**2*x**2 + 1), x) + Integral(b**2*x**3 
*asin(c*x)**2/(c**4*x**4 - 2*c**2*x**2 + 1), x) + Integral(2*a*b*x**3*asin 
(c*x)/(c**4*x**4 - 2*c**2*x**2 + 1), x))/d**2
 

3.2.93.7 Maxima [F]

\[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{3}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]

input

integrate(x^3*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^2,x, algorithm="maxima")
 

output

-1/2*a^2*(1/(c^6*d^2*x^2 - c^4*d^2) - log(c^2*x^2 - 1)/(c^4*d^2)) - 1/2*(b 
^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 - (b^2*c^2*x^2 - b^2)*arct 
an2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2*log(c*x + 1) - (b^2*c^2*x^2 - b^2 
)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2*log(-c*x + 1) - 2*(c^6*d^2* 
x^2 - c^4*d^2)*integrate((2*a*b*c^3*x^3*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c 
*x + 1)) - (b^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) - (b^2*c^2*x^2 
- b^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(c*x + 1) - (b^2*c^2* 
x^2 - 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^7*d^2*x^4 - 2*c^5*d^2*x^2 + c^3*d^2), x))/(c^6 
*d^2*x^2 - c^4*d^2)
 

3.2.93.8 Giac [F]

\[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{3}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]

input

integrate(x^3*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^2,x, algorithm="giac")
 

output

integrate((b*arcsin(c*x) + a)^2*x^3/(c^2*d*x^2 - d)^2, x)
 

3.2.93.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx=\int \frac {x^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]

input

int((x^3*(a + b*asin(c*x))^2)/(d - c^2*d*x^2)^2,x)
 

output

int((x^3*(a + b*asin(c*x))^2)/(d - c^2*d*x^2)^2, x)