3.3.92 \(\int \frac {\arcsin (a x)^3}{c-a^2 c x^2} \, dx\) [292]

3.3.92.1 Optimal result

Integrand size = 20, antiderivative size = 200 \[ \int \frac {\arcsin (a x)^3}{c-a^2 c x^2} \, dx=-\frac {2 i \arcsin (a x)^3 \arctan \left (e^{i \arcsin (a x)}\right )}{a c}+\frac {3 i \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arcsin (a x)}\right )}{a c}-\frac {3 i \arcsin (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arcsin (a x)}\right )}{a c}-\frac {6 \arcsin (a x) \operatorname {PolyLog}\left (3,-i e^{i \arcsin (a x)}\right )}{a c}+\frac {6 \arcsin (a x) \operatorname {PolyLog}\left (3,i e^{i \arcsin (a x)}\right )}{a c}-\frac {6 i \operatorname {PolyLog}\left (4,-i e^{i \arcsin (a x)}\right )}{a c}+\frac {6 i \operatorname {PolyLog}\left (4,i e^{i \arcsin (a x)}\right )}{a c} \]

output

-2*I*arcsin(a*x)^3*arctan(I*a*x+(-a^2*x^2+1)^(1/2))/a/c+3*I*arcsin(a*x)^2* 
polylog(2,-I*(I*a*x+(-a^2*x^2+1)^(1/2)))/a/c-3*I*arcsin(a*x)^2*polylog(2,I 
*(I*a*x+(-a^2*x^2+1)^(1/2)))/a/c-6*arcsin(a*x)*polylog(3,-I*(I*a*x+(-a^2*x 
^2+1)^(1/2)))/a/c+6*arcsin(a*x)*polylog(3,I*(I*a*x+(-a^2*x^2+1)^(1/2)))/a/ 
c-6*I*polylog(4,-I*(I*a*x+(-a^2*x^2+1)^(1/2)))/a/c+6*I*polylog(4,I*(I*a*x+ 
(-a^2*x^2+1)^(1/2)))/a/c
 

3.3.92.2 Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.81 \[ \int \frac {\arcsin (a x)^3}{c-a^2 c x^2} \, dx=-\frac {i \left (2 \arcsin (a x)^3 \arctan \left (e^{i \arcsin (a x)}\right )-3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arcsin (a x)}\right )+3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arcsin (a x)}\right )-6 i \arcsin (a x) \operatorname {PolyLog}\left (3,-i e^{i \arcsin (a x)}\right )+6 i \arcsin (a x) \operatorname {PolyLog}\left (3,i e^{i \arcsin (a x)}\right )+6 \operatorname {PolyLog}\left (4,-i e^{i \arcsin (a x)}\right )-6 \operatorname {PolyLog}\left (4,i e^{i \arcsin (a x)}\right )\right )}{a c} \]

input

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

output

((-I)*(2*ArcSin[a*x]^3*ArcTan[E^(I*ArcSin[a*x])] - 3*ArcSin[a*x]^2*PolyLog 
[2, (-I)*E^(I*ArcSin[a*x])] + 3*ArcSin[a*x]^2*PolyLog[2, I*E^(I*ArcSin[a*x 
])] - (6*I)*ArcSin[a*x]*PolyLog[3, (-I)*E^(I*ArcSin[a*x])] + (6*I)*ArcSin[ 
a*x]*PolyLog[3, I*E^(I*ArcSin[a*x])] + 6*PolyLog[4, (-I)*E^(I*ArcSin[a*x]) 
] - 6*PolyLog[4, I*E^(I*ArcSin[a*x])]))/(a*c)
 

3.3.92.3 Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.88, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {5164, 3042, 4669, 3011, 7163, 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.

\(\displaystyle \int \frac {\arcsin (a x)^3}{c-a^2 c x^2} \, dx\)

\(\Big \downarrow \) 5164

\(\displaystyle \frac {\int \frac {\arcsin (a x)^3}{\sqrt {1-a^2 x^2}}d\arcsin (a x)}{a c}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \arcsin (a x)^3 \csc \left (\arcsin (a x)+\frac {\pi }{2}\right )d\arcsin (a x)}{a c}\)

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

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

output

((-2*I)*ArcSin[a*x]^3*ArcTan[E^(I*ArcSin[a*x])] + 3*(I*ArcSin[a*x]^2*PolyL 
og[2, (-I)*E^(I*ArcSin[a*x])] - (2*I)*((-I)*ArcSin[a*x]*PolyLog[3, (-I)*E^ 
(I*ArcSin[a*x])] + PolyLog[4, (-I)*E^(I*ArcSin[a*x])])) - 3*(I*ArcSin[a*x] 
^2*PolyLog[2, I*E^(I*ArcSin[a*x])] - (2*I)*((-I)*ArcSin[a*x]*PolyLog[3, I* 
E^(I*ArcSin[a*x])] + PolyLog[4, I*E^(I*ArcSin[a*x])])))/(a*c)
 

3.3.92.3.1 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]
 

rule 7163

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

3.3.92.4 Maple [F]

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

input

int(arcsin(a*x)^3/(-a^2*c*x^2+c),x)
 

output

int(arcsin(a*x)^3/(-a^2*c*x^2+c),x)
 

3.3.92.5 Fricas [F]

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

input

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

output

integral(-arcsin(a*x)^3/(a^2*c*x^2 - c), x)
 

3.3.92.6 Sympy [F]

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

input

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

output

-Integral(asin(a*x)**3/(a**2*x**2 - 1), x)/c
 

3.3.92.7 Maxima [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.18 \[ \int \frac {\arcsin (a x)^3}{c-a^2 c x^2} \, dx=\frac {1}{2} \, {\left (\frac {\log \left (a x + 1\right )}{a c} - \frac {\log \left (a x - 1\right )}{a c}\right )} \arcsin \left (a x\right )^{3} \]

input

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

output

1/2*(log(a*x + 1)/(a*c) - log(a*x - 1)/(a*c))*arcsin(a*x)^3
 

3.3.92.8 Giac [F]

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

input

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

output

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

3.3.92.9 Mupad [F(-1)]

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

input

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

output

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