3.3.71 \(\int \frac {\arcsin (a x)^2}{x^3 \sqrt {1-a^2 x^2}} \, dx\) [271]

3.3.71.1 Optimal result

Integrand size = 24, antiderivative size = 163 \[ \int \frac {\arcsin (a x)^2}{x^3 \sqrt {1-a^2 x^2}} \, dx=-\frac {a \arcsin (a x)}{x}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 x^2}-a^2 \arcsin (a x)^2 \text {arctanh}\left (e^{i \arcsin (a x)}\right )-a^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )+i a^2 \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-i a^2 \arcsin (a x) \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-a^2 \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+a^2 \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right ) \]

output

-a*arcsin(a*x)/x-a^2*arcsin(a*x)^2*arctanh(I*a*x+(-a^2*x^2+1)^(1/2))-a^2*a 
rctanh((-a^2*x^2+1)^(1/2))+I*a^2*arcsin(a*x)*polylog(2,-I*a*x-(-a^2*x^2+1) 
^(1/2))-I*a^2*arcsin(a*x)*polylog(2,I*a*x+(-a^2*x^2+1)^(1/2))-a^2*polylog( 
3,-I*a*x-(-a^2*x^2+1)^(1/2))+a^2*polylog(3,I*a*x+(-a^2*x^2+1)^(1/2))-1/2*a 
rcsin(a*x)^2*(-a^2*x^2+1)^(1/2)/x^2
 

3.3.71.2 Mathematica [A] (verified)

Time = 1.16 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.19 \[ \int \frac {\arcsin (a x)^2}{x^3 \sqrt {1-a^2 x^2}} \, dx=\frac {1}{8} a^2 \left (-4 \arcsin (a x) \cot \left (\frac {1}{2} \arcsin (a x)\right )-\arcsin (a x)^2 \csc ^2\left (\frac {1}{2} \arcsin (a x)\right )+4 \arcsin (a x)^2 \left (\log \left (1-e^{i \arcsin (a x)}\right )-\log \left (1+e^{i \arcsin (a x)}\right )\right )+8 \log \left (\tan \left (\frac {1}{2} \arcsin (a x)\right )\right )+8 i \arcsin (a x) \left (\operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-\operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )\right )+8 \left (-\operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+\operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )\right )+\arcsin (a x)^2 \sec ^2\left (\frac {1}{2} \arcsin (a x)\right )-4 \arcsin (a x) \tan \left (\frac {1}{2} \arcsin (a x)\right )\right ) \]

input

Integrate[ArcSin[a*x]^2/(x^3*Sqrt[1 - a^2*x^2]),x]
 

output

(a^2*(-4*ArcSin[a*x]*Cot[ArcSin[a*x]/2] - ArcSin[a*x]^2*Csc[ArcSin[a*x]/2] 
^2 + 4*ArcSin[a*x]^2*(Log[1 - E^(I*ArcSin[a*x])] - Log[1 + E^(I*ArcSin[a*x 
])]) + 8*Log[Tan[ArcSin[a*x]/2]] + (8*I)*ArcSin[a*x]*(PolyLog[2, -E^(I*Arc 
Sin[a*x])] - PolyLog[2, E^(I*ArcSin[a*x])]) + 8*(-PolyLog[3, -E^(I*ArcSin[ 
a*x])] + PolyLog[3, E^(I*ArcSin[a*x])]) + ArcSin[a*x]^2*Sec[ArcSin[a*x]/2] 
^2 - 4*ArcSin[a*x]*Tan[ArcSin[a*x]/2]))/8
 

3.3.71.3 Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {5204, 5138, 243, 73, 221, 5218, 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.

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

\(\Big \downarrow \) 5204

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 5218

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} a^2 \int \arcsin (a x)^2 \csc (\arcsin (a x))d\arcsin (a x)+a \left (-a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\arcsin (a x)}{x}\right )-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 x^2}\)

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

\(\displaystyle \frac {1}{2} a^2 \left (-2 \arcsin (a x)^2 \text {arctanh}\left (e^{i \arcsin (a x)}\right )+2 \left (i \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-\operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )\right )-2 \left (i \arcsin (a x) \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-\operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )\right )\right )+a \left (-a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\arcsin (a x)}{x}\right )-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 x^2}\)

input

Int[ArcSin[a*x]^2/(x^3*Sqrt[1 - a^2*x^2]),x]
 

output

-1/2*(Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/x^2 + a*(-(ArcSin[a*x]/x) - a*ArcTa 
nh[Sqrt[1 - a^2*x^2]]) + (a^2*(-2*ArcSin[a*x]^2*ArcTanh[E^(I*ArcSin[a*x])] 
 + 2*(I*ArcSin[a*x]*PolyLog[2, -E^(I*ArcSin[a*x])] - PolyLog[3, -E^(I*ArcS 
in[a*x])]) - 2*(I*ArcSin[a*x]*PolyLog[2, E^(I*ArcSin[a*x])] - PolyLog[3, E 
^(I*ArcSin[a*x])])))/2
 

3.3.71.3.1 Defintions of rubi rules used

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

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 5138

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

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]
 

3.3.71.4 Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.56

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \arcsin \left (a x \right ) \left (a^{2} x^{2} \arcsin \left (a x \right )-2 a x \sqrt {-a^{2} x^{2}+1}-\arcsin \left (a x \right )\right )}{2 \left (a^{2} x^{2}-1\right ) x^{2}}-\frac {a^{2} \left (\arcsin \left (a x \right )^{2} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )-\arcsin \left (a x \right )^{2} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-2 i \arcsin \left (a x \right ) \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+2 i \arcsin \left (a x \right ) \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 \,\operatorname {arctanh}\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2}\)	\(254\)

input

int(arcsin(a*x)^2/x^3/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
 

output

-1/2*(-a^2*x^2+1)^(1/2)/(a^2*x^2-1)/x^2*arcsin(a*x)*(a^2*x^2*arcsin(a*x)-2 
*a*x*(-a^2*x^2+1)^(1/2)-arcsin(a*x))-1/2*a^2*(arcsin(a*x)^2*ln(1+I*a*x+(-a 
^2*x^2+1)^(1/2))-arcsin(a*x)^2*ln(1-I*a*x-(-a^2*x^2+1)^(1/2))-2*I*arcsin(a 
*x)*polylog(2,-I*a*x-(-a^2*x^2+1)^(1/2))+2*I*arcsin(a*x)*polylog(2,I*a*x+( 
-a^2*x^2+1)^(1/2))+4*arctanh(I*a*x+(-a^2*x^2+1)^(1/2))+2*polylog(3,-I*a*x- 
(-a^2*x^2+1)^(1/2))-2*polylog(3,I*a*x+(-a^2*x^2+1)^(1/2)))
 

3.3.71.5 Fricas [F]

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

input

integrate(arcsin(a*x)^2/x^3/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
 

output

integral(-sqrt(-a^2*x^2 + 1)*arcsin(a*x)^2/(a^2*x^5 - x^3), x)
 

3.3.71.6 Sympy [F]

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

input

integrate(asin(a*x)**2/x**3/(-a**2*x**2+1)**(1/2),x)
 

output

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

3.3.71.7 Maxima [F]

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

input

integrate(arcsin(a*x)^2/x^3/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
 

output

integrate(arcsin(a*x)^2/(sqrt(-a^2*x^2 + 1)*x^3), x)
 

3.3.71.8 Giac [F]

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

input

integrate(arcsin(a*x)^2/x^3/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
 

output

integrate(arcsin(a*x)^2/(sqrt(-a^2*x^2 + 1)*x^3), x)
 

3.3.71.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\arcsin (a x)^2}{x^3 \sqrt {1-a^2 x^2}} \, dx=\int \frac {{\mathrm {asin}\left (a\,x\right )}^2}{x^3\,\sqrt {1-a^2\,x^2}} \,d x \]

input

int(asin(a*x)^2/(x^3*(1 - a^2*x^2)^(1/2)),x)
 

output

int(asin(a*x)^2/(x^3*(1 - a^2*x^2)^(1/2)), x)