3.1.29 \(\int \frac {\arcsin (a x)^3}{x^3} \, dx\) [29]

3.1.29.1 Optimal result

Integrand size = 10, antiderivative size = 102 \[ \int \frac {\arcsin (a x)^3}{x^3} \, dx=-\frac {3}{2} i a^2 \arcsin (a x)^2-\frac {3 a \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 x}-\frac {\arcsin (a x)^3}{2 x^2}+3 a^2 \arcsin (a x) \log \left (1-e^{2 i \arcsin (a x)}\right )-\frac {3}{2} i a^2 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right ) \]

output

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

3.1.29.2 Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.90 \[ \int \frac {\arcsin (a x)^3}{x^3} \, dx=-\frac {\arcsin (a x) \left (3 a x \left (i a x+\sqrt {1-a^2 x^2}\right ) \arcsin (a x)+\arcsin (a x)^2-6 a^2 x^2 \log \left (1-e^{2 i \arcsin (a x)}\right )\right )}{2 x^2}-\frac {3}{2} i a^2 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right ) \]

input

Integrate[ArcSin[a*x]^3/x^3,x]
 

output

-1/2*(ArcSin[a*x]*(3*a*x*(I*a*x + Sqrt[1 - a^2*x^2])*ArcSin[a*x] + ArcSin[ 
a*x]^2 - 6*a^2*x^2*Log[1 - E^((2*I)*ArcSin[a*x])]))/x^2 - ((3*I)/2)*a^2*Po 
lyLog[2, E^((2*I)*ArcSin[a*x])]
 

3.1.29.3 Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5138, 5186, 5136, 3042, 25, 4200, 25, 2620, 2715, 2838}

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.

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 5186

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

\(\Big \downarrow \) 5136

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3}{2} a \left (2 a \int -\arcsin (a x) \tan \left (\arcsin (a x)+\frac {\pi }{2}\right )d\arcsin (a x)-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{x}\right )-\frac {\arcsin (a x)^3}{2 x^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3}{2} a \left (-2 a \int \arcsin (a x) \tan \left (\arcsin (a x)+\frac {\pi }{2}\right )d\arcsin (a x)-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{x}\right )-\frac {\arcsin (a x)^3}{2 x^2}\)

\(\Big \downarrow \) 4200

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

input

Int[ArcSin[a*x]^3/x^3,x]
 

output

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

3.1.29.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 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 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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4200

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))), x] 
, x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]
 

rule 5136

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[( 
a + b*x)^n*Cot[x], x], x, ArcSin[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 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 5186

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[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*A 
rcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^ 
2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
 

3.1.29.4 Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.58

method	result	size
derivativedivides	\(a^{2} \left (-\frac {\arcsin \left (a x \right )^{2} \left (-3 i a^{2} x^{2}+3 a x \sqrt {-a^{2} x^{2}+1}+\arcsin \left (a x \right )\right )}{2 a^{2} x^{2}}+3 \arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )+3 \arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )-3 i \arcsin \left (a x \right )^{2}-3 i \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )-3 i \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )\right )\)	\(161\)
default	\(a^{2} \left (-\frac {\arcsin \left (a x \right )^{2} \left (-3 i a^{2} x^{2}+3 a x \sqrt {-a^{2} x^{2}+1}+\arcsin \left (a x \right )\right )}{2 a^{2} x^{2}}+3 \arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )+3 \arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )-3 i \arcsin \left (a x \right )^{2}-3 i \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )-3 i \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )\right )\)	\(161\)

input

int(arcsin(a*x)^3/x^3,x,method=_RETURNVERBOSE)
 

output

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

3.1.29.5 Fricas [F]

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

input

integrate(arcsin(a*x)^3/x^3,x, algorithm="fricas")
 

output

integral(arcsin(a*x)^3/x^3, x)
 

3.1.29.6 Sympy [F]

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

input

integrate(asin(a*x)**3/x**3,x)
 

output

Integral(asin(a*x)**3/x**3, x)
 

3.1.29.7 Maxima [F]

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

input

integrate(arcsin(a*x)^3/x^3,x, algorithm="maxima")
 

output

-1/2*(6*a*x^2*integrate(1/2*sqrt(a*x + 1)*sqrt(-a*x + 1)*arctan2(a*x, sqrt 
(a*x + 1)*sqrt(-a*x + 1))^2/(a^2*x^4 - x^2), x) + arctan2(a*x, sqrt(a*x + 
1)*sqrt(-a*x + 1))^3)/x^2
 

3.1.29.8 Giac [F]

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

input

integrate(arcsin(a*x)^3/x^3,x, algorithm="giac")
 

output

integrate(arcsin(a*x)^3/x^3, x)
 

3.1.29.9 Mupad [F(-1)]

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

input

int(asin(a*x)^3/x^3,x)
 

output

int(asin(a*x)^3/x^3, x)