3.1.31 \(\int \frac {\arcsin (a x)^3}{x^5} \, dx\) [31]

3.1.31.1 Optimal result

Integrand size = 10, antiderivative size = 169 \[ \int \frac {\arcsin (a x)^3}{x^5} \, dx=-\frac {a^3 \sqrt {1-a^2 x^2}}{4 x}-\frac {a^2 \arcsin (a x)}{4 x^2}-\frac {1}{2} i a^4 \arcsin (a x)^2-\frac {a \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 x^3}-\frac {a^3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 x}-\frac {\arcsin (a x)^3}{4 x^4}+a^4 \arcsin (a x) \log \left (1-e^{2 i \arcsin (a x)}\right )-\frac {1}{2} i a^4 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right ) \]

output

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

3.1.31.2 Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.69 \[ \int \frac {\arcsin (a x)^3}{x^5} \, dx=\frac {1}{4} \left (-\frac {\arcsin (a x)^3}{x^4}+a^4 \left (-\frac {\sqrt {1-a^2 x^2} \left (1+\left (2+\frac {1}{a^2 x^2}\right ) \arcsin (a x)^2\right )}{a x}-\arcsin (a x) \left (\frac {1}{a^2 x^2}+2 i \arcsin (a x)-4 \log \left (1-e^{2 i \arcsin (a x)}\right )\right )-2 i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )\right )\right ) \]

input

Integrate[ArcSin[a*x]^3/x^5,x]
 

output

(-(ArcSin[a*x]^3/x^4) + a^4*(-((Sqrt[1 - a^2*x^2]*(1 + (2 + 1/(a^2*x^2))*A 
rcSin[a*x]^2))/(a*x)) - ArcSin[a*x]*(1/(a^2*x^2) + (2*I)*ArcSin[a*x] - 4*L 
og[1 - E^((2*I)*ArcSin[a*x])]) - (2*I)*PolyLog[2, E^((2*I)*ArcSin[a*x])])) 
/4
 

3.1.31.3 Rubi [A] (verified)

Time = 0.93 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.07, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.300, Rules used = {5138, 5204, 5138, 242, 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.

\(\displaystyle \int \frac {\arcsin (a x)^3}{x^5} \, dx\)

\(\Big \downarrow \) 5138

\(\displaystyle \frac {3}{4} a \int \frac {\arcsin (a x)^2}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {\arcsin (a x)^3}{4 x^4}\)

\(\Big \downarrow \) 5204

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 242

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

\(\Big \downarrow \) 5186

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

\(\Big \downarrow \) 5136

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3}{4} a \left (\frac {2}{3} a^2 \left (2 a \int -\arcsin (a x) \tan \left (\arcsin (a x)+\frac {\pi }{2}\right )d\arcsin (a x)-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{x}\right )+\frac {2}{3} a \left (-\frac {a \sqrt {1-a^2 x^2}}{2 x}-\frac {\arcsin (a x)}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{3 x^3}\right )-\frac {\arcsin (a x)^3}{4 x^4}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3}{4} a \left (\frac {2}{3} a^2 \left (-2 a \int \arcsin (a x) \tan \left (\arcsin (a x)+\frac {\pi }{2}\right )d\arcsin (a x)-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{x}\right )+\frac {2}{3} a \left (-\frac {a \sqrt {1-a^2 x^2}}{2 x}-\frac {\arcsin (a x)}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{3 x^3}\right )-\frac {\arcsin (a x)^3}{4 x^4}\)

\(\Big \downarrow \) 4200

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

input

Int[ArcSin[a*x]^3/x^5,x]
 

output

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

3.1.31.3.1 Defintions of rubi rules used

rule 25

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

rule 242

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x 
] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
 

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]
 

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]
 

3.1.31.4 Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.37

method	result	size
derivativedivides	\(a^{4} \left (-\frac {-2 i \arcsin \left (a x \right )^{2} a^{4} x^{4}+2 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-i a^{4} x^{4}+\arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a x +a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}+\arcsin \left (a x \right )^{3}+a^{2} x^{2} \arcsin \left (a x \right )}{4 a^{4} x^{4}}+\arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )+\arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )-i \arcsin \left (a x \right )^{2}-i \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )\right )\)	\(231\)
default	\(a^{4} \left (-\frac {-2 i \arcsin \left (a x \right )^{2} a^{4} x^{4}+2 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-i a^{4} x^{4}+\arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a x +a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}+\arcsin \left (a x \right )^{3}+a^{2} x^{2} \arcsin \left (a x \right )}{4 a^{4} x^{4}}+\arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )+\arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )-i \arcsin \left (a x \right )^{2}-i \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )\right )\)	\(231\)

input

int(arcsin(a*x)^3/x^5,x,method=_RETURNVERBOSE)
 

output

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

3.1.31.5 Fricas [F]

\[ \int \frac {\arcsin (a x)^3}{x^5} \, dx=\int { \frac {\arcsin \left (a x\right )^{3}}{x^{5}} \,d x } \]

input

integrate(arcsin(a*x)^3/x^5,x, algorithm="fricas")
 

output

integral(arcsin(a*x)^3/x^5, x)
 

3.1.31.6 Sympy [F]

\[ \int \frac {\arcsin (a x)^3}{x^5} \, dx=\int \frac {\operatorname {asin}^{3}{\left (a x \right )}}{x^{5}}\, dx \]

input

integrate(asin(a*x)**3/x**5,x)
 

output

Integral(asin(a*x)**3/x**5, x)
 

3.1.31.7 Maxima [F]

\[ \int \frac {\arcsin (a x)^3}{x^5} \, dx=\int { \frac {\arcsin \left (a x\right )^{3}}{x^{5}} \,d x } \]

input

integrate(arcsin(a*x)^3/x^5,x, algorithm="maxima")
 

output

-1/4*(12*a*x^4*integrate(1/4*sqrt(a*x + 1)*sqrt(-a*x + 1)*arctan2(a*x, sqr 
t(a*x + 1)*sqrt(-a*x + 1))^2/(a^2*x^6 - x^4), x) + arctan2(a*x, sqrt(a*x + 
 1)*sqrt(-a*x + 1))^3)/x^4
 

3.1.31.8 Giac [F(-2)]

Exception generated. \[ \int \frac {\arcsin (a x)^3}{x^5} \, dx=\text {Exception raised: TypeError} \]

input

integrate(arcsin(a*x)^3/x^5,x, algorithm="giac")
 

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

3.1.31.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\arcsin (a x)^3}{x^5} \, dx=\int \frac {{\mathrm {asin}\left (a\,x\right )}^3}{x^5} \,d x \]

input

int(asin(a*x)^3/x^5,x)
 

output

int(asin(a*x)^3/x^5, x)