\(\int \frac {(a+b \arcsin (c x))^3}{x^5} \, dx\) [29]

Optimal result

Integrand size = 14, antiderivative size = 208 \[ \int \frac {(a+b \arcsin (c x))^3}{x^5} \, dx=-\frac {b^3 c^3 \sqrt {1-c^2 x^2}}{4 x}-\frac {b^2 c^2 (a+b \arcsin (c x))}{4 x^2}-\frac {1}{2} i b c^4 (a+b \arcsin (c x))^2-\frac {b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{4 x^3}-\frac {b c^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 x}-\frac {(a+b \arcsin (c x))^3}{4 x^4}+b^2 c^4 (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )-\frac {1}{2} i b^3 c^4 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.38 \[ \int \frac {(a+b \arcsin (c x))^3}{x^5} \, dx=-\frac {a^3+a b^2 c^2 x^2+a^2 b c x \sqrt {1-c^2 x^2}+2 a^2 b c^3 x^3 \sqrt {1-c^2 x^2}+b^3 c^3 x^3 \sqrt {1-c^2 x^2}+b^2 \left (3 a+b c x \left (2 i c^3 x^3+\sqrt {1-c^2 x^2}+2 c^2 x^2 \sqrt {1-c^2 x^2}\right )\right ) \arcsin (c x)^2+b^3 \arcsin (c x)^3+b \arcsin (c x) \left (3 a^2+b^2 c^2 x^2+2 a b c x \sqrt {1-c^2 x^2} \left (1+2 c^2 x^2\right )-4 b^2 c^4 x^4 \log \left (1-e^{2 i \arcsin (c x)}\right )\right )-4 a b^2 c^4 x^4 \log (c x)+2 i b^3 c^4 x^4 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{4 x^4} \] Input:

Integrate[(a + b*ArcSin[c*x])^3/x^5,x]
 

Output:

-1/4*(a^3 + a*b^2*c^2*x^2 + a^2*b*c*x*Sqrt[1 - c^2*x^2] + 2*a^2*b*c^3*x^3* 
Sqrt[1 - c^2*x^2] + b^3*c^3*x^3*Sqrt[1 - c^2*x^2] + b^2*(3*a + b*c*x*((2*I 
)*c^3*x^3 + Sqrt[1 - c^2*x^2] + 2*c^2*x^2*Sqrt[1 - c^2*x^2]))*ArcSin[c*x]^ 
2 + b^3*ArcSin[c*x]^3 + b*ArcSin[c*x]*(3*a^2 + b^2*c^2*x^2 + 2*a*b*c*x*Sqr 
t[1 - c^2*x^2]*(1 + 2*c^2*x^2) - 4*b^2*c^4*x^4*Log[1 - E^((2*I)*ArcSin[c*x 
])]) - 4*a*b^2*c^4*x^4*Log[c*x] + (2*I)*b^3*c^4*x^4*PolyLog[2, E^((2*I)*Ar 
cSin[c*x])])/x^4
 

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.02, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, 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.

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 5204

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 242

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

\(\Big \downarrow \) 5186

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

\(\Big \downarrow \) 5136

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4200

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

Int[(a + b*ArcSin[c*x])^3/x^5,x]
 

Output:

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

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]
 

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.87

method	result	size
derivativedivides	\(c^{4} \left (-\frac {a^{3}}{4 c^{4} x^{4}}+b^{3} \left (-\frac {-2 i \arcsin \left (c x \right )^{2} c^{4} x^{4}+2 \arcsin \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}\, c^{3} x^{3}-i c^{4} x^{4}+\arcsin \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}\, c x +c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )^{3}+c^{2} x^{2} \arcsin \left (c x \right )}{4 c^{4} x^{4}}+\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \arcsin \left (c x \right )^{2}-i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )+3 a \,b^{2} \left (-\frac {\arcsin \left (c x \right )^{2}}{4 c^{4} x^{4}}-\frac {\arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{6 c^{3} x^{3}}-\frac {1}{12 c^{2} x^{2}}-\frac {\arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{3 c x}+\frac {\ln \left (c x \right )}{3}\right )+3 a^{2} b \left (-\frac {\arcsin \left (c x \right )}{4 c^{4} x^{4}}-\frac {\sqrt {-c^{2} x^{2}+1}}{12 c^{3} x^{3}}-\frac {\sqrt {-c^{2} x^{2}+1}}{6 c x}\right )\right )\)	\(389\)
default	\(c^{4} \left (-\frac {a^{3}}{4 c^{4} x^{4}}+b^{3} \left (-\frac {-2 i \arcsin \left (c x \right )^{2} c^{4} x^{4}+2 \arcsin \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}\, c^{3} x^{3}-i c^{4} x^{4}+\arcsin \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}\, c x +c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )^{3}+c^{2} x^{2} \arcsin \left (c x \right )}{4 c^{4} x^{4}}+\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \arcsin \left (c x \right )^{2}-i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )+3 a \,b^{2} \left (-\frac {\arcsin \left (c x \right )^{2}}{4 c^{4} x^{4}}-\frac {\arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{6 c^{3} x^{3}}-\frac {1}{12 c^{2} x^{2}}-\frac {\arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{3 c x}+\frac {\ln \left (c x \right )}{3}\right )+3 a^{2} b \left (-\frac {\arcsin \left (c x \right )}{4 c^{4} x^{4}}-\frac {\sqrt {-c^{2} x^{2}+1}}{12 c^{3} x^{3}}-\frac {\sqrt {-c^{2} x^{2}+1}}{6 c x}\right )\right )\)	\(389\)
parts	\(-\frac {a^{3}}{4 x^{4}}+b^{3} c^{4} \left (-\frac {-2 i \arcsin \left (c x \right )^{2} c^{4} x^{4}+2 \arcsin \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}\, c^{3} x^{3}-i c^{4} x^{4}+\arcsin \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}\, c x +c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )^{3}+c^{2} x^{2} \arcsin \left (c x \right )}{4 c^{4} x^{4}}+\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \arcsin \left (c x \right )^{2}-i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )+3 a^{2} b \,c^{4} \left (-\frac {\arcsin \left (c x \right )}{4 c^{4} x^{4}}-\frac {\sqrt {-c^{2} x^{2}+1}}{12 c^{3} x^{3}}-\frac {\sqrt {-c^{2} x^{2}+1}}{6 c x}\right )+3 a \,b^{2} c^{4} \left (-\frac {\arcsin \left (c x \right )^{2}}{4 c^{4} x^{4}}-\frac {\arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{6 c^{3} x^{3}}-\frac {1}{12 c^{2} x^{2}}-\frac {\arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{3 c x}+\frac {\ln \left (c x \right )}{3}\right )\)	\(391\)

Input:

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

Output:

c^4*(-1/4*a^3/c^4/x^4+b^3*(-1/4*(-2*I*arcsin(c*x)^2*c^4*x^4+2*arcsin(c*x)^ 
2*(-c^2*x^2+1)^(1/2)*c^3*x^3-I*c^4*x^4+arcsin(c*x)^2*(-c^2*x^2+1)^(1/2)*c* 
x+c^3*x^3*(-c^2*x^2+1)^(1/2)+arcsin(c*x)^3+c^2*x^2*arcsin(c*x))/c^4/x^4+ar 
csin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+ 
1)^(1/2))-I*arcsin(c*x)^2-I*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))-I*polylog( 
2,-I*c*x-(-c^2*x^2+1)^(1/2)))+3*a*b^2*(-1/4/c^4/x^4*arcsin(c*x)^2-1/6*arcs 
in(c*x)*(-c^2*x^2+1)^(1/2)/c^3/x^3-1/12/c^2/x^2-1/3*arcsin(c*x)/c/x*(-c^2* 
x^2+1)^(1/2)+1/3*ln(c*x))+3*a^2*b*(-1/4/c^4/x^4*arcsin(c*x)-1/12/c^3/x^3*( 
-c^2*x^2+1)^(1/2)-1/6/c/x*(-c^2*x^2+1)^(1/2)))
 

Fricas [F]

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

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

Output:

integral((b^3*arcsin(c*x)^3 + 3*a*b^2*arcsin(c*x)^2 + 3*a^2*b*arcsin(c*x) 
+ a^3)/x^5, x)
 

Sympy [F]

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

integrate((a+b*asin(c*x))**3/x**5,x)
                                                                                    
                                                                                    
 

Output:

Integral((a + b*asin(c*x))**3/x**5, x)
 

Maxima [F]

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

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

Output:

-1/4*((2*sqrt(-c^2*x^2 + 1)*c^2/x + sqrt(-c^2*x^2 + 1)/x^3)*c + 3*arcsin(c 
*x)/x^4)*a^2*b + 1/4*((4*c^2*log(x) - 1/x^2)*c^2 - 2*(2*sqrt(-c^2*x^2 + 1) 
*c^2/x + sqrt(-c^2*x^2 + 1)/x^3)*c*arcsin(c*x))*a*b^2 - 3/4*a*b^2*arcsin(c 
*x)^2/x^4 - 1/4*(12*c*x^4*integrate(1/4*sqrt(c*x + 1)*sqrt(-c*x + 1)*arcta 
n2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2/(c^2*x^6 - x^4), x) + arctan2(c*x, 
 sqrt(c*x + 1)*sqrt(-c*x + 1))^3)*b^3/x^4 - 1/4*a^3/x^4
 

Giac [F(-2)]

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

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(a+b \arcsin (c x))^3}{x^5} \, dx=\frac {-3 \mathit {asin} \left (c x \right ) a^{2} b -2 \sqrt {-c^{2} x^{2}+1}\, a^{2} b \,c^{3} x^{3}-\sqrt {-c^{2} x^{2}+1}\, a^{2} b c x +4 \left (\int \frac {\mathit {asin} \left (c x \right )^{3}}{x^{5}}d x \right ) b^{3} x^{4}+12 \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{x^{5}}d x \right ) a \,b^{2} x^{4}-a^{3}}{4 x^{4}} \] Input:

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

Output:

( - 3*asin(c*x)*a**2*b - 2*sqrt( - c**2*x**2 + 1)*a**2*b*c**3*x**3 - sqrt( 
 - c**2*x**2 + 1)*a**2*b*c*x + 4*int(asin(c*x)**3/x**5,x)*b**3*x**4 + 12*i 
nt(asin(c*x)**2/x**5,x)*a*b**2*x**4 - a**3)/(4*x**4)