\(\int \frac {(a+b \arcsin (c x))^3}{x^3} \, dx\) [27]

Optimal result

Integrand size = 14, antiderivative size = 126 \[ \int \frac {(a+b \arcsin (c x))^3}{x^3} \, dx=-\frac {3}{2} i b c^2 (a+b \arcsin (c x))^2-\frac {3 b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 x}-\frac {(a+b \arcsin (c x))^3}{2 x^2}+3 b^2 c^2 (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )-\frac {3}{2} i b^3 c^2 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.42 \[ \int \frac {(a+b \arcsin (c x))^3}{x^3} \, dx=-\frac {3 b^2 \left (a+b c x \left (i c x+\sqrt {1-c^2 x^2}\right )\right ) \arcsin (c x)^2+b^3 \arcsin (c x)^3+3 b \arcsin (c x) \left (a \left (a+2 b c x \sqrt {1-c^2 x^2}\right )-2 b^2 c^2 x^2 \log \left (1-e^{2 i \arcsin (c x)}\right )\right )+a \left (a \left (a+3 b c x \sqrt {1-c^2 x^2}\right )-6 b^2 c^2 x^2 \log (c x)\right )+3 i b^3 c^2 x^2 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 x^2} \] Input:

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

Output:

-1/2*(3*b^2*(a + b*c*x*(I*c*x + Sqrt[1 - c^2*x^2]))*ArcSin[c*x]^2 + b^3*Ar 
cSin[c*x]^3 + 3*b*ArcSin[c*x]*(a*(a + 2*b*c*x*Sqrt[1 - c^2*x^2]) - 2*b^2*c 
^2*x^2*Log[1 - E^((2*I)*ArcSin[c*x])]) + a*(a*(a + 3*b*c*x*Sqrt[1 - c^2*x^ 
2]) - 6*b^2*c^2*x^2*Log[c*x]) + (3*I)*b^3*c^2*x^2*PolyLog[2, E^((2*I)*ArcS 
in[c*x])])/x^2
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.02, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, 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.

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 5186

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

\(\Big \downarrow \) 5136

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4200

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

-1/2*(a + b*ArcSin[c*x])^3/x^2 + (3*b*c*(-((Sqrt[1 - c^2*x^2]*(a + b*ArcSi 
n[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))))/2
 

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]
 

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.10

method	result	size
derivativedivides	\(c^{2} \left (-\frac {a^{3}}{2 c^{2} x^{2}}+b^{3} \left (-\frac {\arcsin \left (c x \right )^{2} \left (-3 i c^{2} x^{2}+3 c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )\right )}{2 c^{2} x^{2}}+3 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+3 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 i \arcsin \left (c x \right )^{2}-3 i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )+3 a \,b^{2} \left (-\frac {\arcsin \left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{c x}+\ln \left (c x \right )\right )+3 a^{2} b \left (-\frac {\arcsin \left (c x \right )}{2 c^{2} x^{2}}-\frac {\sqrt {-c^{2} x^{2}+1}}{2 c x}\right )\right )\)	\(265\)
default	\(c^{2} \left (-\frac {a^{3}}{2 c^{2} x^{2}}+b^{3} \left (-\frac {\arcsin \left (c x \right )^{2} \left (-3 i c^{2} x^{2}+3 c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )\right )}{2 c^{2} x^{2}}+3 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+3 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 i \arcsin \left (c x \right )^{2}-3 i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )+3 a \,b^{2} \left (-\frac {\arcsin \left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{c x}+\ln \left (c x \right )\right )+3 a^{2} b \left (-\frac {\arcsin \left (c x \right )}{2 c^{2} x^{2}}-\frac {\sqrt {-c^{2} x^{2}+1}}{2 c x}\right )\right )\)	\(265\)
parts	\(-\frac {a^{3}}{2 x^{2}}+b^{3} c^{2} \left (-\frac {\arcsin \left (c x \right )^{2} \left (-3 i c^{2} x^{2}+3 c x \sqrt {-c^{2} x^{2}+1}+\arcsin \left (c x \right )\right )}{2 c^{2} x^{2}}+3 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+3 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 i \arcsin \left (c x \right )^{2}-3 i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )+3 a^{2} b \,c^{2} \left (-\frac {\arcsin \left (c x \right )}{2 c^{2} x^{2}}-\frac {\sqrt {-c^{2} x^{2}+1}}{2 c x}\right )+3 a \,b^{2} c^{2} \left (-\frac {\arcsin \left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{c x}+\ln \left (c x \right )\right )\)	\(267\)

Input:

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

Output:

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

Fricas [F]

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

integrate((a+b*arcsin(c*x))^3/x^3,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^3, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

3*(c^2*log(x) - sqrt(-c^2*x^2 + 1)*c*arcsin(c*x)/x)*a*b^2 - 3/2*a^2*b*(sqr 
t(-c^2*x^2 + 1)*c/x + arcsin(c*x)/x^2) - 3/2*a*b^2*arcsin(c*x)^2/x^2 - 1/2 
*(6*c*x^2*integrate(1/2*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x 
 + 1)*sqrt(-c*x + 1))^2/(c^2*x^4 - x^2), x) + arctan2(c*x, sqrt(c*x + 1)*s 
qrt(-c*x + 1))^3)*b^3/x^2 - 1/2*a^3/x^2
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {(a+b \arcsin (c x))^3}{x^3} \, dx=\frac {-3 \mathit {asin} \left (c x \right ) a^{2} b -3 \sqrt {-c^{2} x^{2}+1}\, a^{2} b c x +2 \left (\int \frac {\mathit {asin} \left (c x \right )^{3}}{x^{3}}d x \right ) b^{3} x^{2}+6 \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{x^{3}}d x \right ) a \,b^{2} x^{2}-a^{3}}{2 x^{2}} \] Input:

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

Output:

( - 3*asin(c*x)*a**2*b - 3*sqrt( - c**2*x**2 + 1)*a**2*b*c*x + 2*int(asin( 
c*x)**3/x**3,x)*b**3*x**2 + 6*int(asin(c*x)**2/x**3,x)*a*b**2*x**2 - a**3) 
/(2*x**2)