\(\int \frac {(a+b \arcsin (c x))^3}{x^2} \, dx\) [26]

Optimal result

Integrand size = 14, antiderivative size = 137 \[ \int \frac {(a+b \arcsin (c x))^3}{x^2} \, dx=-\frac {(a+b \arcsin (c x))^3}{x}-6 b c (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )+6 i b^2 c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-6 i b^2 c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-6 b^3 c \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )+6 b^3 c \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right ) \] Output:

-(a+b*arcsin(c*x))^3/x-6*b*c*(a+b*arcsin(c*x))^2*arctanh(I*c*x+(-c^2*x^2+1 
)^(1/2))+6*I*b^2*c*(a+b*arcsin(c*x))*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))- 
6*I*b^2*c*(a+b*arcsin(c*x))*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))-6*b^3*c*po 
lylog(3,-I*c*x-(-c^2*x^2+1)^(1/2))+6*b^3*c*polylog(3,I*c*x+(-c^2*x^2+1)^(1 
/2))
 

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(283\) vs. \(2(137)=274\).

Time = 0.21 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.07 \[ \int \frac {(a+b \arcsin (c x))^3}{x^2} \, dx=-\frac {a^3}{x}-\frac {3 a^2 b \arcsin (c x)}{x}+3 a^2 b c \log (x)-3 a^2 b c \log \left (1+\sqrt {1-c^2 x^2}\right )+3 a b^2 c \left (-\arcsin (c x) \left (\frac {\arcsin (c x)}{c x}-2 \log \left (1-e^{i \arcsin (c x)}\right )+2 \log \left (1+e^{i \arcsin (c x)}\right )\right )+2 i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-2 i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )+b^3 c \left (-\frac {\arcsin (c x)^3}{c x}+3 \arcsin (c x)^2 \log \left (1-e^{i \arcsin (c x)}\right )-3 \arcsin (c x)^2 \log \left (1+e^{i \arcsin (c x)}\right )+6 i \arcsin (c x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-6 i \arcsin (c x) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-6 \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )+6 \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )\right ) \] Input:

Integrate[(a + b*ArcSin[c*x])^3/x^2,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5138, 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.

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 5218

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

\(\Big \downarrow \) 3042

\(\displaystyle 3 b c \int (a+b \arcsin (c x))^2 \csc (\arcsin (c x))d\arcsin (c x)-\frac {(a+b \arcsin (c x))^3}{x}\)

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

Int[(a + b*ArcSin[c*x])^3/x^2,x]
 

Output:

-((a + b*ArcSin[c*x])^3/x) + 3*b*c*(-2*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I* 
ArcSin[c*x])] + 2*b*(I*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*ArcSin[c*x])] 
- b*PolyLog[3, -E^(I*ArcSin[c*x])]) - 2*b*(I*(a + b*ArcSin[c*x])*PolyLog[2 
, E^(I*ArcSin[c*x])] - b*PolyLog[3, E^(I*ArcSin[c*x])]))
 

Defintions of rubi rules used

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 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]
 

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.55

method	result	size
parts	\(-\frac {a^{3}}{x}+b^{3} c \left (-\frac {\arcsin \left (c x \right )^{3}}{c x}+3 \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )+3 a^{2} b c \left (-\frac {\arcsin \left (c x \right )}{c x}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )+3 a \,b^{2} c \left (-\frac {\arcsin \left (c x \right )^{2}}{c x}+2 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )\)	\(349\)
derivativedivides	\(c \left (-\frac {a^{3}}{c x}+b^{3} \left (-\frac {\arcsin \left (c x \right )^{3}}{c x}+3 \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )+3 a \,b^{2} \left (-\frac {\arcsin \left (c x \right )^{2}}{c x}+2 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )+3 a^{2} b \left (-\frac {\arcsin \left (c x \right )}{c x}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\)	\(351\)
default	\(c \left (-\frac {a^{3}}{c x}+b^{3} \left (-\frac {\arcsin \left (c x \right )^{3}}{c x}+3 \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )+3 a \,b^{2} \left (-\frac {\arcsin \left (c x \right )^{2}}{c x}+2 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )+3 a^{2} b \left (-\frac {\arcsin \left (c x \right )}{c x}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\)	\(351\)

Input:

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

Output:

-a^3/x+b^3*c*(-1/c/x*arcsin(c*x)^3+3*arcsin(c*x)^2*ln(1-I*c*x-(-c^2*x^2+1) 
^(1/2))-6*I*arcsin(c*x)*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))+6*polylog(3,I* 
c*x+(-c^2*x^2+1)^(1/2))-3*arcsin(c*x)^2*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+6*I 
*arcsin(c*x)*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-6*polylog(3,-I*c*x-(-c^2 
*x^2+1)^(1/2)))+3*a^2*b*c*(-1/c/x*arcsin(c*x)-arctanh(1/(-c^2*x^2+1)^(1/2) 
))+3*a*b^2*c*(-1/c/x*arcsin(c*x)^2+2*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^( 
1/2))-2*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+2*I*dilog(1+I*c*x+(-c^2 
*x^2+1)^(1/2))-2*I*dilog(1-I*c*x-(-c^2*x^2+1)^(1/2)))
 

Fricas [F]

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

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

Sympy [F]

\[ \int \frac {(a+b \arcsin (c x))^3}{x^2} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{3}}{x^{2}}\, dx \] Input:

integrate((a+b*asin(c*x))**3/x**2,x)
 

Output:

Integral((a + b*asin(c*x))**3/x**2, x)
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {(a+b \arcsin (c x))^3}{x^2} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((a+b*arcsin(c*x))^3/x^2,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. \[ \int \frac {(a+b \arcsin (c x))^3}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^3}{x^2} \,d x \] Input:

int((a + b*asin(c*x))^3/x^2,x)
 

Output:

int((a + b*asin(c*x))^3/x^2, x)
 

Reduce [F]

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

int((a+b*asin(c*x))^3/x^2,x)
 

Output:

( - 3*asin(c*x)*a**2*b + int(asin(c*x)**3/x**2,x)*b**3*x + 3*int(asin(c*x) 
**2/x**2,x)*a*b**2*x + 3*log(tan(asin(c*x)/2))*a**2*b*c*x - a**3)/x