\(\int \frac {(a+b \arcsin (c x))^3}{x^4} \, dx\) [28]

Optimal result

Integrand size = 14, antiderivative size = 223 \[ \int \frac {(a+b \arcsin (c x))^3}{x^4} \, dx=-\frac {b^2 c^2 (a+b \arcsin (c x))}{x}-\frac {b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 x^2}-\frac {(a+b \arcsin (c x))^3}{3 x^3}-b c^3 (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )-b^3 c^3 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )+i b^2 c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i b^2 c^3 (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-b^3 c^3 \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )+b^3 c^3 \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right ) \] Output:

-b^2*c^2*(a+b*arcsin(c*x))/x-1/2*b*c*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^ 
2/x^2-1/3*(a+b*arcsin(c*x))^3/x^3-b*c^3*(a+b*arcsin(c*x))^2*arctanh(I*c*x+ 
(-c^2*x^2+1)^(1/2))-b^3*c^3*arctanh((-c^2*x^2+1)^(1/2))+I*b^2*c^3*(a+b*arc 
sin(c*x))*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-I*b^2*c^3*(a+b*arcsin(c*x)) 
*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))-b^3*c^3*polylog(3,-I*c*x-(-c^2*x^2+1) 
^(1/2))+b^3*c^3*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. \(567\) vs. \(2(223)=446\).

Time = 7.18 (sec) , antiderivative size = 567, normalized size of antiderivative = 2.54 \[ \int \frac {(a+b \arcsin (c x))^3}{x^4} \, dx=-\frac {a^3}{3 x^3}-\frac {a^2 b c \sqrt {1-c^2 x^2}}{2 x^2}-\frac {a^2 b \arcsin (c x)}{x^3}+\frac {1}{2} a^2 b c^3 \log (x)-\frac {1}{2} a^2 b c^3 \log \left (1+\sqrt {1-c^2 x^2}\right )+\frac {1}{8} a b^2 c^3 \left (8 i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-\frac {2 \left (2+4 \arcsin (c x)^2-2 \cos (2 \arcsin (c x))-3 c x \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )+3 c x \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )+4 i c^3 x^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )+2 \arcsin (c x) \sin (2 \arcsin (c x))+\arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right ) \sin (3 \arcsin (c x))-\arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right ) \sin (3 \arcsin (c x))\right )}{c^3 x^3}\right )+\frac {1}{48} b^3 c^3 \left (-24 \arcsin (c x) \cot \left (\frac {1}{2} \arcsin (c x)\right )-4 \arcsin (c x)^3 \cot \left (\frac {1}{2} \arcsin (c x)\right )-6 \arcsin (c x)^2 \csc ^2\left (\frac {1}{2} \arcsin (c x)\right )-c x \arcsin (c x)^3 \csc ^4\left (\frac {1}{2} \arcsin (c x)\right )+24 \arcsin (c x)^2 \log \left (1-e^{i \arcsin (c x)}\right )-24 \arcsin (c x)^2 \log \left (1+e^{i \arcsin (c x)}\right )+48 \log \left (\tan \left (\frac {1}{2} \arcsin (c x)\right )\right )+48 i \arcsin (c x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-48 i \arcsin (c x) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-48 \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )+48 \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )+6 \arcsin (c x)^2 \sec ^2\left (\frac {1}{2} \arcsin (c x)\right )-\frac {16 \arcsin (c x)^3 \sin ^4\left (\frac {1}{2} \arcsin (c x)\right )}{c^3 x^3}-24 \arcsin (c x) \tan \left (\frac {1}{2} \arcsin (c x)\right )-4 \arcsin (c x)^3 \tan \left (\frac {1}{2} \arcsin (c x)\right )\right ) \] Input:

Integrate[(a + b*ArcSin[c*x])^3/x^4,x]
 

Output:

-1/3*a^3/x^3 - (a^2*b*c*Sqrt[1 - c^2*x^2])/(2*x^2) - (a^2*b*ArcSin[c*x])/x 
^3 + (a^2*b*c^3*Log[x])/2 - (a^2*b*c^3*Log[1 + Sqrt[1 - c^2*x^2]])/2 + (a* 
b^2*c^3*((8*I)*PolyLog[2, -E^(I*ArcSin[c*x])] - (2*(2 + 4*ArcSin[c*x]^2 - 
2*Cos[2*ArcSin[c*x]] - 3*c*x*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 3*c* 
x*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + (4*I)*c^3*x^3*PolyLog[2, E^(I*A 
rcSin[c*x])] + 2*ArcSin[c*x]*Sin[2*ArcSin[c*x]] + ArcSin[c*x]*Log[1 - E^(I 
*ArcSin[c*x])]*Sin[3*ArcSin[c*x]] - ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] 
*Sin[3*ArcSin[c*x]]))/(c^3*x^3)))/8 + (b^3*c^3*(-24*ArcSin[c*x]*Cot[ArcSin 
[c*x]/2] - 4*ArcSin[c*x]^3*Cot[ArcSin[c*x]/2] - 6*ArcSin[c*x]^2*Csc[ArcSin 
[c*x]/2]^2 - c*x*ArcSin[c*x]^3*Csc[ArcSin[c*x]/2]^4 + 24*ArcSin[c*x]^2*Log 
[1 - E^(I*ArcSin[c*x])] - 24*ArcSin[c*x]^2*Log[1 + E^(I*ArcSin[c*x])] + 48 
*Log[Tan[ArcSin[c*x]/2]] + (48*I)*ArcSin[c*x]*PolyLog[2, -E^(I*ArcSin[c*x] 
)] - (48*I)*ArcSin[c*x]*PolyLog[2, E^(I*ArcSin[c*x])] - 48*PolyLog[3, -E^( 
I*ArcSin[c*x])] + 48*PolyLog[3, E^(I*ArcSin[c*x])] + 6*ArcSin[c*x]^2*Sec[A 
rcSin[c*x]/2]^2 - (16*ArcSin[c*x]^3*Sin[ArcSin[c*x]/2]^4)/(c^3*x^3) - 24*A 
rcSin[c*x]*Tan[ArcSin[c*x]/2] - 4*ArcSin[c*x]^3*Tan[ArcSin[c*x]/2]))/48
 

Rubi [A] (verified)

Time = 1.53 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.94, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {5138, 5204, 5138, 243, 73, 221, 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.

\(\displaystyle \int \frac {(a+b \arcsin (c x))^3}{x^4} \, dx\)

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 5204

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 5218

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

Int[(a + b*ArcSin[c*x])^3/x^4,x]
 

Output:

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

Defintions of rubi rules used

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

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 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]
 

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 = 0.30 (sec) , antiderivative size = 454, normalized size of antiderivative = 2.04

method	result	size
derivativedivides	\(c^{3} \left (-\frac {a^{3}}{3 c^{3} x^{3}}+b^{3} \left (-\frac {\arcsin \left (c x \right ) \left (3 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x +2 \arcsin \left (c x \right )^{2}+6 c^{2} x^{2}\right )}{6 c^{3} x^{3}}+\frac {\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{2}-i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{2}+i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \,\operatorname {arctanh}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+3 a \,b^{2} \left (-\frac {\arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x +\arcsin \left (c x \right )^{2}+c^{2} x^{2}}{3 c^{3} x^{3}}+\frac {\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}\right )+3 a^{2} b \left (-\frac {\arcsin \left (c x \right )}{3 c^{3} x^{3}}-\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}\right )\right )\)	\(454\)
default	\(c^{3} \left (-\frac {a^{3}}{3 c^{3} x^{3}}+b^{3} \left (-\frac {\arcsin \left (c x \right ) \left (3 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x +2 \arcsin \left (c x \right )^{2}+6 c^{2} x^{2}\right )}{6 c^{3} x^{3}}+\frac {\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{2}-i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{2}+i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \,\operatorname {arctanh}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+3 a \,b^{2} \left (-\frac {\arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x +\arcsin \left (c x \right )^{2}+c^{2} x^{2}}{3 c^{3} x^{3}}+\frac {\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}\right )+3 a^{2} b \left (-\frac {\arcsin \left (c x \right )}{3 c^{3} x^{3}}-\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}\right )\right )\)	\(454\)
parts	\(-\frac {a^{3}}{3 x^{3}}+b^{3} c^{3} \left (-\frac {\arcsin \left (c x \right ) \left (3 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x +2 \arcsin \left (c x \right )^{2}+6 c^{2} x^{2}\right )}{6 c^{3} x^{3}}+\frac {\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{2}-i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{2}+i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 \,\operatorname {arctanh}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+3 a^{2} b \,c^{3} \left (-\frac {\arcsin \left (c x \right )}{3 c^{3} x^{3}}-\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}\right )+3 a \,b^{2} c^{3} \left (-\frac {\arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x +\arcsin \left (c x \right )^{2}+c^{2} x^{2}}{3 c^{3} x^{3}}+\frac {\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}\right )\)	\(456\)

Input:

int((a+b*arcsin(c*x))^3/x^4,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

\[ \int \frac {(a+b \arcsin (c x))^3}{x^4} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{3}}{x^{4}} \,d x } \] Input:

integrate((a+b*arcsin(c*x))^3/x^4,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^4, x)
 

Sympy [F]

\[ \int \frac {(a+b \arcsin (c x))^3}{x^4} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{3}}{x^{4}}\, dx \] Input:

integrate((a+b*asin(c*x))**3/x**4,x)
                                                                                    
                                                                                    
 

Output:

Integral((a + b*asin(c*x))**3/x**4, x)
 

Maxima [F]

\[ \int \frac {(a+b \arcsin (c x))^3}{x^4} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{3}}{x^{4}} \,d x } \] Input:

integrate((a+b*arcsin(c*x))^3/x^4,x, algorithm="maxima")
 

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {(a+b \arcsin (c x))^3}{x^4} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((a+b*arcsin(c*x))^3/x^4,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. \[ \int \frac {(a+b \arcsin (c x))^3}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^3}{x^4} \,d x \] Input:

int((a + b*asin(c*x))^3/x^4,x)
 

Output:

int((a + b*asin(c*x))^3/x^4, x)
 

Reduce [F]

\[ \int \frac {(a+b \arcsin (c x))^3}{x^4} \, dx=\frac {-6 \mathit {asin} \left (c x \right ) a^{2} b -3 \sqrt {-c^{2} x^{2}+1}\, a^{2} b c x +6 \left (\int \frac {\mathit {asin} \left (c x \right )^{3}}{x^{4}}d x \right ) b^{3} x^{3}+18 \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{x^{4}}d x \right ) a \,b^{2} x^{3}+3 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right )\right ) a^{2} b \,c^{3} x^{3}-2 a^{3}}{6 x^{3}} \] Input:

int((a+b*asin(c*x))^3/x^4,x)
 

Output:

( - 6*asin(c*x)*a**2*b - 3*sqrt( - c**2*x**2 + 1)*a**2*b*c*x + 6*int(asin( 
c*x)**3/x**4,x)*b**3*x**3 + 18*int(asin(c*x)**2/x**4,x)*a*b**2*x**3 + 3*lo 
g(tan(asin(c*x)/2))*a**2*b*c**3*x**3 - 2*a**3)/(6*x**3)