\(\int \frac {\arcsin (a x)^3}{x^4} \, dx\) [30]

Optimal result

Integrand size = 10, antiderivative size = 179 \[ \int \frac {\arcsin (a x)^3}{x^4} \, dx=-\frac {a^2 \arcsin (a x)}{x}-\frac {a \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 x^2}-\frac {\arcsin (a x)^3}{3 x^3}-a^3 \arcsin (a x)^2 \text {arctanh}\left (e^{i \arcsin (a x)}\right )-a^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )+i a^3 \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-i a^3 \arcsin (a x) \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-a^3 \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+a^3 \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4723, 4789, 4803, 4268, 2611, 2320, 6724, 272, 65, 214} \[ \int \frac {\arcsin (a x)^3}{x^4} \, dx=a^3 \left (-\arcsin (a x)^2\right ) \text {arctanh}\left (e^{i \arcsin (a x)}\right )+i a^3 \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-i a^3 \arcsin (a x) \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-a^3 \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+a^3 \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )-\frac {a \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 x^2}-\frac {a^2 \arcsin (a x)}{x}-a^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\arcsin (a x)^3}{3 x^3} \]

[In]

Int[ArcSin[a*x]^3/x^4,x]

[Out]

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{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 2611

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] + Dist[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 4268

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] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[d*(m/f), Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]

Rule 4723

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSi
n[c*x])^n/(d*(m + 1))), x] - Dist[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 4789

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] + (Dist[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] - Dist[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]) /; Free
Q[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]

Rule 4803

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[(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] /; Free
Q[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> 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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\arcsin (a x)^3}{3 x^3}+a \int \frac {\arcsin (a x)^2}{x^3 \sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {a \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 x^2}-\frac {\arcsin (a x)^3}{3 x^3}+a^2 \int \frac {\arcsin (a x)}{x^2} \, dx+\frac {1}{2} a^3 \int \frac {\arcsin (a x)^2}{x \sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {a^2 \arcsin (a x)}{x}-\frac {a \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 x^2}-\frac {\arcsin (a x)^3}{3 x^3}+\frac {1}{2} a^3 \text {Subst}\left (\int x^2 \csc (x) \, dx,x,\arcsin (a x)\right )+a^3 \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {a^2 \arcsin (a x)}{x}-\frac {a \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 x^2}-\frac {\arcsin (a x)^3}{3 x^3}-a^3 \arcsin (a x)^2 \text {arctanh}\left (e^{i \arcsin (a x)}\right )+\frac {1}{2} a^3 \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )-a^3 \text {Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (a x)\right )+a^3 \text {Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (a x)\right ) \\ & = -\frac {a^2 \arcsin (a x)}{x}-\frac {a \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 x^2}-\frac {\arcsin (a x)^3}{3 x^3}-a^3 \arcsin (a x)^2 \text {arctanh}\left (e^{i \arcsin (a x)}\right )+i a^3 \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-i a^3 \arcsin (a x) \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-a \text {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )-\left (i a^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\arcsin (a x)\right )+\left (i a^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx,x,\arcsin (a x)\right ) \\ & = -\frac {a^2 \arcsin (a x)}{x}-\frac {a \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 x^2}-\frac {\arcsin (a x)^3}{3 x^3}-a^3 \arcsin (a x)^2 \text {arctanh}\left (e^{i \arcsin (a x)}\right )-a^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )+i a^3 \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-i a^3 \arcsin (a x) \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-a^3 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{i \arcsin (a x)}\right )+a^3 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{i \arcsin (a x)}\right ) \\ & = -\frac {a^2 \arcsin (a x)}{x}-\frac {a \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 x^2}-\frac {\arcsin (a x)^3}{3 x^3}-a^3 \arcsin (a x)^2 \text {arctanh}\left (e^{i \arcsin (a x)}\right )-a^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )+i a^3 \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-i a^3 \arcsin (a x) \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-a^3 \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+a^3 \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 2.30 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.59 \[ \int \frac {\arcsin (a x)^3}{x^4} \, dx=\frac {1}{48} a^3 \left (-24 \arcsin (a x) \cot \left (\frac {1}{2} \arcsin (a x)\right )-4 \arcsin (a x)^3 \cot \left (\frac {1}{2} \arcsin (a x)\right )-6 \arcsin (a x)^2 \csc ^2\left (\frac {1}{2} \arcsin (a x)\right )-a x \arcsin (a x)^3 \csc ^4\left (\frac {1}{2} \arcsin (a x)\right )+24 \arcsin (a x)^2 \log \left (1-e^{i \arcsin (a x)}\right )-24 \arcsin (a x)^2 \log \left (1+e^{i \arcsin (a x)}\right )+48 \log \left (\tan \left (\frac {1}{2} \arcsin (a x)\right )\right )+48 i \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-48 i \arcsin (a x) \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-48 \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+48 \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )+6 \arcsin (a x)^2 \sec ^2\left (\frac {1}{2} \arcsin (a x)\right )-\frac {16 \arcsin (a x)^3 \sin ^4\left (\frac {1}{2} \arcsin (a x)\right )}{a^3 x^3}-24 \arcsin (a x) \tan \left (\frac {1}{2} \arcsin (a x)\right )-4 \arcsin (a x)^3 \tan \left (\frac {1}{2} \arcsin (a x)\right )\right ) \]

[In]

Integrate[ArcSin[a*x]^3/x^4,x]

[Out]

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

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.31

method	result	size
derivativedivides	\(a^{3} \left (-\frac {\arcsin \left (a x \right ) \left (3 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x +2 \arcsin \left (a x \right )^{2}+6 a^{2} x^{2}\right )}{6 a^{3} x^{3}}+\frac {\arcsin \left (a x \right )^{2} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )}{2}-i \arcsin \left (a x \right ) \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+\operatorname {polylog}\left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )-\frac {\arcsin \left (a x \right )^{2} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )}{2}+i \arcsin \left (a x \right ) \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-\operatorname {polylog}\left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-2 \,\operatorname {arctanh}\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )\)	\(234\)
default	\(a^{3} \left (-\frac {\arcsin \left (a x \right ) \left (3 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x +2 \arcsin \left (a x \right )^{2}+6 a^{2} x^{2}\right )}{6 a^{3} x^{3}}+\frac {\arcsin \left (a x \right )^{2} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )}{2}-i \arcsin \left (a x \right ) \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+\operatorname {polylog}\left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )-\frac {\arcsin \left (a x \right )^{2} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )}{2}+i \arcsin \left (a x \right ) \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-\operatorname {polylog}\left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-2 \,\operatorname {arctanh}\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )\)	\(234\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

integrate(arcsin(a*x)^3/x^4,x, algorithm="fricas")

[Out]

integral(arcsin(a*x)^3/x^4, x)

Sympy [F]

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

[In]

integrate(asin(a*x)**3/x**4,x)

[Out]

Integral(asin(a*x)**3/x**4, x)

Maxima [F]

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

[In]

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

[Out]

-1/3*(3*a*x^3*integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2/(a^2*x^5 - x
^3), x) + arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^3)/x^3

Giac [F]

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

[In]

integrate(arcsin(a*x)^3/x^4,x, algorithm="giac")

[Out]

integrate(arcsin(a*x)^3/x^4, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\arcsin (a x)^3}{x^4} \, dx=\int \frac {{\mathrm {asin}\left (a\,x\right )}^3}{x^4} \,d x \]

[In]

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

[Out]

int(asin(a*x)^3/x^4, x)