\(\int \frac {1}{\arcsin (a x)^3} \, dx\) [64]

Optimal result

Integrand size = 6, antiderivative size = 51 \[ \int \frac {1}{\arcsin (a x)^3} \, dx=-\frac {\sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}+\frac {x}{2 \arcsin (a x)}-\frac {\operatorname {CosIntegral}(\arcsin (a x))}{2 a} \]

[Out]

1/2*x/arcsin(a*x)-1/2*Ci(arcsin(a*x))/a-1/2*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^2

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4717, 4807, 4719, 3383} \[ \int \frac {1}{\arcsin (a x)^3} \, dx=-\frac {\sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}-\frac {\operatorname {CosIntegral}(\arcsin (a x))}{2 a}+\frac {x}{2 \arcsin (a x)} \]

[In]

Int[ArcSin[a*x]^(-3),x]

[Out]

-1/2*Sqrt[1 - a^2*x^2]/(a*ArcSin[a*x]^2) + x/(2*ArcSin[a*x]) - CosIntegral[ArcSin[a*x]]/(2*a)

Rule 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 4717

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Sqrt[1 - c^2*x^2]*((a + b*ArcSin[c*x])^(n + 1)/
(b*c*(n + 1))), x] + Dist[c/(b*(n + 1)), Int[x*((a + b*ArcSin[c*x])^(n + 1)/Sqrt[1 - c^2*x^2]), x], x] /; Free
Q[{a, b, c}, x] && LtQ[n, -1]

Rule 4719

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Cos[-a/b + x/b], x], x,
a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x]

Rule 4807

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
((f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] - Dist[f*(m/(b
*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]], Int[(f*x)^(m - 1)*(a + b*ArcSin[c*x])^(n + 1), x], x] /;
FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}-\frac {1}{2} a \int \frac {x}{\sqrt {1-a^2 x^2} \arcsin (a x)^2} \, dx \\ & = -\frac {\sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}+\frac {x}{2 \arcsin (a x)}-\frac {1}{2} \int \frac {1}{\arcsin (a x)} \, dx \\ & = -\frac {\sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}+\frac {x}{2 \arcsin (a x)}-\frac {\text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\arcsin (a x)\right )}{2 a} \\ & = -\frac {\sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}+\frac {x}{2 \arcsin (a x)}-\frac {\operatorname {CosIntegral}(\arcsin (a x))}{2 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\arcsin (a x)^3} \, dx=-\frac {\sqrt {1-a^2 x^2}-a x \arcsin (a x)+\arcsin (a x)^2 \operatorname {CosIntegral}(\arcsin (a x))}{2 a \arcsin (a x)^2} \]

[In]

Integrate[ArcSin[a*x]^(-3),x]

[Out]

-1/2*(Sqrt[1 - a^2*x^2] - a*x*ArcSin[a*x] + ArcSin[a*x]^2*CosIntegral[ArcSin[a*x]])/(a*ArcSin[a*x]^2)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.84

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{2 \arcsin \left (a x \right )^{2}}+\frac {a x}{2 \arcsin \left (a x \right )}-\frac {\operatorname {Ci}\left (\arcsin \left (a x \right )\right )}{2}}{a}\)	\(43\)
default	\(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{2 \arcsin \left (a x \right )^{2}}+\frac {a x}{2 \arcsin \left (a x \right )}-\frac {\operatorname {Ci}\left (\arcsin \left (a x \right )\right )}{2}}{a}\)	\(43\)

[In]

int(1/arcsin(a*x)^3,x,method=_RETURNVERBOSE)

[Out]

1/a*(-1/2/arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)+1/2*a*x/arcsin(a*x)-1/2*Ci(arcsin(a*x)))

Fricas [F]

\[ \int \frac {1}{\arcsin (a x)^3} \, dx=\int { \frac {1}{\arcsin \left (a x\right )^{3}} \,d x } \]

[In]

integrate(1/arcsin(a*x)^3,x, algorithm="fricas")

[Out]

integral(arcsin(a*x)^(-3), x)

Sympy [F]

\[ \int \frac {1}{\arcsin (a x)^3} \, dx=\int \frac {1}{\operatorname {asin}^{3}{\left (a x \right )}}\, dx \]

[In]

integrate(1/asin(a*x)**3,x)

[Out]

Integral(asin(a*x)**(-3), x)

Maxima [F]

\[ \int \frac {1}{\arcsin (a x)^3} \, dx=\int { \frac {1}{\arcsin \left (a x\right )^{3}} \,d x } \]

[In]

integrate(1/arcsin(a*x)^3,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\arcsin (a x)^3} \, dx=\frac {x}{2 \, \arcsin \left (a x\right )} - \frac {\operatorname {Ci}\left (\arcsin \left (a x\right )\right )}{2 \, a} - \frac {\sqrt {-a^{2} x^{2} + 1}}{2 \, a \arcsin \left (a x\right )^{2}} \]

[In]

integrate(1/arcsin(a*x)^3,x, algorithm="giac")

[Out]

1/2*x/arcsin(a*x) - 1/2*cos_integral(arcsin(a*x))/a - 1/2*sqrt(-a^2*x^2 + 1)/(a*arcsin(a*x)^2)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\arcsin (a x)^3} \, dx=\int \frac {1}{{\mathrm {asin}\left (a\,x\right )}^3} \,d x \]

[In]

int(1/asin(a*x)^3,x)

[Out]

int(1/asin(a*x)^3, x)