\(\int \frac {1}{\arcsin (a x)^2} \, dx\) [57]

Optimal result

Integrand size = 6, antiderivative size = 36 \[ \int \frac {1}{\arcsin (a x)^2} \, dx=-\frac {\sqrt {1-a^2 x^2}}{a \arcsin (a x)}-\frac {\text {Si}(\arcsin (a x))}{a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4717, 4809, 3380} \[ \int \frac {1}{\arcsin (a x)^2} \, dx=-\frac {\sqrt {1-a^2 x^2}}{a \arcsin (a x)}-\frac {\text {Si}(\arcsin (a x))}{a} \]

[In]

Int[ArcSin[a*x]^(-2),x]

[Out]

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

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - 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 4809

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

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\arcsin (a x)^2} \, dx=-\frac {\frac {\sqrt {1-a^2 x^2}}{\arcsin (a x)}+\text {Si}(\arcsin (a x))}{a} \]

[In]

Integrate[ArcSin[a*x]^(-2),x]

[Out]

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

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.92

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{\arcsin \left (a x \right )}-\operatorname {Si}\left (\arcsin \left (a x \right )\right )}{a}\)	\(33\)
default	\(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{\arcsin \left (a x \right )}-\operatorname {Si}\left (\arcsin \left (a x \right )\right )}{a}\)	\(33\)

[In]

int(1/arcsin(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int \frac {1}{\arcsin (a x)^2} \, dx=\int { \frac {1}{\arcsin \left (a x\right )^{2}} \,d x } \]

[In]

integrate(1/arcsin(a*x)^2,x, algorithm="fricas")

[Out]

integral(arcsin(a*x)^(-2), x)

Sympy [F]

\[ \int \frac {1}{\arcsin (a x)^2} \, dx=\int \frac {1}{\operatorname {asin}^{2}{\left (a x \right )}}\, dx \]

[In]

integrate(1/asin(a*x)**2,x)

[Out]

Integral(asin(a*x)**(-2), x)

Maxima [F]

\[ \int \frac {1}{\arcsin (a x)^2} \, dx=\int { \frac {1}{\arcsin \left (a x\right )^{2}} \,d x } \]

[In]

integrate(1/arcsin(a*x)^2,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\arcsin (a x)^2} \, dx=-\frac {\operatorname {Si}\left (\arcsin \left (a x\right )\right )}{a} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a \arcsin \left (a x\right )} \]

[In]

integrate(1/arcsin(a*x)^2,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\arcsin (a x)^2} \, dx=\int \frac {1}{{\mathrm {asin}\left (a\,x\right )}^2} \,d x \]

[In]

int(1/asin(a*x)^2,x)

[Out]

int(1/asin(a*x)^2, x)