\(\int \frac {x^2}{\arcsin (a x)^2} \, dx\) [55]

Optimal result

Integrand size = 10, antiderivative size = 55 \[ \int \frac {x^2}{\arcsin (a x)^2} \, dx=-\frac {x^2 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}-\frac {\text {Si}(\arcsin (a x))}{4 a^3}+\frac {3 \text {Si}(3 \arcsin (a x))}{4 a^3} \]

[Out]

-1/4*Si(arcsin(a*x))/a^3+3/4*Si(3*arcsin(a*x))/a^3-x^2*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4727, 3380} \[ \int \frac {x^2}{\arcsin (a x)^2} \, dx=-\frac {\text {Si}(\arcsin (a x))}{4 a^3}+\frac {3 \text {Si}(3 \arcsin (a x))}{4 a^3}-\frac {x^2 \sqrt {1-a^2 x^2}}{a \arcsin (a x)} \]

[In]

Int[x^2/ArcSin[a*x]^2,x]

[Out]

-((x^2*Sqrt[1 - a^2*x^2])/(a*ArcSin[a*x])) - SinIntegral[ArcSin[a*x]]/(4*a^3) + (3*SinIntegral[3*ArcSin[a*x]])
/(4*a^3)

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 4727

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^m*Sqrt[1 - c^2*x^2]*((a + b*ArcSin
[c*x])^(n + 1)/(b*c*(n + 1))), x] - Dist[1/(b^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^(n + 1), Sin[
-a/b + x/b]^(m - 1)*(m - (m + 1)*Sin[-a/b + x/b]^2), x], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c}, x]
&& IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}+\frac {\text {Subst}\left (\int \left (-\frac {\sin (x)}{4 x}+\frac {3 \sin (3 x)}{4 x}\right ) \, dx,x,\arcsin (a x)\right )}{a^3} \\ & = -\frac {x^2 \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 )}{4 a^3}+\frac {3 \text {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\arcsin (a x)\right )}{4 a^3} \\ & = -\frac {x^2 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}-\frac {\text {Si}(\arcsin (a x))}{4 a^3}+\frac {3 \text {Si}(3 \arcsin (a x))}{4 a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.91 \[ \int \frac {x^2}{\arcsin (a x)^2} \, dx=-\frac {\frac {4 a^2 x^2 \sqrt {1-a^2 x^2}}{\arcsin (a x)}+\text {Si}(\arcsin (a x))-3 \text {Si}(3 \arcsin (a x))}{4 a^3} \]

[In]

Integrate[x^2/ArcSin[a*x]^2,x]

[Out]

-1/4*((4*a^2*x^2*Sqrt[1 - a^2*x^2])/ArcSin[a*x] + SinIntegral[ArcSin[a*x]] - 3*SinIntegral[3*ArcSin[a*x]])/a^3

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.04

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{4 \arcsin \left (a x \right )}-\frac {\operatorname {Si}\left (\arcsin \left (a x \right )\right )}{4}+\frac {\cos \left (3 \arcsin \left (a x \right )\right )}{4 \arcsin \left (a x \right )}+\frac {3 \,\operatorname {Si}\left (3 \arcsin \left (a x \right )\right )}{4}}{a^{3}}\)	\(57\)
default	\(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{4 \arcsin \left (a x \right )}-\frac {\operatorname {Si}\left (\arcsin \left (a x \right )\right )}{4}+\frac {\cos \left (3 \arcsin \left (a x \right )\right )}{4 \arcsin \left (a x \right )}+\frac {3 \,\operatorname {Si}\left (3 \arcsin \left (a x \right )\right )}{4}}{a^{3}}\)	\(57\)

[In]

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

[Out]

1/a^3*(-1/4/arcsin(a*x)*(-a^2*x^2+1)^(1/2)-1/4*Si(arcsin(a*x))+1/4/arcsin(a*x)*cos(3*arcsin(a*x))+3/4*Si(3*arc
sin(a*x)))

Fricas [F]

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

[In]

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

[Out]

integral(x^2/arcsin(a*x)^2, x)

Sympy [F]

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

[In]

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

[Out]

Integral(x**2/asin(a*x)**2, x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.24 \[ \int \frac {x^2}{\arcsin (a x)^2} \, dx=\frac {3 \, \operatorname {Si}\left (3 \, \arcsin \left (a x\right )\right )}{4 \, a^{3}} - \frac {\operatorname {Si}\left (\arcsin \left (a x\right )\right )}{4 \, a^{3}} + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{3} \arcsin \left (a x\right )} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{3} \arcsin \left (a x\right )} \]

[In]

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

[Out]

3/4*sin_integral(3*arcsin(a*x))/a^3 - 1/4*sin_integral(arcsin(a*x))/a^3 + (-a^2*x^2 + 1)^(3/2)/(a^3*arcsin(a*x
)) - sqrt(-a^2*x^2 + 1)/(a^3*arcsin(a*x))

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^2/asin(a*x)^2, x)