\(\int \frac {x}{\arcsin (a x)^2} \, dx\) [56]

Optimal result

Integrand size = 8, antiderivative size = 38 \[ \int \frac {x}{\arcsin (a x)^2} \, dx=-\frac {x \sqrt {1-a^2 x^2}}{a \arcsin (a x)}+\frac {\operatorname {CosIntegral}(2 \arcsin (a x))}{a^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4727, 3383} \[ \int \frac {x}{\arcsin (a x)^2} \, dx=\frac {\operatorname {CosIntegral}(2 \arcsin (a x))}{a^2}-\frac {x \sqrt {1-a^2 x^2}}{a \arcsin (a x)} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.76 \[ \int \frac {x}{\arcsin (a x)^2} \, dx=\frac {\operatorname {CosIntegral}(2 \arcsin (a x))-\frac {\sin (2 \arcsin (a x))}{2 \arcsin (a x)}}{a^2} \]

[In]

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

[Out]

(CosIntegral[2*ArcSin[a*x]] - Sin[2*ArcSin[a*x]]/(2*ArcSin[a*x]))/a^2

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.74

method	result	size
derivativedivides	\(\frac {-\frac {\sin \left (2 \arcsin \left (a x \right )\right )}{2 \arcsin \left (a x \right )}+\operatorname {Ci}\left (2 \arcsin \left (a x \right )\right )}{a^{2}}\)	\(28\)
default	\(\frac {-\frac {\sin \left (2 \arcsin \left (a x \right )\right )}{2 \arcsin \left (a x \right )}+\operatorname {Ci}\left (2 \arcsin \left (a x \right )\right )}{a^{2}}\)	\(28\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(x/arcsin(a*x)^2, x)

Sympy [F]

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

[In]

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

[Out]

Integral(x/asin(a*x)**2, x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.95 \[ \int \frac {x}{\arcsin (a x)^2} \, dx=-\frac {\sqrt {-a^{2} x^{2} + 1} x}{a \arcsin \left (a x\right )} + \frac {\operatorname {Ci}\left (2 \, \arcsin \left (a x\right )\right )}{a^{2}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

int(x/asin(a*x)^2, x)