3.56 \(\int \frac{x}{\sin ^{-1}(a x)^2} \, dx\)

Optimal. Leaf size=38 \[ \frac{\text{CosIntegral}\left (2 \sin ^{-1}(a x)\right )}{a^2}-\frac{x \sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)} \]

[Out]

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

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Rubi [A]  time = 0.0247518, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4631, 3302} \[ \frac{\text{CosIntegral}\left (2 \sin ^{-1}(a x)\right )}{a^2}-\frac{x \sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)} \]

Antiderivative was successfully verified.

[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 4631

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*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a + b*x)^(n + 1)
, Sin[x]^(m - 1)*(m - (m + 1)*Sin[x]^2), x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && G
eQ[n, -2] && LtQ[n, -1]

Rule 3302

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]

Rubi steps

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

Mathematica [A]  time = 0.0032364, size = 32, normalized size = 0.84 \[ \frac{\text{CosIntegral}\left (2 \sin ^{-1}(a x)\right )}{a^2}-\frac{\sin \left (2 \sin ^{-1}(a x)\right )}{2 a^2 \sin ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.024, size = 28, normalized size = 0.7 \begin{align*}{\frac{1}{{a}^{2}} \left ( -{\frac{\sin \left ( 2\,\arcsin \left ( ax \right ) \right ) }{2\,\arcsin \left ( ax \right ) }}+{\it Ci} \left ( 2\,\arcsin \left ( ax \right ) \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x}{\arcsin \left (a x\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x/arcsin(a*x)^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\operatorname{asin}^{2}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/asin(a*x)**2, x)

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Giac [A]  time = 1.37361, size = 49, normalized size = 1.29 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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