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

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

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 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]

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]

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*}

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Mathematica [A]  time = 0.00, size = 32, normalized size = 0.84 \[ \frac {\text {Ci}\left (2 \sin ^{-1}(a x)\right )}{a^2}-\frac {\sin \left (2 \sin ^{-1}(a x)\right )}{2 a^2 \sin ^{-1}(a x)} \]

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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fricas [F]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x}{\arcsin \left (a x\right )^{2}}, x\right ) \]

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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giac [A]  time = 0.26, size = 36, normalized size = 0.95 \[ -\frac {\sqrt {-a^{2} x^{2} + 1} x}{a \arcsin \left (a x\right )} + \frac {\operatorname {Ci}\left (2 \, \arcsin \left (a x\right )\right )}{a^{2}} \]

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

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maple [A]  time = 0.04, size = 28, normalized size = 0.74 \[ \frac {-\frac {\sin \left (2 \arcsin \left (a x \right )\right )}{2 \arcsin \left (a x \right )}+\Ci \left (2 \arcsin \left (a x \right )\right )}{a^{2}} \]

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]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {-\sqrt {a x + 1} \sqrt {-a x + 1} x + {\left (\int \frac {2 \, \sqrt {-a x + 1} a^{2} x^{2}}{\sqrt {a x + 1} a x \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right ) - \sqrt {a x + 1} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )}\,{d x} + \int -\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1} a x \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right ) - \sqrt {a x + 1} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )}\,{d x}\right )} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )}{a \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x}{{\mathrm {asin}\left (a\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x/asin(a*x)^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\operatorname {asin}^{2}{\left (a x \right )}}\, dx \]

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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