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

Optimal. Leaf size=9 \[ \frac{\log \left (\sin ^{-1}(a x)\right )}{a} \]

[Out]

Log[ArcSin[a*x]]/a

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Rubi [A]  time = 0.032975, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {4639} \[ \frac{\log \left (\sin ^{-1}(a x)\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[ArcSin[a*x]]/a

Rule 4639

Int[1/(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*Sqrt[(d_) + (e_.)*(x_)^2]), x_Symbol] :> Simp[Log[a + b*ArcSin[c*x]]
/(b*c*Sqrt[d]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)} \, dx &=\frac{\log \left (\sin ^{-1}(a x)\right )}{a}\\ \end{align*}

Mathematica [A]  time = 0.0185966, size = 9, normalized size = 1. \[ \frac{\log \left (\sin ^{-1}(a x)\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[ArcSin[a*x]]/a

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Maple [A]  time = 0.003, size = 10, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( \arcsin \left ( ax \right ) \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(arcsin(a*x))/a

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Maxima [A]  time = 1.87578, size = 12, normalized size = 1.33 \begin{align*} \frac{\log \left (\arcsin \left (a x\right )\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(arcsin(a*x))/a

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Fricas [A]  time = 1.85105, size = 28, normalized size = 3.11 \begin{align*} \frac{\log \left (-\arcsin \left (a x\right )\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(-arcsin(a*x))/a

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Sympy [A]  time = 0.496362, size = 7, normalized size = 0.78 \begin{align*} \frac{\log{\left (\operatorname{asin}{\left (a x \right )} \right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(asin(a*x))/a

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Giac [A]  time = 1.2914, size = 14, normalized size = 1.56 \begin{align*} \frac{\log \left ({\left | \arcsin \left (a x\right ) \right |}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(arcsin(a*x)))/a