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

Optimal. Leaf size=26 \[ \text{Unintegrable}\left (\frac{\sin ^{-1}(a x)^n}{x^2 \sqrt{1-a^2 x^2}},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.101614, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sin ^{-1}(a x)^n}{x^2 \sqrt{1-a^2 x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.93575, size = 0, normalized size = 0. \[ \int \frac{\sin ^{-1}(a x)^n}{x^2 \sqrt{1-a^2 x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.127, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \arcsin \left ( ax \right ) \right ) ^{n}}{{x}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{n}}{a^{2} x^{4} - x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-a^2*x^2 + 1)*arcsin(a*x)^n/(a^2*x^4 - x^2), x)

________________________________________________________________________________________

Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asin}^{n}{\left (a x \right )}}{x^{2} \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(asin(a*x)**n/(x**2*sqrt(-(a*x - 1)*(a*x + 1))), x)

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arcsin \left (a x\right )^{n}}{\sqrt{-a^{2} x^{2} + 1} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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