\(\int \frac {\arcsin (a x)^n}{x^2 \sqrt {1-a^2 x^2}} \, dx\) [503]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {\arcsin (a x)^n}{x^2 \sqrt {1-a^2 x^2}} \, dx=\text {Int}\left (\frac {\arcsin (a x)^n}{x^2 \sqrt {1-a^2 x^2}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\arcsin (a x)^n}{x^2 \sqrt {1-a^2 x^2}} \, dx=\int \frac {\arcsin (a x)^n}{x^2 \sqrt {1-a^2 x^2}} \, dx \]

[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*} \text {integral}& = \int \frac {\arcsin (a x)^n}{x^2 \sqrt {1-a^2 x^2}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.86 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\arcsin (a x)^n}{x^2 \sqrt {1-a^2 x^2}} \, dx=\int \frac {\arcsin (a x)^n}{x^2 \sqrt {1-a^2 x^2}} \, dx \]

[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 [N/A] (verified)

Not integrable

Time = 0.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

\[\int \frac {\arcsin \left (a x \right )^{n}}{x^{2} \sqrt {-a^{2} x^{2}+1}}d x\]

[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)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int \frac {\arcsin (a x)^n}{x^2 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\arcsin \left (a x\right )^{n}}{\sqrt {-a^{2} x^{2} + 1} x^{2}} \,d x } \]

[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 [N/A]

Not integrable

Time = 1.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\arcsin (a x)^n}{x^2 \sqrt {1-a^2 x^2}} \, dx=\int \frac {\operatorname {asin}^{n}{\left (a x \right )}}{x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]

[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)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\arcsin (a x)^n}{x^2 \sqrt {1-a^2 x^2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\arcsin (a x)^n}{x^2 \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\arcsin \left (a x\right )^{n}}{\sqrt {-a^{2} x^{2} + 1} x^{2}} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\arcsin (a x)^n}{x^2 \sqrt {1-a^2 x^2}} \, dx=\int \frac {{\mathrm {asin}\left (a\,x\right )}^n}{x^2\,\sqrt {1-a^2\,x^2}} \,d x \]

[In]

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

[Out]

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