\(\int \frac {x^m \arcsin (a x)^n}{\sqrt {1-a^2 x^2}} \, dx\) [497]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.07 (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 = {} \[ \int \frac {x^m \arcsin (a x)^n}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^m \arcsin (a x)^n}{\sqrt {1-a^2 x^2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.55 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^m \arcsin (a x)^n}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^m \arcsin (a x)^n}{\sqrt {1-a^2 x^2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {x^m \arcsin (a x)^n}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{m} \arcsin \left (a x\right )^{n}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 4.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^m \arcsin (a x)^n}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^{m} \operatorname {asin}^{n}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [F(-1)]

Timed out. \[ \int \frac {x^m \arcsin (a x)^n}{\sqrt {1-a^2 x^2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^m \arcsin (a x)^n}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^m\,{\mathrm {asin}\left (a\,x\right )}^n}{\sqrt {1-a^2\,x^2}} \,d x \]

[In]

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

[Out]

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