\(\int \frac {1}{x^2 \sqrt {1-a^2 x^2} \arcsin (a x)} \, dx\) [348]

Optimal result

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

[Out]

Unintegrable(1/x^2/arcsin(a*x)/(-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 = {} \[ \int \frac {1}{x^2 \sqrt {1-a^2 x^2} \arcsin (a x)} \, dx=\int \frac {1}{x^2 \sqrt {1-a^2 x^2} \arcsin (a x)} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^2 \sqrt {1-a^2 x^2} \arcsin (a x)} \, dx=\int \frac {1}{x^2 \sqrt {1-a^2 x^2} \arcsin (a x)} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int \frac {1}{x^2 \sqrt {1-a^2 x^2} \arcsin (a x)} \, dx=\int { \frac {1}{\sqrt {-a^{2} x^{2} + 1} x^{2} \arcsin \left (a x\right )} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^2 \sqrt {1-a^2 x^2} \arcsin (a x)} \, dx=\int \frac {1}{x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {asin}{\left (a x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt {1-a^2 x^2} \arcsin (a x)} \, dx=\int { \frac {1}{\sqrt {-a^{2} x^{2} + 1} x^{2} \arcsin \left (a x\right )} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt {1-a^2 x^2} \arcsin (a x)} \, dx=\int { \frac {1}{\sqrt {-a^{2} x^{2} + 1} x^{2} \arcsin \left (a x\right )} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

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

[In]

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

[Out]

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