\(\int \frac {1}{x^2 \sqrt {a+b \arcsin (c x)}} \, dx\) [192]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \frac {1}{x^2 \sqrt {a+b \arcsin (c x)}} \, dx=\text {Int}\left (\frac {1}{x^2 \sqrt {a+b \arcsin (c x)}},x\right ) \]

[Out]

Unintegrable(1/x^2/(a+b*arcsin(c*x))^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 16, 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 = {} \[ \int \frac {1}{x^2 \sqrt {a+b \arcsin (c x)}} \, dx=\int \frac {1}{x^2 \sqrt {a+b \arcsin (c x)}} \, dx \]

[In]

Int[1/(x^2*Sqrt[a + b*ArcSin[c*x]]),x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 4.55 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x^2 \sqrt {a+b \arcsin (c x)}} \, dx=\int \frac {1}{x^2 \sqrt {a+b \arcsin (c x)}} \, dx \]

[In]

Integrate[1/(x^2*Sqrt[a + b*ArcSin[c*x]]),x]

[Out]

Integrate[1/(x^2*Sqrt[a + b*ArcSin[c*x]]), x]

Maple [N/A] (verified)

Not integrable

Time = 0.10 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88

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

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{x^2 \sqrt {a+b \arcsin (c x)}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [N/A]

Not integrable

Time = 0.53 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^2 \sqrt {a+b \arcsin (c x)}} \, dx=\int \frac {1}{x^{2} \sqrt {a + b \operatorname {asin}{\left (c x \right )}}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.63 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt {a+b \arcsin (c x)}} \, dx=\int { \frac {1}{\sqrt {b \arcsin \left (c x\right ) + a} x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.65 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt {a+b \arcsin (c x)}} \, dx=\int { \frac {1}{\sqrt {b \arcsin \left (c x\right ) + a} x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt {a+b \arcsin (c x)}} \, dx=\int \frac {1}{x^2\,\sqrt {a+b\,\mathrm {asin}\left (c\,x\right )}} \,d x \]

[In]

int(1/(x^2*(a + b*asin(c*x))^(1/2)),x)

[Out]

int(1/(x^2*(a + b*asin(c*x))^(1/2)), x)